# questions

tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-2$</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-c-o-s-2-8-c-o-s-8-8-c-o-s-17-8-s-i-n-20-8-s-i-n-3-4-c-o-s-23-6-s-i-n-5-6-s-i-n-12-4

slug:
what-is-the-value-of-c-o-s-2-8-c-o-s-8-8-c-o-s-17-8-s-i-n-20-8-s-i-n-3-4-c-o-s-23-6-s-i-n-5-6-s-i-n-12-4-34715

question:
<div class="question">What is the value of <br/>$\frac{\sin 34^{\circ} \cos 236^{\circ}-\sin 56^{\circ} \sin 124^{\circ}}{\cos 28^{\circ} \cos 88^{\circ}+\cos 178^{\circ} \sin 208^{\circ}} ?$</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">$-1$</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">$\underline{1}$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} \frac{\sin 34^{\circ} \cos 236^{\circ}-\sin 56^{\circ} \sin 124^{\circ}}{\cos 28^{\circ} \cos 88^{\circ}+\cos 178^{\circ} \sin 208^{\circ}} \\= \frac{\sin 34^{\circ}\left(-\cos 56^{\circ}\right)-\sin 56^{\circ} \cos 34^{\circ}}{\cos 28^{\circ} \sin 2^{\circ}+\cos 2^{\circ} \sin 28^{\circ}} \\= \frac{-\sin 34^{\circ} \cos 56^{\circ}-\sin 56^{\circ} \cos 34^{\circ}}{\sin \left(28^{\circ}+2^{\circ}\right)} \\= \frac{\sin \left(34^{\circ}+56^{\circ}\right)}{\sin 30^{\circ}}=\frac{-\sin 90^{\circ}}{\sin 30^{\circ}}=\frac{-1}{\frac{1}{2}}=-2 \\ \end{aligned}$</div>


title: what-is-the-value-of-c-o-s-3-1-5-c-o-s-3-4-5-c-o-s-1-5-c-o-s-4-5
slug: what-is-the-value-of-c-o-s-3-1-5-c-o-s-3-4-5-c-o-s-1-5-c-o-s-4-5-36039
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\frac{\cos 15^{\circ}+\cos 45^{\circ}}{\cos ^{3} 15^{\circ}+\cos ^{3} 45^{\circ}}$ ?</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=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</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">Consider $\frac{\cos 15^{\circ}+\cos 45^{\circ}}{\cos ^{3} 15^{\circ}+\cos ^{3} 45^{\circ}}$<br/>$=\frac{\cos 15^{\circ}+\cos 45^{\circ}}{\left(\cos 15^{\circ}+\cos 45^{\circ}\right)\left(\cos ^{2} 45^{\circ}+\cos ^{2} 15^{\circ}-\cos 45^{\circ} \cos 15^{\circ}\right)}$<br/>$\left(\because a^{3}+b^{3}=(a+b)\left(a^{2}+b^{2}-a b\right)\right)$<br/>$=\frac{1}{\left(\cos ^{2} 45^{\circ}+\cos ^{2} 15^{\circ}-\cos 45^{\circ} \cos 15^{\circ}\right)}$<br/>$=\frac{1}{\frac{1}{2}+\left(\cos 45^{\circ} \cos 30^{\circ}+\sin 45^{\circ} \sin 30^{\circ}\right)^{2}-\frac{\cos 15^{\circ}}{\sqrt{2}}}$<br/>$\left[\because \cos 15^{\circ}=\cos \left(45^{\circ}-30^{\circ}\right)\right]$<br/>$=\frac{1}{\frac{1}{2}+\left(\frac{\sqrt{3}}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}\right)^{2}-\frac{1}{\sqrt{2}}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)}$<br/>$=\frac{1}{\frac{1}{2}+\frac{3+1+2 \sqrt{3}}{8}-\frac{\sqrt{3}+1}{4}}$<br/>$=\frac{1}{\frac{4+4+2 \sqrt{3}-2 \sqrt{3}-2}{8}}=\frac{8}{6}=\frac{4}{3}$</div>


title: what-is-the-value-of-c-o-t-2-s-e-c-2-s-e-c-3-2-cosec-c-o-t-9-2-c-o-s-cc-2-2
slug: what-is-the-value-of-c-o-t-2-s-e-c-2-s-e-c-3-2-cosec-c-o-t-9-2-c-o-s-cc-2-2-33897
tags: ['Mathematics', 'Trigonometric Ratios & Identities', '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_P4.db
question:
<div class="question">What is the value of<br/>$\frac{\operatorname{cosec}(\pi+\theta) \cot \{(9 \pi / 2-\theta)\} \cos \operatorname{cc}^{2}(2 \pi-\theta)}{\cot (2 \pi-\theta) \sec ^{2}(\pi-\theta) \sec \{(3 \pi / 2)+\theta\}}$</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">$\infty$</span> </li> </ul>

answer:
<div class="solution">The expression<br/>$\frac{\cos \operatorname{cc}(\pi+\theta) \cot \left\{\frac{9 \pi}{2}-\theta\right\} \operatorname{cosec}^{2}(2 \pi-\theta)}{\cot (2 \pi-\theta) \sec ^{2}(\pi-\theta) \sec \left\{\left(\frac{3 \pi}{2}\right)+\theta\right\}}$<br/>$=\frac{-\cos \operatorname{cc} \theta \cdot \tan \theta \cos \operatorname{cc}^{2} \theta}{-\cot \theta \sec ^{2} \theta \operatorname{cosec} \theta}$<br/>$=\frac{\tan ^{2} \theta \cos \operatorname{cc}^{2} \theta}{\sec ^{2} \theta}=\tan ^{2} \theta \times \frac{\cos ^{2} \theta}{\sin ^{2} \theta}$<br/>$=\tan ^{2} \theta \times \frac{1}{\tan ^{2} \theta}=1$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-consider-the-function-f-x-2-x-c-o-s-x-3-if-if-x-2-x-2-which-is-continuous-at-x-2-where-is-a-constant

slug:
what-is-the-value-of-consider-the-function-f-x-2-x-c-o-s-x-3-if-if-x-2-x-2-which-is-continuous-at-x-2-where-is-a-constant-82345

question:
<div class="question">What is the value of $\alpha ?$<br/>Consider the function $f(x)=\left\{\begin{array}{ccc}\frac{\alpha \cos x}{\pi-2 x} &amp; \text { If } &amp; x \neq \frac{\pi}{2} \\ 3 &amp; \text { If } &amp; x=\frac{\pi}{2}\end{array}\right.$<br/>Which is continuous at $x=\frac{\pi}{2}$, where $\alpha$ is a constant.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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><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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ccc}\frac{\alpha \cos \mathrm{x}}{\pi-2 \mathrm{x}} &amp; \text { if } &amp; \mathrm{x} \neq \frac{\pi}{2} \\ 3 &amp; \text { if } &amp; \mathrm{x}=\frac{\pi}{2}\end{array}\right.$<br/>For continuity at $\mathrm{x}=\frac{\pi}{2}$<br/>L.H.L. $=\lim _{\mathrm{x} \rightarrow \frac{\pi^{-}}{2}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow \frac{\pi^{-}}{2}} \frac{\alpha \cos \mathrm{x}}{\pi-2 \mathrm{x}}$<br/>Put $\mathrm{x}=\frac{\pi}{2}-\mathrm{h}$ where $\mathrm{x} \rightarrow \frac{\pi}{2}$, then $\mathrm{h} \rightarrow 0$<br/>$\therefore \quad$ L.H.L. $=\lim _{\mathrm{h} \rightarrow 0} \frac{\alpha \cos \left(\frac{\pi}{2}-\mathrm{h}\right)}{\pi-2\left(\frac{\pi}{2}-\mathrm{h}\right)}=\lim _{\mathrm{h} \rightarrow 0} \frac{\alpha \sin \mathrm{h}}{\pi-\pi+2 \mathrm{~h}}$</div>


title: what-is-the-value-of-cos-1-0-cos-11-0-cos-13-0
slug: what-is-the-value-of-cos-1-0-cos-11-0-cos-13-0-87526
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_P4.db
question:
<div class="question">What is the value of $\cos 10^{\circ}+\cos 110^{\circ}+\cos 130^{\circ} ?$</div>

answer:
<div class="solution">Consider $\cos 10^{\circ}+\cos 110^{\circ}+\cos 130^{\circ}$<br/>$=\cos 130^{\circ}+\cos 10^{\circ}+\cos 110^{\circ}$<br/>$=2 \cos \left(\frac{130+10}{2}\right) \cos \left(\frac{130-10}{2}\right)+\cos 110^{\circ}$<br/>$=2 \cos \left(\frac{140}{2}\right) \cos \left(\frac{120}{2}\right)+\cos 110^{\circ}$<br/>$=2 \cos 60^{\circ} \cos 70^{\circ}+\cos 110^{\circ}$<br/>$=\cos 70^{\circ}+\cos 110^{\circ} \quad\left(\because \cos 60^{\circ}=\frac{1}{2}\right)$<br/>$=\cos \left(180^{\circ}-110^{\circ}\right)+\cos 110^{\circ}$<br/>$=-\cos 110^{\circ}+\cos 110^{\circ}=0\left(\because \cos \left(180^{\circ}-\theta\right)=-\cos \theta\right)$</div>


title: what-is-the-value-of-cos-1-5-cos-4-5-sin-1-5-sin-4-5-cos-4-5-sin-4-5-cos-1-5-sin-1-5
slug: what-is-the-value-of-cos-1-5-cos-4-5-sin-1-5-sin-4-5-cos-4-5-sin-4-5-cos-1-5-sin-1-5-80030
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\left|\begin{array}{ll}\cos 15^{\circ} &amp; \sin 15^{\circ} \\ \cos 45^{\circ} &amp; \sin 45^{\circ}\end{array}\right| \times\left|\begin{array}{ll}\cos 45^{\circ} &amp; \cos 15^{\circ} \\ \sin 45^{\circ} &amp; \sin 15^{\circ}\end{array}\right| ?$</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">$\frac{\sqrt{3}}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{1}{4}$</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">$-\frac{3}{4}$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\left|\begin{array}{lll}\cos 15^{\circ} &amp; \sin 15^{\circ} \\ \cos 45^{\circ} &amp; \sin 45^{\circ}\end{array}\right| \times\left|\begin{array}{cc}\cos 45^{\circ} &amp; \cos 15^{\circ} \\ \sin 45^{\circ} &amp; \sin 15^{\circ}\end{array}\right|$<br/>$=\left(\sin 45^{\circ} \cos 15^{\circ}-\cos 45^{\circ} \sin 15^{\circ}\right)$<br/>$\times\left(\cos 45^{\circ} \sin 15^{\circ}-\sin 45^{\circ} \cos 15^{\circ}\right)$<br/>$=-\sin \left(45^{\circ}-15^{\circ}\right) \times \sin \left(45^{\circ}-15^{\circ}\right)$<br/>$($ using $\sin (A-B)=-\cos A \sin B+\sin A \cos B)$<br/>$=-\sin 30^{\circ} \times \sin 30^{\circ}=-\frac{1}{2} \times \frac{1}{2}=-\frac{1}{4}$</div>


title: what-is-the-value-of-cos-1-5
slug: what-is-the-value-of-cos-1-5-83092
question: <div class="question">What is the value of $\cos 15^{\circ}$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}(\sqrt{2-\sqrt{3}})$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}(\sqrt{2+\sqrt{3}})$</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">$\sqrt{2}+\sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{2}-\sqrt{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2}(\sqrt{2+\sqrt{3}})$</span> </div>

answer:
<div class="solution">$\cos 2 \theta=2 \cos ^{2} \theta-1$<br/>Put $\theta=15^{\circ}$<br/>$\therefore \cos 30^{\circ}=2 \cos ^{2} 15^{\circ}-1$<br/>$\Rightarrow \frac{\sqrt{3}}{2}+1=2 \cos ^{2} 15^{\circ}$<br/>$\Rightarrow \cos ^{2} 15^{\circ}=\frac{\sqrt{3}+2}{4}$<br/>$\Rightarrow \cos 15^{\circ}=\frac{1}{2}(\sqrt{2+\sqrt{3}})$</div>


title: what-is-the-value-of-cos-2-cos-1-08
slug: what-is-the-value-of-cos-2-cos-1-08-55437
question: <div class="question">What is the value of $\cos \left(2 \cos ^{-1}(0.8) ?\right.$</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.28$</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.81$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.56$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.48$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.28$</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">We know that<br/>$2 \cos ^{-1} x=\cos ^{-1}\left(2 x^{2}-1\right)$<br/>here, $x=0.8$<br/>$\therefore \quad 2 \cos ^{-1}(0.8)=\cos ^{-1}\left(2(0.8)^{2}-1\right)$<br/>$=\cos ^{-1}(0.28)$<br/>Now, $\cos \left(\cos ^{-1}(x)\right)=x$<br/>$\therefore \cos \left(\cos ^{-1}(0.28)\right)=0.28$</div>


title: what-is-the-value-of-cos-3-6
slug: what-is-the-value-of-cos-3-6-70160
question: <div class="question">What is the value of $\cos 36^{\circ} ?$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P4.db
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">$\frac{\sqrt{10+2 \sqrt{5}}}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{10-2 \sqrt{5}}}{4}$</span> </li> </ul>

