# questions

tags: ['Physics', 'Waves and Sound', 'NEET', 'NEET 2014']
source_db: MarksBatch2_P1.db
title:
if-n-1-n-2-and-n-3-are-the-fundamental-frequencies-of-three-segments-into-which-a-string-is-divided-then-the-original-fundamental-frequency-n-of-the-s

slug:
if-n-1-n-2-and-n-3-are-the-fundamental-frequencies-of-three-segments-into-which-a-string-is-divided-then-the-original-fundamental-frequency-n-of-the-s-50398

question:
<div class="question">If <math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi> </mi><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and <math><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></math> are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency <math><mi>n</mi></math> of the string is given by:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></msqrt></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></msqrt></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></msqrt></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><msqrt><mi>n</mi></msqrt><mo>=</mo><msqrt><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></msqrt><mo>+</mo><msqrt><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></msqrt><mo>+</mo><msqrt><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></msqrt></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mi>n</mi><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/neet/75b5a93b-454f-4875-94cd-5f5466932eaf-image16.png"/><br/>For first segment we have,</p><p><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></msqrt></math></p><p>For second segment we have,<br/><math><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></msqrt></math></p><p>For third segment we have,<br/><math><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></msqrt></math></p><p>For the whole string we have,<br/><math><mi>n</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></msqrt></math></p><p>Total length of the string is,<br/><math><mi>l</mi><mo>=</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p><p>Thus, the fundamental frequency for the whole string is,<br/><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mfrac></math></p></div>


title: if-n-2017-then-what-is-2018-1-l-o-g-2-n-1-l-o-g-3-n-1-l-o-g-4-n-1-l-o-g-2017-n-1-equal-to
slug: if-n-2017-then-what-is-2018-1-l-o-g-2-n-1-l-o-g-3-n-1-l-o-g-4-n-1-l-o-g-2017-n-1-equal-to-62748
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $n=(2017) !$, then what is $[2018-1]$<br/>$\frac{1}{\log _{2} n}+\frac{1}{\log _{3} n}+\frac{1}{\log _{4} n}+\ldots .+\frac{1}{\log _{2017} n}$<br/>equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{n}}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{n}$</span> </li> </ul>

answer:
<div class="solution">$n=(2017) !$<br/>$\frac{1}{\log _{2} n}+\frac{1}{\log _{3} n}+\frac{1}{\log _{4} n}+\ldots .+\frac{1}{\log _{2017} n}$<br/>$=\frac{1}{\frac{\log _{n}}{\log _{2}}}+\frac{1}{\frac{\log _{n}}{\log _{3}}}+\frac{1}{\frac{\log _{n}}{\log _{4}}}+\ldots .+\frac{1}{\frac{\log _{n}}{\log _{2017}}}$<br/>$\left(\because \log _{\mathrm{a}} \mathrm{b}=\frac{\log _{\mathrm{b}}}{\log _{\mathrm{a}}}\right)$<br/>$=\frac{\log _{2}}{\log _{n}}+\frac{\log _{3}}{\log _{n}}+\frac{\log _{4}}{\log _{n}}+\ldots .+\frac{\log _{2017}}{\log _{n}}$<br/>$=\frac{\log _{2}+\log _{3}+\log _{4}+\ldots .+\log _{2017}}{\log _{n}}$<br/>$=\frac{\log (2.3 .4 \ldots .2017)}{\log _{\mathrm{n}}}$<br/>$(\because \log a+\log b+\log c+\ldots=\log a . b . c \ldots \ldots)$<br/>$=\frac{\log (2017 !)}{\log _{\mathrm{n}}}=\frac{\log _{\mathrm{n}}}{\log _{\mathrm{n}}}=1$</div>


title: if-n-a-115-n-b-326-n-a-b-47-then-what-is-n-a-b-equal-to
slug: if-n-a-115-n-b-326-n-a-b-47-then-what-is-n-a-b-equal-to-39864
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">373</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $n(A)=115, n(B)=326, n(A-B)=47$, then what is<br/>$\begin{array}{ll}n(A \cup B) \text { equal to? } &amp;\end{array}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">373</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">165</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">370</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">394</span> </li> </ul>

answer:
<div class="solution">We know, for two sets $\mathrm{A}$ and $\mathrm{B}$ $\mathrm{A}-\mathrm{B}=\mathrm{A}-(\mathrm{A} \cap \mathrm{B})$<br/>$n(A-B)=n(A)-n(A \cap B)$<br/>Given, $n(A)=115, n(B)=326$ and $n(A-B)=47$.<br/>$\Rightarrow 47=115-n(A \cap B)$<br/>$\Rightarrow n(A \cap B)=68$<br/>Consider $n(A \cup B)=n(A)+n(B)-n(A \cap B)$<br/>$=115+326-68=373$</div>


title: if-n-a-a-x-x-n-then-what-is-n-12-n-8-equal-to
slug: if-n-a-a-x-x-n-then-what-is-n-12-n-8-equal-to-56651
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $N_{a}=\{a x \mid x \in N\}$, then what is $N_{12} \cap N_{8}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$N_{12}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$N_{24}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$N_{20}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$N_{48}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$N_{20}$</span> </div>

answer:
<div class="solution">Given, $N_{a}=\{a x \mid x \in N$ $N_{12}=\{12,24,36,48, \ldots$<br/>and $N_{8}=\{8,16,24, .$<br/>$N_{12} \cap N_{8}=\{24,48, \ldots\}$</div>


title: if-n-and-r-are-two-positive-integers-such-that-n-r-then-n-c-r-1-n-c-r
slug: if-n-and-r-are-two-positive-integers-such-that-n-r-then-n-c-r-1-n-c-r-25901
tags: ['Mathematics', 'Permutation Combination', 'JEE Main']
answer: <div class="solution">${ }^n C_{r-1}+{ }^n C_r={ }^{n+1} C_r$ is a standard formula.</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $n$ and $r$ are two positive integers such that $n \geq r$, then ${ }^n C_{r-1}+{ }^n C_r=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^n C_{n-r}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">${ }^n C_r$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">${ }^{n-1} C_{\text {, }}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">${ }^{n+1} C_r$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{n+1} C_r$</span> </div>


title: if-n-n-then-12-1-n-2-5-n-190-0-n-4-n-is-divisible-by-which-one-of-the-following
slug: if-n-n-then-12-1-n-2-5-n-190-0-n-4-n-is-divisible-by-which-one-of-the-following-17478
tags: ['Mathematics', 'Mathematical Induction', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2000</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $n \in \mathrm{N}$, then $121^{\mathrm{n}}-25^{\mathrm{n}}+1900^{\mathrm{n}}-(-4)^{\mathrm{n}}$ is divisible by which one of the following?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1904</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2000</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2002</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2006</span> </li> </ul>

answer:
<div class="solution">So, $z=1+\sqrt{5}$<br/>$121^{n}-25^{n}+1900^{n}-\left(-4^{n}\right)$<br/>Let us substitute $\mathrm{n}=1$ We get,$(121)^{1}-(25)^{1}+(1900)^{1}-\left(-4^{1}\right)$<br/>$=121-25+1900+4$<br/>$=2025-25$<br/>$=2000$<br/>So, given expression is divisible by 2000</div>


tags: ['Mathematics', 'Ellipse', 'JEE Main', 'JEE Main 2014 (09 Apr Online)']
source_db: MarksBatch2_P1.db
title:
if-o-b-is-the-semiminor-axis-of-an-ellipse-f-1-and-f-2-are-its-focii-and-the-angle-between-f-1-b-and-f-2-b-is-a-right-angle-then-the-square-of-the-ecc

slug:
if-o-b-is-the-semiminor-axis-of-an-ellipse-f-1-and-f-2-are-its-focii-and-the-angle-between-f-1-b-and-f-2-b-is-a-right-angle-then-the-square-of-the-ecc-26555

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>B</mi></math> is the semi-minor axis of an ellipse, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub></math> are its focii and the angle between <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub><mi>B</mi></math> is a right angle, then the square of the eccentricity of the ellipse is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> </math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> </math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math></span> </div>

answer:
<div class="solution"><p>In standard ellipse  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p>Semi-major axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>A</mi><mo>=</mo><mi>a</mi></math></p><p>Semi-minor axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>B</mi><mo>=</mo><mi>b</mi></math></p><p>Focus <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub><mo>≡</mo><mfenced><mrow><mo>-</mo><mi>a</mi><mi>e</mi><mo>,</mo><mn>0</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub><mo>≡</mo><mfenced><mrow><mi>a</mi><mi>e</mi><mo>,</mo><mn>0</mn></mrow></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/97267906-9fe5-4c22-8e5c-e302e59d48ac-image.png" style="width: 350px; height: 191px;"/><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∠</mo><msub><mi>F</mi><mn>1</mn></msub><mi>B</mi><msub><mi>F</mi><mn>2</mn></msub><mo>=</mo><mn>9</mn><mn>0</mn><mo>°</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><msub><mi>F</mi><mn>1</mn></msub><mo>=</mo><mi>O</mi><msub><mi>F</mi><mn>2</mn></msub><mo>⇒</mo><mo>∠</mo><msub><mi>F</mi><mn>1</mn></msub><mi>B</mi><mi>O</mi><mo>=</mo><mo>∠</mo><msub><mi>F</mi><mn>2</mn></msub><mi>B</mi><mi>O</mi><mo>=</mo><mn>4</mn><mn>5</mn><mo>°</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>a</mi><mi>e</mi></mrow><mi>b</mi></mfrac><mo>=</mo><mo>tan</mo><mn>4</mn><mn>5</mn><mo>°</mo><mo>=</mo><mn>1</mn></math> </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>b</mi><mo>=</mo><mi>a</mi><mi>e</mi></math></p><p>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>e</mi><mn>2</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>e</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>e</mi><mn>2</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><msup><mi>e</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>e</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p></div>


title: if-o-be-the-origin-and-a-x-1-y-1-b-x-2-y-2-are-two-points-then-what-is-oa-ob-cos-aob
slug: if-o-be-the-origin-and-a-x-1-y-1-b-x-2-y-2-are-two-points-then-what-is-oa-ob-cos-aob-64259
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{O}$ be the origin and $\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)$ are two points,<br/>then what is $(\mathrm{OA})(\mathrm{OB}) \cos \angle \mathrm{AOB}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}_{1}^{2}+\mathrm{x}_{2}^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{y}_{1}^{2}+\mathrm{y}_{2}^{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}_{1} \mathrm{x}_{2}+\mathrm{y}_{1} \mathrm{y}_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{x}_{1} \mathrm{y}_{1}+\mathrm{x}_{2} \mathrm{y}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{x}_{1} \mathrm{x}_{2}+\mathrm{y}_{1} \mathrm{y}_{2}$</span> </div>

answer:
<div class="solution">Let $\mathrm{O}(0,0), \mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ and $\mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)$ be three points<br/>$\mathrm{OA}=\sqrt{\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}}, \mathrm{OB}=\sqrt{\mathrm{x}_{2}^{2}+\mathrm{y}_{2}^{2}}$<br/>$\mathrm{AB}=\sqrt{\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}}$<br/>In $\Delta \mathrm{AOB}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/8bS8VzsLmzi7-I0AGzji600a9pzwXUMN3_jFrGUY3RA.original.fullsize.png"/><br/><br/>$\cos \angle \mathrm{AOB}=\frac{\mathrm{OA}^{2}+\mathrm{OB}^{2}-\mathrm{AB}^{2}}{2 . \mathrm{OA} \cdot \mathrm{OB}}$<br/>$\Rightarrow \quad \mathrm{OA} \cdot \mathrm{OB} \cos \angle \mathrm{AOB}=\frac{\mathrm{OA}^{2}+\mathrm{OB}^{2}-\mathrm{AB}^{2}}{2}$<br/>$=\frac{\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}+\mathrm{x}_{2}^{2}+\mathrm{y}_{2}^{2}-\left\{\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}\right\}}{2}$<br/>$\Rightarrow \quad \mathrm{OA} \cdot \mathrm{OB} \cdot \cos \angle \mathrm{AOB}$<br/>$=\frac{\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}+\mathrm{x}_{2}^{2}+\mathrm{y}_{2}^{2}-\left\{\mathrm{x}_{2}^{2}+\mathrm{x}_{1}^{2}-2 \mathrm{x}_{1} \mathrm{x}_{2}+\mathrm{y}_{2}^{2}+\mathrm{y}_{1}^{2}-2 \mathrm{y}_{1} \mathrm{y}_{2}\right\}}{2}$<br/>$=\frac{2\left(\mathrm{x}_{1} \mathrm{x}_{2}+\mathrm{y}_{1} \mathrm{y}_{2}\right)}{2}=\mathrm{x}_{1} \mathrm{x}_{2}+\mathrm{y}_{1} \mathrm{y}_{2}$</div>


title: if-one-debarks-a-tree-what-parts-of-the-plant-is-being-removed
slug: if-one-debarks-a-tree-what-parts-of-the-plant-is-being-removed-81877
question: <div class="question">If one debarks a tree, what parts of the plant is being removed?</div>
tags: ['Biology', 'Anatomy of Flowering Plants']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
answer:
<div class="solution">Debarking refers to removal of bark, i.e., all tissues exterior to the vascular cambium, including secondary phloem. Bark includes periderm (phellogen, phellem and phelloderm) and secondary phloem.</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-one-of-the-cube-roots-of-1-be-omega-then-leftbeginarrayccc1-1omega2-omega2-1i-1-omega21-i-1omega-1endarrayright

slug:
if-one-of-the-cube-roots-of-1-be-omega-then-leftbeginarrayccc1-1omega2-omega2-1i-1-omega21-i-1omega-1endarrayright-55963

question:
<div class="question">If one of the cube roots of 1 be \(\omega\), then<br/>\(\left|\begin{array}{ccc}1 &amp; 1+\omega^2 &amp; \omega^2 \\ 1-i &amp; -1 &amp; \omega^2-1 \\ -i &amp; -1+\omega &amp; -1\end{array}\right|=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\omega\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Hints: \(\mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_3\)<br/>\(\begin{aligned}<br/>&amp; \mathrm{C}_3 \rightarrow \mathrm{C}_3+\mathrm{C}_2 \\<br/>&amp; \mathrm{C}_3 \rightarrow \mathrm{C}_3+\omega \mathrm{C}_1 \\<br/>&amp; \mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_1<br/>\end{aligned}\)</div>


title: if-one-of-the-roots-of-the-equation-a-b-c-x-2-b-c-a-x-c-a-b-0-is-1-what-is-the-second-root
slug: if-one-of-the-roots-of-the-equation-a-b-c-x-2-b-c-a-x-c-a-b-0-is-1-what-is-the-second-root-86966
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If one of the roots of the equation $a(b-c) x^{2}+b(c-a) x+$ $c(a-b)=0$ is 1, what is the second root?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{b(c-a)}{a(b-c)}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{b(c-a)}{a(b-c)}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{c(a-b)}{a(b-c)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{c(a-b)}{a(b-c)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{c(a-b)}{a(b-c)}$</span> </div>

answer:
<div class="solution">Givenequation is<br/>$a(b-c) x^{2}+b(c-a) x+o(a-b)=0$<br/>Let $\alpha$ be the second root.<br/>$\mathrm{So},(\alpha)(1)=\frac{c(a-b)}{a(b-c)}$<br/>Hence, $\alpha=$ second root $=\frac{c(a-b)}{a(b-c)}$</div>


title: if-one-of-the-roots-of-the-equation-x-2-a-x-6-0-is-1-then-what-is-a-6-equal-to
slug: if-one-of-the-roots-of-the-equation-x-2-a-x-6-0-is-1-then-what-is-a-6-equal-to-14249
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If one of the roots of the equation $x^{2}+a x-6=0$ is 1 , then what is $(\mathrm{a}-6)$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-2</span> </li> </ul>

answer:
<div class="solution">Since 1 is the root of given equation $\therefore$ it satisfies the equation. $\therefore(1)^{2}+a(1)-6=0$<br/>$\Rightarrow a-6=-1$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-one-root-of-the-equation-1-m-x-2-1-x-1-0-is-double-the-other-and-1-is-real-then-what-is-the-greatest-value-of-m

slug:
if-one-root-of-the-equation-1-m-x-2-1-x-1-0-is-double-the-other-and-1-is-real-then-what-is-the-greatest-value-of-m-38959

question:
<div class="question">If one root of the equation $(1-\mathrm{m}) \mathrm{x}^{2}+1 \mathrm{x}+1=0$ is double the other and 1 is real, then what is the greatest value of $\mathrm{m}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{9}{8}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{9}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{8}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{8}{9}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{9}{8}$</span> </div>

answer:
<div class="solution">Given equation is<br/>$(\ell-m) x^{2}+\ell x+1=0$<br/>Roots are $\alpha, \beta$. One root is double the other. $\beta=2 \alpha$<br/>Sum of roots $=\alpha+\beta$<br/>$\begin{array}{ll}3 \alpha=\frac{-\ell}{\ell-m} &amp; \alpha(2 \alpha)=\frac{1}{(\ell-m)} \\ \Rightarrow \alpha^{2}=\frac{\ell^{2}}{9(\ell-m)^{2}} \quad 2 \alpha^{2}=\frac{1}{\ell-m} \\ \Rightarrow 2 \frac{\ell^{2}}{9(\ell-m)^{2}}=\frac{1}{(\ell-m)} \\ \Rightarrow \frac{2 \ell^{2}}{9(\ell-m)}=1 \\ \Rightarrow 2 \ell^{2}=9(\ell-m) \Rightarrow 2 \ell^{2}-9 \ell+9 m=0\end{array}$<br/>For $\ell$ to be real discriminant should be $b^{2}-4 a c \geq 0$ $81-4 \times 2 \times 9 m \geq 0$<br/>$m \leq \frac{9}{8}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$a=c$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-one-root-of-the-equation-a-x-2-b-x-c-0-a-0-is-reciprocal-ofthe-other-root-then-which-one-of-the-following-is-correct

slug:
if-one-root-of-the-equation-a-x-2-b-x-c-0-a-0-is-reciprocal-ofthe-other-root-then-which-one-of-the-following-is-correct-39300

question:
<div class="question">If one root of the equation $a x^{2}+b x+c=0, a \neq 0$ is reciprocal ofthe other root, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$a=c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$b=c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a=-c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$b=0$</span> </li> </ul>

answer:
<div class="solution">Let the roots of $\alpha x^{2}+b x+c=0, a \neq 0$ be $\alpha$ and $\frac{1}{\alpha}$<br/>eroduct of roots $=\alpha \cdot \frac{1}{\alpha}=\frac{c}{a}$<br/>$\Rightarrow c=a$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-p-0-and-the-conditions-p-q-p-r-and-p-q-p-r-hold-simultaneously-then-which-one-of-the-following-is-correct

slug:
if-p-0-and-the-conditions-p-q-p-r-and-p-q-p-r-hold-simultaneously-then-which-one-of-the-following-is-correct-85060

question:
<div class="question">If $\overrightarrow{\mathrm{p}} \neq \overrightarrow{0}$ and the conditions $\overrightarrow{\mathrm{p}} \overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{r}}$ and $\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{r}}$ hold<br/>simultaneously, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\overrightarrow{\mathrm{q}} \neq \overrightarrow{\mathrm{r}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\overrightarrow{\mathrm{q}}=\overrightarrow{-\mathrm{r}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\overrightarrow{\text { q. } \cdot \mathrm{r}}=0 \quad$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{r}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{r}}$</span> </div>

answer:
<div class="solution">Given that $\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{r}}$<br/>$\Rightarrow \overrightarrow{\mathrm{p}} \cdot(\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{r}})=0$<br/>$\Rightarrow \overrightarrow{\mathrm{p}}$ is perpendicular to $\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{r}}$<br/>Also, $\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{r}}$ (given).<br/>$\Rightarrow \overrightarrow{\mathrm{p}} \times(\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{r}})=0$<br/>$\Rightarrow \overrightarrow{\mathrm{p}}$ is parallel to $\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{r}}$<br/>Which is not possible simultaneously unless either $\mathrm{p}$ or $\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{r}}$ is zero, since $\overrightarrow{\mathrm{p}} \neq 0, \Rightarrow \overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{r}}=0$<br/>Thus, the given conditions hold simultaneously if $\overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{r}}$</div>


title: if-p-0-q-q-p-0-0-q-p-0-then-which-one-of-the-following-is-correct
slug: if-p-0-q-q-p-0-0-q-p-0-then-which-one-of-the-following-is-correct-23808
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{ccc}p &amp; -q &amp; 0 \\ 0 &amp; p &amp; q \\ q &amp; 0 &amp; p\end{array}\right|=0$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p$ is one of the cube roots of unity</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$q$ is one of the cube roots of unity</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{p}{q}$ is one of the cube roots of unity</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{p}{q}$ is one of the cube roots of unity</span> </div>

answer:
<div class="solution">Let $\left|\begin{array}{ccc}p &amp; -q &amp; 0 \\ 0 &amp; p &amp; q \\ q &amp; 0 &amp; p\end{array}\right|=0$<br/>$\Rightarrow p\left(p^{2}\right)+q\left(-q^{2}\right)+0=0$<br/>$\Rightarrow p^{3}-q^{3}=0$<br/>$\Rightarrow p^{3}=q^{3}$<br/>$\Rightarrow \frac{p^{3}}{q^{3}}=1 \Rightarrow\left(\frac{p}{q}\right)^{3}=1$<br/>$\Rightarrow \frac{p}{q}$ is one of the cube roots of unity.</div>


title: if-p-32-6-k-c-32-6-then-what-is-the-value-of-k
slug: if-p-32-6-k-c-32-6-then-what-is-the-value-of-k-61734
question: <div class="question">If $P(32,6)=k C(32,6)$, then what is the value of $k$ ?</div>
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">720</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">32</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">120</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">720</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Since ${ }^{32} \mathrm{P}_{6}=\mathrm{k}^{32} \mathrm{C}_{6}$<br/>$\Rightarrow \frac{32 !}{(32-6) !}=k \cdot \frac{32 !}{6 !(32-6) !}$<br/>$\Rightarrow \mathrm{k}=6 !=720$</div>


title: if-p-77-31-x-and-c-77-31-y-then-which-one-of-the-following-is-correct
slug: if-p-77-31-x-and-c-77-31-y-then-which-one-of-the-following-is-correct-54713
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{P}(77,31)=\mathrm{x}$ and $\mathrm{C}(77,31)=\mathrm{y}$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=y$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x=y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$77 x=31 y$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x&gt;y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x&gt;y$</span> </div>

