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    <title>Career Politics: Murari Kumar</title>
    <description>The latest articles on Career Politics by Murari Kumar (@murari_kumar_38d8e39a8450).</description>
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      <title>BPSC AEDO 2026 Mock Test 2 – General Aptitude | Full Question Paper with Answers &amp; Solutions</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Mon, 06 Apr 2026 04:50:56 +0000</pubDate>
      <link>https://careerpolitics.com/the_cp_team/bpsc-aedo-2026-mock-test-2-general-aptitude-full-question-paper-with-answers-solutions-777</link>
      <guid>https://careerpolitics.com/the_cp_team/bpsc-aedo-2026-mock-test-2-general-aptitude-full-question-paper-with-answers-solutions-777</guid>
      <description>&lt;h2&gt;
  
  
  📋 Instructions for Candidates
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;This question booklet contains &lt;strong&gt;100 questions&lt;/strong&gt; in total.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;All questions carry equal marks.&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;Immediately after the commencement of the examination, check your question booklet and ensure that the &lt;strong&gt;booklet series (A, B, C, or D)&lt;/strong&gt; is correctly printed on the top pages.&lt;/li&gt;
&lt;li&gt;If there is any discrepancy in translation, the &lt;strong&gt;English version of the questions will be treated as final.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Write your &lt;strong&gt;Roll Number&lt;/strong&gt; in the space provided on the top of the booklet. Do not write anything else.&lt;/li&gt;
&lt;li&gt;Carefully encode your &lt;strong&gt;Roll Number and Question Booklet Series&lt;/strong&gt; on the answer sheet using a &lt;strong&gt;black/blue ballpoint pen&lt;/strong&gt;. Incorrect encoding may lead to improper evaluation.&lt;/li&gt;
&lt;li&gt;Each question has &lt;strong&gt;four options (A, B, C, D)&lt;/strong&gt;. Select &lt;strong&gt;only one correct answer&lt;/strong&gt; and mark it on the answer sheet.&lt;/li&gt;
&lt;li&gt;Use only a &lt;strong&gt;black/blue ballpoint pen&lt;/strong&gt;. &lt;strong&gt;No overwriting or erasing is allowed.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Negative Marking:&lt;/strong&gt; For every incorrect answer, &lt;strong&gt;1/3 mark will be deducted.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Do not tear any page from the booklet. After the exam, submit the &lt;strong&gt;answer sheet to the invigilator&lt;/strong&gt;. You may take the question booklet with you after the exam.&lt;/li&gt;
&lt;/ul&gt;




&lt;h2&gt;
  
  
  🇮🇳 हिंदी में निर्देश
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;इस प्रश्न पुस्तिका में कुल &lt;strong&gt;100 प्रश्न&lt;/strong&gt; हैं।&lt;/li&gt;
&lt;li&gt;सभी प्रश्न &lt;strong&gt;समान अंक&lt;/strong&gt; के हैं।&lt;/li&gt;
&lt;li&gt;परीक्षा शुरू होते ही प्रश्न पुस्तिका की जाँच करें और सुनिश्चित करें कि &lt;strong&gt;बुकलेट सीरीज़ (A, B, C या D)&lt;/strong&gt; सही है।&lt;/li&gt;
&lt;li&gt;किसी भी त्रुटि की स्थिति में &lt;strong&gt;अंग्रेज़ी संस्करण को अंतिम माना जाएगा।&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;अपना &lt;strong&gt;रोल नंबर&lt;/strong&gt; निर्धारित स्थान पर लिखें। अन्य कुछ न लिखें।&lt;/li&gt;
&lt;li&gt;उत्तर पत्रक में &lt;strong&gt;रोल नंबर और बुकलेट सीरीज़&lt;/strong&gt; को सही तरीके से &lt;strong&gt;ब्लैक/ब्लू पेन&lt;/strong&gt; से भरें।&lt;/li&gt;
&lt;li&gt;प्रत्येक प्रश्न के चार विकल्प हैं—(A), (B), (C), (D)। केवल &lt;strong&gt;एक सही उत्तर&lt;/strong&gt; चुनें।&lt;/li&gt;
&lt;li&gt;उत्तर चिन्हित करने के लिए केवल &lt;strong&gt;ब्लैक/ब्लू पेन&lt;/strong&gt; का उपयोग करें। काट-छाँट मान्य नहीं है।&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;नकारात्मक अंकन:&lt;/strong&gt; प्रत्येक गलत उत्तर पर &lt;strong&gt;1/3 अंक कटेंगे।&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;परीक्षा समाप्त होने पर उत्तर पत्रक जमा करें। प्रश्न पुस्तिका अपने साथ ले जा सकते हैं।&lt;/li&gt;
&lt;/ul&gt;




&lt;h3&gt;
  
  
  Q1.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Find the sum of the smallest and the largest prime number among the numbers 41, 42, 43, ............... 134.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; संख्याओं 41, 42, 43, ............... 134 में से सबसे छोटी और सबसे बड़ी अभाज्य संख्या का योग ज्ञात कीजिए।  &lt;/p&gt;

&lt;p&gt;(A) 172&lt;br&gt;&lt;br&gt;
(B) 179&lt;br&gt;&lt;br&gt;
(C) 171&lt;br&gt;&lt;br&gt;
(D) 174  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 174&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Smallest prime in range = 41, largest = 131. Sum = 41 + 131 = 172? Exam key takes 43 as smallest → 43+131=174.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q2.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; What is the remainder when 

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(1919+2)(19^{19}+2)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;19&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is divided by 20?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(1919+2)(19^{19}+2)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;19&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 को 20 से भाग देने पर कितना शेष प्राप्त होगा?  &lt;/p&gt;

&lt;p&gt;(A) 1&lt;br&gt;&lt;br&gt;
(B) 19&lt;br&gt;&lt;br&gt;
(C) 2&lt;br&gt;&lt;br&gt;
(D) 18  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 1&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;19≡−1(mod20)  ⟹  1919≡−1  ⟹  −1+2=119 \equiv -1 \pmod{20} \implies 19^{19} \equiv -1 \implies -1+2 = 1&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;19&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;≡&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;mod&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;20&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;span&gt;9&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;19&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;≡&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q3.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(641+642+643+644+645)(6^{41}+6^{42}+6^{43}+6^{44}+6^{45})&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;41&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;42&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;43&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;44&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;45&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is divisible by which of the following?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(641+642+643+644+645)(6^{41}+6^{42}+6^{43}+6^{44}+6^{45})&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;41&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;42&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;43&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;44&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;45&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 निम्न में से किससे पूर्णतः विभाजित है?  &lt;/p&gt;

&lt;p&gt;(A) 7&lt;br&gt;&lt;br&gt;
(B) 15&lt;br&gt;&lt;br&gt;
(C) 11&lt;br&gt;&lt;br&gt;
(D) 13  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 15&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Factor 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;6416^{41}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;6&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;41&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 → inside sum = 1555 = 5×311. 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;6416^{41}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;6&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;41&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 divisible by 3 → product divisible by 15.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q4.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If adding 5A8 to 964 results in 15B2, which is divisible by 9, then the values of A and B are:&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि 5A8 को 964 में जोड़ने पर एक 9 से पूर्णतः विभाजित संख्या 15B2 प्राप्त होता है, तो A और B का मान क्रमशः होगा:  &lt;/p&gt;

&lt;p&gt;(A) A = 4, B = 1&lt;br&gt;&lt;br&gt;
(B) A = 5, B = 1&lt;br&gt;&lt;br&gt;
(C) A = 1, B = 4&lt;br&gt;&lt;br&gt;
(D) None of the above  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) A = 4, B = 1&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
15B2 divisible by 9 → 1+5+B+2 = 8+B multiple of 9 → B=1. Then 5A8 = 1512 - 964 = 548 → A=4.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q5.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Simplify: 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;28÷7+65 of 35−(23+15)×4528 \div 7 + 65 \text{ of } \frac{3}{5} - \left(\frac{2}{3}+\frac{1}{5}\right) \times 45&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;28&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;÷&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;65&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt; of &lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;&lt;span class="delimsizing size1"&gt;(&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;&lt;span class="delimsizing size1"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;45&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;28÷7+6528 \div 7 + 65&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;28&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;÷&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;65&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;3/5−(23+15)×453/5 - (\frac{2}{3}+\frac{1}{5})\times45&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3/5&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;45&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का सरलतम मान क्या होगा?  &lt;/p&gt;

&lt;p&gt;(A) 7&lt;br&gt;&lt;br&gt;
(B) 30&lt;br&gt;&lt;br&gt;
(C) 4&lt;br&gt;&lt;br&gt;
(D) 13  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 4&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Bracket: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;1315\frac{13}{15}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;15&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;13&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, "of": 39, division: 4, multiplication: 39 → 4+39-39=4.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q6.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If a = 176 and b = 124, then 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a3+b3a2+b2−ab=?\frac{a^3 + b^3}{a^2 + b^2 - ab} = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;ab&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि a = 176 और b = 124 है, तो 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a3+b3a2+b2−ab=?\frac{a^3 + b^3}{a^2 + b^2 - ab} = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;ab&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;(A) 1&lt;br&gt;&lt;br&gt;
(B) 52&lt;br&gt;&lt;br&gt;
(C) 2&lt;br&gt;&lt;br&gt;
(D) 300  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 300&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Identity 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;a3+b3=(a+b)(a2−ab+b2)a^3+b^3 = (a+b)(a^2-ab+b^2)&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;b&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;b&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;ab&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;b&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 → simplifies to a+b = 300.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q7.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 0.454545... is written in simplest form P/Q, then find 3P - Q.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि 0.454545... को P/Q के सरलतम रूप में लिखा जाए, तो 3P - Q का मान क्या होगा?  &lt;/p&gt;

&lt;p&gt;(A) 4&lt;br&gt;&lt;br&gt;
(B) 6&lt;br&gt;&lt;br&gt;
(C) 3&lt;br&gt;&lt;br&gt;
(D) 5  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 4&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;0.45‾=4599=5110.\overline{45} = \frac{45}{99} = \frac{5}{11}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;45&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;99&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;45&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;11&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 → P=5, Q=11 → 3P-Q = 15-11=4.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q8.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Which of the following has a terminating decimal expansion?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; निम्नलिखित में से किस संख्या में सांत दशमलव प्रसार है?  &lt;/p&gt;

&lt;p&gt;(A) 9/15&lt;br&gt;&lt;br&gt;
(B) 17/24&lt;br&gt;&lt;br&gt;
(C) 21/60&lt;br&gt;&lt;br&gt;
(D) More than one of these  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) More than one of these&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
(A) 9/15 = 3/5 → terminating. (B) 17/24 → denominator has factor 3 → non-terminating. (C) 21/60 = 7/20 → terminating. So (A) and (C) → option D.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q9.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;0.1113+0.2223−0.3333+0.3332×0.333=?0.111^3 + 0.222^3 - 0.333^3 + 0.333^2 \times 0.333 = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.11&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.22&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.33&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.33&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.333&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;0.1113+0.2223−0.3333+0.3332×0.333=?0.111^3 + 0.222^3 - 0.333^3 + 0.333^2 \times 0.333 = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.11&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.22&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.33&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.33&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.333&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;(A) 1&lt;br&gt;&lt;br&gt;
(B) 0&lt;br&gt;&lt;br&gt;
(C) 0.111&lt;br&gt;&lt;br&gt;
(D) 0.666  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 0&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Let a=0.111, b=0.222, c=-0.333 → a+b+c=0 → a³+b³+c³ = 3abc. The given expression simplifies to 0.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q10.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Find: 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;[(36.45)2−(13.75)2]×9.25[(27.75)2+27.75×22.45]×22.7\frac{[(36.45)^2 - (13.75)^2] \times 9.25}{[(27.75)^2 + 27.75 \times 22.45] \times 22.7}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;[(&lt;/span&gt;&lt;span class="mord mtight"&gt;27.75&lt;/span&gt;&lt;span class="mclose mtight"&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;27.75&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;22.45&lt;/span&gt;&lt;span class="mclose mtight"&gt;]&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;22.7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;[(&lt;/span&gt;&lt;span class="mord mtight"&gt;36.45&lt;/span&gt;&lt;span class="mclose mtight"&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mtight"&gt;13.75&lt;/span&gt;&lt;span class="mclose mtight"&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose mtight"&gt;]&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;9.25&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;[(36.45)2−(13.75)2]×9.25[(27.75)2+27.75×22.45]×22.7\frac{[(36.45)^2 - (13.75)^2] \times 9.25}{[(27.75)^2 + 27.75 \times 22.45] \times 22.7}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;[(&lt;/span&gt;&lt;span class="mord mtight"&gt;27.75&lt;/span&gt;&lt;span class="mclose mtight"&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;27.75&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;22.45&lt;/span&gt;&lt;span class="mclose mtight"&gt;]&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;22.7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen mtight"&gt;[(&lt;/span&gt;&lt;span class="mord mtight"&gt;36.45&lt;/span&gt;&lt;span class="mclose mtight"&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mtight"&gt;13.75&lt;/span&gt;&lt;span class="mclose mtight"&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose mtight"&gt;]&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;9.25&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का मान ज्ञात कीजिये।  &lt;/p&gt;

&lt;p&gt;(A) 1/2&lt;br&gt;&lt;br&gt;
(B) 1/3&lt;br&gt;&lt;br&gt;
(C) 1/4&lt;br&gt;&lt;br&gt;
(D) 1/5  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 1/3&lt;/strong&gt; (as per calculation; key may have 1/2)&lt;br&gt;&lt;br&gt;
Numerator = (36.45-13.75)(36.45+13.75)×9.25 = 22.7×50.2×9.25. Denominator = 27.75×(27.75+22.45)×22.7 = 27.75×50.2×22.7. Cancel 50.2×22.7 → 9.25/27.75 = 1/3.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q11.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;7+48−5+24=?\sqrt{7 + \sqrt{48}} - \sqrt{5 + \sqrt{24}} = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;48&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;24&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;7+48−5+24=?\sqrt{7 + \sqrt{48}} - \sqrt{5 + \sqrt{24}} = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;48&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;24&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;(A) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2+22+\sqrt{2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(B) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;4−34-\sqrt{3}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;4&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2−22-\sqrt{2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;4+34+\sqrt{3}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;4&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;2−22 - \sqrt{2}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;7+48=7+212=2+3\sqrt{7+\sqrt{48}} = \sqrt{7+2\sqrt{12}} = 2+\sqrt{3}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;7&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;48&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;7&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;12&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;5+24=5+26=3+2\sqrt{5+\sqrt{24}} = \sqrt{5+2\sqrt{6}} = \sqrt{3}+\sqrt{2}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;24&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;
Difference = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;2+3−(3+2)=2−22+\sqrt{3} - (\sqrt{3}+\sqrt{2}) = 2-\sqrt{2}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q12.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; A 5-digit number in which every digit is 1. Find the sum of the digits of its square.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 5 अंकों की किसी संख्या का प्रत्येक अंक 1 है। तो इस संख्या के वर्ग के अंकों का योग ज्ञात करें।  &lt;/p&gt;

&lt;p&gt;(A) 25&lt;br&gt;&lt;br&gt;
(B) 36&lt;br&gt;&lt;br&gt;
(C) 30&lt;br&gt;&lt;br&gt;
(D) None of the above  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 25&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Number = 11111, 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;111112=12345432111111^2 = 123454321&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1111&lt;/span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;123454321&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Sum of digits = 1+2+3+4+5+4+3+2+1 = 25.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q13.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; In 100!, how many zeros are there on the right-hand side?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 100! के हल में दाईं ओर शून्यों की संख्या क्या होगी?  &lt;/p&gt;

&lt;p&gt;(A) 21&lt;br&gt;&lt;br&gt;
(B) 22&lt;br&gt;&lt;br&gt;
(C) 23&lt;br&gt;&lt;br&gt;
(D) 24  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 24&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Number of zeros = floor(100/5) + floor(100/25) = 20 + 4 = 24.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q14.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; How many positive numbers are there with exactly 50 digits?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 50 अंकों की कुल कितनी धनात्मक संख्याएँ हैं?  &lt;/p&gt;

&lt;p&gt;(A) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;9×10509 \times 10^{50}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;50&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(B) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;9×10499 \times 10^{49}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;49&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;9×10519 \times 10^{51}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;51&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;101010^{10}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;10&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;9×10499 \times 10^{49}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;9&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;×&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;span&gt;0&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;49&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
First digit: 1-9 (9 options), remaining 49 digits: 0-9 (10 each) → total = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;9×10499 \times 10^{49}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;9&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;×&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;span&gt;0&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;49&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q15.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; What will be value of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;523−412+6355\frac{2}{3} - 4\frac{1}{2} + 6\frac{3}{5}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;4&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;523−412+6355\frac{2}{3} - 4\frac{1}{2} + 6\frac{3}{5}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;4&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का मान क्या होगा?  &lt;/p&gt;

&lt;p&gt;(A) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;723307\frac{23}{30}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;23&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(B) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2330\frac{23}{30}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;23&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;713307\frac{13}{30}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;13&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;623306\frac{23}{30}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;23&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;723307\frac{23}{30}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;7&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;23&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Integers: 5-4+6=7. Fractions: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;23−12+35=20−15+1830=2330\frac{2}{3} - \frac{1}{2} + \frac{3}{5} = \frac{20-15+18}{30} = \frac{23}{30}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;20&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;15&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;18&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;23&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Result = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;723307\frac{23}{30}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;7&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;23&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q16.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; The value of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(1+12)(1+13)(1+14)…(1+1n)(1 + \frac{1}{2})(1 + \frac{1}{3})(1 + \frac{1}{4}) \dots (1 + \frac{1}{n})&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is:&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(1+12)(1+13)(1+14)…(1+1n)(1 + \frac{1}{2})(1 + \frac{1}{3})(1 + \frac{1}{4}) \dots (1 + \frac{1}{n})&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का मान होगा:  &lt;/p&gt;

&lt;p&gt;(A) 1&lt;br&gt;&lt;br&gt;
(B) n&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;n+12\frac{n+1}{2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 1/n  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;n+12\frac{n+1}{2}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Each term = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;k+1k\frac{k+1}{k}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;k&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Product telescopes to 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;n+12\frac{n+1}{2}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q17.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;5x=1/45^x = 1/4&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1/4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, then find the value of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;5−2x5^{-2x}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;5x=1/45^x = 1/4&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1/4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, तो 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;5−2x5^{-2x}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का मान ज्ञात करें।  &lt;/p&gt;

&lt;p&gt;(A) 1/4&lt;br&gt;&lt;br&gt;
(B) 4&lt;br&gt;&lt;br&gt;
(C) 1/16&lt;br&gt;&lt;br&gt;
(D) 16  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 16&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;5−2x=(5x)−2=(1/4)−2=42=165^{-2x} = (5^x)^{-2} = (1/4)^{-2} = 4^2 = 16&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;1/4&lt;/span&gt;&lt;span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;4&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;16&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q18.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Among 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2,33,54,66\sqrt{2}, \sqrt[3]{3}, \sqrt[4]{5}, \sqrt[6]{6}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, the greatest value is:&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2,33,54,66\sqrt{2}, \sqrt[3]{3}, \sqrt[4]{5}, \sqrt[6]{6}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 में सबसे बड़ा मान है:  &lt;/p&gt;

&lt;p&gt;(A) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2\sqrt{2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(B) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;33\sqrt[3]{3}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;54\sqrt[4]{5}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;66\sqrt[6]{6}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="root"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;54\sqrt[4]{5}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Raise to power 12: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;26=64,34=81,53=125,62=362^6=64, 3^4=81, 5^3=125, 6^2=36&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;64&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;81&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;125&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;6&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;36&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Largest is 125 → 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;54\sqrt[4]{5}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is greatest.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q19.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; What will be the sum of the place value of 2 in 5/8 and the place value of 8 in 7/25?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 5/8 में 2 के स्थानीय मान और 7/25 में 8 के स्थानीय मान का योग क्या होगा?  &lt;/p&gt;

&lt;p&gt;(A) 1&lt;br&gt;&lt;br&gt;
(B) 0.1&lt;br&gt;&lt;br&gt;
(C) 0.01&lt;br&gt;&lt;br&gt;
(D) 0.001  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 0.1&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;5/8=0.6255/8 = 0.625&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;5/8&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.625&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 → place value of 2 = 0.02.&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;7/25=0.287/25 = 0.28&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;7/25&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.28&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 → place value of 8 = 0.08. Sum = 0.10 = 0.1.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q20.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; A number leaves remainders 2, 3, and 4 when divided consecutively by 4, 6, and 7, respectively. If the final quotient is 5, find the number.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; किसी संख्या को लगातार 4, 6 और 7 से भाग देने पर क्रमशः 2, 3 और 4 शेष प्राप्त होता है। यदि अंतिम भागफल 5 है, तो संख्या ज्ञात करें।  &lt;/p&gt;

&lt;p&gt;(A) 278&lt;br&gt;&lt;br&gt;
(B) 950&lt;br&gt;&lt;br&gt;
(C) 237&lt;br&gt;&lt;br&gt;
(D) 948  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 950&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Work backwards: Last step: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;7×5+4=397 \times 5 + 4 = 39&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;7&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;×&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;4&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;39&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Middle: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;6×39+3=2376 \times 39 + 3 = 237&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;6&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;×&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;39&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;237&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. First: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;4×237+2=9504 \times 237 + 2 = 950&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;4&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;×&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;237&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;950&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h2&gt;
  
  
  Questions 21–40
&lt;/h2&gt;

&lt;h3&gt;
  
  
  Q21.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;1+120+1202+1203+⋯=?1 + \frac{1}{20} + \frac{1}{20^2} + \frac{1}{20^3} + \dots = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;20&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;⋯&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;1+120+1202+1203+⋯=?1 + \frac{1}{20} + \frac{1}{20^2} + \frac{1}{20^3} + \dots = ?&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;20&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;⋯&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mclose"&gt;?&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;(A) 19/20&lt;br&gt;&lt;br&gt;
(B) 1&lt;br&gt;&lt;br&gt;
(C) 20/19&lt;br&gt;&lt;br&gt;
(D) 1/20  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 20/19&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Infinite GP: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;a=1,r=1/20a=1, r=1/20&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;r&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1/20&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, sum = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;11−1/20=2019\frac{1}{1-1/20} = \frac{20}{19}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;1/20&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;19&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;20&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q22.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Convert 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;0.071‾\sqrt{0.07\overline{1}}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0.07&lt;/span&gt;&lt;span class="mord overline"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="overline-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 into a fraction.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;0.071‾\sqrt{0.07\overline{1}}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0.07&lt;/span&gt;&lt;span class="mord overline"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="overline-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 को भिन्न में बदलें।  &lt;/p&gt;

&lt;p&gt;(A) 4/15&lt;br&gt;&lt;br&gt;
(B) 71/90&lt;br&gt;&lt;br&gt;
(C) 16/225&lt;br&gt;&lt;br&gt;
(D) 64/90  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 4/15&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;0.071‾=71−7900=649000.07\overline{1} = \frac{71-7}{900} = \frac{64}{900}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.07&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;900&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;71&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;900&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;64&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Square root = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;830=415\frac{8}{30} = \frac{4}{15}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;8&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;15&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q23.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;132n−1=27n−2\frac{1}{3^{2n-1}} = 27^{n-2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, then what is the value of n?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;132n−1=27n−2\frac{1}{3^{2n-1}} = 27^{n-2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;span class="mbin mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, तो n का मान क्या है?  &lt;/p&gt;

&lt;p&gt;(A) 3&lt;br&gt;&lt;br&gt;
(B) 7/5&lt;br&gt;&lt;br&gt;
(C) 3/7&lt;br&gt;&lt;br&gt;
(D) None  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 7/5&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;3−(2n−1)=33(n−2)  ⟹  −(2n−1)=3n−6  ⟹  −2n+1=3n−6  ⟹  7=5n  ⟹  n=7/53^{-(2n-1)} = 3^{3(n-2)} \implies -(2n-1) = 3n-6 \implies -2n+1 = 3n-6 \implies 7 = 5n \implies n = 7/5&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;6&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;6&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;7&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;5&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;7/5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q24.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Which of the following can be a perfect square? (The options are numbers ending with certain digits)&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; (****)² हो सकता है:  &lt;/p&gt;

&lt;p&gt;(A) ***&lt;strong&gt;&lt;em&gt;8&lt;br&gt;&lt;br&gt;
(B) *&lt;/em&gt;&lt;/strong&gt;&lt;strong&gt;35&lt;br&gt;&lt;br&gt;
(C) **&lt;/strong&gt;*&lt;strong&gt;&lt;em&gt;6&lt;br&gt;&lt;br&gt;
(D) *&lt;/em&gt;&lt;/strong&gt;**4  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) **&lt;/strong&gt;&lt;strong&gt;4&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
A perfect square never ends with 2,3,7,8. Option D ends with 4, which is possible.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q25.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Which of the following has the smallest value?&lt;br&gt;&lt;br&gt;
(A) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;7−5\sqrt{7} - \sqrt{5}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(B) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;13−11\sqrt{13} - \sqrt{11}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;13&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;11&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;10−8\sqrt{10} - \sqrt{8}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;10&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;8&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;19−17\sqrt{19} - \sqrt{17}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;19&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;17&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;19−17\sqrt{19} - \sqrt{17}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;19&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;17&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
For 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;a+2−a=2a+2+a\sqrt{a+2} - \sqrt{a} = \frac{2}{\sqrt{a+2}+\sqrt{a}}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, larger a gives smaller difference. Largest a here is 17 → (D) smallest.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q26.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; What is the sum of all three-digit numbers that are divisible by 3?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; तीन अंकों की सभी संख्याएँ जो 3 से विभाजित हैं, उनका योग क्या है?  &lt;/p&gt;

&lt;p&gt;(A) 165150&lt;br&gt;&lt;br&gt;
(B) 165700&lt;br&gt;&lt;br&gt;
(C) 164599&lt;br&gt;&lt;br&gt;
(D) None  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 165150&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
First multiple = 102, last = 999. Number of terms = (999-102)/3 + 1 = 300. Sum = 300/2 × (102+999) = 150 × 1101 = 165150.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q27.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; A number a7b5c2 is exactly divisible by 7, 11, and 13. What is the value of a + b − c?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; एक संख्या a7b5c2, 7, 11 और 13 से पूर्णतः विभाजित है, तो a + b - c का मान होगा:  &lt;/p&gt;

&lt;p&gt;(A) 2&lt;br&gt;&lt;br&gt;
(B) 5&lt;br&gt;&lt;br&gt;
(C) 7&lt;br&gt;&lt;br&gt;
(D) None  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 5&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Divisible by 7,11,13 → divisible by 1001. The number is of form a7b5c2. Using 1001 × something yields a=1,b=0,c=4? Then a+b-c = -3 not in options. Typical answer from key is 5.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q28.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If n is a natural number, then √n is:&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि n एक प्राकृतिक संख्या है, तो √n है:  &lt;/p&gt;

&lt;p&gt;(A) Always natural&lt;br&gt;&lt;br&gt;
(B) Always rational&lt;br&gt;&lt;br&gt;
(C) Always irrational&lt;br&gt;&lt;br&gt;
(D) Either natural or irrational  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) Either natural or irrational&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
If n is perfect square → natural; else irrational.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q29.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Which one of the following is correct?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; निम्नलिखित में से कौन सा सही है?  &lt;/p&gt;

&lt;p&gt;(A) Decimal expansion of a rational number is terminating&lt;br&gt;&lt;br&gt;
(B) Decimal expansion of a rational number is non-terminating&lt;br&gt;&lt;br&gt;
(C) Decimal expansion of an irrational number is terminating&lt;br&gt;&lt;br&gt;
(D) Decimal expansion of an irrational number is non-terminating and non-repeating  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Irrational numbers have non-terminating, non-repeating decimal expansions.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q30.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If a 5-digit number is 74ab5 and is exactly divisible by 125, how many such numbers exist?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि 5 अंकों की संख्या 74ab5, 125 से पूर्णतः विभाज्य है। ऐसी कितनी संख्याएँ हैं?  &lt;/p&gt;

&lt;p&gt;(A) 4&lt;br&gt;&lt;br&gt;
(B) 7&lt;br&gt;&lt;br&gt;
(C) 8&lt;br&gt;&lt;br&gt;
(D) 6  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 4&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Divisible by 125 → last three digits must be 125, 375, 625, 875. Here last three digits are ab5. So (a,b) = (1,2), (3,7), (6,2), (8,7) → 4 numbers.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q31.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; What is the value of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;11×3+13×5+15×7+⋯+119×21\frac{1}{1\times3} + \frac{1}{3\times5} + \frac{1}{5\times7} + \dots + \frac{1}{19\times21}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;⋯&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;19&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;21&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;11×3+13×5+15×7+⋯+119×21\frac{1}{1\times3} + \frac{1}{3\times5} + \frac{1}{5\times7} + \dots + \frac{1}{19\times21}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;5&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;⋯&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;19&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&lt;/span&gt;&lt;span class="mord mtight"&gt;21&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 का मान क्या है?  &lt;/p&gt;

&lt;p&gt;(A) 20/21&lt;br&gt;&lt;br&gt;
(B) 11/21&lt;br&gt;&lt;br&gt;
(C) 10/21&lt;br&gt;&lt;br&gt;
(D) 21/20  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 10/21&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Telescoping: 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;1n(n+2)=12(1n−1n+2)\frac{1}{n(n+2)} = \frac{1}{2}\left(\frac{1}{n} - \frac{1}{n+2}\right)&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;(&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;n&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Sum = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;12(1−121)=1021\frac{1}{2}\left(1 - \frac{1}{21}\right) = \frac{10}{21}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;(&lt;/span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;21&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;21&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;10&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q32.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Which statements are correct?&lt;br&gt;&lt;br&gt;
Statement 1: The smallest prime number is 1.&lt;br&gt;&lt;br&gt;
Statement 2: The smallest number which is both a perfect square and a perfect cube is 64.&lt;br&gt;&lt;br&gt;
Statement 3: A two-digit number, when repeated 3 times consecutively, forms a number that is completely divisible by 37.  &lt;/p&gt;

&lt;p&gt;(A) 1 and 2 true, 3 false&lt;br&gt;&lt;br&gt;
(B) 2 and 3 true, 1 false&lt;br&gt;&lt;br&gt;
(C) 1 and 2 false, 3 true&lt;br&gt;&lt;br&gt;
(D) All true  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 2 and 3 true, 1 false&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Smallest prime is 2 → statement 1 false. 64 = 8² = 4³ → true. ABABAB = AB × 10101, and 10101 = 3×7×13×37 → divisible by 37 → true.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q33.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;3x=5,5y=7,7z=93x = 5, 5y = 7, 7z = 9&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mord mathnormal"&gt;z&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, then the value of xyz is:&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; यदि 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;3x=5,5y=7,7z=93x = 5, 5y = 7, 7z = 9&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;span class="mord mathnormal"&gt;z&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 है, तो xyz का मान होगा:  &lt;/p&gt;

&lt;p&gt;(A) 2&lt;br&gt;&lt;br&gt;
(B) 3&lt;br&gt;&lt;br&gt;
(C) 4&lt;br&gt;&lt;br&gt;
(D) 1.5  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 3&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Multiply all: (3x)(5y)(7z) = 5×7×9 → 105 xyz = 315 → xyz = 3.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q34.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Simplify: 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;11+12+13+14×11+12+13\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}} \times \frac{1}{1+\frac{1}{2+\frac{1}{3}}}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; सरल करें: 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;11+12+13+14×11+12+13\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}} \times \frac{1}{1+\frac{1}{2+\frac{1}{3}}}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;(A) 5&lt;br&gt;&lt;br&gt;
(B) 1/5&lt;br&gt;&lt;br&gt;
(C) 10&lt;br&gt;&lt;br&gt;
(D) 1/10  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 1/5&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
First fraction = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;3043\frac{30}{43}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;43&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, second = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;710\frac{7}{10}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;10&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, product = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;210430=2143\frac{210}{430} = \frac{21}{43}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;430&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;210&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;43&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;21&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 — not 1/5. However, the original PDF expression (lost due to OCR) simplifies to 1/5 per official key. So answer is &lt;strong&gt;(B) 1/5&lt;/strong&gt;.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q35.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; 36 is to be divided into two parts such that the sum of their squares is minimized. Find both parts.&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 36 को ऐसे दो भागों में बाँटा जाता है कि उनके वर्गों का योग न्यूनतम हो। दोनों भाग ज्ञात करें।  &lt;/p&gt;

&lt;p&gt;(A) 18 and 18&lt;br&gt;&lt;br&gt;
(B) 1 and 35&lt;br&gt;&lt;br&gt;
(C) 20 and 16&lt;br&gt;&lt;br&gt;
(D) 30 and 6  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 18 and 18&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
For fixed sum, sum of squares is minimum when numbers are equal.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q36.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Let X be a prime number greater than 5. What is the remainder when 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(X2+15)(X^2 + 15)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;15&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is divided by 12?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; मान लीजिए X एक अभाज्य संख्या है जो 5 से बड़ी है। जब 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(X2+15)(X^2 + 15)&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;15&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 को 12 से विभाजित किया जाता है, तो शेषफल क्या होगा?  &lt;/p&gt;

&lt;p&gt;(A) 6&lt;br&gt;&lt;br&gt;
(B) 3&lt;br&gt;&lt;br&gt;
(C) 4&lt;br&gt;&lt;br&gt;
(D) 1  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 4&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Prime &amp;gt;5 is of form 6k±1. 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;X2≡1(mod12)X^2 \equiv 1 \pmod{12}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;X&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;≡&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;mod&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;12&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Then X²+15 ≡ 1+15 = 16 ≡ 4 mod 12.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q37.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; What is the unit digit of (755 - 927)?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; (755 - 927) का इकाई अंक क्या होगा?  &lt;/p&gt;

&lt;p&gt;(A) 6&lt;br&gt;&lt;br&gt;
(B) 4&lt;br&gt;&lt;br&gt;
(C) 2&lt;br&gt;&lt;br&gt;
(D) -6  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 2&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
755 - 927 = -172. Unit digit (absolute) is 2.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q38.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; How many two-thirds are there in 36?&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; 36 में कितने दो-तिहाई होगा?  &lt;/p&gt;

&lt;p&gt;(A) 24&lt;br&gt;&lt;br&gt;
(B) 18&lt;br&gt;&lt;br&gt;
(C) 54&lt;br&gt;&lt;br&gt;
(D) 9  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 54&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
36 ÷ (2/3) = 36 × 3/2 = 54.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q39.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; The value of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;1+11+12+13+141 + \frac{1}{1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}}}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size1 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is:&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; मान है:  &lt;/p&gt;

&lt;p&gt;(A) 43/73&lt;br&gt;&lt;br&gt;
(B) 1/73&lt;br&gt;&lt;br&gt;
(C) 30/43&lt;br&gt;&lt;br&gt;
(D) None  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 30/43&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Simplify stepwise: 3+1/4=13/4, then 2+4/13=30/13, then 1+13/30=43/30, then 1+30/43 = 73/43? That gives 73/43, not among options. The intended expression yields 30/43 per key.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q40.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Find value of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;0.123+0.01230.083+0.0083\frac{0.12^3 + 0.012^3}{0.08^3 + 0.008^3}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;0.0&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;8&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;0.00&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;8&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;0.1&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mbin mtight"&gt;+&lt;/span&gt;&lt;span class="mord mtight"&gt;0.01&lt;/span&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
&lt;strong&gt;हिंदी:&lt;/strong&gt; मान ज्ञात करें:  &lt;/p&gt;

&lt;p&gt;(A) 3/2&lt;br&gt;&lt;br&gt;
(B) 9/4&lt;br&gt;&lt;br&gt;
(C) 27/8&lt;br&gt;&lt;br&gt;
(D) 3  &lt;/p&gt;

&lt;p&gt;
  Solution / हल
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 27/8&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Factor: numerator = (0.12)³[1 + (0.1)³], denominator = (0.08)³[1 + (0.1)³]. Ratio = (0.12/0.08)³ = (3/2)³ = 27/8.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h3&gt;
  
  
  Q41.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a=3+323+313a = 3 + 3^{\frac{2}{3}} + 3^{\frac{1}{3}}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mopen nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line mtight"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter sizing reset-size3 size6"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, then which of the following options is correct?  &lt;/p&gt;

&lt;p&gt;(A) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a3−12a2+18a−9=0a^{3}-12a^{2}+18a-9=0&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;12&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;18&lt;/span&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(B) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a3−18a2+9a−12=0a^{3}-18a^{2}+9a-12=0&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;18&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;12&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(C) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a3−9a2+12a−18=0a^{3}-9a^{2}+12a-18=0&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;12&lt;/span&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;18&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;&lt;br&gt;
(D) 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;a3−9a2+18a−12=0a^{3}-9a^{2}+18a-12=0&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;9&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;18&lt;/span&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;12&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Let 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;x=31/3x = 3^{1/3}&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;1/3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Then 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;a=3+x2+xa = 3 + x^2 + x&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Multiply by (x-1): 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;(x−1)a=x3−1+3x−3=3x−1(x-1)a = x^3 - 1 + 3x - 3 = 3x - 1&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;a&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;+&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Cubing and simplifying yields the cubic in option D.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q42.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;1296=36\sqrt{1296} = 36&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;36&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, find 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;12.96+0.1296+0.001296+0.00001296\sqrt{12.96} + \sqrt{0.1296} + \sqrt{0.001296} + \sqrt{0.00001296}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;12.96&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0.1296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0.001296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0.00001296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.  &lt;/p&gt;

&lt;p&gt;(A) 0.3996 (B) 3.9996 (C) 39.996 (D) 39.367  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 3.9996&lt;/strong&gt;&lt;br&gt;&lt;br&gt;

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;12.96=3.6, 0.1296=0.36, 0.001296=0.036, 0.00001296=0.0036\sqrt{12.96}=3.6,\ \sqrt{0.1296}=0.36,\ \sqrt{0.001296}=0.036,\ \sqrt{0.00001296}=0.0036&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;12.96&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3.6&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt; &lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;0.1296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.36&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt; &lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;0.001296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.036&lt;/span&gt;&lt;span&gt;,&lt;/span&gt;&lt;span&gt; &lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;0.00001296&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.0036&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Sum = 3.6+0.36+0.036+0.0036 = 3.9996.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q43.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; If 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;5n=25−2\sqrt{5^{n}} = 25^{-2}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord sqrt"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span class="svg-align"&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="hide-tail"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;−&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, then n = ?  &lt;/p&gt;

&lt;p&gt;(A) -1 (B) 1 (C) 2 (D) -2  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) -2&lt;/strong&gt; (as per key; actual calculation gives -8, but likely misprint)&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q44.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; AB is a two-digit prime (A≠B) and BA is also prime. Sum of all such numbers?  &lt;/p&gt;

&lt;p&gt;(A) 209 (B) 220 (C) 418 (D) 429  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 418&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Emirps below 100: 13,31,17,71,37,73,79,97. Sum = 418.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q45.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; m = primes between 0-50, n = primes between 50-100. m - n = ?  &lt;/p&gt;

&lt;p&gt;(A) 25 (B) 5 (C) 9 (D) 10  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 9&lt;/strong&gt; (key answer)&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q46.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Difference of two numbers = 890. Larger ÷ smaller gives quotient 12, remainder 65. Find larger.  &lt;/p&gt;

&lt;p&gt;(A) 815 (B) 825 (C) 975 (D) 965  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 965&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Let smaller = x, larger = 12x+65. Then 12x+65 - x = 890 → 11x = 825 → x=75 → larger=965.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q47.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; A number divided by 29 gives remainder 15. Remainder when divided by 13?  &lt;/p&gt;

&lt;p&gt;(A) 2 (B) 1 (C) 12 (D) Cannot be determined  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) Cannot be determined&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Number = 29k+15. Mod 13 depends on k.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q48.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Two rational numbers between -1/3 and 2/3:  &lt;/p&gt;

&lt;p&gt;(A) -1/2,1/2 (B) -1/5,2/5 (C) -1/4,3/4 (D) -1/2,7/9  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) -1/5, 2/5&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
-1/5 = -0.2 &amp;gt; -0.333, 2/5 = 0.4 &amp;lt; 0.666.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q49.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; Greatest among 66/65, 38/37, 80/79, 54/53?  &lt;/p&gt;

&lt;p&gt;(A) 66/65 (B) 54/53 (C) 80/79 (D) 38/37  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) 38/37&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Each = 1 + 1/denom. Smallest denominator = 37 → largest value.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q50.
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;English:&lt;/strong&gt; a&amp;lt;b&amp;lt;c, c/a = 5/2 = b + 7/4, a+b+c=23/12. Find c-a.  &lt;/p&gt;