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


title: what-is-the-value-of-cos-4-6-cos-4-7-cos-4-8-cos-4-9-cos-5-0-cos-13-5
slug: what-is-the-value-of-cos-4-6-cos-4-7-cos-4-8-cos-4-9-cos-5-0-cos-13-5-43589
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\cos 46^{\circ} \cos 47^{\circ} \cos 48^{\circ} \cos 49^{\circ} \cos$<br/>$50^{\circ} \ldots \cos 135^{\circ} ?$</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">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">Greater than 1</span> </li> </ul>

answer:
<div class="solution">$\cos 46^{\circ} \cos 47^{\circ} \cos 48^{\circ} \cos 49^{\circ} \ldots . . \cos 135^{\circ}$<br/>We know, $\cos 90^{\circ}=0$<br/>$\therefore$ Given expression has $\cos 90^{\circ}$ and so it's value is 0 .</div>


title: what-is-the-value-of-cos-9-cos-3-cos-5-9-cos-7-9
slug: what-is-the-value-of-cos-9-cos-3-cos-5-9-cos-7-9-48423
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\cos (\pi / 9)+\cos (\pi / 3)+\cos (5 \pi / 9)+\cos (7 \pi / 9) ?$</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=""> <span class="option-label">C</span> <span class="option-data">$-1 / 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 2$</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> </ul>

answer:
<div class="solution">Consider $\cos \frac{\pi}{9}+\cos \frac{\pi}{3}+\cos \frac{5 \pi}{9}+\cos \frac{7 \pi}{9}$<br/>$=\cos \frac{\pi}{9}+\frac{1}{2}+\cos \frac{5 \pi}{9}+\cos \frac{7 \pi}{9}\left(\because \cos \frac{\pi}{3}=\frac{1}{2}\right)$<br/>$=\frac{1}{2}+\left(\cos \frac{\pi}{9}+\cos \frac{5 \pi}{9}\right)+\cos \frac{7 \pi}{9}$<br/>$=\frac{1}{2}+\left[2 \cos \frac{6 \pi}{18} \cos \frac{4 \pi}{18}\right]+\cos \frac{7 \pi}{9}$<br/>$\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$<br/>$=\frac{1}{2}+\left[2 \cos \frac{\pi}{3} \cos \frac{2 \pi}{9}\right]+\cos \frac{7 \pi}{9}$<br/>$=\frac{1}{2}+\left[2 \cdot \frac{1}{2} \cos \frac{2 \pi}{9}\right]+\cos \frac{7 \pi}{9}$<br/>$=\frac{1}{2}+\cos \frac{2 \pi}{9}+\cos \frac{7 \pi}{9}=\frac{1}{2}+2 \cos \left(\frac{9 \pi}{18}\right) \cos \left(\frac{5 \pi}{18}\right)$<br/>$=\frac{1}{2}+2 \cos \frac{\pi}{2} \cos \frac{5 \pi}{18}=\frac{1}{2}\left(\because \cos \frac{\pi}{2}=0\right)$</div>


title: what-is-the-value-of-cos-30-6-cos-23-4-cos-16-2-cos-1-8
slug: what-is-the-value-of-cos-30-6-cos-23-4-cos-16-2-cos-1-8-55037
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\cos 306^{\circ}+\cos 234^{\circ}+\cos 162^{\circ}+\cos 18^{\circ} ?$</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">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">2</span> </li> </ul>

answer:
<div class="solution">Given expression is:<br/>$\cos 306^{\circ}+\cos 234^{\circ}+\cos 162^{\circ}+\cos 18^{\circ}=\mathrm{x}($ Let $)$<br/>$\Rightarrow \mathrm{x}=\cos \left(360^{\circ}-54^{\circ}\right)+\cos \left(180^{\circ}+54^{\circ}\right)$<br/>$+\cos \left(180^{\circ}-18^{\circ}\right)+\cos 18^{\circ}$<br/>$=\cos 54^{\circ}-\cos 54^{\circ}-\cos 18^{\circ}+\cos 18^{\circ}=0$</div>


title: what-is-the-value-of-cos-cos-1-5-4-cos-1-13-12
slug: what-is-the-value-of-cos-cos-1-5-4-cos-1-13-12-34752
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\cos \left\{\cos ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}\right\} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$63 / 65$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$33 / 65$</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">$22 / 65$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$11 / 65$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$33 / 65$</span> </div>

answer:
<div class="solution">Let $\cos ^{-1} \frac{4}{5}=\mathrm{A} \Rightarrow \cos \mathrm{A}=\frac{4}{5} \Rightarrow \sin \mathrm{A}=\frac{3}{5}$<br/>Let $\cos ^{-1} \frac{12}{13}=\mathrm{B} \Rightarrow \cos \mathrm{B}=\frac{12}{13} \Rightarrow \sin \mathrm{B}=\frac{5}{13}$<br/>Now, $\cos \left(\cos ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}\right)=\cos (\mathrm{A}+\mathrm{B})$<br/>$=\cos \mathrm{A} \cos \mathrm{B}-\sin \mathrm{A} \sin \mathrm{B}$<br/>$=\left(\frac{4}{5}\right)\left(\frac{12}{13}\right)-\left(\frac{3}{5}\right)\left(\frac{5}{13}\right)=\frac{33}{65}$</div>


title: what-is-the-value-of-cos-tan-1-tan-4-15
slug: what-is-the-value-of-cos-tan-1-tan-4-15-25085
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of :<br/>$\cos \left[\tan ^{-1}\left\{\tan \left(\frac{15 \pi}{4}\right)\right\}\right] ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\sqrt{2}}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2 \sqrt{2}}$</span> </li> </ul>

answer:
<div class="solution">The given trigonometric expression is:<br/>$\cos \left[\tan ^{-1}\left\{\tan \left(\frac{15 \pi}{4}\right)\right\}\right]$<br/>$=\cos \left[\tan ^{-1}\left\{\tan \left(4 \pi-\frac{\pi}{4}\right)\right\}\right]$<br/>$=\cos \left[\tan ^{-1}\left\{-\tan \frac{\pi}{4}\right\}\right]=\cos \left[\tan ^{-1} \tan \left(\frac{-\pi}{4}\right)\right]$<br/>$\left(\right.$ Since $\tan ^{-1} \theta$ is defined for $\left.\frac{-\pi}{2}   &lt; \theta &lt; \frac{-\pi}{2}\right)$<br/>$=\cos \left(\frac{-\pi}{4}\right)$<br/>$\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} \quad[$ since $\cos (-\theta)=\cos \theta]$</div>


title: what-is-the-value-of-cosec-13-12
slug: what-is-the-value-of-cosec-13-12-53369
question: <div class="question">What is the value of $\operatorname{cosec}(13 \pi / 12) ?$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{6}+\sqrt{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\sqrt{6}+\sqrt{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{6}-\sqrt{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-\sqrt{6}-\sqrt{2}$</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">$-\sqrt{6}-\sqrt{2}$</span> </div>

answer:
<div class="solution">$\operatorname{cosec}\left(\frac{13 \pi}{12}\right)=\operatorname{cosec}\left(\pi+\frac{\pi}{12}\right)$<br/>$=-\operatorname{cosec} \frac{\pi}{12}=-\cos \operatorname{cc} 15^{\circ}$<br/>$=-\sqrt{1+\cot ^{2} 15^{\circ}}$<br/>$=-\sqrt{1+(2+\sqrt{3})^{2}} \quad\left[\because \cot 15^{\circ}=2+\sqrt{3}\right]$<br/>$=-\sqrt{1+4+3+4 \sqrt{3}}$<br/>$=-\sqrt{8+4 \sqrt{3}}=-\sqrt{6+2+2 \sqrt{12}}$<br/>$=-\sqrt{(\sqrt{6})^{2}+(\sqrt{2})^{2}+2(\sqrt{6})(\sqrt{2})}$<br/>$=-\sqrt{(\sqrt{6}+\sqrt{2})^{2}}=-\sqrt{6}-\sqrt{2}$</div>


title: what-is-the-value-of-cot-87-0
slug: what-is-the-value-of-cot-87-0-59473
question: <div class="question">What is the value of cot $\left(-870^{\circ}\right) ?$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{3}$</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">B</span> <span class="option-data">$\frac{1}{\sqrt{3}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{\sqrt{3}}</span> </li> </ul>

answer:
<div class="solution">$\cot \left(-870^{\circ}\right)=-\cot \left(2 \times 360^{\circ}+150^{\circ}\right)$<br/>$=-\cot 150^{\circ}=-\cot \left(180^{\circ}-30^{\circ}\right)=\cot 30^{\circ}=\sqrt{3} .$</div>


title: what-is-the-value-of-for-which-i-j-k-3-i-2-j-4-k-2-i-11-j-7-k
slug: what-is-the-value-of-for-which-i-j-k-3-i-2-j-4-k-2-i-11-j-7-k-50459
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lambda$ for which<br/>$(\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}) \times(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=(2 \hat{\mathrm{i}}-11 \hat{\mathrm{j}}-7 \hat{\mathrm{k}})$ ?</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">$-2$</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">7</span> </li> </ul>

answer:
<div class="solution">Given $(\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}) \times(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=(2 \hat{\mathrm{i}}-11 \hat{\mathrm{j}}-7 \hat{\mathrm{k}})$<br/>$\Rightarrow \quad 2 \hat{\mathrm{i}}-\hat{\mathrm{j}}(4 \lambda+3)+\hat{\mathrm{k}}(-2 \lambda-3)=2 \hat{\mathrm{i}}-11 \hat{\mathrm{j}}-7 \hat{\mathrm{k}}$<br/>$\Rightarrow \quad-4 \lambda-3=-11$<br/>$\Rightarrow \quad 4 \lambda=8 \Rightarrow \lambda=2$</div>


title: what-is-the-value-of-for-which-the-vectors-3-i-4-j-k-and-2-i-j-10-k-are-perpendicular
slug: what-is-the-value-of-for-which-the-vectors-3-i-4-j-k-and-2-i-j-10-k-are-perpendicular-38973
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lambda$ for which the vectors<br/>$3 \hat{i}+4 \hat{j}-\hat{k}$ and $-2 \hat{i}+\lambda \hat{j}+10 \hat{k}$ are perpendicular?</div>

answer:
<div class="solution">Given vectors are $3 \hat{i}+4 \hat{j}-\hat{k}$ and $-2 \hat{i}+\lambda \hat{j}+10 \hat{k}$.<br/>If these are perpendicular, dot product is $0 .$<br/>$(3 \hat{i}+4 \hat{j}-\hat{k}) \cdot(-2 \hat{i}+\lambda \hat{j}+10 \hat{k})=0$<br/>$\Rightarrow(3)(-2)+(4)(\lambda)+(-10)=0$<br/>$\Rightarrow-6+4 \lambda-10=0$<br/>$\Rightarrow 4 \lambda=16$<br/>$\Rightarrow \lambda=4$</div>


title: what-is-the-value-of-for-which-the-vectors-i-j-k-2-i-j-k-and-i-j-k-are-coplanar
slug: what-is-the-value-of-for-which-the-vectors-i-j-k-2-i-j-k-and-i-j-k-are-coplanar-28960
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lambda$ for which the vectors<br/>$\hat{i}-\hat{j}+\hat{k}, 2 \hat{i}+\hat{j}-\hat{k}$ and $\lambda \hat{i}-\hat{j}+\lambda \hat{k}$ are co-planar?</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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Given vectors are $\hat{i}-\hat{j}+\hat{k}, 2 \hat{i}+\hat{j}-\hat{k}$ and $\lambda \hat{i}-\hat{j}+\lambda \hat{k}$<br/>We know given vectors are coplanar, if $\left|\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; 1 &amp; -1 \\ \lambda &amp; -1 &amp; \lambda\end{array}\right|=0$<br/>$\Rightarrow 1(\lambda-1)+1(2 \lambda+\lambda)+1(-2-\lambda)=0$<br/>$\Rightarrow \lambda-1+3 \lambda-2-\lambda=0$<br/>$\Rightarrow 3 \lambda=3 \Rightarrow \lambda=1$</div>


title: what-is-the-value-of-i-3-i-3-200-i-3-i-3-200-1
slug: what-is-the-value-of-i-3-i-3-200-i-3-i-3-200-1-78887
tags: ['Mathematics', 'Complex Number', '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_P4.db
question:
<div class="question">What is the value of $\left(\frac{i+\sqrt{3}}{-i+\sqrt{3}}\right)^{200}+\left(\frac{i-\sqrt{3}}{i+\sqrt{3}}\right)^{200}+1$ ?</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">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>

answer:
<div class="solution">Consider,<br/>$\frac{i+\sqrt{3}}{-i+\sqrt{3}}=\frac{(i+\sqrt{3})^{2}}{(\sqrt{3}-i)(\sqrt{3}+i)}($ By Rationalizing $)$<br/>$=\frac{i^{2}+3+2 \sqrt{3} i}{3+1}\left(\right.$ using $\left.(a-b)(a+b)=a^{2}-b^{2}\right)$<br/>$=\frac{-1+3+2 \sqrt{3} i}{4}=\frac{1+\sqrt{3} i}{2}$<br/>$=\frac{-(-1-\sqrt{3} i)}{2}=-\omega^{2} \quad\left(\because i^{2}=-1\right)$<br/>and consider<br/>$\frac{i-\sqrt{3}}{i+\sqrt{3}}=\frac{(i-\sqrt{3})^{2}}{i^{2}-(\sqrt{3})^{2}}($ By Rationalizing)<br/>$=\frac{i^{2}+3-2 i \sqrt{3}}{-4}=\frac{2-2 i \sqrt{3}}{-4}\left(\because i^{2}=-1\right)$<br/>$=\frac{-1+i \sqrt{3}}{2}=\omega$<br/>$\therefore\left(\frac{i+\sqrt{3}}{-i+\sqrt{3}}\right)^{200}+\left(\frac{i-\sqrt{3}}{i+\sqrt{3}}\right)^{200}+1$<br/>$=\left(-\omega^{2}\right)^{200}+\omega^{200}+1$<br/>$=\omega^{400}+\omega^{200}+1=\omega^{3 \times 133+1}+\omega^{3 \times 66+2}+1$<br/>$=\left(\omega^{3}\right)^{133} \omega+\left(\omega^{3}\right)^{66} \omega^{2}+1$<br/>$=\omega+\omega^{2}+1 \quad\left(\because \omega^{3}=1\right)$<br/>$=0$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-if-the-function-is-continuous-at-x-2-consider-the-function-f-x-2-x-2-1-s-i-n-x-where-x-2-and-f-2

slug:
what-is-the-value-of-if-the-function-is-continuous-at-x-2-consider-the-function-f-x-2-x-2-1-s-i-n-x-where-x-2-and-f-2-84139

question:
<div class="question">What is the value of $\lambda$ if the function is continuous at<br/>$x=\frac{\pi}{2} ?$<br/>Consider the function $f(x)=\frac{1-\sin x}{(\pi-2 x)^{2}}$<br/>Where $x \neq \frac{\pi}{2}$ and $f\left(\frac{\pi}{2}\right)=\lambda$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 8$</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">B</span> <span class="option-data">$1 / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 2$</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">$1 / 8$</span> </div>