answer:
<div class="solution">As we know $\mathrm{P}(\mathrm{n}, \mathrm{r})=\mathrm{r} ! \mathrm{C}(\mathrm{n}, \mathrm{r})$<br/>$\therefore \quad$ From the question, we have $\mathrm{x}=\mathrm{r} !(\mathrm{y})$<br/>Here $\mathrm{r}=31$<br/>$\therefore \quad x=(31) ! . y \rightarrow x&gt;y$</div>


title: if-p-a-3-2-p-b-5-2-and-p-a-b-p-a-b-5-2-then-what-is-p-a-b-equal-to
slug: if-p-a-3-2-p-b-5-2-and-p-a-b-p-a-b-5-2-then-what-is-p-a-b-equal-to-94019
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $P(A)=\frac{2}{3}, P(B)=\frac{2}{5}$ and $P(A \cup B)-P(A \cap B)=\frac{2}{5}$, then<br/>what is $P(A \cap B)$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{5}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5}{11}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$\quad P(A \cup B)-P(A \cap B)=\frac{2}{5}$<br/>We know that, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$\therefore \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5}$<br/>$\frac{2}{3}+\frac{2}{5}-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5}$<br/>$-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5}-\frac{2}{3}-\frac{2}{5}$<br/>$-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{6-16}{15} \Rightarrow \frac{-10}{15}=-\frac{2}{3} \times \frac{-1}{2}=\frac{1}{3}$</div>


title: if-p-a-08-p-b-09-p-a-b-p-which-one-ofthe-following-is-correct
slug: if-p-a-08-p-b-09-p-a-b-p-which-one-ofthe-following-is-correct-22616
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $P(A)=0.8, P(B)=0.9, P(A B)=p$, which one ofthe following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.72 \leq p \leq 0.8$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\quad 0.7 \leq p \leq 0.8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.72   &lt; p &lt; 0.8$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.7   &lt; p &lt; 0.8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 0.7 \leq p \leq 0.8$</span> </div>

answer:
<div class="solution">We know,<br/>$P(A \cup B)=P(A)+P(B)-P(A \cap B)$<br/>$\Rightarrow 0.8+0.9-p \leq 1$<br/>$\Rightarrow \quad 1.7-p \leq 1$<br/>$\Rightarrow \quad 0.7 \leq p$<br/>Now, $\quad P(A)   &lt; P(B)$<br/>$\therefore \quad P(A \cap B) \leq P(A)$<br/>$\Rightarrow \quad p \leq 0.8$<br/>Hence, $0.7 \leq p \leq 0.8$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P1.db
title:
if-p-a-denotes-the-power-set-of-a-and-a-is-the-void-set-then-what-is-number-of-elements-in-p-p-p-p-a

slug:
if-p-a-denotes-the-power-set-of-a-and-a-is-the-void-set-then-what-is-number-of-elements-in-p-p-p-p-a-35681

question:
<div class="question">If $P(A)$ denotes the power set of $A$ and $A$ is the void set, then what is number of elements in $P\{P\{P\{P(A)\}\}\}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Since, $\mathrm{A}$ is void set therefore the number of elements in power set of $A$ is 1 . $P\{P(A)\}=2^{1}=2$<br/>because If set $A$ has $n$ elements then $P(A)$ has $2^{n}$ elements.<br/>$\Rightarrow P\{P\{P(A)\}\}=2^{2}=4$<br/>$\Rightarrow P\left\{P\{P\{P(A)\}\}=2^{4}=16\right.$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-p-and-q-are-positive-integers-then-which-one-of-the-following-equations-has-p-q-as-one-of-its-roots

slug:
if-p-and-q-are-positive-integers-then-which-one-of-the-following-equations-has-p-q-as-one-of-its-roots-60868

question:
<div class="question">If $p$ and $q$ are positive integers, then which one of the<br/>following equations has $p-\sqrt{q}$ as one of its roots?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}-2 p x-\left(p^{2}-q\right)=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}-2 p x+\left(p^{2}-q\right)=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+2 p x-\left(p^{2}-q\right)=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+2 p x+\left(p^{2}-q\right)=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}-2 p x+\left(p^{2}-q\right)=0$</span> </div>

answer:
<div class="solution">If any equation has $p-\sqrt{q}$ as a root, then another root will be $p+\sqrt{q}$.<br/>So, sum of roots $=p-\sqrt{q}+p+\sqrt{q}=2 p$<br/>and product of roots $=(p-\sqrt{q})(p+\sqrt{q})=p^{2}-q$<br/>Now, required equation is $x^{2}-($ sum of roots $) x+($ product of roots $)=0$ $\Rightarrow x^{2}-2 p x+\left(p^{2}-q\right)=0$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1,0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-p-and-q-are-the-roots-of-the-equation-x-2-p-x-q-0-then-what-are-the-values-of-p-and-q-respectively

slug:
if-p-and-q-are-the-roots-of-the-equation-x-2-p-x-q-0-then-what-are-the-values-of-p-and-q-respectively-20565

question:
<div class="question">If $p$ and $q$ are the roots of the equation $x^{2}-p x+q=0$, then what are the values of $p$ and $q$ respectively?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1,0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\overline{0}, 1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-2,0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-2,1$</span> </li> </ul>

answer:
<div class="solution">Given quadratic equation is $x^{2}-p x+q=0$<br/>$p$ and $q$ are the roots of $x^{2}-p x+q=0$ : Sum of roots $=p+q=p \Rightarrow q=p-p=$ and product of roots $=p q=q \Rightarrow p=q / q=$<br/>$\Rightarrow q(p-1)=0 \Rightarrow p=1, q=0$</div>


title: if-p-and-q-are-two-complex-numbers-then-the-modulus-of-the-quotient-of-p-and-q-is
slug: if-p-and-q-are-two-complex-numbers-then-the-modulus-of-the-quotient-of-p-and-q-is-98786
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $P$ and $Q$ are two complex numbers, then the modulus of the quotient of $P$ and $Q$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Greater than the quotient of their moduli</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Less than the quotient of their moduli</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Less than or equal to the quotient of their moduli</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Equal to the quotient of their moduli</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Equal to the quotient of their moduli</span> </div>

answer:
<div class="solution">Thetwo complex numbers are $\mathrm{P}=\mathrm{x}+$ iy and $\mathrm{Q}=\alpha+\mathrm{i} \beta$<br/>Quotient $=\frac{\mathrm{P}}{\mathrm{Q}}=\frac{\mathrm{x}+\mathrm{iy}}{\alpha+\mathrm{i} \beta},\left|\frac{\mathrm{P}}{\mathrm{Q}}\right|=\left|\frac{\mathrm{x}+\mathrm{iy}}{\alpha+\mathrm{i} \beta}\right|$<br/>$=\frac{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}{\sqrt{\alpha^{2}+\beta^{2}}}=\sqrt{\frac{\mathrm{x}^{2}+\mathrm{y}^{2}}{\alpha^{2}+\beta^{2}}}=\frac{\left|\mathrm{Z}_{1}\right|}{\left|\mathrm{Z}_{2}\right|}=\left|\frac{\mathrm{Z}_{1}}{\mathrm{Z}_{2}}\right|$<br/>Hence, the quotient of their modulus is equal to the quotient of their moduli.</div>


title: if-p-b-4-3-p-a-b-c-3-1-and-p-a-b-c-3-1-then-what-is-p-b-c-equal-to
slug: if-p-b-4-3-p-a-b-c-3-1-and-p-a-b-c-3-1-then-what-is-p-b-c-equal-to-96197
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{P}(\mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}$ and $\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}$,<br/>then what is $\mathrm{P}(\mathrm{B} \cap \mathrm{C})$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{15}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{9}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{12}$</span> </div>

answer:
<div class="solution">$\mathrm{P}(\mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}, \mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}$.<br/>We know. $\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})=\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})+\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})$<br/>$=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$<br/>$\mathrm{P}(\mathrm{B})=\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})$<br/>$\therefore \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})$<br/>$=\frac{3}{4}-\frac{2}{3}=\frac{9-8}{12}=\frac{1}{12} .$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-p-be-the-length-of-the-perpendicular-from-the-orgin-on-the-straight-line-x-2-by-2-p-0-then-what-is-the-value-of-b

slug:
if-p-be-the-length-of-the-perpendicular-from-the-orgin-on-the-straight-line-x-2-by-2-p-0-then-what-is-the-value-of-b-49028

question:
<div class="question">If p be the length of the perpendicular from the orgin on the straight line $\mathrm{x}+2 \mathrm{by}+2 \mathrm{p}=0$, then what is the value of b?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\mathrm{p}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{3}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3}}{2}$</span> </div>

answer:
<div class="solution">Length of perpendicular from the origin on the straight line $x+2 b y-2 p=0$ is<br/>$\left|\frac{0+2 b \times 0-2 p}{\sqrt{1^{2}+(2 b)^{2}}}\right|=p$<br/>or $\mathrm{p}=\left|\frac{-2 \mathrm{p}}{\sqrt{1^{2}+4 \mathrm{~b}^{2}}}\right|$<br/>or $p^{2}=\frac{4 p^{2}}{1+4 b^{2}}$<br/>$\frac{4}{1+4 b^{2}}=1$<br/>$\Rightarrow 1+4 b^{2}=4$ or $4 b^{2}=3$<br/>$\Rightarrow \quad b^{2}=\frac{3}{4} \Rightarrow b=\pm \frac{\sqrt{3}}{2} \Rightarrow b=\frac{\sqrt{3}}{2}$ matches with the<br/>given option.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-p-be-the-length-of-the-perpendicular-from-the-origin-on-the-straight-line-ax-by-p-and-b-2-3-then-what-is-the-angle-between-the-perpendicular-and-th

slug:
if-p-be-the-length-of-the-perpendicular-from-the-origin-on-the-straight-line-ax-by-p-and-b-2-3-then-what-is-the-angle-between-the-perpendicular-and-th-62180

question:
<div class="question">If p be the length of the perpendicular from the origin on the<br/>straight line $\mathrm{ax}+\mathrm{by}=\mathrm{p}$ and $\mathrm{b}=\frac{\sqrt{3}}{2}$, then what is the angle between the perpendicular and the positive direction of $\begin{array}{ll}\text { x-axis? } &amp; {}\end{array}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$60^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$90^{\circ}$</span> </li> </ul>

answer:
<div class="solution">Equation of line is $a x+b y-p=0$, then length of perpendicular, from the origin.<br/>$p=\left|\frac{a \times 0+b \times 0-p}{\sqrt{a^{2}+b^{2}}}\right|$ or $p=\left|\frac{-p}{\sqrt{a^{2}+b^{2}}}\right|$<br/>or $\left|\frac{1}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}\right|=1$ or $\mathrm{a}^{2}+\mathrm{b}^{2}=1$<br/>$\mathrm{b}=\frac{\sqrt{3}}{2}$ or $\mathrm{b}^{2}=\frac{3}{4}$<br/>$a^{2}+\frac{3}{4}=1$<br/>$\mathrm{a}^{2}=\frac{1}{4} \Rightarrow \mathrm{a}=\frac{1}{2} \quad\left[\mathrm{a}=-\frac{1}{2}\right.$ not taken since<br/>angle is with $+$ ve direction to $\mathrm{x}$ -axis. $]$ Equation is $\frac{1}{2} x+\frac{\sqrt{3}}{2} y=p$ or $x \cos 60^{\circ}+y \sin 60^{\circ}=p$<br/>Angle $=60^{\circ}$</div>


title: if-p-cosec-cot-and-q-cosec-cot-1-then-which-one-of-the-following-is-correct
slug: if-p-cosec-cot-and-q-cosec-cot-1-then-which-one-of-the-following-is-correct-16517
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}=\operatorname{cosec} \theta-\cot \theta$ and $\mathrm{q}=(\operatorname{cosec} \theta+\cot \theta)^{-1}$, then which<br/>one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{pq}=1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}=\mathrm{q}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$p+q=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{p}+\mathrm{q}=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{p}=\mathrm{q}$</span> </div>

answer:
<div class="solution">&amp; \mathrm{p}=\operatorname{cosec} \theta-\cot \theta \\<br/>&amp; \mathrm{q}=(\operatorname{cosec} \theta+\cot \theta)^{-1} \\<br/>&amp; \Rightarrow \frac{1}{q}=\operatorname{cosec} \theta+\cot \theta \\<br/>&amp; \text { We know, } \operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1 \\<br/>&amp; \Rightarrow(\operatorname{cosec} \theta+\cot \theta)(\operatorname{cosec} \theta-\cot \theta)=1 \\<br/>&amp; \Rightarrow\left(\frac{1}{q}\right)(p)=1 \\<br/>&amp; \Rightarrow \mathrm{p}=\mathrm{q}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-p-is-the-length-of-the-perpendicular-drawn-from-the-origin-to-the-line-a-x-b-y-1-then-which-one-of-the-following-is-correct

slug:
if-p-is-the-length-of-the-perpendicular-drawn-from-the-origin-to-the-line-a-x-b-y-1-then-which-one-of-the-following-is-correct-34298

question:
<div class="question">If $p$ is the length of the perpendicular drawn from the<br/>origin to the line $\frac{x}{a}+\frac{y}{b}=1$, then which one of the following<br/>is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}-\frac{1}{\mathrm{~b}^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{p}=\frac{1}{a}+\frac{1}{b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{p}=\frac{1}{a}-\frac{1}{b}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$</span> </div>

answer:
<div class="solution">Given equation of line is $\frac{x}{a}+\frac{y}{b}=1$<br/>$\Rightarrow b x+a y-a b=0$<br/>$p=\frac{|b \cdot 0+a \cdot 0-a b|}{\sqrt{b^{2}+a^{2}}}$<br/>$p=\frac{a b}{\sqrt{b^{2}+a^{2}}}$<br/>on squaring both side, we get<br/>$p^{2}=\frac{a^{2} b^{2}}{a^{2}+b^{2}} \Rightarrow \frac{1}{p^{2}}=\frac{a^{2}+b^{2}}{a^{2} b^{2}}=\frac{1}{b^{2}}+\frac{1}{a^{2}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/IJuO6meqVobqxR7ZmIcWdaB2803bpSbvNVcQYQZeNMg.original.fullsize.png"/></div>


title: if-p-q-and-r-are-rational-numbers-then-the-roots-of-the-equation-x-2-2-p-x-p-2-q-2-2-q-r-r-2-0-are
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-p-q-and-r-are-rational-numbers-then-the-roots-of-the-equation-x-2-2-p-x-p-2-q-2-2-q-r-r-2-0-are-50467

question:
<div class="question">If $p, q$ and $r$ are rational numbers, then the roots of the equation $x^{2}-2 p x+p^{2}-q^{2}+2 q r-r^{2}=0$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">complex</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">pure imaginary</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">irrational</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">rational</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">rational</span> </div>

answer:
<div class="solution">The given equation is $x^{2}-2 p x+p^{2}-q^{2}+2 q r-r^{2}=0$ Now, discriminant $=(-2 p)^{2}-4(1)\left(p^{2}-q^{2}+2 q r-r^{2}\right)$<br/>$=4 p^{2}-4 p^{2}+4 q^{2}-8 q r+4 r^{2}$<br/>$=(2 q-2 r)^{2}=4(q-r)^{2}$<br/>which is always greater than zero. Therefore, the roots of given equation are rational</div>


title: if-p-q-and-r-are-three-noncollinear-points-then-what-is-pq-pr-equal-to
slug: if-p-q-and-r-are-three-noncollinear-points-then-what-is-pq-pr-equal-to-62108
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $P, Q$ and $R$ are three non-collinear points, then what is<br/>$P Q \cap P R$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Null set</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\{P\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\{P, Q, R\}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\{Q, R\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\{P\}$</span> </div>

answer:
<div class="solution">Since, $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are three non-collinear points.<br/>We have $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are like as<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/cEet6lyIaRTRoZOnbjceBzHDRnQAC2E9H_YgoWmmbvk.original.fullsize.png"/></div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-p-q-is-the-point-on-the-x-axis-equidistant-from-the-points-1-2-and-2-3-then-which-one-of-the-following-is-correct

slug:
if-p-q-is-the-point-on-the-x-axis-equidistant-from-the-points-1-2-and-2-3-then-which-one-of-the-following-is-correct-92664

question:
<div class="question">If $(p, q)$ is the point on the $x$ -axis equidistant from the points $(1,2)$ and $(2,3)$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p=0, q=4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$p=4, q=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$p=3 / 2, q=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$p=1, q=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p=4, q=0$</span> </div>

answer:
<div class="solution">Since $(p, q)$ is the point on the $\mathrm{x}$ -axis $\therefore q=0$<br/>Let $P=(p, 0)$<br/>$A=(1,2)$ and $B=(2,3)$<br/>Given: $P A=P B$ $\Rightarrow P A^{2}=P B^{2}$<br/>$\Rightarrow(1-p)^{2}+4=(2-p)^{2}+9$<br/>$\Rightarrow 1+p^{2}-2 p-4-p^{2}+4 p=5$<br/>$\Rightarrow 2 p=8$<br/>$\Rightarrow p=4$<br/>Hence, $p=4, q=0$</div>


title: if-p-q-r-a-b-c-0-then-the-determinant-p-a-q-c-r-b-q-b-r-a-p-c-rc-p-b-q-a-equals
slug: if-p-q-r-a-b-c-0-then-the-determinant-p-a-q-c-r-b-q-b-r-a-p-c-rc-p-b-q-a-equals-95888
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}+\mathrm{q}+\mathrm{r}=\mathrm{a}+\mathrm{b}+\mathrm{c}=0$, then the determinant $\left|\begin{array}{lll}p a &amp; q b &amp; r c \\ q c &amp; r a &amp; p b \\ r b &amp; p c &amp; q a\end{array}\right|$ equals</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{pa}+\mathrm{qb}+\mathrm{rc}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{pa}+\mathrm{qb}+\mathrm{rc}+\mathrm{a}+\mathrm{b}+\mathrm{c}$</span> </li> </ul>

answer:
<div class="solution">$\left|\begin{array}{lll}\mathrm{pa} &amp; \mathrm{qb} &amp; \mathrm{rc} \\ \mathrm{qc} &amp; \mathrm{ra} &amp; \mathrm{pb} \\ \mathrm{rb} &amp; \mathrm{pc} &amp; \mathrm{qa}\end{array}\right|=\mathrm{pa}\left(\mathrm{rqa}^{2}-\mathrm{p}^{2} \mathrm{bc}\right)-\mathrm{qb}\left(\mathrm{q}^{2} \mathrm{ac}-\mathrm{prb}^{2}\right)+$<br/>$\mathrm{rc}\left(\mathrm{qpc}^{2}-\mathrm{r}^{2} \mathrm{ab}\right)$<br/>$=\mathrm{pqra}^{3}-\mathrm{p}^{3} \mathrm{abc}-\mathrm{q}^{3} \mathrm{abc}+\mathrm{pqrb}^{3}+\mathrm{pqrc}^{3}-\mathrm{r}^{3} \mathrm{abc}$<br/>$=\mathrm{pqr}\left(\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}\right)-\mathrm{abc}\left(\mathrm{p}^{3}+\mathrm{q}^{3}+\mathrm{r}^{3}\right)$<br/>Given, $\mathrm{p}+\mathrm{q}+\mathrm{r}=\mathrm{a}+\mathrm{b}+\mathrm{c}=0$<br/>$\Rightarrow \mathrm{p}^{3}+\mathrm{q}^{3}+\mathrm{r}^{3}=3$ pqr and $\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}=3 \mathrm{abc} .$<br/>$\therefore \mathrm{pqr}\left(\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}\right)-\mathrm{abc}\left(\mathrm{p}^{3}+\mathrm{q}^{3}+\mathrm{r}^{3}\right)=\mathrm{pqr}(3 \mathrm{abc})-$<br/>$\mathrm{abc}(3 \mathrm{pqr})=0$</div>


title: if-p-q-r-are-in-ap-as-well-as-gp-then-which-one-of-the-following-is-correct
slug: if-p-q-r-are-in-ap-as-well-as-gp-then-which-one-of-the-following-is-correct-69333
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in AP as well as G.P., then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{p}=\mathrm{q} \neq \mathrm{r}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad p \neq q \neq r$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{p} \neq \mathrm{q}=\mathrm{r}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{p}=\mathrm{q}=\mathrm{r}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{p}=\mathrm{q}=\mathrm{r}$</span> </div>

answer:
<div class="solution">Since $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in A.P. <br/>$\therefore 2 \mathrm{q}=\mathrm{p}+\mathrm{r}$ ...(i)<br/>$\mathrm{q}^{2}=\mathrm{pr} \Rightarrow \mathrm{q}=\sqrt{\mathrm{pr}}$<br/>$\therefore 2 \sqrt{\mathrm{pr}}=\mathrm{p}+\mathrm{r}$<br/>$(\sqrt{\mathrm{p}})^{2}+(\sqrt{\mathrm{r}})^{2}-2 \sqrt{\mathrm{p}} \cdot \sqrt{\mathrm{r}}=0$<br/>$\Rightarrow(\sqrt{\mathrm{p}}-\sqrt{\mathrm{r}})^{2}=0$<br/>$\Rightarrow \sqrt{\mathrm{p}}=\sqrt{\mathrm{r}}$<br/>$\Rightarrow \mathrm{p}=\mathrm{r}$ ...(ii)<br/>$2 q=2 p$<br/>$\Rightarrow q=p$...(iii)<br/>from (2) and (3)<br/>$p=q=r$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-p-q-r-are-in-one-geometric-progression-and-a-b-c-are-in-another-geometric-progression-then-ap-bq-cr-are-in

slug:
if-p-q-r-are-in-one-geometric-progression-and-a-b-c-are-in-another-geometric-progression-then-ap-bq-cr-are-in-88793

question:
<div class="question">If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in one geometric progression and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in another geometric progression, then ap, bq, cr are in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Arithmetic progression</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Geometric progression</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Harmonic progression</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Geometric progression</span> </div>