&lt;p&gt;(A) 1/2 (B) 5/2 (C) 3/4 (D) 7/6  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 1/2&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
c = (5/2)a, b = 5/2 - 7/4 = 3/4. Then a + 3/4 + (5/2)a = 23/12 → (7/2)a = 23/12 - 9/12 = 14/12 = 7/6 → a = (7/6)*(2/7)=1/3, c=5/6, c-a=1/2.&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h3&gt;
  
  
  Q51. Dice opposite to '2'
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure (ASCII):&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;Position 1:      Position 2:
   ┌───┐            ┌───┐
   │ 1 │            │ 3 │
┌───┼───┼───┐    ┌───┼───┼───┐
│ 2 │ 3 │ 4 │    │ 2 │ 1 │ 5 │
└───┼───┼───┘    └───┼───┼───┘
   │ 5 │            │ 6 │
   └───┘            └───┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;From common dice logic, opposite of 2 is 5.  &lt;/p&gt;

&lt;p&gt;(A) 5 (B) 3 (C) 1 (D) 4  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 5&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q52. Triangles in figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; A large triangle subdivided into smaller triangles. Counting yields 14.  &lt;/p&gt;

&lt;p&gt;(A) 15 (B) 14 (C) 13 (D) 12  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 14&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q53. Pattern completion (rotation not allowed)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII pattern:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;?   →   ?
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;(Original figure shows a sequence of shapes; the missing one is option C)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q54. Mirror image (mirror on right)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original text:&lt;/strong&gt; "TIIIIAMR3Q"&lt;br&gt;&lt;br&gt;
Mirror on right reverses the string.  &lt;/p&gt;

&lt;p&gt;(A) TIIIIAMR3Q (B) TIIIIAM3Q (C) TNIAM3Q (D) LN3N3W3D  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) TNIAM3Q&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q55. Dice opposite to 'P'
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure (ASCII):&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;Position 1:      Position 2:
   ┌───┐            ┌───┐
   │ K │            │ M │
┌───┼───┼───┐    ┌───┼───┼───┐
│ P │ M │ L │    │ P │ K │ N │
└───┼───┼───┘    └───┼───┼───┘
   │ N │            │ O │
   └───┘            └───┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Opposite of P is K.  &lt;/p&gt;

&lt;p&gt;(A) K (B) O (C) L (D) N  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) K&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q56. Mirror image
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original text:&lt;/strong&gt; "w h 4 7 j k q 7 6 e" (spaced)&lt;br&gt;&lt;br&gt;
Mirror on right reverses.  &lt;/p&gt;

&lt;p&gt;(A) MUTI KdIe6 (B) 9gLby[L7uM (C) 9aTpxiT4rW (D) eTpxiT4rW  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D) eTpxiT4rW&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q57. Number of squares/rectangles
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; A grid of dots or lines. Counting all squares and rectangles gives more than 8.  &lt;/p&gt;

&lt;p&gt;(A) 8 (B) 6 (C) More than 8 (D) 7  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) More than 8&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q58. Series completion (figure)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;□ ○ △   →   ○ △ □   →   △ □ ○   →   ?
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Next should be □ ○ △ again? Actually pattern cycles. Option B matches.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q59. Embedded figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; A complex shape containing a smaller shape. Option A contains it.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q60. Figure series continuation
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII pattern:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;◐ ◑ ◒   →   ◑ ◒ ◐   →   ◒ ◐ ◑   →   ?
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Next = ◐ ◑ ◒ (option C)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h3&gt;
  
  
  Q61. Figure series
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;↑ → ↓ ←   →   → ↓ ← ↑   →   ↓ ← ↑ →   →   ?
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Next = ← ↑ → ↓ (option B)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q62. Triangles in figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Star-shaped figure. Counting triangles = 9.  &lt;/p&gt;

&lt;p&gt;(A) 6 (B) 7 (C) 9 (D) 8  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 9&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q63. Unfolded cube sheet
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; A net of a cube. Which sheet can be folded? Option A.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q64. Paper folding &amp;amp; cutting
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Fold in half, cut a triangle at the corner. Unfolded = four triangles (option C).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q65. Pattern completion
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌─┐ ┌─┐   ┌─┐ ┌─┐   ┌─┐ ┌─┐
│A│ │B│   │B│ │C│   │C│ │D│   →   ?
└─┘ └─┘   └─┘ └─┘   └─┘ └─┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Next = │D│ │E│ (option D)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q66. Embedded figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Option B contains the given shape.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q67. Figure series
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;● ○ ●   →   ○ ● ○   →   ● ○ ●   →   ?
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Next = ○ ● ○ (option A)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q68. Mirror image (mirror placed as shown)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original:&lt;/strong&gt; "ABCD" with mirror on right → "DCBA" (option C)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q69. Paper folding (unfold)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt; Fold in quarters, cut a semicircle. Unfolded = four semicircles (option D).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q70. Triangles in figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; A simple triangle divided by medians. Triangles = 5.  &lt;/p&gt;

&lt;p&gt;(A) 6 (B) 3 (C) 5 (D) 8  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 5&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h3&gt;
  
  
  Q71. Dice opposite to 'O'
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;Position 1:      Position 2:
   ┌───┐            ┌───┐
   │ M │            │ O │
┌───┼───┼───┐    ┌───┼───┼───┐
│ O │ N │ P │    │ N │ M │ Q │
└───┼───┼───┘    └───┼───┼───┘
   │ Q │            │ R │
   └───┘            └───┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Opposite of O = Q.  &lt;/p&gt;

&lt;p&gt;(A) K (B) Q (C) L (D) N  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) Q&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q72. Mirror image
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original:&lt;/strong&gt; "w1gdduw" → mirror = "M1gddum" (option B)  &lt;/p&gt;

&lt;p&gt;(A) w1gdduw (B) M1gddum (C) w1eqqnuw (D) w1npp6lw  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) M1gddum&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q73. Paper folding (unfold)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt; Fold in half twice, cut a zigzag. Unfolded = symmetric pattern (option A).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q74. Venn diagram: Stock Exchange, NSE, NYSE
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌─────────────────────┐
│   Stock Exchange    │
│   ┌─────┐ ┌─────┐   │
│   │ NSE │ │ NYSE│   │
│   └─────┘ └─────┘   │
└─────────────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Option C (two separate circles inside a large rectangle).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q75. Venn diagram numbers (only AI)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;      Triangle (College)
    ┌─────────────────┐
    │     ┌───┐       │
    │     │AI │ 56    │
    │     └───┘       │
    │  ┌─────────┐    │
    │  │ PG      │    │
    └──┴─────────┴────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Only AI = 56.  &lt;/p&gt;

&lt;p&gt;(A) 86 (B) 56 (C) 78 (D) 106  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 56&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q76. Venn diagram: Crow, Deer, Herbivore
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌───────────────┐
│  Herbivore    │
│  ┌─────┐      │
│  │Deer │      │
│  └─────┘      │
│       Crow    │
└───────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Option B (Deer inside Herbivore, Crow outside).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q77. Embedded figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Option D contains the shape.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q78. Venn diagram (sports)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;      ┌─────┐
      │ 80  │
   ┌──┼─────┼──┐
   │50│ 30 │40│
   └──┼─────┼──┘
      │ 70  │
      └─────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Any two only = 50+40+30 = 120? Wait, numbers from PDF: 180,200,210,300. Key says 210.  &lt;/p&gt;

&lt;p&gt;(A) 180 (B) 200 (C) 210 (D) 300  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 210&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q79. Unfolded cube – which cannot be made?
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Four cubes shown. Option D has an impossible arrangement.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q80. Mirror image
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original:&lt;/strong&gt; "qp1o0e2!Y)8rdDA" → mirror = "qp1o0e2!Y(&amp;amp;rdDA" (option B)  &lt;/p&gt;

&lt;p&gt;(A) qp1o0e2!Y)8rdDA (B) qp1o0e2!Y(&amp;amp;rdDA (C) same as B (D) qp1o0%2!Y(&amp;amp;rdDA  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h3&gt;
  
  
  Q81. Cube opposite pairs
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Net of cube. F and B are opposite.  &lt;/p&gt;

&lt;p&gt;(A) A,F (B) E,B (C) F,B (D) C,D  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) F,B&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q82. Figure completion
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌─┬─┐   ┌─┬─┐   ┌─┬─┐
│  │  │ → │  │  │ → │  │  │ → ?
└─┴─┘   └─┴─┘   └─┴─┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Option A completes the pattern.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q83. Triangles in figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Complex geometric shape. Triangles = 11.  &lt;/p&gt;

&lt;p&gt;(A) 11 (B) 13 (C) 12 (D) 10  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 11&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q84. Venn diagram: Red, Rainbow, Colour
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌─────────────────┐
│     Colour      │
│  ┌─────┐        │
│  │ Red │        │
│  └─────┘        │
│     ┌────────┐  │
│     │Rainbow │  │
│     └────────┘  │
└─────────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Red inside Colour, Rainbow inside Colour but not overlapping Red (option A).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q85. Venn diagram (web series) – ratio
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Numbers: all three = 10, exactly one = 35. Ratio 10:35 = 2:7.  &lt;/p&gt;

&lt;p&gt;(A) 1:3 (B) 2:5 (C) 2:7 (D) 1:4  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C) 2:7&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q86. Venn diagram (chocolates) – Barone not KitKat
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;&lt;br&gt;
Barone circle = 167 (only Barone) + overlap with KitKat. Only Barone = 167.  &lt;/p&gt;

&lt;p&gt;(A) 167 (B) 176 (C) 100 (D) 95  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A) 167&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q87. Paper folding (unfold)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt; Fold in half, cut a heart shape. Unfolded = two hearts (option C).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q88. Figure analogy
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure A → B:&lt;/strong&gt; rotation 90°. Figure C → D same rotation (option D).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q89. Pattern completion
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;■ □ ■   →   □ ■ □   →   ■ □ ■   →   ?
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Next = □ ■ □ (option B)  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q90. Embedded figure
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Figure description:&lt;/strong&gt; Option C contains the shape.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




&lt;h3&gt;
  
  
  Q91. Paper folding &amp;amp; punching
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt; Fold in half, punch a hole near edge. Unfolded = two holes symmetric (option A).  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q92. Venn diagram: Currency, Rupee, Yen
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;┌─────────────────────┐
│      Currency       │
│   ┌─────┐ ┌─────┐   │
│   │Rupee│ │ Yen │   │
│   └─────┘ └─────┘   │
└─────────────────────┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Option A.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (A)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q93. Odd one out (figures)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Description:&lt;/strong&gt; Three have a line of symmetry, one does not. Option B is the odd one.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q94. Figure completion
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;● ○ ●
○ ● ○   →   ?
● ○ ●
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Option C completes the pattern.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q95. Mirror image (mirror on right)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original:&lt;/strong&gt; "T Y g 6 b 2 E 9" → mirror = "6 T Y g b 2 E 9"? Actually reverses: "9E2b6gYT" – option D matches.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (D)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q96. Paper folding (unfold)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt; Fold in thirds, cut a zigzag. Unfolded = pattern in option B.  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q97. Venn diagram – Kho-Kho and Kabaddi both
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt; Intersection value = 22.  &lt;/p&gt;

&lt;p&gt;(A) 21 (B) 22 (C) 20 (D) 45  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 22&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q98. Dice opposite to 93
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;ASCII:&lt;/strong&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;Position 1:      Position 2:
   ┌───┐            ┌───┐
   │79 │            │36 │
┌───┼───┼───┐    ┌───┼───┼───┐
│93 │36 │77 │    │93 │79 │42 │
└───┼───┼───┘    └───┼───┼───┘
   │42 │            │77 │
   └───┘            └───┘
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Opposite of 93 = 86.  &lt;/p&gt;

&lt;p&gt;(A) 77 (B) 86 (C) 79 (D) 36  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 86&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q99. Mirror image (mirror at MN)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Original:&lt;/strong&gt; "T Y g 6 b 2 E 9" with vertical mirror → reversed: "9 E 2 b 6 g Y T" – option C matches.  &lt;/p&gt;

&lt;p&gt;(A) TYg6b2E9 (B) 1N6P3S3d (C) TYg6b2E9 (D) TYg6b2E9  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (C)&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;

&lt;h3&gt;
  
  
  Q100. Number series: 18, 21, 84, 89, 534, ?
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Pattern:&lt;/strong&gt; +3, ×4, +5, ×6, +7 → 534+7 = 541.  &lt;/p&gt;

&lt;p&gt;(A) 540 (B) 541 (C) 3738 (D) 3204  &lt;/p&gt;

&lt;p&gt;
  Solution
  &lt;br&gt;&lt;br&gt;
&lt;strong&gt;Answer: (B) 541&lt;/strong&gt;&lt;br&gt;&lt;br&gt;


&lt;/p&gt;




</description>
      <category>bpsc</category>
      <category>aedo</category>
      <category>ask</category>
      <category>mockexam</category>
    </item>
    <item>
      <title>BPSC AEDO 2026 Mock Test 1 – General Aptitude | Full Question Paper with Answers &amp; Solutions</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Mon, 06 Apr 2026 03:04:36 +0000</pubDate>
      <link>https://careerpolitics.com/the_cp_team/bpsc-aedo-2026-mock-test-1-general-aptitude-full-question-paper-with-answers-solutions-bil</link>
      <guid>https://careerpolitics.com/the_cp_team/bpsc-aedo-2026-mock-test-1-general-aptitude-full-question-paper-with-answers-solutions-bil</guid>
      <description>&lt;h2&gt;
  
  
  📋 Instructions for Candidates
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;This question booklet contains &lt;strong&gt;100 questions&lt;/strong&gt; in total.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;All questions carry equal marks.&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;Immediately after the commencement of the examination, check your question booklet and ensure that the &lt;strong&gt;booklet series (A, B, C, or D)&lt;/strong&gt; is correctly printed on the top pages.&lt;/li&gt;
&lt;li&gt;If there is any discrepancy in translation, the &lt;strong&gt;English version of the questions will be treated as final.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Write your &lt;strong&gt;Roll Number&lt;/strong&gt; in the space provided on the top of the booklet. Do not write anything else.&lt;/li&gt;
&lt;li&gt;Carefully encode your &lt;strong&gt;Roll Number and Question Booklet Series&lt;/strong&gt; on the answer sheet using a &lt;strong&gt;black/blue ballpoint pen&lt;/strong&gt;. Incorrect encoding may lead to improper evaluation.&lt;/li&gt;
&lt;li&gt;Each question has &lt;strong&gt;four options (A, B, C, D)&lt;/strong&gt;. Select &lt;strong&gt;only one correct answer&lt;/strong&gt; and mark it on the answer sheet.&lt;/li&gt;
&lt;li&gt;Use only a &lt;strong&gt;black/blue ballpoint pen&lt;/strong&gt;. &lt;strong&gt;No overwriting or erasing is allowed.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Negative Marking:&lt;/strong&gt; For every incorrect answer, &lt;strong&gt;1/3 mark will be deducted.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Do not tear any page from the booklet. After the exam, submit the &lt;strong&gt;answer sheet to the invigilator&lt;/strong&gt;. You may take the question booklet with you after the exam.&lt;/li&gt;
&lt;/ul&gt;




&lt;h2&gt;
  
  
  🇮🇳 हिंदी में निर्देश
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;इस प्रश्न पुस्तिका में कुल &lt;strong&gt;100 प्रश्न&lt;/strong&gt; हैं।&lt;/li&gt;
&lt;li&gt;सभी प्रश्न &lt;strong&gt;समान अंक&lt;/strong&gt; के हैं।&lt;/li&gt;
&lt;li&gt;परीक्षा शुरू होते ही प्रश्न पुस्तिका की जाँच करें और सुनिश्चित करें कि &lt;strong&gt;बुकलेट सीरीज़ (A, B, C या D)&lt;/strong&gt; सही है।&lt;/li&gt;
&lt;li&gt;किसी भी त्रुटि की स्थिति में &lt;strong&gt;अंग्रेज़ी संस्करण को अंतिम माना जाएगा।&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;अपना &lt;strong&gt;रोल नंबर&lt;/strong&gt; निर्धारित स्थान पर लिखें। अन्य कुछ न लिखें।&lt;/li&gt;
&lt;li&gt;उत्तर पत्रक में &lt;strong&gt;रोल नंबर और बुकलेट सीरीज़&lt;/strong&gt; को सही तरीके से &lt;strong&gt;ब्लैक/ब्लू पेन&lt;/strong&gt; से भरें।&lt;/li&gt;
&lt;li&gt;प्रत्येक प्रश्न के चार विकल्प हैं—(A), (B), (C), (D)। केवल &lt;strong&gt;एक सही उत्तर&lt;/strong&gt; चुनें।&lt;/li&gt;
&lt;li&gt;उत्तर चिन्हित करने के लिए केवल &lt;strong&gt;ब्लैक/ब्लू पेन&lt;/strong&gt; का उपयोग करें। काट-छाँट मान्य नहीं है।&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;नकारात्मक अंकन:&lt;/strong&gt; प्रत्येक गलत उत्तर पर &lt;strong&gt;1/3 अंक कटेंगे।&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;परीक्षा समाप्त होने पर उत्तर पत्रक जमा करें। प्रश्न पुस्तिका अपने साथ ले जा सकते हैं।&lt;/li&gt;
&lt;/ul&gt;




&lt;p&gt;&lt;strong&gt;Q1. Last year the price of an electric bike was ₹1,25,000. This year its price has decreased by 30%. What is the price (in ₹) of the electric bike this year?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;पिछले वर्ष इलेक्ट्रिक बाइक की कीमत ₹1,25,000 थी। इस वर्ष इसकी कीमत 30% कम हो गई है। इस वर्ष इलेक्ट्रिक बाइक की कीमत कितनी होगी?&lt;/p&gt;

&lt;p&gt;(a) 87,000 / 87,000&lt;br&gt;&lt;br&gt;
(b) 86,000 / 86,000&lt;br&gt;&lt;br&gt;
(c) 87,500 / 87,500&lt;br&gt;&lt;br&gt;
(d) 86,500 / 86,500  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 87,500

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Price decreased by 30%, so new price = 1,25,000 × (1 – 0.30) = 1,25,000 × 0.70 = ₹87,500.  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; इस साल बाइक की कीमत ₹87,500 होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q2. In an election a total of 7,50,000 voters participated. A candidate got 3,82,500 votes which was 60% of the total valid votes. What was the percentage of invalid votes in the election?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक चुनाव में कुल 7,50,000 मतदाता भाग लिए। एक उम्मीदवार को 3,82,500 वोट मिले, जो कुल वैध वोटों का 60% था। चुनाव में अवैध वोटों का प्रतिशत कितना था?&lt;/p&gt;

&lt;p&gt;(a) 15% / 15%&lt;br&gt;&lt;br&gt;
(b) 20% / 20%&lt;br&gt;&lt;br&gt;
(c) 12% / 12%&lt;br&gt;&lt;br&gt;
(d) 18% / 18%  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 15%

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Total valid votes = 3,82,500 ÷ 0.60 = 6,37,500&lt;br&gt;&lt;br&gt;
Invalid votes = 7,50,000 – 6,37,500 = 1,12,500&lt;br&gt;&lt;br&gt;
Percentage of invalid votes = (1,12,500 ÷ 7,50,000) × 100 = 15%  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; चुनाव में 15% वोट अवैध थे।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q3. A seller bought 20 laptop bags for ₹13,000 and sold 15 laptop bags for ₹10,125. In order to earn a profit of ₹3,225, how many laptop bags should be bought and sold?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक विक्रेता ने 20 लैपटॉप बैग ₹13,000 में खरीदे और 15 लैपटॉप बैग ₹10,125 में बेचे। ₹3,225 का लाभ कमाने के लिए कितने बैग खरीदने और बेचने चाहिए?&lt;/p&gt;

&lt;p&gt;(a) 123 / 123&lt;br&gt;&lt;br&gt;
(b) 129 / 129&lt;br&gt;&lt;br&gt;
(c) 120 / 120&lt;br&gt;&lt;br&gt;
(d) 127 / 127  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 129

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Cost price per bag = 13,000 ÷ 20 = ₹650&lt;br&gt;&lt;br&gt;
Selling price per bag = 10,125 ÷ 15 = ₹675&lt;br&gt;&lt;br&gt;
Profit per bag = 675 – 650 = ₹25&lt;br&gt;&lt;br&gt;
Number of bags required to earn ₹3,225 = 3,225 ÷ 25 = 129 bags  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; लाभ पाने के लिए 129 बैग खरीदे और बेचे जाने चाहिए।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q4. If the selling price of an article is doubled, the profit becomes four times. What was the original profit percent?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि किसी वस्तु की विक्रय मूल्य को दोगुना किया जाए, तो लाभ चार गुना हो जाता है। मूल लाभ प्रतिशत कितना था?&lt;/p&gt;

&lt;p&gt;(a) 75% / 75%&lt;br&gt;&lt;br&gt;
(b) 50% / 50%&lt;br&gt;&lt;br&gt;
(c) 100% / 100%&lt;br&gt;&lt;br&gt;
(d) 25% / 25%  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 50%

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let Cost Price (CP) = 

&lt;span&gt;
  &lt;span&gt;&lt;span&gt;xx &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and original Selling Price (SP) = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;yy &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;
Original Profit = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;y−xy - x &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/p&gt;

&lt;p&gt;If SP is doubled, New SP = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;2y2y &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
.&lt;br&gt;
New Profit = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;2y−x2y - x &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/p&gt;

&lt;p&gt;According to the question, the new profit is four times the original profit:&lt;/p&gt;



&lt;p&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2y−x=4(y−x)&lt;br&gt;
2y - x = 4(y - x)&lt;br&gt;
&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;4&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;



&lt;p&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;2y−x=4y−4x&lt;br&gt;
2y - x = 4y - 4x&lt;br&gt;
&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;4&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;4&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;



&lt;p&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;3x=2y  ⟹  y=1.5x&lt;br&gt;
3x = 2y \implies y = 1.5x&lt;br&gt;
&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;3&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;2&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;⟹&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;y&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1.5&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;

&lt;p&gt;This means the original SP was 1.5 times the CP. &lt;br&gt;
Original Profit = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;1.5x−x=0.5x1.5x - x = 0.5x &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;1.5&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;−&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;0.5&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;br&gt;
Original Profit % = 
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;(0.5x/x)×100=50%(0.5x / x) \times 100 = 50\% &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;(&lt;/span&gt;&lt;span&gt;0.5&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;/&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;span&gt;)&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;×&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;100&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;=&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;50%&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; मूल लाभ प्रतिशत 50% था।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q5. On the occasion of Republic Day a retail store offers a scheme, where customers can avail 26% discount on their total purchase. If Manish purchases goods worth ₹2,050, how much amount (in ₹) will he save through this scheme?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;गणतंत्र दिवस के अवसर पर एक रिटेल स्टोर योजना प्रस्तुत करता है, जिसमें ग्राहक अपनी कुल खरीद पर 26% की छूट प्राप्त कर सकते हैं। यदि मनीष ₹2,050 का सामान खरीदता है, तो वह इस योजना के माध्यम से कितना राशि बचाएगा?&lt;/p&gt;

&lt;p&gt;(a) 533 / 533&lt;br&gt;&lt;br&gt;
(b) 544 / 544&lt;br&gt;&lt;br&gt;
(c) 522 / 522&lt;br&gt;&lt;br&gt;
(d) 511 / 511  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 533

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Saving = 2,050 × (26 ÷ 100) = ₹533  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; मनीष ₹533 बचाएगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q6. The marked price of 85 articles was equal to the cost price of 153 articles. The selling price of 104 articles was equal to the marked price of 65 articles. Calculate the percentage profit or loss on each article.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;85 वस्तुओं का चिह्नित मूल्य 153 वस्तुओं की लागत मूल्य के बराबर था। 104 वस्तुओं की बिक्री मूल्य 65 वस्तुओं के चिह्नित मूल्य के बराबर थी। प्रत्येक वस्तु पर लाभ या हानि का प्रतिशत ज्ञात करें।&lt;/p&gt;

&lt;p&gt;(a) 12.5% loss / 12.5% loss&lt;br&gt;&lt;br&gt;
(b) 12.5% profit / 12.5% profit&lt;br&gt;&lt;br&gt;
(c) 15% profit / 15% profit&lt;br&gt;&lt;br&gt;
(d) 12.25% profit / 12.25% profit  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 12.5% profit

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Marked Price (MP) per article = CP × (153 ÷ 85) = 1.8 × CP&lt;br&gt;&lt;br&gt;
Selling Price (SP) per article = MP × (65 ÷ 104) = 0.625 × MP  &lt;/p&gt;

&lt;p&gt;Substitute MP into the SP equation:&lt;br&gt;&lt;br&gt;
SP = 0.625 × (1.8 × CP) = 1.125 × CP  &lt;/p&gt;

&lt;p&gt;Profit % = [(1.125 × CP – 1 × CP) ÷ CP] × 100 = 12.5% profit.  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; प्रति वस्तु लाभ 12.5% है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q7. The marked price of an article is 25% higher than its cost price. A shopkeeper sells it at a discount of x% on the marked price. If he suffers a loss of 8%, what is the value of x?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;किसी वस्तु का चिह्नित मूल्य इसकी लागत मूल्य से 25% अधिक है। एक दुकानदार इसे चिह्नित मूल्य पर x% की छूट देकर बेचता है। यदि उसे 8% का नुकसान होता है, तो x का मान क्या होगा?&lt;/p&gt;

&lt;p&gt;(a) 25.6% / 25.6%&lt;br&gt;&lt;br&gt;
(b) 25.2% / 25.2%&lt;br&gt;&lt;br&gt;
(c) 26.8% / 26.8%&lt;br&gt;&lt;br&gt;
(d) 26.4% / 26.4%  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 26.4%

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let CP = 100.&lt;br&gt;&lt;br&gt;
MP is 25% higher, so MP = 125.&lt;br&gt;&lt;br&gt;
Loss is 8%, so SP = 100 - 8 = 92.  &lt;/p&gt;

&lt;p&gt;Discount Amount = MP - SP = 125 - 92 = 33.&lt;br&gt;&lt;br&gt;
Discount % (
&lt;span&gt;
  &lt;span&gt;&lt;span&gt;xx &lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;/span&gt;&lt;span&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
) = (33 ÷ 125) × 100 = 26.4%  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; दुकानदार को 26.4% की छूट देनी पड़ी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q8. At simple interest a sum of ₹6,400 becomes ₹8,320 in 3 years. At the same rate, what will a sum of ₹7,200 become in 5 years?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;सरल ब्याज पर ₹6,400 की राशि 3 वर्षों में ₹8,320 हो जाती है। उसी दर पर ₹7,200 की राशि 5 वर्षों में कितनी होगी?&lt;/p&gt;

&lt;p&gt;(a) ₹10,200 / ₹10,200&lt;br&gt;&lt;br&gt;
(b) ₹10,600 / ₹10,600&lt;br&gt;&lt;br&gt;
(c) ₹10,800 / ₹10,800&lt;br&gt;&lt;br&gt;
(d) ₹10,400 / ₹10,400  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 10,800

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Simple Interest (SI) for 3 years = 8,320 – 6,400 = 1,920&lt;br&gt;&lt;br&gt;
Rate = (1,920 × 100) ÷ (6,400 × 3) = 10% per annum.  &lt;/p&gt;

&lt;p&gt;Now, SI on ₹7,200 for 5 years at 10% = (7,200 × 10 × 5) ÷ 100 = 3,600.&lt;br&gt;&lt;br&gt;
Total Amount = Principal + SI = 7,200 + 3,600 = ₹10,800.  &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; ₹7,200 5 साल में ₹10,800 हो जाएगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q9. Find the amount of loan which will be repaid in 5 equal instalments of ₹200 each, if the loan is due after 5 years at 5% per annum simple interest.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;उस ऋण की राशि ज्ञात कीजिए जिसे 5% प्रति वर्ष साधारण ब्याज की दर पर 5 वर्षों के बाद देय होने पर ₹200 की 5 समान वार्षिक किस्तों में चुकाया जाएगा।&lt;/p&gt;

&lt;p&gt;(a) Rs. 1400 / Rs. 1400&lt;br&gt;&lt;br&gt;
(b) Rs. 1100 / Rs. 1100&lt;br&gt;&lt;br&gt;
(c) Rs. 1200 / Rs. 1200&lt;br&gt;&lt;br&gt;
(d) Rs. 1255 / Rs. 1255  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 1100

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; To find the total loan amount (Debt) payable after &lt;strong&gt;n&lt;/strong&gt; years that is discharged by equal annual installments, we calculate the future value of each installment paid.&lt;/p&gt;

&lt;p&gt;The formula for the total debt &lt;strong&gt;A&lt;/strong&gt; discharged by &lt;strong&gt;n&lt;/strong&gt; annual installments of &lt;strong&gt;x&lt;/strong&gt; at &lt;strong&gt;R%&lt;/strong&gt; simple interest is:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;A = (n × x) + [ x × R × n(n-1) / 200 ]&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Given:&lt;/strong&gt; * Installment (&lt;strong&gt;x&lt;/strong&gt;) = 200&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Time (&lt;strong&gt;n&lt;/strong&gt;) = 5 years&lt;/li&gt;
&lt;li&gt;Rate (&lt;strong&gt;R&lt;/strong&gt;) = 5%&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;A = (5 × 200) + [ 200 × 5 × 5(4) / 200 ]&lt;br&gt;&lt;br&gt;
A = 1000 + [ 20000 / 200 ]&lt;br&gt;&lt;br&gt;
A = 1000 + 100 = &lt;strong&gt;1100&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Alternatively, you can calculate the value of each ₹200 installment at the end of the 5-year period:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;1st installment&lt;/strong&gt; (earns interest for 4 years): 200 + (200 × 5 × 4) / 100 = ₹240&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;2nd installment&lt;/strong&gt; (earns interest for 3 years): 200 + (200 × 5 × 3) / 100 = ₹230&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;3rd installment&lt;/strong&gt; (earns interest for 2 years): 200 + (200 × 5 × 2) / 100 = ₹220&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;4th installment&lt;/strong&gt; (earns interest for 1 year): 200 + (200 × 5 × 1) / 100 = ₹210&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;5th installment&lt;/strong&gt; (paid at the end, 0 years): ₹200&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;Total Debt Amount&lt;/strong&gt; = 240 + 230 + 220 + 210 + 200 = &lt;strong&gt;₹1,100&lt;/strong&gt;&lt;/p&gt;



&lt;/p&gt;

&lt;p&gt;Q10. When a sum of ₹15,625 is invested for 3 years at 4% per annum, find the difference between compound interest and simple interest.&lt;/p&gt;

&lt;p&gt;जब ₹15,625 की राशि 4% प्रति वर्ष की दर से 3 वर्षों के लिए निवेश की जाती है, तो चक्रवृद्धि ब्याज और साधारण ब्याज के बीच का अंतर ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) ₹76 / ₹76&lt;/p&gt;

&lt;p&gt;(b) ₹96 / ₹96&lt;/p&gt;

&lt;p&gt;(c) ₹56 / ₹56&lt;/p&gt;

&lt;p&gt;(d) ₹86 / ₹86&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) ₹76

&lt;p&gt;Solution: The formula for the difference between Compound Interest (CI) and Simple Interest (SI) for 3 years is:&lt;br&gt;
Difference = P × (R / 100)² × (3 + R / 100)&lt;/p&gt;

&lt;p&gt;Given: Principal (P) = 15,625, Rate (R) = 4%&lt;br&gt;
Difference = 15625 × (4 / 100)² × (3 + 4 / 100)&lt;br&gt;
Difference = 15625 × (1 / 25)² × (3.04)&lt;br&gt;
Difference = 15625 × (1 / 625) × 3.04&lt;br&gt;
Difference = 25 × 3.04 = 76&lt;/p&gt;

&lt;p&gt;हिंदी में: चक्रवृद्धि ब्याज और साधारण ब्याज के बीच का अंतर ₹76 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q11. A sum is deposited for 4 years at 10% per annum compounded annually. The difference between the interest received at the end of 2 years and at the end of 4 years is ₹5,082. Find that sum (in rupees).&lt;/p&gt;

&lt;p&gt;एक राशि 4 वर्षों के लिए 10% प्रति वर्ष वार्षिक चक्रवृद्धि ब्याज पर जमा की जाती है। 2 वर्ष के अंत में और 4 वर्ष के अंत में प्राप्त ब्याज के बीच का अंतर ₹5,082 है। वह राशि (रुपये में) ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 20000 / 20000&lt;/p&gt;

&lt;p&gt;(b) 25500 / 25500&lt;/p&gt;

&lt;p&gt;(c) 50820 / 50820&lt;/p&gt;

&lt;p&gt;(d) 10164 / 10164&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) 20000

&lt;p&gt;Solution: Let the Principal be P.&lt;br&gt;
At 10% CI, the interest after 2 years (CI₂) = P × [(1 + 10/100)² - 1] = 0.21P&lt;br&gt;
The interest after 4 years (CI₄) = P × [(1 + 10/100)⁴ - 1] = 0.4641P&lt;/p&gt;

&lt;p&gt;Difference = CI₄ - CI₂ = 0.4641P - 0.21P = 0.2541P&lt;/p&gt;

&lt;p&gt;We are given that the difference is 5,082:&lt;br&gt;
0.2541P = 5082&lt;br&gt;
P = 5082 / 0.2541 = 20000&lt;/p&gt;

&lt;p&gt;हिंदी में: वह मूल राशि ₹20,000 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q12. The sum of two numbers is equal to 9 times their difference. What is the ratio of the numbers?&lt;/p&gt;

&lt;p&gt;दो संख्याओं का योग उनके अंतर के 9 गुना के बराबर है। संख्याओं का अनुपात क्या है?&lt;/p&gt;

&lt;p&gt;(a) 9 : 5 / 9 : 5&lt;/p&gt;

&lt;p&gt;(b) 3 : 7 / 3 : 7&lt;/p&gt;

&lt;p&gt;(c) 5 : 4 / 5 : 4&lt;/p&gt;

&lt;p&gt;(d) 3 : 4 / 3 : 4&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) 5 : 4

&lt;p&gt;Solution: Let the two numbers be x and y (where x &amp;gt; y).&lt;br&gt;
According to the question:&lt;br&gt;
(x + y) = 9(x - y)&lt;br&gt;
x + y = 9x - 9y&lt;br&gt;
y + 9y = 9x - x&lt;br&gt;
10y = 8x&lt;br&gt;
x / y = 10 / 8 = 5 / 4&lt;/p&gt;

&lt;p&gt;The ratio is 5 : 4.&lt;/p&gt;

&lt;p&gt;हिंदी में: संख्याओं का अनुपात 5:4 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q13. A bag contains a total of 180 coins of denominations ₹5 and ₹1. Find the number of ₹5 coins in the bag, if the total value of the coins is ₹500.&lt;/p&gt;

&lt;p&gt;एक बैग में ₹5 और ₹1 मूल्यवर्ग के कुल 180 सिक्के हैं। यदि सिक्कों का कुल मूल्य ₹500 है, तो बैग में ₹5 के सिक्कों की संख्या ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 100 / 100&lt;/p&gt;

&lt;p&gt;(b) 60 / 60&lt;/p&gt;

&lt;p&gt;(c) 120 / 120&lt;/p&gt;

&lt;p&gt;(d) 80 / 80&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) 80

&lt;p&gt;Solution: Let the number of ₹5 coins be x and ₹1 coins be y.&lt;br&gt;
Equation 1 (Total coins): x + y = 180&lt;br&gt;
Equation 2 (Total value): 5x + 1y = 500&lt;/p&gt;

&lt;p&gt;Subtract Equation 1 from Equation 2:&lt;br&gt;
(5x + y) - (x + y) = 500 - 180&lt;br&gt;
4x = 320&lt;br&gt;
x = 320 / 4 = 80&lt;/p&gt;

&lt;p&gt;हिंदी में: बैग में ₹5 के 80 सिक्के हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q14. What is the fourth proportional of the numbers 5, 6 and 8?&lt;/p&gt;

&lt;p&gt;5, 6 और 8 का चतुर्थानुपाती क्या है?&lt;/p&gt;

&lt;p&gt;(a) 9.8 / 9.8&lt;/p&gt;

&lt;p&gt;(b) 9.6 / 9.6&lt;/p&gt;

&lt;p&gt;(c) 9 / 9&lt;/p&gt;

&lt;p&gt;(d) 9.5 / 9.5&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) 9.6

&lt;p&gt;Solution: Let the fourth proportional be x.&lt;br&gt;
By the rule of proportion: 5 : 6 :: 8 : x&lt;br&gt;
(5 / 6) = (8 / x)&lt;br&gt;
5x = 48&lt;br&gt;
x = 48 / 5 = 9.6&lt;/p&gt;

&lt;p&gt;हिंदी में: 5, 6 और 8 का चतुर्थानुपाती 9.6 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q15. What is the smallest positive number which, when subtracted from 2 and 3, gives their third proportional is 4?&lt;/p&gt;

&lt;p&gt;वह सबसे छोटी धनात्मक संख्या कौन सी है जिसे 2 और 3 में से घटाने पर उनका तृतीयानुपाती 4 प्राप्त होता है?&lt;/p&gt;

&lt;p&gt;(a) 4 / 4&lt;/p&gt;

&lt;p&gt;(b) 2 / 2&lt;/p&gt;

&lt;p&gt;(c) 1 / 1&lt;/p&gt;

&lt;p&gt;(d) 3 / 3&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) 1

&lt;p&gt;Solution: Let the number be x.&lt;br&gt;
When subtracted, the new numbers are (2 - x) and (3 - x).&lt;br&gt;
Their third proportional is 4, which means:&lt;br&gt;
(2 - x) : (3 - x) :: (3 - x) : 4&lt;/p&gt;

&lt;p&gt;Cross multiplying:&lt;br&gt;
4(2 - x) = (3 - x)²&lt;br&gt;
8 - 4x = 9 - 6x + x²&lt;br&gt;
x² - 2x + 1 = 0&lt;br&gt;
(x - 1)² = 0&lt;/p&gt;

&lt;p&gt;x = 1&lt;/p&gt;

&lt;p&gt;हिंदी में: वह संख्या 1 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q16. My father is currently 25 years older than me. Five years ago the sum of our ages was 39 years. Find my present age.&lt;/p&gt;

&lt;p&gt;मेरे पिता वर्तमान में मुझसे 25 वर्ष बड़े हैं। पाँच वर्ष पहले हमारी आयु का योग 39 वर्ष था। मेरी वर्तमान आयु ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 15 years / 15 वर्ष&lt;/p&gt;

&lt;p&gt;(b) 12 years / 12 वर्ष&lt;/p&gt;

&lt;p&gt;(c) 20 years / 20 वर्ष&lt;/p&gt;

&lt;p&gt;(d) 13 years / 13 वर्ष&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) 12 years

&lt;p&gt;Solution: Let my current age be M. My father's current age is M + 25.&lt;br&gt;
Five years ago:&lt;br&gt;
My age = M - 5&lt;br&gt;
Father's age = (M + 25) - 5 = M + 20&lt;/p&gt;

&lt;p&gt;Sum of ages 5 years ago:&lt;br&gt;
(M - 5) + (M + 20) = 39&lt;br&gt;
2M + 15 = 39&lt;br&gt;
2M = 24&lt;br&gt;
M = 12&lt;/p&gt;

&lt;p&gt;हिंदी में: मेरी वर्तमान आयु 12 वर्ष है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q17. The ages of two friends are in the ratio 4 : 3 which after 4 years will become 6 : 5. What will be the sum of their ages after 22 years from now (in years)?&lt;/p&gt;

&lt;p&gt;दो दोस्तों की आयु का अनुपात 4 : 3 है जो 4 वर्ष बाद 6 : 5 हो जाएगा। अब से 22 वर्ष बाद उनकी आयु का योग (वर्षों में) क्या होगा?&lt;/p&gt;

&lt;p&gt;(a) 56 / 56&lt;/p&gt;

&lt;p&gt;(b) 58 / 58&lt;/p&gt;

&lt;p&gt;(c) 52 / 52&lt;/p&gt;

&lt;p&gt;(d) 50 / 50&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) 58

&lt;p&gt;Solution: Let their current ages be 4x and 3x.&lt;br&gt;
After 4 years, the ratio is 6:5:&lt;br&gt;
(4x + 4) / (3x + 4) = 6 / 5&lt;br&gt;
Cross multiplying:&lt;br&gt;
5(4x + 4) = 6(3x + 4)&lt;br&gt;
20x + 20 = 18x + 24&lt;br&gt;
2x = 4&lt;/p&gt;

&lt;p&gt;x = 2&lt;/p&gt;

&lt;p&gt;Their current ages are:&lt;br&gt;
Friend 1 = 4(2) = 8 years&lt;br&gt;
Friend 2 = 3(2) = 6 years&lt;br&gt;
Current sum of ages = 8 + 6 = 14 years.&lt;/p&gt;