answer:
<div class="solution">Function is continuous at $\mathrm{x}=\frac{\pi}{2}$<br/>$\mathrm{f}\left(\frac{\pi}{2}\right)=\lim _{\mathrm{x} \rightarrow \frac{\pi}{2}} \frac{1-\sin \mathrm{x}}{(\pi-2 \mathrm{x})^{2}}=\frac{1}{8}$</div>


title: what-is-the-value-of-if-the-straight-line-2-x-3-y-4-6-x-y-12-0-is-parallel-to-yaxis
slug: what-is-the-value-of-if-the-straight-line-2-x-3-y-4-6-x-y-12-0-is-parallel-to-yaxis-68283
tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lambda$ if the straight line $(2 x+3 y+4)+\lambda$ $(6 \mathrm{x}-\mathrm{y}+12)=0$ is parallel to y-axis?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">$-6$</span> </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>

answer:
<div class="solution">The given line is, $(2 x+3 y+4)+\lambda(6 x-y+12)=0$<br/>$\Rightarrow(2+6 \lambda) x+(3-\lambda) y+4+12 \lambda=0 \quad \ldots .(1)$<br/>Since line (i) is parallel to y-axis. $\therefore \quad$ Coefficient of $\mathrm{y}=0$<br/>$3-\lambda=0 \Rightarrow \lambda=3$</div>


title: what-is-the-value-of-if-the-triangle-whose-vertices-are-i-j-and-i-j-k-will-be-right-angled
slug: what-is-the-value-of-if-the-triangle-whose-vertices-are-i-j-and-i-j-k-will-be-right-angled-10842
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lambda$ if the triangle whose vertices are<br/>$\hat{i}, \hat{j}$ and $\hat{i}+\hat{j}+\lambda \hat{k}$ will be right angled?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Let us consider triangle $\mathrm{ABC}$. Suppose $\hat{i}, \hat{j}$ and<br/>$\hat{i}+\hat{j}+\lambda \hat{k}$ are the position vector of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$.<br/>Then $\overrightarrow{A B}=\hat{j}-\hat{i}, \overrightarrow{A C}=\hat{j}+\lambda \hat{k}, \overrightarrow{B C}=\hat{i}+\lambda \hat{k}$<br/>$|\overrightarrow{A B}|=\sqrt{(-1)^{2}+(1)^{2}}=\sqrt{2}$<br/>$|\overrightarrow{B C}|=\sqrt{(1)^{2}+(\lambda)^{2}}=\sqrt{1+\lambda^{2}}$<br/>$|\overrightarrow{A C}|=\sqrt{(1)^{2}+(\lambda)^{2}}=\sqrt{1+\lambda^{2}}$<br/>To be $\Delta \mathrm{ABC}$ is a right angled triangle, $\angle C$ should be right angle,<br/>i.e., $\overrightarrow{B C} \cdot \overrightarrow{A C}=0$<br/>$\Rightarrow(\hat{i}+\lambda \hat{k}) \cdot(\hat{j}+\lambda \hat{k})=0$<br/>$\Rightarrow 0+0+\lambda^{2}=0$<br/>$\therefore \quad \lambda=0$</div>


title: what-is-the-value-of-integral-i-0-1-x-1-x-9-dx
slug: what-is-the-value-of-integral-i-0-1-x-1-x-9-dx-30725
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of integral $\mathrm{I}=\int_{0}^{1} \mathrm{x}(1-\mathrm{x})^{9} \mathrm{dx}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{110}$</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">$\frac{1}{111}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{112} \quad$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{119} \quad$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{110}$</span> </div>

answer:
<div class="solution">Let the given integral be,<br/>$\mathrm{I}=\int_{0}^{1} \mathrm{x}(1-\mathrm{x})^{9} \mathrm{~d} \mathrm{x}$<br/>Put $1-\mathrm{x}=\mathrm{t} \Rightarrow \mathrm{dx}=-\mathrm{dt}$ and $\mathrm{x}=1-\mathrm{t}$<br/>when $x=0, t=1$ and when $x=1, t=0$<br/>$\Rightarrow \quad I=\int_{1}^{0}(1-i) i^{9}(-d l)$<br/>$=-\int_{1}^{0}\left(-\mathrm{t}^{10}+\mathrm{t}^{9}\right) \mathrm{dt}=-\int_{1}^{0}\left(-\mathrm{t}^{10}+\mathrm{t}^{9}\right) \mathrm{dt}$<br/>$\left[\frac{-\mathrm{t}^{11}}{11}+\frac{\mathrm{t}^{10}}{10}\right]_{0}^{1}=\frac{-1}{11}+\frac{1}{10}=\frac{-10+11}{110}=\frac{1}{110}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-k-for-which-the-following-function-f-x-is-continuous-for-all-x-f-x-x-1-2-x-3-3-x-2-k-for-x-1-for-x-1

slug:
what-is-the-value-of-k-for-which-the-following-function-f-x-is-continuous-for-all-x-f-x-x-1-2-x-3-3-x-2-k-for-x-1-for-x-1-83166

question:
<div class="question">What is the value of $k$ for which the following function $f(x)$ is continuous for all $x ?$<br/>$f(x)=\left\{\begin{array}{ll}\frac{x^{3}-3 x+2}{(x-1)^{2}} &amp; , \text { for } x \neq 1 \\ k &amp; , \text { for } \quad x=1\end{array}\right.$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">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">-1</span> </li> </ul>

answer:
<div class="solution">Let $f(x)=\left\{\begin{array}{cl}\frac{x^{3}-3 x+2}{(x-1)^{2}}, &amp; \forall x \neq 1 \\ k, &amp; \forall x=1\end{array}\right.$<br/>and $f(x)$ is continuous.<br/>$\therefore \quad \lim _{x \rightarrow 1} f(x)=k$<br/>$\Rightarrow \quad \lim _{x \rightarrow 1} \frac{x^{3}-3 x+2}{(x-1)^{2}}=k$<br/>$\Rightarrow \quad k=\lim _{x \rightarrow 1} \frac{3 x^{2}-3}{2(x-1)} \quad[$ By $L$ 'Hospitals rule $]$<br/>$\Rightarrow \quad k=\lim _{x \rightarrow 1} \frac{6 x}{2} \quad[$ ByL'Hospitals rule $]$<br/>$\Rightarrow k=3$</div>


title: what-is-the-value-of-k-if-k-k-k-b-c-c-a-a-b-b-2-c-2-c-2-a-2-a-2-b-2-a-b-b-c-c-a
slug: what-is-the-value-of-k-if-k-k-k-b-c-c-a-a-b-b-2-c-2-c-2-a-2-a-2-b-2-a-b-b-c-c-a-90906
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $k$, if<br/>$\left|\begin{array}{lll}k &amp; b+c &amp; b^{2}+c^{2} \\ k &amp; c+a &amp; c^{2}+a^{2} \\ k &amp; a+b &amp; a^{2}+b^{2}\end{array}\right|=(a-b)(b-c)(c-a) ?$</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">0</span> </li> </ul>

answer:
<div class="solution">Let $\left|\begin{array}{ccc}k &amp; b+c &amp; b^{2}+c^{2} \\ k &amp; c+a &amp; c^{2}+a^{2} \\ k &amp; a+b &amp; a^{2}+b^{2}\end{array}\right|=\Delta$<br/>$=k\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ b+c &amp; c+a &amp; a+b \\ b^{2}+c^{2} &amp; c^{2}+a^{2} &amp; a^{2}+b^{2}\end{array}\right|$<br/>Applying $\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{1}, \mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1}$<br/>$=k\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ b+c &amp; a-b &amp; a-c \\ b^{2}+c^{2} &amp; a^{2}-b^{2} &amp; a^{2}-c^{2}\end{array}\right|$<br/>$=k\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ b+c &amp; a-b &amp; a-c \\ b^{2}+c^{2} &amp; (a-b)(a+b) &amp; (a-c)(a+c)\end{array}\right|$<br/>$=k(a-b)(a-c)\left|\begin{array}{cc}1 &amp; 1 \\ a+b &amp; a+c\end{array}\right|$<br/>$=k(a-b)(a-c)(a+c-a-b)$<br/>$=k(a-b)(b-c)(c-a)$<br/>But given $\Delta=(a-b)(b-c)(c-a)$<br/>Thus, $k=1$</div>


title: what-is-the-value-of-k-if-the-area-bounded-by-the-curve-y-sin-k-x-y-0-x-k-x-3-k-is-3-sq-unit
slug: what-is-the-value-of-k-if-the-area-bounded-by-the-curve-y-sin-k-x-y-0-x-k-x-3-k-is-3-sq-unit-81885
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $k$ if the area bounded by the curve $y=\sin k x, y=0, x=\pi / k, x=\pi /(3 k)$ is 3 sq unit?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 2$</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">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Given $x=\pi / k, x=\frac{\pi}{3 k}$ and $y=\sin k x$<br/>Let the required area be $A$ So, $\mathrm{A}=3$ (Given)<br/>ThereforeArea, $A=\int_{\pi / 3 k}^{\pi / k} \sin k x d x$<br/>$\Rightarrow 3=-\left[\frac{\cos k x}{k}\right]_{\pi / 3 k}^{\pi / k}$<br/>$\Rightarrow 3=-\frac{1}{k}\left[\cos \pi-\cos \frac{\pi}{3}\right]$<br/>$\Rightarrow 3=-\frac{1}{k}\left[-1-\frac{1}{2}\right]$<br/>$(\because \cos \pi=\cos (\pi / 2+\pi / 2)$<br/>$\quad=-\sin \pi / 2$ and $\left.\cos 60^{\circ}=\frac{1}{2}\right)$<br/>$\Rightarrow 3=\frac{3}{2 k} \Rightarrow k=\frac{1}{2}$</div>


title: what-is-the-value-of-l-o-g-4-2-l-o-g-27-9-l-o-g-16-64-1
slug: what-is-the-value-of-l-o-g-4-2-l-o-g-27-9-l-o-g-16-64-1-25924
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\frac{\log _{27} 9 \times \log _{16} 64}{\log _{4} \sqrt{2}} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{6}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4}$</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">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> </ul>

answer:
<div class="solution">The given logarithm expression<br/>$\frac{\log _{27} 9 \log _{16} 64}{\log _{4} \sqrt{2}}$<br/>is simplified as:<br/>$\frac{\log 9}{\log 27} \times \frac{\log 64}{\log 16} \times \frac{\log 4}{\log \sqrt{2}}$<br/>$=\frac{2 \log 3}{3 \log 3} \times \frac{6 \log 2}{4 \log 2} \times \frac{2 \log 2}{\frac{1}{2} \log 2}$<br/>$=\frac{2}{3} \times \frac{6}{4} \times 4=4$</div>


title: what-is-the-value-of-l-o-g-4-2-l-o-g-27-9-l-o-g-16-64
slug: what-is-the-value-of-l-o-g-4-2-l-o-g-27-9-l-o-g-16-64-98606
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\frac{\left(\log _{27} 9\right)\left(\log _{16} 64\right)}{\log _{4} \sqrt{2}}$ ?</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">2</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Consider $\frac{\left(\log _{27} 9\right)\left(\log _{16} 64\right)}{\log _{4} \sqrt{2}}$<br/>$=\frac{\log _{3^{3}}\left(3^{2}\right) \log _{4^{2}}(4)^{3}}{\log _{2^{2}}\left(2^{1 / 2}\right)}$<br/>$=\frac{\frac{2}{3} \log _{3} 3 \times \frac{3}{2} \log _{4} 4}{\frac{1}{2 \times 2} \log _{2} 2}=\frac{1}{\frac{1}{4}}=4$</div>


title: what-is-the-value-of-l-o-g-h-l-o-g-h
slug: what-is-the-value-of-l-o-g-h-l-o-g-h-33732
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\frac{\log \sqrt{\alpha \beta}(H)}{\log \sqrt{\alpha \beta \gamma}(H)}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\log _{\alpha \beta}(\alpha)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\log _{\alpha \beta \gamma}(\alpha \beta)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\log _{\alpha \beta}(\alpha \beta \gamma)$</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">$\log _{\alpha \beta}(\beta)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{\alpha \beta}(\alpha \beta \gamma)$</span> </div>

answer:
<div class="solution">Consider $\frac{\log _{\sqrt{\alpha \rho}} H}{\log _{\sqrt{a \gamma}} H}=\frac{\log _{H} \sqrt{a_{n} r}}{\log _{H} \sqrt{x \beta}}$</div>


title: what-is-the-value-of-lim-x-0-s-i-n-2-4-x-x-s-i-n-5-x
slug: what-is-the-value-of-lim-x-0-s-i-n-2-4-x-x-s-i-n-5-x-76033
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lim _{x \rightarrow 0} \frac{x \sin 5 x}{\sin ^{2} 4 x}$</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">$\frac{5}{4}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5}{16}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{25}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{16}$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{x \sin 5 x}{\sin ^{2} 4 x}$<br/>[multiply denominator and number with $\mathrm{x}]$ We get,<br/>$\lim _{x \rightarrow 0} \frac{x^{2} \sin 5 x}{x \sin ^{2} 4 x}=\lim _{x \rightarrow 0} \frac{\sin 5 x}{x} \cdot \frac{x^{2}}{\sin ^{2} 4 x}$<br/>Rearranging to bring a standard form, we get,<br/>$\lim _{x \rightarrow 0} \frac{5 \sin 5 x}{5 x} \cdot \frac{(4 x)^{2}}{16 \sin ^{2} 4 x}$<br/>$=\frac{5}{16}\left(\lim _{x \rightarrow 0} \frac{\sin 5 x}{5 x}\right) \cdot \frac{1}{\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{4 x}\right)^{2}}=\frac{5}{16}$</div>


title: what-is-the-value-of-lim-x-0-x-2-c-o-s-a-x-c-o-s-b-x
slug: what-is-the-value-of-lim-x-0-x-2-c-o-s-a-x-c-o-s-b-x-61641
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\operatorname{Lim}_{x \rightarrow 0} \frac{\cos (a x)-\cos (b x)}{x^{2}}$ ?</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">$\mathrm{a}+\mathrm{b}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{b^{2}-a^{2}}{2}$</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">$\frac{b^{2}+a^{2}}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{x^{2}}$<br/>$=\lim _{x \rightarrow 0} \frac{-a \sin a x+b \sin b x}{2 x}$<br/>(using L' Hospital's rule)<br/>$=\lim _{x \rightarrow 0} \frac{-a^{2} \cos a x+b^{2} \cos b x}{2}$<br/>(using L' Hospital's rule)<br/>$=\frac{b^{2}-a^{2}}{2}(\because \cos 0=1)$</div>