answer:
<div class="solution">Let the common ratio be $\mathrm{K}_{1}$ for $\mathrm{p}, \mathrm{q}$ and $\mathrm{r}$. $\therefore \quad \mathrm{q}=\mathrm{K}_{1} \mathrm{p}$<br/>$\mathrm{\&amp;} \mathrm{r}=\left(\mathrm{K}_{1}\right)^{2} \mathrm{p}$<br/>Let the common ratio be $\mathrm{K}_{2}$ for $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$<br/>$\begin{array}{ll}\therefore &amp; \mathrm{b}=\mathrm{K}_{2} \mathrm{a} \\ \ &amp; \mathrm{c}=\left(\mathrm{K}_{2}\right)^{2} \mathrm{a} \\ \therefore &amp; \mathrm{bq}=\left(\mathrm{K}_{1} \mathrm{~K}_{2}\right) \mathrm{ap} \\ \ &amp; \mathrm{Cr}=\left(\mathrm{K}_{1} \mathrm{~K}_{2}\right)^{2} \text { ap } \\ \text { So } \mathrm{ap}, \mathrm{bq}, \mathrm{cr} \text { are in G.P. }\end{array}$</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-p-q-r-are-positive-integers-and-is-the-cube-root-of-unity-and-f-x-x-3-p-x-3-q-1-x-3-r-2-then-what-is-f-equal-to

slug:
if-p-q-r-are-positive-integers-and-is-the-cube-root-of-unity-and-f-x-x-3-p-x-3-q-1-x-3-r-2-then-what-is-f-equal-to-18348

question:
<div class="question">If $p, q, r$ are positive integers and $\omega$ is the cube root of unity and $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3 \mathrm{p}}+\mathrm{x}^{3 \mathrm{q}}+1+\mathrm{x}^{3 \mathrm{r}+2}$, then what is $\mathrm{f}(\omega)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\omega$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\omega^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\omega$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Since $\omega$ is a cube root of unity $\therefore \omega^{3}=1$ and $1+\omega+\omega^{2}=0$<br/>Let $\mathrm{f}(\mathrm{x})=\left(x^{3}\right) p+\left(x^{3}\right)^{\mathrm{q}} \cdot x+\left(x^{3}\right)^{r} \cdot x^{2}$<br/>Now, put $x=\omega$ $\mathrm{f}(\omega)=\left(\omega^{3}\right)^{p}+\left(\omega^{3}\right)^{q} . \omega+\left(\omega^{3}\right)^{r} \cdot \omega^{2}$<br/>$=1^{p}+1^{q} \cdot \omega+1^{r} . \omega^{2}=1+\omega+\omega^{2}=0$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-p-q-r-are-the-mid-points-of-the-sides-a-b-bc-c-a-respectively-of-a-triangle-a-bc-and-if-a-p-q-are-the-position-vector-of-a-p-q-respectively-then-wh

slug:
if-p-q-r-are-the-mid-points-of-the-sides-a-b-bc-c-a-respectively-of-a-triangle-a-bc-and-if-a-p-q-are-the-position-vector-of-a-p-q-respectively-then-wh-30637

question:
<div class="question">If $P, Q, R$ are the mid points of the sides $A B, B C, C A$, respectively of a triangle $A B C$ and if $\vec{a}, \vec{p}, \vec{q}$ are the position vector of $A, P, Q$ respectively, then what is the position vector of $R ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \vec{a}-(\vec{p}-\vec{q})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(\vec{p}-\vec{q})-2 \vec{a}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad \vec{a}-(\vec{p}-\vec{q})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\vec{a} / 2-(\vec{p}-\vec{q}) / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \vec{a}-(\vec{p}-\vec{q})$</span> </div>

answer:
<div class="solution">Let the position vectors of $B, C$ and $R$ are $\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$ and<br/>$\overrightarrow{\mathbf{r}}$ respectively.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/lVnKefJLn5gTAvOuRuhfC0p-SQy5xWbLC9QdJWlKT30.original.fullsize.png"/><br/><br/>$\therefore \overrightarrow{\mathrm{p}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}}{2} \Rightarrow \overrightarrow{\mathrm{b}}=2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{a}}$<br/>$\overrightarrow{\mathrm{q}}=\frac{\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}$<br/>$\Rightarrow \vec{c}=2 \vec{q}-\vec{b}$<br/>$\Rightarrow \overrightarrow{\mathrm{c}}=2 \overrightarrow{\mathrm{q}}-(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{a}})=2 \overrightarrow{\mathrm{q}}-2 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{a}}$<br/>and $\overrightarrow{\mathrm{r}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}}{2}$<br/>$=\frac{2 \overrightarrow{\mathrm{q}}-2 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{a}}}{2}=\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{a}}-(\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}})$</div>


title: if-p-sin-98-9-cos-99-1-then-which-one-of-the-following-is-correct
slug: if-p-sin-98-9-cos-99-1-then-which-one-of-the-following-is-correct-35156
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $p=\sin \left(989^{\circ}\right) \cos \left(991^{\circ}\right)$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p$ is finite and positive</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$p$ is finite and negative</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$p=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$p$ is undefined</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p$ is finite and negative</span> </div>

answer:
<div class="solution">Given, $p=\sin \left(989^{\circ}\right) \cos \left(991^{\circ}\right)$<br/>Which can be written as $=\sin \left(1080^{\circ}-91^{\circ}\right) \cos \left(1080^{\circ}-89^{\circ}\right)$<br/>$=-\sin 91^{\circ} \cos 89^{\circ}$<br/>$=-\sin \left(90^{\circ}+1^{\circ}\right) \cos 89^{\circ}$<br/>$=-\cos 1^{\circ} \cos 89^{\circ}$<br/>As $\cos 1^{\circ}$ and $\cos 89^{\circ}$ are positive. therefore their product is also $+\mathrm{ve}$ Hence, $p$ is finite and negative.</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch2_P1.db
title:
if-p-tan-6-11-q-tan-4-21-and-r-cot-6-283-then-which-of-the-following-isare-correct-1-the-value-of-p-r-is-2-2-p-q-andr-are-in-gp-select-the-correct-ans

slug:
if-p-tan-6-11-q-tan-4-21-and-r-cot-6-283-then-which-of-the-following-isare-correct-1-the-value-of-p-r-is-2-2-p-q-andr-are-in-gp-select-the-correct-ans-27460

question:
<div class="question">If $\mathrm{p}=\tan \left(-\frac{11 \pi}{6}\right), \mathrm{q}=\tan \left(\frac{21 \pi}{4}\right)$ and $\mathrm{r}=\cot \left(\frac{283 \pi}{6}\right)$,<br/>then which of the following is/are correct?<br/>1. The value of $p \times r$ is 2 .<br/>2. $\mathrm{p}, \mathrm{q}$ andr are in G.P. Select the correct answer using the code given below:</div>

answer:
<div class="solution">\begin{array}{l}<br/>\mathrm{p}=\tan \left(-\frac{11 \pi}{6}\right) \\<br/>\mathrm{p}=-\tan \left(2 \pi-\frac{\pi}{6}\right) \\<br/>\mathrm{p}=\tan \frac{\pi}{6}=\frac{1}{\sqrt{3}} \\<br/>\mathrm{q}=\tan \left(\frac{21 \pi}{4}\right)=\tan \left(6 \pi-\frac{3 \pi}{4}\right) \\<br/>\mathrm{q}=-\tan \frac{3 \pi}{4}=+\tan \frac{\pi}{4}=1 \\<br/>\mathrm{r}=\cot \left(\frac{283 \pi}{6}\right)=\cot \left(46 \pi+\frac{7 \pi}{6}\right)=\cot \left(\pi+\frac{\pi}{6}\right) \\<br/>\mathrm{r}=\cot \frac{\pi}{6}=\sqrt{3} \\<br/>\mathrm{p} \times \mathrm{r}=\frac{1}{\sqrt{3}} \times \sqrt{3}=1 \\<br/>\therefore \text { Statement(1) is incorrect. }<br/>\end{array}<br/>also <br/>$\frac{\mathrm{p}}{\mathrm{q}}=\frac{\mathrm{q}}{\mathrm{r}}=\frac{1}{\sqrt{3}}$<br/>So p, q, rare in G.P. $\therefore$ Statement $(2)$ is correct.</div>


title: if-p-th-term-of-an-ap-is-q-and-its-q-th-term-is-p-then-what-is-the-common-difference
slug: if-p-th-term-of-an-ap-is-q-and-its-q-th-term-is-p-then-what-is-the-common-difference-21247
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}^{\text {th }}$ term of an $\mathrm{AP}$ is $\mathrm{q}$, and its $\mathrm{q}^{\text {th }}$ term is $\mathrm{p}$, then what is the common difference?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Let first term and common difference of an AP are a and d respectively. Its $\mathrm{P}^{\text {th }}$ term $=\mathrm{a}+(\mathrm{p}-1) \mathrm{d}=\mathrm{q}$ ...(i) <br/>and $q^{\text {th }}$ term $=a+(q-1) d=p$ ...(ii) <br/>Solving Eqs. (i) and (ii), we find<br/>$a=p+q-1, d=-1$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-p-times-the-p-th-term-of-an-ap-is-q-times-the-q-th-term-then-what-is-the-p-q-th-term-equal-to-2010-i

slug:
if-p-times-the-p-th-term-of-an-ap-is-q-times-the-q-th-term-then-what-is-the-p-q-th-term-equal-to-2010-i-73297

question:
<div class="question">If $p$ times the $p$ th term of an AP is $q$ times the $q$ th term, then what is the $(p+q)$ th term equal to? $[2010-I]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p+q$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$p q$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let a and d be the first term and common difference of an AP respectively $\mathrm{p}^{\text {th }}$ term $=\mathrm{a}+(\mathrm{p}-1) \mathrm{d}$<br/>and $q^{\text {th }}$ term $=\mathrm{a}+(\mathrm{q}-1) \mathrm{d}$<br/>According to question. <br/>$p[a+(p-1) d]=q[a+(q-1) d]$<br/>$\Rightarrow p a+\left(p^{2}-p\right) d=q a+\left(q^{2}-q\right) d$<br/>$\Rightarrow(p-q) a=\left(q^{2}-p^{2}+p-q\right) d$<br/>$\Rightarrow(p-q) a=(p-q)(-p-q+1) d$<br/>$\Rightarrow a=-(p+q-1) d$<br/>Now, $(p+q)^{t h}$ term $=\mathrm{a}+(\mathrm{p}+\mathrm{q}-1) d$<br/>$\quad=-(p+q-1) d+(p+q-1) d=0$</div>


title: if-p-x-cos-y-sin-q-x-sin-y-cos-and-p-2-4-pq-q-2-a-x-2-by-2-0-0-2-what-is-the-value-of-a
slug: if-p-x-cos-y-sin-q-x-sin-y-cos-and-p-2-4-pq-q-2-a-x-2-by-2-0-0-2-what-is-the-value-of-a-28563
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}=\mathrm{X} \cos \theta-\mathrm{Y} \sin \theta, \mathrm{q}=\mathrm{X} \sin \theta+\mathrm{Y} \cos \theta$ and $\mathrm{p}^{2}+4 \mathrm{pq}+\mathrm{q}^{2}=$<br/>$\mathrm{A} \mathrm{X}^{2}+\mathrm{BY}^{2}, 0 \leq 0 \leq \frac{\pi}{2}$<br/>What is the value of A?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">\quad \begin{array}{l}\mathrm{p}=\mathrm{x} \cos \theta-\mathrm{y} \sin \theta \\ \mathrm{q}=\mathrm{x} \sin \theta+\mathrm{y} \cos \theta \\ \text { Given, } \mathrm{p}^{2}+4 \mathrm{pq}+\mathrm{q}^{2}=\mathrm{Ax}^{2}+\mathrm{By}^{2}\end{array} \end{aligned}$<br/>Let us take $\theta=\frac{\pi}{4}$.<br/>$p=x \cos \frac{\pi}{4}-y \cos \frac{\pi}{4}=\frac{x-y}{\sqrt{2}}$<br/>$$<br/>\begin{array}{l}<br/>q=x \sin \frac{\pi}{4}+y \cos \frac{\pi}{4}=\frac{x+y}{\sqrt{2}} \\<br/>p q=\frac{x^{2}-y^{2}}{2} \Rightarrow 2 p q=x^{2}-y^{2} \\<br/>\Rightarrow 4 \mathrm{pq}=2 \mathrm{x}^{2}-2 \mathrm{y}^{2} \\<br/>\text { Now, } \mathrm{p}^{2}+\mathrm{q}^{2}=\mathrm{x}^{2} \cos ^{2} \theta+\mathrm{y}^{2} \sin ^{2} \theta-2 \mathrm{xy} \cos \theta \sin \theta \\<br/>+\mathrm{x}^{2} \sin ^{2} \theta+\mathrm{y}^{2} \cos ^{2} \theta+2 \mathrm{x} \mathrm{sin} \theta \cos \theta=\mathrm{x}^{2}+\mathrm{y}^{2} \\<br/>\text { From }(1),(2), \mathrm{p}^{2}+\mathrm{q}^{2}+4 \mathrm{pq}=\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{x}^{2}-2 \mathrm{y}^{2} \\<br/>=3 \mathrm{x}^{2}-\mathrm{y}^{2}<br/>\end{array}<br/>$$<br/>Comparing this with the given form, we get<br/>$$<br/>\theta=\frac{\pi}{4}, A=3, B=-1<br/>$$</div>


title: if-p-x-cos-y-sin-q-x-sin-y-cos-and-p-2-4-pq-q-2-a-x-2-by-2-0-0-2-what-is-the-value-of-b
slug: if-p-x-cos-y-sin-q-x-sin-y-cos-and-p-2-4-pq-q-2-a-x-2-by-2-0-0-2-what-is-the-value-of-b-87325
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}=\mathrm{X} \cos \theta-\mathrm{Y} \sin \theta, \mathrm{q}=\mathrm{X} \sin \theta+\mathrm{Y} \cos \theta$ and $\mathrm{p}^{2}+4 \mathrm{pq}+\mathrm{q}^{2}=$<br/>$\mathrm{A} \mathrm{X}^{2}+\mathrm{BY}^{2}, 0 \leq 0 \leq \frac{\pi}{2}$<br/>What is the value of B ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">\quad \begin{array}{l}\mathrm{p}=\mathrm{x} \cos \theta-\mathrm{y} \sin \theta \\ \mathrm{q}=\mathrm{x} \sin \theta+\mathrm{y} \cos \theta \\ \text { Given, } \mathrm{p}^{2}+4 \mathrm{pq}+\mathrm{q}^{2}=\mathrm{Ax}^{2}+\mathrm{By}^{2}\end{array} \end{aligned}$<br/>Let us take $\theta=\frac{\pi}{4}$.<br/>$p=x \cos \frac{\pi}{4}-y \cos \frac{\pi}{4}=\frac{x-y}{\sqrt{2}}$<br/>$$<br/>\begin{array}{l}<br/>q=x \sin \frac{\pi}{4}+y \cos \frac{\pi}{4}=\frac{x+y}{\sqrt{2}} \\<br/>p q=\frac{x^{2}-y^{2}}{2} \Rightarrow 2 p q=x^{2}-y^{2} \\<br/>\Rightarrow 4 \mathrm{pq}=2 \mathrm{x}^{2}-2 \mathrm{y}^{2} \\<br/>\text { Now, } \mathrm{p}^{2}+\mathrm{q}^{2}=\mathrm{x}^{2} \cos ^{2} \theta+\mathrm{y}^{2} \sin ^{2} \theta-2 \mathrm{xy} \cos \theta \sin \theta \\<br/>+\mathrm{x}^{2} \sin ^{2} \theta+\mathrm{y}^{2} \cos ^{2} \theta+2 \mathrm{x} \mathrm{sin} \theta \cos \theta=\mathrm{x}^{2}+\mathrm{y}^{2} \\<br/>\text { From }(1),(2), \mathrm{p}^{2}+\mathrm{q}^{2}+4 \mathrm{pq}=\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{x}^{2}-2 \mathrm{y}^{2} \\<br/>=3 \mathrm{x}^{2}-\mathrm{y}^{2}<br/>\end{array}<br/>$$<br/>Comparing this with the given form, we get<br/>$$<br/>\theta=\frac{\pi}{4}, A=3, B=-1<br/>$$</div>


title: if-p-x-cos-y-sin-q-x-sin-y-cos-and-p-2-4-pq-q-2-a-x-2-by-2-0-0-2-what-is-the-value-of-theta
slug: if-p-x-cos-y-sin-q-x-sin-y-cos-and-p-2-4-pq-q-2-a-x-2-by-2-0-0-2-what-is-the-value-of-theta-82292
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}=\mathrm{X} \cos \theta-\mathrm{Y} \sin \theta, \mathrm{q}=\mathrm{X} \sin \theta+\mathrm{Y} \cos \theta$ and $\mathrm{p}^{2}+4 \mathrm{pq}+\mathrm{q}^{2}=$<br/>$\mathrm{A} \mathrm{X}^{2}+\mathrm{BY}^{2}, 0 \leq 0 \leq \frac{\pi}{2}$<br/>What is the value of  \theta</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{4}$</span> </div>

answer:
<div class="solution">\quad \begin{array}{l}\mathrm{p}=\mathrm{x} \cos \theta-\mathrm{y} \sin \theta \\ \mathrm{q}=\mathrm{x} \sin \theta+\mathrm{y} \cos \theta \\ \text { Given, } \mathrm{p}^{2}+4 \mathrm{pq}+\mathrm{q}^{2}=\mathrm{Ax}^{2}+\mathrm{By}^{2}\end{array} \end{aligned}$<br/>Let us take $\theta=\frac{\pi}{4}$.<br/>$p=x \cos \frac{\pi}{4}-y \cos \frac{\pi}{4}=\frac{x-y}{\sqrt{2}}$<br/>$$<br/>\begin{array}{l}<br/>q=x \sin \frac{\pi}{4}+y \cos \frac{\pi}{4}=\frac{x+y}{\sqrt{2}} \\<br/>p q=\frac{x^{2}-y^{2}}{2} \Rightarrow 2 p q=x^{2}-y^{2} \\<br/>\Rightarrow 4 \mathrm{pq}=2 \mathrm{x}^{2}-2 \mathrm{y}^{2} \\<br/>\text { Now, } \mathrm{p}^{2}+\mathrm{q}^{2}=\mathrm{x}^{2} \cos ^{2} \theta+\mathrm{y}^{2} \sin ^{2} \theta-2 \mathrm{xy} \cos \theta \sin \theta \\<br/>+\mathrm{x}^{2} \sin ^{2} \theta+\mathrm{y}^{2} \cos ^{2} \theta+2 \mathrm{x} \mathrm{sin} \theta \cos \theta=\mathrm{x}^{2}+\mathrm{y}^{2} \\<br/>\text { From }(1),(2), \mathrm{p}^{2}+\mathrm{q}^{2}+4 \mathrm{pq}=\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{x}^{2}-2 \mathrm{y}^{2} \\<br/>=3 \mathrm{x}^{2}-\mathrm{y}^{2}<br/>\end{array}<br/>$$<br/>Comparing this with the given form, we get<br/>$$<br/>\theta=\frac{\pi}{4}, A=3, B=-1<br/>$$</div>


tags: ['Physics', 'Units and Dimensions']
source_db: MarksBatch2_P1.db
title:
if-plancks-constant-h-and-speed-of-light-in-vacuum-c-are-taken-as-two-fundamental-quantities-which-one-of-the-following-can-in-addition-be-taken-to-ex

slug:
if-plancks-constant-h-and-speed-of-light-in-vacuum-c-are-taken-as-two-fundamental-quantities-which-one-of-the-following-can-in-addition-be-taken-to-ex-35582

question:
<div class="question">If Planck's constant $(h)$ and speed of light in vacuum $(c)$ are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>Mass of electron $\left(m_e\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Universal gravitational constant $(G)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Charge of electron $(e)$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>Mass of proton $\left(m_p\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>Mass of electron $\left(m_e\right)$<br/>, <br/>Universal gravitational constant $(G)$<br/>, <br/>Mass of proton $\left(m_p\right)$</span> </div>

answer:
<div class="solution">As we know that dimension of $[h]$<br/>$$<br/>=\frac{E}{v}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]<br/>$$<br/>Dimension of $[c]=\frac{s}{t}=\frac{[\mathrm{L}]}{[\mathrm{T}]}$ $=\left[\mathrm{LT}^{-1}\right]$<br/>Dimension of mass of electron $\left[m_e\right]=\mathrm{M}$<br/>$$<br/>\begin{aligned}<br/>&amp;{[G]=\frac{F r^2}{M_1 M_2}=\frac{\left[\mathrm{ML}^3 \mathrm{~T}^{-2}\right]}{[\mathrm{M}][\mathrm{M}]}=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]} \\<br/>&amp;{[e]=[A T]}<br/>\end{aligned}<br/>$$<br/>Dimension of mass of proton $\left[m_p\right]=[\mathbf{M}]$<br/>$$<br/>\begin{aligned}<br/>&amp;{\left[\frac{h c}{G}\right]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]\left[\mathrm{LT}^{-1}\right]}{\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]}=\left[\mathrm{M}^2\right]} \\<br/>&amp;\Rightarrow M=\sqrt{\frac{h c}{G}}<br/>\end{aligned}<br/>$$<br/>Similarly,<br/>$$<br/>\frac{h}{c}=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]}{\left[\mathrm{LT}^{-1}\right]}=[\mathrm{ML}]<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp;L=\frac{h}{c M}=\frac{h}{c} \sqrt{\frac{G}{h c}}=\frac{\sqrt{G h}}{c^{3 / 2}} \\<br/>&amp;\text { As, } c=\left[\mathrm{LT}^{-1}\right] \\<br/>&amp;\Rightarrow[T]=\frac{[L]}{[c]}=\frac{\sqrt{G h}}{c^{3 / 2} \cdot c}=\frac{\sqrt{G h}}{c^{5 / 2}}<br/>\end{aligned}<br/>$$<br/>Hence physical quantities (a), (b) or (d) any can be used to represent L, M and T in terms of three chosen fundamental quantities.</div>


tags: ['Physics', 'Work Power Energy', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">25</span> </div>
source_db: MarksBatch2_P1.db
title:
if-potential-energy-function-for-the-force-between-two-atoms-in-a-diatomic-molecule-is-approximately-given-by-u-x-a-x-8-b-x-4-where-a-and-b-are-consta

slug:
if-potential-energy-function-for-the-force-between-two-atoms-in-a-diatomic-molecule-is-approximately-given-by-u-x-a-x-8-b-x-4-where-a-and-b-are-consta-82571