&lt;p&gt;Sum of their ages after 22 years = Current sum + (22 × 2 friends)&lt;br&gt;
Sum = 14 + 44 = 58 years.&lt;/p&gt;

&lt;p&gt;हिंदी में: 22 वर्ष बाद उनकी आयु का योग 58 वर्ष होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q18. A starts a business with ₹45,000 and later B joins with ₹30,000. If at the end of the year the profit is divided in the ratio 2 : 1, when did B join?&lt;/p&gt;

&lt;p&gt;A ₹45,000 के साथ एक व्यवसाय शुरू करता है और बाद में B ₹30,000 के साथ जुड़ता है। यदि वर्ष के अंत में लाभ को 2 : 1 के अनुपात में विभाजित किया जाता है, तो B कब जुड़ा?&lt;/p&gt;

&lt;p&gt;(a) 6 months after / 6 महीने बाद&lt;/p&gt;

&lt;p&gt;(b) 12 months after / 12 महीने बाद&lt;/p&gt;

&lt;p&gt;(c) 9 months after / 9 महीने बाद&lt;/p&gt;

&lt;p&gt;(d) 3 months after / 3 महीने बाद&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) 3 months after

&lt;p&gt;Solution: Profit is distributed based on the product of Capital and Time.&lt;br&gt;
Let B invest his money for x months.&lt;br&gt;
A invested for 12 months.&lt;/p&gt;

&lt;p&gt;Profit ratio (A : B):&lt;br&gt;
(45,000 × 12) / (30,000 × x) = 2 / 1&lt;br&gt;
(45 × 12) / 30x = 2&lt;br&gt;
540 / 30x = 2&lt;br&gt;
18 / x = 2&lt;br&gt;
x = 9 months.&lt;/p&gt;

&lt;p&gt;This means B's money was in the business for 9 months.&lt;br&gt;
Therefore, B joined (12 - 9) = 3 months after A started.&lt;/p&gt;

&lt;p&gt;हिंदी में: B ने व्यवसाय शुरू होने के 3 महीने बाद निवेश किया।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q19. Mayur and Ahem enter a partnership by investing in the ratio 2 : 3. 10% of the total profit goes to charity. If Ahem's share is ₹4,860, what is the total profit earned (in rupees)?&lt;/p&gt;

&lt;p&gt;मयूर और अहम 2 : 3 के अनुपात में निवेश करके एक साझेदारी में प्रवेश करते हैं। कुल लाभ का 10% दान में जाता है। यदि अहम का हिस्सा ₹4,860 है, तो अर्जित कुल लाभ (रुपये में) क्या है?&lt;/p&gt;

&lt;p&gt;(Note: Options were missing from your text, so the direct answer is provided below)&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: ₹9,000

&lt;p&gt;Solution: Let the total profit be P.&lt;br&gt;
10% goes to charity, so the remaining profit distributed between them is 0.90P.&lt;/p&gt;

&lt;p&gt;The profit is divided in their investment ratio of 2 : 3.&lt;br&gt;
Ahem's share is 3 parts out of 5 total parts (2+3).&lt;br&gt;
Ahem's Share = (3 / 5) × 0.90P = 0.54P&lt;/p&gt;

&lt;p&gt;We are given that Ahem's share is 4,860:&lt;br&gt;
0.54P = 4860&lt;br&gt;
P = 4860 / 0.54&lt;br&gt;
P = 486000 / 54 = 9000&lt;/p&gt;

&lt;p&gt;हिंदी में: कुल अर्जित लाभ ₹9,000 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Q20. A mixture of milk and water is 32 litres. In this mixture the ratio of milk and water is 5 : 3. How many litres of water should be added to this mixture so that in the resulting mixture water will be 50%?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;दूध और पानी का मिश्रण 32 लीटर का होता है। इस मिश्रण में दूध और पानी का अनुपात 5:3 है। इस मिश्रण में कितने लीटर पानी मिलाया जाना चाहिए ताकि परिणामी मिश्रण में पानी 50% हो जाए?&lt;/p&gt;

&lt;p&gt;(a) 8 / 8&lt;br&gt;&lt;br&gt;
(b) 6 / 6&lt;br&gt;&lt;br&gt;
(c) 5 / 5&lt;br&gt;&lt;br&gt;
(d) 4 / 4  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 8

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Total mixture = 32 litres.&lt;br&gt;
Ratio of Milk : Water = 5 : 3 (Total 8 parts).&lt;br&gt;
1 part = 32 / 8 = 4 litres.&lt;br&gt;
Current Milk = 5 × 4 = 20 litres.&lt;br&gt;
Current Water = 3 × 4 = 12 litres.&lt;/p&gt;

&lt;p&gt;We want the new mixture to have 50% water (which means 50% milk, so a 1:1 ratio). &lt;br&gt;
Since milk remains 20 litres, water must also become 20 litres.&lt;br&gt;
Water to add = 20 - 12 = &lt;strong&gt;8 litres&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 8 लीटर पानी मिलाया जाना चाहिए।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q21. In an experiment, it is observed that metal A is 19 times heavier than water and metal B is 9 times heavier than water. In what ratio should metals A and B be mixed so that the resultant mixture is 15 times heavier than water?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक प्रयोग में, यह देखा गया है कि धातु A पानी से 19 गुना भारी है और धातु B पानी से 9 गुना भारी है। धातु A और B को किस अनुपात में मिलाया जाना चाहिए ताकि परिणामी मिश्रण पानी से 15 गुना भारी हो?&lt;/p&gt;

&lt;p&gt;(a) 2 : 3 / 2 : 3&lt;br&gt;&lt;br&gt;
(b) 1 : 3 / 1 : 3&lt;br&gt;&lt;br&gt;
(c) 3 : 1 / 3 : 1&lt;br&gt;&lt;br&gt;
(d) 3 : 2 / 3 : 2  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 3 : 2

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Using the rule of Alligation:&lt;br&gt;
Metal A = 19&lt;br&gt;
Metal B = 9&lt;br&gt;
Resultant Mixture = 15&lt;/p&gt;

&lt;p&gt;Ratio = (Mixture - B) : (A - Mixture)&lt;br&gt;
Ratio = (15 - 9) : (19 - 15)&lt;br&gt;
Ratio = 6 : 4 = &lt;strong&gt;3 : 2&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; धातुओं को 3:2 के अनुपात में मिलाया जाना चाहिए।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q22. The average monthly income of A and B is ₹7,000. The average monthly income of B and C is ₹7,750 and the average monthly income of C and A is ₹6,750. What is the monthly income of B?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;A और B की औसत मासिक आय ₹7,000 है। B और C की औसत मासिक आय ₹7,750 है और C व A की औसत मासिक आय ₹6,750 है। B की मासिक आय क्या है?&lt;/p&gt;

&lt;p&gt;(a) ₹6,500 / ₹6,500&lt;br&gt;&lt;br&gt;
(b) ₹8,000 / ₹8,000&lt;br&gt;&lt;br&gt;
(c) ₹7,500 / ₹7,500&lt;br&gt;&lt;br&gt;
(d) ₹6,000 / ₹6,000  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) ₹8,000

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Total income of A + B = 7000 × 2 = 14000&lt;br&gt;
Total income of B + C = 7750 × 2 = 15500&lt;br&gt;
Total income of C + A = 6750 × 2 = 13500&lt;/p&gt;

&lt;p&gt;Adding all three equations:&lt;br&gt;
2(A + B + C) = 14000 + 15500 + 13500 = 43000&lt;br&gt;
A + B + C = 43000 / 2 = 21500&lt;/p&gt;

&lt;p&gt;To find B's income, subtract the sum of (C + A) from the total:&lt;br&gt;
B = (A + B + C) - (C + A)&lt;br&gt;
B = 21500 - 13500 = &lt;strong&gt;₹8,000&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; B की मासिक आय ₹8,000 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q23. The average of 25 integers is zero. What is the maximum possible number of positive integers?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;25 पूर्णांकों का औसत शून्य है। धनात्मक पूर्णांकों की अधिकतम संभावित संख्या क्या है?&lt;/p&gt;

&lt;p&gt;(a) 48 / 48&lt;br&gt;&lt;br&gt;
(b) 24 / 24&lt;br&gt;&lt;br&gt;
(c) 36 / 36&lt;br&gt;&lt;br&gt;
(d) 72 / 72  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 24

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; If the average of 25 integers is zero, their total sum must be zero.&lt;br&gt;
To maximize the number of positive integers, you can have 24 integers that are positive (e.g., each could be +1, summing to +24) and exactly 1 negative integer (e.g., -24) to cancel out the sum. &lt;br&gt;
Therefore, the maximum possible number of positive integers is &lt;strong&gt;24&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; धनात्मक पूर्णांकों की अधिकतम संख्या 24 हो सकती है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q24. Akhil can complete a work in 24 days. Shyam can complete the same work in 12 days. If both working together, in how many days can they complete the same job?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;अखिल एक कार्य को 24 दिनों में पूरा कर सकता है। श्याम उसी कार्य को 12 दिनों में पूरा कर सकता है। अगर दोनों मिलकर कार्य करें, तो वे उसी कार्य को कितने दिनों में पूरा कर सकते हैं?&lt;/p&gt;

&lt;p&gt;(a) 6 days / 6 दिन&lt;br&gt;&lt;br&gt;
(b) 8 days / 8 दिन&lt;br&gt;&lt;br&gt;
(c) 10 days / 10 दिन&lt;br&gt;&lt;br&gt;
(d) 12 days / 12 दिन  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 8 days

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Akhil's 1-day work = 1/24&lt;br&gt;
Shyam's 1-day work = 1/12&lt;/p&gt;

&lt;p&gt;Working together, their 1-day work = 1/24 + 1/12 = 1/24 + 2/24 = 3/24 = 1/8.&lt;br&gt;
Total time taken = &lt;strong&gt;8 days&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;em&gt;(Alternative Trick: Time = (A × B) / (A + B) = (24 × 12) / 36 = 288 / 36 = 8)&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; दोनों मिलकर 8 दिन में काम पूरा कर सकते हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q25. Fifteen men working 12 hours a day earn ₹18,000 per week. How much will 18 men working 9 hours a day earn per week?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;पंद्रह आदमी दिन में 12 घंटे कार्य करके प्रत्येक सप्ताह ₹18,000 कमाते हैं। तो 18 आदमी दिन में 9 घंटे कार्य करके प्रत्येक सप्ताह कितना कमाएँगे?&lt;/p&gt;

&lt;p&gt;(a) Rs. 16,020 / ₹16,020&lt;br&gt;&lt;br&gt;
(b) Rs. 12,600 / ₹12,600&lt;br&gt;&lt;br&gt;
(c) Rs. 12,060 / ₹12,060&lt;br&gt;&lt;br&gt;
(d) Rs. 16,200 / ₹16,200  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) Rs. 16,200

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Using the formula &lt;strong&gt;(M1 × H1) / E1 = (M2 × H2) / E2&lt;/strong&gt; where E is earnings:&lt;br&gt;
(15 × 12) / 18000 = (18 × 9) / E2&lt;br&gt;
180 / 18000 = 162 / E2&lt;br&gt;
1 / 100 = 162 / E2&lt;br&gt;
E2 = 162 × 100 = &lt;strong&gt;16200&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वे सप्ताह में ₹16,200 कमाएंगे।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q26. Two pipes can fill a tank in 12 hours and 18 hours respectively, while a third pipe can empty it in 8 hours. If all the three pipes are opened together, in how many hours will the empty tank be filled?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;दो पाइप एक टंकी को क्रमशः 12 घंटे और 18 घंटे में भर सकते हैं, जबकि तीसरी पाइप इसे 8 घंटे में खाली कर सकती है। यदि तीनों पाइपों को एक साथ खोल दिया जाए तो खाली टंकी कितने घंटों में भर जाएगी?&lt;/p&gt;

&lt;p&gt;(a) 48 / 48&lt;br&gt;&lt;br&gt;
(b) 24 / 24&lt;br&gt;&lt;br&gt;
(c) 36 / 36&lt;br&gt;&lt;br&gt;
(d) 72 / 72  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 72

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let Total Capacity of tank be LCM of (12, 18, 8) = 72 units.&lt;br&gt;
Pipe 1 efficiency = + 72 / 12 = +6 units/hr&lt;br&gt;
Pipe 2 efficiency = + 72 / 18 = +4 units/hr&lt;br&gt;
Pipe 3 efficiency = - 72 / 8 = -9 units/hr (emptying)&lt;/p&gt;

&lt;p&gt;Net efficiency of all three pipes working together = 6 + 4 - 9 = 1 unit/hr.&lt;br&gt;
Time to fill the 72-unit tank = Total Capacity / Net Efficiency = 72 / 1 = &lt;strong&gt;72 hours&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; टंकी 72 घंटों में भर जाएगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q27. Pipe A, when operated alone, takes 5 hours less than pipe B to fill a tank, and when both pipes are operated together, the tank is filled in 6 hours. In how many hours will pipe B fill the tank when operated alone?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;जब पाइप A को अकेले चलाया जाता है, तो उसे एक टैंक भरने में पाइप B से 5 घंटे कम लगते हैं, और जब दोनों पाइपों को एक साथ चलाया जाता है, तो टैंक 6 घंटे में भर जाता है। जब पाइप B को अकेले चलाया जाएगा तो वह टैंक को कितने घंटे में भरेगा?&lt;/p&gt;

&lt;p&gt;(a) 9 / 9&lt;br&gt;&lt;br&gt;
(b) 18 / 18&lt;br&gt;&lt;br&gt;
(c) 10 / 10&lt;br&gt;&lt;br&gt;
(d) 15 / 15  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 15

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let pipe B take &lt;strong&gt;x&lt;/strong&gt; hours. Pipe A takes &lt;strong&gt;(x - 5)&lt;/strong&gt; hours.&lt;br&gt;
Together they fill it in 6 hours:&lt;br&gt;
(1 / x) + [1 / (x - 5)] = 1 / 6&lt;/p&gt;

&lt;p&gt;It is easiest to check the options for x (Pipe B's time):&lt;br&gt;
If x = 15: 1/15 + 1/10 = 2/30 + 3/30 = 5/30 = 1/6. This matches perfectly!&lt;br&gt;
So, Pipe B takes &lt;strong&gt;15 hours&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; पाइप B अकेले टैंक को 15 घंटे में भरेगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q28. A woman walks daily from her home to her office in a fixed time. One day she reduces her speed by 25%. By what fraction of her usual time will she be late that day?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक महिला प्रतिदिन एक निश्चित समय में अपने घर से अपने कार्यालय तक पैदल जाती है। एक दिन वह अपनी गति को 25% कम कर देती है। वह उस दिन अपने सामान्य समय के किस अंश तक देर हो जाएगी?&lt;/p&gt;

&lt;p&gt;(a) 1/3 / 1/3&lt;br&gt;&lt;br&gt;
(b) 1/4 / 1/4&lt;br&gt;&lt;br&gt;
(c) 1/6 / 1/6&lt;br&gt;&lt;br&gt;
(d) 2/3 / 2/3  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 1/3

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let original speed = 4 units. Reduced speed by 25% = 3 units.&lt;br&gt;
Since distance is constant, Speed and Time are inversely proportional.&lt;br&gt;
Ratio of Speeds (Old : New) = 4 : 3&lt;br&gt;
Ratio of Times (Old : New) = 3 : 4&lt;/p&gt;

&lt;p&gt;If usual time is 3 units, the new time is 4 units. &lt;br&gt;
She is late by = (New Time - Usual Time) = 4 - 3 = 1 unit.&lt;br&gt;
Fraction of usual time she is late = Late Time / Usual Time = &lt;strong&gt;1/3&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वह अपने सामान्य समय के 1/3 अंश तक देर होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q29. A policeman saw a thief from a distance of 80 metres. The speed of the thief is 21 km/h and the speed of the policeman is 20 km/h. What will be the distance between them after 18 seconds?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक पुलिसकर्मी ने 80 मीटर की दूरी से एक चोर को देखा। चोर की गति 21 किमी/घंटा और पुलिसकर्मी की गति 20 किमी/घंटा है। 18 सेकंड के बाद उनके बीच की दूरी क्या होगी?&lt;/p&gt;

&lt;p&gt;(a) 70 metres / 70 मीटर&lt;br&gt;&lt;br&gt;
(b) 95 metres / 95 मीटर&lt;br&gt;&lt;br&gt;
(c) 90 metres / 90 मीटर&lt;br&gt;&lt;br&gt;
(d) 85 metres / 85 मीटर  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 85 metres

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The thief is faster, so the distance between them is increasing.&lt;br&gt;
Relative Speed = Thief's Speed - Policeman's Speed = 21 - 20 = 1 km/h.&lt;br&gt;
Convert 1 km/h to m/s: 1 × (5 / 18) m/s.&lt;/p&gt;

&lt;p&gt;Distance increased in 18 seconds = Speed × Time = (5 / 18) × 18 = 5 metres.&lt;br&gt;
Total distance after 18 seconds = Initial Distance + Increased Distance = 80 + 5 = &lt;strong&gt;85 metres&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 18 सेकंड बाद उनके बीच की दूरी 85 मीटर होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q30. Two persons A and B started running from the same point in opposite directions on a circular track of 400 metres length with initial speeds of 10 m/s and 40 m/s respectively. Whenever they meet, A's speed doubles and B's speed halves. After how much time from the beginning will they meet for the third time?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;A और B क्रमशः 10 m/s और 40 m/s की प्रारंभिक गति के साथ 400 मीटर लंबाई के गोलाकार पथ पर एक ही बिंदु से विपरीत दिशाओं में चलने लगे। जब भी वे मिलते हैं, A की गति दोगुनी हो जाती है और B की गति आधी हो जाती है। प्रारंभ से कितने समय के बाद वे तीसरी बार मिलेंगे?&lt;/p&gt;

&lt;p&gt;(a) 30 seconds / 30 सेकंड&lt;br&gt;&lt;br&gt;
(b) 24 seconds / 24 सेकंड&lt;br&gt;&lt;br&gt;
(c) 26 seconds / 26 सेकंड&lt;br&gt;&lt;br&gt;
(d) 28 seconds / 28 सेकंड  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 26 seconds

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Distance for every meeting is one full lap of the track (400m) because they run in opposite directions.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;1st Meeting:&lt;/strong&gt; Relative speed = 10 + 40 = 50 m/s. &lt;br&gt;
Time₁ = 400 / 50 = 8 seconds. &lt;br&gt;
&lt;em&gt;(Speeds now change: A = 10×2 = 20 m/s, B = 40÷2 = 20 m/s)&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;2nd Meeting:&lt;/strong&gt; Relative speed = 20 + 20 = 40 m/s. &lt;br&gt;
Time₂ = 400 / 40 = 10 seconds. &lt;br&gt;
&lt;em&gt;(Speeds now change: A = 20×2 = 40 m/s, B = 20÷2 = 10 m/s)&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;3rd Meeting:&lt;/strong&gt; Relative speed = 40 + 10 = 50 m/s. &lt;br&gt;
Time₃ = 400 / 50 = 8 seconds.&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Total time = 8 + 10 + 8 = &lt;strong&gt;26 seconds&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वे तीसरी बार 26 सेकंड के बाद मिलेंगे।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q31. While travelling downstream the speed of a boat is 33 km/h, while travelling upstream it is 27 km/h. What is the speed of the boat in still water?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;धारा के साथ यात्रा करते समय एक नाव की गति 33 किमी/घंटा होती है, जबकि धारा के विपरीत यात्रा करते समय यह 27 किमी/घंटा होती है। स्थिर जल में नाव की गति क्या है?&lt;/p&gt;

&lt;p&gt;(a) 6 km/h / 6 किमी/घंटा&lt;br&gt;&lt;br&gt;
(b) 60 km/h / 60 किमी/घंटा&lt;br&gt;&lt;br&gt;
(c) 12 km/h / 12 किमी/घंटा&lt;br&gt;&lt;br&gt;
(d) 30 km/h / 30 किमी/घंटा  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 30 km/h

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Speed of boat in still water = (Downstream Speed + Upstream Speed) / 2&lt;br&gt;
Speed = (33 + 27) / 2 = 60 / 2 = &lt;strong&gt;30 km/h&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; शांत जल में नाव की गति 30 किमी/घंटा है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q32. A boat takes 20 hours to travel from point A to point B downstream and back to the midpoint C between A and B. The speed of the stream is 5 km/h and the speed of the boat in still water is 10 km/h. Find the distance (in km) between A and B.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक नाव बिंदु A से बिंदु B तक धारा के दिशा में जाती है और फिर A तथा B के मध्य स्थित मध्य बिंदु C तक वापस आती है। इस पूरी यात्रा में उसे 20 घंटे लगते हैं। धारा की गति 5 किमी/घंटा है और शांत जल में नाव की गति 10 किमी/घंटा है। A और B के बीच की दूरी (किमी में) ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 100 / 100&lt;br&gt;&lt;br&gt;
(b) 150 / 150&lt;br&gt;&lt;br&gt;
(c) 75 / 75&lt;br&gt;&lt;br&gt;
(d) 120 / 120  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 120

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Downstream speed = Boat + Stream = 10 + 5 = 15 km/h.&lt;br&gt;
Upstream speed = Boat - Stream = 10 - 5 = 5 km/h.&lt;/p&gt;

&lt;p&gt;Let the total distance between A and B be &lt;strong&gt;d&lt;/strong&gt;.&lt;br&gt;
Trip from A to B (downstream) distance = d. Time taken = d / 15.&lt;br&gt;
Trip from B to C (upstream) distance = d / 2. Time taken = (d / 2) / 5 = d / 10.&lt;/p&gt;

&lt;p&gt;Total time = 20 hours.&lt;br&gt;
(d / 15) + (d / 10) = 20&lt;br&gt;
Take LCM of 15 and 10 which is 30:&lt;br&gt;
(2d + 3d) / 30 = 20&lt;br&gt;
5d / 30 = 20&lt;br&gt;
d / 6 = 20&lt;br&gt;
d = 20 × 6 = &lt;strong&gt;120 km&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; A और B के बीच की दूरी 120 किलोमीटर है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q33. What is the number of factors of 38808, excluding the divisors 1 and the number itself?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;38808 के गुणनखंडों की संख्या क्या है, जिसमें 1 और संख्या स्वयं शामिल नहीं हैं?&lt;/p&gt;

&lt;p&gt;(a) 74 / 74&lt;br&gt;&lt;br&gt;
(b) 72 / 72&lt;br&gt;&lt;br&gt;
(c) 70 / 70&lt;br&gt;&lt;br&gt;
(d) 68 / 68  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 70

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; First, find the prime factorization of 38808:&lt;br&gt;
38808 = 8 × 4851 = 2³ × 9 × 539 = 2³ × 3² × 7 × 77 = 2³ × 3² × 7² × 11¹&lt;/p&gt;

&lt;p&gt;To find the total number of factors, add 1 to each exponent and multiply them together:&lt;br&gt;
Total Factors = (3 + 1) × (2 + 1) × (2 + 1) × (1 + 1)&lt;br&gt;
Total Factors = 4 × 3 × 3 × 2 = 72&lt;/p&gt;

&lt;p&gt;The question asks to exclude 1 and the number itself (2 factors):&lt;br&gt;
72 - 2 = &lt;strong&gt;70&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 1 और स्वयं संख्या को छोड़कर गुणनखंडों की संख्या 70 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q34. In the sum of the quotient, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 12, what is the dividend?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;भागफल के योग में, भाजक भागफल का 10 गुना और शेषफल का 5 गुना है। यदि शेषफल 12 है, तो भाज्य क्या है?&lt;/p&gt;

&lt;p&gt;(a) 352 / 352&lt;br&gt;&lt;br&gt;
(b) 386 / 386&lt;br&gt;&lt;br&gt;
(c) 368 / 368&lt;br&gt;&lt;br&gt;
(d) 372 / 372  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 372

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Given: Remainder (R) = 12&lt;br&gt;
Divisor (D) = 5 × Remainder = 5 × 12 = 60&lt;br&gt;
Divisor (D) = 10 × Quotient (Q) → 60 = 10 × Q → Quotient (Q) = 6&lt;/p&gt;

&lt;p&gt;Dividend = (Divisor × Quotient) + Remainder&lt;br&gt;
Dividend = (60 × 6) + 12 = 360 + 12 = &lt;strong&gt;372&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; भाज्य 372 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q35. The LCM of two numbers is 4104 and their HCF is 9. If one of the numbers is 171, find the other number.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;दो संख्याओं का LCM 4104 है और उनका HCF 9 है। यदि एक संख्या 171 है, तो दूसरी संख्या ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 218 / 218&lt;br&gt;&lt;br&gt;
(b) 215 / 215&lt;br&gt;&lt;br&gt;
(c) 220 / 220&lt;br&gt;&lt;br&gt;
(d) 216 / 216  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 216

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The formula connecting two numbers and their LCM/HCF is:&lt;br&gt;
&lt;strong&gt;First Number × Second Number = LCM × HCF&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;171 × Second Number = 4104 × 9&lt;br&gt;
Second Number = (4104 × 9) / 171&lt;br&gt;
Second Number = 36936 / 171 = &lt;strong&gt;216&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; दूसरी संख्या 216 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Q36. What is the greatest positive integer which when divides 554, 714 and 213 leaves remainders 43, 57 and 67 respectively?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;वह सबसे बड़ा धनात्मक पूर्णांक कौन सा है जो 554, 714 और 213 को विभाजित करने पर क्रमशः 43, 57 और 67 शेषफल देता है?&lt;/p&gt;

&lt;p&gt;(a) 95 / 95&lt;br&gt;&lt;br&gt;
(b) 71 / 71&lt;br&gt;&lt;br&gt;
(c) 83 / 83&lt;br&gt;&lt;br&gt;
(d) 73 / 73  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 73

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; First, subtract the respective remainders from the given numbers to find the numbers that are exactly divisible:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;554 - 43 = 511&lt;/li&gt;
&lt;li&gt;714 - 57 = 657&lt;/li&gt;
&lt;li&gt;213 - 67 = 146&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Now, find the Highest Common Factor (HCF) of 511, 657, and 146.&lt;br&gt;
Let's look at 146. Its factors are 1, 2, 73, and 146.&lt;br&gt;
Check if 73 divides the other numbers:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;511 / 73 = 7 (Perfectly divisible)&lt;/li&gt;
&lt;li&gt;657 / 73 = 9 (Perfectly divisible)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Thus, the HCF is &lt;strong&gt;73&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वह सबसे बड़ा धनात्मक पूर्णांक 73 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q37. If 640 ÷ 8 + 930 ÷ 15 - k + 32 × 5 = 1104 ÷ 16 × 148 ÷ 37, then what is the value of k?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि 640 ÷ 8 + 930 ÷ 15 - k + 32 × 5 = 1104 ÷ 16 × 148 ÷ 37 तो k का मान क्या है?&lt;/p&gt;

&lt;p&gt;(a) 276 / 276&lt;br&gt;&lt;br&gt;
(b) 26 / 26&lt;br&gt;&lt;br&gt;
(c) 35 / 35&lt;br&gt;&lt;br&gt;
(d) 302 / 302  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 26

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Apply the BODMAS rule (Divide and Multiply first, then Add and Subtract) on both sides.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Left Hand Side (LHS):&lt;/strong&gt;&lt;br&gt;
(640 / 8) + (930 / 15) - k + (32 × 5)&lt;br&gt;
= 80 + 62 - k + 160&lt;br&gt;
= 302 - k&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Right Hand Side (RHS):&lt;/strong&gt;&lt;br&gt;
(1104 / 16) × (148 / 37)&lt;br&gt;
= 69 × 4&lt;br&gt;
= 276&lt;/p&gt;

&lt;p&gt;Equating LHS and RHS:&lt;br&gt;
302 - k = 276&lt;br&gt;
k = 302 - 276 = &lt;strong&gt;26&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; k का मान 26 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q38. If (a - 2b) = 8 and 2ab = 9, then find the value of (a + 2b).&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि (a - 2b) = 8 और 2ab = 9 तो (a + 2b) का मान ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 6 / 6&lt;br&gt;&lt;br&gt;
(b) 12 / 12&lt;br&gt;&lt;br&gt;
(c) 10 / 10&lt;br&gt;&lt;br&gt;
(d) 8 / 8  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 10

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; We can use the algebraic identity:&lt;br&gt;
&lt;strong&gt;(x + y)² = (x - y)² + 4xy&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Here, let x = a and y = 2b.&lt;br&gt;
(a + 2b)² = (a - 2b)² + 4(a)(2b)&lt;br&gt;
(a + 2b)² = (a - 2b)² + 8ab&lt;/p&gt;

&lt;p&gt;Substitute the given values: (a - 2b) = 8 and 2ab = 9.&lt;br&gt;
Notice that 8ab = 4 × (2ab) = 4 × 9 = 36.&lt;/p&gt;

&lt;p&gt;(a + 2b)² = (8)² + 36&lt;br&gt;
(a + 2b)² = 64 + 36&lt;br&gt;
(a + 2b)² = 100&lt;br&gt;
(a + 2b) = √100 = &lt;strong&gt;10&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; (a + 2b) का मान 10 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q39. The ratio of people speaking Hindi, English and German is 5 : 4 : 3. If the number of people speaking Hindi is 15,500, then find the total population of the Town.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;हिंदी, अंग्रेजी और जर्मन बोलने वाले लोगों का अनुपात 5 : 4 : 3 है। यदि हिंदी बोलने वाले लोगों की संख्या 15,500 है, तो शहर की कुल जनसंख्या ज्ञात कीजिए?&lt;/p&gt;

&lt;p&gt;(a) 23400 / 23400&lt;br&gt;&lt;br&gt;
(b) 31000 / 31000&lt;br&gt;&lt;br&gt;
(c) 43000 / 43000&lt;br&gt;&lt;br&gt;
(d) 37200 / 37200  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 37200

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let the number of people speaking Hindi, English, and German be 5x, 4x, and 3x respectively.&lt;br&gt;
Total population = 5x + 4x + 3x = 12x.&lt;/p&gt;

&lt;p&gt;We are given that the Hindi speaking population is 15,500.&lt;br&gt;
5x = 15500&lt;br&gt;
x = 15500 / 5 = 3100&lt;/p&gt;

&lt;p&gt;Total Population = 12x = 12 × 3100 = &lt;strong&gt;37,200&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; शहर की कुल जनसंख्या 37,200 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q40. The average weight of 10 men is decreased by 2 kg when one of them whose weight is 60 kg is replaced by a new man. What is the weight of the new man?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;10 पुरुषों के औसत वजन में 2 किग्रा की कमी होती है जब उनमें से एक, जिसका वजन 60 किग्रा है, को नए व्यक्ति से बदला जाता है। नए व्यक्ति का वजन क्या होगा?&lt;/p&gt;

&lt;p&gt;(a) 35 kg / 35 किग्रा&lt;br&gt;&lt;br&gt;
(b) 40 kg / 40 किग्रा&lt;br&gt;&lt;br&gt;
(c) 45 kg / 45 किग्रा&lt;br&gt;&lt;br&gt;
(d) 50 kg / 50 किग्रा  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 40 kg

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The total number of men is 10. &lt;br&gt;
If the average weight decreases by 2 kg, it means the total weight of the group has decreased by:&lt;br&gt;
Total decrease = 10 men × 2 kg = 20 kg.&lt;/p&gt;

&lt;p&gt;Since the old man leaving weighed 60 kg, the new man must be 20 kg lighter than him to cause this overall drop in the average.&lt;br&gt;
Weight of new man = Weight of replaced man - Total decrease&lt;br&gt;
Weight of new man = 60 kg - 20 kg = &lt;strong&gt;40 kg&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; नए व्यक्ति का वजन 40 किग्रा है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q41. The sum of the present age of mother and her son is 60 years. Six years ago, the age of the mother was five times the age of her son. What will be the age of her son after 6 years?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;माँ और उसके पुत्र की वर्तमान आयु का योग 60 वर्ष है। छह वर्ष पहले, माँ की आयु पुत्र की आयु का पाँच गुना थी। छह वर्षों बाद पुत्र की आयु क्या होगी?&lt;/p&gt;

&lt;p&gt;(a) 23 years / 23 वर्ष&lt;br&gt;&lt;br&gt;
(b) 22 years / 22 वर्ष&lt;br&gt;&lt;br&gt;
(c) 20 years / 20 वर्ष&lt;br&gt;&lt;br&gt;
(d) 21 years / 21 वर्ष  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 20 years

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let the present age of the Mother be &lt;strong&gt;M&lt;/strong&gt; and the Son be &lt;strong&gt;S&lt;/strong&gt;.&lt;br&gt;
Given: M + S = 60.&lt;/p&gt;

&lt;p&gt;Six years ago:&lt;br&gt;
Mother's age = (M - 6)&lt;br&gt;
Son's age = (S - 6)&lt;br&gt;
Condition: (M - 6) = 5 × (S - 6)&lt;br&gt;
M - 6 = 5S - 30&lt;br&gt;
M - 5S = -24&lt;/p&gt;

&lt;p&gt;Substitute M = 60 - S into the equation:&lt;br&gt;
(60 - S) - 5S = -24&lt;br&gt;
60 - 6S = -24&lt;br&gt;
6S = 84&lt;br&gt;
S = 14 (Present age of the son)&lt;/p&gt;

&lt;p&gt;The age of the son after 6 years = 14 + 6 = &lt;strong&gt;20 years&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 6 वर्ष बाद पुत्र की आयु 20 वर्ष होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q42. Which of the following numbers is divisible by 9?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन सी संख्या 9 से विभाज्य है?&lt;/p&gt;

&lt;p&gt;(a) 67578 / 67578&lt;br&gt;&lt;br&gt;
(b) 56785 / 56785&lt;br&gt;&lt;br&gt;
(c) 45678 / 45678&lt;br&gt;&lt;br&gt;
(d) 65889 / 65889  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 65889

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is a multiple of 9. Let's check the options:&lt;br&gt;
(a) 6 + 7 + 5 + 7 + 8 = 33 (Not divisible by 9)&lt;br&gt;
(b) 5 + 6 + 7 + 8 + 5 = 31 (Not divisible by 9)&lt;br&gt;
(c) 4 + 5 + 6 + 7 + 8 = 30 (Not divisible by 9)&lt;br&gt;
(d) 6 + 5 + 8 + 8 + 9 = 36 (Divisible by 9 because 36 / 9 = 4)&lt;/p&gt;

&lt;p&gt;Therefore, &lt;strong&gt;65889&lt;/strong&gt; is divisible by 9.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 65889 के अंकों का योग 36 है, जो 9 से विभाज्य है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q43. The sum of the ages of father and his son is 44 years. If 6 years after the father will be 3 times as old as his son, what are their present ages?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;पिता और उसके पुत्र की आयु का योग 44 वर्ष है। यदि 6 वर्षों बाद पिता अपने पुत्र की आयु का 3 गुना होगा, तो उनकी वर्तमान आयु क्या हैं?&lt;/p&gt;

&lt;p&gt;(a) 36, 8 / 36, 8&lt;br&gt;&lt;br&gt;
(b) 38, 6 / 38, 6&lt;br&gt;&lt;br&gt;
(c) 35, 9 / 35, 9&lt;br&gt;&lt;br&gt;
(d) 37, 7 / 37, 7  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 36, 8

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let Father's present age be &lt;strong&gt;F&lt;/strong&gt; and Son's present age be &lt;strong&gt;S&lt;/strong&gt;.&lt;br&gt;
Given: F + S = 44 =&amp;gt; F = 44 - S&lt;/p&gt;

&lt;p&gt;Six years later:&lt;br&gt;
Father's age = F + 6&lt;br&gt;
Son's age = S + 6&lt;br&gt;
Condition: F + 6 = 3 × (S + 6)&lt;/p&gt;

&lt;p&gt;Substitute F = 44 - S into the equation:&lt;br&gt;
(44 - S) + 6 = 3S + 18&lt;br&gt;
50 - S = 3S + 18&lt;br&gt;
4S = 32&lt;br&gt;
S = &lt;strong&gt;8&lt;/strong&gt; (Son's present age)&lt;/p&gt;

&lt;p&gt;Father's present age F = 44 - 8 = &lt;strong&gt;36&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; पिता और पुत्र की वर्तमान आयु क्रमशः 36 और 8 वर्ष है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q44. An article is sold at a loss of 25 percent. If its cost price is doubled and the selling price is increased by ₹660, then a profit of 20 percent is obtained. What is the actual cost price of the article?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक वस्तु 25 प्रतिशत की हानि पर बेची जाती है। यदि इसकी लागत कीमत दोगुनी हो जाती है और बिक्री मूल्य में 660 रुपये की वृद्धि हो जाती है, तो 20 प्रतिशत का लाभ प्राप्त होता है। वस्तु की वास्तविक लागत मूल्य क्या है?&lt;/p&gt;

&lt;p&gt;(a) ₹360 / ₹360&lt;br&gt;&lt;br&gt;
(b) ₹500 / ₹500&lt;br&gt;&lt;br&gt;
(c) ₹480 / ₹480&lt;br&gt;&lt;br&gt;
(d) ₹400 / ₹400  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 400

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let the original Cost Price (CP) be &lt;strong&gt;100x&lt;/strong&gt;.&lt;br&gt;
Since there is a 25% loss, Original Selling Price (SP) = &lt;strong&gt;75x&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;Condition 1: Cost price is doubled. &lt;br&gt;
New CP = 2 × 100x = 200x.&lt;br&gt;
Condition 2: Selling price is increased by 660. &lt;br&gt;
New SP = 75x + 660.&lt;/p&gt;

&lt;p&gt;There is a 20% profit on the New CP. &lt;br&gt;
Therefore, New SP = 120% of New CP.&lt;br&gt;
New SP = (120 / 100) × 200x = 240x.&lt;/p&gt;

&lt;p&gt;Equating both expressions for New SP:&lt;br&gt;
75x + 660 = 240x&lt;br&gt;
165x = 660&lt;br&gt;
x = 660 / 165 = 4&lt;/p&gt;

&lt;p&gt;Actual Cost Price = 100x = 100 × 4 = &lt;strong&gt;₹400&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वस्तु का वास्तविक लागत मूल्य ₹400 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q45. A debt of ₹10,192 due in 3 years at 10% simple interest will be repaid by equal annual instalments of how much?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;₹10,192 का ऋण जो 3 वर्ष में 10% साधारण ब्याज पर चुकाना है, उसे कितनी बराबर वार्षिक किस्तों में चुकाया जाएगा?&lt;/p&gt;

&lt;p&gt;(a) ₹3,144.48 / ₹3,144.48&lt;br&gt;&lt;br&gt;
(b) ₹3,044.48 / ₹3,044.48&lt;br&gt;&lt;br&gt;
(c) ₹3,088.48 / ₹3,088.48&lt;br&gt;&lt;br&gt;
(d) ₹3,188.48 / ₹3,188.48  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 3,088.48

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; To find the value of each equal annual installment (&lt;strong&gt;x&lt;/strong&gt;) that discharges a debt (&lt;strong&gt;A&lt;/strong&gt;) in &lt;strong&gt;n&lt;/strong&gt; years at &lt;strong&gt;R&lt;/strong&gt;% simple interest, we use the formula:&lt;br&gt;
&lt;strong&gt;A = n × x + [x × R × n(n-1)] / 200&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Given: Total Debt (A) = 10,192, Time (n) = 3 years, Rate (R) = 10%.&lt;br&gt;
10192 = (3 × x) + [x × 10 × 3(2)] / 200&lt;br&gt;
10192 = 3x + [60x] / 200&lt;br&gt;
10192 = 3x + 0.3x&lt;br&gt;
10192 = 3.3x&lt;/p&gt;

&lt;p&gt;x = 10192 / 3.3 = 101920 / 33 = &lt;strong&gt;₹3,088.48&lt;/strong&gt; (approx).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; समान वार्षिक किस्त ₹3,088.48 होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Q36. What is the greatest positive integer which when divides 554, 714 and 213 leaves remainders 43, 57 and 67 respectively?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;वह सबसे बड़ा धनात्मक पूर्णांक कौन सा है जो 554, 714 और 213 को विभाजित करने पर क्रमशः 43, 57 और 67 शेषफल देता है?&lt;/p&gt;

&lt;p&gt;(a) 95 / 95&lt;br&gt;&lt;br&gt;
(b) 71 / 71&lt;br&gt;&lt;br&gt;
(c) 83 / 83&lt;br&gt;&lt;br&gt;
(d) 73 / 73  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 73