title: what-is-the-value-of-lim-x-0-x-s-i-n-x
slug: what-is-the-value-of-lim-x-0-x-s-i-n-x-72340
question: <div class="question">What is the value of $\lim _{x \rightarrow 0} \frac{\sin |x|}{x}$ ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
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=""> <span class="option-label">C</span> <span class="option-data">$\infty$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Limit does not exist</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">Limit does not exist</span> </div>

answer:
<div class="solution">$\lim _{\mathrm{x} \rightarrow 0} \frac{\sin |\mathrm{x}|}{\mathrm{x}}, \mathrm{LHL}$ is limit when $\mathrm{x}   &lt; 0$<br/>$\mathrm{LHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (-\mathrm{x})}{\mathrm{x}}=-\lim _{\mathrm{x} \rightarrow 0} \frac{\sin \mathrm{x}}{\mathrm{x}}=-1$<br/>RHL is limit when $\mathrm{x}&gt;0$<br/>$\mathrm{RHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (\mathrm{x})}{\mathrm{x}}=1$<br/>So, LHL $\neq$ RHL Hence, Limit does not exist.</div>


title: what-is-the-value-of-lim-x-1-x-1-x-1-2
slug: what-is-the-value-of-lim-x-1-x-1-x-1-2-96069
tags: ['Mathematics', 'Limits', '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_P4.db
question:
<div class="question">What is the value of $\lim _{x \rightarrow 1} \frac{(x-1)^{2}}{|x-1|} ?$</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">The limit does not exist</span> </li> </ul>

answer:
<div class="solution">Let $f(x)=\frac{(x-1)^{2}}{|x-1|}=\left\{\begin{array}{l}(x-1), x \geq 1 \\ -(x-1), x   &lt; 1\end{array}\right.$<br/>Now, LHL $\begin{aligned} &amp;=\lim _{h \rightarrow 0} f(1-h) \\=&amp; \lim _{h \rightarrow 0}[-(1-h-1)]=\lim _{h \rightarrow 0} h=0 \end{aligned}$<br/>and $\mathrm{RHL}=\lim _{h \rightarrow 0} f(1+h)$<br/>$\quad=\lim _{h \rightarrow 0}(1+h-1)=\lim _{h \rightarrow 0} h=0$<br/>$\therefore \mathrm{LHL}=\mathrm{RHL}$<br/>$\therefore \quad \lim _{x \rightarrow 0} f(x)=\mathrm{LHL}=\mathrm{RHL}$<br/>$\Rightarrow \quad \lim _{x \rightarrow 0} \frac{(x-1)^{2}}{|x-1|}=0$</div>


title: what-is-the-value-of-lim-x-a-3-x-2-x-2-x-3-x
slug: what-is-the-value-of-lim-x-a-3-x-2-x-2-x-3-x-46343
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lim _{x \rightarrow a} \frac{\sqrt{\alpha+2 x}-\sqrt{3 x}}{\sqrt{3 \alpha+x}-2 \sqrt{x}}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{\sqrt{3}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{(3 \sqrt{3})}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{(3 \sqrt{3})}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{\sqrt{3}}$</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> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp; \lim _{x \rightarrow \infty} \frac{\sqrt{\alpha+2 x}-\sqrt{3 x}}{\sqrt{3 \alpha+x}-2 \sqrt{x}} \\ &amp;=\lim _{x \rightarrow \alpha} \frac{(\sqrt{\alpha+2 x})^{2}-(\sqrt{3 x})^{2}}{\sqrt{\alpha+2 x}+\sqrt{3 x}} \times \frac{\sqrt{3 \alpha+x}+2 \sqrt{x}}{(\sqrt{3 \alpha+x})^{2}-(2 \sqrt{x})^{2}} \\ &amp;=\lim _{x \rightarrow \alpha} \frac{\alpha+2 x-3 x}{\sqrt{\alpha+2 x}+\sqrt{3 x}} \times \frac{\sqrt{3 \alpha+x}+2 \sqrt{x}}{3 \alpha+x-4 x} \\ &amp;=\lim _{x \rightarrow \alpha} \frac{\sqrt{3 \alpha+x}+2 \sqrt{x}}{\sqrt{\alpha+2 x}+\sqrt{3 x}} \times \frac{(\alpha-x)}{3(\alpha-x)} \\ &amp;=\frac{1}{3} \lim _{x \rightarrow \alpha} \frac{\sqrt{3 \alpha+x}+2 \sqrt{x}}{\sqrt{\alpha+2 x}+\sqrt{3 x}} \\ &amp;=\frac{1}{3} \frac{\sqrt{4 \alpha}+2 \sqrt{\alpha}}{\sqrt{3 \alpha}+\sqrt{3 \alpha}}=\frac{1}{2}\left(\frac{4 \sqrt{\alpha}}{2 \sqrt{3} \sqrt{\alpha}}\right)=\frac{1}{\sqrt{3}} \end{aligned}$</div>


title: what-is-the-value-of-lim-x-x-1-x-6-x-4
slug: what-is-the-value-of-lim-x-x-1-x-6-x-4-34301
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">e</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$e^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$e^{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$e^{5}$</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">$e^{5}$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}=\lim _{x \rightarrow \infty}\left(\frac{x+5+1}{x+1}\right)^{x+4}=\lim _{x \rightarrow \infty}\left(\frac{x+1}{x+1}+\frac{5}{x+1}\right)^{x+4}$<br/>$=\lim _{x \rightarrow \infty}\left(1+\frac{5}{x+1}\right)^{\frac{x+4}{5} \times \frac{5(x+1)}{x+1}}$<br/>$=\lim _{x \rightarrow \infty}\left[\left(1+\frac{5}{x+1}\right)^{\frac{x+1}{5}}\right]^{5\left(\frac{x+4}{x+1}\right)}$<br/>$=e^{5 \lim _{x \rightarrow \infty} \frac{1+\frac{4}{x}}{x+\frac{1}{x}}}=e^{5} \quad\left(\because \lim _{x \rightarrow \infty} \frac{1+4 / x}{1+1 / x}=1\right)$</div>


title: what-is-the-value-of-lim-x-x-2-x-2-x-2
slug: what-is-the-value-of-lim-x-x-2-x-2-x-2-50678
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\lim _{x \rightarrow \infty}\left(\frac{x-2}{x+2}\right)^{x+2}$ ?</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">$\mathrm{e}^{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{e}^{-2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{e}^{-4}$</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> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow 0}\left(\frac{x-2}{x+2}\right)^{x+2}$<br/>can be written as<br/>$\lim _{x \rightarrow 0}\left(\frac{x+2-4}{x+2}\right)^{x+2}$<br/>or, $\lim _{x \rightarrow \infty}\left\{1-\frac{4}{(x+2)}\right\}^{x+2}$<br/>Putting $\mathrm{x}+2=\mathrm{t}$<br/>when $\mathrm{x} \rightarrow \infty, \mathrm{t} \rightarrow \infty$<br/>So, $\lim _{t \rightarrow \infty}\left\{1-\frac{4}{t}\right\}^{t}$<br/>or, $\lim _{t \rightarrow \infty}\left\{1-\frac{4}{t}\right\}^{\frac{t}{4} \times 4}$<br/>or, $\lim _{x \rightarrow \infty}\left\{\left(1-\frac{4}{t}\right)^{\frac{t}{4}}\right\}^{4}=\left(e^{-1}\right)^{4}=e^{-4}$</div>


title: what-is-the-value-of-lim-x-x-s-i-n-x
slug: what-is-the-value-of-lim-x-x-s-i-n-x-53180
question: <div class="question">What is the value of $\lim _{x \rightarrow \infty} \frac{\sin x}{x}$ ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
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">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">$\infty$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1</span> </li> </ul>

answer:
<div class="solution">Given limit is: $\lim _{x \rightarrow \infty} \frac{\sin x}{x}$<br/>Let, $\mathrm{x}=\frac{1}{\mathrm{~h}}$ and as $\mathrm{x} \rightarrow \infty, \mathrm{h} \rightarrow 0$<br/>$\therefore \lim _{x \rightarrow \infty} \frac{\sin x}{x}$ change to : $\lim _{h \rightarrow 0} h \sin \frac{1}{h}$<br/>$=0$ (value of $\sin \frac{1}{\mathrm{~h}}$ is finite as it lies between $-1$ and 1 )</div>


title: what-is-the-value-of-lo-g-2-lo-g-3-81
slug: what-is-the-value-of-lo-g-2-lo-g-3-81-56328
question: <div class="question">What is the value of $\log _{2}\left(\log _{3} 81\right)$ ?</div>
tags: ['Mathematics', 'Sets and Relations', '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_P4.db
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">3</span> </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">9</span> </li> </ul>

answer:
<div class="solution">Let $\log _{2}\left(\log _{3} 81\right)=x$<br/>$\Rightarrow \log _{2}\left(\log _{3} 3^{4}\right)=x$<br/>$\Rightarrow \log _{2}\left(4 \log _{3} 3\right)=x$<br/>$\Rightarrow \log _{2}(4)=x$<br/>$\left(\because \log _{a} a=1\right)$<br/>$\Rightarrow 4=2$<br/>$\Rightarrow 2^{2}=2^{x} \Rightarrow x=2$</div>


title: what-is-the-value-of-lo-g-7-lo-g-7-7-7-7-equal-to
slug: what-is-the-value-of-lo-g-7-lo-g-7-7-7-7-equal-to-94414
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\log _{7} \log _{7} \sqrt{7 \sqrt{7} \sqrt{7}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \log _{2} 7$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1-3 \log _{2} 7$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1-3 \log _{7} 2$</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">$\frac{7}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1-3 \log _{7} 2$</span> </div>

answer:
<div class="solution">Let $\log _{7} \log _{7} \sqrt{7 \sqrt{7 \sqrt{7}}}=x$<br/>then $7^{x}=\log _{7} \sqrt{7 \sqrt{7 \sqrt{7}}}$<br/>$$<br/>\begin{aligned}<br/>7^{x} &amp;=\frac{1}{2} \cdot \log _{7} 7 \sqrt{7 \sqrt{7}} \\<br/>7^{x} &amp;=\frac{1}{2}\left[\log _{7} 7+\frac{1}{2} \log _{7} 7 \sqrt{7}\right] \\<br/>&amp;=\frac{1}{2}\left[1+\frac{1}{2} \log _{7} 7+\frac{1}{2} \log _{7} \sqrt{7}\right] \\<br/>&amp;=\frac{1}{2}\left[1+\frac{1}{2}+\frac{1}{4}\right] \\<br/>7^{x} &amp;=\frac{7}{8} \\<br/>x &amp;=\log _{7}\left(\frac{7}{8}\right) \\<br/>x &amp;=\log _{7} 7-\log _{7} 8 \\<br/>x &amp;=1-3 \log _{7} 2<br/>\end{aligned}<br/>$$</div>


title: what-is-the-value-of-lo-g-9-27-lo-g-8-32
slug: what-is-the-value-of-lo-g-9-27-lo-g-8-32-98821
question: <div class="question">What is the value of $\log _{9} 27+\log _{8} 32$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{19}{6}$</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><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">7</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{19}{6}$</span> </div>

answer:
<div class="solution">$\log _{9} 27+\log _{8} 32$<br/>$\quad=\log _{9} 3^{3}+\log _{8} 2^{5}$<br/>$=3 \log _{9} 3+5 \log _{8} 2$<br/>$\left.3 \log _{(3}^{2}\right)^{ \left.3+5 \log _{(2} 3\right)^{3}}$<br/>$\frac{3}{2} \log _{3} 3+\frac{5}{3} \log _{2} 2$<br/>$\frac{3}{2}+\frac{5}{3}=\frac{19}{6}$</div>


title: what-is-the-value-of-lo-g-10-8-9-lo-g-10-32-27-lo-g-10-4-3
slug: what-is-the-value-of-lo-g-10-8-9-lo-g-10-32-27-lo-g-10-4-3-36411
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\log _{10}\left(\frac{9}{8}\right)-\log _{10}\left(\frac{27}{32}\right)+\log _{10}\left(\frac{3}{4}\right)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </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">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">Consider, $\log \frac{9}{8}-\log \frac{27}{32}+\log \frac{3}{4}$<br/>$=\log \frac{9}{8}+\log \frac{32}{27}+\log \frac{3}{4}$<br/>$=\log \left(\frac{9}{8} \times \frac{32}{27}\right)+\log \frac{3}{4}$<br/>$=\log \left(\frac{4}{3}\right)+\log \frac{3}{4}=\log \left(\frac{4}{3} \times \frac{3}{4}\right)=\log 1=0$</div>


title: what-is-the-value-of-lo-g-y-x-5-lo-g-x-y-2-lo-g-z-z
slug: what-is-the-value-of-lo-g-y-x-5-lo-g-x-y-2-lo-g-z-z-51789
question: <div class="question">What is the value of $\log _{y} x^{5} \log _{x} y^{2} \log _{z} z$</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">30</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">20</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">30</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">60</span> </li> </ul>

answer:
<div class="solution">$\log _{y} x^{5} \times \log _{x} y^{2} \log _{z} z^{3}$<br/>$=\frac{5 \ln x}{\ln y} \times \frac{2 \ln y}{\ln x} \times \frac{3 \ln z}{\ln z}=30$</div>