question:
<div class="question">If potential energy function for the force between two atoms in a diatomic molecule is approximately given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mfenced separators="|"><mi>x</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">a</mi><msup><mi>x</mi><mn>8</mn></msup></mfrac><mo>-</mo><mfrac><mi mathvariant="normal">b</mi><msup><mi>x</mi><mn>4</mn></msup></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi></math> are constants in standard SI units and <math><mi>x</mi></math> is in metres. Find the dissociation energy of the molecule (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">J</mi></math>). [Take <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>=</mo><mn>4</mn><mo> </mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mn>8</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi><mo>=</mo><mn>20</mn><mo> </mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mn>4</mn></msup></math>]</div>

answer:
<div class="solution"><math><mi>U</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>8</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>20</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac></math><br/><math><mo>â´</mo><mi> </mi><mfrac><mrow><mi>d</mi><mi>U</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mo>â</mo><mi> </mi><mi> </mi><mfrac><mrow><mo>-</mo><mn>8</mn><mo>Ã</mo><mn>4</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>9</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>20</mn><mo>Ã</mo><mn>4</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></mfrac></math><br/><math><mo>â</mo><mi> </mi><mi> </mi><mfrac><mrow><mn>4</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></mfrac><mfenced separators="|"><mrow><mn>20</mn><mo>-</mo><mfrac><mrow><mn>8</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn></math><br/><math><mi>x</mi><mo>=</mo><mi>â</mi></math> and <math><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>=</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>20</mn></mrow></mfrac></math><br/>Also <math><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>U</mi></mrow><mrow><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mo>+</mo><mi mathvariant="normal">v</mi><mi mathvariant="normal">e</mi><mi> </mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi> </mi><mi> </mi><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>=</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>20</mn></mrow></mfrac></math><br/><math><mo>â´</mo><mi> </mi><mi> </mi><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac><mo>-</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><mo>Ã</mo><mn>20</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mn>20</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>-</mo><mn>5</mn></mrow></mfenced></math><br/><math><mo>â</mo><mi> </mi><mi> </mi><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mn>4</mn><mo>Ã</mo><mn>20</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>Ã</mo><mfrac><mrow><mn>20</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>=</mo><mo>-</mo><mn>25</mn><mi> </mi><mi> </mi><mi mathvariant="normal">J</mi></math><br/><math><mo>â´</mo></math> Dissociation energy <math><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi></mrow></msub><mo>-</mo><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn>25</mn><mi> </mi><mi mathvariant="normal">J</mi></math></div>


tags: ['Physics', 'Units and Dimensions', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-pressure-p-velocity-v-and-time-t-are-taken-as-fundamental-physical-quantities-the-dimensional-formula-of-force-is-1

slug:
if-pressure-p-velocity-v-and-time-t-are-taken-as-fundamental-physical-quantities-the-dimensional-formula-of-force-is-1-29527

question:
<div class="question">If pressure <math><mi>P</mi></math>, velocity <math><mi>V</mi></math> and time <math><mi>T</mi></math> are taken as fundamental physical quantities, the dimensional formula of force is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mi>P</mi><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><msup><mrow><mi>P</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mi>P</mi><mi>V</mi><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><msup><mrow><mi>P</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>V</mi><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mi>P</mi><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> </div>

answer:
<div class="solution">Let <math><mi>F</mi><mo>∝</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>x</mi></mrow></msup><msup><mrow><mi>V</mi></mrow><mrow><mi>y</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mi>z</mi></mrow></msup></math><br/><br/>by substituting the following dimensions :<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>P</mi><mo>]</mo><mo>=</mo><mo>[</mo><msup><mi>ML</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>]</mo><mo>, </mo><mo>[</mo><mi>V</mi><mo>]</mo><mo>=</mo><mo>[</mo><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>,</mo><mi mathvariant="normal"> </mi><mo>[</mo><mi>T</mi><mo>]</mo><mo>=</mo><mo>[</mo><mi mathvariant="normal">T</mi><mo>]</mo></math><br/><br/>and comparing the dimension of both sides<br/><br/><math><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi>z</mi><mo>=</mo><mn>2</mn></math> <math><mi>F</mi><mo>=</mo><mi>P</mi><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-pressure-p-velocity-v-and-time-t-are-taken-as-fundamental-physical-quantities-the-dimensional-formula-of-force-is

slug:
if-pressure-p-velocity-v-and-time-t-are-taken-as-fundamental-physical-quantities-the-dimensional-formula-of-force-is-18722

question:
<div class="question">If pressure <math> <mi>P</mi></math>, velocity <math> <mi>V</mi> </math> and time <math> <mi>T</mi> </math> are taken as fundamental physical quantities, the dimensional formula of force is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <math> <mrow> <mi>P</mi><msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </math> </span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <math> <mrow> <msup> <mi>P</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> </mrow> </math> </span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <math> <mrow> <mi>P</mi><mi>V</mi><msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </math> </span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <math> <mrow> <msup> <mi>P</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <mi>V</mi><msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </math> </span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <math> <mrow> <mi>P</mi><msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </math> </span> </div>

answer:
<div class="solution">Let  <math> <mrow> <mi>F</mi><mo>â</mo><msup> <mi>P</mi> <mi>x</mi> </msup> <msup> <mi>V</mi> <mi>y</mi> </msup> <msup> <mi>T</mi> <mi>z</mi> </msup> </mrow> </math>  by substituting the following dimensions :  <math> <mrow> <mo stretchy="false">[</mo><mi>P</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>M</mi><msup> <mi>L</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <msup> <mi>T</mi> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> <mo stretchy="false">]</mo></mrow> </math> <math> <mrow> <mo stretchy="false">[</mo><mi>V</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>L</mi><msup> <mi>T</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <mo stretchy="false">]</mo><mo>,</mo><mtext> </mtext><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow> </math>  and comparing the dimension of both sides  <math> <mrow> <mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mtext> </mtext><mi>z</mi><mo>=</mo><mn>2</mn></mrow> </math> , so  <math> <mrow> <mi>F</mi><mo>=</mo><mi>P</mi><msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </math> </div>


title: if-progesterone-receptors-are-disrupted-then
slug: if-progesterone-receptors-are-disrupted-then-65372
question: <div class="question">If progesterone receptors are disrupted then</div>
tags: ['Biology', 'Chemical Coordination and Integration', 'NEET']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">secondary sexual characters would not be maintained.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">pituitary gonadotropins will not be secreted.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">the uterus will not be able to support pregnancy.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the uterine endometrium will not begin to proliferate.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the uterus will not be able to support pregnancy.</span> </div>

answer:
<div class="solution">After zygote formation, embryonic development begins. This includes cleavage, morula formation, blastula formation, implantation, gastrulation and organogenesis. Implantation happens on 8<sup>th</sup> day after fertilisation and this is considered to be the beginning of pregnancy. This continues till parturition begins. This is maintained because the shedding of the endometrium is prevented as progesterone and estrogen levels are maintained and this is due to secretion of human chorionic gonadotrophin. Thus if progesterone receptors are disrupted then uterus will not be able to support pregnancy as it will not respond to progesterone.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-px-2-qx-r-p-x-x-and-p-3-pq-r-0-p-q-and-r-being-real-numbers-then-which-of-the-following-is-not-possible

slug:
if-px-2-qx-r-p-x-x-and-p-3-pq-r-0-p-q-and-r-being-real-numbers-then-which-of-the-following-is-not-possible-22974

question:
<div class="question">If $\mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}=\mathrm{p}(\mathrm{x}-\alpha)(\mathrm{x}-\beta)$, and $\mathrm{p}^{3}+\mathrm{pq}+\mathrm{r}=0 ; \mathrm{p}, \mathrm{q}$ and r being real numbers, then which of the following is not possible?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\alpha=\beta=\mathrm{p}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\alpha \neq \beta=\mathrm{p}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha=\beta \neq \mathrm{p}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\beta \neq \alpha=\mathrm{p}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha=\beta=\mathrm{p}$</span> </div>

answer:
<div class="solution">Given equation is $\mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}=\mathrm{p}(\mathrm{x}-\alpha)(\mathrm{x}-\beta)$<br/>$\Rightarrow p x^{2}+q x+r=p x^{2}-p(\alpha+\beta) x+\alpha \beta p$<br/>$\Rightarrow \alpha \beta \mathrm{p}=\mathrm{r}$ and $\mathrm{q}=-(\alpha+\beta) \mathrm{p}$...(1)<br/>Also given that $\mathrm{p}^{3}+\mathrm{pq}+\mathrm{r}=0$<br/>Putting value of $q$ and $r$ from $(1)$ $\Rightarrow p^{3}-p^{2}(\alpha+\beta)+\alpha \beta p=0$<br/>$\Rightarrow \mathrm{p}^{2}-\mathrm{p}(\alpha+\beta)+\alpha \beta=0$<br/>$\Rightarrow(p-\alpha)(p-\beta)=0 \Rightarrow \alpha=\beta=p$</div>


tags: ['Mathematics', 'Matrices', 'AP EAMCET', 'AP EAMCET 2020 (18 Sep Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
title:
if-q-is-the-inverse-of-a-when-aleftbeginarrayccc1-1-1-2-1-3-1-1-1endarrayright-and-10-times-qleftbeginarrayccc4-2-2-5-0-x-1-2-3endarrayright-find-x

slug:
if-q-is-the-inverse-of-a-when-aleftbeginarrayccc1-1-1-2-1-3-1-1-1endarrayright-and-10-times-qleftbeginarrayccc4-2-2-5-0-x-1-2-3endarrayright-find-x-19502

question:
<div class="question">If \(Q\) is the inverse of \(A\), when \(A=\left[\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; 1 &amp; -3 \\ 1 &amp; 1 &amp; 1\end{array}\right]\) and \(10 \times Q=\left[\begin{array}{ccc}4 &amp; 2 &amp; 2 \\ -5 &amp; 0 &amp; x \\ 1 &amp; -2 &amp; 3\end{array}\right]\), find \(x=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, \(Q=A^{-1}\)<br/>\(\begin{aligned}<br/>A &amp; =\left[\begin{array}{ccc}<br/>1 &amp; -1 &amp; 1 \\<br/>2 &amp; 1 &amp; -3 \\<br/>1 &amp; 1 &amp; 1<br/>\end{array}\right] \\<br/>|A| &amp; =1(1+3)+1(2+3)+1(2-1)=4+5+1 \\<br/>|A| &amp; =10<br/>\end{aligned}\)<br/>\(\operatorname{Adj} A=\left[\begin{array}{ccc}4 &amp; -5 &amp; 1 \\ 2 &amp; 0 &amp; -2 \\ 2 &amp; 5 &amp; 3\end{array}\right]^{\mathrm{T}}\)<br/>Adj \(A=\left[\begin{array}{ccc}4 &amp; 2 &amp; 2 \\ -5 &amp; 0 &amp; 5 \\ 1 &amp; -2 &amp; 3\end{array}\right]\)<br/>\(\begin{aligned}<br/>&amp; A^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{Adj} A \\<br/>&amp; A^{-1}=\frac{1}{10}\left[\begin{array}{rrr}<br/>4 &amp; 2 &amp; 2 \\<br/>-5 &amp; 0 &amp; 5 \\<br/>1 &amp; -2 &amp; 3<br/>\end{array}\right] \\<br/>&amp; 10 Q=\left[\begin{array}{rrr}<br/>4 &amp; 2 &amp; 2 \\<br/>-5 &amp; 0 &amp; x \\<br/>1 &amp; -2 &amp; 3<br/>\end{array}\right] \\<br/>&amp; 10 A^{-1}=\left[\begin{array}{rrr}<br/>4 &amp; 2 &amp; 2 \\<br/>-5 &amp; 0 &amp; x \\<br/>1 &amp; -2 &amp; 3<br/>\end{array}\right] \\<br/>&amp; {\left[\begin{array}{rrr}<br/>4 &amp; 2 &amp; 2 \\<br/>-5 &amp; 0 &amp; 5 \\<br/>1 &amp; -2 &amp; 3<br/>\end{array}\right]=\left[\begin{array}{rrr}<br/>4 &amp; 2 &amp; 2 \\<br/>-5 &amp; 0 &amp; x \\<br/>1 &amp; -2 &amp; 3<br/>\end{array}\right]} \\<br/>&amp; \therefore \quad x=5 \\<br/>\end{aligned}\)<br/>Hence, option (d) is correct.</div>


title: if-r-1-i-2-j-k-r-2-i-2-j-2-k-are-such-that-r-1-r-2-then-satisfies-which-one-of-the-following
slug: if-r-1-i-2-j-k-r-2-i-2-j-2-k-are-such-that-r-1-r-2-then-satisfies-which-one-of-the-following-24122
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{r}_{1}=\lambda \hat{i}+2 \hat{j}+\hat{k}, \vec{r}_{2}=\hat{i}+(2-\lambda) \hat{j}+2 \hat{k}$ are such that<br/>$\left|\vec{r}_{1}\right|&gt;\left|\vec{r}_{2}\right|$, then $\lambda$ satisfies which one of the following?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\lambda=0$ only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \lambda=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\lambda   &lt; 1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\lambda \geq 1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\lambda \geq 1$</span> </div>

answer:
<div class="solution">Given, $\overrightarrow{\mathbf{r}}_{1}=\lambda \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$<br/>$\begin{aligned} &amp; \text { and } \overrightarrow{\mathbf{r}}_{1}=\hat{\mathbf{i}}+(2-\lambda) \hat{\mathbf{j}}+2 \hat{\mathbf{k}} \\ &amp; \therefore\left|\overrightarrow{\mathbf{r}}_{1}\right|&gt;\left|\overrightarrow{\mathbf{r}}_{2}\right| \\ \Rightarrow &amp; \sqrt{\lambda^{2}+(2)^{2}+(1)^{2}}&gt;\sqrt{(1)^{2}+(2-\lambda)^{2}+(2)^{2}} \\ \Rightarrow &amp; \lambda^{2}+4+1&gt;1+4+\lambda^{2}-4 \lambda+4 \\ \Rightarrow &amp; 5&gt;9-4 \lambda \\ \Rightarrow &amp; 4 \lambda&gt;4 \\ \Rightarrow &amp; \lambda&gt;1 \end{aligned}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-r-1-r-2-r-3-are-the-position-vectors-of-three-collinear-points-and-scalars-m-and-n-exist-such-that-r-3-m-r-1-n-r-2-then-what-is-the-value-of-m-n

slug:
if-r-1-r-2-r-3-are-the-position-vectors-of-three-collinear-points-and-scalars-m-and-n-exist-such-that-r-3-m-r-1-n-r-2-then-what-is-the-value-of-m-n-31735

question:
<div class="question">If $\overrightarrow{\mathrm{r}}_{1}, \overrightarrow{\mathrm{r}_{2}}, \overrightarrow{\mathrm{r}_{3}}$ are the position vectors of three collinear points<br/>and scalars $\mathrm{m}$ and $\mathrm{n}$ exist such that $\overrightarrow{\mathrm{r}}_{3}=\mathrm{m} \overrightarrow{\mathrm{r}}_{1}+\mathrm{n} \overrightarrow{\mathrm{r}}_{2}$, then<br/>what is the value of $(\mathrm{m}+\mathrm{n})$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Since $\overrightarrow{\mathrm{r}}_{1}, \overrightarrow{\mathrm{r}}_{2}$ and $\overrightarrow{\mathrm{r}}_{3}$ are the position vector of three collinear points. Thus $\vec{r}_{3}$ is the position vector of the point which divides the joining of points whose<br/>position vectors are $\overrightarrow{\mathrm{r}}_{1}$ and $\overrightarrow{\mathrm{r}}_{2}$ in the ratio $\mathrm{m}: \mathrm{n}$.<br/>So, $\vec{r}_{3}=\frac{\mathrm{mr}_{1}+\mathrm{n} \overrightarrow{\mathrm{r}}_{2}}{\mathrm{~m}+\mathrm{n}}$<br/>But as given, $\overrightarrow{\mathrm{r}}_{3}=\mathrm{mr}_{1}+\overrightarrow{\mathrm{n}} \overrightarrow{\mathrm{r}}_{2}$<br/>So, $\frac{\overline{\mathrm{mr}_{1}}+\overline{\mathrm{nr}_{2}}}{\mathrm{~m}+\mathrm{n}}=\mathrm{mr}_{1}+\mathrm{nr}_{2}$<br/>$\Rightarrow \mathrm{m}+\mathrm{n}=1$</div>


title: if-r-13-r-13-1-3-for-a-real-number-r-0-then-what-is-r-1-equal-to
slug: if-r-13-r-13-1-3-for-a-real-number-r-0-then-what-is-r-1-equal-to-85833
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">18</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{r}^{1 / 3}+\frac{1}{r^{1 / 3}}=3$ for a real number $\mathrm{r} \neq 0$, then what is<br/>$\mathrm{r}+1$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">27</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">36</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">18</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given equation is<br/>$\mathrm{r}^{1 / 3}+\frac{1}{\mathrm{r}^{1 / 3}}=3$<br/>Cubing both sides, we get<br/>$\left(r^{1 / 3}+\frac{1}{r^{1 / 3}}\right)^{3}=3^{3}\left[(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\right]$<br/>$\Rightarrow \mathrm{r}+\frac{1}{\mathrm{r}}+3\left(\mathrm{r}^{1 / 3}+\frac{1}{\mathrm{r}^{1 / 3}}\right)=27$<br/>$\Rightarrow \mathrm{r}+\frac{1}{\mathrm{r}}+3.3=27 \Rightarrow \mathrm{r}+\frac{1}{\mathrm{r}}+27-9=18$</div>


title: if-r-and-s-are-roots-of-x-2-p-x-q-0-then-what-is-the-value-of-1-r-2-1-s-2
slug: if-r-and-s-are-roots-of-x-2-p-x-q-0-then-what-is-the-value-of-1-r-2-1-s-2-85783
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $r$ and $s$ are roots of $x^{2}+p x+q=0$, then what is the value $\operatorname{of}\left(1 / r^{2}\right)+\left(1 / s^{2}\right) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p^{2}-4 q$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{p^{2}-4 q}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{p^{2}-4 q}{q^{2}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{p^{2}-2 q}{q^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{p^{2}-2 q}{q^{2}}$</span> </div>

answer:
<div class="solution">Since $r$ and $s$ are the roots of $x^{2}+p x+q=0$.<br/>Then, $r+s=-p$ and $r s=q$ Now, $\frac{1}{r^{2}}+\frac{1}{s^{2}}=\frac{r^{2}+s^{2}}{(r s)^{2}}=\frac{(r+s)^{2}-2 r s}{(r s)^{2}}$<br/>$=\frac{(-p)^{2}-2 q}{q^{2}}=\frac{p^{2}-2 q}{q^{2}}$</div>


tags: ['Mathematics', 'Sets and Relations']
source_db: MarksBatch2_P1.db
title:
if-r-be-the-set-of-points-inside-a-rectangle-of-sides-a-and-b-a-b-1-with-two-sides-along-the-positive-direction-of-x-axis-and-y-axis-then

slug:
if-r-be-the-set-of-points-inside-a-rectangle-of-sides-a-and-b-a-b-1-with-two-sides-along-the-positive-direction-of-x-axis-and-y-axis-then-74202

question:
<div class="question">If $\mathrm{R}$ be the set of points inside a rectangle of sides a and $b$ $(\mathrm{a}, \mathrm{b}&gt;1)$ with two sides along the positive direction of $\mathrm{X}$-axis and $\mathrm{Y}$-axis. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{R}=\{(x, y): 0 \leq x \leq a, 0 \leq y \leq b\}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{R}=\{(x, y): 0 \leq x &lt; a, 0 \leq y \leq b\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{R}=\{(x, y): 0 \leq x \leq a, 0 &lt; y &lt; b\}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{R}=\{(x, y): 0 &lt; x &lt; a, 0 &lt; y &lt; b\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{R}=\{(x, y): 0 &lt; x &lt; a, 0 &lt; y &lt; b\}$</span> </div>

answer:
<div class="solution">Here, $R$ be the set of points inside a rectangle of sides a and b. So, $\mathrm{R}=\{(x, y): 0 &lt; x &lt; a$ and $0 &lt; y &lt; b\}$<img src="https://cdn-question-pool.getmarks.app/modules/cbse/RP_BbZox8LZJXnuHJD077WKIFASTanDAhSXOmDJFpng.original.fullsize.png"/><br/></div>


title: if-r-x-i-y-j-z-k-then-what-is-r-i-j-k-equal-to
slug: if-r-x-i-y-j-z-k-then-what-is-r-i-j-k-equal-to-83237
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{r}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$, then what is $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{x}+\mathrm{y}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-(x+y+z)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(x+y+z)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(x+y+z)$</span> </div>

answer:
<div class="solution">$\quad(x \hat{i}+y \hat{j}+z \hat{k}) \cdot(\hat{i}+\hat{j}+\hat{k})=x+y+z$</div>


title: if-re-z-1-z-1-0-where-z-x-iy-is-a-complex-number-then-which-one-of-the-following-is-correct
slug: if-re-z-1-z-1-0-where-z-x-iy-is-a-complex-number-then-which-one-of-the-following-is-correct-95468
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\operatorname{Re}\left(\frac{z-1}{z+1}\right)=0$, where $z=x+$ iy is a complex number, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$z=1+i$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|z|=2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$z=1-i$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$|z|=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|z|=1$</span> </div>

answer:
<div class="solution">$\frac{z-1}{z+1}=\frac{x+i y-1}{x+i y+1}$<br/>$\frac{z-1}{z+1}=\frac{x^{2}+y^{2}-1+2 i y}{x^{2}+y^{2}+2 x+1}$<br/>$\Rightarrow \operatorname{Re}\left(\frac{z-1}{z+1}\right)=\frac{x^{2}+y^{2}-1}{x^{2}+y^{2}+2 x+1}=0$<br/>$\Rightarrow x^{2}+y^{2}-1=0$<br/>$\Rightarrow x^{2}+y^{2}=1$<br/>Also, $z \bar{z}=x^{2}+y^{2}=1$<br/>and $z \bar{z}=|z|^{2}$<br/>$\Rightarrow|z|^{2}=1$<br/>$\Rightarrow|z|=1$</div>