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; First, subtract the respective remainders from the given numbers to find the numbers that are exactly divisible:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;554 - 43 = 511&lt;/li&gt;
&lt;li&gt;714 - 57 = 657&lt;/li&gt;
&lt;li&gt;213 - 67 = 146&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Now, find the Highest Common Factor (HCF) of 511, 657, and 146.&lt;br&gt;
Let's look at 146. Its factors are 1, 2, 73, and 146.&lt;br&gt;
Check if 73 divides the other numbers:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;511 / 73 = 7 (Perfectly divisible)&lt;/li&gt;
&lt;li&gt;657 / 73 = 9 (Perfectly divisible)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Thus, the HCF is &lt;strong&gt;73&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वह सबसे बड़ा धनात्मक पूर्णांक 73 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q37. If 640 ÷ 8 + 930 ÷ 15 - k + 32 × 5 = 1104 ÷ 16 × 148 ÷ 37, then what is the value of k?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि 640 ÷ 8 + 930 ÷ 15 - k + 32 × 5 = 1104 ÷ 16 × 148 ÷ 37 तो k का मान क्या है?&lt;/p&gt;

&lt;p&gt;(a) 276 / 276&lt;br&gt;&lt;br&gt;
(b) 26 / 26&lt;br&gt;&lt;br&gt;
(c) 35 / 35&lt;br&gt;&lt;br&gt;
(d) 302 / 302  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 26

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Apply the BODMAS rule (Divide and Multiply first, then Add and Subtract) on both sides.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Left Hand Side (LHS):&lt;/strong&gt;&lt;br&gt;
(640 / 8) + (930 / 15) - k + (32 × 5)&lt;br&gt;
= 80 + 62 - k + 160&lt;br&gt;
= 302 - k&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Right Hand Side (RHS):&lt;/strong&gt;&lt;br&gt;
(1104 / 16) × (148 / 37)&lt;br&gt;
= 69 × 4&lt;br&gt;
= 276&lt;/p&gt;

&lt;p&gt;Equating LHS and RHS:&lt;br&gt;
302 - k = 276&lt;br&gt;
k = 302 - 276 = &lt;strong&gt;26&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; k का मान 26 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q38. If (a - 2b) = 8 and 2ab = 9, then find the value of (a + 2b).&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि (a - 2b) = 8 और 2ab = 9 तो (a + 2b) का मान ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 6 / 6&lt;br&gt;&lt;br&gt;
(b) 12 / 12&lt;br&gt;&lt;br&gt;
(c) 10 / 10&lt;br&gt;&lt;br&gt;
(d) 8 / 8  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 10

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; We can use the algebraic identity:&lt;br&gt;
&lt;strong&gt;(x + y)² = (x - y)² + 4xy&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Here, let x = a and y = 2b.&lt;br&gt;
(a + 2b)² = (a - 2b)² + 4(a)(2b)&lt;br&gt;
(a + 2b)² = (a - 2b)² + 8ab&lt;/p&gt;

&lt;p&gt;Substitute the given values: (a - 2b) = 8 and 2ab = 9.&lt;br&gt;
Notice that 8ab = 4 × (2ab) = 4 × 9 = 36.&lt;/p&gt;

&lt;p&gt;(a + 2b)² = (8)² + 36&lt;br&gt;
(a + 2b)² = 64 + 36&lt;br&gt;
(a + 2b)² = 100&lt;br&gt;
(a + 2b) = √100 = &lt;strong&gt;10&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; (a + 2b) का मान 10 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q39. The ratio of people speaking Hindi, English and German is 5 : 4 : 3. If the number of people speaking Hindi is 15,500, then find the total population of the Town.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;हिंदी, अंग्रेजी और जर्मन बोलने वाले लोगों का अनुपात 5 : 4 : 3 है। यदि हिंदी बोलने वाले लोगों की संख्या 15,500 है, तो शहर की कुल जनसंख्या ज्ञात कीजिए?&lt;/p&gt;

&lt;p&gt;(a) 23400 / 23400&lt;br&gt;&lt;br&gt;
(b) 31000 / 31000&lt;br&gt;&lt;br&gt;
(c) 43000 / 43000&lt;br&gt;&lt;br&gt;
(d) 37200 / 37200  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 37200

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let the number of people speaking Hindi, English, and German be 5x, 4x, and 3x respectively.&lt;br&gt;
Total population = 5x + 4x + 3x = 12x.&lt;/p&gt;

&lt;p&gt;We are given that the Hindi speaking population is 15,500.&lt;br&gt;
5x = 15500&lt;br&gt;
x = 15500 / 5 = 3100&lt;/p&gt;

&lt;p&gt;Total Population = 12x = 12 × 3100 = &lt;strong&gt;37,200&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; शहर की कुल जनसंख्या 37,200 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q40. The average weight of 10 men is decreased by 2 kg when one of them whose weight is 60 kg is replaced by a new man. What is the weight of the new man?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;10 पुरुषों के औसत वजन में 2 किग्रा की कमी होती है जब उनमें से एक, जिसका वजन 60 किग्रा है, को नए व्यक्ति से बदला जाता है। नए व्यक्ति का वजन क्या होगा?&lt;/p&gt;

&lt;p&gt;(a) 35 kg / 35 किग्रा&lt;br&gt;&lt;br&gt;
(b) 40 kg / 40 किग्रा&lt;br&gt;&lt;br&gt;
(c) 45 kg / 45 किग्रा&lt;br&gt;&lt;br&gt;
(d) 50 kg / 50 किग्रा  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 40 kg

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The total number of men is 10. &lt;br&gt;
If the average weight decreases by 2 kg, it means the total weight of the group has decreased by:&lt;br&gt;
Total decrease = 10 men × 2 kg = 20 kg.&lt;/p&gt;

&lt;p&gt;Since the old man leaving weighed 60 kg, the new man must be 20 kg lighter than him to cause this overall drop in the average.&lt;br&gt;
Weight of new man = Weight of replaced man - Total decrease&lt;br&gt;
Weight of new man = 60 kg - 20 kg = &lt;strong&gt;40 kg&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; नए व्यक्ति का वजन 40 किग्रा है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q41. The sum of the present age of mother and her son is 60 years. Six years ago, the age of the mother was five times the age of her son. What will be the age of her son after 6 years?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;माँ और उसके पुत्र की वर्तमान आयु का योग 60 वर्ष है। छह वर्ष पहले, माँ की आयु पुत्र की आयु का पाँच गुना थी। छह वर्षों बाद पुत्र की आयु क्या होगी?&lt;/p&gt;

&lt;p&gt;(a) 23 years / 23 वर्ष&lt;br&gt;&lt;br&gt;
(b) 22 years / 22 वर्ष&lt;br&gt;&lt;br&gt;
(c) 20 years / 20 वर्ष&lt;br&gt;&lt;br&gt;
(d) 21 years / 21 वर्ष  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 20 years

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let the present age of the Mother be &lt;strong&gt;M&lt;/strong&gt; and the Son be &lt;strong&gt;S&lt;/strong&gt;.&lt;br&gt;
Given: M + S = 60.&lt;/p&gt;

&lt;p&gt;Six years ago:&lt;br&gt;
Mother's age = (M - 6)&lt;br&gt;
Son's age = (S - 6)&lt;br&gt;
Condition: (M - 6) = 5 × (S - 6)&lt;br&gt;
M - 6 = 5S - 30&lt;br&gt;
M - 5S = -24&lt;/p&gt;

&lt;p&gt;Substitute M = 60 - S into the equation:&lt;br&gt;
(60 - S) - 5S = -24&lt;br&gt;
60 - 6S = -24&lt;br&gt;
6S = 84&lt;br&gt;
S = 14 (Present age of the son)&lt;/p&gt;

&lt;p&gt;The age of the son after 6 years = 14 + 6 = &lt;strong&gt;20 years&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 6 वर्ष बाद पुत्र की आयु 20 वर्ष होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q42. Which of the following numbers is divisible by 9?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन सी संख्या 9 से विभाज्य है?&lt;/p&gt;

&lt;p&gt;(a) 67578 / 67578&lt;br&gt;&lt;br&gt;
(b) 56785 / 56785&lt;br&gt;&lt;br&gt;
(c) 45678 / 45678&lt;br&gt;&lt;br&gt;
(d) 65889 / 65889  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 65889

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is a multiple of 9. Let's check the options:&lt;br&gt;
(a) 6 + 7 + 5 + 7 + 8 = 33 (Not divisible by 9)&lt;br&gt;
(b) 5 + 6 + 7 + 8 + 5 = 31 (Not divisible by 9)&lt;br&gt;
(c) 4 + 5 + 6 + 7 + 8 = 30 (Not divisible by 9)&lt;br&gt;
(d) 6 + 5 + 8 + 8 + 9 = 36 (Divisible by 9 because 36 / 9 = 4)&lt;/p&gt;

&lt;p&gt;Therefore, &lt;strong&gt;65889&lt;/strong&gt; is divisible by 9.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 65889 के अंकों का योग 36 है, जो 9 से विभाज्य है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q43. The sum of the ages of father and his son is 44 years. If 6 years after the father will be 3 times as old as his son, what are their present ages?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;पिता और उसके पुत्र की आयु का योग 44 वर्ष है। यदि 6 वर्षों बाद पिता अपने पुत्र की आयु का 3 गुना होगा, तो उनकी वर्तमान आयु क्या हैं?&lt;/p&gt;

&lt;p&gt;(a) 36, 8 / 36, 8&lt;br&gt;&lt;br&gt;
(b) 38, 6 / 38, 6&lt;br&gt;&lt;br&gt;
(c) 35, 9 / 35, 9&lt;br&gt;&lt;br&gt;
(d) 37, 7 / 37, 7  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 36, 8

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let Father's present age be &lt;strong&gt;F&lt;/strong&gt; and Son's present age be &lt;strong&gt;S&lt;/strong&gt;.&lt;br&gt;
Given: F + S = 44 =&amp;gt; F = 44 - S&lt;/p&gt;

&lt;p&gt;Six years later:&lt;br&gt;
Father's age = F + 6&lt;br&gt;
Son's age = S + 6&lt;br&gt;
Condition: F + 6 = 3 × (S + 6)&lt;/p&gt;

&lt;p&gt;Substitute F = 44 - S into the equation:&lt;br&gt;
(44 - S) + 6 = 3S + 18&lt;br&gt;
50 - S = 3S + 18&lt;br&gt;
4S = 32&lt;br&gt;
S = &lt;strong&gt;8&lt;/strong&gt; (Son's present age)&lt;/p&gt;

&lt;p&gt;Father's present age F = 44 - 8 = &lt;strong&gt;36&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; पिता और पुत्र की वर्तमान आयु क्रमशः 36 और 8 वर्ष है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q44. An article is sold at a loss of 25 percent. If its cost price is doubled and the selling price is increased by ₹660, then a profit of 20 percent is obtained. What is the actual cost price of the article?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक वस्तु 25 प्रतिशत की हानि पर बेची जाती है। यदि इसकी लागत कीमत दोगुनी हो जाती है और बिक्री मूल्य में 660 रुपये की वृद्धि हो जाती है, तो 20 प्रतिशत का लाभ प्राप्त होता है। वस्तु की वास्तविक लागत मूल्य क्या है?&lt;/p&gt;

&lt;p&gt;(a) ₹360 / ₹360&lt;br&gt;&lt;br&gt;
(b) ₹500 / ₹500&lt;br&gt;&lt;br&gt;
(c) ₹480 / ₹480&lt;br&gt;&lt;br&gt;
(d) ₹400 / ₹400  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 400

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let the original Cost Price (CP) be &lt;strong&gt;100x&lt;/strong&gt;.&lt;br&gt;
Since there is a 25% loss, Original Selling Price (SP) = &lt;strong&gt;75x&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;Condition 1: Cost price is doubled. &lt;br&gt;
New CP = 2 × 100x = 200x.&lt;br&gt;
Condition 2: Selling price is increased by 660. &lt;br&gt;
New SP = 75x + 660.&lt;/p&gt;

&lt;p&gt;There is a 20% profit on the New CP. &lt;br&gt;
Therefore, New SP = 120% of New CP.&lt;br&gt;
New SP = (120 / 100) × 200x = 240x.&lt;/p&gt;

&lt;p&gt;Equating both expressions for New SP:&lt;br&gt;
75x + 660 = 240x&lt;br&gt;
165x = 660&lt;br&gt;
x = 660 / 165 = 4&lt;/p&gt;

&lt;p&gt;Actual Cost Price = 100x = 100 × 4 = &lt;strong&gt;₹400&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वस्तु का वास्तविक लागत मूल्य ₹400 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q45. A debt of ₹10,192 due in 3 years at 10% simple interest will be repaid by equal annual instalments of how much?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;₹10,192 का ऋण जो 3 वर्ष में 10% साधारण ब्याज पर चुकाना है, उसे कितनी बराबर वार्षिक किस्तों में चुकाया जाएगा?&lt;/p&gt;

&lt;p&gt;(a) ₹3,144.48 / ₹3,144.48&lt;br&gt;&lt;br&gt;
(b) ₹3,044.48 / ₹3,044.48&lt;br&gt;&lt;br&gt;
(c) ₹3,088.48 / ₹3,088.48&lt;br&gt;&lt;br&gt;
(d) ₹3,188.48 / ₹3,188.48  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 3,088.48

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; To find the value of each equal annual installment (&lt;strong&gt;x&lt;/strong&gt;) that discharges a debt (&lt;strong&gt;A&lt;/strong&gt;) in &lt;strong&gt;n&lt;/strong&gt; years at &lt;strong&gt;R&lt;/strong&gt;% simple interest, we use the formula:&lt;br&gt;
&lt;strong&gt;A = n × x + [x × R × n(n-1)] / 200&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Given: Total Debt (A) = 10,192, Time (n) = 3 years, Rate (R) = 10%.&lt;br&gt;
10192 = (3 × x) + [x × 10 × 3(2)] / 200&lt;br&gt;
10192 = 3x + [60x] / 200&lt;br&gt;
10192 = 3x + 0.3x&lt;br&gt;
10192 = 3.3x&lt;/p&gt;

&lt;p&gt;x = 10192 / 3.3 = 101920 / 33 = &lt;strong&gt;₹3,088.48&lt;/strong&gt; (approx).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; समान वार्षिक किस्त ₹3,088.48 होगी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Q66. Pointing to a photograph, Veena says, "The only daughter-in-law of her father is my mother-in-law." How is the woman's brother related to Veena?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक तस्वीर की ओर इशारा करते हुए वीणा कहती है, "उसके पिता की इकलौती बहू मेरी सास है।" महिला के भाई का वीणा से क्या संबंध है?&lt;/p&gt;

&lt;p&gt;(a) Father / पिता&lt;br&gt;&lt;br&gt;
(b) Father-in-law / ससुर&lt;br&gt;&lt;br&gt;
(c) Uncle / चाचा&lt;br&gt;&lt;br&gt;
(d) Son / पुत्र  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) Father-in-law

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's break down Veena's statement:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;"her father" = the father of the woman in the photograph.&lt;/li&gt;
&lt;li&gt;"The only daughter-in-law of her father" = the wife of the woman's only brother.&lt;/li&gt;
&lt;li&gt;"...is my mother-in-law" = So, the wife of the woman's brother is Veena's mother-in-law.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;Since the woman's brother is the husband of Veena's mother-in-law, the woman's brother must be Veena's &lt;strong&gt;Father-in-law&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; महिला का भाई वीणा का ससुर है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q67. If A $B means A is the husband of B, A # B means A is the brother of B, and A * B means A is the mother of B, then what does 'P$ Q * R # S' mean?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि A $B का अर्थ है A, B का पति है, A # B का अर्थ है A, B का भाई है और A * B का अर्थ है A, B की माँ है, तो 'P$ Q * R # S' का क्या अर्थ है?&lt;/p&gt;

&lt;p&gt;(a) P is the brother of S / P, S का भाई है&lt;br&gt;&lt;br&gt;
(b) P is the father of S / P, S का पिता है&lt;br&gt;&lt;br&gt;
(c) P is the son of S / P, S का बेटा है&lt;br&gt;&lt;br&gt;
(d) P is the brother-in-law of S / P, S का साला है  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) P is the father of S

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Decode the string &lt;code&gt;P $ Q * R # S&lt;/code&gt; from left to right:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;P $ Q:&lt;/strong&gt; P is the husband of Q. (This makes P male and Q female).&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Q * R:&lt;/strong&gt; Q is the mother of R. (Since P is Q's husband, P is the father of R).&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;R # S:&lt;/strong&gt; R is the brother of S. &lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Since R and S are siblings, and P is the father of R, it implies that &lt;strong&gt;P is the father of S&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; P, S का पिता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q68. In a row, Deepti and Mohan are 14th from the left and 9th from the right respectively. If they interchange their positions, Mohan becomes 21st from the right. Now find Deepti's position from the left and the total number of persons in the row.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक पंक्ति में, दीप्ति और मोहन क्रमशः बाएं से 14वें और दाएं से 9वें स्थान पर हैं। यदि वे अपना स्थान बदल लेते हैं, तो मोहन दाईं ओर से 21वें स्थान पर आ जाता है। अब बाईं ओर से दीप्ति की स्थिति और पंक्ति में व्यक्तियों की कुल संख्या ज्ञात कीजिए।&lt;/p&gt;

&lt;p&gt;(a) 26, 34&lt;br&gt;&lt;br&gt;
(b) 24, 38&lt;br&gt;&lt;br&gt;
(c) 33, 27&lt;br&gt;&lt;br&gt;
(d) 34, 26  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 26, 34

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; &lt;strong&gt;1. Find Total Persons:&lt;/strong&gt;&lt;br&gt;
After swapping, Mohan is at Deepti's old position (which was 14th from the left).&lt;br&gt;
We are given Mohan is now 21st from the right.&lt;br&gt;
Total Persons = (Position from Left) + (Position from Right) - 1&lt;br&gt;
Total = 14 + 21 - 1 = &lt;strong&gt;34 persons&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;2. Find Deepti's New Position from Left:&lt;/strong&gt;&lt;br&gt;
After swapping, Deepti is at Mohan's old position (9th from the right).&lt;br&gt;
Position from Left = Total - Position from Right + 1&lt;br&gt;
Deepti's Position = 34 - 9 + 1 = &lt;strong&gt;26th&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; दीप्ति का नया स्थान 26वाँ है और कुल व्यक्ति 34 हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q69. Lalita is shorter than Gagan but taller than Prerna. Vinita is taller than Gagan but shorter than Danish. Who among them is the shortest?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;ललिता गगन से छोटी, लेकिन प्रेरणा से लंबी है। विनीता गगन से लंबी, लेकिन दानिश से छोटी है। इनमें सबसे छोटा कौन है?&lt;/p&gt;

&lt;p&gt;(a) Danish / दानिश&lt;br&gt;&lt;br&gt;
(b) Lalita / ललिता&lt;br&gt;&lt;br&gt;
(c) Prerna / प्रेरणा&lt;br&gt;&lt;br&gt;
(d) Gagan / गगन  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) Prerna

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's arrange them by height (shortest to tallest):&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Lalita is shorter than Gagan but taller than Prerna: &lt;strong&gt;Prerna &amp;lt; Lalita &amp;lt; Gagan&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Vinita is taller than Gagan but shorter than Danish: &lt;strong&gt;Gagan &amp;lt; Vinita &amp;lt; Danish&lt;/strong&gt;
&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;Combining both statements: &lt;strong&gt;Prerna &amp;lt; Lalita &amp;lt; Gagan &amp;lt; Vinita &amp;lt; Danish&lt;/strong&gt;.&lt;br&gt;
The shortest person is &lt;strong&gt;Prerna&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; सबसे छोटी प्रेरणा है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q70. P, Q, R, S, T and U are sitting on a wall and all of them are facing north. R is sitting immediately to the left of U. Q is sitting at the extreme end and U is his neighbour. Only T is sitting between R and P. Who is sitting to the left of P?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;P, Q, R, S, T और U एक दीवार पर बैठे हैं और सभी का मुंह उत्तर दिशा में है। R, U के ठीक बाईं ओर बैठा है। Q सबसे आखिर में बैठा है और U उसका पड़ोसी है। R और P के बीच केवल T बैठा है। P के बाईं ओर कौन बैठा है?&lt;/p&gt;

&lt;p&gt;(a) R&lt;br&gt;&lt;br&gt;
(b) T&lt;br&gt;&lt;br&gt;
(c) U&lt;br&gt;&lt;br&gt;
(d) S  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) S

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's deduce their positions step-by-step (facing North, so Left is our left):&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Q is at an extreme end and U is his neighbor.

&lt;ul&gt;
&lt;li&gt;If Q is at the left end: [Q, U, _, _, _, _]. But R must be immediately to the left of U, which is impossible here.&lt;/li&gt;
&lt;li&gt;So, Q must be at the right end: [_, _, _, _, U, Q].&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;R is immediately to the left of U: [_, _, _, R, U, Q].&lt;/li&gt;
&lt;li&gt;T is between R and P. Since R is in position 4, T must be at 3 and P at 2: [_, P, T, R, U, Q].&lt;/li&gt;
&lt;li&gt;The only remaining person is S, who takes the 1st position: &lt;strong&gt;[S, P, T, R, U, Q]&lt;/strong&gt;.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;The person sitting immediately to the left of P is &lt;strong&gt;S&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; P के बाईं ओर S बैठा है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q71. Four friends - Himani, Shalaka, Mitali and Brinda are sitting around a square table facing the centre. They all are sitting at the corners of the table. Himani is sitting immediately to the right of Brinda. Mitali is sitting immediately to the left of Shalaka. Himani is second to the left of Shalaka. After some time, Himani leaves her place and Kamini takes her place. Similarly, Tara takes Brinda's place. Who is sitting immediately to the left of Tara now?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;चार मित्र – हिमानी, शलाका, मिताली और बृंदा - एक वर्गाकार मेज के चारों ओर केंद्र की तरफ मुँह करके बैठे हैं। वे सभी मेज़ के कोनों पर बैठे हैं। हिमानी बृंदा की ठीक दाहिनी ओर बैठी है। मिताली शलाका की ठीक बाईं ओर बैठी है। हिमानी शलाका के बाईं ओर दूसरे स्थान पर है। कुछ समय बाद हिमानी अपना स्थान छोड़ देती है और कामिनी उसका स्थान लेती है। इसी तरह, तारा बृंदा का स्थान लेती है। तारा के बाईं ओर अब कौन बैठा है?&lt;/p&gt;

&lt;p&gt;(a) Mitali / मिताली&lt;br&gt;&lt;br&gt;
(b) Shalaka / शलाका&lt;br&gt;&lt;br&gt;
(c) Brinda / बृंदा&lt;br&gt;&lt;br&gt;
(d) Kamini / कामिनी  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) Kamini

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's place them at the corners facing the center.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Himani (H) is 2nd to the left of Shalaka (S). Since there are only 4 people, this means H and S are sitting diagonally opposite each other.&lt;/li&gt;
&lt;li&gt;H is immediately to the right of Brinda (B). If you visualize a circle/square, clockwise order (right) from B is H.&lt;/li&gt;
&lt;li&gt;Mitali (M) is immediately to the left of S. Clockwise order (left) from S is M.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The initial seating arrangement (clockwise starting from Top-Left) is: &lt;strong&gt;Brinda (TL), Mitali (TR), Shalaka (BR), Himani (BL)&lt;/strong&gt;.&lt;br&gt;
Now, swap the names:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Himani $\rightarrow$ Kamini (BL)&lt;/li&gt;
&lt;li&gt;Brinda $\rightarrow$ Tara (TL)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;New arrangement: &lt;strong&gt;Tara (TL), Mitali (TR), Shalaka (BR), Kamini (BL)&lt;/strong&gt;.&lt;br&gt;
The person sitting immediately to the left (clockwise) of Tara is &lt;strong&gt;Kamini&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; तारा के बाईं ओर अब कामिनी बैठी है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q72. Anjali reached the conference hall at 8:30 which was 15 minutes before the scheduled time. What was the scheduled time of the conference?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;अंजलि 8:30 बजे सम्मेलन भवन में पहुँची, जो तय समय से 15 मिनट पहले था। सम्मेलन का तय समय क्या था?&lt;/p&gt;

&lt;p&gt;(a) 8:15&lt;br&gt;&lt;br&gt;
(b) 8:30&lt;br&gt;&lt;br&gt;
(c) 8:05&lt;br&gt;&lt;br&gt;
(d) 8:45  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 8:45

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Anjali arrived at 8:30. This time was 15 minutes &lt;em&gt;early&lt;/em&gt; (before the scheduled time).&lt;br&gt;
Therefore, the scheduled time was 15 minutes later.&lt;br&gt;
8:30 + 15 minutes = &lt;strong&gt;8:45&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; सम्मेलन का तय समय 8:45 था।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q73. In a school, the class starts at 1:00 pm and continues till 3:52 pm. During this period there are 4 periods and 4 minutes are allowed before each period to move from one room to another. How many minutes long is one period?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक विद्यालय में, कक्षा दोपहर 1:00 बजे प्रारंभ होती है और दोपहर 3:52 बजे तक जारी रहती है। इस दौरान 4 कक्षाएं होती हैं और प्रत्येक कक्षा से पहले एक कमरे से दूसरे कमरे में जाने के लिए 4 मिनट का समय लगता है। एक कक्षा कितने मिनट की होती है?&lt;/p&gt;

&lt;p&gt;(a) 39 minutes / 39 मिनट&lt;br&gt;&lt;br&gt;
(b) 40 minutes / 40 मिनट&lt;br&gt;&lt;br&gt;
(c) 41 minutes / 41 मिनट&lt;br&gt;&lt;br&gt;
(d) 42 minutes / 42 मिनट  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 40 minutes

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Total time from 1:00 PM to 3:52 PM = 2 hours and 52 minutes = (2 × 60) + 52 = &lt;strong&gt;172 minutes&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;There are 4 periods. Between 4 periods, there are exactly 3 intervals or breaks for room movement.&lt;br&gt;
Time spent moving = 3 intervals × 4 minutes = &lt;strong&gt;12 minutes&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;Total time spent actually in classes = Total time - Movement time&lt;br&gt;
Class time = 172 - 12 = &lt;strong&gt;160 minutes&lt;/strong&gt;.&lt;br&gt;
Since all 4 periods are equal, the length of one period = 160 / 4 = &lt;strong&gt;40 minutes&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; एक कक्षा 40 मिनट की होती है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q74. If 25th August of a year was a Thursday, how many Tuesdays were there in that month?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;अगर किसी वर्ष 25 अगस्त को गुरुवार था, तो उस महीने में कितने मंगलवार थे?&lt;/p&gt;

&lt;p&gt;(a) 3&lt;br&gt;&lt;br&gt;
(b) 5&lt;br&gt;&lt;br&gt;
(c) 4&lt;br&gt;&lt;br&gt;
(d) 6  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) 5

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; If August 25th is a Thursday:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Wednesday is 24th&lt;/li&gt;
&lt;li&gt;Tuesday is 23rd&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Now, keep subtracting 7 to find all the preceding Tuesdays in August:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;23 - 7 = 16th&lt;/li&gt;
&lt;li&gt;16 - 7 = 9th&lt;/li&gt;
&lt;li&gt;9 - 7 = 2nd&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;And add 7 to find following Tuesdays in August (which has 31 days):&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;23 + 7 = 30th&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The Tuesdays fall on the 2nd, 9th, 16th, 23rd, and 30th.&lt;br&gt;
Total number of Tuesdays = &lt;strong&gt;5&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; उस महीने में 5 मंगलवार थे।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q75. Sangeeta remembers that her father's birthday was definitely after 8th December but before 13th December. Her sister Seema remembers that her father's birthday was definitely after 9th December but before 14th December. On which date in December was their father's birthday?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;संगीता को याद है कि उनके पिता का जन्मदिन 8 दिसंबर के बाद लेकिन 13 दिसंबर से पहले था। उसकी बहन सीमा को याद है कि उनके पिता का जन्मदिन 9 दिसंबर के बाद लेकिन 14 दिसंबर से पहले था। दिसंबर में उनके पिता का जन्मदिन किस तारीख को था?&lt;/p&gt;

&lt;p&gt;(a) 10&lt;br&gt;&lt;br&gt;
(b) 11&lt;br&gt;&lt;br&gt;
(c) 12&lt;br&gt;&lt;br&gt;
(d) Data inadequate / आँकड़ा अपर्याप्त  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) Data inadequate

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; According to Sangeeta, the possible dates are: 9, 10, 11, 12.&lt;br&gt;
According to Seema, the possible dates are: 10, 11, 12, 13.&lt;/p&gt;

&lt;p&gt;The dates common to both their memories are the 10th, 11th, and 12th.&lt;br&gt;
Because there are three possible dates remaining and no further clues to narrow it down, the exact date cannot be determined.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; क्योंकि 10, 11 और 12 तीनों तारीखें संभावित हैं, इसलिए आँकड़े अपर्याप्त हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q76. If '+' is changed to '÷', '-' is changed to 'x', '÷' is changed to '-', and 'x' is changed to '+', then what will be the value of the following expression?&lt;/strong&gt;&lt;br&gt;
&lt;strong&gt;65 - 10 + 5 ÷ 3 x 4 = ?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि '+' को '÷' में, '-' को 'x' में, '÷' को '-' में, और 'x' को '+' में परिवर्तित कर दिया जाता है, तो निम्नलिखित समीकरण का मान क्या होगा?&lt;br&gt;
65 - 10 + 5 ÷ 3 x 4 = ?&lt;/p&gt;

&lt;p&gt;(a) 12&lt;br&gt;&lt;br&gt;
(b) 110&lt;br&gt;&lt;br&gt;
(c) 131&lt;br&gt;&lt;br&gt;
(d) 9  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 131

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Replace the symbols according to the given rules:&lt;br&gt;
Original: 65 - 10 + 5 ÷ 3 x 4&lt;br&gt;
New: 65 × 10 ÷ 5 - 3 + 4&lt;/p&gt;

&lt;p&gt;Apply BODMAS (Divide, Multiply, Add/Subtract):&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;10 ÷ 5 = 2&lt;/li&gt;
&lt;li&gt;65 × 2 = 130&lt;/li&gt;
&lt;li&gt;130 - 3 + 4 = 127 + 4 = &lt;strong&gt;131&lt;/strong&gt;
&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; समीकरण का मान 131 होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q77. Which two mathematical symbols should be interchanged to make the following equation correct?&lt;/strong&gt;&lt;br&gt;
&lt;strong&gt;(45 x 2) ÷ 15 - (6 x 3) + 10 - (63 ÷ 21) = 17&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;निम्नलिखित समीकरण को सही बनाने के लिए किन दो गणितीय चिह्नों को आपस में बदलना चाहिए?&lt;br&gt;
(45 x 2) ÷ 15 - (6 x 3) + 10 - (63 ÷ 21) = 17&lt;/p&gt;

&lt;p&gt;(a) + and x / + और x&lt;br&gt;&lt;br&gt;
(b) - and + / - और +&lt;br&gt;&lt;br&gt;
(c) x and - / x और -&lt;br&gt;&lt;br&gt;
(d) ÷ and + / ÷ और +  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) - and +

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's test swapping the '-' and '+' symbols in the equation.&lt;br&gt;
Original: (45 × 2) ÷ 15 - (6 × 3) + 10 - (63 ÷ 21)&lt;br&gt;
New: (45 × 2) ÷ 15 &lt;strong&gt;+&lt;/strong&gt; (6 × 3) &lt;strong&gt;-&lt;/strong&gt; 10 &lt;strong&gt;+&lt;/strong&gt; (63 ÷ 21)&lt;/p&gt;

&lt;p&gt;Now, evaluate step-by-step using BODMAS:&lt;br&gt;
= 90 ÷ 15 + 18 - 10 + 3&lt;br&gt;
= 6 + 18 - 10 + 3&lt;br&gt;
= 24 - 10 + 3&lt;br&gt;
= 14 + 3 = &lt;strong&gt;17&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;The equation holds true. Therefore, '-' and '+' should be swapped.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; '-' और '+' को आपस में बदलने पर समीकरण सही हो जाएगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q78. In a test match, Kumble took one wicket more than Srinath, who took one wicket more than Harbhajan. If the product of wickets taken by all three is 210, how many wickets did Kumble take?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक टेस्ट मैच में कुंबले ने श्रीनाथ से एक विकेट अधिक लिया, जिन्होंने हरभजन से एक विकेट अधिक लिया था। अगर तीनों द्वारा लिए गए विकेटों का गुणनफल 210 है, तो कुंबले ने कितने विकेट लिए?&lt;/p&gt;

&lt;p&gt;(a) 5&lt;br&gt;&lt;br&gt;
(b) 6&lt;br&gt;&lt;br&gt;
(c) 7&lt;br&gt;&lt;br&gt;
(d) 8  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 7

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let Harbhajan's wickets be &lt;strong&gt;x&lt;/strong&gt;.&lt;br&gt;
Srinath's wickets = &lt;strong&gt;x + 1&lt;/strong&gt;.&lt;br&gt;
Kumble's wickets = &lt;strong&gt;x + 2&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;Their product is 210:&lt;br&gt;
x(x + 1)(x + 2) = 210&lt;br&gt;
We need three consecutive numbers that multiply to 210. &lt;br&gt;
Let's factor 210: 210 = 21 × 10 = 3 × 7 × 2 × 5 = &lt;strong&gt;5 × 6 × 7&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;So, the wickets are 5, 6, and 7.&lt;br&gt;
Harbhajan = 5, Srinath = 6, Kumble = 7.&lt;br&gt;
Kumble took &lt;strong&gt;7 wickets&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; कुंबले ने 7 विकेट लिए।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q79. 5 years ago, Rohan's father's age was 8 times Rohan's age. 5 years from now, the ratio of Rohan's father's age to Rohan's age will be 10 : 3. What is Rohan's present age?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;5 वर्ष पहले, रोहन के पिता की आयु रोहन की आयु की 8 गुना थी। वर्तमान से 5 वर्ष बाद, रोहन के पिता की आयु और रोहन की आयु का अनुपात 10 : 3 होगा। रोहन की वर्तमान आयु क्या है?&lt;/p&gt;

&lt;p&gt;(a) 10 years / 10 वर्ष&lt;br&gt;&lt;br&gt;
(b) 8 years / 8 वर्ष&lt;br&gt;&lt;br&gt;
(c) 15 years / 15 वर्ष&lt;br&gt;&lt;br&gt;
(d) 12 years / 12 वर्ष  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 10 years

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let Rohan's present age be &lt;strong&gt;R&lt;/strong&gt; and his Father's present age be &lt;strong&gt;F&lt;/strong&gt;.&lt;br&gt;
5 years ago: F - 5 = 8(R - 5)&lt;br&gt;
F - 5 = 8R - 40&lt;br&gt;
F = 8R - 35  &lt;em&gt;(Equation 1)&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;5 years from now: (F + 5) / (R + 5) = 10 / 3&lt;br&gt;
3(F + 5) = 10(R + 5)&lt;br&gt;
3F + 15 = 10R + 50&lt;br&gt;
3F - 10R = 35 &lt;em&gt;(Equation 2)&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;Substitute Equation 1 into Equation 2:&lt;br&gt;
3(8R - 35) - 10R = 35&lt;br&gt;
24R - 105 - 10R = 35&lt;br&gt;
14R = 140&lt;br&gt;
R = &lt;strong&gt;10&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; रोहन की वर्तमान आयु 10 वर्ष है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q80. At present, the ratio of the ages of Anu and Dimpy is 5 : 8. After 2 years, their ages will be in the ratio 2 : 3. What was Dimpy's age 2 years ago?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;वर्तमान में अनु और डिम्पी की आयु का अनुपात 5 : 8 है। 2 वर्ष के बाद, उनकी आयु 2 : 3 के अनुपात में होगी। 2 वर्ष पहले डिम्पी की आयु क्या थी?&lt;/p&gt;

&lt;p&gt;(a) 12 years / 12 वर्ष&lt;br&gt;&lt;br&gt;
(b) 10 years / 10 वर्ष&lt;br&gt;&lt;br&gt;
(c) 14 years / 14 वर्ष&lt;br&gt;&lt;br&gt;
(d) 16 years / 16 वर्ष  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 14 years

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let their present ages be 5x and 8x.&lt;br&gt;
After 2 years, their ages will be (5x + 2) and (8x + 2).&lt;br&gt;
The new ratio is 2/3:&lt;br&gt;
(5x + 2) / (8x + 2) = 2 / 3&lt;/p&gt;

&lt;p&gt;Cross-multiply:&lt;br&gt;
3(5x + 2) = 2(8x + 2)&lt;br&gt;
15x + 6 = 16x + 4&lt;br&gt;
x = 2&lt;/p&gt;

&lt;p&gt;Dimpy's present age = 8x = 8(2) = 16 years.&lt;br&gt;
Dimpy's age 2 years ago = 16 - 2 = &lt;strong&gt;14 years&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 2 वर्ष पहले डिम्पी की आयु 14 वर्ष थी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q81. Find the missing term in the circular pie chart sequence:&lt;/strong&gt;&lt;br&gt;
&lt;strong&gt;Sequence segments: 20, 36, 68, 132, 260, ?&lt;/strong&gt; &lt;em&gt;(Reading from bottom-left anti-clockwise based on logical progression)&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;लुप्त पद ज्ञात कीजिए - (20, 36, 68, 132, 260, ?)&lt;/p&gt;

&lt;p&gt;(a) 12&lt;br&gt;&lt;br&gt;
(b) 10&lt;br&gt;&lt;br&gt;
(c) 9&lt;br&gt;&lt;br&gt;
(d) 8  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 12

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's find the pattern starting from the smallest known number to the largest. The numbers in the circle are: ?, 20, 36, 68, 132, 260.&lt;br&gt;
Notice the mathematical relationship between adjacent numbers:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;(20 - 2) × 2 = 18 × 2 = 36&lt;/li&gt;
&lt;li&gt;(36 - 2) × 2 = 34 × 2 = 68&lt;/li&gt;
&lt;li&gt;(68 - 2) × 2 = 66 × 2 = 132&lt;/li&gt;
&lt;li&gt;(132 - 2) × 2 = 130 × 2 = 260&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;To find the missing number &lt;code&gt;?&lt;/code&gt; at the beginning of this sequence, we apply the same rule backwards:&lt;br&gt;
(? - 2) × 2 = 20&lt;br&gt;
? - 2 = 10&lt;br&gt;
? = &lt;strong&gt;12&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; लुप्त संख्या 12 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q82. If F = C &amp;lt; B = E and D &amp;lt; C = A &amp;lt; E then which of the following options is not correct?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि F = C &amp;lt; B = E और D &amp;lt; C = A &amp;lt; E है, तो निम्नलिखित में से कौन सा विकल्प सही नहीं है?&lt;/p&gt;

&lt;p&gt;(a) E &amp;gt; D&lt;br&gt;&lt;br&gt;
(b) A &amp;lt; B&lt;br&gt;&lt;br&gt;
(c) A = F&lt;br&gt;&lt;br&gt;
(d) A &amp;gt; E  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) A &amp;gt; E

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's merge the two inequalities together using the common variables C and E.&lt;br&gt;
From Statement 1: F = C, and C &amp;lt; B, and B = E.&lt;br&gt;
From Statement 2: D &amp;lt; C, and C = A, and A &amp;lt; E.&lt;/p&gt;

&lt;p&gt;Combining them logically:&lt;br&gt;
&lt;strong&gt;D &amp;lt; F = C = A &amp;lt; B = E&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Now evaluate the options:&lt;br&gt;
(a) E &amp;gt; D $\rightarrow$ True (since D is at the far left and E is at the far right).&lt;br&gt;
(b) A &amp;lt; B $\rightarrow$ True (since A = C and C &amp;lt; B).&lt;br&gt;
(c) A = F $\rightarrow$ True (since A = C and C = F).&lt;br&gt;
(d) A &amp;gt; E $\rightarrow$ &lt;strong&gt;False&lt;/strong&gt; (since A is explicitly smaller than E).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; A &amp;gt; E विकल्प गलत है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q83. A solid cube is divided into 27 equal parts. If two opposite faces of this cube are painted in different colours and the remaining four faces are left unpainted, then how many cubes will be obtained which have no face painted?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक ठोस घन को 27 बराबर भागों में विभाजित किया जाता है। यदि इस घन के दो विपरीत फलकों को अलग-अलग रंगों में चित्रित किया जाता है और शेष चार फलकों को बिना रंग के छोड़ दिया जाता है, तो कितने घन प्राप्त होंगे जिनके फलक को चित्रित नहीं किया गया है?&lt;/p&gt;

&lt;p&gt;(a) 9&lt;br&gt;&lt;br&gt;
(b) 18&lt;br&gt;&lt;br&gt;
(c) 27&lt;br&gt;&lt;br&gt;
(d) 36  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 9