title: what-is-the-value-of-m-if-the-vectors-2-i-j-k-i-2-j-3-k-and-3-i-m-j-5-k-are-coplanar
slug: what-is-the-value-of-m-if-the-vectors-2-i-j-k-i-2-j-3-k-and-3-i-m-j-5-k-are-coplanar-91975
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-4$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\mathrm{m}$ if the vectors<br/>$2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $3 \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ are coplanar?</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">2</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Three vectors $x_{1} \vec{a}+y_{1} \vec{b}+z_{1} \vec{c}, x_{2} \vec{a}+y_{2} \vec{b}+z_{2} \vec{c}$ and<br/>$\mathrm{x}_{3} \overrightarrow{\mathrm{a}}+\mathrm{y}_{3} \overrightarrow{\mathrm{b}}+\mathrm{z}_{3} \overrightarrow{\mathrm{c}}$ will be coplanar iff<br/>$\left|\begin{array}{lll}\mathrm{x}_{1} &amp; \mathrm{y}_{1} &amp; \mathrm{z}_{1} \\ \mathrm{x}_{2} &amp; \mathrm{y}_{2} &amp; \mathrm{z}_{2} \\ \mathrm{x}_{3} &amp; \mathrm{y}_{3} &amp; \mathrm{z}_{3}\end{array}\right|=0$<br/>Here, $\mathrm{x}_{1}=2, \mathrm{y}_{1}=-1, \mathrm{z}_{1}=1$<br/>$\mathrm{x}_{2}=1, \mathrm{y}_{2}=2, \mathrm{z}_{2}=-3$<br/>$\mathrm{x}_{3}=3, \mathrm{y}_{3}=\mathrm{m}, \mathrm{z}_{3}=5$<br/>$\therefore\left|\begin{array}{ccc}2 &amp; -1 &amp; 1 \\ 1 &amp; 2 &amp; -3 \\ 3 &amp; \mathrm{~m} &amp; 5\end{array}\right|=0$<br/>$\Rightarrow 2(10+3 \mathrm{~m})+1(5+9)+1(\mathrm{~m}-6)=0$<br/>$\Rightarrow 7 \mathrm{~m}+28=0 \Rightarrow \mathrm{m}=-4$</div>


title: what-is-the-value-of-n-for-which-the-numbers-1-2-3-n-have-variance-2
slug: what-is-the-value-of-n-for-which-the-numbers-1-2-3-n-have-variance-2-69043
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $n$ for which the numbers $1,2,3, \ldots, n$ have variance $2 ?$</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">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">6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Mean of the numbers $=\frac{\frac{n(n+1)}{2}}{n}$<br/>$=\frac{n+1}{2}$<br/>$\therefore$ Variance $=\frac{\left(1-\frac{n+1}{2}\right)^{2}+\left(2-\frac{n+1}{2}\right)^{2}+\left(3-\frac{n+1}{2}\right)^{2}+\ldots}{n}$<br/>$2=\frac{\left(1^{2}+2^{2}+3^{2}+\ldots\right)+n\left(\frac{n+1}{2}\right)^{2}-2\left(\frac{n+1}{2}\right)[1+2+3+\ldots .]}{n}$<br/>$2 n=\frac{1}{6} n(n+1)(2 n+1)+\frac{n(n+1)^{2}}{4}-2\left(\frac{n+1}{2}\right)\left\{\frac{n(n+1)}{2}\right\}$<br/>$2 n=n(n+1)\left[\frac{2 n+1}{6}+\frac{n+1}{4}-\frac{n+1}{2}\right]$<br/>$2=(n+1)\left[\frac{4 n+2-3 n-3}{12}\right]$<br/>$\begin{array}{ll}\Rightarrow 24=(n+1)(n-1) &amp; \Rightarrow n^{2}-1=24 \\ \Rightarrow n^{2}=25 &amp; \Rightarrow n=\pm 5\end{array}$</div>


title: what-is-the-value-of-n-if-p-15-n-1-p-16-n-2-3-4
slug: what-is-the-value-of-n-if-p-15-n-1-p-16-n-2-3-4-92817
question: <div class="question">What is the value of $n$, if $P(15, n-1): P(16, n-2)$ $=3: 4 ?</div>
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">14</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">15</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{P}(15, n-1): \mathrm{P}(16, n-2)=3: 4$<br/>$\Rightarrow \frac{15}{16}{ }^{15} \mathrm{P}_{n-1}=\frac{3}{4}$<br/>$\Rightarrow \quad \frac{15 !}{(15-n+1) !} \times \frac{(16-n+2) !}{16 !}=\frac{3}{4}$<br/>$\Rightarrow \quad \frac{15 !}{(16-n) !} \times \frac{(18-n) !}{16 !}=\frac{3}{4}$<br/>$\Rightarrow \quad \frac{(18-n) !}{16(16-n) !}=\frac{3}{4}$<br/>$\Rightarrow \quad \frac{(18-n)(17-n)(16-n) !}{16(16-n) !}=\frac{3}{4}$<br/>$\Rightarrow \quad(18-n)(17-n)=12$<br/>$\Rightarrow \quad 306-17 n-18 n+n^{2}=12$<br/>$\Rightarrow n^{2}-35 n+294=0$<br/>$\Rightarrow \quad(n-14)(n-21)=0$<br/>$\Rightarrow \quad n=14 \quad(\because n \neq 21)$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-n-so-that-the-angle-between-the-lines-having-direction-ratios-1-1-1-and-1-1-n-is-6-0

slug:
what-is-the-value-of-n-so-that-the-angle-between-the-lines-having-direction-ratios-1-1-1-and-1-1-n-is-6-0-81347

question:
<div class="question">What is the value of $n$ so that the angle between the lines having direction ratios $(1,1,1)$ and $(1,-1, n)$ is $60^{\circ}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{6}$</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><li class=""> <span class="option-label">C</span> <span class="option-data">3</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">$\sqrt{6}$</span> </div>

answer:
<div class="solution">If $\left(l_{1}, m_{1}, n_{1}\right)$ and $\left(l_{2}, m_{2}, n_{2}\right)$ are the direction ratios then angle between the lines is<br/>$\cos \theta=\frac{l_{l} l_{2}+m_{1} m_{2}+n_{1} n_{2}}{\sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2}} \sqrt{l_{2}^{2}+m_{2}^{2}+n_{2}^{2}}}$<br/>Here $l_{1}=1, m_{1}=1, n_{1}=1$ and<br/>$l_{2}=1, m_{2}=-1, n_{2}=n$<br/>and $\theta=60^{\circ}$<br/>$\cos 60^{\circ}=\frac{1 \times 1+1 \times(-1)+1 \times n}{\sqrt{1^{2}+1^{2}+1^{2} \times \sqrt{1^{2}+1^{2}+n^{2}}}}$<br/>$\Rightarrow \frac{1}{2}=\frac{n}{\sqrt{3} \sqrt{2+n^{2}}} \Rightarrow 3\left(2+n^{2}\right)=4 n^{2}$<br/>$\Rightarrow n^{2}=6 \Rightarrow n=\pm \sqrt{6}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-p-for-which-the-function-f-x-p-sin-x-3-s-i-n-3-x-ha-s-an-e-x-t-re-m-u-ma-t-xfracpi3

slug:
what-is-the-value-of-p-for-which-the-function-f-x-p-sin-x-3-s-i-n-3-x-ha-s-an-e-x-t-re-m-u-ma-t-xfracpi3-96471

question:
<div class="question">What is the value of $p$ for which the function<br/>$$<br/>f(x)=p \sin x+\frac{\sin 3 x}{3}<br/>$$<br/>has an extremum at $x=\frac{\pi}{3} ?$</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">$-1$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Let $\mathrm{f}(\mathrm{x})=\mathrm{p} \sin \mathrm{x}+\frac{\sin 3 \mathrm{x}}{3}$<br/>Differentiate both side w.r.t $(\mathrm{x})$.<br/>$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\mathrm{p} \cos \mathrm{x}+\frac{3 \cos 3 \mathrm{x}}{3}=\mathrm{p} \cos \mathrm{x}+\cos 3 \mathrm{x}$<br/>It is given that $\mathrm{f}(\mathrm{x})$ has extremum value at $\mathrm{x}=\pi / 3$ $\therefore \quad f^{\prime}\left(\frac{\pi}{3}\right)=0$<br/>$\Rightarrow \quad p \cos \frac{\pi}{3}+\cos \pi=0$<br/>$\Rightarrow \frac{\mathrm{p}}{2}-1=0 \quad \Rightarrow \mathrm{p}=2$</div>


title: what-is-the-value-of-p-for-which-the-vector-p-2-i-j-2-k-is-of-3-units-length
slug: what-is-the-value-of-p-for-which-the-vector-p-2-i-j-2-k-is-of-3-units-length-79629
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $P$ for which the vector $p(2 \hat{i}-\hat{j}+2 \hat{k})$ is of 3 units length ?</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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">$|\mathrm{P}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})|=3$<br/>$\mathrm{P}\left|\sqrt{2^{2}+(-1)^{2}+2^{2}}\right|=3$<br/>$3 \mathrm{P}=3 \Rightarrow \mathrm{P}=1$</div>


title: what-is-the-value-of-r-if-p5-rp6-r1
slug: what-is-the-value-of-r-if-p5-rp6-r1-23456
question: <div class="question">What is the value of $r$, if $P(5, r)=P(6, r-1) ?</div>
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Given $P(5, r)=P(6, r-1)$<br/>$\Rightarrow{ }^{5} P_{r}={ }^{6} P_{r-1}$<br/>$\Rightarrow \frac{5 !}{(5-r) !}=\frac{6 !}{(6-r+1) !}$<br/>$\Rightarrow \frac{5 !}{(5-r) !}=\frac{6 !}{(7-r) !}$<br/>$\Rightarrow \frac{5 !}{(5-r) !}=\frac{6 \times 5 !}{(7-r)(6-r)(5-r) !}$<br/>$\Rightarrow(7-r)(6-r)=6$<br/>$\Rightarrow 42-13 r+r^{2}=6$<br/>$\Rightarrow r^{2}-13 r+36=0$<br/>$\Rightarrow r^{2}-9 r-4 r+36=0$<br/>$\Rightarrow(r-9)(r-4)=0$<br/>$\Rightarrow \mathrm{r}=4 \quad(\because \mathrm{r} \neq 9)$</div>


title: what-is-the-value-of-s-e-c-cosec-c-o-t-s-i-n-c-o-s-t-a-n-when-4-3
slug: what-is-the-value-of-s-e-c-cosec-c-o-t-s-i-n-c-o-s-t-a-n-when-4-3-90073
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\frac{\sin \theta+\cos \theta-\tan \theta}{\sec \theta+\operatorname{cosec}-\cot \theta}$, when $\theta=\frac{3 \pi}{4}$ ?</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 $[2010-I I]$</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">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 $[2010-I I]$</span> </div>

answer:
<div class="solution">Consider \end{array}$<br/>$\frac{\sin \theta+\cos \theta-\tan \theta}{\sec \theta+\cos e c \theta-\cot \theta}$<br/>Now, put value of $\theta=\frac{3 \pi}{4}$, we get<br/>$\frac{\sin \frac{3 \pi}{4}+\cos \frac{3 \pi}{4}-\tan \frac{3 \pi}{4}}{\sec \frac{3 \pi}{4}+\cos e c \frac{3 \pi}{4}-\cot \frac{3 \pi}{4}}$<br/>$=\frac{\sin \frac{\pi}{4}-\cos \frac{\pi}{4}+\tan \frac{\pi}{4}}{-\frac{1}{\cos \frac{\pi}{4}}+\frac{1}{\sin \frac{\pi}{4}}+\frac{1}{\tan \frac{\pi}{4}}}=\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}+1}{-\sqrt{2}+\sqrt{2}+1}=1$</div>


title: what-is-the-value-of-s-i-n-2-xe-x-e-x-1-x-dx
slug: what-is-the-value-of-s-i-n-2-xe-x-e-x-1-x-dx-99487
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\int \frac{\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})}{\sin ^{2}\left(\mathrm{xe}^{\mathrm{x}}\right)} \mathrm{dx}$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-e^{x} \cot x+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos ^{2}\left(x e^{x}\right)+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\log \sin \left(x e^{x}\right)+c$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-\cot \left(x e^{x}\right)+c$</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> </ul>

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

answer:
<div class="solution">Let the given integral be $\mathrm{I}=\int \frac{\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})}{\sin ^{2}\left(\mathrm{xe}^{\mathrm{x}}\right)} \mathrm{dx}$<br/>Put $x e^{x}=t$ and $e^{x}(1+x) d x=d t$<br/>$\Rightarrow \quad \mathrm{I}=\int \frac{\mathrm{dt}}{\sin ^{2} \mathrm{t}}=\int \operatorname{cosec}^{2} \mathrm{t} \mathrm{dt}$<br/>$=-\cot t+c=-\cot \left(x e^{x}\right)+c$</div>


title: what-is-the-value-of-sec-2-tan-1-11-5
slug: what-is-the-value-of-sec-2-tan-1-11-5-17436
question: <div class="question">What is the value of $\sec ^{2} \tan ^{-1}\left(\frac{5}{11}\right) ?$</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$121 / 96$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$211 / 921$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$146 / 121$</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">$267 / 121$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$146 / 121$</span> </div>

answer:
<div class="solution">Let $\sec ^{2}\left(\tan ^{-1}\left(\frac{5}{11}\right)\right)$<br/>$=1+\tan ^{2}\left(\tan ^{-1}\left(\frac{5}{11}\right)\right) \quad\left(\because \sec ^{2} \theta-\tan ^{2} \theta=1\right)$<br/>$=1+\left[\tan \left(\tan ^{-1}\left(\frac{5}{11}\right)\right)\right]^{2}=1+\left(\frac{5}{11}\right)^{2}$<br/>$=1+\frac{25}{121}=\frac{146}{121}$</div>


title: what-is-the-value-of-sec-cos-cosec-sin-cot-tan
slug: what-is-the-value-of-sec-cos-cosec-sin-cot-tan-68110
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $(\sec \theta-\cos \theta)(\operatorname{cosec} \theta-\sin \theta)(\cot \theta+\tan \theta) ?$</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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\sin \theta &amp; \text</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\cos \theta</span> </li> </ul>

answer:
<div class="solution">The given expression is:<br/>$(\sec \theta-\cos \theta)(\operatorname{cosec} \theta-\sin \theta)(\cot \theta+\tan \theta)$<br/>$=\left(\frac{1}{\cos \theta}-\cos \theta\right)\left(\frac{1}{\sin \theta}-\sin \theta\right)\left(\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}\right)$<br/>$=\left(\frac{1-\cos ^{2} \theta}{\cos \theta}\right)\left(\frac{1-\sin ^{2} \theta}{\sin \theta}\right)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}\right)$<br/>$=\frac{\sin ^{2} \theta}{\cos \theta} \cdot \frac{\cos ^{2} \theta}{\sin \theta} \times \frac{1}{\sin \theta \cos \theta}=\frac{\sin ^{2} \theta \cdot \cos ^{2} \theta}{\cos ^{2} \theta \cdot \sin ^{2} \theta}=1$</div>