title: if-roots-of-an-equation-a-x-2-b-x-c-0-are-positive-then-which-one-of-the-following-is-correct
slug: if-roots-of-an-equation-a-x-2-b-x-c-0-are-positive-then-which-one-of-the-following-is-correct-59492
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If roots of an equation $a x^{2}+b x+c=0$ are positive, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Signs of a and $c$ should be like</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Signs of $\mathrm{b}$ and $\mathrm{c}$ should be like</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Signs of a and b should be like</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Signs of a and $c$ should be like</span> </div>

answer:
<div class="solution">For roots of an equation $a x^{2}+b x+c=0$ to be positive sign of a and c should be like.</div>


title: if-s-i-n-x-y-s-i-n-x-y-a-b-a-b-then-what-is-t-a-n-y-t-a-n-x-equal-to-1
slug: if-s-i-n-x-y-s-i-n-x-y-a-b-a-b-then-what-is-t-a-n-y-t-a-n-x-equal-to-1-82426
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}$, then what is $\frac{\tan x}{\tan y}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{a}}{\mathrm{b}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{b}}{\mathrm{a}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{a+b}{a-b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{a-b}{a+b}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{a}}{\mathrm{b}}$</span> </div>

answer:
<div class="solution">\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}<br/>\end{array}\\<br/>\text { Applying componendo and dividendo, we get }\\<br/>\frac{\sin (x+y)+\sin (x-y)}{\sin (x+y)-\sin (x-y)}=\frac{a+b+a-b}{a+b-a+b}<br/>\end{array}<br/>$$<br/>$\Rightarrow \frac{2 \sin \left(\frac{x+y+x-y}{2}\right) \cos \left(\frac{x+y-x+y}{2}\right)}{2 \cos \left(\frac{x+y+x-y}{2}\right) \sin \left(\frac{x+y-x+y}{2}\right)}=\frac{2 a}{2 b}$<br/>$\Rightarrow \frac{\sin x \cdot \cos y}{\cos x \cdot \sin y}=\frac{\mathrm{a}}{\mathrm{b}}$<br/>$\Rightarrow \frac{\tan x}{\tan y}=\frac{a}{b}$</div>


title: if-s-i-n-x-y-s-i-n-x-y-a-b-a-b-then-what-is-t-a-n-y-t-a-n-x-equal-to
slug: if-s-i-n-x-y-s-i-n-x-y-a-b-a-b-then-what-is-t-a-n-y-t-a-n-x-equal-to-26998
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}$, then what is $\frac{\tan x}{\tan y}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{b}{a}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{a}{b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a \bar{b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a}{b}$</span> </div>

answer:
<div class="solution">$\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}$<br/>Applying componendo and dividendo, we get<br/>$\frac{\sin (x+y)+\sin (x-y)}{\sin (x+y)-\sin (x-y)}=\frac{(a+b)+(a-b)}{(a+b)-(a-b)}$<br/>$\Rightarrow \frac{2 \sin x \cdot \cos y}{2 \cos x \cdot \sin y}=\frac{2 a}{2 b} \Rightarrow \tan x \cdot \cot y=\frac{a}{b}$<br/>$\therefore \frac{\tan x}{\tan y}=\frac{a}{b}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$Q$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-s-n-np-2-n-n-1-q-where-s-n-denotes-the-sum-of-the-first-n-terms-of-an-ap-then-the-common-difference-is-2017-ii

slug:
if-s-n-np-2-n-n-1-q-where-s-n-denotes-the-sum-of-the-first-n-terms-of-an-ap-then-the-common-difference-is-2017-ii-12412

question:
<div class="question">If $\mathrm{S}_{\mathrm{n}}=\mathrm{nP}+\frac{\mathrm{n}(\mathrm{n}-1) \mathrm{Q}}{2}$, where $\mathrm{S}_{\mathrm{n}}$ denotes the sum of the<br/>first $\mathrm{n}$ terms of an AP, then the common difference is $[2017-I I]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{P}+\mathrm{Q}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{P}+3 \mathrm{Q}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 Q$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$Q$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\mathrm{S}_{\mathrm{n}}=\mathrm{np}+\frac{\mathrm{n}(\mathrm{n}-1) \mathrm{Q}}{2}$<br/>We know, $\mathrm{T}_{1}=\mathrm{S}_{1}$ and $\mathrm{T}_{2}=\mathrm{S}_{2}-\mathrm{S}_{1}$<br/>Common difference $(\mathrm{d})=\mathrm{T}_{2}-\mathrm{T}_{1}$<br/>$\therefore \mathrm{S}_{1}=(1) \mathrm{P}+\frac{1(1-1) \mathrm{Q}}{2}=\mathrm{P}+0=\mathrm{P}$<br/>$\mathrm{S}_{2}=(2) \mathrm{P}+\frac{2(2-1) \mathrm{Q}}{2}=2 \mathrm{P}+\frac{2 \mathrm{Q}}{2}=2 \mathrm{P}+\mathrm{Q}$<br/>$\therefore \mathrm{T}_{1}=\mathrm{P} ; \mathrm{T}_{2}=2 \mathrm{P}+\mathrm{Q}-\mathrm{P}=\mathrm{P}+\mathrm{Q}$<br/>$\therefore$ Common difference $(\mathrm{d})=\mathrm{T}_{2}-\mathrm{T}_{1}$<br/>$=\mathrm{P}+\mathrm{Q}-\mathrm{P}=\mathrm{Q}$</div>


title: if-s-t-2-1-then-dt-2-d-2-s-is-equal-to
slug: if-s-t-2-1-then-dt-2-d-2-s-is-equal-to-11314
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{s}=\sqrt{\mathrm{t}^{2}+1}$, then $\frac{\mathrm{d}^{2} \mathrm{~s}}{\mathrm{dt}^{2}}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\mathrm{~s}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{s^{2}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{s^{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{s^{4}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{s^{3}}$</span> </div>

answer:
<div class="solution">$\mathrm{s}=\sqrt{\mathrm{t}^{2}+1}$<br/>$\Rightarrow \frac{\mathrm{ds}}{\mathrm{dt}}=\frac{\mathrm{t}}{\sqrt{\mathrm{t}^{2}+1}}$<br/>$\frac{\mathrm{d}^{2} \mathrm{~s}}{\mathrm{dt}^{2}}=\frac{1}{\sqrt{\left(\mathrm{t}^{2}+1\right)^{3}}}$<br/>$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{~s}}{\mathrm{dt}^{2}}=\frac{1}{\mathrm{~s}^{3}}$</div>


title: if-s-x-x-2-l-0-x-is-real-then-s-is
slug: if-s-x-x-2-l-0-x-is-real-then-s-is-73325
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If S $=\left(\mathrm{x}: \mathrm{x}^{2}+\mathrm{l}=0, \mathrm{x}\right.$ is real $)$, then $\mathrm{S}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{-1\}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\{1\}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">an empty set</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">an empty set</span> </div>

answer:
<div class="solution">$\mathrm{S}=\left\{\mathrm{x}: \mathrm{x}^{2}+1=0, \mathrm{x}: 5\right.$ real\}<br/>$\mathrm{x}^{2}+1=0 \Rightarrow \mathrm{x}^{2}=-1 \Rightarrow \mathrm{x}=\sqrt{-1} \rightarrow$ complex number<br/>No real numbers. So, $\mathrm{S}$ is empty set</div>


tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-sample-a-contains-100-observations-101-102-200-and-sample-b-contains-100-obsections-151-152-ldots-250-then-ratio-of-variance-v-a-v-b

slug:
if-sample-a-contains-100-observations-101-102-200-and-sample-b-contains-100-obsections-151-152-ldots-250-then-ratio-of-variance-v-a-v-b-18905

question:
<div class="question">If sample A contains 100 observations $101,102, \ldots$ 200 and sample B contains 100 obsections $151,152,$ \ldots 250, then ratio of variance $\mathrm{v}_{\mathrm{A}} / \mathrm{v}_{\mathrm{B}}=$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}$</span> </li> </ul>

answer:
<div class="solution">$\because \quad \delta_{\mathrm{x}}^{2}=\frac{\Sigma \mathrm{d}^{2} \mathrm{i}}{\mathrm{n}}$
<br/>But both $A$ and $B$ have 100 observations, then both the sample $A$ and $B$ have same standard deviation and the same variance.
<br/>Hence, $\frac{\mathrm{V}_{\mathrm{A}}}{\mathrm{V}_{\mathrm{B}}}=1$</div>


title: if-sec-5-13-where-27-0-36-0-then-what-is-sin-equal-to
slug: if-sec-5-13-where-27-0-36-0-then-what-is-sin-equal-to-78929
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sec \alpha=\frac{13}{5}$ where $270^{\circ}   &lt; \alpha &lt; 360^{\circ}$ then what is $\sin \alpha$ equal<br/>to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{13}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{12}{13}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{12}{13}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{13}{12}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{12}{13}$</span> </div>

answer:
<div class="solution">We have sec $\alpha=\frac{13}{5}$<br/>Since $\frac{3 \pi}{2}   &lt; \alpha &lt; 2 \pi$<br/>$\therefore \sin \alpha &lt; 0$<br/>Now, $\sin \alpha=-\sqrt{1-\frac{1}{\sec ^{2} \alpha}}$<br/>$=-\sqrt{1-\frac{25}{169}}=-\sqrt{\frac{144}{169}}=-\frac{12}{13}$<br/>133. (b) $\tan \left(-585^{\circ}\right)=-\tan 585^{\circ}$<br/>$=-\tan \left[540^{\circ}+45^{\circ}\right]=-\tan \left[3 \pi+\frac{\pi}{4}\right]$<br/>$=-\left[\frac{\tan 3 \pi+\tan \frac{\pi}{4}}{1-\tan 3 \pi \tan \frac{\pi}{4}}\right]=-\left[\frac{0+1}{1-0 \times 1}\right]=-1$</div>


title: if-sec-a-tan-a-p-then-what-is-the-value-of-sin-a-2008-ii
slug: if-sec-a-tan-a-p-then-what-is-the-value-of-sin-a-2008-ii-26310
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sec A+\tan A=p$, then what is the value of $\sin A$ ? $[2008-\mathrm{II}]$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{p^{2}-1}{p^{2}+1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{p^{2}+1}{p^{2}-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{p^{2}-1}{p^{2}+1}$</span> </div>

answer:
<div class="solution">$\sec A+\tan A=P$<br/>$\Rightarrow \frac{1}{\cos A}+\frac{\sin A}{\cos A}=p$<br/>$\Rightarrow \frac{1+\sin A}{\cos A}=p$<br/>$\Rightarrow \frac{(1+\sin A)^{2}}{\cos ^{2} A}=p^{2}$<br/>$\Rightarrow \frac{(1+\sin A)^{2}}{1-\sin ^{2} A}=p^{2}$<br/>$\Rightarrow \frac{(1+\sin A)^{2}}{(1+\sin A)(1-\sin A)}=p^{2}$</div>


title: if-sec-cosec-3-4-then-what-is-sin-cos-equal-to
slug: if-sec-cosec-3-4-then-what-is-sin-cos-equal-to-51439
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sec \theta-\operatorname{cosec} \theta=\frac{4}{3}$, then what is $(\sin \theta-\cos \theta)$ equal<br/>to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2$ only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$ only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both $-2$ and $\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither $\frac{1}{2}$ nor $-2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2}$ only</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \sec \theta-\operatorname{cosec} \theta=\frac{4}{3} \Rightarrow \frac{1}{\cos \theta}-\frac{1}{\sin \theta}=\frac{4}{3} \\ &amp; \Rightarrow \frac{\sin \theta-\cos \theta}{\sin \theta \cos \theta}=\frac{4}{3} \\ &amp; \Rightarrow \frac{(\sin \theta-\cos \theta)^{2}}{(\sin \theta \cos \theta)^{2}}=\frac{16}{9} \\ &amp; \Rightarrow \frac{\sin ^{2} \theta+\cos ^{2} \theta-2 \sin \theta \cos \theta}{\sin ^{2} \theta \cos ^{2} \theta}=\frac{16}{9} \\ &amp; \Rightarrow \frac{1-2 \sin \theta \cos \theta}{\sin ^{2} \theta \cos ^{2} \theta}=\frac{16}{9} \\ &amp; \text { Let } \sin \theta \cos \theta=x \Rightarrow \frac{1-2 x}{x^{2}}=\frac{16}{9} \\ &amp; \Rightarrow 16 x^{2}+18 x-9=0 \\ &amp; \Rightarrow(8 x-3)(2 x+3)=0 \\ &amp; \Rightarrow x=\frac{3}{8}, x=\frac{-3}{2} \\ &amp; \therefore \sin \theta \cos \theta=\frac{3}{8} \\ &amp; \text { from(i), } \sin \theta-\cos \theta=\frac{4}{3} \times \frac{3}{8}=\frac{4}{8}=\frac{1}{2} \end{aligned}$</div>


title: if-sec-sec-and-sec-are-in-ap-where-cos-1-then-what-is-the-value-of-sin-2-cos-2018-ii
slug: if-sec-sec-and-sec-are-in-ap-where-cos-1-then-what-is-the-value-of-sin-2-cos-2018-ii-89996
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If sec $(\theta-\alpha)$, sec $\theta$ and sec $(\theta+\alpha)$ are in AP, where cos $\alpha \neq 1$, then what is the value of $\sin ^{2} \theta+\cos \alpha$ ? $[2018-\mathrm{II}]$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 2$</span> </li> </ul>

answer:
<div class="solution">$\frac{2}{\cos \theta}=\frac{\cos (\theta+\alpha)+\cos (\theta-\alpha)}{\cos (\theta+\alpha) \cos (\theta-\alpha)}=\frac{2 \cos \theta \cdot \cos \alpha}{\cos ^{2} \theta-\sin ^{2} \alpha}$<br/>$\Rightarrow \cos ^{2} \theta \cos \alpha=\cos ^{2} \theta-\sin ^{2} \alpha$<br/>$\Rightarrow \sin ^{2} \alpha=\cos ^{2} \theta(1-\cos \alpha)$<br/>$\Rightarrow \cos ^{2} \theta=\frac{\sin ^{2} \alpha}{1-\cos \alpha}=1+\cos \alpha$<br/>$\Rightarrow \quad 1-\sin ^{2} \theta=1+\cos \alpha$<br/>$\Rightarrow \sin ^{2} \theta+\cos \alpha=0$</div>


title: if-sec-x-cosec-x-d-x-lo-g-g-x-c-then-what-is-g-x-equal-to
slug: if-sec-x-cosec-x-d-x-lo-g-g-x-c-then-what-is-g-x-equal-to-27733
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int \sec x \operatorname{cosec} x d x=\log |g(x)|+c$, then what is $g(x)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin x \cos x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sec ^{2} x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\tan x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log |\tan x|$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan x$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int \sec x \cdot \operatorname{cosec} x \mathrm{~d} x=\int \frac{2}{2 \sin x \cos x} \mathrm{dx}$<br/>$=2 \int \frac{1}{\sin 2 x} d x=2 \int \frac{1}{\frac{2 \tan x}{1+\tan ^{2} x}}\left[\because \sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}\right]$<br/>$=\int \frac{\sec ^{2} x}{\tan x} d x$<br/>Let $\tan x=t \Rightarrow \sec ^{2} \mathrm{~d} x=\mathrm{dt}$<br/>So, $I=\int \frac{\mathrm{dt}}{\mathrm{t}}=\log |t|+\mathrm{c}=\log |\tan x|+\mathrm{c}$<br/>$g(x)=\tan x$<br/>But $\int \sec x \operatorname{cosec} x d x=\log |g(x)|+c$<br/>$\therefore \quad g(x)=\tan x$</div>


title: if-sin-1-1-a-2-2-a-cos-1-1-b-2-1-b-2-tan-1-1-x-2-2-x-then-what-is-the-value-of-x
slug: if-sin-1-1-a-2-2-a-cos-1-1-b-2-1-b-2-tan-1-1-x-2-2-x-then-what-is-the-value-of-x-99637
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1}\left(\frac{2 a}{1+a^{2}}\right)-\cos ^{-1}\left(\frac{1-b^{2}}{1+b^{2}}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$,<br/>then what is the value of $x ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a / b$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a b$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$b / a$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{a-b}{1+a b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a-b}{1+a b}$</span> </div>

answer:
<div class="solution">Given,<br/>$\sin ^{-1}\left(\frac{2 a}{1+a^{2}}\right)-\cos ^{-1}\left(\frac{1-b^{2}}{1+b^{2}}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$<br/>$\therefore \quad 2 \tan ^{-1} a-2 \tan ^{-1} b=2 \tan ^{-1} x$<br/>$\Rightarrow \tan ^{-1} a-\tan ^{-1} b=\tan ^{-1} x$<br/>$\Rightarrow \tan ^{-1}\left(\frac{a-b}{1+a b}\right)=\tan ^{-1} x$<br/>$\Rightarrow x=\frac{a-b}{1+a b}$</div>


title: if-sin-1-1-a-2-2-a-sin-1-1-b-2-2-b-2-tan-1-x-then-x-is-equal-to
slug: if-sin-1-1-a-2-2-a-sin-1-1-b-2-2-b-2-tan-1-x-then-x-is-equal-to-70543
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1}\left(\frac{2 \mathrm{a}}{1+\mathrm{a}^{2}}\right)+\sin ^{-1}\left(\frac{2 \mathrm{~b}}{1+\mathrm{b}^{2}}\right)=2 \tan ^{-1} \mathrm{x}$, then $\mathrm{x}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{a-b}{1+a b}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{a-b}{1-a b}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2 a b}{1+a b}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{a+b}{1-a b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a+b}{1-a b}$</span> </div>

answer:
<div class="solution">Since a $&gt;0, b&gt;0$ and $2 \tan ^{-1} x=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$<br/>$\therefore$ Given expression is $2 \tan ^{-1} \mathrm{a}+2 \tan ^{-1} \mathrm{~b}=2 \tan ^{-1} \mathrm{x}$<br/>$\Rightarrow 2 \tan ^{-1}\left(\frac{a+b}{1-a b}\right)=2 \tan ^{-1} x$<br/>$\Rightarrow x=\frac{a+b}{1-a b}$</div>


title: if-sin-1-1-p-2-2-p-cos-1-1-q-2-1-q-2-tan-1-1-x-2-2-x-then-what-is-x-equal-to
slug: if-sin-1-1-p-2-2-p-cos-1-1-q-2-1-q-2-tan-1-1-x-2-2-x-then-what-is-x-equal-to-90063
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} \frac{2 p}{1+p^{2}}-\cos ^{-1} \frac{1-q^{2}}{1+q^{2}}=\tan ^{-1} \frac{2 x}{1-x^{2}}$, then what is $\mathrm{x}$<br/>equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{p+q}{1+p q}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{p-q}{1+p q}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{p q}{1+p q}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{p+q}{1-p q}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{p-q}{1+p q}$</span> </div>

answer:
<div class="solution">$\sin ^{-1} \frac{2 p}{1+p^{2}}-\cos ^{-1} \frac{1-q^{2}}{1+q^{2}}=\tan ^{-1} \frac{2 x}{1-x^{2}}$<br/>$\Rightarrow 2 \tan ^{-1} \mathrm{p}-2 \tan ^{-1} \mathrm{q}=2 \tan ^{-1} \mathrm{x}$<br/>$\Rightarrow \tan ^{-1} \mathrm{p}-\tan ^{-1} \mathrm{q}=\tan ^{-1} \mathrm{x}$<br/>$\Rightarrow \tan ^{-1} \frac{p-q}{1+p q}=\tan ^{-1} \mathrm{x}$<br/>$\Rightarrow \mathrm{x}=\frac{p-q}{1+p q}$</div>


title: if-sin-1-1-sin-1-5-4-sin-1-x-then-what-is-x-equal-to
slug: if-sin-1-1-sin-1-5-4-sin-1-x-then-what-is-x-equal-to-84460
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} 1+\sin ^{-1} \frac{4}{5}=\sin ^{-1} x$, then what is $x$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 / 5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 / 5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 / 5$</span> </div>

answer:
<div class="solution">Let $\sin ^{-1}(1)+\sin ^{-1}\left(\frac{4}{5}\right)=\sin ^{-1} \mathrm{x}$<br/>$\operatorname{Let} \sin ^{-1}(1)=\theta \Rightarrow \sin \theta=1 \Rightarrow \cos \theta=0$<br/>and $\sin ^{-1}\left(\frac{4}{5}\right)=\phi \Rightarrow \sin \phi=\left(\frac{4}{5}\right) \Rightarrow \cos \phi=\sqrt{1-\frac{16}{25}}$<br/>$\begin{aligned} &amp;=\sqrt{\frac{9}{25}}=\frac{3}{5} \\ \therefore &amp; \sin ^{-1} \mathrm{x}=\theta+\phi \\ \Rightarrow &amp; \mathrm{x}=\sin (\theta+\phi)=\sin \theta \cos \phi+\cos \theta \sin \phi \\ &amp;=1 \times \frac{3}{5}+0 \times \frac{4}{5} \\ \Rightarrow &amp; \mathrm{x}=\frac{3}{5} \end{aligned}$</div>


title: if-sin-1-8-4-5-1-then-what-is-the-value-of-sin-8-1
slug: if-sin-1-8-4-5-1-then-what-is-the-value-of-sin-8-1-40082
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$, then what is the value of $\sin 81^{\circ}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3-\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}}}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/VcDzvSA1h2Bw_NUQKYaH17ryC0HTRgSGd7CJ1FHGf8E.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>\because &amp; \sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \\<br/>&amp; x^{2}=4^{2}-(\sqrt{5}-1)^{2} \\<br/>\Rightarrow &amp; x=\sqrt{10+2 \sqrt{5}} \\<br/>\Rightarrow &amp; \cos 18^{\circ}=\frac{\sqrt{10+2 \sqrt{5}}}{4} \\<br/>\Rightarrow &amp; 2 \cos ^{2} 9-1=\frac{\sqrt{10+2 \sqrt{5}}}{4} \\<br/>&amp; \cos ^{2} 9=\frac{\sqrt{10+2 \sqrt{5}}+4}{8} \\<br/>\Rightarrow &amp; \sin ^{2} 81=\frac{4+\sqrt{10+2 \sqrt{5}}}{8}<br/>\end{aligned}<br/>$$<br/>After squaring all the options available, we come to a conclusion that option (a) is correct.</div>


title: if-sin-1-sin-2-sin-3-sin-4-4-then-what-is-the-value-of-cos-theta1cos-theta2cos-theta3cos-theta4
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-sin-1-sin-2-sin-3-sin-4-4-then-what-is-the-value-of-cos-theta1cos-theta2cos-theta3cos-theta4-50242