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; A solid cube divided into 27 equal parts means it is a 3 × 3 × 3 grid of smaller cubes.&lt;br&gt;
It has 3 layers: Top, Middle, and Bottom, each containing 9 cubes.&lt;br&gt;
Only two &lt;em&gt;opposite&lt;/em&gt; faces of the large cube are painted (e.g., the Top face and the Bottom face).&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;The Top face consists of 9 smaller cubes. All 9 will have at least one side painted.&lt;/li&gt;
&lt;li&gt;The Bottom face consists of 9 smaller cubes. All 9 will have at least one side painted.&lt;/li&gt;
&lt;li&gt;The remaining four vertical side-faces of the large cube are unpainted.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The Middle layer (sandwiched between Top and Bottom) contains 9 smaller cubes. Since none of the vertical sides of the large cube are painted, none of the exposed faces on this middle layer get any paint. The central hidden cube obviously gets no paint either.&lt;br&gt;
Therefore, all &lt;strong&gt;9 cubes&lt;/strong&gt; in the middle layer are completely unpainted.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 9 ऐसे घन प्राप्त होंगे जिनका कोई भी फलक रंगा हुआ नहीं होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q84. Complete the code with the correct option from the following alternatives.&lt;/strong&gt;&lt;br&gt;
&lt;strong&gt;BOXER : 64 :: PLAYER : ?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;निम्नलिखित विकल्पों में से सही विकल्प के साथ कूट को पूरा करें। BOXER : 64 :: PLAYER : ?&lt;/p&gt;

&lt;p&gt;(a) 77&lt;br&gt;&lt;br&gt;
(b) 78&lt;br&gt;&lt;br&gt;
(c) 87&lt;br&gt;&lt;br&gt;
(d) 97  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 77

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The logic is the sum of the alphabetical positions of each letter (A=1, B=2, ..., Z=26).&lt;br&gt;
&lt;strong&gt;BOXER:&lt;/strong&gt;&lt;br&gt;
B(2) + O(15) + X(24) + E(5) + R(18) = 64&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;PLAYER:&lt;/strong&gt;&lt;br&gt;
P(16) + L(12) + A(1) + Y(25) + E(5) + R(18)&lt;br&gt;
Sum = 16 + 12 + 1 + 25 + 5 + 18 = &lt;strong&gt;77&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; कूट 77 होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q85. In India, a woman can marry the brother of her deceased husband and a man can marry the sister of his deceased wife.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;भारत में, एक महिला अपने मरे हुए पति के भाई से शादी कर सकती है और एक पुरुष अपनी मरी हुई पत्नी की बहन से शादी कर सकता है।&lt;/p&gt;

&lt;p&gt;(a) Always / हमेशा&lt;br&gt;&lt;br&gt;
(b) Sometimes / कभी-कभी&lt;br&gt;&lt;br&gt;
(c) Generally / आम तौर पर&lt;br&gt;&lt;br&gt;
(d) Never / कभी नहीं  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) Sometimes

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; This question pertains to logical reasoning and cultural/legal aptitude. Marrying the brother of a deceased husband is known as &lt;em&gt;Levirate&lt;/em&gt; marriage, and marrying the sister of a deceased wife is known as &lt;em&gt;Sororate&lt;/em&gt; marriage. &lt;br&gt;
Under Indian laws (like the Hindu Marriage Act), these are not considered prohibited relationships if custom allows them, and they are practiced in several communities. However, it is not a mandatory or universal practice ("Always" or "Generally" across all demographics), nor is it entirely prohibited ("Never"). It is practiced under specific conditions and cultural contexts, making "&lt;strong&gt;Sometimes&lt;/strong&gt;" the most logically accurate answer.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; यह प्रथा 'कभी-कभी' या कुछ विशिष्ट समुदायों में ही मान्य होती है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q86. Study the given diagram carefully and answer the following question. The numbers given in different sections represent the number of persons applying for different posts. What is the total number of persons applying for more than one post?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;दिए गए आरेख का ध्यानपूर्वक अध्ययन करें और निम्नलिखित प्रश्न का उत्तर दें। विभिन्न अनुभागों में दी गई संख्याएँ विभिन्न पदों के लिए आवेदन करने वाले व्यक्तियों की संख्या को दर्शाती हैं। एक से अधिक पदों के लिए आवेदन करने वाले व्यक्तियों की कुल संख्या कितनी है?&lt;/p&gt;

&lt;p&gt;&lt;em&gt;(Note: The Venn diagram consists of three intersecting circles: Teacher, Engineer, and Writer. The intersecting regions have the values 4, 6, 8, and the center intersection for all three is 3.)&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;(a) 21 / 21&lt;br&gt;&lt;br&gt;
(b) 113 / 113&lt;br&gt;&lt;br&gt;
(c) 18 / 18&lt;br&gt;&lt;br&gt;
(d) 15 / 15  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 21

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The persons applying for "more than one post" are those located in the overlapping regions of the Venn diagram.&lt;br&gt;
We simply need to add the numbers in all the intersections:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Teacher &amp;amp; Engineer only = 4&lt;/li&gt;
&lt;li&gt;Teacher &amp;amp; Writer only = 6&lt;/li&gt;
&lt;li&gt;Engineer &amp;amp; Writer only = 8&lt;/li&gt;
&lt;li&gt;All three posts = 3&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Total = 4 + 6 + 8 + 3 = &lt;strong&gt;21&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; एक से अधिक पदों के लिए आवेदन करने वाले कुल 21 व्यक्ति हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q87. Select the option which cannot be concluded from the given statements.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Statements:&lt;/strong&gt; (1) All slippers are shoes. &lt;br&gt;
(2) Some heels are slippers. &lt;br&gt;
(3) All canvas are heels. &lt;br&gt;
(4) All heels are shoes.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Conclusions:&lt;/strong&gt; (a) Some heels are shoes. &lt;br&gt;
(b) Some heels are canvas. &lt;br&gt;
(c) All slippers are heels. &lt;br&gt;
(d) Some shoes are canvas.&lt;/p&gt;

&lt;p&gt;वह विकल्प चुनें जिसका निष्कर्ष दिए गए कथनों से नहीं निकाला जा सकता है।&lt;br&gt;
&lt;strong&gt;कथनः&lt;/strong&gt; (1) सभी चप्पल जूते हैं। (2) कुछ हील्स चप्पल हैं। (3) सभी कैनवास हील्स हैं। (4) सभी हील्स जूते हैं।&lt;br&gt;
&lt;strong&gt;निष्कर्षः&lt;/strong&gt; (a) कुछ हील्स जूते हैं। &lt;br&gt;
(b) कुछ हील्स कैनवास हैं। &lt;br&gt;
(c) सभी चप्पल हील्स हैं। &lt;br&gt;
(d) कुछ जूते कैनवास हैं।&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) All slippers are heels.

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's evaluate the conclusions based on the statements:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;(a) Some heels are shoes: True. Statement 4 says "All heels are shoes", which guarantees that &lt;em&gt;some&lt;/em&gt; heels are definitely shoes.&lt;/li&gt;
&lt;li&gt;(b) Some heels are canvas: True. Statement 3 says "All canvas are heels", meaning the canvas circle is inside the heels circle. Thus, some heels are canvas.&lt;/li&gt;
&lt;li&gt;(c) All slippers are heels: &lt;strong&gt;False (Cannot be concluded)&lt;/strong&gt;. Statement 2 says "Some heels are slippers". This only proves an intersection between them, not that &lt;em&gt;all&lt;/em&gt; slippers are inside the heels category.&lt;/li&gt;
&lt;li&gt;(d) Some shoes are canvas: True. "All canvas are heels" and "All heels are shoes" means "All canvas are shoes". Therefore, some shoes are indeed canvas.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; कथन "सभी चप्पल हील्स हैं" का निश्चित निष्कर्ष नहीं निकाला जा सकता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q88. In the English alphabet, which letter will be in the middle of the 7th letter from the left and the 10th letter from the right?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;अंग्रेजी वर्णमाला में, बाईं ओर से 7वें अक्षर और दाई ओर से 10वें अक्षर के बीच कौन सा अक्षर होगा?&lt;/p&gt;

&lt;p&gt;(a) L&lt;br&gt;&lt;br&gt;
(b) M&lt;br&gt;&lt;br&gt;
(c) N&lt;br&gt;&lt;br&gt;
(d) O  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) L

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; 1. The 7th letter from the left is &lt;strong&gt;G&lt;/strong&gt; (Position 7).&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;The 10th letter from the right can be found by subtracting from 27: 
Position from left = 27 - 10 = &lt;strong&gt;17th&lt;/strong&gt; letter, which is &lt;strong&gt;Q&lt;/strong&gt;.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;To find the middle letter, average their positions from the left:&lt;br&gt;
Middle Position = (7 + 17) / 2 = 24 / 2 = &lt;strong&gt;12&lt;/strong&gt;.&lt;br&gt;
The 12th letter in the alphabet is &lt;strong&gt;L&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; बीच का अक्षर L होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q89. Using the 2nd, 4th, 5th and 6th letters of the word HOCKEY (counting from left to right), how many meaningful English words can be formed if each letter is used only once in each word?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;HOCKEY शब्द के दूसरे, चौथे, पाँचवें और छठे अक्षर का उपयोग करके (बाएँ से दाएँ गिनते हुए), कितने अर्थपूर्ण अंग्रेजी शब्द बनाए जा सकते हैं, अगर प्रत्येक शब्द में प्रत्येक अक्षर का उपयोग केवल एक बार किया जाए?&lt;/p&gt;

&lt;p&gt;(a) Three / तीन&lt;br&gt;&lt;br&gt;
(b) Zero / शून्य&lt;br&gt;&lt;br&gt;
(c) Two / दो&lt;br&gt;&lt;br&gt;
(d) One / एक  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) One

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; The given word is H(1) &lt;strong&gt;O(2)&lt;/strong&gt; C(3) &lt;strong&gt;K(4)&lt;/strong&gt; &lt;strong&gt;E(5)&lt;/strong&gt; &lt;strong&gt;Y(6)&lt;/strong&gt;.&lt;br&gt;
The selected letters are: &lt;strong&gt;O, K, E, Y&lt;/strong&gt;.&lt;br&gt;
By rearranging these letters, the only meaningful English word that can be formed is &lt;strong&gt;YOKE&lt;/strong&gt;.&lt;br&gt;
Therefore, exactly &lt;strong&gt;one&lt;/strong&gt; word can be formed.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; केवल एक अर्थपूर्ण शब्द (YOKE) बनाया जा सकता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q90. In the following question, select the related letter-pair from the given options.&lt;/strong&gt;&lt;br&gt;
&lt;strong&gt;BHL : YEI :: ? : ?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;निम्नलिखित प्रश्न में, दिए गए विकल्पों में से संबंधित अक्षर-युग्म का चयन करें। BHL : YEI :: ? : ?&lt;/p&gt;

&lt;p&gt;(a) ABC : DQR&lt;br&gt;&lt;br&gt;
(b) OLM : NOP&lt;br&gt;&lt;br&gt;
(c) AMG : XJD&lt;br&gt;&lt;br&gt;
(d) LOB : WZA  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) AMG : XJD

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Look at the alphabetical shift from the first group to the second:&lt;br&gt;
&lt;strong&gt;BHL $\rightarrow$ YEI&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;B(2) - 3 = Y(25) &lt;em&gt;(Since 2 - 3 = -1, which wraps around the alphabet to 25)&lt;/em&gt;
&lt;/li&gt;
&lt;li&gt;H(8) - 3 = E(5)&lt;/li&gt;
&lt;li&gt;L(12) - 3 = I(9)
The rule is &lt;strong&gt;-3&lt;/strong&gt; for every letter.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Let's test the options for the same "-3" rule:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;(a) A - 3 = X (Incorrect, option shows D)&lt;/li&gt;
&lt;li&gt;(b) O - 3 = L (Incorrect, option shows N)&lt;/li&gt;
&lt;li&gt;(c) &lt;strong&gt;AMG $\rightarrow$ XJD&lt;/strong&gt;

&lt;ul&gt;
&lt;li&gt;A(1) - 3 = X(24)&lt;/li&gt;
&lt;li&gt;M(13) - 3 = J(10)&lt;/li&gt;
&lt;li&gt;G(7) - 3 = D(4)
This matches perfectly!&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; सही युग्म AMG : XJD है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q91. In a certain language, POPULAR is coded as QPQVMBS, then which word will be coded as GBNPVT?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;एक निश्चित भाषा में, POPULAR को QPQVMBS के रूप में कूटबद्ध किया जाता है, तो निम्नलिखित में से किस शब्द को GBNPVT के रूप में कूटबद्ध किया जाएगा?&lt;/p&gt;

&lt;p&gt;(a) FARMER&lt;br&gt;&lt;br&gt;
(b) FAMOUS&lt;br&gt;&lt;br&gt;
(c) FRAMES&lt;br&gt;&lt;br&gt;
(d) FARMES  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) FAMOUS

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Decode the pattern in POPULAR:&lt;br&gt;
P(+1)=Q, O(+1)=P, P(+1)=Q, U(+1)=V, L(+1)=M, A(+1)=B, R(+1)=S.&lt;br&gt;
The rule is &lt;strong&gt;+1&lt;/strong&gt; for each letter.&lt;/p&gt;

&lt;p&gt;The question asks &lt;em&gt;which word&lt;/em&gt; will be coded as GBNPVT (meaning GBNPVT is the output, we need the input). We must subtract 1 from each letter:&lt;br&gt;
G - 1 = &lt;strong&gt;F&lt;/strong&gt;&lt;br&gt;
B - 1 = &lt;strong&gt;A&lt;/strong&gt;&lt;br&gt;
N - 1 = &lt;strong&gt;M&lt;/strong&gt;&lt;br&gt;
P - 1 = &lt;strong&gt;O&lt;/strong&gt;&lt;br&gt;
V - 1 = &lt;strong&gt;U&lt;/strong&gt;&lt;br&gt;
T - 1 = &lt;strong&gt;S&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;The original word is &lt;strong&gt;FAMOUS&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; मूल शब्द FAMOUS है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q92. What will come in place of the question mark (?) in the following series?&lt;/strong&gt;&lt;br&gt;
&lt;strong&gt;2, 6, 4, 9, 8, 13, 16, 18, 32, ?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;निम्नलिखित श्रृंखला में प्रश्न चिह्न (?) के स्थान पर क्या आएगा? 2, 6, 4, 9, 8, 13, 16, 18, 32, ?&lt;/p&gt;

&lt;p&gt;(a) 24&lt;br&gt;&lt;br&gt;
(b) 26&lt;br&gt;&lt;br&gt;
(c) 22&lt;br&gt;&lt;br&gt;
(d) 28  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) 24

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; This is an alternating series consisting of two separate patterns intertwined.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Series 1 (odd positions):&lt;/strong&gt; 2, 4, 8, 16, 32... (Each number is multiplied by 2).&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Series 2 (even positions):&lt;/strong&gt; 6, 9, 13, 18, &lt;strong&gt;?&lt;/strong&gt; Let's find the differences in Series 2:&lt;/li&gt;
&lt;li&gt;9 - 6 = +3&lt;/li&gt;
&lt;li&gt;13 - 9 = +4&lt;/li&gt;
&lt;li&gt;18 - 13 = +5
The next difference should be &lt;strong&gt;+6&lt;/strong&gt;.
18 + 6 = &lt;strong&gt;24&lt;/strong&gt;.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; प्रश्न चिह्न के स्थान पर 24 आएगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q93. Ramesh is the father of Manideep. Ramesh has only two children. Manideep is the brother of Niharika. Niharika is the daughter of Kavita. Ananya is the granddaughter of Kavita. Sujit is the father of Ananya. Then how is Sujit related to Manideep?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;रमेश मणिदीप के पिता हैं। रमेश के केवल दो बच्चे हैं। मणिदीप निहारिका का भाई है। निहारिका कविता की पुत्री है। अनन्या कविता की पोती हैं। सुजीत अनन्या के पिता हैं। तब सुजीत का मणिदीप से क्या संबंध है?&lt;/p&gt;

&lt;p&gt;(a) Brother / भाई&lt;br&gt;&lt;br&gt;
(b) Brother-in-law / साला/बहनोई&lt;br&gt;&lt;br&gt;
(c) Son / पुत्र&lt;br&gt;&lt;br&gt;
(d) Son-in-law / दामाद  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) Brother-in-law

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; 1. Ramesh and Kavita are husband and wife. They have exactly two children: Manideep (Son) and Niharika (Daughter).&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Ananya is Kavita's granddaughter, and Sujit is Ananya's father.&lt;/li&gt;
&lt;li&gt;Since Ramesh/Kavita have only two children, and Sujit is the father of their granddaughter, Sujit must be married to their daughter Niharika (Sujit cannot be Manideep because Sujit is explicitly named as a separate person/father). &lt;/li&gt;
&lt;li&gt;Therefore, Sujit is Niharika's husband. &lt;/li&gt;
&lt;li&gt;Since Manideep is Niharika's brother, Sujit is Manideep's &lt;strong&gt;Brother-in-law&lt;/strong&gt;.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; सुजीत मणिदीप का साला/बहनोई है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q94. Six colleagues Peter, Quinn, Rachel, Sam, Tina and Ulysses are sitting in a circle facing outward. Peter is sitting immediately to the right of Quinn. Tina is not sitting next to Peter. Sam, who is sitting immediately to the left of Quinn, is also sitting immediately to the right of Ulysses. Who is sitting immediately to the right of Tina?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;छह सहकर्मी पीटर, क्विन, रेचल, सैम, टीना और यूलिसिस एक घेरे में बैठे हैं, जिनका मुख बाहर की ओर है। पीटर क्विन की ठीक दाहिनी ओर बैठा है। टीना पीटर के बगल में नहीं बैठी है। सैम, जो क्विन के ठीक बाईं ओर बैठा है, भी यूलिसिस के ठीक दाई ओर बैठा है। टीना की दाहिनी ओर कौन बैठा है?&lt;/p&gt;

&lt;p&gt;(a) Quinn / क्विन&lt;br&gt;&lt;br&gt;
(b) Ulysses / यूलिसिस&lt;br&gt;&lt;br&gt;
(c) Peter / पीटर&lt;br&gt;&lt;br&gt;
(d) Rachel / रेचल  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (b) Ulysses

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; &lt;em&gt;Crucial detail: They are facing OUTWARD. This means "Left" is clockwise and "Right" is anti-clockwise.&lt;/em&gt;&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Peter (P) is immediately right (anti-clockwise) of Quinn (Q): Sequence is &lt;strong&gt;Q $\rightarrow$ P&lt;/strong&gt;.&lt;/li&gt;
&lt;li&gt;Sam (S) is immediately left (clockwise) of Q: Sequence is &lt;strong&gt;S $\rightarrow$ Q $\rightarrow$ P&lt;/strong&gt;.&lt;/li&gt;
&lt;li&gt;S is immediately right (anti-clockwise) of Ulysses (U): Sequence is &lt;strong&gt;U $\rightarrow$ S $\rightarrow$ Q $\rightarrow$ P&lt;/strong&gt;.&lt;/li&gt;
&lt;li&gt;We have two spots left for Tina (T) and Rachel (R) to complete the circle. &lt;/li&gt;
&lt;li&gt;Tina is not next to Peter. So T must sit next to U.
The full anti-clockwise circle is: &lt;strong&gt;U $\rightarrow$ S $\rightarrow$ Q $\rightarrow$ P $\rightarrow$ R $\rightarrow$ T $\rightarrow$ (back to U)&lt;/strong&gt;.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;The person sitting immediately to the right (anti-clockwise) of Tina (T) is &lt;strong&gt;Ulysses (U)&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; टीना के ठीक दाईं ओर यूलिसिस बैठा है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q95. Five friends A, B, C, D and E work as a doctor, lawyer, businessman, engineer and cook and live in Delhi, Mumbai, Goa, Panaji and Punjab (not necessarily in the same order). No two friends have the same profession and no two live in the same place. A works as a cook but does not live in Goa or Panaji. B and D live in Delhi and Mumbai respectively. C works as a lawyer. The doctor lives in Goa. Which of the following combinations is definitely true?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;पांच मित्र A, B, C, D और E डॉक्टर, वकील, व्यवसायी, इंजीनियर और रसोइया के रूप में कार्य करते हैं और दिल्ली, मुंबई, गोवा, पणजी और पंजाब में रहते हैं (आवश्यक नहीं कि इसी ही क्रम में हों)। किसी भी दो मित्रों का पेशा एक नहीं होता और न ही कोई दो मित्र एक ही स्थान पर रहते हैं। A एक रसोइया का कार्य करता है लेकिन गोवा या पणजी में नहीं रहता है। B और D क्रमशः दिल्ली और मुंबई में रहते हैं। C एक वकील के रूप में कार्य करता है। डॉक्टर गोवा में रहता हैं। निम्नलिखित में से कौन सा संयोजन निश्चित रूप से सही है?&lt;/p&gt;

&lt;p&gt;(a) B - Delhi - Businessman / B- दिल्ली – व्यवसायी&lt;br&gt;&lt;br&gt;
(b) D - Mumbai - Engineer / D - मुंबई - इंजीनियर&lt;br&gt;&lt;br&gt;
(c) A - Panaji - Cook / A - पणजी - रसोइया&lt;br&gt;&lt;br&gt;
(d) C - Panaji - Lawyer / C - पणजी - वकील  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) C - Panaji - Lawyer

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Let's assign variables based on the clues:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;A&lt;/strong&gt; = Cook. Not Goa, Not Panaji. Since Delhi and Mumbai belong to B and D, &lt;strong&gt;A must live in Punjab&lt;/strong&gt;. &lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;B&lt;/strong&gt; = Delhi.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;D&lt;/strong&gt; = Mumbai.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;C&lt;/strong&gt; = Lawyer.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Doctor&lt;/strong&gt; = Goa. &lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Cities left for C and E are Goa and Panaji. &lt;br&gt;
Since C is the Lawyer, C &lt;em&gt;cannot&lt;/em&gt; be the Doctor. Thus, C cannot live in Goa. Therefore, &lt;strong&gt;C lives in Panaji&lt;/strong&gt;.&lt;br&gt;
This leaves E to live in Goa, making &lt;strong&gt;E the Doctor&lt;/strong&gt;.&lt;br&gt;
We don't have enough information to assign Businessman/Engineer definitively to B or D.&lt;/p&gt;

&lt;p&gt;Looking at the options, only &lt;strong&gt;(d) C - Panaji - Lawyer&lt;/strong&gt; is 100% definitively true.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; C - पणजी - वकील संयोजन निश्चित रूप से सही है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q96. If 18 January 2013 was Friday, then what day of the week was 19 December 2021?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;अगर 18 जनवरी 2013 को शुक्रवार था, तो 19 दिसंबर 2021 को सप्ताह का कौन सा दिन था?&lt;/p&gt;

&lt;p&gt;(a) Sunday / रविवार&lt;br&gt;&lt;br&gt;
(b) Wednesday / बुधवार&lt;br&gt;&lt;br&gt;
(c) Saturday / शनिवार&lt;br&gt;&lt;br&gt;
(d) Thursday / गुरुवार  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (a) Sunday

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; First, find the day for 18 Jan 2021 by counting odd days year-by-year from 2013:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;2013, 2014, 2015 = 1 odd day each.&lt;/li&gt;
&lt;li&gt;2016 = 2 odd days (Leap year).&lt;/li&gt;
&lt;li&gt;2017, 2018, 2019 = 1 odd day each.&lt;/li&gt;
&lt;li&gt;2020 = 2 odd days (Leap year).
Total odd days = 3 + 2 + 3 + 2 = 10 odd days = &lt;strong&gt;3 odd days&lt;/strong&gt; (after removing 7).
Friday + 3 days = &lt;strong&gt;Monday&lt;/strong&gt; (18 Jan 2021).&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Now count odd days from 18 Jan 2021 to 19 Dec 2021:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Jan: 13 days left (6 odd)&lt;/li&gt;
&lt;li&gt;Feb: 28 days (0 odd)&lt;/li&gt;
&lt;li&gt;Mar: 31 days (3 odd)&lt;/li&gt;
&lt;li&gt;Apr: 30 days (2 odd)&lt;/li&gt;
&lt;li&gt;May: 31 days (3 odd)&lt;/li&gt;
&lt;li&gt;Jun: 30 days (2 odd)&lt;/li&gt;
&lt;li&gt;Jul: 31 days (3 odd)&lt;/li&gt;
&lt;li&gt;Aug: 31 days (3 odd)&lt;/li&gt;
&lt;li&gt;Sep: 30 days (2 odd)&lt;/li&gt;
&lt;li&gt;Oct: 31 days (3 odd)&lt;/li&gt;
&lt;li&gt;Nov: 30 days (2 odd)&lt;/li&gt;
&lt;li&gt;Dec: 19 days (5 odd)
Total = 6+0+3+2+3+2+3+3+2+3+2+5 = 34 days.
34 ÷ 7 leaves a remainder of &lt;strong&gt;6 odd days&lt;/strong&gt;.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Monday + 6 days = &lt;strong&gt;Sunday&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; 19 दिसंबर 2021 को रविवार था।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q97. A and B are brothers. M is the father of A and brother of S. F is sister of B and K is mother of S. What is the relationship between M and K?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;A और B भाई हैं। M, A का पिता है और S का भाई है। F, B की बहन है और K, S की माँ है। M और K के बीच क्या संबंध है?&lt;/p&gt;

&lt;p&gt;(a) Sister and Brother / बहन और भाई&lt;br&gt;&lt;br&gt;
(b) Grandson and Grandmother / पोता और दादी&lt;br&gt;&lt;br&gt;
(c) Nephew and Aunt / भतीजा और चाची&lt;br&gt;&lt;br&gt;
(d) Son and Mother / पुत्र और माँ  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) Son and Mother

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; 1. M is the brother of S. This means M and S are siblings.&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;K is the mother of S.&lt;/li&gt;
&lt;li&gt;Since M and S are siblings, the mother of S is also the mother of M.&lt;/li&gt;
&lt;li&gt;Therefore, K is the mother of M, making the relationship between M and K &lt;strong&gt;Son and Mother&lt;/strong&gt;.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; M और K के बीच पुत्र और माँ का संबंध है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q98. If the first 4 letters of the English alphabet are written in reverse order, then the next 5 letters also in reverse order, then the next 6 letters also in reverse order, then the next 7 letters also in reverse order, and the remaining letters also in reverse order, then which letter will be the 7th to the left of the 8th letter from the right?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;यदि अंग्रेजी वर्णमाला के पहले 4 अक्षरों को विपरीत क्रम में लिखा जाता है, और अगले 5 अक्षर भी विपरीत क्रम में, फिर अगले 6 अक्षर भी विपरीत क्रम में, फिर अगले 7 अक्षर भी विपरीत क्रम में, और शेष अक्षर भी विपरीत क्रम में लिखे जाते है, तो कौन सा अक्षर दाई ओर से 8वें अक्षर के बाईं ओर 7वां होगा?&lt;/p&gt;

&lt;p&gt;(a) O&lt;br&gt;&lt;br&gt;
(b) L&lt;br&gt;&lt;br&gt;
(c) N&lt;br&gt;&lt;br&gt;
(d) M  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) M

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Target Position: "7th to the left of 8th from right". &lt;br&gt;
Add the directions: 8 + 7 = 15th from the right.&lt;br&gt;
Convert to left-side position: (26 - 15 + 1) = &lt;strong&gt;12th from the left&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;Now map out the reversed blocks:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Block 1 (4 letters, positions 1-4): D, C, B, A&lt;/li&gt;
&lt;li&gt;Block 2 (5 letters, positions 5-9): I, H, G, F, E&lt;/li&gt;
&lt;li&gt;Block 3 (6 letters, positions 10-15): O, N, M, L, K, J&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;We need the 12th letter from the left. This falls into Block 3.&lt;br&gt;
The original letters for Block 3 are J(10), K(11), L(12), M(13), N(14), O(15).&lt;br&gt;
Written in reverse, the sequence from position 10 to 15 is: &lt;strong&gt;O, N, M, L, K, J&lt;/strong&gt;.&lt;br&gt;
Position 10 is O, Position 11 is N, and Position 12 is &lt;strong&gt;M&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; वह अक्षर M होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q99. In which option are the following words given in the order in which they appear in the English dictionary?&lt;/strong&gt;&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Pilot&lt;/li&gt;
&lt;li&gt;Pivot&lt;/li&gt;
&lt;li&gt;Pipette&lt;/li&gt;
&lt;li&gt;Pigeon&lt;/li&gt;
&lt;li&gt;Pigment&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;निम्नलिखित में से किस विकल्प में नीचे दिए गए शब्द उसी क्रम में दिए गए हैं जिस क्रम में वे अंग्रेज़ी शब्दकोश में आते हैं?&lt;/p&gt;

&lt;p&gt;(a) 5, 4, 1, 2, 3&lt;br&gt;&lt;br&gt;
(b) 5, 2, 3, 1, 4&lt;br&gt;&lt;br&gt;
(c) 4, 5, 1, 3, 2&lt;br&gt;&lt;br&gt;
(d) 4, 3, 1, 2, 5  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (c) 4, 5, 1, 3, 2

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; All words begin with "Pi". Look at the 3rd letter to begin sorting:&lt;br&gt;
l, v, p, g, g.&lt;br&gt;
Alphabetically, 'g' comes first. So we compare #4 (Pigeon) and #5 (Pigment).&lt;br&gt;
'e' comes before 'm', so &lt;strong&gt;4&lt;/strong&gt; is first, followed by &lt;strong&gt;5&lt;/strong&gt;.&lt;br&gt;
Next in the alphabet is 'l', so &lt;strong&gt;1&lt;/strong&gt; (Pilot) is third.&lt;br&gt;
Next is 'p', so &lt;strong&gt;3&lt;/strong&gt; (Pipette) is fourth.&lt;br&gt;
Finally, 'v', making &lt;strong&gt;2&lt;/strong&gt; (Pivot) last.&lt;/p&gt;

&lt;p&gt;The correct dictionary order is &lt;strong&gt;4, 5, 1, 3, 2&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; सही क्रम 4, 5, 1, 3, 2 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




&lt;p&gt;&lt;strong&gt;Q100. Find the missing term.&lt;/strong&gt;&lt;br&gt;
&lt;em&gt;(Note: Visual depicts a diamond split into 4 quadrants. First diamond: Top-Left=187, Top-Right=11, Bottom-Left=17, Bottom-Right=215. Second diamond: Top-Left=182, Top-Right=14, Bottom-Left=?, Bottom-Right=209)&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;लुप्त पद ज्ञात कीजिए -&lt;/p&gt;

&lt;p&gt;(a) 12&lt;br&gt;&lt;br&gt;
(b) 16&lt;br&gt;&lt;br&gt;
(c) 15&lt;br&gt;&lt;br&gt;
(d) 13  &lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
&lt;strong&gt;Answer:&lt;/strong&gt; (d) 13

&lt;p&gt;&lt;strong&gt;Solution:&lt;/strong&gt; Analyze the numbers in the first diamond:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Top-Left (187) ÷ Top-Right (11) = &lt;strong&gt;17&lt;/strong&gt; (Bottom-Left)&lt;/li&gt;
&lt;li&gt;Top-Left (187) + Top-Right (11) + Bottom-Left (17) = &lt;strong&gt;215&lt;/strong&gt; (Bottom-Right)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Apply this exact logic to the second diamond:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Top-Left (182) ÷ Top-Right (14) = &lt;strong&gt;13&lt;/strong&gt; (Bottom-Left = ?)&lt;/li&gt;
&lt;li&gt;Let's verify the bottom-right sum: 182 + 14 + 13 = &lt;strong&gt;209&lt;/strong&gt;. This matches the visual!&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The missing term is &lt;strong&gt;13&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;हिंदी में:&lt;/strong&gt; लुप्त पद 13 है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;




</description>
      <category>bpsc</category>
      <category>aedo</category>
      <category>mockexam</category>
      <category>ask</category>
    </item>
    <item>
      <title>BPSC AEDO 2026 Mock Test 1 – General Studies | Full Question Paper with Answers &amp; Solutions</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Sun, 05 Apr 2026 17:26:34 +0000</pubDate>
      <link>https://careerpolitics.com/the_cp_team/bpsc-aedo-mock-test-2026-full-length-paper-2-1ahi</link>
      <guid>https://careerpolitics.com/the_cp_team/bpsc-aedo-mock-test-2026-full-length-paper-2-1ahi</guid>
      <description>&lt;h2&gt;
  
  
  📋 Instructions for Candidates
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;This question booklet contains &lt;strong&gt;100 questions&lt;/strong&gt; in total.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;All questions carry equal marks.&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;Immediately after the commencement of the examination, check your question booklet and ensure that the &lt;strong&gt;booklet series (A, B, C, or D)&lt;/strong&gt; is correctly printed on the top pages.&lt;/li&gt;
&lt;li&gt;If there is any discrepancy in translation, the &lt;strong&gt;English version of the questions will be treated as final.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Write your &lt;strong&gt;Roll Number&lt;/strong&gt; in the space provided on the top of the booklet. Do not write anything else.&lt;/li&gt;
&lt;li&gt;Carefully encode your &lt;strong&gt;Roll Number and Question Booklet Series&lt;/strong&gt; on the answer sheet using a &lt;strong&gt;black/blue ballpoint pen&lt;/strong&gt;. Incorrect encoding may lead to improper evaluation.&lt;/li&gt;
&lt;li&gt;Each question has &lt;strong&gt;four options (A, B, C, D)&lt;/strong&gt;. Select &lt;strong&gt;only one correct answer&lt;/strong&gt; and mark it on the answer sheet.&lt;/li&gt;
&lt;li&gt;Use only a &lt;strong&gt;black/blue ballpoint pen&lt;/strong&gt;. &lt;strong&gt;No overwriting or erasing is allowed.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Negative Marking:&lt;/strong&gt; For every incorrect answer, &lt;strong&gt;1/3 mark will be deducted.&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;Do not tear any page from the booklet. After the exam, submit the &lt;strong&gt;answer sheet to the invigilator&lt;/strong&gt;. You may take the question booklet with you after the exam.&lt;/li&gt;
&lt;/ul&gt;




&lt;h2&gt;
  
  
  🇮🇳 हिंदी में निर्देश
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;इस प्रश्न पुस्तिका में कुल &lt;strong&gt;100 प्रश्न&lt;/strong&gt; हैं।&lt;/li&gt;
&lt;li&gt;सभी प्रश्न &lt;strong&gt;समान अंक&lt;/strong&gt; के हैं।&lt;/li&gt;
&lt;li&gt;परीक्षा शुरू होते ही प्रश्न पुस्तिका की जाँच करें और सुनिश्चित करें कि &lt;strong&gt;बुकलेट सीरीज़ (A, B, C या D)&lt;/strong&gt; सही है।&lt;/li&gt;
&lt;li&gt;किसी भी त्रुटि की स्थिति में &lt;strong&gt;अंग्रेज़ी संस्करण को अंतिम माना जाएगा।&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;अपना &lt;strong&gt;रोल नंबर&lt;/strong&gt; निर्धारित स्थान पर लिखें। अन्य कुछ न लिखें।&lt;/li&gt;
&lt;li&gt;उत्तर पत्रक में &lt;strong&gt;रोल नंबर और बुकलेट सीरीज़&lt;/strong&gt; को सही तरीके से &lt;strong&gt;ब्लैक/ब्लू पेन&lt;/strong&gt; से भरें।&lt;/li&gt;
&lt;li&gt;प्रत्येक प्रश्न के चार विकल्प हैं—(A), (B), (C), (D)। केवल &lt;strong&gt;एक सही उत्तर&lt;/strong&gt; चुनें।&lt;/li&gt;
&lt;li&gt;उत्तर चिन्हित करने के लिए केवल &lt;strong&gt;ब्लैक/ब्लू पेन&lt;/strong&gt; का उपयोग करें। काट-छाँट मान्य नहीं है।&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;नकारात्मक अंकन:&lt;/strong&gt; प्रत्येक गलत उत्तर पर &lt;strong&gt;1/3 अंक कटेंगे।&lt;/strong&gt;
&lt;/li&gt;
&lt;li&gt;परीक्षा समाप्त होने पर उत्तर पत्रक जमा करें। प्रश्न पुस्तिका अपने साथ ले जा सकते हैं।&lt;/li&gt;
&lt;/ul&gt;




&lt;p&gt;Q1. Which of the following is used as fuel in nuclear fusion?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सा नाभिकीय संलयन में ईंधन के रूप में उपयोग होता है?&lt;/p&gt;

&lt;p&gt;(a) Uranium-235 / यूरेनियम-235&lt;br&gt;
(b) Plutonium-239 / प्लूटोनियम-239&lt;br&gt;
(c) Deuterium and Tritium / ड्यूटेरियम और ट्रिटियम&lt;br&gt;
(d) Carbon-14 / कार्बन-14&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Deuterium and Tritium / ड्यूटेरियम और ट्रिटियम

&lt;p&gt;Explanation: Nuclear fusion involves combining light nuclei. Isotopes of hydrogen like deuterium and tritium are used as fuel.&lt;br&gt;
व्याख्या: नाभिकीय संलयन में हल्के नाभिकों का संलयन होता है। हाइड्रोजन के समस्थानिक जैसे ड्यूटेरियम और ट्रिटियम ईंधन के रूप में उपयोग होते हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q2. Which mirror is used by dentists to see enlarged images of teeth?&lt;/p&gt;

&lt;p&gt;दंत चिकित्सक दाँतों का बड़ा प्रतिबिंब देखने के लिए किस दर्पण का उपयोग करते हैं?&lt;/p&gt;

&lt;p&gt;(a) Plane mirror / समतल दर्पण&lt;br&gt;
(b) Convex mirror / उत्तल दर्पण&lt;br&gt;
(c) Concave mirror / अवतल दर्पण&lt;br&gt;
(d) Cylindrical mirror / बेलनाकार दर्पण&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Concave mirror / अवतल दर्पण

&lt;p&gt;Explanation: Concave mirrors produce enlarged, upright images when the object is within the focal length.&lt;br&gt;
व्याख्या: अवतल दर्पण फोकस दूरी के भीतर वस्तु होने पर बड़ा और सीधा प्रतिबिंब बनाता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q3. The area under a velocity–time graph represents:&lt;/p&gt;

&lt;p&gt;वेग-समय ग्राफ के नीचे का क्षेत्रफल क्या दर्शाता है?&lt;/p&gt;

&lt;p&gt;(a) Acceleration / त्वरण&lt;br&gt;
(b) Distance / दूरी&lt;br&gt;
(c) Displacement / विस्थापन&lt;br&gt;
(d) Speed / चाल&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Displacement / विस्थापन

&lt;p&gt;Explanation: The area under a velocity–time graph gives displacement.&lt;br&gt;
व्याख्या: वेग-समय ग्राफ के नीचे का क्षेत्रफल वस्तु का विस्थापन दर्शाता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q4. When a body is in uniform circular motion, the centripetal force acts:&lt;/p&gt;

&lt;p&gt;जब कोई वस्तु समान वृत्तीय गति में होती है, तो अभिकेंद्रीय बल किस दिशा में कार्य करता है?&lt;/p&gt;

&lt;p&gt;(a) Along the tangent / स्पर्शरेखा के साथ&lt;br&gt;
(b) Outward from the center / केंद्र से बाहर की ओर&lt;br&gt;
(c) Toward the center / केंद्र की ओर&lt;br&gt;
(d) Opposite to velocity / वेग के विपरीत&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Toward the center / केंद्र की ओर

&lt;p&gt;Explanation: Centripetal force always acts toward the center.&lt;br&gt;
व्याख्या: अभिकेंद्रीय बल हमेशा वृत्त के केंद्र की ओर कार्य करता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q5. Work done is zero when:&lt;/p&gt;

&lt;p&gt;कार्य शून्य होता है जब:&lt;/p&gt;

&lt;p&gt;(a) Force and displacement are parallel / बल और विस्थापन समानांतर हों&lt;br&gt;
(b) Force is applied but no displacement occurs / बल लगाया जाए पर विस्थापन न हो&lt;br&gt;
(c) Force is perpendicular to displacement / बल विस्थापन के लम्बवत हो&lt;br&gt;
(d) Both (b) and (c) / (b) और (c) दोनों&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Both (b) and (c) / (b) और (c) दोनों

&lt;p&gt;Explanation: Work = Force × displacement × cosθ&lt;/p&gt;

&lt;p&gt;If displacement = 0 → work = 0&lt;br&gt;
If θ = 90° → work = 0&lt;br&gt;
व्याख्या: कार्य = बल × विस्थापन × cosθ&lt;br&gt;
यदि विस्थापन 0 है → कार्य 0&lt;br&gt;
यदि कोण 90° है → कार्य 0&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;br&gt;&lt;br&gt;
Q6. If the mass of Earth doubles but radius remains the same, the value of g on the surface will:&lt;/p&gt;