title: what-is-the-value-of-sin-1-0-sin-8-0-cos-1-0-cos-8-0
slug: what-is-the-value-of-sin-1-0-sin-8-0-cos-1-0-cos-8-0-59436
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left|\begin{array}{rr}\sin 10^{\circ} &amp; -\cos 10^{\circ} \\ \sin 80^{\circ} &amp; \cos 80^{\circ}\end{array}\right| ?$</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">$1 / 2$</span> </li> </ul>

answer:
<div class="solution">$\left|\begin{array}{lr}\sin 10^{\circ} &amp; -\cos 10^{\circ} \\ \sin 80^{\circ} &amp; \cos 80^{\circ}\end{array}\right|$<br/>$=\sin 10^{\circ} \cos 80^{\circ}+\sin 80^{\circ} \cos 10^{\circ}$<br/>$=\sin 10^{\circ} \sin 10^{\circ}+\cos 10^{\circ} \cos 10^{\circ}$<br/>$=\sin ^{2} 10^{\circ}+\cos ^{2} 10^{\circ}=1$</div>


title: what-is-the-value-of-sin-1-5-4-2-tan-1-3-1
slug: what-is-the-value-of-sin-1-5-4-2-tan-1-3-1-66255
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}$</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">C</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{6}$</span> </li> </ul>

answer:
<div class="solution">Consider $\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3}$<br/>$=\tan ^{-1}\left(\frac{4 / 5}{\sqrt{1-16 / 25}}\right)+\tan ^{-1}\left(\frac{2 / 3}{1-1 / 9}\right)$<br/>$\left[\because \sin ^{-1} x=\tan ^{-1} \frac{x}{\sqrt{1-x^{2}}}\right.$ and $\left.2 \tan ^{-1} x=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\right]$<br/>$=\tan ^{-1}\left(\frac{4 / 5}{3 / 5}\right)+\tan ^{-1}\left(\frac{2 / 3}{8 / 9}\right)$<br/>$=\tan ^{-1}\left(\frac{4}{3}\right)+\tan ^{-1}\left(\frac{3}{4}\right)$<br/>$=\tan ^{-1}\left(\frac{4}{3}\right)+\cot ^{-1}\left(\frac{4}{3}\right)=\frac{\pi}{2}$<br/>$\left(\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right)$</div>


title: what-is-the-value-of-sin-1-5-4-sec-1-4-5-2
slug: what-is-the-value-of-sin-1-5-4-sec-1-4-5-2-46161
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\sin ^{-1} \frac{4}{5}+\sec ^{-1} \frac{5}{4}-\frac{\pi}{2} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi$</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">$\begin{aligned} &amp; \sin ^{-1} \frac{4}{5}+\sec ^{-1} \frac{5}{4}-\frac{\pi}{2}=\sin ^{-1} \frac{4}{5}+\cos ^{-1} \frac{4}{5}-\frac{\pi}{2} \\=&amp; \frac{\pi}{2}-\frac{\pi}{2}=0 \end{aligned}$</div>


title: what-is-the-value-of-sin-1-5-sin-7-5
slug: what-is-the-value-of-sin-1-5-sin-7-5-75862
question: <div class="question">What is the value of $\sin 15^{\circ} \sin 75^{\circ}$ ? $\quad$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1.4$</span> </div>
source_db: MarksBatch2_P4.db
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">$1 / 8$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 16$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Consider $\sin 15^{\circ} \sin 75^{\circ}$<br/>$=\sin \left(45^{\circ}-30^{\circ}\right) \sin \left(45^{\circ}+30^{\circ}\right)$<br/>$=\left(\sin 45^{\circ} \cos 30^{\circ}-\cos 45^{\circ} \sin 30^{\circ}\right)\left(\sin 45^{\circ} \cos 30^{\circ}\right.$<br/>$\left.\quad+\cos 45^{\circ} \sin 30^{\circ}\right)$<br/>$($ using $\sin (\mathrm{A}+\mathrm{B})=\sin A \cos B$<br/>$+\cos A \sin B$ and $\sin (A-B) \sin A \cos B-\cos A \sin B)$<br/>$=\left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right)\left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right)$<br/>$=\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)=\frac{3-1}{8}=\frac{2}{8}=\frac{1}{4}$</div>


title: what-is-the-value-of-sin-1-5
slug: what-is-the-value-of-sin-1-5-78683
question: <div class="question">What is the value of $\sin 15^{\circ} ?$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3}-1}{2 \sqrt{2}}$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{3}+1}{2 \sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}-1}{\sqrt{3}+1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{3}+1}{\sqrt{3}-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3}-1}{2 \sqrt{2}}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \sin 15^{\circ}=\sin \left[45^{\circ}-30^{\circ}\right] \\ &amp;=\sin 45^{\circ} \cdot \cos 30^{\circ}-\cos 45^{\circ} \sin 30^{\circ} \\ &amp;=\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \times \frac{1}{2} \\ &amp;=\frac{\sqrt{3}}{2 \sqrt{2}}-\frac{1}{2 \sqrt{2}}=\frac{\sqrt{3}-1}{2 \sqrt{2}} \\ \text { 130. (c) } &amp; 4 \sin ^{2} \theta=1 \\ &amp; \Rightarrow \sin ^{2} \theta=\frac{1}{4} \Rightarrow \sin \theta=\pm \frac{1}{2} \Rightarrow \theta=\pm \frac{\pi}{6} \end{aligned}$<br/>Hence $\theta$ take 4 values. (one value for each quadrant)</div>


title: what-is-the-value-of-sin-1-8-cos-3-6-equal-to
slug: what-is-the-value-of-sin-1-8-cos-3-6-equal-to-19269
question: <div class="question">What is the value of $\sin 18^{\circ} \cos 36^{\circ}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </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">1</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">We know that $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$<br/>Now, $\cos 36^{\circ}=1-2 \sin ^{2} 18^{\circ}$<br/>$=1-\frac{2}{16}(\sqrt{5}-1)^{2}=1-\frac{(3-\sqrt{5})}{4}$<br/>$\cos 36^{\circ}=\frac{\sqrt{5}+1}{4}$<br/>Now, $\sin 18^{\circ} \cos 36^{\circ}=\frac{(\sqrt{5})^{2}-(1)^{2}}{16}=\frac{4}{16}=\frac{1}{4}$</div>


title: what-is-the-value-of-sin-1-sin-3-2
slug: what-is-the-value-of-sin-1-sin-3-2-36452
question: <div class="question">What is the value of $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right) ?$</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2 \pi}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{2 \pi}{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{3}$</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> </ul>

answer:
<div class="solution">$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\sin ^{-1}\left[\sin \left(\pi-\frac{2 \pi}{3}\right)\right]=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$<br/>$=\frac{\pi}{3} \quad\left[\right.$ Since, $\left.\sin \frac{2 \pi}{3}=\sin \left(\pi-\frac{\pi}{3}\right)\right]$</div>


title: what-is-the-value-of-sin-4-cos-4-1-cosec-2
slug: what-is-the-value-of-sin-4-cos-4-1-cosec-2-75003
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_P4.db
question:
<div class="question">What is the value of $\left(\sin ^{4} \theta-\cos ^{4} \theta+1\right) \operatorname{cosec}^{2} \theta$ ?</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">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">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>

answer:
<div class="solution">Consider $\left(\sin ^{4} \theta-\cos ^{4} \theta+1\right) \operatorname{cosec}^{2} \theta .$<br/>$\quad=\left[\left(\sin ^{2} \theta\right)^{2}-\left(\cos ^{2} \theta\right)^{2}+1\right] \operatorname{cosec}^{2} \theta$<br/>$=\left[\left(\sin ^{2} \theta-\cos ^{2} \theta\right)\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+1\right] \operatorname{cosec}^{2} \theta$<br/>$=\left[1-\cos ^{2} \theta+\sin ^{2} \theta\right] \operatorname{cosec}^{2} \theta$<br/>$=\left(\sin ^{2} \theta+\sin ^{2} \theta\right) \operatorname{cosec}^{2} \theta$<br/>$=2 \sin ^{2} \theta \times \frac{1}{\sin ^{2} \theta}=2$</div>


title: what-is-the-value-of-sin-5-0-sin-7-0-sin-1-0
slug: what-is-the-value-of-sin-5-0-sin-7-0-sin-1-0-73036
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left(\sin 50^{\circ}-\sin 70^{\circ}+\sin 10^{\circ}\right) ?$</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">$\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}}{2}$</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">Consider $\sin 50^{\circ}-\sin 70^{\circ}+\sin 10^{\circ}$<br/>$=2 \cos \frac{70^{\circ}+50^{\circ}}{2} \cdot \sin \frac{50^{\circ}-70^{\circ}}{2}+\sin 10^{\circ}$<br/>$\left[\because \operatorname{Sin} A-\operatorname{Sin} B=2 \cos \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right) \operatorname{Sin}\left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)\right]$<br/>$=-2 \cos 60^{\circ} \sin 10^{\circ}+\sin 10^{\circ}=-\sin 10^{\circ}+\sin 10^{\circ}=0$<br/>$\left(\because \cos 60^{\circ}=\frac{1}{2}\right)$</div>


title: what-is-the-value-of-sin-12-5
slug: what-is-the-value-of-sin-12-5-61150
question: <div class="question">What is the value of $\sin \left(\frac{5 \pi}{12}\right)$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3}+1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{6}+\sqrt{2}}{4}$</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">$\frac{\sqrt{3}+\sqrt{2}}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{6}+1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{6}+\sqrt{2}}{4}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \sin \frac{5 \pi}{12}=\sin 75^{\circ} \\ &amp;=\sin \left(45^{\circ}+30^{\circ}\right) \\ &amp;=\sin 45^{\circ} \cos 30^{\circ}+\cos 45^{\circ} \sin 30^{\circ} \\ &amp;=\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \cdot \frac{1}{2}=\frac{1}{\sqrt{2}}\left(\frac{\sqrt{3}+1}{2}\right) \\ &amp;=\frac{\sqrt{3}+1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{4} \end{aligned}$</div>


title: what-is-the-value-of-sin-22-2-1-cos-22-2-1-4
slug: what-is-the-value-of-sin-22-2-1-cos-22-2-1-4-80956
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left(\sin 22 \frac{1^{\circ}}{2}+\cos 22 \frac{1^{\circ}}{2}\right)^{4}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3+2 \sqrt{2}}{2}$</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">$\frac{1+2 \sqrt{2}}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3 \sqrt{2}+2}{2}$</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">$\frac{3+2 \sqrt{2}}{2}$</span> </div>

answer:
<div class="solution">Let $x=\left(\sin 22 \frac{1^{\circ}}{2}+\cos ^{2} 22 \frac{1^{\circ}}{2}\right)^{4}$<br/>$=\left\{\left(\sin 22 \frac{1^{\circ}}{2}+\cos 22 \frac{1}{2}^{-}\right)^{2}\right\}^{2}$<br/>$=\left(\sin ^{2} 22 \frac{1^{\circ}}{2}+\cos ^{2} 22 \frac{1^{\circ}}{2}+2 \sin 22 \frac{1^{\circ}}{2} \cos 22 \frac{1^{\circ}}{2}\right)^{2}$<br/>$=\left(1+\sin 45^{\circ}\right)^{2}$<br/>$(\because 2 \sin \theta \cos \theta=\sin 2 \theta)$<br/>$=\left(1+\frac{1}{\sqrt{2}}\right)^{2}=\left(\frac{\sqrt{2}+1}{\sqrt{2}}\right)^{2}$<br/>$=\frac{2+1+2 \sqrt{2}}{2}=\frac{3+2 \sqrt{2}}{2}$</div>


title: what-is-the-value-of-sin-192-0
slug: what-is-the-value-of-sin-192-0-76891
question: <div class="question">What is the value of $\sin \left(1920^{\circ}\right) ?$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\sqrt{2}}$</span> </li><li class="correct"> <span class="option-label">C</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><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 3$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \sin \left(1920^{\circ}\right)=\sin \left(360 \times 5+120^{\circ}\right) \\ &amp;=\sin 120^{\circ} \quad\left(\because \sin \left(360^{\circ}+\theta\right)=\sin \theta\right) \\ \text { 122. (b) } &amp; \sin (\mathrm{A}+\mathrm{B})=1 \\ &amp; \Rightarrow \sin (\mathrm{A}+\mathrm{B})=\sin 90^{\circ} \\ &amp; \Rightarrow \mathrm{A}+\mathrm{B}=90^{\circ} \\ &amp; \text { Given } \sin (\mathrm{A}-\mathrm{B})=\frac{1}{2}=\sin 30^{\circ} \\ &amp; \Rightarrow \mathrm{A}-\mathrm{B}=30^{\circ} \end{aligned}$<br/>On solving (1) and (2), we get $\mathrm{A}=60$<br/>$\mathrm{B}=30$</div>


title: what-is-the-value-of-sin-292-2-1
slug: what-is-the-value-of-sin-292-2-1-81824
question: <div class="question">What is the value of $\sin 292 \frac{1}{2}^{\circ}$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3} \sqrt{2+\sqrt{3}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{3} \sqrt{2-\sqrt{3}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2} \sqrt{2+\sqrt{2}}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{2} \sqrt{2+\sqrt{2}}$</span> </li> </ul>