question:
<div class="question">If $\sin \theta_{1}+\sin \theta_{2}+\sin \theta_{3}+\sin \theta_{4}=4$, then what is the value<br/>of $\cos \theta_{1}+\cos \theta_{2}+\cos \theta_{3}+\cos \theta_{4}=?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\underline{0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\underline{0}$</span> </div>

answer:
<div class="solution">\sin \theta_{1}+\sin \theta_{2}+\sin \theta_{3}+\sin \theta_{4}=4 &amp; \ldots(1) \end{aligned}$<br/>Since max. value of $\sin \theta=1$ We have four terms in LHS of eq. (1). $\Rightarrow$ Every term should be $=1$ $\Rightarrow \sin \theta_{1}=\sin \theta_{2}=\sin \theta_{3}=\sin \theta_{4}=1$<br/>$\Rightarrow \theta_{1}=\theta_{2}=\theta_{3}=\theta_{4}=90^{\circ}$<br/>Now, $\cos \theta_{1}+\cos \theta_{2}+\cos \theta_{3}+\cos \theta_{4}=\left(\cos 90^{\circ}\right) \times 4=0$</div>


title: if-sin-1-x-5-sin-1-x-12-2-then-what-is-the-value-of-x
slug: if-sin-1-x-5-sin-1-x-12-2-then-what-is-the-value-of-x-52871
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">13</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} \frac{5}{x}+\sin ^{-1} \frac{12}{x}=\frac{\pi}{2}$, then what is the value of $x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">13</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">17</span> </li> </ul>

answer:
<div class="solution">$\sin ^{-1} \frac{5}{x}+\sin ^{-1} \frac{12}{x}=\frac{\pi}{2}$<br/>$\Rightarrow \sin ^{-1} \frac{5}{x}+\cos ^{-1} \frac{\sqrt{x^{2}-144}}{x}=\frac{\pi}{2}$<br/>$\left(\because \sin ^{-1} x=\cos ^{-1} \sqrt{1-x^{2}}\right)$<br/>But we know $\sin ^{-1} y+\cos ^{-1} y=\frac{\pi}{2}$<br/>$\therefore \quad \frac{5}{x}=\frac{\sqrt{x^{2}-144}}{x}$<br/>$\Rightarrow 5=\sqrt{x^{2}-144}$<br/>$\Rightarrow 25=x^{2}-144 \Rightarrow x^{2}=169$<br/>$\Rightarrow \quad x=13$</div>


title: if-sin-1-x-cos-1-x-6-then-what-is-the-value-of-x
slug: if-sin-1-x-cos-1-x-6-then-what-is-the-value-of-x-91894
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} x-\cos ^{-1} x=\frac{\pi}{6}$, then what is the value of $x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}=-\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}=\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{x}=\frac{\sqrt{3}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{x}=\frac{\sqrt{3}}{2}$</span> </div>

answer:
<div class="solution">As given:<br/>$\sin ^{-1} x-\cos ^{-1} x=\frac{\pi}{6}$ ...(1)<br/>and we know that $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$ ...(2)<br/>On adding Eqs. (1) and (2) we get<br/>$2 \sin ^{-1} x=\frac{\pi}{6}+\frac{\pi}{2}=\frac{2 \pi}{3}$<br/>$\Rightarrow \sin ^{-1} x=\frac{\pi}{3} \Rightarrow x=\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}$</div>


title: if-sin-1-x-cot-1-12-2-then-what-is-the-value-of-x
slug: if-sin-1-x-cot-1-12-2-then-what-is-the-value-of-x-89018
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} x+\cot ^{-1}(1 / 2)=\pi / 2$, then what is the value of $x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 / \sqrt{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 / \sqrt{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{3} / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / \sqrt{5}$</span> </div>

answer:
<div class="solution">Let $\sin ^{-1} x+\cot ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}$<br/>As we know $\cot ^{-1} x=\sin ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)$<br/>$\therefore \sin ^{-1} x+\cot ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}$<br/>$\Rightarrow \sin ^{-1} x+\sin ^{-1}\left(\frac{1}{\sqrt{1+\frac{1}{4}}}\right)=\frac{\pi}{2}$<br/>$\Rightarrow \sin ^{-1} x+\sin ^{-1}\left(\frac{2}{\sqrt{5}}\right)=\frac{\pi}{2}$<br/>Now, $\sin ^{-1} x=\cos ^{-1} \sqrt{1-x^{2}}$<br/>$\therefore \sin ^{-1}\left(\frac{2}{\sqrt{5}}\right)=\cos ^{-1} \sqrt{1-\frac{4}{5}}=\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)$<br/>From equation (1), we have<br/>$\sin ^{-1} x+\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)=\frac{\pi}{2}$<br/>since, $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$<br/>$x=\frac{1}{\sqrt{5}}$</div>


title: if-sin-1-x-sin-1-y-2-and-cos-1-x-cos-1-y-0-then-values-x-and-y-are-respectively
slug: if-sin-1-x-sin-1-y-2-and-cos-1-x-cos-1-y-0-then-values-x-and-y-are-respectively-25157
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} x+\sin ^{-1} y=\pi / 2$ and $\cos ^{-1} x-\cos ^{-1} y=0$, then<br/>values $x$ and $y$ are respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}, \frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2},-\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$</span> </div>

answer:
<div class="solution">Given, $\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{2}$ ...(i)<br/>and $\cos ^{-1} x-\cos ^{-1} y=0$<br/>$\Rightarrow\left(\frac{\pi}{2}-\sin ^{-1} x\right)-\left(\frac{\pi}{2}-\sin ^{-1} y\right)=0$<br/>$\Rightarrow \sin ^{-1} y-\sin ^{-1} x=0$<br/>$\Rightarrow \sin ^{-1} y=\sin ^{-1} x$ ...(ii)<br/>From equations (i) and (ii), we get<br/>$2 \sin ^{-1} x=\frac{\pi}{2}$<br/>$\Rightarrow \sin ^{-1} x=\frac{\pi}{4}$<br/>$\Rightarrow x=\frac{1}{\sqrt{2}}$<br/>From equation (ii)<br/>$y=\frac{1}{\sqrt{2}}$</div>


title: if-sin-1-x-tan-1-y-what-is-the-value-of-x-2-1-y-2-1
slug: if-sin-1-x-tan-1-y-what-is-the-value-of-x-2-1-y-2-1-12953
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} \mathrm{x}=\tan ^{-1} \mathrm{y}$, what is the value of $\frac{1}{\mathrm{x}^{2}}-\frac{1}{\mathrm{y}^{2}}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Let, $\sin ^{-1} x=\tan ^{-1} y=\theta$<br/>$\Rightarrow \mathrm{x}=\sin \theta$ and $\mathrm{y}=\tan \theta$<br/>$\frac{1}{x^{2}}=\frac{1}{\sin ^{2} \theta}=\operatorname{cosec}^{2} \theta$<br/>and $\frac{1}{\mathrm{y}^{2}}=\frac{1}{\tan ^{2} \theta}=\cot ^{2} \theta$.<br/>$\Rightarrow \frac{1}{\mathrm{x}^{2}}-\frac{1}{\mathrm{y}^{2}}=\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$</div>


title: if-sin-2-1-and-tan-3-1-then-in-which-quadrant-does-lie
slug: if-sin-2-1-and-tan-3-1-then-in-which-quadrant-does-lie-25648
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Third</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta=-\frac{1}{2}$ and $\tan \theta=\frac{1}{\sqrt{3}}$, then in which quadrant does<br/>$\theta$ lie?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">First</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Second</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Third</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Fourth</span> </li> </ul>

answer:
<div class="solution">$\sin \theta=\frac{-1}{2}, \tan \theta=\frac{1}{\sqrt{3}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/16UsZHDtLNFi0Onvjywpgak6PsrYJsO5N88iYx4VPrA.original.fullsize.png"/><br/><br/>$\sin \theta$ is negative, $\tan \theta$ is positive<br/>$\theta$ lies in third quadrant.</div>


title: if-sin-2-a-5-4-then-what-is-the-value-of-tan-a-0-a-4
slug: if-sin-2-a-5-4-then-what-is-the-value-of-tan-a-0-a-4-85388
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin 2 \mathrm{~A}=\frac{4}{5}$, then what is the value of tan $\mathrm{A}\left(0 \leq \mathrm{A} \leq \frac{\pi}{4}\right)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">As given : $\sin 2 \mathrm{~A}=\frac{4}{5}$<br/>$\sin 2 \mathrm{~A}=\frac{2 \tan \mathrm{A}}{1+\tan ^{2} \mathrm{~A}}$<br/>$\Rightarrow \frac{2 \tan \mathrm{A}}{1+\tan ^{2} \mathrm{~A}}=\frac{4}{5}$<br/>$\Rightarrow 10 \tan \mathrm{A}=4+4 \tan ^{2} \mathrm{~A}$<br/>$\Rightarrow 5 \tan \mathrm{A}=2+2 \tan ^{2} \mathrm{~A}$<br/>$\Rightarrow 2 \tan ^{2} \mathrm{~A}-5 \tan \mathrm{A}+2=0$<br/>$\Rightarrow 2 \tan ^{2} \mathrm{~A}-4 \tan \mathrm{A}-\tan \mathrm{A}+2=0$<br/>$\Rightarrow 2 \tan \mathrm{A}(\tan \mathrm{A}-2)-1(\tan \mathrm{A}-2)=0$<br/>$\Rightarrow(2 \tan \mathrm{A}-1)(\tan \mathrm{A}-2)=0$<br/>$\Rightarrow \tan \mathrm{A}=\frac{1}{2}$ (since $\left.\mathrm{A} \leq \frac{\pi}{4} \Rightarrow \tan \mathrm{A} \neq 2\right)$</div>


title: if-sin-2-cos-1-then-what-is-2-sin-cos-equal-to
slug: if-sin-2-cos-1-then-what-is-2-sin-cos-equal-to-28956
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta+2 \cos \theta=1$, then what is $2 \sin \theta-\cos \theta$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Let $\sin \theta+2 \cos \theta=1 \quad$....<br/>.... Consider $2 \sin \theta-\cos \theta=\alpha$ (let)<br/>squaring and adding $\sin ^{2} \theta+4 \cos ^{2} \theta+4 \sin \theta \cos \theta+4 \sin ^{2} \theta+\cos ^{2} \theta-$<br/>$4 \sin \theta \cos \theta=1+\alpha^{2}$<br/>$\Rightarrow\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+4\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=1+\alpha^{2}$<br/>$\Rightarrow 1+4=1+\alpha^{2} \Rightarrow \alpha^{2}=4 \Rightarrow \alpha=2$</div>


title: if-sin-2-cos-3-where-0-2-then-what-is-sin-equal-to
slug: if-sin-2-cos-3-where-0-2-then-what-is-sin-equal-to-83742
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin 2 \theta=\cos 3 \theta$, where $0   &lt; \theta &lt; \frac{\pi}{2}$, then what is $\sin \theta$ equal<br/>to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{5}+1}{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{5}-1}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{5}+1}{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{5}-1}{16}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{5}-1}{4}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \sin 2 \theta=\cos 3 \theta \\ \Rightarrow \sin 2 \theta=\sin \left(90^{\circ}-3 \theta\right) \\ 2 \theta=90^{\circ}-3 \theta \quad \therefore \sin \theta=\sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \\ &amp; \Rightarrow 5 \theta=90^{\circ} \\ &amp; \Rightarrow \theta=18^{\circ} \\ &amp; \Rightarrow\left(\frac{1}{q}\right)(p)=1 \end{aligned}$<br/>$\Rightarrow \mathrm{p}=\mathrm{q}$</div>


title: if-sin-2-x-sin-2-y-1-then-what-is-the-value-of-cot-x-y
slug: if-sin-2-x-sin-2-y-1-then-what-is-the-value-of-cot-x-y-64082
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{2} x+\sin ^{2} y=1$, then what is the value of $\cot (x+y)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / \sqrt{3}$</span> </li> </ul>

answer:
<div class="solution">Given $\sin ^{2} x+\sin ^{2} y=1$<br/>$\Rightarrow \sin ^{2} x=1-\sin ^{2} y$<br/>$\Rightarrow \sin ^{2} x=\cos ^{2} y$<br/>$\Rightarrow \sin x=\cos y$<br/>Similarly By considering $\sin ^{2} y=1-\sin ^{2} x$, we have<br/>$\cos x=\sin y$<br/>Now, Consider $\cot (x+y)=\frac{\cos (x+y)}{\sin (x+y)}$<br/>$=\frac{\cos x \cos y-\sin x \sin y}{\sin (x+y)}$<br/>$=\frac{\cos x \cos y-\cos x \cos y}{\sin (x+y)}=0$</div>


title: if-sin-3-a-1-then-how-many-distinct-values-can-sin-a-assume
slug: if-sin-3-a-1-then-how-many-distinct-values-can-sin-a-assume-14888
question: <div class="question">If $\sin 3 A=1$, then how many distinct values can $\sin A$ assume?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
answer:
<div class="solution">Let $\sin 3 A=1$<br/>$\Rightarrow 3 \sin A-4 \sin ^{3} A=1$<br/>$\Rightarrow 4 \sin ^{3} A-3 \sin A+1=0$<br/>$\Rightarrow(\sin A+1)\left(4 \sin ^{2} A-4 \sin A+1\right)=0$<br/>$\Rightarrow(\sin A+1)(2 \sin A-1)^{2}=0$<br/>$\Rightarrow \sin A=-1$ or $\frac{1}{2}$<br/>Hence, $\sin A$ can take two distinct values</div>


title: if-sin-3-cos-3-0-then-what-is-the-value-of
slug: if-sin-3-cos-3-0-then-what-is-the-value-of-74026
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{3} \theta+\cos ^{3} \theta=0$, then what is the value of $\theta$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{-\pi}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-\pi}{4}$</span> </div>

answer:
<div class="solution">Since, $\sin ^{3} \theta+\cos ^{3} \theta=0$<br/>$\Rightarrow(\sin \theta+\cos \theta)\left(\sin ^{2} \theta-\sin \theta \cos \theta+\cos ^{2} \theta\right)=0$<br/>$\left(\because a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right)$<br/>$\Rightarrow(\sin \theta+\cos \theta)(1-\sin \theta \cos \theta)=0$<br/>$\Rightarrow(\sin \theta+\cos \theta)\left(1-\frac{\sin 2 \theta}{2}\right)=0$<br/>$\Rightarrow \sin \theta+\cos \theta=0$<br/>or $\sin 2 \theta=2$<br/>(discarded since $\sin ^{2} \theta=2$ is not possible) $\Rightarrow \sin \theta+\cos \theta=0 \Rightarrow \sin \theta=-\cos \theta$<br/>$\Rightarrow \tan \theta=-1 \Rightarrow \theta=-\pi / 4$</div>


title: if-sin-3-sin-2-then-the-value-of-tan-2-tan-is-equal-to
slug: if-sin-3-sin-2-then-the-value-of-tan-2-tan-is-equal-to-12846
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta=3 \sin (\theta+2 \alpha)$, then the value of $\tan (\theta+\alpha)+$<br/>$2 \tan \alpha$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\underline{0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\underline{0}$</span> </div>

answer:
<div class="solution">Given, $\sin \theta=3 \sin (\theta+2 \alpha)$<br/>$\Rightarrow \frac{\sin (\theta+2 \alpha)}{\sin \theta}=\frac{1}{3}$<br/>Apply componendo and divide do rule $\Rightarrow \frac{\sin (\theta+2 \alpha)+\sin \theta}{\sin (\theta+2 \alpha)-\sin \theta}=\frac{1+3}{1-3}$<br/>$\Rightarrow \frac{2 \sin (\theta+\alpha) \cos \alpha}{2 \cos (\theta+\alpha) \sin \alpha}=\frac{4}{-2}=-2$<br/>$\Rightarrow \frac{\tan (\theta+\alpha)}{\tan \alpha}=-2$<br/>$\Rightarrow \tan (\theta+\alpha)=-2 \tan \alpha \Rightarrow \tan (\theta+\alpha)+2 \tan \alpha=0$</div>


title: if-sin-4-x-cos-4-x-p-then-which-one-of-the-following-is-correct
slug: if-sin-4-x-cos-4-x-p-then-which-one-of-the-following-is-correct-66233
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{4} x-\cos ^{4} x=p$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{p}=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$p=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$|\mathrm{p}|&gt;1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$|\mathrm{p}| \leq 1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|\mathrm{p}| \leq 1$</span> </div>

answer:
<div class="solution">Consider $\sin ^{4} x-\cos ^{4} x=p$<br/>$\Rightarrow\left(\sin ^{2} x\right)^{2}-\left(\cos ^{2} x\right)^{2}=p$<br/>$\Rightarrow\left(\sin ^{2} x-\cos ^{2} x\right)\left(\sin ^{2} x+\cos ^{2} x\right)=p$<br/>$\Rightarrow \sin ^{2} x-\cos ^{2} x=p\left(\because \sin ^{2} x+\cos ^{2} x=1\right)$<br/>$\Rightarrow-\cos 2 x=p \quad\left(\because \cos ^{2} x-\sin ^{2} x=\cos 2 x\right)$<br/>$\Rightarrow \cos 2 x=-p$<br/>$\therefore \quad|p| \leq 1$</div>


title: if-sin-a-1-5-cos-b-3-10-a-b-being-positive-acute-angles-then-what-is-a-b-equal-to
slug: if-sin-a-1-5-cos-b-3-10-a-b-being-positive-acute-angles-then-what-is-a-b-equal-to-66651
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If sin $A=1 / \sqrt{5}, \cos B=3 / \sqrt{10} ; A, B$ being positive acute angles, then what is $(A+B)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi / 6$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\pi / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi / 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi / 2$</span> </li> </ul>

answer:
<div class="solution">Given, $\sin A=\frac{1}{\sqrt{5}}$ and $\cos B=\frac{3}{\sqrt{10}}$<br/>$\therefore \sin (A+B)=\sin A \cos B+\cos A \sin B$<br/>$=\frac{1}{\sqrt{5}} \times \frac{3}{\sqrt{10}}+\sqrt{1-\frac{1}{5}} \times \sqrt{1-\frac{9}{10}}$<br/>$=\frac{3}{\sqrt{50}}+\frac{2}{\sqrt{5}} \times \frac{1}{\sqrt{10}}=\frac{3+2}{\sqrt{50}}=\frac{1}{\sqrt{2}}=\sin \frac{\pi}{4}$<br/>$\Rightarrow A+B=\frac{\pi}{4}$</div>


title: if-sin-a-5-2-and-cos-b-10-1-where-a-and-b-are-acute-angles-then-what-is-a-b-equal-to-2012-ii
slug: if-sin-a-5-2-and-cos-b-10-1-where-a-and-b-are-acute-angles-then-what-is-a-b-equal-to-2012-ii-18122
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \mathrm{A}=\frac{2}{\sqrt{5}}$ and $\cos \mathrm{B}=\frac{1}{\sqrt{10}}$ where $\mathrm{A}$ and $\mathrm{B}$ are acute<br/>$\begin{array}{ll}\text { angles, then what is } \mathrm{A}+\mathrm{B} \text { equal to } ? &amp; {[2012-\mathrm{II}]}\end{array}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$135^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$90^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$75^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\overline{60^{\circ}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$135^{\circ}$</span> </div>

answer:
<div class="solution">Given, $\sin \mathrm{A}=\frac{2}{\sqrt{5}}$ and $\cos \mathrm{B}=\frac{2}{\sqrt{10}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/fqKixogW04NpWnYd8MGOxqrFcN6dqKIJX00Koynby_0.original.fullsize.png"/><br/><br/>$\therefore \quad \cos \mathrm{A}=\frac{1}{\sqrt{5}} \quad \sin \mathrm{B}=\frac{3}{\sqrt{10}}$<br/>$\therefore \sin (\mathrm{A}+\mathrm{B})=\sin \mathrm{A} \cos \mathrm{B}+\cos \mathrm{A} \sin \mathrm{B}$<br/>$=\left(\frac{2}{\sqrt{15}}\right)\left(\frac{1}{\sqrt{10}}\right)+\left(\frac{1}{\sqrt{5}}\right)\left(\frac{3}{\sqrt{10}}\right)$<br/>$=\frac{2}{\sqrt{50}}+\frac{3}{\sqrt{50}}=\frac{5}{\sqrt{50}}=\frac{5}{\sqrt{25 \times 2}}=\frac{5}{5 \sqrt{2}}=\frac{1}{\sqrt{2}}$<br/>$\sin (\mathrm{A}+\mathrm{B})=\frac{1}{\sqrt{2}} \Rightarrow \sin (90+45)=\frac{1}{\sqrt{2}}$<br/>$\left(\because \sin (90+45)=\cos 45^{\circ}\right)$<br/>$\therefore \quad A=90^{\circ}$ and $B=45^{\circ}$<br/>$\therefore \quad A+B=90^{\circ}+45^{\circ}=135^{\circ}$</div>


title: if-sin-a-5-3-where-45-0-a-54-0-then-cos-2-a-is-equal-to
slug: if-sin-a-5-3-where-45-0-a-54-0-then-cos-2-a-is-equal-to-45166
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin A=\frac{3}{5}$, where $450^{\circ}   &lt; \mathrm{A} &lt; 540^{\circ}$, then $\cos \frac{\mathrm{A}}{2}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\sqrt{10}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\sqrt{\frac{3}{10}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}}{\sqrt{10}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of the above</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\operatorname{Sin} A=\frac{3}{5} ; 450^{\circ}   &lt; \mathrm{A} &lt; 540^{\circ}$<br/>$\Rightarrow 225^{\circ} &lt; \frac{\mathrm{A}}{2} &lt; 270^{\circ}$<br/>$\cos \mathrm{A}=\frac{-4}{5} \quad(\because \mathrm{A}$ lies in Q2 $)$<img src="https://cdn-question-pool.getmarks.app/pyq/nda/ZRaMyaK8g0-RRDQi-Wmpm9rmWdBzoZXd__CK1UwIpMg.original.fullsize.png"/><br/><br/>$\therefore \cos ^{2} \frac{\mathrm{A}}{2}=\frac{1+\cos \mathrm{A}}{2}=\frac{1}{10}$<br/>$\Rightarrow \cos \frac{\mathrm{A}}{2}=\frac{-1}{\sqrt{10}}\left(\because \frac{\mathrm{A}}{2}\right.$ lies in $\left.\mathrm{Q} 3\right)$</div>