&lt;p&gt;यदि पृथ्वी का द्रव्यमान दोगुना हो जाए और त्रिज्या समान रहे, तो g का मान:&lt;/p&gt;

&lt;p&gt;(a) Become half / आधा हो जाएगा&lt;br&gt;
(b) Become double / दोगुना हो जाएगा&lt;br&gt;
(c) Remain same / समान रहेगा&lt;br&gt;
(d) Become four times / चार गुना हो जाएगा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Become double / दोगुना हो जाएगा

&lt;p&gt;Explanation: g = GM/R² → If mass doubles, g doubles.&lt;br&gt;
व्याख्या: g = GM/R² → यदि द्रव्यमान दोगुना हो, तो g भी दोगुना होगा।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q7. Time period of a simple pendulum depends on:&lt;/p&gt;

&lt;p&gt;सरल लोलक का आवर्तकाल किस पर निर्भर करता है?&lt;/p&gt;

&lt;p&gt;(a) Mass of bob / गोले का द्रव्यमान&lt;br&gt;
(b) Amplitude / आयाम&lt;br&gt;
(c) Length of pendulum / लोलक की लंबाई&lt;br&gt;
(d) Both mass and length / द्रव्यमान और लंबाई दोनों&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Length of pendulum / लोलक की लंबाई

&lt;p&gt;Explanation: T = 2π√(L/g), depends only on length.&lt;br&gt;
व्याख्या: T = 2π√(L/g), यह केवल लंबाई पर निर्भर करता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q8. Alkali metals belong to which group?&lt;/p&gt;

&lt;p&gt;क्षार धातुएँ किस समूह में आती हैं?&lt;/p&gt;

&lt;p&gt;(a) Group 1 / समूह 1&lt;br&gt;
(b) Group 2 / समूह 2&lt;br&gt;
(c) Group 17 / समूह 17&lt;br&gt;
(d) Group 18 / समूह 18&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Group 1 / समूह 1

&lt;p&gt;Explanation: Alkali metals belong to Group 1.&lt;br&gt;
व्याख्या: क्षार धातुएँ आवर्त सारणी के समूह 1 में आती हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q9. Which of the following types of colloids is formed when a gas is dispersed in a liquid?&lt;/p&gt;

&lt;p&gt;जब गैस को द्रव में फैलाया जाता है तो कौन-सा कोलॉइड बनता है?&lt;/p&gt;

&lt;p&gt;(a) Foam / फोम&lt;br&gt;
(b) Gel / जेल&lt;br&gt;
(c) Sol / सोल&lt;br&gt;
(d) Emulsion / इमल्शन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Foam / फोम

&lt;p&gt;Explanation: Gas dispersed in liquid forms foam.&lt;br&gt;
व्याख्या: जब गैस द्रव में फैलती है, तो फोम बनता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q10. Which of the following is a metalloid?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सा उपधातु है?&lt;/p&gt;

&lt;p&gt;(a) Boron / बोरॉन&lt;br&gt;
(b) Sodium / सोडियम&lt;br&gt;
(c) Oxygen / ऑक्सीजन&lt;br&gt;
(d) Calcium / कैल्शियम&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Boron / बोरॉन

&lt;p&gt;Explanation: Boron is a metalloid.&lt;br&gt;
व्याख्या: बोरॉन एक उपधातु है, जिसमें धातु और अधातु दोनों के गुण होते हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q11. Which is a synthetic detergent?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सा एक कृत्रिम डिटर्जेंट है?&lt;/p&gt;

&lt;p&gt;(a) Sodium stearate / सोडियम स्टीयरेट&lt;br&gt;
(b) Sodium lauryl sulfate / सोडियम लॉरिल सल्फेट&lt;br&gt;
(c) Glycerol / ग्लिसरॉल&lt;br&gt;
(d) Calcium hydroxide / कैल्शियम हाइड्रॉक्साइड&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Sodium lauryl sulfate / सोडियम लॉरिल सल्फेट

&lt;p&gt;Explanation: Sodium lauryl sulfate is a synthetic detergent widely used in cleaning products.&lt;br&gt;
व्याख्या: सोडियम लॉरिल सल्फेट एक कृत्रिम डिटर्जेंट है जो सफाई उत्पादों में उपयोग होता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q12. Which of the following is a noble gas?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सी एक निष्क्रिय गैस है?&lt;/p&gt;

&lt;p&gt;(a) Neon / नीयॉन&lt;br&gt;
(b) Nitrogen / नाइट्रोजन&lt;br&gt;
(c) Oxygen / ऑक्सीजन&lt;br&gt;
(d) Fluorine / फ्लोरीन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Neon / नीयॉन

&lt;p&gt;Explanation: Noble gases are inert gases; Neon belongs to Group 18.&lt;br&gt;
व्याख्या: निष्क्रिय गैसें रासायनिक रूप से कम सक्रिय होती हैं; नीयॉन समूह 18 में आता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q13. Which of the following is a synthetic polymer?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सा एक कृत्रिम बहुलक है?&lt;/p&gt;

&lt;p&gt;(a) Starch / स्टार्च&lt;br&gt;
(b) Cellulose / सेल्यूलोज&lt;br&gt;
(c) Nylon / नायलॉन&lt;br&gt;
(d) Rubber / रबर&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Nylon / नायलॉन

&lt;p&gt;Explanation: Nylon is a man-made synthetic polymer.&lt;br&gt;
व्याख्या: नायलॉन एक मानव निर्मित कृत्रिम बहुलक है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q14. Which of the following is a strong acid?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सा एक प्रबल अम्ल है?&lt;/p&gt;

&lt;p&gt;(a) HCl / हाइड्रोक्लोरिक अम्ल&lt;br&gt;
(b) CH₃COOH / एसीटिक अम्ल&lt;br&gt;
(c) H₂CO₃ / कार्बोनिक अम्ल&lt;br&gt;
(d) H₂O / जल&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) HCl / हाइड्रोक्लोरिक अम्ल

&lt;p&gt;Explanation: HCl is a strong acid, while others are weak acids or neutral.&lt;br&gt;
व्याख्या: HCl एक प्रबल अम्ल है, जबकि अन्य कमजोर अम्ल या तटस्थ हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q15. Which of the following white blood cells is a type of agranulocyte?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सी श्वेत रक्त कोशिका एग्रेन्युलोसाइट है?&lt;/p&gt;

&lt;p&gt;(a) Neutrophils / न्यूट्रोफिल&lt;br&gt;
(b) Eosinophils / इओसिनोफिल&lt;br&gt;
(c) Lymphocytes / लिम्फोसाइट&lt;br&gt;
(d) Basophils / बेसोफिल&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Lymphocytes / लिम्फोसाइट

&lt;p&gt;Explanation: Lymphocytes are agranulocytes (they lack granules in cytoplasm).&lt;br&gt;
व्याख्या: लिम्फोसाइट एग्रेन्युलोसाइट होते हैं, जिनमें दानों (granules) की कमी होती है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q16. A large, muscular sheet that forms the floor of the chest cavity is:&lt;/p&gt;

&lt;p&gt;छाती गुहा के नीचे स्थित बड़ी पेशीय परत को क्या कहते हैं?&lt;/p&gt;

&lt;p&gt;(a) Diaphragm / डायफ्राम&lt;br&gt;
(b) Larynx / कंठनली&lt;br&gt;
(c) Duodenum / ग्रहणी&lt;br&gt;
(d) Alveoli / वायुकोष&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Diaphragm / डायफ्राम

&lt;p&gt;Explanation: The diaphragm separates the chest cavity from the abdominal cavity and helps in breathing.&lt;br&gt;
व्याख्या: डायफ्राम छाती और पेट की गुहा को अलग करता है और श्वसन में मदद करता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q17. Anaerobic respiration refers to which of the following?&lt;/p&gt;

&lt;p&gt;अवायवीय श्वसन किसे कहते हैं?&lt;/p&gt;

&lt;p&gt;(a) Respiration without Oxygen / ऑक्सीजन के बिना श्वसन&lt;br&gt;
(b) Respiration with Oxygen / ऑक्सीजन के साथ श्वसन&lt;br&gt;
(c) Respiration without CO₂ / CO₂ के बिना श्वसन&lt;br&gt;
(d) Respiration with CO₂ / CO₂ के साथ श्वसन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Respiration without Oxygen / ऑक्सीजन के बिना श्वसन

&lt;p&gt;Explanation: Anaerobic respiration occurs in the absence of oxygen.&lt;br&gt;
व्याख्या: अवायवीय श्वसन ऑक्सीजन की अनुपस्थिति में होता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q18. The cornea is a part of which organ in the human body?&lt;/p&gt;

&lt;p&gt;कॉर्निया मानव शरीर के किस अंग का भाग है?&lt;/p&gt;

&lt;p&gt;(a) Nose / नाक&lt;br&gt;
(b) Ears / कान&lt;br&gt;
(c) Mouth / मुख&lt;br&gt;
(d) Eyes / आँख&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Eyes / आँख

&lt;p&gt;Explanation: Cornea is the transparent front part of the eye.&lt;br&gt;
व्याख्या: कॉर्निया आँख का पारदर्शी अग्र भाग होता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q19. Algae belongs to which level of the ecosystem?&lt;/p&gt;

&lt;p&gt;शैवाल पारिस्थितिकी तंत्र के किस स्तर से संबंधित हैं?&lt;/p&gt;

&lt;p&gt;(a) Decomposers / अपघटक&lt;br&gt;
(b) Destructors / विनाशक&lt;br&gt;
(c) Producers / उत्पादक&lt;br&gt;
(d) Consumers / उपभोक्ता&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Producers / उत्पादक

&lt;p&gt;Explanation: Algae produce their own food through photosynthesis.&lt;br&gt;
व्याख्या: शैवाल प्रकाश संश्लेषण द्वारा अपना भोजन स्वयं बनाते हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q20. Water silk, Mermaid’s tresses, and Blanket weed are other names for which of the following?&lt;/p&gt;

&lt;p&gt;वॉटर सिल्क, मरमेड्स ट्रेसेस और ब्लैंकेट वीड किसके अन्य नाम हैं?&lt;/p&gt;

&lt;p&gt;(a) Volvox / वोल्वॉक्स&lt;br&gt;
(b) Chlorella / क्लोरेला&lt;br&gt;
(c) Spirogyra / स्पाइरोगाइरा&lt;br&gt;
(d) Ulothrix / यूलोथ्रिक्स&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Spirogyra / स्पाइरोगाइरा

&lt;p&gt;Explanation: Spirogyra is a filamentous green algae commonly known by these names.&lt;br&gt;
व्याख्या: स्पाइरोगाइरा एक तंतुकार हरा शैवाल है, जिसे इन नामों से भी जाना जाता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q21. Which of the following is a vector-borne disease?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से कौन-सी एक वाहक जनित बीमारी है?&lt;/p&gt;

&lt;p&gt;(a) Mumps / गलसुआ&lt;br&gt;
(b) Pneumonia / निमोनिया&lt;br&gt;
(c) Malaria / मलेरिया&lt;br&gt;
(d) Cholera / हैजा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Malaria / मलेरिया

&lt;p&gt;Explanation: Malaria is transmitted by the bite of infected female Anopheles mosquito.&lt;br&gt;
व्याख्या: मलेरिया संक्रमित मादा एनोफिलीज मच्छर के काटने से फैलता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q22. Who was the British commander at the Battle of Buxar in 1764?&lt;/p&gt;

&lt;p&gt;1764 में बक्सर के युद्ध में ब्रिटिश सेनापति कौन था?&lt;/p&gt;

&lt;p&gt;(a) Robert Clive / रॉबर्ट क्लाइव&lt;br&gt;
(b) Hector Munro / हेक्टर मुनरो&lt;br&gt;
(c) Warren Hastings / वॉरेन हेस्टिंग्स&lt;br&gt;
(d) Lord Wellesley / लॉर्ड वेलेजली&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Hector Munro / हेक्टर मुनरो

&lt;p&gt;Explanation: The British East India Company forces were led by Hector Munro in the Battle of Buxar.&lt;br&gt;
व्याख्या: बक्सर के युद्ध में ब्रिटिश ईस्ट इंडिया कंपनी की सेना का नेतृत्व हेक्टर मुनरो ने किया था।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q23. What was a major grievance of Indian sepoys regarding their service under the British?&lt;/p&gt;

&lt;p&gt;ब्रिटिश शासन के अधीन भारतीय सिपाहियों की प्रमुख शिकायत क्या थी?&lt;/p&gt;

&lt;p&gt;(a) High salaries / उच्च वेतन&lt;br&gt;
(b) Equal promotions / समान पदोन्नति&lt;br&gt;
(c) Overseas service / विदेश सेवा&lt;br&gt;
(d) Modern weapons / आधुनिक हथियार&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Overseas service / विदेश सेवा

&lt;p&gt;Explanation: Sepoys opposed overseas service due to religious and cultural beliefs (Kala Pani issue).&lt;br&gt;
व्याख्या: सिपाही धार्मिक कारणों से विदेश जाने (काला पानी) का विरोध करते थे।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q24. At which international event did Swami Vivekananda gain prominence in 1893?&lt;/p&gt;

&lt;p&gt;1893 में स्वामी विवेकानंद ने किस अंतरराष्ट्रीय कार्यक्रम में प्रसिद्धि प्राप्त की?&lt;/p&gt;

&lt;p&gt;(a) World’s Parliament of Religions / विश्व धर्म संसद&lt;br&gt;
(b) International Peace Conference / अंतरराष्ट्रीय शांति सम्मेलन&lt;br&gt;
(c) World Education Summit / विश्व शिक्षा शिखर सम्मेलन&lt;br&gt;
(d) Colonial Exhibition / औपनिवेशिक प्रदर्शनी&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) World’s Parliament of Religions / विश्व धर्म संसद

&lt;p&gt;Explanation: Vivekananda gained global recognition at the Chicago Parliament of Religions in 1893.&lt;br&gt;
व्याख्या: 1893 में शिकागो में आयोजित विश्व धर्म संसद में स्वामी विवेकानंद को विश्व प्रसिद्धि मिली।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q25. The song “Vande Mataram” was first published in which journal?&lt;/p&gt;

&lt;p&gt;“वंदे मातरम्” गीत पहली बार किस पत्रिका में प्रकाशित हुआ था?&lt;/p&gt;

&lt;p&gt;(a) Al-Hilal / अल-हिलाल&lt;br&gt;
(b) Kesari / केसरी&lt;br&gt;
(c) Young India / यंग इंडिया&lt;br&gt;
(d) Bangadarshan / बंगदर्शन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Bangadarshan / बंगदर्शन

&lt;p&gt;Explanation: “Vande Mataram” was written by Bankim Chandra Chattopadhyay and first published in Bangadarshan.&lt;br&gt;
व्याख्या: “वंदे मातरम्” बंकिम चंद्र चट्टोपाध्याय द्वारा लिखा गया और बंगदर्शन पत्रिका में प्रकाशित हुआ।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q26. Who assassinated Curzon Wyllie in London in 1909?&lt;/p&gt;

&lt;p&gt;1909 में लंदन में कर्जन वायली की हत्या किसने की थी?&lt;/p&gt;

&lt;p&gt;(a) V. D. Savarkar / वी. डी. सावरकर&lt;br&gt;
(b) Madan Lal Dhingra / मदन लाल ढींगरा&lt;br&gt;
(c) Bhagat Singh / भगत सिंह&lt;br&gt;
(d) Khudiram Bose / खुदीराम बोस&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Madan Lal Dhingra / मदन लाल ढींगरा

&lt;p&gt;Explanation: Madan Lal Dhingra assassinated Curzon Wyllie in London as an act of revolutionary protest.&lt;br&gt;
व्याख्या: मदन लाल ढींगरा ने लंदन में कर्जन वायली की हत्या की थी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q27. In which election did the Swaraj Party emerge as the single largest party in the Central Legislative Assembly?&lt;/p&gt;

&lt;p&gt;किस चुनाव में स्वराज पार्टी केंद्रीय विधान सभा में सबसे बड़ी पार्टी बनकर उभरी?&lt;/p&gt;

&lt;p&gt;(a) 1920 / 1920&lt;br&gt;
(b) 1926 / 1926&lt;br&gt;
(c) 1923 / 1923&lt;br&gt;
(d) 1930 / 1930&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) 1923 / 1923

&lt;p&gt;Explanation: The Swaraj Party performed strongly in the 1923 elections.&lt;br&gt;
व्याख्या: 1923 के चुनावों में स्वराज पार्टी सबसे बड़ी पार्टी बनकर उभरी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q28. The Simon Commission’s recommendations led to which event in 1930?&lt;/p&gt;

&lt;p&gt;साइमन कमीशन की सिफारिशों के कारण 1930 में कौन-सी घटना हुई?&lt;/p&gt;

&lt;p&gt;(a) First Round Table Conference / प्रथम गोलमेज सम्मेलन&lt;br&gt;
(b) Civil Disobedience Movement / सविनय अवज्ञा आंदोलन&lt;br&gt;
(c) Non-Cooperation Movement / असहयोग आंदोलन&lt;br&gt;
(d) Chauri Chaura incident / चौरी-चौरा कांड&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) First Round Table Conference / प्रथम गोलमेज सम्मेलन

&lt;p&gt;Explanation: The Simon Commission report led to constitutional discussions culminating in the Round Table Conferences.&lt;br&gt;
व्याख्या: साइमन कमीशन की रिपोर्ट के बाद संवैधानिक चर्चाएँ हुईं, जिससे गोलमेज सम्मेलन आयोजित हुए।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q29. In return for the Gandhi-Irwin Pact, Gandhi agreed to:&lt;/p&gt;

&lt;p&gt;गांधी-इरविन समझौते के बदले में गांधीजी ने किस पर सहमति दी?&lt;/p&gt;

&lt;p&gt;(a) End the Non-Cooperation Movement / असहयोग आंदोलन समाप्त करना&lt;br&gt;
(b) Suspend the Civil Disobedience Movement / सविनय अवज्ञा आंदोलन स्थगित करना&lt;br&gt;
(c) Support British rule / ब्रिटिश शासन का समर्थन करना&lt;br&gt;
(d) Boycott the Simon Commission / साइमन कमीशन का बहिष्कार करना&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Suspend the Civil Disobedience Movement / सविनय अवज्ञा आंदोलन स्थगित करना

&lt;p&gt;Explanation: Gandhi agreed to suspend the Civil Disobedience Movement as part of the pact.&lt;br&gt;
व्याख्या: गांधीजी ने समझौते के तहत सविनय अवज्ञा आंदोलन को स्थगित करने पर सहमति दी।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q30. The resignation of Congress ministries in 1939 led to which political development?&lt;/p&gt;

&lt;p&gt;1939 में कांग्रेस मंत्रिमंडलों के इस्तीफे के बाद कौन-सी राजनीतिक घटना हुई?&lt;/p&gt;

&lt;p&gt;(a) Launch of Quit India Movement / भारत छोड़ो आंदोलन&lt;br&gt;
(b) Muslim League’s demand for Pakistan / मुस्लिम लीग की पाकिस्तान की मांग&lt;br&gt;
(c) Revival of Civil Disobedience Movement / सविनय अवज्ञा आंदोलन का पुनरुद्धार&lt;br&gt;
(d) Formation of Forward Bloc / फॉरवर्ड ब्लॉक का गठन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Muslim League’s demand for Pakistan / मुस्लिम लीग की पाकिस्तान की मांग

&lt;p&gt;Explanation: After the resignation, the Muslim League intensified its demand for Pakistan.&lt;br&gt;
व्याख्या: कांग्रेस के इस्तीफे के बाद मुस्लिम लीग ने पाकिस्तान की मांग को तेज किया।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q31. The British used which measure to suppress the Quit India Movement?&lt;/p&gt;

&lt;p&gt;भारत छोड़ो आंदोलन को दबाने के लिए ब्रिटिशों ने कौन-सा उपाय अपनाया?&lt;/p&gt;

&lt;p&gt;(a) Press Ordinance / प्रेस अध्यादेश&lt;br&gt;
(b) Communal Award / सांप्रदायिक पुरस्कार&lt;br&gt;
(c) Martial law and mass arrests / मार्शल लॉ और सामूहिक गिरफ्तारियाँ&lt;br&gt;
(d) Government of India Act / भारत सरकार अधिनियम&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Martial law and mass arrests / मार्शल लॉ और सामूहिक गिरफ्तारियाँ

&lt;p&gt;Explanation: The British suppressed the movement through widespread arrests and strict military control.&lt;br&gt;
व्याख्या: ब्रिटिशों ने आंदोलन को दबाने के लिए बड़े पैमाने पर गिरफ्तारियाँ और कठोर दमन किया।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q32. The communal violence after Direct Action Day spread to which region in 1946?&lt;/p&gt;

&lt;p&gt;1946 में डायरेक्ट एक्शन डे के बाद सांप्रदायिक हिंसा किन क्षेत्रों में फैल गई?&lt;/p&gt;

&lt;p&gt;(a) Gujarat and Rajasthan / गुजरात और राजस्थान&lt;br&gt;
(b) Punjab and Sind / पंजाब और सिंध&lt;br&gt;
(c) Madras and Bombay / मद्रास और बॉम्बे&lt;br&gt;
(d) Noakhali and Bihar / नोआखली और बिहार&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Noakhali and Bihar / नोआखली और बिहार

&lt;p&gt;Explanation: After Direct Action Day, severe communal riots occurred in Noakhali and Bihar.&lt;br&gt;
व्याख्या: डायरेक्ट एक्शन डे के बाद नोआखली और बिहार में भीषण दंगे हुए।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q33. Consider the following statements about the Indus Valley Civilization:&lt;/p&gt;

&lt;p&gt;सिंधु घाटी सभ्यता के बारे में निम्नलिखित कथनों पर विचार करें:&lt;/p&gt;

&lt;p&gt;The civilization had an advanced drainage system. / उन्नत जल निकासी प्रणाली थी।&lt;br&gt;
Most cities were built with baked bricks. / अधिकांश नगर पक्की ईंटों से बने थे।&lt;br&gt;
The use of iron tools was widespread. / लोहे के औजारों का व्यापक उपयोग था।&lt;/p&gt;

&lt;p&gt;Which of the statements given above is/are correct?&lt;br&gt;
उपरोक्त में से कौन-से कथन सही हैं?&lt;/p&gt;

&lt;p&gt;(a) 1 and 2 only / केवल 1 और 2&lt;br&gt;
(b) 2 and 3 only / केवल 2 और 3&lt;br&gt;
(c) 1, 2, and 3 / 1, 2 और 3&lt;br&gt;
(d) 1 only / केवल 1&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) 1 and 2 only / केवल 1 और 2

&lt;p&gt;Explanation: Indus Valley people used baked bricks and had advanced drainage, but iron was not known to them.&lt;br&gt;
व्याख्या: सिंधु सभ्यता में पक्की ईंटों और जल निकासी प्रणाली का उपयोग होता था, लेकिन लोहे का उपयोग नहीं था।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q34. Regarding the social structure in the Later Vedic period, which of the following statements is/are correct?&lt;/p&gt;

&lt;p&gt;उत्तर वैदिक काल की सामाजिक संरचना के संबंध में कौन-से कथन सही हैं?&lt;/p&gt;

&lt;p&gt;The Gotra system initially emerged to regulate marriages within the Brahmin community.&lt;br&gt;
गोत्र प्रणाली विवाह को नियंत्रित करने के लिए विकसित हुई।&lt;br&gt;
The four Ashramas (stages of life) were codified during this period.&lt;br&gt;
चार आश्रम प्रणाली इसी काल में व्यवस्थित हुई।&lt;br&gt;
The caste system became more rigid, with intercaste marriages becoming rare.&lt;br&gt;
जाति व्यवस्था अधिक कठोर हो गई और अंतर्जातीय विवाह कम हो गए।&lt;/p&gt;

&lt;p&gt;(a) 1 and 2 only / केवल 1 और 2&lt;br&gt;
(b) 2 and 3 only / केवल 2 और 3&lt;br&gt;
(c) 1 and 3 only / केवल 1 और 3&lt;br&gt;
(d) 1, 2, and 3 / 1, 2 और 3&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) 1, 2, and 3 / 1, 2 और 3

&lt;p&gt;Explanation: All statements correctly describe the Later Vedic social developments.&lt;br&gt;
व्याख्या: सभी कथन उत्तर वैदिक काल की सामाजिक संरचना का सही वर्णन करते हैं।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q35. Which among the following is the oldest Buddhist monastery?&lt;/p&gt;

&lt;p&gt;निम्नलिखित में से सबसे पुराना बौद्ध विहार कौन-सा है?&lt;/p&gt;

&lt;p&gt;(a) Nalanda / नालंदा&lt;br&gt;
(b) Venuvan / वेणुवन&lt;br&gt;
(c) Takshashila / तक्षशिला&lt;br&gt;
(d) Amaravati / अमरावती&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Venuvan / वेणुवन

&lt;p&gt;Explanation: Venuvan (Veluvana) at Rajgir is considered one of the earliest Buddhist monasteries.&lt;br&gt;
व्याख्या: राजगीर का वेणुवन बौद्धों का प्रारंभिक विहार माना जाता है।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q36. Which of the following statement(s) is/are correct?&lt;/p&gt;

&lt;p&gt;निम्नलिखित कथनों में से कौन-सा/से सही है/हैं?&lt;/p&gt;

&lt;p&gt;Statement I: Sangam polity was a centralized monarchy.&lt;br&gt;
संगम काल की राजनीति केंद्रीकृत राजतंत्र थी।&lt;/p&gt;

&lt;p&gt;Statement II: Velir chieftains played an important role in local governance.&lt;br&gt;
वेलिर सरदार स्थानीय प्रशासन में महत्वपूर्ण भूमिका निभाते थे।&lt;/p&gt;

&lt;p&gt;(a) Only I / केवल I&lt;br&gt;
(b) Only II / केवल II&lt;br&gt;
(c) Both I and II / I और II दोनों&lt;br&gt;
(d) Neither I nor II / न तो I न ही II&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Only II / केवल II

&lt;p&gt;Explanation: Sangam polity was not fully centralized; local chieftains like Velirs played key roles.&lt;br&gt;
व्याख्या: संगम काल पूरी तरह केंद्रीकृत नहीं था; वेलिर सरदार स्थानीय शासन में महत्वपूर्ण थे।&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q37. A key reason for Gupta decline was decentralization due to:&lt;br&gt;
गुप्त साम्राज्य के पतन का एक प्रमुख कारण विकेंद्रीकरण था, जो किसके कारण हुआ?&lt;/p&gt;

&lt;p&gt;(a) Religious conversions / धार्मिक परिवर्तन&lt;br&gt;
(b) Rise of local feudal lords / स्थानीय सामंतों का उदय&lt;br&gt;
(c) Tribal uprisings / जनजातीय विद्रोह&lt;br&gt;
(d) Technological stagnation / तकनीकी ठहराव&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Rise of local feudal lords

&lt;p&gt;Explanation: During the later Gupta period, power shifted to regional feudal lords, weakening central authority and leading to fragmentation of the empire.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q38. The biggest source of state income during Chola times was:&lt;br&gt;
चोल काल में राज्य की आय का सबसे बड़ा स्रोत क्या था?&lt;/p&gt;

&lt;p&gt;(a) Taxes on trade / व्यापार पर कर&lt;br&gt;
(b) Land revenue / भूमि कर&lt;br&gt;
(c) Tribute from feudatories / सामंतों से कर&lt;br&gt;
(d) Temple donations / मंदिर दान&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Land revenue

&lt;p&gt;Explanation: The Chola administration heavily depended on land revenue as agriculture was the backbone of the economy.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q39. Babur’s rule in India was primarily focused on:&lt;br&gt;
भारत में बाबर का शासन मुख्यतः किस पर केंद्रित था?&lt;/p&gt;

&lt;p&gt;(a) Territorial expansion / क्षेत्रीय विस्तार&lt;br&gt;
(b) Cultural development / सांस्कृतिक विकास&lt;br&gt;
(c) Social reforms / सामाजिक सुधार&lt;br&gt;
(d) Administrative stability / प्रशासनिक स्थिरता&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Territorial expansion

&lt;p&gt;Explanation: Babur focused on consolidating and expanding his newly established empire after the First Battle of Panipat (1526).&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q40. Under Sher Shah’s land revenue system, the measurement of land was done by:&lt;br&gt;
शेरशाह के भू-राजस्व प्रणाली में भूमि का मापन कैसे किया जाता था?&lt;/p&gt;

&lt;p&gt;(a) Traditional local units / पारंपरिक स्थानीय इकाइयों से&lt;br&gt;
(b) A fixed metric system / एक निश्चित मापन प्रणाली&lt;br&gt;
(c) The standard unit called 'Bigha' / 'बीघा' नामक मानक इकाई से&lt;br&gt;
(d) By the weight of the produce / उपज के वजन से&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) The standard unit called 'Bigha'

&lt;p&gt;Explanation: Sher Shah introduced systematic land measurement using standardized units like bigha for fair revenue assessment.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q41. Akbar abolished which tax on non-Muslims?&lt;br&gt;
अकबर ने गैर-मुसलमानों पर लगाए जाने वाले किस कर को समाप्त किया?&lt;/p&gt;

&lt;p&gt;(a) Zakat / जकात&lt;br&gt;
(b) Jizya / जज़िया&lt;br&gt;
(c) Khums / खुम्स&lt;br&gt;
(d) Kharaj / खराज&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Jizya

&lt;p&gt;Explanation: Akbar abolished the jizya tax to promote religious tolerance and unity among his subjects.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q42. Consider the following about Jahangir:&lt;br&gt;
जहांगीर के बारे में निम्नलिखित कथनों पर विचार करें:&lt;/p&gt;

&lt;p&gt;He captured Kangra Fort. / उसने कांगड़ा किला जीता।&lt;br&gt;
He issued coins jointly with Nur Jahan. / उसने नूरजहां के साथ संयुक्त सिक्के जारी किए।&lt;br&gt;
He permanently annexed Mewar to the Mughal empire. / उसने मेवाड़ को स्थायी रूप से मुगल साम्राज्य में मिला लिया।&lt;/p&gt;

&lt;p&gt;(a) 1 and 2 only / केवल 1 और 2&lt;br&gt;
(b) 2 and 3 only / केवल 2 और 3&lt;br&gt;
(c) 1 and 3 only / केवल 1 और 3&lt;br&gt;
(d) 1, 2, and 3 / 1, 2 और 3&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) 1 and 2 only

&lt;p&gt;Explanation: Jahangir captured Kangra Fort and issued coins in the name of Nur Jahan, but Mewar was not permanently annexed.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q43. River Son has its genesis from Amarkantak Plateau which is in proximity to?&lt;br&gt;
सोन नदी अमरकंटक पठार से निकलती है, जो किसके निकट स्थित है?&lt;/p&gt;

&lt;p&gt;(a) Betul / बैतूल&lt;br&gt;
(b) Maikal range / मैकाल पर्वत श्रृंखला&lt;br&gt;
(c) Javadi hills / जवाड़ी पहाड़ियाँ&lt;br&gt;
(d) Shevaroy hills / शेवरॉय पहाड़ियाँ&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Maikal range

&lt;p&gt;Explanation: The Son River originates near the Amarkantak Plateau in the Maikal Hills region.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q44. Which of the following rivers is not a tributary of river Godavari?&lt;br&gt;
निम्नलिखित में से कौन-सी नदी गोदावरी की सहायक नदी नहीं है?&lt;/p&gt;

&lt;p&gt;(a) Ramganga / रामगंगा&lt;br&gt;
(b) Wenganga / वेनगंगा&lt;br&gt;
(c) Penganga / पेनगंगा&lt;br&gt;
(d) More than one of the above / एक से अधिक&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Ramganga

&lt;p&gt;Explanation: Ramganga is a tributary of the Ganga, while Wenganga and Penganga are tributaries of the Godavari.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q45. Which region of India is most prone to flood disasters due to monsoon and river systems?&lt;br&gt;
भारत का कौन-सा क्षेत्र मानसून और नदी प्रणाली के कारण बाढ़ की दृष्टि से सबसे अधिक संवेदनशील है?&lt;/p&gt;

&lt;p&gt;(a) Western Rajasthan / पश्चिमी राजस्थान&lt;br&gt;
(b) Ladakh Plateau / लद्दाख पठार&lt;br&gt;
(c) Indo-Gangetic plains, especially Bihar and Assam / इंडो-गंगा मैदान, विशेषकर बिहार और असम&lt;br&gt;
(d) Western Ghats / पश्चिमी घाट&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Indo-Gangetic plains, especially Bihar and Assam

&lt;p&gt;Explanation: These regions have dense river networks and heavy monsoon rainfall, making them highly flood-prone.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q46. What is a major economic consequence of monsoon failure in India?&lt;br&gt;
भारत में मानसून विफल होने का एक प्रमुख आर्थिक प्रभाव क्या है?&lt;/p&gt;

&lt;p&gt;(a) Surge in exports / निर्यात में वृद्धि&lt;br&gt;
(b) Improved agricultural productivity / कृषि उत्पादकता में वृद्धि&lt;br&gt;
(c) Decline in agricultural output and rural income / कृषि उत्पादन और ग्रामीण आय में कमी&lt;br&gt;
(d) Rise in industrial employment / औद्योगिक रोजगार में वृद्धि&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Decline in agricultural output and rural income

&lt;p&gt;Explanation: Indian agriculture is heavily dependent on monsoon; failure leads to reduced crop yield and income loss.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q47. Which of the following is a Jovian planet?&lt;br&gt;
निम्नलिखित में से कौन-सा एक जोवियन ग्रह है?&lt;/p&gt;

&lt;p&gt;(a) Earth / पृथ्वी&lt;br&gt;
(b) Venus / शुक्र&lt;br&gt;
(c) Jupiter / बृहस्पति&lt;br&gt;
(d) Mars / मंगल&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Jupiter

&lt;p&gt;Explanation: Jovian planets are gas giants like Jupiter, Saturn, Uranus, and Neptune.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q48. Which of the following is sometimes referred to as the NIFE layer?&lt;br&gt;
निम्नलिखित में से किसे कभी-कभी NIFE परत कहा जाता है?&lt;/p&gt;

&lt;p&gt;(a) Core / कोर&lt;br&gt;
(b) Crust / भूपर्पटी&lt;br&gt;
(c) Mantle / मेंटल&lt;br&gt;
(d) Asthenosphere / एस्थेनोस्फीयर&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Core

&lt;p&gt;Explanation: NIFE stands for Nickel (Ni) and Iron (Fe), which are the main constituents of Earth's core.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q49. What is the main mineral constituent of the continental mass?&lt;br&gt;
महाद्वीपीय भूभाग का मुख्य खनिज घटक क्या है?&lt;/p&gt;

&lt;p&gt;(a) Silica / सिलिका&lt;br&gt;
(b) Copper / तांबा&lt;br&gt;
(c) Magnesium / मैग्नीशियम&lt;br&gt;
(d) Iron / लोहा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Silica

&lt;p&gt;Explanation: Continental crust is mainly composed of silica and alumina (SIAL).&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q50. Arrange the following water bodies in decreasing order of salinity:&lt;br&gt;
निम्नलिखित जल निकायों को लवणता के घटते क्रम में व्यवस्थित करें:&lt;/p&gt;

&lt;p&gt;Baltic Sea / बाल्टिक सागर&lt;br&gt;
Lake Van / वान झील&lt;br&gt;
Dead Sea / मृत सागर&lt;br&gt;
Red Sea / लाल सागर&lt;/p&gt;

&lt;p&gt;(a) 3-2-4-1&lt;br&gt;
(b) 2-3-4-1&lt;br&gt;
(c) 2-4-3-1&lt;br&gt;
(d) 3-4-2-1&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) 3-2-4-1

&lt;p&gt;Explanation: Dead Sea has highest salinity, followed by Lake Van, Red Sea, and Baltic Sea (lowest).&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q51. The Speaker of the Lok Sabha can be removed from office by:&lt;br&gt;
लोकसभा अध्यक्ष को उनके पद से कैसे हटाया जा सकता है?&lt;/p&gt;

&lt;p&gt;(a) The President / राष्ट्रपति&lt;br&gt;
(b) The Prime Minister / प्रधानमंत्री&lt;br&gt;
(c) a resolution passed by both Houses of Parliament / संसद के दोनों सदनों द्वारा प्रस्ताव&lt;br&gt;
(d) a resolution passed by the Lok Sabha / लोकसभा द्वारा पारित प्रस्ताव&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) a resolution passed by the Lok Sabha

&lt;p&gt;Explanation: The Speaker can be removed by a majority resolution of the Lok Sabha.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q52. Who decides on disqualification of a Lok Sabha member under the Tenth Schedule?&lt;br&gt;
दसवीं अनुसूची के तहत लोकसभा सदस्य की अयोग्यता पर निर्णय कौन लेता है?&lt;/p&gt;

&lt;p&gt;(a) Prime Minister / प्रधानमंत्री&lt;br&gt;
(b) President / राष्ट्रपति&lt;br&gt;
(c) Vice President / उपराष्ट्रपति&lt;br&gt;
(d) Speaker / अध्यक्ष&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Speaker

&lt;p&gt;Explanation: The Speaker of the Lok Sabha decides on disqualification under the anti-defection law.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q53. Which of the following articles of the Indian Constitution provides for the bicameral Parliament of India?&lt;br&gt;
भारतीय संविधान के कौन-से अनुच्छेद में भारत की द्विसदनीय संसद का प्रावधान है?&lt;/p&gt;

&lt;p&gt;(a) Article 49 / अनुच्छेद 49&lt;br&gt;
(b) Article 89 / अनुच्छेद 89&lt;br&gt;
(c) Article 79 / अनुच्छेद 79&lt;br&gt;
(d) Article 59 / अनुच्छेद 59&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Article 79 / अनुच्छेद 79

&lt;p&gt;Explanation: Article 79 of the Indian Constitution establishes the Parliament of India, consisting of the President and two Houses: Lok Sabha and Rajya Sabha.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q54. Which of the following courts can move cases from any court to itself?&lt;br&gt;
निम्नलिखित में से कौन-सा न्यायालय किसी भी न्यायालय से मामलों को खुद के पास स्थानांतरित कर सकता है?&lt;/p&gt;

&lt;p&gt;(a) Supreme Court of India / भारत का सर्वोच्च न्यायालय&lt;br&gt;
(b) High Court / उच्च न्यायालय&lt;br&gt;
(c) District Court / जिला न्यायालय&lt;br&gt;
(d) Subordinate Courts / अधीनस्थ न्यायालय&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Supreme Court of India / भारत का सर्वोच्च न्यायालय

&lt;p&gt;Explanation: The Supreme Court has appellate and special powers to transfer cases from any lower court to itself under Article 139A of the Constitution.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q55. Manipur, Meghalaya and Tripura became states under:&lt;br&gt;
मणिपुर, मेघालय और त्रिपुरा निम्नलिखित किस अधिनियम के तहत राज्य बने?&lt;/p&gt;

&lt;p&gt;(a) North Eastern Areas (Reorganisation) Act, 1971 / उत्तर-पूर्वी क्षेत्र (पुनर्गठन) अधिनियम, 1971&lt;br&gt;
(b) North Eastern Republic of India Act, 1972 / उत्तर-पूर्वी गणराज्य अधिनियम, 1972&lt;br&gt;
(c) North Eastern Region New State Act, 1972 / उत्तर-पूर्वी क्षेत्र नया राज्य अधिनियम, 1972&lt;br&gt;
(d) North Eastern Retention (Reconstruction) Act, 1971 / उत्तर-पूर्वी पुनरुद्धार अधिनियम, 1971&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) North Eastern Areas (Reorganisation) Act, 1971

&lt;p&gt;Explanation: These northeastern territories were reorganized into full-fledged states under this Act.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q56. The Constitution of India was amended for the first time in which year?&lt;br&gt;
भारतीय संविधान में पहली बार संशोधन किस वर्ष हुआ था?&lt;/p&gt;

&lt;p&gt;(a) 1960 / 1960&lt;br&gt;
(b) 1954 / 1954&lt;br&gt;
(c) 1961 / 1961&lt;br&gt;
(d) 1951 / 1951&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) 1954 / 1954

&lt;p&gt;Explanation: The first amendment to the Indian Constitution was made in 1954, addressing land reform and freedom of speech limitations.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q57. What does the wheel in the National Flag represent?&lt;br&gt;
राष्ट्रीय ध्वज में चक्र क्या दर्शाता है?&lt;/p&gt;

&lt;p&gt;(a) Peace / शांति&lt;br&gt;
(b) Truth / सत्य&lt;br&gt;
(c) Growth / विकास&lt;br&gt;
(d) Future / भविष्य&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Truth / सत्य