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

answer:
<div class="solution">$\quad \sin \left(292 \frac{1}{2}\right)^{\circ}=\sin \frac{585^{\circ}}{2}$<br/>$=\sqrt{\frac{1-\cos 585^{\circ}}{2}}$<br/>$=\left(\because \cos 2 \theta=1-2 \sin ^{2} \theta \Rightarrow \sin \theta=\sqrt{\frac{1-\cos 2 \theta}{2}}\right)$<br/>$=\sqrt{\frac{1-\cos \left(360^{\circ}+225^{\circ}\right)}{2}}=\sqrt{\frac{1-\cos 225^{\circ}}{2}}$<br/>$=\sqrt{\frac{1-\cos \left(180^{\circ}+45^{\circ}\right)}{2}}$<br/>96. (b) We know that, $1^{\circ}   &lt; 1$ radian $\quad\left(\because 1^{\circ}=\frac{\pi}{180}\right.$ radian $)$<br/>$\Rightarrow \sin 1^{\circ} &lt; \sin 1$</div>


title: what-is-the-value-of-sin-420circ-cdot-cos-390circcos-left300circright-cdot-sin-left330circright
tags: ['Mathematics', 'Trigonometric Ratios & Identities', '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_P4.db
slug:
what-is-the-value-of-sin-420circ-cdot-cos-390circcos-left300circright-cdot-sin-left330circright-71119

question:
<div class="question">What is the value of $\sin 420^{\circ} \cdot \cos 390^{\circ}+\cos \left(-300^{\circ}\right) \cdot \sin \left(-330^{\circ}\right) ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\underline{0}$</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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-1$</span> </li> </ul>

answer:
<div class="solution">Given expression $=\sin \left(360^{\circ}+60^{\circ}\right) \cdot \cos \left(360^{\circ}+30^{\circ}\right)+\cos \left(360^{\circ}-60^{\circ}\right)$<br/>$\left(-\sin \left(360^{\circ}-30^{\circ}\right)\right.$<br/>$=\sin 60^{\circ} \cdot \cos 30^{\circ}+\cos 60^{\circ} \sin 30^{\circ}=\sin \left(60^{\circ}+30^{\circ}\right)$<br/>$=\sin 90^{\circ}=1$</div>


title: what-is-the-value-of-sin-a-b-sin-a-b-sin-b-c-sin-b-c-sin-c-a-sin-mathrmcmathrma
slug: what-is-the-value-of-sin-a-b-sin-a-b-sin-b-c-sin-b-c-sin-c-a-sin-mathrmcmathrma-13107
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\sin (A+B) \sin (A-B)+\sin (B+C) \sin (B-C)+\sin (C+A)$<br/>$\sin (\mathrm{C}-\mathrm{A}) ?</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">$\sin A+\sin B+\sin C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\cos A+\cos B+\cos C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})$<br/>$=\frac{1}{2}\{2 \sin (\mathrm{A}+\mathrm{B}) \cdot \sin (\mathrm{A}+\mathrm{B})\}$<br/>$=\frac{1}{2}\{\cos (\mathrm{A}-\mathrm{B}-\mathrm{A}-\mathrm{B})-\cos (\mathrm{A}-\mathrm{B}+\mathrm{A}+\mathrm{B})\}$<br/>$[$ Since $2 \sin \mathrm{X} \sin \mathrm{Y}=\cos (\mathrm{X}-\mathrm{Y})-\cos (\mathrm{X}+\mathrm{Y})]$<br/>$=\frac{1}{2}\{\cos 2 \mathrm{~B}-\cos 2 \mathrm{~A}\}$<br/>Also, $\sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{C})=\frac{1}{2}\{\cos 2 \mathrm{C}-\cos 2 \mathrm{~B}\}$<br/>and $\sin (C+A) \cdot \sin (C-A)$<br/>$\begin{aligned}=\frac{1}{2}\{\cos 2 \mathrm{~A}-\cos 2 \mathrm{C}\} &amp; \\ \therefore \quad \sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})+\sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{c}) \\ &amp;+\sin (\mathrm{C}+\mathrm{A}) \cdot \sin (\mathrm{C}-\mathrm{A}) \\=\frac{1}{2}\{\cos 2 \mathrm{C}-\cos 2 \mathrm{~B}+\cos 2 \mathrm{~A}-\cos 2 \mathrm{C}\\+&amp;\cos 2 \mathrm{~B}-\cos 2 \mathrm{~A}\}=0 \end{aligned}$</div>


title: what-is-the-value-of-sumr1n-fracpn-rr
slug: what-is-the-value-of-sumr1n-fracpn-rr-26198
question: <div class="question">What is the value of $\sum_{r=1}^{n} \frac{P(n, r)}{r !} ?</div>
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2^{n}-1$</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">$2^{n}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{n-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2^{n}+1$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\sum_{r=1}^{n} \frac{P(n, r)}{r !} \equiv \sum_{r=1}^{n} C(n, r)$<br/>Now, consider $(1+1)^{n}={ }^{n} \mathrm{C}_{0}+{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{2}+\ldots \ldots . .+{ }^{n} \mathrm{C}_{n}$<br/>$\Rightarrow 2^{n}=1+{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{2}+\ldots \ldots . .+{ }^{n} \mathrm{C}_{n}$<br/>$\Rightarrow 2^{n}-1={ }^{n} C_{1}+{ }^{n} C_{2}+\ldots \ldots . .+{ }^{n} C_{n}$<br/>$\Rightarrow 2^{n}-1=\sum_{r=1}^{n}{ }^{n} C_{r} \equiv \sum_{r=1}^{n} \frac{P(n, r)}{r !}$</div>


title: what-is-the-value-of-tan-1-5-tan-19-5
slug: what-is-the-value-of-tan-1-5-tan-19-5-44839
question: <div class="question">What is the value of $\tan 15^{\circ} \cdot \tan 195^{\circ} ?$</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$7-4 \sqrt{3}$</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">B</span> <span class="option-data">$7+4 \sqrt{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$7+2 \sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7+6 \sqrt{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7-4 \sqrt{3}$</span> </div>

answer:
<div class="solution">Consider $\tan 15^{\circ}$ tan $195^{\circ}$ $=\tan 15^{\circ} \tan \left(180+15^{\circ}\right)$<br/>$=\tan 15^{\circ} \tan 15^{\circ} \quad(\because \tan (180+\theta)=\tan \theta)$<br/>$\begin{aligned} &amp;=\left(\tan 15^{\circ}\right)^{2} \\ &amp;=(2-\sqrt{3})^{2} \quad\left(\because \tan 15^{\circ}=2-\sqrt{3}\right) \\ &amp;=4+3-4 \sqrt{3}=7-4 \sqrt{3} \\ \text { 109. (b) } &amp; \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{\sin ^{2} x+1+\cos ^{2} x+2 \cos x}{(1+\cos x)(\sin x)} \\ &amp;=\frac{2+2 \cos x}{(1+\cos x)(\sin x)}=\frac{2(1+\cos x)}{(1+\cos x)(\sin x)} \\ &amp;=\frac{2}{\sin x}=2 \operatorname{cosec} x \\ &amp; \Rightarrow 3 \sin A-4 \sin ^{3} A=1 \\ &amp; \Rightarrow 4 \sin ^{3} A-3 \sin A+1=0 \\ &amp; \Rightarrow(\sin A+1)\left(4 \sin ^{2} A-4 \sin A+1\right)=0 \\ &amp; \Rightarrow(\sin A+1)(2 \sin A-1)^{2}=0 \\ &amp; \Rightarrow \sin A=-1 \text { or } \frac{1}{2} \end{aligned}$<br/>Hence, $\sin A$ can take two distinct values.</div>


title: what-is-the-value-of-tan-1-n-m-tan-1-m-n-m-n
slug: what-is-the-value-of-tan-1-n-m-tan-1-m-n-m-n-59265
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\tan ^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right)-\tan ^{-1}\left(\frac{\mathrm{m}-\mathrm{n}}{\mathrm{m}+\mathrm{n}}\right) ?$</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">$\frac{\pi}{3}$</span> </li> </ul>

answer:
<div class="solution">The given expression is:<br/>$\tan ^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right)-\tan ^{-1}\left(\frac{\mathrm{m}-\mathrm{n}}{\mathrm{m}+\mathrm{n}}\right)$<br/>$=\tan ^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right)-\tan ^{-1}\left(\frac{1-\frac{\mathrm{n}}{\mathrm{m}}}{1+\frac{\mathrm{n}}{\mathrm{m}}}\right)$<br/>$=\tan ^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right)-\tan ^{-1}(1)+\tan ^{-1}\left(\frac{\mathrm{n}}{\mathrm{m}}\right)$<br/>$=\tan ^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right)+\cot ^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right)-\frac{\pi}{4}=\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}$</div>


title: what-is-the-value-of-tan-9-tan-2-7-tan-6-3-tan-8-1
slug: what-is-the-value-of-tan-9-tan-2-7-tan-6-3-tan-8-1-23265
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ} ?$</div>

answer:
<div class="solution">Given expression is $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$<br/>$=\tan 9^{\circ}-\tan 27^{\circ}-\tan \left(90^{\circ}-27^{\circ}\right)+\tan \left(90^{\circ}-9^{\circ}\right)$<br/>$=\tan 9^{\circ}-\tan 27^{\circ}-\cot 27^{\circ}+\cot 9^{\circ}$<br/>$=\left(\tan 9^{\circ}+\cot 9^{\circ}\right)-\left(\tan 27^{\circ}+\cot 27^{\circ}\right)$<br/>$=\frac{1}{\sin 9^{\circ} \cos 9^{\circ}}-\frac{1}{\sin 27^{\circ} \cos 27^{\circ}}$<br/>$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}}$<br/>$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin \left(90^{\circ}-36^{\circ}\right)}$<br/>$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\cos 36^{\circ}}$<br/>$=2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right]=8 \times \frac{2}{4}=4$<br/>(By putting value of $\sin 18^{\circ}$ and $\cos 36^{\circ}$ )</div>


title: what-is-the-value-of-tan-10-5
slug: what-is-the-value-of-tan-10-5-87035
question: <div class="question">What is the value of tan $105^{\circ}$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3}+1}{\sqrt{3}-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{3}+1}{1-\sqrt{3}}$</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">$\frac{\sqrt{3}-1}{\sqrt{3}+1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{3}+2}{\sqrt{3}-1}$</span> </li> </ul>

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

answer:
<div class="solution">Consider tan $\left(105^{\circ}\right)=\tan \left(60+45^{\circ}\right)$<br/>$=\frac{\tan 60^{\circ}+\tan 45^{\circ}}{1-\tan 60^{\circ} \tan 45^{\circ}}$<br/>$=\frac{\sqrt{3}+1}{1-\sqrt{3}} \quad\left(\because \tan 60^{\circ}=\sqrt{3}\right.$ and $\left.\tan 45^{\circ}=1\right)$</div>


title: what-is-the-value-of-tan-15circcot-15circ
slug: what-is-the-value-of-tan-15circcot-15circ-52649
question: <div class="question">What is the value of $\tan 15^{\circ}+\cot 15^{\circ} ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{3}$</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">$\quad$ Consider $\tan 15^{\circ}+\cot 15^{\circ}=\frac{\sin 15^{\circ}}{\cos 15^{\circ}}+\frac{\cos 15^{\circ}}{\sin 15^{\circ}}$<br/>$=\frac{\sin ^{2} 15^{\circ}+\cos ^{2} 15^{\circ}}{\cos 15^{\circ} \sin 15^{\circ}}=\frac{2 \times 1}{2 \cos 15^{\circ} \sin 15^{\circ}}$<br/>$\begin{aligned} &amp;\left(\because \sin ^{2} \theta+\cos ^{2} \theta=1\right) \\=\frac{2}{\sin 30^{\circ}}=4\left(\because \sin 30^{\circ}=\frac{1}{2}\right) &amp; \end{aligned}$</div>


title: what-is-the-value-of-tan-18circ
slug: what-is-the-value-of-tan-18circ-74781
question: <div class="question">What is the value of \tan 18^{\circ} ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{5}-1}{\sqrt{10+2 \sqrt{5}}}$</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">B</span> <span class="option-data">$\frac{\sqrt{5}-1}{\sqrt{10+\sqrt{5}}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{10+2 \sqrt{5}}}{\sqrt{5}-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{10+\sqrt{5}}}{\sqrt{5}-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{5}-1}{\sqrt{10+2 \sqrt{5}}}$</span> </div>

answer:
<div class="solution">$\tan 18^{\circ}=\frac{\sin 18^{\circ}}{\cos 18^{\circ}}=\frac{\frac{\sqrt{5}-1}{4}}{\frac{\sqrt{10+2 \sqrt{5}}}{4}}=\frac{\sqrt{5}-1}{\sqrt{10+2 \sqrt{5}}}$</div>


title: what-is-the-value-of-tan-left1575circright
slug: what-is-the-value-of-tan-left1575circright-11883
question: <div class="question">What is the value of \tan \left(-1575^{\circ}\right) ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
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">$1 / 2$</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">$-1$</span> </li> </ul>

answer:
<div class="solution">$\tan \left(-1575^{\circ}\right)=-\tan \left(4 \times 360^{\circ}+135^{\circ}\right)$<br/>$=-\tan 135^{\circ}=-\tan \left(90^{\circ}+45^{\circ}\right)=\cot 45^{\circ}=1$</div>


title: what-is-the-value-of-tan-tan-1-x-tan-1-y-tan-1-z-cot-cot-1-x-cot-1-y-cot-1-z
slug: what-is-the-value-of-tan-tan-1-x-tan-1-y-tan-1-z-cot-cot-1-x-cot-1-y-cot-1-z-16909
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)-\cot \left(\cot ^{-1} x+\cot ^{-1} y+\cot ^{-1} z\right) ?$</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">$2(\mathrm{x}+\mathrm{y}+\mathrm{z})$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3 \pi}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3 \pi}{2}+\mathrm{x}+\mathrm{y}+\mathrm{z}$</span> </li> </ul>

answer:
<div class="solution">$\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)$<br/>$-\cot \left(\cot ^{-1} x+\cot ^{-1} y+\cot ^{-1} z\right)$<br/>$=\tan \left(\tan ^{-1} x+\tan ^{-1} \mathrm{y}+\tan ^{-1} z\right)$<br/>$-\cot \left(\frac{\pi}{2}-\tan ^{-1} x+\frac{\pi}{2}-\tan ^{-1} y+\frac{\pi}{2}-\tan ^{-1} z\right)$<br/>$\quad\left(\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right)$<br/>$=\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)$<br/>$=\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)$<br/>$-\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)=0$</div>


title: what-is-the-value-of-the-determinant-1-1-1-1-1-xyz-1-1-1-1-xyz
slug: what-is-the-value-of-the-determinant-1-1-1-1-1-xyz-1-1-1-1-xyz-70365
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of the determinant <br/>$\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; 1+\mathrm{xyz} &amp; 1 \\ 1 &amp; 1 &amp; 1+\mathrm{xyz}\end{array}\right| ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+x+y+z$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x y z$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}^{2} \mathrm{y}^{2} \mathrm{z}^{2}$</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">$2 x^{2} y^{2} z^{2}$</span> </li> </ul>