title: if-sin-a-sin-6-0-a-sin-6-0-a-k-sin-3-a-then-what-isk-equal-to
slug: if-sin-a-sin-6-0-a-sin-6-0-a-k-sin-3-a-then-what-isk-equal-to-54063
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \mathrm{A} \sin \left(60^{\circ}-\mathrm{A}\right) \sin \left(60^{\circ}+\mathrm{A}\right)=\mathrm{k} \sin 3 \mathrm{~A}$, then what isk<br/>equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">$\sin A \cdot \sin \left(60^{\circ}-\mathrm{A}\right) \sin \left(60^{\circ}+\mathrm{A}\right)=\mathrm{k} \sin 3 \mathrm{~A}$<br/>$\begin{aligned} \Rightarrow &amp; \sin \mathrm{A} \cdot \frac{\sin 3 \mathrm{~A}}{4 \sin \mathrm{A}}=\mathrm{k} \cdot \sin 3 \mathrm{~A} \\ &amp; \quad\left[\because \sin \left(60^{\circ}+\mathrm{A}\right) \cdot \sin \left(60^{\circ}-\mathrm{A}\right)=\frac{\sin 3 \mathrm{~A}}{4 \sin \mathrm{A}}\right] \\ \Rightarrow &amp; \frac{\sin 3 \mathrm{~A}}{4}=\mathrm{k} \cdot \sin 3 \mathrm{~A} \\ \therefore \mathrm{k} &amp;=\frac{1}{4} \end{aligned}$<br/>$$</div>


title: if-sin-a-sin-b-and-cos-a-cos-b-then-which-one-of-the-following-is-correct
slug: if-sin-a-sin-b-and-cos-a-cos-b-then-which-one-of-the-following-is-correct-95624
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \mathrm{A}=\sin \mathrm{B}$ and $\cos \mathrm{A}=\cos \mathrm{B}$, then which one of the<br/>following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad B=n \pi+A$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{A}=2 \mathrm{n} \pi-\mathrm{B}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{A}=2 \mathrm{n} \pi+\mathrm{B}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{B}=\mathrm{n} \pi-\mathrm{A} \quad$ (n is an integer)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad B=n \pi+A$</span> </div>

answer:
<div class="solution">As given:<br/>$\sin A=\sin B$<br/>and $\cos A=\cos B$<br/>$\Rightarrow \frac{\sin A}{\cos A}=\frac{\sin B}{\cos B}$<br/>$\Rightarrow \tan A=\tan B$ or $\tan B=\tan A=\tan (n \pi+\mathrm{A})$<br/>$\Rightarrow B=n \pi+A$</div>


title: if-sin-a-sin-b-sin-c-3-then-what-is-cos-a-cos-b-cos-c-equal-to
slug: if-sin-a-sin-b-sin-c-3-then-what-is-cos-a-cos-b-cos-c-equal-to-89325
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin A+\sin B+\sin C=3$ then what is $\cos A+\cos B+\cos$<br/> C equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\underline{0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\underline{0}$</span> </div>

answer:
<div class="solution">Let  \sin A+\sin B+&amp; \sin C=3 \end{aligned}$<br/>$\Rightarrow \sin A=\sin B=\sin C=1 \quad(\because$ max value of sin is 1$)$<br/>$\therefore \cos A=\sqrt{1-\sin ^{2} A}=\sqrt{1-1}=0$<br/>Similarly, $\cos B=0=\cos C$<br/>Hence, $\cos A+\cos B+\cos C=0+0+0=0$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-sin-and-cos-are-the-roots-of-a-x-2-b-x-c-0-then-constants-a-b-c-will-satisfy-which-one-of-the-following-conditions

slug:
if-sin-and-cos-are-the-roots-of-a-x-2-b-x-c-0-then-constants-a-b-c-will-satisfy-which-one-of-the-following-conditions-42391

question:
<div class="question">If $\sin \theta$ and $\cos \theta$ are the roots of $a x^{2}+b x+c=0$, then constants $\mathrm{a}, \mathrm{b}, \mathrm{c}$ will satisfy which one of the following conditions?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a^{2}+b^{2}+2 a c=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{a}^{2}+\mathrm{b}^{2}-2 \mathrm{ac}=0^{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$a^{2}-b^{2}+2 a c=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-a^{2}+b^{2}+2 a c=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a^{2}-b^{2}+2 a c=0$</span> </div>

answer:
<div class="solution">As given $\sin \theta$ and $\cos \theta$ are the roots of the equation<br/>So, sum of roots $=\sin \theta+\cos \theta=-\frac{b}{a}$...(1)<br/>and product of roots $=\sin \theta \cos \theta=\frac{\mathrm{c}}{a}$...(2)<br/>On squaring both sides in Eq. (1), we get $(\sin \theta+\cos \theta)^{2}=\frac{b^{2}}{a^{2}}$<br/>$\Rightarrow \sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}$<br/>$\Rightarrow 1+\frac{2 \mathrm{c}}{\mathrm{a}}=\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}} \quad$ using Eq. $(2)$<br/>$\Rightarrow a^{2}+2 c a=b^{2}$<br/>$\Rightarrow a^{2}-b^{2}+2 a c=0$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-sin-and-cos-are-the-roots-of-the-equation-p-x-2-q-x-1-0-then-which-one-of-the-following-is-correct

slug:
if-sin-and-cos-are-the-roots-of-the-equation-p-x-2-q-x-1-0-then-which-one-of-the-following-is-correct-75093

question:
<div class="question">If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $p x^{2}+q x+1$ $=0$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p^{2}+q^{2}-2 p r=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$p^{2}-q^{2}+2 p r=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(p+r)^{2}=2\left(p^{2}+r^{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(p-r)^{2}=q^{2}+r^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p^{2}-q^{2}+2 p r=0$</span> </div>

answer:
<div class="solution">$\sin \alpha$ and $\cos \alpha$ are the roots of the equation $p x^{2}+q x+r=0$<br/>$\therefore \sin \alpha+\cos \alpha=\frac{-q}{p}\cdots. (i)$<br/>and $\sin \alpha \cos \alpha=\frac{r}{p}$ ...(ii)<br/>From equation (i)<br/>$(\sin \alpha+\cos \alpha)^{2}=\frac{q^{2}}{p^{2}}$<br/>$\Rightarrow \sin ^{2} \alpha+\cos ^{2} \alpha+2 \sin \alpha \cos \alpha=\frac{q^{2}}{p^{2}}$<br/>$\Rightarrow 1+2 \cdot \frac{r}{p}=\frac{q^{2}}{p^{2}}$<br/>$\Rightarrow p^{2}+2 p r=q^{2}$<br/>$p^{2}-q^{2}+2 p r=0$</div>


title: if-sin-cos-2-cos-then-what-is-cos-sin-equal-to
slug: if-sin-cos-2-cos-then-what-is-cos-sin-equal-to-24022
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta+\cos \theta=\sqrt{2} \cos \theta$, then what is $(\cos \theta-\sin \theta)$ equal<br/>to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\sqrt{2} \cos \theta$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\sqrt{2} \sin \theta$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{2} \sin \theta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sin \theta$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{2} \sin \theta$</span> </div>

answer:
<div class="solution">$\sin \theta+\cos \theta=\sqrt{2} \cos \theta$<br/>Let $\cos \theta-\sin \theta=\mathrm{P}$<br/>$(1)^{2}+(2)^{2} \Rightarrow \sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta+\cos ^{2} \theta+\sin ^{2} \theta$<br/>$-2 \sin \theta \cos \theta=2 \cos ^{2} \theta+\mathrm{p}^{2}$<br/>$\Rightarrow 2=2 \cos ^{2} \theta+\mathrm{p}^{2}$<br/>$\Rightarrow \mathrm{p}^{2}=2\left(1-\cos ^{2} \theta\right)=2 \sin ^{2} \theta$<br/>$\Rightarrow p=\sqrt{2} \sin \theta$</div>


title: if-sin-cos-2-then-what-is-cos-2-1-cos-2-equal-to
slug: if-sin-cos-2-then-what-is-cos-2-1-cos-2-equal-to-56172
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta=\cos ^{2} \theta$, then what is $\cos ^{2} \theta\left(1+\cos ^{2} \theta\right)$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\cos ^{2} \theta$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sin \theta$</span> </li> </ul>

answer:
<div class="solution">Let $\sin \theta=\cos ^{2} \theta$<br/>$\Rightarrow \sin ^{2} \theta=\cos ^{4} \theta$<br/>Consider $\cos ^{2} \theta\left(1+\cos ^{2} \theta\right)=\cos ^{2} \theta+\cos ^{4} \theta$<br/>$=\cos ^{2} \theta+\sin ^{2} \theta \quad$ (using 1)<br/>$=1$</div>


title: if-sin-cos-p-then-what-is-cos-2-2-equal-to
slug: if-sin-cos-p-then-what-is-cos-2-2-equal-to-46174
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \alpha+\cos \alpha=\mathrm{p}$, then what is $\cos ^{2}(2 \alpha)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{P}^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}^{2}-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{P}^{2}\left(2-\mathrm{p}^{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{p}^{2}+1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}^{2}\left(2-\mathrm{p}^{2}\right)$</span> </div>

answer:
<div class="solution">\sin \alpha+\cos \alpha=p \\ &amp; \Rightarrow(\sin \alpha+\cos \alpha)^{2}=p^{2} \\ &amp; \Rightarrow \sin ^{2} \alpha+\cos ^{2} \alpha+2 \sin \alpha \cos \alpha=p^{2} \\ &amp; \Rightarrow 1+\sin 2 \alpha=p^{2} \\ &amp; \Rightarrow \sin 2 \alpha=\mathrm{p}^{2}-1 \\ &amp; \cos ^{2} 2 \alpha=1-\sin ^{2} 2 \alpha=1-\left(\mathrm{p}^{2}-1\right)^{2} \\ &amp;=1-\left(\mathrm{p}^{4}+1-2 \mathrm{p}^{2}\right)=-\mathrm{p}^{4}+2 \mathrm{p}^{2} \\ &amp;=\mathrm{p}^{2}\left(2-\mathrm{p}^{2}\right) \\</div>


title: if-sin-cos-x-cos-sin-x-then-what-is-one-of-the-values-of-sin-2-x
slug: if-sin-cos-x-cos-sin-x-then-what-is-one-of-the-values-of-sin-2-x-96276
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin (\pi \cos x)=\cos (\pi \sin x)$, then what is one of the values<br/>of $\sin 2 \mathrm{x}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{3}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{3}{4}$</span> </div>

answer:
<div class="solution">Given that : $\sin (\pi \cos x)=\cos (\pi \sin x)$<br/>$3 u, \cos \left(\frac{\pi}{2}-\pi \cos x\right)=\cos (\pi \sin x)$<br/>$\Rightarrow \frac{\pi}{2}-\pi \cos x=\pi \sin x$<br/>$\Rightarrow \sin x+\cos x=\frac{1}{2}$<br/>Squaring both sides, we get<br/>$\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=\frac{1}{4}$<br/>$\Rightarrow \sin 2 x=\frac{1}{4}-1=-\frac{3}{4}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-sin-is-the-harmonic-mean-of-sin-and-cos-and-sin-is-the-arithmetic-mean-of-sin-and-cos-then-which-of-the-following-isare-correct-2018-ii-1-2-sin-4-s

slug:
if-sin-is-the-harmonic-mean-of-sin-and-cos-and-sin-is-the-arithmetic-mean-of-sin-and-cos-then-which-of-the-following-isare-correct-2018-ii-1-2-sin-4-s-27847

question:
<div class="question">If $\sin \beta$ is the harmonic mean of $\sin \alpha$ and $\cos \alpha$, and $\sin \theta$ is the arithmetic mean of $\sin \alpha$ and $\cos \alpha$, then which of the following is/are correct? $[2018-I I]$<br/>1) $\sqrt{2} \sin \left(\alpha+\frac{\pi}{4}\right) \sin \beta=\sin 2 \alpha$<br/>2) $\sqrt{2} \sin \theta=\cos \left(\alpha-\frac{\pi}{4}\right)$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">From question, $\sin \alpha, \sin \beta$ and $\cos \alpha$ are in $H . P$<br/>then, $\sin \beta=\frac{2 \sin \alpha \cos \alpha}{\sin \alpha+\cos \alpha}$<br/>and $\sin \alpha, \sin \theta$ and $\cos \alpha$ are in A.P.<br/>then, $2 \sin \theta=\sin \alpha+\cos \alpha$<br/>Statement 1: $\sqrt{2} \sin \left(\alpha+\frac{\pi}{4}\right) \cdot \sin \beta$<br/>$=(\sin \alpha+\cos \alpha) \cdot \frac{2 \sin \alpha \cos \alpha}{\sin \alpha+\cos \alpha}$<br/>$=2 \sin \alpha \cos \alpha=\sin 2 \alpha$<br/>Statement $2: \cos \left(\alpha-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}(\sin \alpha+\cos \alpha)$<br/>$=\frac{2 \sin \theta}{\sqrt{2}}=\sqrt{2} \sin \theta$</div>


title: if-sin-sin-0-cos-cos-where-0-2-then-which-one-of-the-following-is-correct
slug: if-sin-sin-0-cos-cos-where-0-2-then-which-one-of-the-following-is-correct-15592
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \alpha+\sin \beta=0=\cos \alpha+\cos \beta$, where $0   &lt; \beta &lt; \alpha &lt; 2 \pi$, then<br/>which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \alpha=\pi-\beta$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\alpha=\pi+\beta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \alpha=2 \pi-\beta$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \alpha=\pi+2 \beta$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha=\pi+\beta$</span> </div>

answer:
<div class="solution">$\sin \alpha+\sin \beta=0=\cos \alpha+\cos \beta$<br/>$\sin \alpha+\sin \beta=0$<br/>$\Rightarrow \sin \alpha=-\sin \beta$<br/>$\Rightarrow \sin \alpha=\sin (\pi+\beta)$<br/>$\Rightarrow \alpha=\pi+\beta$</div>


title: if-sin-sin-1-5-1-cos-1-x-1-then-what-is-x-equal-to
slug: if-sin-sin-1-5-1-cos-1-x-1-then-what-is-x-equal-to-10249
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$, then what is $x$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{5}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{5}$</span> </div>

answer:
<div class="solution">Let $\sin \left[\sin ^{-1}\left(\frac{1}{5}\right)+\cos ^{-1} x\right]=1$<br/>$\Rightarrow \sin ^{-1}\left(\frac{1}{5}\right)+\cos ^{-1} x=\sin ^{-1} 1$<br/>$\Rightarrow \sin ^{-1}\left(\frac{1}{5}\right)+\cos ^{-1} x=\frac{\pi}{2}$<br/>$\Rightarrow x=\frac{1}{5}$<br/>$\left(\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right)$</div>


title: if-sin-x-5-1-sin-y-10-1-where-0-x-2-0-y-2-then-what-is-x-y-equal-to
slug: if-sin-x-5-1-sin-y-10-1-where-0-x-2-0-y-2-then-what-is-x-y-equal-to-40238
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin x=\frac{1}{\sqrt{5}}, \sin y=\frac{1}{\sqrt{10}}$, where $0   &lt; x &lt; \frac{\pi}{2}, 0 &lt; y &lt; \frac{\pi}{2}$,<br/>then what is $(\mathrm{x}+\mathrm{y})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">$\sin x=\frac{1}{\sqrt{5}}, \sin y=\frac{1}{\sqrt{10}}, 0   &lt; x &lt; \frac{\pi}{2}, 0 &lt; y &lt; \frac{\pi}{2}$<br/>$\cos x=\sqrt{1-\sin ^{2} x} \quad \cos y=\sqrt{1-\sin ^{2} y}$<br/>$=\sqrt{1-\frac{1}{5}} \quad=\sqrt{1-\frac{1}{10}}$<br/>$=\sqrt{\frac{4}{5}}=\frac{2}{\sqrt{5}} \quad=\sqrt{\frac{9}{10}}=\frac{3}{\sqrt{10}} .$<br/>$\sin (x+y)=\sin x \cos y+\cos x \sin y$<br/>$=\frac{1}{\sqrt{5}} \cdot \frac{3}{\sqrt{10}}+\frac{2}{\sqrt{5}} \cdot \frac{1}{\sqrt{10}}$<br/>$=\frac{5}{\sqrt{5} \cdot \sqrt{10}}=\frac{\sqrt{5} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{10}}=\sqrt{\frac{5}{10}}=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}} .$<br/>$\therefore \quad x+y=\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4} .$</div>


title: if-sin-x-cos-y-2-1-then-what-is-the-value-of-dx-2-d-2-y-at-4-4
slug: if-sin-x-cos-y-2-1-then-what-is-the-value-of-dx-2-d-2-y-at-4-4-77600
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-4$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin x \cos y=\frac{1}{2}$, then what is the value of $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-6$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">$\sin x \cos y=\frac{1}{2}$<br/>Differentiating both the sides<br/>$\sin x(-\sin y) \frac{d y}{d x}+\cos y \cos x=0$<br/>$\frac{d y}{d x}=\frac{\cos x \cos y}{\sin x \sin y}=\cot x \cot y$<br/>$\frac{d^{2} y}{d x^{2}}=\cot x\left(-\operatorname{cosec}^{2} y\right) \frac{d y}{d x}+\cot y\left(-\operatorname{cosec}^{2} x\right)$<br/>when the point is $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$<br/>then $\mathrm{x}=\frac{\pi}{4}$ and $\mathrm{y}=\frac{\pi}{4}$<br/>$\frac{d y}{d x}=\cot \frac{\pi}{4} \cdot \cot \frac{\pi}{4}=1$<br/>So, $\frac{d^{2} y}{d x^{2}}=\cot \frac{\pi}{4}\left(-\cos e c^{2} \frac{\pi}{4}\right) 1+\cot \frac{\pi}{4}\left(-\cos e c^{2} \frac{\pi}{4}\right)$<br/>$=1(-2)+1 .(-2)=-4$</div>


title: if-sin-x-cosec-x-2-cos-x-sec-x-2-k-tan-2-x-cot-2-x-then-what-is-the-value-of-k
slug: if-sin-x-cosec-x-2-cos-x-sec-x-2-k-tan-2-x-cot-2-x-then-what-is-the-value-of-k-94420
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $(\sin x+\operatorname{cosec} x)^{2}+(\cos x+\sec x)^{2}=k+\tan ^{2} x+\cot ^{2} x$<br/>then what is the value of $k ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\&amp;</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\underline{7}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\underline{7}$</span> </div>

answer:
<div class="solution">Given, $(\sin x+\operatorname{cosec} x)^{2}+(\cos x+\sec x)^{2}$<br/>$=k+\tan ^{2} x+\cot ^{2} x$<br/>$\Rightarrow \sin ^{2} x+\operatorname{cosec}^{2} x+2 \sin x \operatorname{cosec} x+\cos ^{2} x+\sec ^{2} x$<br/>$+2 \sec x \cos x=k+\tan ^{2} x+\cot ^{2} x$<br/>$\Rightarrow \sin ^{2} x+\operatorname{cosec}^{2} x+2+\cos ^{2} x+\sec ^{2} x+2$<br/>$\quad=k+\tan ^{2} x+\cot ^{2} x$<br/>$(\because \sin x \operatorname{cosec} x=1$ and $\sec x \cos x=1)$<br/>$\Rightarrow 1+\operatorname{cosec}^{2} x-\cot ^{2} x+\sec ^{2} x-\tan ^{2} x+4=k$<br/>$\Rightarrow 1+1+1+4=k \Rightarrow k=7$</div>


title: if-sin-x-cosec-x-2-then-what-is-the-value-of-sin-4-x-cosec-4-x
slug: if-sin-x-cosec-x-2-then-what-is-the-value-of-sin-4-x-cosec-4-x-71212
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin x+\operatorname{cosec} x=2$, then what is the value of $\sin ^{4} x+\operatorname{cosec}^{4} x ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">Given $\sin x+\operatorname{cosec} x=2$<br/>$\begin{aligned} \text { Consider } \sin ^{4} x+\operatorname{cosec}^{4} x=\left(\sin ^{2} x+\operatorname{cosec}^{2} x\right)^{2} \\ &amp;-2\left(\sin ^{2} x \cos ^{2} x\right) \\=&amp; \left.\left[(\sin x+\operatorname{cosec} x)^{2}-2\right)\right]^{2}-2 \\=(4-2)^{2}-2=2 &amp; \end{aligned}$</div>


title: if-sin-x-sin-y-a-and-cos-x-cos-y-b-then-tan-2-2-x-y-tan-2-2-x-y-is-equal-to
slug: if-sin-x-sin-y-a-and-cos-x-cos-y-b-then-tan-2-2-x-y-tan-2-2-x-y-is-equal-to-79741
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin x+\sin y=a$ and $\cos x+\cos y=b$, then<br/>$\tan ^{2}\left(\frac{x+y}{2}\right)+\tan ^{2}\left(\frac{x-y}{2}\right)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{a^{4}+b^{4}+4 b^{2}}{a^{2} b^{2}+b^{4}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{a^{4}-b^{4}+4 b^{2}}{a^{2} b^{2}+b^{4}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{a^{4}-b^{4}+4 a^{2}}{a^{2} b^{2}+a^{4}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a^{4}-b^{4}+4 b^{2}}{a^{2} b^{2}+b^{4}}$</span> </div>