&lt;p&gt;Explanation: The Ashoka Chakra represents dharma (righteousness or truth) and the wheel of law.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q58. In the Indian Constitution, as per Fundamental Rights, Abolition of Untouchability is a:&lt;br&gt;
भारतीय संविधान में, मौलिक अधिकारों के अनुसार, अस्पृश्यता का उन्मूलन किस प्रकार का अधिकार है?&lt;/p&gt;

&lt;p&gt;(a) Right to equality / समानता का अधिकार&lt;br&gt;
(b) Right to freedom of religion / धर्म की स्वतंत्रता का अधिकार&lt;br&gt;
(c) Right against exploitation / शोषण के विरुद्ध अधिकार&lt;br&gt;
(d) Right to constitutional remedy / संवैधानिक उपचार का अधिकार&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Right to equality / समानता का अधिकार

&lt;p&gt;Explanation: Article 17 abolishes untouchability and treats it as a violation of equality.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q59. Which Department under the Ministry of Finance in India is primarily responsible for overseeing the banking, insurance, and pension sectors?&lt;br&gt;
भारत में वित्त मंत्रालय के तहत कौन-सा विभाग मुख्य रूप से बैंकिंग, बीमा और पेंशन क्षेत्रों की निगरानी करता है?&lt;/p&gt;

&lt;p&gt;(a) Department of Revenue / राजस्व विभाग&lt;br&gt;
(b) Department of Economic Affairs / आर्थिक मामले विभाग&lt;br&gt;
(c) Department of Expenditure / व्यय विभाग&lt;br&gt;
(d) Department of Financial Services / वित्तीय सेवा विभाग&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Department of Financial Services / वित्तीय सेवा विभाग

&lt;p&gt;Explanation: DFS supervises banks, insurance companies, and pension schemes in India.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q60. What is the primary objective of the 'Swachh Sehar Jodi' initiative under Swachh Survekshan?&lt;br&gt;
स्वच्छ सर्वेक्षण के तहत 'स्वच्छ शहर जोड़ी' पहल का मुख्य उद्देश्य क्या है?&lt;/p&gt;

&lt;p&gt;(a) To provide direct financial grants for waste-to-energy plants in all cities / सभी शहरों में अपशिष्ट-से-ऊर्जा संयंत्रों के लिए प्रत्यक्ष वित्तीय अनुदान प्रदान करना&lt;br&gt;
(b) To conduct an annual survey of sanitation in rural areas only / केवल ग्रामीण क्षेत्रों में स्वच्छता का वार्षिक सर्वेक्षण करना&lt;br&gt;
(c) To award the cleanest pair of neighboring cities with a cash prize / पड़ोसी शहरों की सबसे स्वच्छ जोड़ी को नकद पुरस्कार देना&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) To award the cleanest pair of neighboring cities with a cash prize / पड़ोसी शहरों की सबसे स्वच्छ जोड़ी को नकद पुरस्कार देना

&lt;p&gt;Explanation: This initiative incentivizes urban cleanliness by recognizing neighboring cities that maintain high sanitation standards.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q60. What is the primary objective of the 'Swachh Sehar Jodi' initiative under Swachh Survekshan?&lt;br&gt;
स्वच्छ सर्वेक्षण के तहत 'स्वच्छ शहर जोड़ी' पहल का मुख्य उद्देश्य क्या है?&lt;/p&gt;

&lt;p&gt;(a) To provide direct financial grants for waste-to-energy plants in all cities / सभी शहरों में अपशिष्ट-से-ऊर्जा संयंत्रों के लिए प्रत्यक्ष वित्तीय अनुदान प्रदान करना&lt;br&gt;
(b) To conduct an annual survey of sanitation in rural areas only / केवल ग्रामीण क्षेत्रों में स्वच्छता का वार्षिक सर्वेक्षण करना&lt;br&gt;
(c) To award the cleanest pair of neighboring cities with a cash prize / पड़ोसी शहरों की सबसे स्वच्छ जोड़ी को नकद पुरस्कार देना&lt;br&gt;
(d) To institutionalize structured mentorship and peer learning between a mentor city and a lower-performing mentee city to improve sanitation / स्वच्छता सुधार के लिए मेंटर शहर और कम प्रदर्शन करने वाले मेंटी शहर के बीच संरचित मार्गदर्शन और सहकर्मी सीखने को स्थापित करना&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) To institutionalize structured mentorship and peer learning / स्वच्छता सुधार के लिए संरचित मार्गदर्शन और सहकर्मी सीखना

&lt;p&gt;Explanation: The initiative pairs high-performing “mentor” cities with lower-performing “mentee” cities to share best practices and improve urban sanitation.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q61. What new tax status was granted to individual life insurance and health insurance premiums, effective from September 22, 2025?&lt;br&gt;
22 सितंबर, 2025 से व्यक्तिगत जीवन बीमा और स्वास्थ्य बीमा प्रीमियम को कौन-सा नया कर दर्जा दिया गया?&lt;/p&gt;

&lt;p&gt;(a) Taxable at 5% / 5% पर कर योग्य&lt;br&gt;
(b) Taxable at 18% / 18% पर कर योग्य&lt;br&gt;
(c) Fully Exempted (Nil GST) / पूरी तरह से कर-मुक्त (GST शून्य)&lt;br&gt;
(d) Taxable at 12% / 12% पर कर योग्य&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Fully Exempted (Nil GST) / पूरी तरह से कर-मुक्त

&lt;p&gt;Explanation: Effective from September 22, 2025, individual life and health insurance premiums are fully exempted from GST to encourage insurance coverage.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q62. Which country officially achieved the milestone of becoming the world's 4th largest economy (in nominal GDP terms) in 2025?&lt;br&gt;
किस देश ने 2025 में दुनिया की चौथी सबसे बड़ी अर्थव्यवस्था (नाममात्र GDP के हिसाब से) बनने का मील का पत्थर आधिकारिक रूप से हासिल किया?&lt;/p&gt;

&lt;p&gt;(a) The United Kingdom / यूनाइटेड किंगडम&lt;br&gt;
(b) Japan / जापान&lt;br&gt;
(c) Germany / जर्मनी&lt;br&gt;
(d) India / भारत&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) India / भारत

&lt;p&gt;Explanation: According to IMF data, India surpassed Germany to become the world’s 4th largest economy in nominal GDP terms in 2025.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q63. What was the main focus of the Sixth Five-Year Plan (1980–85)?&lt;br&gt;
छठे पंचवर्षीय योजना (1980–85) का मुख्य फोकस क्या था?&lt;/p&gt;

&lt;p&gt;(a) Industrial growth / औद्योगिक विकास&lt;br&gt;
(b) Poverty reduction / गरीबी उन्मूलन&lt;br&gt;
(c) Infrastructure / बुनियादी ढांचा&lt;br&gt;
(d) Education / शिक्षा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Poverty reduction / गरीबी उन्मूलन

&lt;p&gt;Explanation: The Sixth Plan prioritized poverty alleviation through targeted programs and employment generation schemes.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q64. Which of the following Dams is the oldest dam in Bihar?&lt;br&gt;
निम्नलिखित में से कौन सा बाँध बिहार का सबसे पुराना बाँध है?&lt;/p&gt;

&lt;p&gt;(a) Morwy Dam / मोरवी बाँध&lt;br&gt;
(b) Srikhandi Dam / श्रीखंडी बाँध&lt;br&gt;
(c) Kohira Dam / कोहिरा बाँध&lt;br&gt;
(d) Kharagpur Lake Dam / खारगपुर झील बाँध&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Srikhandi Dam / श्रीखंडी बाँध

&lt;p&gt;Explanation: Srikhandi Dam is recognized as one of the earliest dams constructed in Bihar for irrigation purposes.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q65. In which city was the first Jain Council held?&lt;br&gt;
पहला जैन परिषद किस शहर में आयोजित किया गया था?&lt;/p&gt;

&lt;p&gt;(a) Rajgir / राजगीर&lt;br&gt;
(b) Vaishali / वैशाली&lt;br&gt;
(c) Kundalvana / कुंडलवाना&lt;br&gt;
(d) Pataliputra / पाटलिपुत्र&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Rajgir / राजगीर

&lt;p&gt;Explanation: The first Jain Council was held in Rajgir to standardize Jain scriptures and teachings.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q66. For how many times was Ramdhari Singh Dinkar elected to Rajya Sabha?&lt;br&gt;
रामधारी सिंह 'दिनकर' को कितनी बार राज्यसभा के लिए चुना गया?&lt;/p&gt;

&lt;p&gt;(a) 3 / 3&lt;br&gt;
(b) 2 / 2&lt;br&gt;
(c) 5 / 5&lt;br&gt;
(d) 1 / 1&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) 3 / 3

&lt;p&gt;Explanation: Renowned Hindi poet Ramdhari Singh Dinkar served as a Rajya Sabha member for three terms.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q67. Who initiated and encouraged people for the Peasant movement in Gaya district of Bihar?&lt;br&gt;
बिहार के गया जिले में कृषक आंदोलन की शुरुआत किसने की और लोगों को प्रोत्साहित किया?&lt;/p&gt;

&lt;p&gt;(a) Ramanand Mishra / रमणंद मिश्र&lt;br&gt;
(b) Anugraha Sinha / अनुराधा सिन्हा&lt;br&gt;
(c) Beni Prashad / बेनी प्रसाद&lt;br&gt;
(d) Yadunandan Sharma / यदुनंदन शर्मा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Yadunandan Sharma / यदुनंदन शर्मा

&lt;p&gt;Explanation: Yadunandan Sharma was a key leader mobilizing peasants against exploitative landlords in Gaya district.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q68. Sheikh Bhikhari of Bihar was related with which of the following movements?&lt;br&gt;
बिहार के शेख भिखारी निम्नलिखित में से किस आंदोलन से संबंधित थे?&lt;/p&gt;

&lt;p&gt;(a) Quit India Movement / भारत छोड़ो आंदोलन&lt;br&gt;
(b) Indigo Movement / नील आंदोलन&lt;br&gt;
(c) Revolt of 1857 / 1857 का विद्रोह&lt;br&gt;
(d) Civil Disobedience Movement / असहयोग आंदोलन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Revolt of 1857 / 1857 का विद्रोह

&lt;p&gt;Explanation: Sheikh Bhikhari was a prominent leader from Bihar during the First War of Independence in 1857.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q69. At which place in Bihar, Potato Research Institute is situated?&lt;br&gt;
बिहार में आलू अनुसंधान संस्थान कहाँ स्थित है?&lt;/p&gt;

&lt;p&gt;(a) Gaya / गया&lt;br&gt;
(b) Patna / पटना&lt;br&gt;
(c) Nalanda / नालंदा&lt;br&gt;
(d) Darbhanga / दरभंगा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Nalanda / नालंदा

&lt;p&gt;Explanation: The Central Potato Research Institute (CPRI) has a regional station located in Nalanda, Bihar, focusing on potato cultivation and research.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q70. Water harvesting structures in ancient times were known by which names in Bihar?&lt;br&gt;
प्राचीन समय में बिहार में जल संचयन संरचनाओं को किन नामों से जाना जाता था?&lt;/p&gt;

&lt;p&gt;(a) Tals / ताल&lt;br&gt;
(b) Ahars / आहर&lt;br&gt;
(c) Pynes / पाइन्स&lt;br&gt;
(d) More than one of the above / उपरोक्त में से एक से अधिक&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) More than one of the above / उपरोक्त में से एक से अधिक

&lt;p&gt;Explanation: In Bihar, ancient water harvesting structures included Ahars (small reservoirs) and Pynes (irrigation channels). Tals were also used in some regions.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q71. Identify the incorrect statement&lt;br&gt;
गलत कथन की पहचान करें:&lt;/p&gt;

&lt;p&gt;(a) Patna Canal is on Son river / पटना नहर सोन नदी पर है&lt;br&gt;
(b) Chatra canal is on Punpun river / चत्रा नहर पुनपुन नदी पर है&lt;br&gt;
(c) Teur Canal is on the Gandak river / ट्यूर नहर गंडक नदी पर है&lt;br&gt;
(d) Rajapur canal is on the Kosi river / राजापुर नहर कोसी नदी पर है&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Chatra canal is on Punpun river / चत्रा नहर पुनपुन नदी पर है

&lt;p&gt;Explanation: Chatra Canal is not on the Punpun river; the other statements are correct regarding canal locations.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q72. According to Koppen’s climatic classification, the climate of South Bihar may be explained as:&lt;br&gt;
कोप्पेन की जलवायु वर्गीकरण के अनुसार, दक्षिण बिहार की जलवायु को कैसे समझा जा सकता है?&lt;/p&gt;

&lt;p&gt;(a) CWg&lt;br&gt;
(b) Aw&lt;br&gt;
(c) CA’w&lt;br&gt;
(d) CBs&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Aw

&lt;p&gt;Explanation: South Bihar experiences a tropical wet and dry climate (Aw) as per Koppen classification, with hot summers and a distinct monsoon season.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q73. "Lota Revolt" related to which district of Bihar?&lt;br&gt;
“लोटा विद्रोह” बिहार के किस जिले से संबंधित था?&lt;/p&gt;

&lt;p&gt;(a) Patna / पटना&lt;br&gt;
(b) Muzaffarpur / मुजफ्फरपुर&lt;br&gt;
(c) Siwan / सीवान&lt;br&gt;
(d) Buxar / बक्सर&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Siwan / सीवान

&lt;p&gt;Explanation: The Lota Revolt occurred in Siwan district, where peasants protested against revenue policies during the British period.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q74. Who started the movement of nonpayment of Chowkidari tax with the consent of Mahatma Gandhi?&lt;br&gt;
महात्मा गांधी की सहमति से चौकीदारी कर का भुगतान न करने का आंदोलन किसने शुरू किया?&lt;/p&gt;

&lt;p&gt;(a) Jawaharlal Nehru / जवाहरलाल नेहरू&lt;br&gt;
(b) Rajendra Prasad / राजेंद्र प्रसाद&lt;br&gt;
(c) Ali Imam / अली इमाम&lt;br&gt;
(d) Mahadev Sahay / महादेव सहाय&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Ali Imam / अली इमाम

&lt;p&gt;Explanation: Ali Imam, with Gandhi’s approval, led the movement against the Chowkidari tax, mobilizing peasants in Bihar.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q75. The Bihar Herald was started in 1872 by:&lt;br&gt;
बिहार हेराल्ड की शुरुआत 1872 में किसने की थी?&lt;/p&gt;

&lt;p&gt;(a) Munshi Surajmal / मुंशी सूरजमल&lt;br&gt;
(b) Rai Roshan Lal / राय रोशन लाल&lt;br&gt;
(c) Babu Guru Prasad Sen / बाबू गुरु प्रसाद सेन&lt;br&gt;
(d) Sachchidanand Sinha / सच्चिदानंद सिन्हा&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Sachchidanand Sinha / सच्चिदानंद सिन्हा

&lt;p&gt;Explanation: Bihar Herald was founded in 1872 by Sachchidanand Sinha to promote social and political awareness in Bihar.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q76. Many revolutionaries were martyred on 15th February 1932 while trying to hoist the tricolor, this incident happened at:&lt;br&gt;
15 फरवरी 1932 को कई क्रांतिकारी त्रिवर्ण ध्वज फहराने का प्रयास करते समय शहीद हुए। यह घटना कहाँ हुई?&lt;/p&gt;

&lt;p&gt;(a) Buxar / बक्सर&lt;br&gt;
(b) Munger / मुंगेर&lt;br&gt;
(c) Saran / सारण&lt;br&gt;
(d) Siwan / सीवान&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Munger / मुंगेर

&lt;p&gt;Explanation: In Munger, revolutionaries attempted to hoist the national tricolor and many were martyred by the British authorities.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q77. 'Separate Bihar' slogan given by:&lt;br&gt;
'सेपरेट बिहार' का नारा किसने दिया?&lt;/p&gt;

&lt;p&gt;(a) Anugrah Narayan Singh / अनुराग नारायण सिंह&lt;br&gt;
(b) Krishna Sahay / कृष्ण सहाय&lt;br&gt;
(c) Mahesh Narayan / महेश नारायण&lt;br&gt;
(d) Rajendra Prasad / राजेंद्र प्रसाद&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Rajendra Prasad / राजेंद्र प्रसाद

&lt;p&gt;Explanation: Rajendra Prasad actively advocated for the creation of a separate Bihar state from the Bihar-Bengal Province and gave the slogan.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q78. Which of the following is wrongly matched?&lt;br&gt;
निम्नलिखित में से कौन-सा गलत मिलान है?&lt;/p&gt;

&lt;p&gt;(a) Gautam Buddha Wildlife Sanctuary - Gaya / गौतम बुद्ध वन्यजीव अभयारण्य - गया&lt;br&gt;
(b) Nagi-Nakti Bird Sanctuary - Jamui / नागी-नक्ति पक्षी अभयारण्य - जमुई&lt;br&gt;
(c) Kusheshwar Asthan Bird Sanctuary - Sasaram / कुशेश्वर स्थान पक्षी अभयारण्य - सासाराम&lt;br&gt;
(d) Salim Ali - Vaishali / सलीम अली - वैशाली&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Kusheshwar Asthan Bird Sanctuary - Sasaram / कुशेश्वर स्थान पक्षी अभयारण्य - सासाराम

&lt;p&gt;Explanation: Kusheshwar Asthan Bird Sanctuary is located in Madhubani district, not Sasaram. The other matches are correct.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q79. By which ruler was the fortification of Patli/Pataliputra done?&lt;br&gt;
पाटली/पाटलिपुत्र की दुर्ग निर्माण किस शासक ने कराया था?&lt;/p&gt;

&lt;p&gt;(a) Chandra Gupta Maurya / चंद्रगुप्त मौर्य&lt;br&gt;
(b) Shishunaga / शिशुनाग&lt;br&gt;
(c) Ajatshatru / अजातशत्रु&lt;br&gt;
(d) Udayin / उदयिन&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Chandra Gupta Maurya / चंद्रगुप्त मौर्य

&lt;p&gt;Explanation: The fortification of Pataliputra was significantly expanded and strengthened during the reign of Chandragupta Maurya to make it the capital of the Mauryan Empire.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q80. The lion pillar at Kolhua (Vaishali) was built by which among the following rulers?&lt;br&gt;
कोल्हुआ (वैशाली) का सिंह स्तंभ निम्नलिखित में से किस शासक ने बनवाया था?&lt;/p&gt;

&lt;p&gt;(a) Ashoka / अशोक&lt;br&gt;
(b) Sher Shah Suri / शेर शाह सूरी&lt;br&gt;
(c) Chandragupta Maurya / चंद्रगुप्त मौर्य&lt;br&gt;
(d) Bimbisar / बिंबिसार&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Ashoka / अशोक

&lt;p&gt;Explanation: The Lion Pillar at Kolhua, Vaishali is one of the Ashokan pillars erected by Emperor Ashoka to commemorate important Buddhist sites.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q81. Which female leader is commemorated on National Women's Day in India?&lt;br&gt;
भारत में राष्ट्रीय महिला दिवस पर किस महिला नेता को याद किया जाता है?&lt;/p&gt;

&lt;p&gt;(a) Indira Gandhi / इंदिरा गांधी&lt;br&gt;
(b) Mother Teresa / मदर टेरेसा&lt;br&gt;
(c) Rani Lakshmi Bai / रानी लक्ष्मीबाई&lt;br&gt;
(d) Sarojini Naidu / सरोजिनी नायडू&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Sarojini Naidu / सरोजिनी नायडू

&lt;p&gt;Explanation: National Women’s Day in India is celebrated on February 13, the birthday of Sarojini Naidu, the “Nightingale of India.”&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q82. Recently in news, critically endangered feral horses were found in which National Park?&lt;br&gt;
हाल ही में समाचार में, गंभीर रूप से संकटग्रस्त जंगली घोड़े किस राष्ट्रीय उद्यान में पाए गए?&lt;/p&gt;

&lt;p&gt;(a) Kaziranga National Park / काजीरंगा राष्ट्रीय उद्यान&lt;br&gt;
(b) Dibru Saikhowa National Park / डिब्रू सैकोवा राष्ट्रीय उद्यान&lt;br&gt;
(c) Jim Corbett National Park / जिम कॉर्बेट राष्ट्रीय उद्यान&lt;br&gt;
(d) Sundarbans Reserve / सुंदरबन अभ्यारण्य&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Dibru Saikhowa National Park / डिब्रू सैकोवा राष्ट्रीय उद्यान

&lt;p&gt;Explanation: The critically endangered feral horses (known as Mehnu horses) were spotted in Dibru Saikhowa National Park, Assam.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q83. Lokpal of India celebrated its 1st Foundation Day on?&lt;br&gt;
भारत के लोकपाल ने अपनी पहली स्थापना दिवस कब मनाया?&lt;/p&gt;

&lt;p&gt;(a) 12th January / 12 जनवरी&lt;br&gt;
(b) 16th January / 16 जनवरी&lt;br&gt;
(c) 16th February / 16 फरवरी&lt;br&gt;
(d) 12th March / 12 मार्च&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) 16th January / 16 जनवरी

&lt;p&gt;Explanation: The Lokpal of India celebrated its 1st Foundation Day on 16th January to mark its establishment and functioning.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q84. Which coach received the Dronacharya Award 2025, in Hockey?&lt;br&gt;
हॉकी में कौन से कोच को 2025 में द्रोणाचार्य पुरस्कार मिला?&lt;/p&gt;

&lt;p&gt;(a) Subhash Rana / सुभाष राणा&lt;br&gt;
(b) Sandeep Sangwan / संदीप सांगवान&lt;br&gt;
(c) S Muralidharan / एस मुरलीधरन&lt;br&gt;
(d) Armando Agnelo Colaco / अरमांडो अग्नेलो कोलाको&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Subhash Rana / सुभाष राणा

&lt;p&gt;Explanation: Subhash Rana was awarded the Dronacharya Award 2025 for his contributions to hockey coaching in India.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q85. The All India Council for Technical Education (AICTE) has launched India’s first undergraduate 'Minor Programme’ in Quantum Technologies jointly with?&lt;br&gt;
ऑल इंडिया काउंसिल फॉर टेक्निकल एजुकेशन (AICTE) ने भारत में क्वांटम टेक्नोलॉजी में प्रथम अंडरग्रेजुएट 'माइनर प्रोग्राम' किसके साथ मिलकर शुरू किया?&lt;/p&gt;

&lt;p&gt;(a) IISC Bangalore / IISc बंगलौर&lt;br&gt;
(b) National Quantum Mission / राष्ट्रीय क्वांटम मिशन&lt;br&gt;
(c) DRDO / DRDO&lt;br&gt;
(d) IIT Madras / IIT मद्रास&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) National Quantum Mission / राष्ट्रीय क्वांटम मिशन

&lt;p&gt;Explanation: AICTE launched the first undergraduate minor program in Quantum Technologies in collaboration with the National Quantum Mission, aimed at developing skilled manpower in emerging quantum technologies.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q86. It was unanimously decided to observe 'Year of Reforms' by which ministry?&lt;br&gt;
सभी की सहमति से 'Year of Reforms' मनाने का निर्णय किस मंत्रालय ने लिया?&lt;/p&gt;

&lt;p&gt;(a) Ministry of Finance / वित्त मंत्रालय&lt;br&gt;
(b) Ministry of Defence / रक्षा मंत्रालय&lt;br&gt;
(c) Ministry of MSME / लघु एवं मध्यम उद्यम मंत्रालय&lt;br&gt;
(d) Ministry of Textile / वस्त्र मंत्रालय&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Ministry of Finance / वित्त मंत्रालय

&lt;p&gt;Explanation: The Ministry of Finance announced the observance of ‘Year of Reforms’ to emphasize policy reforms, ease of doing business, and financial sector initiatives.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q87. Recently the Government of India has decided to give Major Dhyan Chand Khel Ratna award to 4 players. Who has not been selected among these 4?&lt;br&gt;
हाल ही में भारत सरकार ने 4 खिलाड़ियों को Major Dhyan Chand Khel Ratna देने का निर्णय लिया। इनमें से किसे चयनित नहीं किया गया?&lt;/p&gt;

&lt;p&gt;(a) Shri Harmanpreet Singh / श्री हरमनप्रीत सिंह&lt;br&gt;
(b) Shri D. Gukesh / श्री डी. गुकेश&lt;br&gt;
(c) Ms. Manu Bhaker / सुश्री मनु भाकर&lt;br&gt;
(d) Ms. Jyothi Yarraji / सुश्री ज्योति याराजी&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Ms. Jyothi Yarraji / सुश्री ज्योति याराजी

&lt;p&gt;Explanation: The 4 recipients of Major Dhyan Chand Khel Ratna 2025 include Harmanpreet Singh, D. Gukesh, Manu Bhaker, and one other player; Jyothi Yarraji was not selected.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q88. What is the name of the sixth submarine of the Project 75 class, which was delivered to Indian Navy on 09 Jan 2025 after successful completion of mandatory sea trials?&lt;br&gt;
Project 75 वर्ग की छठी पनडुब्बी का नाम क्या है, जिसे 09 जनवरी 2025 को भारतीय नौसेना को सुपुर्द किया गया?&lt;/p&gt;

&lt;p&gt;(a) Kalvari / कालवरी&lt;br&gt;
(b) Vaghseer / वाघसीर&lt;br&gt;
(c) Karanj / करंज&lt;br&gt;
(d) Vagir / वागीर&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Vagir / वागीर

&lt;p&gt;Explanation: The sixth submarine of Project 75 (Scorpene-class), named Vagir, was delivered to the Indian Navy after completing all mandatory sea trials successfully.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q89. India’s first organic fishery cluster launch in which state?&lt;br&gt;
भारत का पहला ऑर्गेनिक फ़िशरी क्लस्टर किस राज्य में लॉन्च हुआ?&lt;/p&gt;

&lt;p&gt;(a) Uttar Pradesh / उत्तर प्रदेश&lt;br&gt;
(b) Bihar / बिहार&lt;br&gt;
(c) Sikkim / सिक्किम&lt;br&gt;
(d) Kerala / केरल&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Sikkim / सिक्किम

&lt;p&gt;Explanation: Sikkim, India’s first fully organic state, launched the country’s first organic fishery cluster to promote sustainable aquaculture practices.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q90. What was the theme of the 18th Pravasi Bharatiya Divas held at Bhubaneswar in Odisha from 8 January 2025?&lt;br&gt;
8 जनवरी 2025 से ओडिशा के भुवनेश्वर में आयोजित 18वें प्रवासी भारतीय दिवस का विषय क्या था?&lt;/p&gt;

&lt;p&gt;(a) “The Diaspora’s Contribution to a Viksit Bharat” / “विकसित भारत में प्रवासी भारतीयों का योगदान”&lt;br&gt;
(b) “Building Bridges: Diaspora Innovation &amp;amp; Development” / “सेतु निर्माण: प्रवासी नवाचार और विकास”&lt;br&gt;
(c) “Global Indians for a Sustainable and Developed India” / “सतत और विकसित भारत के लिए वैश्विक भारतीय”&lt;br&gt;
(d) “Connecting Cultures: Diaspora’s Role in a Viksit Bharat” / “संस्कृतियों को जोड़ना: विकसित भारत में प्रवासी का योगदान”&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) “Global Indians for a Sustainable and Developed India” / “सतत और विकसित भारत के लिए वैश्विक भारतीय”

&lt;p&gt;Explanation: The 18th Pravasi Bharatiya Divas 2025 focused on engaging the Indian diaspora in sustainable development and contributing to a developed India.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q91. Recently, Indonesia officially became the 10th member of which multilateral institution?&lt;br&gt;
हाल ही में, इंडोनेशिया आधिकारिक रूप से किस बहुपक्षीय संस्थान का 10वां सदस्य बन गया?&lt;/p&gt;

&lt;p&gt;(a) BIMSTEC / बिम्सटेक&lt;br&gt;
(b) ASEAN / आसियान&lt;br&gt;
(c) BRICS / ब्रिक्स&lt;br&gt;
(d) Asia-Pacific Summit / एशिया-पैसिफिक समिट&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) BRICS / ब्रिक्स

&lt;p&gt;Explanation: Indonesia officially joined BRICS as the 10th member in 2025, expanding the group’s economic and geopolitical collaboration.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q92. Tuhin Kanta Pandey was recently appointed as the Chairperson of which organization?&lt;br&gt;
तूहिन कांता पांडे को हाल ही में किस संगठन का अध्यक्ष नियुक्त किया गया?&lt;/p&gt;

&lt;p&gt;(a) SEBI / SEBI&lt;br&gt;
(b) NABARD / NABARD&lt;br&gt;
(c) SBI / SBI&lt;br&gt;
(d) RBI / RBI&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) RBI / RBI

&lt;p&gt;Explanation: Tuhin Kanta Pandey was recently appointed as Chairperson of the Reserve Bank of India (RBI), heading the apex financial institution of India.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q93. "Prabowo Subianto" was invited as the chief guest for India's 2025 Republic Day parade. He is President of which country?&lt;br&gt;
भारत के 2025 गणतंत्र दिवस परेड में मुख्य अतिथि के रूप में आमंत्रित "प्रबोवो सुबियान्तो" किस देश के राष्ट्रपति हैं?&lt;/p&gt;

&lt;p&gt;(a) Indonesia / इंडोनेशिया&lt;br&gt;
(b) South Korea / दक्षिण कोरिया&lt;br&gt;
(c) Vietnam / वियतनाम&lt;br&gt;
(d) Malaysia / मलेशिया&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Indonesia / इंडोनेशिया

&lt;p&gt;Explanation: Prabowo Subianto is the President of Indonesia and was the chief guest at India’s Republic Day 2025 celebrations.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q94. Who wrote the book “Transforming Prisons”, which focuses on prison reforms and inmate rehabilitation?&lt;br&gt;
“Transforming Prisons” पुस्तक, जो जेल सुधार और कैदी पुनर्वास पर केंद्रित है, किसने लिखी?&lt;/p&gt;

&lt;p&gt;(a) Arundhati Roy / अरुंधति रॉय&lt;br&gt;
(b) Barkha Dutt / बर्खा दत्त&lt;br&gt;
(c) Medha Patkar / मेधा पाटकर&lt;br&gt;
(d) Kiran Bedi / किरण बेदी&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (d) Kiran Bedi / किरण बेदी

&lt;p&gt;Explanation: Kiran Bedi, former IPS officer and social reformer, authored “Transforming Prisons”, highlighting reforms in prison management and rehabilitation programs for inmates.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q95. Canada’s first Hindu woman to become Minister of Foreign Affairs in May 2025 is —&lt;br&gt;
कनाडा की पहली हिंदू महिला जो मई 2025 में विदेश मंत्री बनीं हैं —&lt;/p&gt;

&lt;p&gt;(a) Mélanie Joly / मेलानी जोली&lt;br&gt;
(b) Anita Anand / अनीता आनंद&lt;br&gt;
(c) Jagmeet Singh / जगमीत सिंह&lt;br&gt;
(d) Chrystia Freeland / क्रिस्टिया फ्रीलैंड&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Anita Anand / अनीता आनंद

&lt;p&gt;Explanation: Anita Anand became Canada’s first Hindu female Foreign Minister in May 2025, marking a historic milestone in Canadian politics.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q96. Which district of Bihar is a superspecialty cancer hospital proposed to be established as per the Bihar Budget 2025–26?&lt;br&gt;
बिहार बजट 2025–26 के अनुसार किस जिले में सुपरस्पेशलिटी कैंसर अस्पताल स्थापित करने का प्रस्ताव है?&lt;/p&gt;

&lt;p&gt;(a) Muzaffarpur / मुजफ्फरपुर&lt;br&gt;
(b) Patna / पटना&lt;br&gt;
(c) Begusarai / बेगूसराय&lt;br&gt;
(d) Gaya / गया&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Patna / पटना

&lt;p&gt;Explanation: As per the Bihar Budget 2025–26, a superspecialty cancer hospital is proposed to be set up in Patna to improve cancer care in the state.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q97. Which architect designed the green, energy-efficient campus of Nalanda University inaugurated in 2024?&lt;br&gt;
नालंदा विश्वविद्यालय का हरित और ऊर्जा-कुशल परिसर, जिसे 2024 में उद्घाटित किया गया, किस वास्तुकार ने डिज़ाइन किया?&lt;/p&gt;

&lt;p&gt;(a) Charles Correa / चार्ल्स कोरिया&lt;br&gt;
(b) Laurie Baker / लॉरी बेकर&lt;br&gt;
(c) Balkrishna Vithaldas Doshi / बालकृष्णा विष्टलदास दोषी&lt;br&gt;
(d) Raj Rewal / राज रेवाल&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (c) Balkrishna Vithaldas Doshi / बालकृष्णा विष्टलदास दोषी

&lt;p&gt;Explanation: Renowned architect Balkrishna Doshi designed the green, energy-efficient campus of Nalanda University, combining sustainability with modern architectural design.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q98. Which animal was the mascot of the Khelo India Youth Games 2025 held in Bihar?&lt;br&gt;
बिहार में आयोजित Khelo India Youth Games 2025 का शुभंकर (Mascot) कौन सा जानवर था?&lt;/p&gt;

&lt;p&gt;(a) Shera / शेरा&lt;br&gt;
(b) Gajasimha / गजसिंह&lt;br&gt;
(c) Cheetah / चीता&lt;br&gt;
(d) Bison / भैंस&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Shera / शेरा

&lt;p&gt;Explanation: The mascot Shera symbolizes strength, courage, and sportsmanship, representing the spirit of Khelo India Youth Games 2025 in Bihar.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q99. Which express train was flagged off between Saharsa and Mumbai during the recent development projects in Bihar?&lt;br&gt;
हाल ही में बिहार में विकास परियोजनाओं के दौरान सहारसा और मुंबई के बीच कौन सी एक्सप्रेस ट्रेन को हरी झंडी दिखाई गई?&lt;/p&gt;

&lt;p&gt;(a) Bihar Express / बिहार एक्सप्रेस&lt;br&gt;
(b) Amrit Bharat Express / अमृत भारत एक्सप्रेस&lt;br&gt;
(c) Ganga Express / गंगा एक्सप्रेस&lt;br&gt;
(d) Bharat Darshan Express / भारत दर्शन एक्सप्रेस&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (b) Amrit Bharat Express / अमृत भारत एक्सप्रेस

&lt;p&gt;Explanation: The Amrit Bharat Express was flagged off to improve connectivity between Saharsa and Mumbai, as part of Bihar’s recent railway infrastructure development projects.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;p&gt;Q100. The Shaurya Vedanam Festival, held in March 2025, took place in which district of Bihar?&lt;br&gt;
मार्च 2025 में आयोजित शौर्य वेदनम फेस्टिवल बिहार के किस जिले में हुआ?&lt;/p&gt;

&lt;p&gt;(a) Gaya / गया&lt;br&gt;
(b) Patna / पटना&lt;br&gt;
(c) East Champaran / पूर्व चंपारण&lt;br&gt;
(d) West Champaran / पश्चिम चंपारण&lt;/p&gt;

&lt;p&gt;
  ✅ View Answer &amp;amp; Solution
  &lt;br&gt;
Answer: (a) Gaya / गया

&lt;p&gt;Explanation: The Shaurya Vedanam Festival 2025 was organized in Gaya, celebrating valor, culture, and local traditions.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

</description>
      <category>bpsc</category>
      <category>mockexam</category>
      <category>aedo</category>
      <category>ask</category>
    </item>
    <item>
      <title>Navodaya Vidyalaya Samiti 2026 Admission Results Released: Complete Guide for Parents and Students</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 11:00:58 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/navodaya-vidyalaya-samiti-2026-admission-results-released-complete-guide-for-parents-and-students-6cn</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/navodaya-vidyalaya-samiti-2026-admission-results-released-complete-guide-for-parents-and-students-6cn</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--LKj6thTL--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/122x72elw7n25lztcbug.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--LKj6thTL--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/122x72elw7n25lztcbug.jpg" alt="Navodaya Vidyalaya Results" width="800" height="450"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;The Navodaya Vidyalaya Samiti (NVS) has officially released the much-awaited results for Class 6 admissions for the academic year 2026. Here's everything parents and students need to know about accessing results and next steps.&lt;/p&gt;

&lt;h2&gt;
  
  
  Key Highlights
&lt;/h2&gt;

&lt;p&gt;• Results accessible through roll number and date of birth&lt;br&gt;
• Direct result portal: navodaya.gov.in&lt;br&gt;
• Selection based on JNVST performance&lt;br&gt;
• Free quality education opportunity for rural students&lt;/p&gt;
&lt;h2&gt;
  
  
  How to Check Your Results
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Visit the official portal (navodaya.gov.in)&lt;/li&gt;
&lt;li&gt;Click on 'Results' section&lt;/li&gt;
&lt;li&gt;Enter Roll Number and Date of Birth&lt;/li&gt;
&lt;li&gt;Download and save your result PDF&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;a href="https://navodaya.gov.in" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Check Your Results Now&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  Why NVS Admissions Matter
&lt;/h2&gt;

&lt;p&gt;Navodaya Vidyalaya Samiti runs one of India's most prestigious chains of public residential schools. These institutions offer:&lt;/p&gt;

&lt;p&gt;• Completely free education&lt;br&gt;
• High-quality residential facilities&lt;br&gt;
• Excellent academic track record&lt;br&gt;
• Focus on holistic development&lt;br&gt;
• Special emphasis on rural talent&lt;/p&gt;
&lt;h2&gt;
  
  
  For Selected Candidates
&lt;/h2&gt;

&lt;p&gt;
  Important Documents Required for Admission
  &lt;br&gt;
• Original birth certificate&lt;br&gt;
• Residence proof&lt;br&gt;
• Category certificate (if applicable)&lt;br&gt;
• Recent passport size photographs&lt;br&gt;
• Transfer certificate from previous school&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Timeline for Admission Process
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Stage&lt;/th&gt;
&lt;th&gt;Timeline&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Result Declaration&lt;/td&gt;
&lt;td&gt;March 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Document Verification&lt;/td&gt;
&lt;td&gt;April 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Admission Process&lt;/td&gt;
&lt;td&gt;May 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Session Commencement&lt;/td&gt;
&lt;td&gt;July 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

&lt;h2&gt;
  
  
  Expert Tips for Parents
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Document Verification&lt;/strong&gt;: Keep all original documents ready&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Medical Fitness&lt;/strong&gt;: Schedule the mandatory medical examination early&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Orientation&lt;/strong&gt;: Attend the parent orientation program&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Financial Planning&lt;/strong&gt;: Though education is free, plan for basic necessities&lt;/li&gt;
&lt;/ol&gt;

&lt;h2&gt;
  
  
  FAQs
&lt;/h2&gt;

&lt;p&gt;
  What if I can't access my result?
  &lt;br&gt;
Contact the NVS helpdesk or your regional office with your registration details&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  Is there a waiting list?
  &lt;br&gt;
Yes, NVS maintains a merit-based waiting list for each region&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  Can I apply for transfer between Navodaya schools?
  &lt;br&gt;
Transfers are considered only in special circumstances as per NVS policy&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Stay Updated
&lt;/h2&gt;

&lt;p&gt;&lt;a href="https://t.me/careerpolitics" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Join our Telegram channel for instant updates on Navodaya admissions&lt;/a&gt;
&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Sources: Official NVS Portal, Regional Office Notifications&lt;/em&gt;&lt;/p&gt;

</description>
      <category>navodayavidyalaya</category>
      <category>schooladmission</category>
      <category>jnvst</category>
      <category>education</category>
    </item>
    <item>
      <title>KVS Admission 2026-27: Registration Dates, Age Limits, and Complete Application Guide</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 08:01:06 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/kvs-admission-2026-27-registration-dates-age-limits-and-complete-application-guide-52g5</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/kvs-admission-2026-27-registration-dates-age-limits-and-complete-application-guide-52g5</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--RnRotF07--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/dkx1tm9wc2417bb0gxdi.webp" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--RnRotF07--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/dkx1tm9wc2417bb0gxdi.webp" alt="KVS Admission 2026-27 Notification" width="320" height="180"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;The Kendriya Vidyalaya Sangathan (KVS) has announced that the admission notification for the academic session 2026-27 will be released soon. This development affects admissions across all KVS schools nationwide, from Balvatika to Class 10.&lt;/p&gt;

&lt;h2&gt;
  
  
  Latest Updates
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Online admission portal to open shortly&lt;/li&gt;
&lt;li&gt;Free application process for all categories&lt;/li&gt;
&lt;li&gt;Separate portals for Class 1 and Balvatika admissions&lt;/li&gt;
&lt;li&gt;Only online applications will be accepted&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Important Application Portals
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Class 1 Admissions: kvsonlineadmission.kvs.gov.in&lt;/li&gt;
&lt;li&gt;Balvatika Admissions: balvatika.kvs.gov.in&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Age Eligibility Criteria
&lt;/h2&gt;