answer:
<div class="solution">$\sin x+\sin y=a$<br/>$\Rightarrow 2 \sin \left(\frac{\mathrm{x}+\mathrm{y}}{2}\right) \cos \left(\frac{\mathrm{x}-\mathrm{y}}{2}\right)=\mathrm{a}$<br/>$\cos x+\cos y=b$<br/>$\Rightarrow \quad 2 \cos \left(\frac{\mathrm{x}+\mathrm{y}}{2}\right) \cos \left(\frac{\mathrm{x}-\mathrm{y}}{2}\right)=\mathrm{b}$<br/>dividing eq $(1) \&amp;(2)$ $\tan \left(\frac{x+y}{2}\right)=\frac{a}{b}$<br/>Squaring eq (1) \&amp; (2) and adding $4 \cos ^{2}\left(\frac{\mathrm{x}-\mathrm{y}}{2}\right)=\mathrm{a}^{2}+\mathrm{b}^{2}$<br/>$\sec ^{2}\left(\frac{x-y}{2}\right)=\frac{4}{a^{2}+b^{2}}$<br/>again, $\tan ^{2}\left(\frac{x+y}{2}\right)+\tan ^{2}\left(\frac{x-y}{2}\right)$<br/>$=\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{2}+\sec ^{2}\left(\frac{\mathrm{x}-\mathrm{y}}{2}\right)-1$<br/>$=\frac{\mathrm{a}^{2}}{\mathrm{~b}^{2}}+\frac{4}{\mathrm{a}^{2}+\mathrm{b}^{2}}-1=\frac{\mathrm{a}^{4}-\mathrm{b}^{4}+4 \mathrm{~b}^{2}}{\mathrm{a}^{2} \mathrm{~b}^{2}+\mathrm{b}^{4}}$</div>


title: if-sin-x-sin-y-a-cos-x-cos-y-b-then-what-is-the-value-of-cos-x-y
slug: if-sin-x-sin-y-a-cos-x-cos-y-b-then-what-is-the-value-of-cos-x-y-43044
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin x+\sin y=a, \cos x+\cos y=b$, then what is the value<br/>of $\cos (x-y)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a^{2}-1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$b^{2}-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}\left(a^{2}+b^{2}-2\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}\left(a^{2}+b^{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2}\left(a^{2}+b^{2}-2\right)$</span> </div>

answer:
<div class="solution">Given $\sin x+\sin y=a$<br/>and $\cos x+\cos y=b$<br/>$\therefore a^{2}+b^{2}=(\sin x+\sin y)^{2}+(\cos x+\cos y)^{2}$<br/>$=\sin ^{2} x+\sin ^{2} y+2 \sin x \sin y+\cos ^{2} x+\cos ^{2} y$<br/>$+2 \cos x \cos y$<br/>$=\left(\sin ^{2} x+\cos ^{2} x\right)+\left(\sin ^{2} y+\cos ^{2} y\right)+2(\cos x \cos y$<br/>$+\sin x \sin y)$<br/>$=1+1+2 \cos (x-y)$<br/>$\Rightarrow 2 \cos (x-y)=a^{2}+b^{2}-2$<br/>$\Rightarrow \cos (x-y)=\frac{1}{2}\left(a^{2}+b^{2}-2\right)$</div>


title: if-sin-x-x-a-for-all-x-r-0-then-which-one-of-the-following-is-correct
slug: if-sin-x-x-a-for-all-x-r-0-then-which-one-of-the-following-is-correct-36811
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sin \theta=x+\frac{a}{x}$ for all $x \in R-\{0\}$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a \geq 4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a \geq \frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$a \leq \frac{1}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a \leq \frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a \leq \frac{1}{4}$</span> </div>

answer:
<div class="solution">Given equation is<br/>$\sin \theta=x+\frac{a}{x}, x \in R-\{0\}$<br/>$\Rightarrow x^{2}+a=x \sin \theta$<br/>$\Rightarrow x^{2}-x \sin \theta+a=0$<br/>Now, discriminant $=\sqrt{\sin ^{2} \theta-4 a}$<br/>For $x$ to be real root, discriminant $\geq 0$<br/>$\Rightarrow \sqrt{\sin ^{2} \theta-4 a} \geq 0$<br/>$\Rightarrow \sin ^{2} \theta-4 a \geq 0 \Rightarrow \sin ^{2} \theta \geq 4 a$<br/>$\Rightarrow \frac{1}{\sin ^{2} \theta} \leq \frac{1}{4 a} \Rightarrow a \leq \frac{\sin ^{2} \theta}{4}$<br/>$\Rightarrow a \leq \frac{1}{4}\left(\because \sin ^{2} \theta\right.$ lies between 0 and 1$)$</div>


tags: ['Chemistry', 'Chemistry in Everyday Life']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
title:
if-soap-has-high-alkali-content-it-irritates-skin-how-can-the-amount-of-excess-alkali-be-determined-what-can-be-the-source-of-excess-alkali

slug:
if-soap-has-high-alkali-content-it-irritates-skin-how-can-the-amount-of-excess-alkali-be-determined-what-can-be-the-source-of-excess-alkali-93444

question:
<div class="question">If soap has high alkali content it irritates skin. How can the amount of excess alkali be determined? What can be the source of excess alkali?</div>

answer:
<div class="solution">Acid-base titration can be used to determine the excess amount of alkali in soap. A soap solution is titrated with standard hydrochloric acid, phenolphthalein is used as an indicator. The source of this excess alkali is the alkali left unused when the soap is prepared by hydrolysis of fat.</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-sound-travels-in-air-with-the-speed-of-340-m-s-1-then-the-number-of-tones-present-in-an-open-organ-pipe-of-length-2-m-for-a-maximum-frequency-of-12-1

slug:
if-sound-travels-in-air-with-the-speed-of-340-m-s-1-then-the-number-of-tones-present-in-an-open-organ-pipe-of-length-2-m-for-a-maximum-frequency-of-12-1-36362

question:
<div class="question">If sound travels in air with the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, then the number of tones present in an open organ pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math> for a maximum frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn><mo> </mo><mi>Hz</mi></math>, are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> </div>

answer:
<div class="solution"><p>Given, speed of sound, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/>Length of open organ pipe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>Frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>1200</mn><mo> </mo><mi>Hz</mi></math><br/>For open organ pipe,</p><p>Fundamental frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mn>340</mn><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>85</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Number of tones present in the open organ pipe<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>f</mi><msub><mi>f</mi><mn>0</mn></msub></mfrac><mo>=</mo><mfrac><mn>1200</mn><mn>85</mn></mfrac><mo>=</mo><mn>14</mn><mo>.</mo><mn>11</mn><mo>≃</mo><mn>14</mn></math></p></div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-sound-travels-in-air-with-the-speed-of-340-m-s-1-then-the-number-of-tones-present-in-an-open-organ-pipe-of-length-2-m-for-a-maximum-frequency-of-12

slug:
if-sound-travels-in-air-with-the-speed-of-340-m-s-1-then-the-number-of-tones-present-in-an-open-organ-pipe-of-length-2-m-for-a-maximum-frequency-of-12-44624

question:
<div class="question">If sound travels in air with the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, then the number of tones present in an open organ pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math> for a maximum frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn><mo> </mo><mi>Hz</mi></math>, are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> </div>

answer:
<div class="solution"><p>Given, speed of sound, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>340</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/>Length of open organ pipe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>Frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>1200</mn><mo> </mo><mi>Hz</mi></math><br/>For open organ pipe,</p><p>Fundamental frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mn>340</mn><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>85</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Number of tones present in the open organ pipe<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>f</mi><msub><mi>f</mi><mn>0</mn></msub></mfrac><mo>=</mo><mfrac><mn>1200</mn><mn>85</mn></mfrac><mo>=</mo><mn>14</mn><mo>.</mo><mn>11</mn><mo>≃</mo><mn>14</mn></math></p></div>


tags: ['Physics', 'Units and Dimensions', 'AP EAMCET', 'AP EAMCET 2023 (16 May Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-speed-of-light-in-vacuum-3-1-0-8-ms-1-acceleration-due-to-gravity-10-ms-2-and-mass-of-electron-91-1-0-31-kg-are-taken-as-fundamental-physical-quant

slug:
if-speed-of-light-in-vacuum-3-1-0-8-ms-1-acceleration-due-to-gravity-10-ms-2-and-mass-of-electron-91-1-0-31-kg-are-taken-as-fundamental-physical-quant-48030

question:
<div class="question">If speed of light in vacuum $\left(3 \times 10^8 \mathrm{~ms}^{-1}\right)$, acceleration due to gravity $\left(10 \mathrm{~ms}^{-2}\right)$ and mass of electron $\left(9.1 \times 10^{-31} \mathrm{~kg}\right)$ are taken as fundamental physical quantities, then the unit of time in this system is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \times 10^3 \mathrm{~s}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 \times 10^{-19} \mathrm{~s}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 \times 10^{19} \mathrm{~s}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3 \times 10^7 \mathrm{~s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \times 10^7 \mathrm{~s}$</span> </div>

answer:
<div class="solution">Speed of light, $\left[\mathrm{L}  \mathrm{T}^{-1}\right]=3 \times 10^8 \mathrm{~m} / \mathrm{s}$ ... (1) acceleration due to gravity, $\left[\mathrm{L} \mathrm{T}^{-2}\right]=10 \mathrm{~m} / \mathrm{s}^{-2}$ ... (2)<br/>Divide equation (1) by (2), we have<br/>$\frac{\left[\mathrm{L} \mathrm{T}^{-1}\right]}{\left[\mathrm{LT}^{-2}\right]}=\frac{3 \times 10^8}{10} \Rightarrow[\mathrm{T}]=3 \times 10^7 \mathrm{~s}$</div>


tags: ['Physics', 'Waves and Sound', 'AIIMS', 'AIIMS 2019 (25 May)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$6$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-speed-of-sound-in-air-in-330-m-s-1-then-find-the-number-of-overtones-present-in-an-open-organ-pipe-of-length-1-m-whose-frequency-f-1000

slug:
if-speed-of-sound-in-air-in-330-m-s-1-then-find-the-number-of-overtones-present-in-an-open-organ-pipe-of-length-1-m-whose-frequency-f-1000-78055

question:
<div class="question">If speed of sound in air in $330 \mathrm{~m} \mathrm{~s}^{-1}$, then find the number of overtones present in an open organ pipe of length $1 \mathrm{~m}$ whose frequency $f \leq 1000$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Fundamental frequency,<br/>$\begin{aligned} &amp; v=\frac{v}{2 \ell}=\frac{330}{2 \times 1}=165 \mathrm{~Hz} \\ &amp; \therefore \text { Number of overtones allowed }=\frac{1000}{165}=6\end{aligned}$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2019 (11 Jan Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-speed-v-acceleration-a-and-force-f-are-considered-as-fundamental-units-the-dimension-of-youngs-modulus-will-be-1

slug:
if-speed-v-acceleration-a-and-force-f-are-considered-as-fundamental-units-the-dimension-of-youngs-modulus-will-be-1-80899

question:
<div class="question">If speed (V), acceleration (A) and force (F) are considered
<br/> <br/>as fundamental units, the dimension of Young's modulus
<br/> <br/>will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{V}^{-2} \mathrm{~A}^{2} \mathrm{~F}^{-2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{V}^{-2} \mathrm{~A}^{2} \mathrm{~F}^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{V}^{-4} \mathrm{~A}^{-2} \mathrm{~F}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{V}^{-4} \mathrm{~A}^{2} \mathrm{~F}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{V}^{-4} \mathrm{~A}^{2} \mathrm{~F}$</span> </div>

answer:
<div class="solution">Let [Y] = [V] $^{a}[\mathrm{~F}]^{\mathrm{b}}[\mathrm{A}]^{\mathrm{c}}$
<br/> <br/>$\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\mathrm{LT}^{-1}\right]^{\mathrm{a}}\left[\mathrm{MLT}^{-2}\right]^{\mathrm{b}}\left[\mathrm{LT}^{-2}\right]^{\mathrm{c}}$
<br/> <br/>$\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\mathrm{M}^{b} \mathrm{~L}^{\mathrm{a}+b+\mathrm{c}} \mathrm{T}^{-\mathrm{a}-2 \mathrm{~b}-2 \mathrm{c}}\right]$
<br/> <br/>Comparing power both side of similar terms we get,
<br/> <br/>$b=1, a+b+c=-1,-a-2 b-2 c=-2$
<br/> <br/>solving above equations we get
<br/> <br/>$a=-4, b=1, c=2$
<br/> <br/>so $[\mathrm{Y}]=\left[\mathrm{V}^{-4} \mathrm{FA}^{2}\right]=\left[\mathrm{V}^{-4} \mathrm{~A}^{2} \mathrm{~F}\right]$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-speed-v-acceleration-a-and-force-f-are-considered-as-fundamental-units-the-dimension-of-youngs-modulus-will-be

slug:
if-speed-v-acceleration-a-and-force-f-are-considered-as-fundamental-units-the-dimension-of-youngs-modulus-will-be-91380

question:
<div class="question">If speed <math><mfenced separators="|"><mrow><mi>V</mi></mrow></mfenced><mo>,</mo></math> acceleration <math><mfenced separators="|"><mrow><mi>A</mi></mrow></mfenced></math> and force <math><mfenced separators="|"><mrow><mi>F</mi></mrow></mfenced></math> are considered as fundamental units, the dimension of Young's modulus will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup><msup><mi mathvariant="normal">F</mi><mn>2</mn></msup></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup><msup><mi mathvariant="normal">F</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mi mathvariant="normal">F</mi></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup><mi mathvariant="normal">F</mi></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup><mi mathvariant="normal">F</mi></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Let <math><mi>Y</mi><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>V</mi><mo>,</mo><mi> </mi><mi>F</mi><mo>,</mo><mi> </mi><mi>A</mi></mrow></mfenced></math><br/><br/><math><mi>Y</mi><mo>=</mo><mi>K</mi><msup><mrow><mi>V</mi></mrow><mrow><mi>x</mi></mrow></msup><msup><mrow><mi>F</mi></mrow><mrow><mi>y</mi></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mi>z</mi></mrow></msup><mo>,</mo><mi> </mi><mi>K</mi><mo>→</mo></math> Unit less<br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>Y</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>V</mi></mrow></mfenced></mrow><mrow><mi>x</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>F</mi></mrow></mfenced></mrow><mrow><mi>y</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow><mrow><mi>z</mi></mrow></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi>ML</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mi mathvariant="normal">x</mi></msup><msup><mfenced close="]" open="[" separators="|"><msup><mi>MLT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">y</mi></msup><msup><mfenced close="]" open="[" separators="|"><msup><mi>LT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">z</mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">M</mi></mfenced><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">M</mi></mfenced><mi mathvariant="normal">y</mi></msup><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">L</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mi mathvariant="normal">z</mi></mrow></msup></mfenced><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">z</mi></mrow></msup></mfenced></math><br/><br/><math><mo>⇒</mo><mi> </mi><mi> </mi><mi>y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi> </mi><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>;</mo><mi> </mi><mo>-</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn></math><br/><br/><math><mi>x</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi> </mi><mi> </mi><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>2</mn></math><br/><br/><math><mi>z</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi> </mi><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn><mi> </mi><mi> </mi><mi>x</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>0</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|" separators="|"><mi mathvariant="normal">Y</mi></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup><mi mathvariant="normal">F</mi></mrow></mfenced></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2020 (02 Sep Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-speed-v-area-a-and-force-f-are-chosen-as-fundamental-units-then-the-dimension-of-youngs-modulus-will-be

slug:
if-speed-v-area-a-and-force-f-are-chosen-as-fundamental-units-then-the-dimension-of-youngs-modulus-will-be-40553

question:
<div class="question">If speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">V</mi></math>, area <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and force <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi></math> are chosen as fundamental units, then the dimension of Young's modulus will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>FA</mi><mn>2</mn></msup><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> </span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>FA</mi><mn>2</mn></msup><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>FA</mi><mn>2</mn></msup><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>FA</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">V</mi><mn>0</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>FA</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">V</mi><mn>0</mn></msup></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>∝</mo><msup><mi>F</mi><mi>a</mi></msup><msup><mi>V</mi><mi>b</mi></msup><msup><mi>A</mi><mi>c</mi></msup><mo>  </mo><mi>Y</mi><mo>=</mo><mfenced><mfrac><mi>F</mi><mi>A</mi></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><msup><mi>L</mi><mn>2</mn></msup></mfrac><mo>∝</mo><msup><mfenced><mrow><msup><mi>M</mi><mn>1</mn></msup><msup><mi>L</mi><mn>1</mn></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mi>a</mi></msup><msup><mfenced><mrow><msup><mi>L</mi><mn>1</mn></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mi>b</mi></msup><msup><mfenced><msup><mi>L</mi><mn>2</mn></msup></mfenced><mi>c</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>∝</mo><msup><mi mathvariant="normal">M</mi><mi mathvariant="normal">a</mi></msup><msup><mi>La</mi><mo>+</mo></msup><mi mathvariant="normal">b</mi><mo>+</mo><mn>2</mn><mi>cT</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mi mathvariant="normal">b</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><mi>b</mi><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><mi>c</mi><mo> </mo><mo>=</mo><mo> </mo><mo>–</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>–</mo><mn>2</mn><mi>a</mi><mo> </mo><mo>+</mo><mo> </mo><mi>b</mi><mo> </mo><mo>=</mo><mo> </mo><mo>–</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mi>b</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mi>c</mi><mo> </mo><mo>=</mo><mo> </mo><mo>–</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><msup><mi>F</mi><mn>1</mn></msup><msup><mi>V</mi><mn>0</mn></msup><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-speed-v-force-f-and-acceleration-a-are-chosen-as-the-fundamental-physical-quantities-then-the-dimension-of-youngs-modulus-in-terms-of-v-f-and-a-are-1

slug:
if-speed-v-force-f-and-acceleration-a-are-chosen-as-the-fundamental-physical-quantities-then-the-dimension-of-youngs-modulus-in-terms-of-v-f-and-a-are-1-81628

question:
<div class="question">If speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">V</mi></math>, force <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi></math> and acceleration <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi></math> are chosen as the fundamental physical quantities, then the dimension of Young's modulus in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">V</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi></math> are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mi>Fa</mi></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">F</mi><mn>2</mn></msup><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>Fa</mi><mn>2</mn></msup></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">F</mi><mn>2</mn></msup><mi mathvariant="normal">a</mi></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>Fa</mi><mn>2</mn></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Y</mi><mo>=</mo><mi mathvariant="normal">f</mi><mfenced separators="|"><mrow><mi mathvariant="normal">V</mi><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">F</mi><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">a</mi></mrow></mfenced></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">Y</mi><mo>=</mo><msup><mi>KV</mi><mi mathvariant="normal">x</mi></msup><msup><mi mathvariant="normal">F</mi><mi mathvariant="normal">y</mi></msup><msup><mi mathvariant="normal">a</mi><mi mathvariant="normal">z</mi></msup><mo>,</mo><mi mathvariant="normal"> </mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">K</mi><mo>→</mo></math>Unitless<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">Y</mi></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">V</mi></mfenced><mi mathvariant="normal">x</mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">F</mi></mfenced><mi mathvariant="normal">y</mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">a</mi></mfenced><mi mathvariant="normal">z</mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mi>ML</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mi mathvariant="normal">x</mi></msup><msup><mfenced close="]" open="[" separators="|"><msup><mi>MLT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">y</mi></msup><msup><mfenced close="]" open="[" separators="|"><msup><mi>LT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">z</mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">M</mi></mfenced><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><msup><mo>=</mo><mi mathvariant="normal"> </mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">M</mi></mfenced><mi mathvariant="normal">y</mi></msup><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">L</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mi mathvariant="normal">z</mi></mrow></msup></mfenced><msup><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">z</mi></mrow></msup></mfenced><mi mathvariant="normal"> </mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn><mo>;</mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mi mathvariant="normal">z</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>;</mo><mi mathvariant="normal"> </mi><mo> </mo><mo> </mo><mo> </mo><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">z</mi><mo>=</mo><mn>2</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">z</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>∴</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">z</mi><mo>=</mo><mn>2</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">z</mi><mo>=</mo><mn>0</mn></math>              <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">z</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">x</mi><mo>=</mo><mo>-</mo><mn>4</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">Y</mi></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>Fa</mi><mn>2</mn></msup></mrow></mfenced></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-speed-v-force-f-and-acceleration-a-are-chosen-as-the-fundamental-physical-quantities-then-the-dimension-of-youngs-modulus-in-terms-of-v-f-and-a-are

slug:
if-speed-v-force-f-and-acceleration-a-are-chosen-as-the-fundamental-physical-quantities-then-the-dimension-of-youngs-modulus-in-terms-of-v-f-and-a-are-97299

question:
<div class="question">If speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">V</mi></math>, force <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi></math> and acceleration <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi></math> are chosen as the fundamental physical quantities, then the dimension of Young's modulus in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">V</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi></math> are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mi>Fa</mi></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">F</mi><mn>2</mn></msup><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>Fa</mi><mn>2</mn></msup></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi mathvariant="normal">F</mi><mn>2</mn></msup><mi mathvariant="normal">a</mi></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>Fa</mi><mn>2</mn></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Y</mi><mo>=</mo><mi mathvariant="normal">f</mi><mfenced separators="|"><mrow><mi mathvariant="normal">V</mi><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">F</mi><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">a</mi></mrow></mfenced></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">Y</mi><mo>=</mo><msup><mi>KV</mi><mi mathvariant="normal">x</mi></msup><msup><mi mathvariant="normal">F</mi><mi mathvariant="normal">y</mi></msup><msup><mi mathvariant="normal">a</mi><mi mathvariant="normal">z</mi></msup><mo>,</mo><mi mathvariant="normal"> </mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">K</mi><mo>→</mo></math>Unitless<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">Y</mi></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">V</mi></mfenced><mi mathvariant="normal">x</mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">F</mi></mfenced><mi mathvariant="normal">y</mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">a</mi></mfenced><mi mathvariant="normal">z</mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mi>ML</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mi mathvariant="normal">x</mi></msup><msup><mfenced close="]" open="[" separators="|"><msup><mi>MLT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">y</mi></msup><msup><mfenced close="]" open="[" separators="|"><msup><mi>LT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">z</mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">M</mi></mfenced><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><msup><mo>=</mo><mi mathvariant="normal"> </mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">M</mi></mfenced><mi mathvariant="normal">y</mi></msup><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">L</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mi mathvariant="normal">z</mi></mrow></msup></mfenced><msup><mfenced close="]" open="[" separators="|"><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">z</mi></mrow></msup></mfenced><mi mathvariant="normal"> </mi></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn><mo>;</mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mi mathvariant="normal">z</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>;</mo><mi mathvariant="normal"> </mi><mo> </mo><mo> </mo><mo> </mo><mo>-</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">z</mi><mo>=</mo><mn>2</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">z</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>∴</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">z</mi><mo>=</mo><mn>2</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">z</mi><mo>=</mo><mn>0</mn></math>              <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal">z</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">x</mi><mo>=</mo><mo>-</mo><mn>4</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">Y</mi></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>Fa</mi><mn>2</mn></msup></mrow></mfenced></math></div>