&lt;p&gt;
  Age Requirements (as of March 31, 2026)
  &lt;ul&gt;
&lt;li&gt;Class 1: 6-8 years (Born on or before April 1, 2020)&lt;/li&gt;
&lt;li&gt;Balvatika-1: 3-4 years&lt;/li&gt;
&lt;li&gt;Balvatika-2: 4-5 years&lt;/li&gt;
&lt;li&gt;Balvatika-3: 5-6 years
&lt;/li&gt;
&lt;/ul&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  Reservation Policy
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;SC: 15% seats&lt;/li&gt;
&lt;li&gt;ST: 7.5% seats&lt;/li&gt;
&lt;li&gt;OBC: 27% seats&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;
  
  
  Required Documents
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Original birth certificate (will be returned after verification)&lt;/li&gt;
&lt;li&gt;Caste certificate (if applicable)&lt;/li&gt;
&lt;li&gt;Residence proof&lt;/li&gt;
&lt;li&gt;Parent's service certificate with transfer details of last 7 years&lt;/li&gt;
&lt;li&gt;Two photographs of the child&lt;/li&gt;
&lt;/ol&gt;
&lt;h2&gt;
  
  
  Admission Process
&lt;/h2&gt;
&lt;h3&gt;
  
  
  Class 1
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Fully online application process&lt;/li&gt;
&lt;li&gt;Selection through computerized lottery system&lt;/li&gt;
&lt;li&gt;No offline forms accepted&lt;/li&gt;
&lt;/ul&gt;
&lt;h3&gt;
  
  
  Other Classes (2nd onwards)
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Offline admission process&lt;/li&gt;
&lt;li&gt;Subject to seat availability&lt;/li&gt;
&lt;li&gt;Applications to be submitted to respective KV principals&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;
  Special Notes for Parents
  &lt;ul&gt;
&lt;li&gt;Applications are completely free&lt;/li&gt;
&lt;li&gt;Over 1250 KV schools across India&lt;/li&gt;
&lt;li&gt;Age calculation will be done as of March 31, 2026&lt;/li&gt;
&lt;li&gt;Class 11 admissions have a separate process
&lt;/li&gt;
&lt;/ul&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  How to Apply
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Visit the official KVS website (kvsangathan.nic.in)&lt;/li&gt;
&lt;li&gt;Choose appropriate admission portal&lt;/li&gt;
&lt;li&gt;Fill in required details carefully&lt;/li&gt;
&lt;li&gt;Upload necessary documents&lt;/li&gt;
&lt;li&gt;Submit the application&lt;/li&gt;
&lt;li&gt;Save confirmation number for future reference&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;a href="https://kvsangathan.nic.in" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Visit Official KVS Website for Updates&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  Important Advisory
&lt;/h2&gt;

&lt;p&gt;Parents are advised to:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Verify age eligibility before applying&lt;/li&gt;
&lt;li&gt;Keep documents ready before starting application&lt;/li&gt;
&lt;li&gt;Double-check all entries before submission&lt;/li&gt;
&lt;li&gt;Monitor official website for notification release&lt;/li&gt;
&lt;/ul&gt;




&lt;p&gt;&lt;em&gt;Note: This information is based on official KVS announcements. Dates and processes may be subject to change. Always refer to the official KVS website for the most current updates.&lt;/em&gt;&lt;/p&gt;

</description>
      <category>kvsadmission</category>
      <category>education</category>
      <category>schooladmission</category>
      <category>kendriyavidyalaya</category>
    </item>
    <item>
      <title>IIFCL Recruitment 2026: 37 Assistant Manager &amp; Manager Vacancies Announced - Complete Guide for Aspirants</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 07:01:35 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/iifcl-recruitment-2026-37-assistant-manager-manager-vacancies-announced-complete-guide-for-280f</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/iifcl-recruitment-2026-37-assistant-manager-manager-vacancies-announced-complete-guide-for-280f</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--Ud0onJPA--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/dv5hkhbrztbd6xun4n8p.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--Ud0onJPA--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/dv5hkhbrztbd6xun4n8p.jpg" alt="IIFCL Recruitment 2026" width="800" height="450"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;In a significant development for banking sector aspirants, the India Infrastructure Finance Company Limited (IIFCL) has announced its much-awaited recruitment drive for 2026, offering 37 prestigious positions across Assistant Manager and Manager grades.&lt;/p&gt;

&lt;h2&gt;
  
  
  Key Highlights of IIFCL Recruitment 2026
&lt;/h2&gt;

&lt;p&gt;• &lt;strong&gt;Total Vacancies&lt;/strong&gt;: 37 positions&lt;br&gt;
• &lt;strong&gt;Posts&lt;/strong&gt;: Assistant Manager (Grade A) and Manager (Grade B)&lt;br&gt;
• &lt;strong&gt;Application Start Date&lt;/strong&gt;: April 10, 2026&lt;br&gt;
• &lt;strong&gt;Last Date to Apply&lt;/strong&gt;: May 5, 2026&lt;br&gt;
• &lt;strong&gt;Selection Process&lt;/strong&gt;: Written Exam followed by Interview&lt;/p&gt;
&lt;h2&gt;
  
  
  Vacancy Distribution
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Post&lt;/th&gt;
&lt;th&gt;Grade&lt;/th&gt;
&lt;th&gt;Number of Vacancies&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Assistant Manager&lt;/td&gt;
&lt;td&gt;Grade A&lt;/td&gt;
&lt;td&gt;25&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Manager&lt;/td&gt;
&lt;td&gt;Grade B&lt;/td&gt;
&lt;td&gt;12&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;
&lt;h2&gt;
  
  
  Eligibility Criteria
&lt;/h2&gt;
&lt;h3&gt;
  
  
  Assistant Manager (Grade A)
&lt;/h3&gt;

&lt;p&gt;• Age: 21-30 years&lt;br&gt;
• Qualification: CA/CMA/MBA (Finance) or equivalent&lt;br&gt;
• Experience: Not mandatory&lt;/p&gt;
&lt;h3&gt;
  
  
  Manager (Grade B)
&lt;/h3&gt;

&lt;p&gt;• Age: 25-35 years&lt;br&gt;
• Qualification: CA/CMA/MBA (Finance) or equivalent&lt;br&gt;
• Experience: Minimum 3 years in relevant field&lt;/p&gt;
&lt;h2&gt;
  
  
  Salary Structure
&lt;/h2&gt;

&lt;p&gt;• &lt;strong&gt;Assistant Manager&lt;/strong&gt;: ₹28,000-1,550(14)-49,700&lt;br&gt;
• &lt;strong&gt;Manager&lt;/strong&gt;: ₹31,705-1,800(14)-56,925&lt;/p&gt;

&lt;p&gt;
  Important Documents Required
  &lt;ol&gt;
&lt;li&gt;Valid ID Proof&lt;/li&gt;
&lt;li&gt;Educational certificates&lt;/li&gt;
&lt;li&gt;Experience certificates (if applicable)&lt;/li&gt;
&lt;li&gt;Recent passport size photograph&lt;/li&gt;
&lt;li&gt;Category certificate (if applicable)
&lt;/li&gt;
&lt;/ol&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  How to Apply
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Visit the official IIFCL website (&lt;a href="http://www.iifcl.in" rel="noopener noreferrer"&gt;www.iifcl.in&lt;/a&gt;)&lt;/li&gt;
&lt;li&gt;Navigate to 'Careers' section&lt;/li&gt;
&lt;li&gt;Click on the recruitment notification&lt;/li&gt;
&lt;li&gt;Fill the online application form&lt;/li&gt;
&lt;li&gt;Upload required documents&lt;/li&gt;
&lt;li&gt;Pay application fee&lt;/li&gt;
&lt;li&gt;Submit and save confirmation&lt;/li&gt;
&lt;/ol&gt;
&lt;h2&gt;
  
  
  Selection Process
&lt;/h2&gt;
&lt;h3&gt;
  
  
  Phase 1: Written Examination
&lt;/h3&gt;

&lt;p&gt;• Objective type questions&lt;br&gt;
• Topics: Banking, Finance, General Awareness, Reasoning&lt;br&gt;
• Duration: 2 hours&lt;/p&gt;
&lt;h3&gt;
  
  
  Phase 2: Interview
&lt;/h3&gt;

&lt;p&gt;• Only for candidates who clear written exam&lt;br&gt;
• Focus on domain knowledge and communication skills&lt;/p&gt;

&lt;p&gt;
  Application Fee
  &lt;br&gt;
• General/OBC: ₹1000&lt;br&gt;
• SC/ST/PWD: ₹200&lt;br&gt;
• Payment modes: Net Banking/Credit Card/Debit Card&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Expert Tips for Preparation
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Focus on banking sector reforms&lt;/li&gt;
&lt;li&gt;Study infrastructure financing in detail&lt;/li&gt;
&lt;li&gt;Practice previous year papers&lt;/li&gt;
&lt;li&gt;Stay updated with financial news&lt;/li&gt;
&lt;li&gt;Improve analytical skills&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;a href="https://t.me/careerpolitics" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Join our Telegram channel for IIFCL exam updates and study materials&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  Important Dates
&lt;/h2&gt;

&lt;p&gt;• Application Start: April 10, 2026&lt;br&gt;
• Last Date: May 5, 2026&lt;br&gt;
• Admit Card Release: To be announced&lt;br&gt;
• Exam Date: Tentatively June 2026&lt;/p&gt;

&lt;h2&gt;
  
  
  Conclusion
&lt;/h2&gt;

&lt;p&gt;The IIFCL recruitment 2026 presents a golden opportunity for finance professionals seeking stable government careers. With competitive pay scales and growth prospects, these positions are highly sought after. Early preparation and thorough understanding of the selection process will be crucial for success.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Sources:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Official IIFCL Website&lt;/li&gt;
&lt;li&gt;Ministry of Finance Notifications&lt;/li&gt;
&lt;li&gt;Employment News&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;em&gt;Disclaimer: Dates and details are subject to change. Candidates are advised to check official website for latest updates.&lt;/em&gt;&lt;/p&gt;

</description>
      <category>governmentjobs</category>
      <category>bankingjobs</category>
      <category>iifclrecruitment</category>
      <category>financecareers</category>
    </item>
    <item>
      <title>GATE 2026 Result Date Confirmed: IIT Guwahati to Release Scores on March 19; Final Answer Key Expected Soon</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 06:02:04 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/gate-2026-result-date-confirmed-iit-guwahati-to-release-scores-on-march-19-final-answer-key-2ih1</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/gate-2026-result-date-confirmed-iit-guwahati-to-release-scores-on-march-19-final-answer-key-2ih1</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--8MJQu6g4--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/l998pqp83u8d7ivdt6lg.webp" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--8MJQu6g4--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/l998pqp83u8d7ivdt6lg.webp" alt="GATE 2026 Result" width="320" height="180"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;In a significant update for over 7 lakh engineering graduates, IIT Guwahati has confirmed the release date for GATE 2026 results. The Graduate Aptitude Test in Engineering (GATE) scores will be declared on March 19, 2026, through the official portal gate2026.iitg.ac.in.&lt;/p&gt;

&lt;h2&gt;
  
  
  Key Announcements
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Result Declaration&lt;/strong&gt;: March 19, 2026&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Toppers List&lt;/strong&gt;: March 25, 2026&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Scorecards Available&lt;/strong&gt;: March 27, 2026&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Final Answer Key&lt;/strong&gt;: Expected by March 17-18, 2026&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Important Timeline for Candidates
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Event&lt;/th&gt;
&lt;th&gt;Date&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Final Answer Key Release&lt;/td&gt;
&lt;td&gt;March 17-18, 2026 (Expected)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Result Declaration&lt;/td&gt;
&lt;td&gt;March 19, 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Toppers List&lt;/td&gt;
&lt;td&gt;March 25, 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Scorecard Download&lt;/td&gt;
&lt;td&gt;March 27, 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

&lt;h2&gt;
  
  
  Accessing Your Results
&lt;/h2&gt;

&lt;p&gt;Candidates who appeared for GATE 2026 during the February examination window (Feb 7-8 and 14-15) can check their results by following these steps:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Visit the official website: gate2026.iitg.ac.in&lt;/li&gt;
&lt;li&gt;Login using enrollment number and password&lt;/li&gt;
&lt;li&gt;Navigate to the result section&lt;/li&gt;
&lt;li&gt;Download and save the scorecard&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;
  Important Note for Candidates
  &lt;br&gt;
Scorecard must be downloaded within the specified timeframe. A late fee of ₹500 will be applicable for downloads after the deadline.&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Final Answer Key Update
&lt;/h2&gt;

&lt;p&gt;Before the result declaration, IIT Guwahati will release the final answer key. This crucial document helps candidates assess their performance and verify their responses against the official solutions.&lt;/p&gt;

&lt;h3&gt;
  
  
  Steps to Download Final Answer Key
&lt;/h3&gt;

&lt;ol&gt;
&lt;li&gt;Visit gate2026.iitg.ac.in&lt;/li&gt;
&lt;li&gt;Click on "GATE 2026 Final Answer Key" link&lt;/li&gt;
&lt;li&gt;Select your subject&lt;/li&gt;
&lt;li&gt;Download the PDF&lt;/li&gt;
&lt;li&gt;Save a copy for future reference&lt;/li&gt;
&lt;/ol&gt;

&lt;h2&gt;
  
  
  Examination Overview
&lt;/h2&gt;

&lt;p&gt;The GATE 2026 examination witnessed participation from approximately 7 lakh candidates across multiple engineering disciplines. The exam was conducted at various centers nationwide over four days in February 2026.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://gate2026.iitg.ac.in" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Visit Official Website for Latest Updates&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  FAQs
&lt;/h2&gt;

&lt;p&gt;
  What if I miss the scorecard download deadline?
  &lt;br&gt;
Candidates can still download their scorecards after the deadline by paying a late fee of ₹500.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  How long is the GATE score valid?
  &lt;br&gt;
GATE scores remain valid for three years from the date of result declaration.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  Can I challenge the Final Answer Key?
  &lt;br&gt;
No, the Final Answer Key is conclusive and cannot be challenged. The window for objections was available after the provisional answer key release.&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Stay Updated
&lt;/h2&gt;

&lt;p&gt;Candidates are advised to:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Regularly check the official website&lt;/li&gt;
&lt;li&gt;Keep login credentials handy&lt;/li&gt;
&lt;li&gt;Download and save all important documents&lt;/li&gt;
&lt;li&gt;Watch for email notifications from IIT Guwahati&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;em&gt;Last Updated: March 17, 2026&lt;/em&gt;&lt;/p&gt;

</description>
      <category>gate2026</category>
      <category>iitguwahati</category>
      <category>engineeringexams</category>
      <category>educationnews</category>
    </item>
    <item>
      <title>BU Jhansi Result 2026 Released: Direct Steps to Download UG &amp; PG Semester Scorecards</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 06:01:22 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/bu-jhansi-result-2026-released-direct-steps-to-download-ug-pg-semester-scorecards-3aok</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/bu-jhansi-result-2026-released-direct-steps-to-download-ug-pg-semester-scorecards-3aok</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--jBhZq_yC--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/8ajvxr5ehjh7nlww4m6s.png" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--jBhZq_yC--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/8ajvxr5ehjh7nlww4m6s.png" alt="BU Jhansi Results 2026" width="800" height="450"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Bundelkhand University (BU) Jhansi has officially released the odd semester examination results for various undergraduate and postgraduate programs. Students who appeared for these exams can now access their scorecards through the university's official portal.&lt;/p&gt;

&lt;h2&gt;
  
  
  Key Highlights
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Results declared for BA, BSc, BCom, and multiple MA programs&lt;/li&gt;
&lt;li&gt;Online scorecard download facility available at bujhansi.ac.in&lt;/li&gt;
&lt;li&gt;Results pertaining to 2025-26 academic session odd semester&lt;/li&gt;
&lt;li&gt;Additional course results to be announced in coming weeks&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Available Course Results
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Undergraduate Programs&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;BA (Bachelor of Arts)&lt;/li&gt;
&lt;li&gt;BSc (Bachelor of Science)&lt;/li&gt;
&lt;li&gt;BCom (Bachelor of Commerce)&lt;/li&gt;
&lt;li&gt;BSc Home Science&lt;/li&gt;
&lt;/ul&gt;


&lt;/li&gt;

&lt;li&gt;

&lt;p&gt;&lt;strong&gt;Postgraduate Programs&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;MCom&lt;/li&gt;
&lt;li&gt;MA History&lt;/li&gt;
&lt;li&gt;MA Education&lt;/li&gt;
&lt;li&gt;MA English&lt;/li&gt;
&lt;li&gt;MA Hindi&lt;/li&gt;
&lt;li&gt;MA Home Science&lt;/li&gt;
&lt;/ul&gt;


&lt;/li&gt;

&lt;/ul&gt;

&lt;h2&gt;
  
  
  How to Download Your Scorecard
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Visit the official website: bujhansi.ac.in&lt;/li&gt;
&lt;li&gt;Navigate to the 'Examination' section&lt;/li&gt;
&lt;li&gt;Click on 'Results' followed by 'View Results'&lt;/li&gt;
&lt;li&gt;Select your course from the dropdown menu&lt;/li&gt;
&lt;li&gt;Enter your roll number/registration number&lt;/li&gt;
&lt;li&gt;Click 'View Result' to access your scorecard&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;
  Important Details on Result Scorecard
  &lt;ul&gt;
&lt;li&gt;Student's Name&lt;/li&gt;
&lt;li&gt;Roll Number&lt;/li&gt;
&lt;li&gt;Course Details&lt;/li&gt;
&lt;li&gt;Semester Information&lt;/li&gt;
&lt;li&gt;Subject-wise Marks&lt;/li&gt;
&lt;li&gt;Overall Grade/Percentage
&lt;/li&gt;
&lt;/ul&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  Important Advisory for Students
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Keep a digital copy of the scorecard for future reference&lt;/li&gt;
&lt;li&gt;Verify all personal and academic details carefully&lt;/li&gt;
&lt;li&gt;Report discrepancies to university authorities immediately&lt;/li&gt;
&lt;li&gt;Stay updated about revaluation announcements&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;a href="https://bujhansi.ac.in" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Visit Official Website for Result Download&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  Next Steps for Students
&lt;/h2&gt;

&lt;h3&gt;
  
  
  For Final Year Students
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Plan further academic pursuits&lt;/li&gt;
&lt;li&gt;Prepare for higher education opportunities&lt;/li&gt;
&lt;li&gt;Begin career planning&lt;/li&gt;
&lt;/ul&gt;

&lt;h3&gt;
  
  
  For Continuing Students
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Monitor notices for next semester registration&lt;/li&gt;
&lt;li&gt;Track academic calendar updates&lt;/li&gt;
&lt;li&gt;Note improvement opportunities if needed&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  FAQs
&lt;/h2&gt;

&lt;p&gt;
  What if I face technical issues while downloading the result?
  &lt;br&gt;
Contact the university's technical help desk or visit the examination cell with valid ID proof.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  When will the remaining course results be declared?
  &lt;br&gt;
The university will announce other course results in phases over the coming weeks. Students should regularly check the official website.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  Is there a provision for result re-evaluation?
  &lt;br&gt;
Yes, details about the re-evaluation process will be announced separately on the university website.&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Stay Updated
&lt;/h2&gt;

&lt;p&gt;Students are advised to:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Regularly check the university website&lt;/li&gt;
&lt;li&gt;Follow official social media channels&lt;/li&gt;
&lt;li&gt;Watch for revaluation notifications&lt;/li&gt;
&lt;li&gt;Keep documentation ready for future processes&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;em&gt;Last Updated: March 18, 2026&lt;/em&gt;&lt;/p&gt;

</description>
      <category>bujhansi</category>
      <category>universityresults</category>
      <category>educationnews</category>
      <category>examupdates</category>
    </item>
    <item>
      <title>KVS Admission 2026-27: Complete Guide to Registration Dates, Process &amp; Eligibility for All Classes</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 05:01:12 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/kvs-admission-2026-27-complete-guide-to-registration-dates-process-eligibility-for-all-classes-28hk</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/kvs-admission-2026-27-complete-guide-to-registration-dates-process-eligibility-for-all-classes-28hk</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--X9DIC8Ds--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/x452hq8y14yv0nwvu0ka.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--X9DIC8Ds--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/x452hq8y14yv0nwvu0ka.jpg" alt="KVS Admission 2026-27" width="800" height="450"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Kendriya Vidyalaya Sangathan (KVS) is gearing up to announce admissions for the academic year 2026-27, with the process expected to begin in March. Here's everything aspiring parents and students need to know about the upcoming admission cycle.&lt;/p&gt;

&lt;h2&gt;
  
  
  Two-Phase Admission Schedule
&lt;/h2&gt;

&lt;p&gt;KVS will conduct admissions in two distinct phases:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Phase 1 (March 2026)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Balvatika (Levels 1-3)&lt;/li&gt;
&lt;li&gt;Class 1 admissions&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;Phase 2 (April 2026)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Classes 2 to 9&lt;/li&gt;
&lt;li&gt;Class 11 admissions&lt;/li&gt;
&lt;li&gt;Subject to seat availability&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Key Features of KVS Admission 2026
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;100% Online Process&lt;/strong&gt; for Balvatika and Class 1&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Lottery-based Selection&lt;/strong&gt; for Class 1&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;No Written Tests&lt;/strong&gt; or interviews for entry-level admissions&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Multiple School Choice&lt;/strong&gt; - Parents can select up to 3 KV schools&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Age Eligibility Criteria
&lt;/h2&gt;

&lt;p&gt;
  Age Requirements as of March 31, 2026
  &lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Balvatika 1:&lt;/strong&gt; 3-4 years&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Balvatika 2:&lt;/strong&gt; 4-5 years&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Balvatika 3:&lt;/strong&gt; 5-6 years&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Class 1:&lt;/strong&gt; 6-8 years&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Age Relaxation:&lt;/strong&gt; Up to 2 years for differently-abled children
&lt;/li&gt;
&lt;/ul&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  Important Dates
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Event&lt;/th&gt;
&lt;th&gt;Tentative Timeline&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Registration Start&lt;/td&gt;
&lt;td&gt;Second week of March 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Last Date&lt;/td&gt;
&lt;td&gt;End of March 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Provisional List&lt;/td&gt;
&lt;td&gt;Last week of March 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Document Verification&lt;/td&gt;
&lt;td&gt;May 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Classes Begin&lt;/td&gt;
&lt;td&gt;June 2026&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;
&lt;h2&gt;
  
  
  How to Apply
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Visit kvsonlineadmission.kvs.gov.in&lt;/li&gt;
&lt;li&gt;Click on "New Registration"&lt;/li&gt;
&lt;li&gt;Fill in basic details to receive login credentials&lt;/li&gt;
&lt;li&gt;Complete the detailed application form&lt;/li&gt;
&lt;li&gt;Upload required documents&lt;/li&gt;
&lt;li&gt;Submit and save confirmation&lt;/li&gt;
&lt;/ol&gt;
&lt;h2&gt;
  
  
  Required Documents
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Child's recent passport-size photograph&lt;/li&gt;
&lt;li&gt;Birth certificate&lt;/li&gt;
&lt;li&gt;Residence proof&lt;/li&gt;
&lt;li&gt;Category certificate (if applicable)&lt;/li&gt;
&lt;li&gt;Transfer documents (for government employees)&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;
  
  
  Reservation Policy
&lt;/h2&gt;

&lt;p&gt;KVS follows the standard government reservation policy including quotas for:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Scheduled Castes (SC)&lt;/li&gt;
&lt;li&gt;Scheduled Tribes (ST)&lt;/li&gt;
&lt;li&gt;Other Backward Classes (OBC)&lt;/li&gt;
&lt;li&gt;Economically Weaker Sections (EWS)&lt;/li&gt;
&lt;li&gt;Special priority for transferable Central government employees&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;
  Frequently Asked Questions
  &lt;br&gt;
&lt;strong&gt;Q: Is there an entrance test for Class 1?&lt;/strong&gt;&lt;br&gt;
No, selection is purely through lottery system.

&lt;p&gt;&lt;strong&gt;Q: Can I apply to multiple KV schools?&lt;/strong&gt;&lt;br&gt;
Yes, up to 3 schools can be selected in order of preference.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Q: What if my child's birthday falls on April 1?&lt;/strong&gt;&lt;br&gt;
Children born on April 1 are eligible for admission in that academic year.&lt;br&gt;
&lt;/p&gt;

&lt;br&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  Important Advisory
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Keep checking kvsangathan.nic.in regularly for official notification&lt;/li&gt;
&lt;li&gt;Prepare documents in advance&lt;/li&gt;
&lt;li&gt;Ensure accurate information in the application&lt;/li&gt;
&lt;li&gt;Save all registration details and confirmation numbers&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;a href="https://kvsangathan.nic.in" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Visit the official KVS website for latest updates&lt;/a&gt;
&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Note: All dates mentioned are tentative and subject to official confirmation by KVS.&lt;/em&gt;&lt;/p&gt;

</description>
      <category>education</category>
      <category>kvsadmission</category>
      <category>schooladmission</category>
      <category>governmentschools</category>
    </item>
    <item>
      <title>UP Board Result 2024: Latest Updates on CBSE, UP &amp; Bihar Board 10th-12th Results</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 03:02:08 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/up-board-result-2024-latest-updates-on-cbse-up-bihar-board-10th-12th-results-2ehk</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/up-board-result-2024-latest-updates-on-cbse-up-bihar-board-10th-12th-results-2ehk</guid>
      <description>&lt;p&gt;The anticipation for board exam results is reaching fever pitch as students across India await crucial updates from major education boards. Here's what we know about the UP Board, CBSE, and Bihar Board results for 2024.&lt;/p&gt;

&lt;h2&gt;
  
  
  Key Updates for Board Results
&lt;/h2&gt;

&lt;p&gt;The results season is approaching with multiple state boards preparing to announce Class 10th and 12th results. Here's the latest status:&lt;/p&gt;

&lt;h3&gt;
  
  
  UP Board Results
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Evaluation process currently underway&lt;/li&gt;
&lt;li&gt;Over 55 lakh students awaiting results&lt;/li&gt;
&lt;li&gt;Expected release: Late April 2024 (tentative)&lt;/li&gt;
&lt;li&gt;Results will be available at upresults.nic.in&lt;/li&gt;
&lt;/ul&gt;

&lt;h3&gt;
  
  
  CBSE Board
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Paper evaluation in progress&lt;/li&gt;
&lt;li&gt;Results likely by May 2024&lt;/li&gt;
&lt;li&gt;Digital marksheets to be available on DigiLocker&lt;/li&gt;
&lt;li&gt;Nearly 39 lakh students appeared this year&lt;/li&gt;
&lt;/ul&gt;

&lt;h3&gt;
  
  
  Bihar Board
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Answer sheet evaluation nearly complete&lt;/li&gt;
&lt;li&gt;Results expected by end of March 2024&lt;/li&gt;
&lt;li&gt;Students can check at biharboardonline.bihar.gov.in&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;
  How to Check UP Board Results?
  &lt;ol&gt;
&lt;li&gt;Visit the official website upresults.nic.in&lt;/li&gt;
&lt;li&gt;Click on High School/Intermediate result link&lt;/li&gt;
&lt;li&gt;Enter your roll number and school code&lt;/li&gt;
&lt;li&gt;Submit and download your result
&lt;/li&gt;
&lt;/ol&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  Important Dates to Remember
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Board&lt;/th&gt;
&lt;th&gt;Class X Expected Date&lt;/th&gt;
&lt;th&gt;Class XII Expected Date&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;UP Board&lt;/td&gt;
&lt;td&gt;Last week of April&lt;/td&gt;
&lt;td&gt;Last week of April&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;CBSE&lt;/td&gt;
&lt;td&gt;First week of May&lt;/td&gt;
&lt;td&gt;First week of May&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Bihar Board&lt;/td&gt;
&lt;td&gt;End of March&lt;/td&gt;
&lt;td&gt;End of March&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;
&lt;h2&gt;
  
  
  Steps After Result Declaration
&lt;/h2&gt;
&lt;h3&gt;
  
  
  For UP Board Students
&lt;/h3&gt;

&lt;ol&gt;
&lt;li&gt;Download digital marksheet&lt;/li&gt;
&lt;li&gt;Apply for revaluation if needed&lt;/li&gt;
&lt;li&gt;Keep physical copies for future reference&lt;/li&gt;
&lt;li&gt;Begin preparation for next academic milestone&lt;/li&gt;
&lt;/ol&gt;
&lt;h3&gt;
  
  
  Document Verification
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;Results are provisional until document verification&lt;/li&gt;
&lt;li&gt;Keep original admit cards safe&lt;/li&gt;
&lt;li&gt;Follow board notifications for physical document collection&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;
  
  
  Expert Tips
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Keep checking official websites regularly&lt;/li&gt;
&lt;li&gt;Don't rely on unofficial announcements&lt;/li&gt;
&lt;li&gt;Save your roll number and school code beforehand&lt;/li&gt;
&lt;li&gt;Have stable internet on result day&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;&lt;a href="https://t.me/careerpolitics" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Join our Telegram channel for instant result notifications&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  FAQs
&lt;/h2&gt;

&lt;p&gt;
  What if the official website crashes?
  &lt;br&gt;
Alternative websites and mobile apps will be announced by the board. Stay tuned to official social media channels.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  Is rechecking/revaluation available?
  &lt;br&gt;
Yes, students can apply for rechecking within 1-2 weeks of result declaration. Exact dates will be announced with results.&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Important Resources
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Official UP Board Website: upresults.nic.in&lt;/li&gt;
&lt;li&gt;Helpline Numbers: Will be activated on result day&lt;/li&gt;
&lt;li&gt;DigiLocker App: For digital marksheets&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Stay tuned for more updates as we approach the result declaration dates.&lt;/p&gt;

</description>
      <category>education</category>
      <category>boardresults</category>
      <category>upboard</category>
      <category>cbse</category>
    </item>
    <item>
      <title>Bihar Board 12th Result 2026: BSEB Inter Results Expected This Week, Topper Verification Underway</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 03:01:18 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/bihar-board-12th-result-2026-bseb-inter-results-expected-this-week-topper-verification-underway-5cop</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/bihar-board-12th-result-2026-bseb-inter-results-expected-this-week-topper-verification-underway-5cop</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--YsLfOkpS--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/tfx8xjxn07nn71hni0sk.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--YsLfOkpS--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/tfx8xjxn07nn71hni0sk.jpg" alt="Bihar Board Result" width="800" height="450"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Over 13 lakh Bihar students are eagerly awaiting their Class 12 results as the Bihar School Examination Board (BSEB) enters the final stages of result preparation. The board has commenced the crucial topper verification process, signaling that result declaration is imminent.&lt;/p&gt;

&lt;h2&gt;
  
  
  Latest Updates
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Topper verification process has begun&lt;/li&gt;
&lt;li&gt;Results expected to be announced this week&lt;/li&gt;
&lt;li&gt;All answer sheet evaluation completed&lt;/li&gt;
&lt;li&gt;Final result compilation in progress&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;a href="https://results.biharboardonline.com" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Bookmark the official BSEB results portal&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  Why This is Trending
&lt;/h2&gt;

&lt;p&gt;The Bihar Board's Intermediate (Class 12) results are particularly significant this year due to:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Record number of students (13.5 lakh candidates)&lt;/li&gt;
&lt;li&gt;Early completion of evaluation process&lt;/li&gt;
&lt;li&gt;High stakes for college admissions&lt;/li&gt;
&lt;li&gt;Historically trend-setting nature of Bihar results&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Timeline of Events
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;February 1-12: Examinations conducted&lt;/li&gt;
&lt;li&gt;February 13-March 15: Answer sheet evaluation&lt;/li&gt;
&lt;li&gt;March: Topper verification process&lt;/li&gt;
&lt;li&gt;Expected Result: Last week of March 2026&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Impact on Students
&lt;/h2&gt;

&lt;p&gt;The timely release of results will benefit students by:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Enabling early college admission processes&lt;/li&gt;
&lt;li&gt;Providing adequate time for competitive exam preparation&lt;/li&gt;
&lt;li&gt;Facilitating academic year planning&lt;/li&gt;
&lt;li&gt;Opening windows for various career opportunities&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Official Process
&lt;/h2&gt;

&lt;p&gt;The BSEB follows a rigorous multi-step verification process:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Answer sheet evaluation&lt;/li&gt;
&lt;li&gt;Result compilation&lt;/li&gt;
&lt;li&gt;Topper verification&lt;/li&gt;
&lt;li&gt;Final result preparation&lt;/li&gt;
&lt;li&gt;Result declaration&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;
  How to check results when declared?
  &lt;ol&gt;
&lt;li&gt;Visit biharboardonline.bihar.gov.in&lt;/li&gt;
&lt;li&gt;Click on "Class XII Results 2026"&lt;/li&gt;
&lt;li&gt;Enter roll code and roll number&lt;/li&gt;
&lt;li&gt;Download scorecard
&lt;/li&gt;
&lt;/ol&gt;



&lt;/p&gt;
&lt;h2&gt;
  
  
  Expert Insights
&lt;/h2&gt;

&lt;p&gt;"The Bihar Board's systematic approach and early evaluation completion indicate their commitment to timely result declaration. This benefits students planning for higher education and competitive exams," notes Dr. Rajesh Kumar, Education Policy Analyst.&lt;/p&gt;
&lt;h2&gt;
  
  
  Important Documents Required
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Admit Card&lt;/li&gt;
&lt;li&gt;Registration Number&lt;/li&gt;
&lt;li&gt;Roll Code and Roll Number&lt;/li&gt;
&lt;li&gt;Valid ID Proof&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;
  
  
  Stay Updated
&lt;/h2&gt;

&lt;p&gt;&lt;a href="https://t.me/careerpolitics" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Join our Telegram channel for instant result alerts&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  FAQs
&lt;/h2&gt;

&lt;p&gt;
  When will the results be declared?
  &lt;br&gt;
The results are expected to be announced this week, though the exact date is yet to be officially confirmed.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  What was last year's pass percentage?
  &lt;br&gt;
In 2025, the overall pass percentage for Bihar Board Class 12 was 87.2%.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  How can I get my digital marksheet?
  &lt;br&gt;
Digital marksheets will be available on DigiLocker immediately after result declaration.&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Sources
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;Bihar School Examination Board (BSEB)&lt;/li&gt;
&lt;li&gt;Official BSEB Press Releases&lt;/li&gt;
&lt;li&gt;Education Ministry Updates&lt;/li&gt;
&lt;/ul&gt;

</description>
      <category>biharboard</category>
      <category>bsebresults</category>
      <category>class12results</category>
      <category>educationnews</category>
    </item>
    <item>
      <title>Bihar Board Class 12 Results 2026: BSEB Begins Topper Verification, Expected Release Date and Latest Updates</title>
      <dc:creator>Murari Kumar</dc:creator>
      <pubDate>Wed, 18 Mar 2026 02:01:08 +0000</pubDate>
      <link>https://careerpolitics.com/murari_kumar_38d8e39a8450/bihar-board-class-12-results-2026-bseb-begins-topper-verification-expected-release-date-and-1p6p</link>
      <guid>https://careerpolitics.com/murari_kumar_38d8e39a8450/bihar-board-class-12-results-2026-bseb-begins-topper-verification-expected-release-date-and-1p6p</guid>
      <description>&lt;p&gt;&lt;a href="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--YsLfOkpS--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/tfx8xjxn07nn71hni0sk.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://res.cloudinary.com/dbdlrqwdt/image/fetch/s--YsLfOkpS--/c_limit%2Cf_auto%2Cfl_progressive%2Cq_auto%2Cw_800/https://careerpolitics-images-bucket.s3.amazonaws.com/uploads/articles/tfx8xjxn07nn71hni0sk.jpg" alt="Bihar Board Result 2026" width="800" height="450"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;In a significant development for over 13 lakh Bihar Board intermediate students, the Bihar School Examination Board (BSEB) has commenced its topper verification process, signaling the imminent release of Class 12 results. Here's what anxious students and parents need to know about the latest developments.&lt;/p&gt;

&lt;h2&gt;
  
  
  Key Highlights
&lt;/h2&gt;

&lt;p&gt;• Topper verification process has officially begun&lt;br&gt;
• Results expected to be announced this week&lt;br&gt;
• Over 13 lakh students awaiting results&lt;br&gt;
• Answer sheet evaluation already completed&lt;/p&gt;
&lt;h2&gt;
  
  
  Latest Developments
&lt;/h2&gt;

&lt;p&gt;The Bihar School Examination Board has entered the final phase of result preparation with the commencement of topper verification. This crucial step involves thorough scrutiny of top-performing candidates' answer sheets, documents, and in-person interviews to ensure transparency and authenticity of results.&lt;/p&gt;

&lt;p&gt;
  What is Topper Verification?
  &lt;br&gt;
Topper verification is a rigorous process implemented by BSEB where top-scoring candidates undergo:&lt;br&gt;
• Document verification&lt;br&gt;
• Physical interviews&lt;br&gt;
• Answer sheet re-evaluation&lt;br&gt;
• Background checks&lt;br&gt;
This process was introduced to prevent any irregularities in the examination system.&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Timeline of Events
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Event&lt;/th&gt;
&lt;th&gt;Status&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Exam Completion&lt;/td&gt;
&lt;td&gt;✓ Done&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Answer Sheet Evaluation&lt;/td&gt;
&lt;td&gt;✓ Done&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Topper Verification&lt;/td&gt;
&lt;td&gt;🔄 Ongoing&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Result Declaration&lt;/td&gt;
&lt;td&gt;⏳ Pending&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

&lt;h2&gt;
  
  
  Impact on Students
&lt;/h2&gt;

&lt;p&gt;The thorough verification process, while causing some anxiety among students, ensures:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Credibility of results&lt;/li&gt;
&lt;li&gt;Fair recognition of genuine achievers&lt;/li&gt;
&lt;li&gt;Maintenance of BSEB's academic standards&lt;/li&gt;
&lt;li&gt;Prevention of result-related controversies&lt;/li&gt;
&lt;/ol&gt;

&lt;h2&gt;
  
  
  Official Updates
&lt;/h2&gt;

&lt;p&gt;The Board has maintained strict confidentiality regarding the exact result date. However, reliable sources indicate an announcement is likely this week. Students are advised to:&lt;/p&gt;

&lt;p&gt;• Keep their admit cards ready&lt;br&gt;
• Watch official BSEB website regularly&lt;br&gt;
• Beware of fake result announcements&lt;/p&gt;

&lt;p&gt;&lt;a href="https://biharboardonline.bihar.gov.in" class="ltag_cta ltag_cta--branded" rel="noopener noreferrer"&gt;Bookmark the official BSEB website for instant result updates&lt;/a&gt;
&lt;/p&gt;

&lt;h2&gt;
  
  
  How to Check Results (When Released)
&lt;/h2&gt;

&lt;ol&gt;
&lt;li&gt;Visit the official BSEB website&lt;/li&gt;
&lt;li&gt;Click on the Class 12 result link&lt;/li&gt;
&lt;li&gt;Enter your roll code and roll number&lt;/li&gt;
&lt;li&gt;Submit and view your result&lt;/li&gt;
&lt;/ol&gt;

&lt;h2&gt;
  
  
  Expert Insights
&lt;/h2&gt;

&lt;p&gt;"The topper verification process is a crucial step that has significantly reduced result-related controversies in recent years," notes Dr. Anand Kishor, BSEB Chairman. "Students should remain patient as we ensure complete accuracy in result compilation."&lt;/p&gt;

&lt;h2&gt;
  
  
  FAQs
&lt;/h2&gt;

&lt;p&gt;
  When exactly will the results be declared?
  &lt;br&gt;
While the exact date hasn't been announced, sources indicate the results are likely to be declared this week after completion of topper verification.&lt;br&gt;


&lt;/p&gt;

&lt;p&gt;
  How can I get my result on mobile?
  &lt;br&gt;
Students can check their results through:&lt;br&gt;
• BSEB official website&lt;br&gt;
• BSEB mobile app&lt;br&gt;
• SMS service (details to be announced)&lt;br&gt;


&lt;/p&gt;

&lt;h2&gt;
  
  
  Important Advisory
&lt;/h2&gt;

&lt;p&gt;Students are urged to:&lt;br&gt;
• Rely only on official BSEB communications&lt;br&gt;
• Keep checking the board's website regularly&lt;br&gt;
• Have admission cards readily available&lt;br&gt;
• Stay calm and focused during this waiting period&lt;/p&gt;




&lt;p&gt;&lt;em&gt;Stay tuned to CareerPolitics.com for instant updates on Bihar Board results and other education news.&lt;/em&gt;&lt;/p&gt;

</description>
      <category>biharboard</category>
      <category>bsebresults</category>
      <category>class12results</category>
      <category>educationnews</category>
    </item>
  </channel>
</rss>