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

answer:
<div class="solution">$\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; 1+x y z &amp; 1 \\ 1 &amp; 1 &amp; 1+x y z\end{array}\right| \begin{array}{l}c_{2} \rightarrow c_{2}-c_{1} \\ c_{3} \rightarrow c_{3}-c_{1}\end{array}$<br/>$=\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 1 &amp; x y z &amp; 0 \\ 1 &amp; 0 &amp; x y z\end{array}\right|=1\left(x^{2} y^{2} z^{2}-0\right)=x^{2} y^{2} z^{2}$</div>


title: what-is-the-value-of-the-determinant-1-1-1-b-c-c-a-ab-a-b-c-b-c-a-c-a-b
slug: what-is-the-value-of-the-determinant-1-1-1-b-c-c-a-ab-a-b-c-b-c-a-c-a-b-22012
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of the determinant <br/>$\left|\begin{array}{lll}1 &amp; b c &amp; a(b+c) \\ 1 &amp; c a &amp; b(c+a) \\ 1 &amp; a b &amp; c(a+b)\end{array}\right|$ ?</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">$a b c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a b+b c+c a$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a b c(a+b+c)$</span> </li> </ul>

answer:
<div class="solution">$\left|\begin{array}{lll}1 &amp; b c &amp; a(b+c) \\ 1 &amp; a c &amp; b(c+a) \\ 1 &amp; a b &amp; c(a+b)\end{array}\right|$<br/>Applying $C_{3} \rightarrow C_{2}+C_{3}$<br/>$\left|\begin{array}{lll}1 &amp; \text { bc } &amp; a b+b c+a c \\ 1 &amp; a c &amp; a b+b c+a c \\ 1 &amp; a b &amp; a b+b c+a c\end{array}\right|$<br/>$=(a b+b c+a c)\left|\begin{array}{lll}1 &amp; b c &amp; 1 \\ 1 &amp; a c &amp; 1 \\ 1 &amp; a b &amp; 1\end{array}\right|$<br/>$=(a b+b c+a c) \times 0=0$</div>


title: what-is-the-value-of-the-determinant-a-b-b-c-c-a-b-c-c-a-a-b-a-b-c
slug: what-is-the-value-of-the-determinant-a-b-b-c-c-a-b-c-c-a-a-b-a-b-c-87794
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of the determinant<br/>$\left|\begin{array}{lll}a-b &amp; b+c &amp; a \\ b-c &amp; c+a &amp; b \\ c-a &amp; a+b &amp; c\end{array}\right| ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a^{3}+b^{3}+c^{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 b c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$a^{3}+b^{3}+c^{3}-3 a b c$</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">0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a^{3}+b^{3}+c^{3}-3 a b c$</span> </div>

answer:
<div class="solution">Given $\begin{array}{rrr}\left|\begin{array}{rrr}a-b &amp; b+c &amp; a \\ b-c &amp; c+a &amp; b \\ c-a &amp; a+b &amp; c\end{array}\right| \\ C_{1} \rightarrow C_{1}-C_{3} &amp; \end{array}$<br/>$=\left|\begin{array}{ccc}-b &amp; b+c &amp; a \\ -c &amp; c+a &amp; b \\ -a &amp; a+b &amp; c\end{array}\right|$<br/>$\begin{aligned} &amp; C_{2} \rightarrow C_{2}+C_{1} \\=&amp;\left|\begin{array}{lll}-b &amp; c &amp; a \\ -c &amp; a &amp; b \\ -a &amp; b &amp; c\end{array}\right|=(-1)\left|\begin{array}{lll}b &amp; c &amp; a \\ c &amp; a &amp; b \\ a &amp; b &amp; c\end{array}\right| \\=&amp;-1\left[b\left(a c-b^{2}\right)-c\left(c^{2}-a b\right)+a\left(b c-a^{2}\right)\right] \\=&amp;-\left[a b c-b^{3}-c^{3}+a b c+a b c-a^{3}\right] \\=&amp; a^{3}+b^{3}+c^{3}-3 a b c . \end{aligned}$</div>


title: what-is-the-value-of-the-determinant-x-1-x-3-x-7-x-2-x-5-x-10-x-4-x-8-x-14
slug: what-is-the-value-of-the-determinant-x-1-x-3-x-7-x-2-x-5-x-10-x-4-x-8-x-14-77684
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-2$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of the determinant <br/>$\left|\begin{array}{ccc}x+1 &amp; x+2 &amp; x+4 \\ x+3 &amp; x+5 &amp; x+8 \\ x+7 &amp; x+10 &amp; x+14\end{array}\right|$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Let $\Delta=\left|\begin{array}{ccc}x+1 &amp; x+2 &amp; x+4 \\ x+3 &amp; x+5 &amp; x+8 \\ x+7 &amp; x+10 &amp; x+14\end{array}\right|$<br/>By applying $\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{1}, \mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1}$<br/>$\quad=\left|\begin{array}{lll}x+1 &amp; 1 &amp; 3 \\ x+3 &amp; 2 &amp; 5 \\ x+7 &amp; 3 &amp; 7\end{array}\right|$<br/>By applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ and $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{2}$, we<br/>get<br/>$\begin{array}{l}<br/>=\left|\begin{array}{ccc}<br/>x+1 &amp; 1 &amp; 3 \\<br/>2 &amp; 1 &amp; 2 \\<br/>4 &amp; 1 &amp; 2<br/>\end{array}\right| \\<br/>=(x+1)(0)-1(4-8)+3(2-4) \\<br/>=4-6=-2<br/>\end{array}$</div>


title: what-is-the-value-of-the-integral-1-1-x-d-x
slug: what-is-the-value-of-the-integral-1-1-x-d-x-67942
question: <div class="question">What is the value of the integral $\int_{-1}^{1}|x| d x ? \quad$</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
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 $\mathrm{I}=\int_{-1}^{1}|\mathrm{x}| \mathrm{d} \mathrm{x}=-\int_{-1}^{0} \mathrm{xd} \mathrm{x}+\int_{0}^{1} \mathrm{x} \mathrm{d} \mathrm{x}$<br/>Since, $|\mathrm{x}|=\left\{\begin{array}{ccc}\mathrm{x} &amp; \text { if } &amp; \mathrm{x} \geq 0 \\ -\mathrm{x} &amp; \text { if } &amp; \mathrm{x}   &lt; 0\end{array}\right\}$<br/>therefore $|x|=-x$ when $x$ lies, between $-1$ and 0 . and $|x|=x$ when $x$ lies between 0 and 1 . $=-\left[\frac{\mathrm{x}^{2}}{2}\right]^{0}+\left[\frac{\mathrm{x}^{2}}{2}\right]^{1}=-\left[-\frac{1}{2}\right]+\left[\frac{1}{2}\right]=1$</div>


title: what-is-the-value-of-the-minor-of-the-element-9-in-the-determinant-10-0-9-19-13-24-2-1-2
slug: what-is-the-value-of-the-minor-of-the-element-9-in-the-determinant-10-0-9-19-13-24-2-1-2-82862
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-7$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of the minor of the element 9 in the determinant $\left|\begin{array}{lll}10 &amp; 19 &amp; 2 \\ 0 &amp; 13 &amp; 1 \\ 9 &amp; 24 &amp; 2\end{array}\right| ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-9$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-7$</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">7</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">Minor of element $9=\left|\begin{array}{cc}19 &amp; 2 \\ 13 &amp; 1\end{array}\right|=19-26=-7$</div>


title: what-is-the-value-of-the-sum-2018-i-n-2-11-i-n-i-n-1-where-i-1
slug: what-is-the-value-of-the-sum-2018-i-n-2-11-i-n-i-n-1-where-i-1-98263
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of the sum $\quad[2018-I]$<br/>$\sum_{n=2}^{11}\left(i^{n}+i^{n+1}\right)$, where $i=\sqrt{-1} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">i</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{i}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-2 \mathrm{i}$</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">$1+\mathrm{i}$</span> </li> </ul>

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

answer:
<div class="solution">$\sum_{n=2}^{11}\left(i^{n}+i^{n+1}\right)$<br/>We know, $\mathrm{i}+i^{2}+i^{3}+i^{4}=i-1-i+1=0$<br/>Also, $i^{3}+i^{4}+i^{5}+i^{6}=0$<br/>The sum of 4 consecutive powers of 1 is always 0 .<br/>$\therefore \sum_{n=2}^{11}\left(i^{n}+i^{n+1}\right)=i^{2}+i^{3}+i^{3}+i^{4}+i^{4}+i^{5}+i^{5}$<br/>$+i^{6}+i^{6}+i^{7}+i^{7}+i^{8}+i^{8}+i^{9}+i^{9}+i^{10}+i^{10}$<br/>$+i^{11}+i^{11}+i^{12}$<br/>$=\left(i^{2}+i^{3}+i^{4}+i^{5}+i^{6}+i^{7}+i^{8}+i^{9}+i^{10}+i^{11}\right)+$<br/>$\quad\left(i^{3}+i^{4}+i^{5}+i^{6}+i^{7}+i^{8}+i^{9}+i^{10}+i^{11}+i^{12}\right)$<br/>$=\left(i^{2}+i^{3}+0\right)+\left(i^{3}+i^{4}+0\right)$<br/>$=i^{2}+i^{3}+i^{3}+i^{4}$<br/>$=i^{2}+2 i^{3}+i^{4}$<br/>$=-1+2(-i)+1$<br/>$=-2 i$</div>


title: what-is-the-value-of-x-2-a-2-x-2-b-2-dx
slug: what-is-the-value-of-x-2-a-2-x-2-b-2-dx-68642
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\int \frac{\mathrm{d} \mathrm{x}}{\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}+\mathrm{b}^{2}\right)} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\int \frac{\left[\left\{\tan ^{-1}(x / a)\right\} / a-\left\{\tan ^{-1}(x / b)\right\} / b\right]}{\left(a^{2}+b^{2}\right)}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\int \frac{\left[\left\{\tan ^{-1}(x / a)\right\} / a+\left\{\tan ^{-1}(x / b)\right\} / b\right]}{\left(a^{2}+b^{2}\right)}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\int \frac{\left[\left\{\tan ^{-1}(x / a)\right\} / a+\left\{\tan ^{-1}(x / b)\right\} / b\right]}{\left(b^{2}-a^{2}\right)}+c$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\int \frac{\left[\left\{\tan ^{-1}(x / a)\right\} / a+\left\{\tan ^{-1}(x / b)\right\} / b\right]}{\left(b^{2}-a^{2}\right)}+c$</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> </ul>

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

answer:
<div class="solution">The given integral is $\int \frac{\mathrm{dx}}{\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}+\mathrm{b}^{2}\right)}$<br/>Breaking the expression under integral into partial fraction $\frac{1}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}$<br/>$=\left(\frac{1}{\left(x^{2}+a^{2}\right)}-\frac{1}{\left(x^{2}+b^{2}\right)}\right) \times \frac{1}{b^{2}-a^{2}}$<br/>The given integral is <br/>$\frac{1}{\left(b^{2}-a^{2}\right)} \int\left(\frac{1}{\left(x^{2}+a^{2}\right)}-\frac{1}{\left(x^{2}+b^{2}\right)}\right) d x$<br/>$=\frac{1}{\left(b^{2}-a^{2}\right)} \int\left[\frac{1}{x^{2}+a^{2}} d x-\int \frac{1}{x^{2}+b^{2}} d x\right]$<br/>$=\frac{1}{\left(b^{2}-a^{2}\right)}\left\{\frac{\tan ^{-1}\left(\frac{x}{a}\right)}{a}-\frac{\tan ^{-1}\left(\frac{x}{b}\right)}{b}\right\}+c$</div>


title: what-is-the-value-of-x-at-the-intersection-of-y-x-2-4-8-and-x-y-2
slug: what-is-the-value-of-x-at-the-intersection-of-y-x-2-4-8-and-x-y-2-78891
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $x$ at the intersection of $y=\frac{8}{\left(x^{2}+4\right)}$<br/>and $x+y=2$</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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-1$</span> </li> </ul>

answer:
<div class="solution">Given equations are<br/>$y=\frac{8}{x^{2}+4}$ and $x+y=2$<br/>Putting value of $y$ from 1 st equation into second<br/>equation<br/>$x+\frac{8}{x^{2}+4}=2$<br/>$\Rightarrow x^{3}+4 x+8=2 x^{2}+8$<br/>$\Rightarrow x^{3}-2 x^{2}+4 x=0$<br/>$\Rightarrow x\left(x^{2}-2 x+4\right)=0$<br/>$\mathrm{x}=0$<br/>[The other valueof $x$ is not real $]$</div>


title: what-is-the-value-of-x-satisfying-the-equation-6-a-x-a-x-3-a-x-a-x
slug: what-is-the-value-of-x-satisfying-the-equation-6-a-x-a-x-3-a-x-a-x-41354
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $x$ satisfying the equation<br/>$6\left(\frac{a-x}{a+x}\right)^{3}=\frac{a+x}{a-x} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a / 2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a / 3$</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">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a / 4$</span> </li> </ul>

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

answer:
<div class="solution">Consider $16\left(\frac{a-x}{a+x}\right)^{3}=\frac{a+x}{a-x}$<br/>$\Rightarrow\left(\frac{a-x}{a+x}\right)^{3} \times\left(\frac{a-x}{a+x}\right)=\frac{1}{16}$<br/>$\Rightarrow\left(\frac{a-x}{a+x}\right)^{4}=\left(\frac{1}{2}\right)^{4}$<br/>$\Rightarrow \frac{a-x}{a+x}=\frac{1}{2}$<br/>$\Rightarrow 2 a-2 x=a+x$<br/>$\Rightarrow a=3 x$<br/>$\Rightarrow x=\frac{a}{3}$</div>


title: what-is-the-value-of-x-that-satisfies-the-equation-cos-1-x-2-sin-1-x
slug: what-is-the-value-of-x-that-satisfies-the-equation-cos-1-x-2-sin-1-x-36653
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $x$ that satisfies the equation $\cos ^{-1} x=2 \sin ^{-1} x ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">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">$-\frac{1}{2}$</span> </li> </ul>

answer:
<div class="solution">Given that $\cos ^{-1} \mathrm{x}=2 \sin ^{-1} \mathrm{x}$<br/>$\Rightarrow \frac{\pi}{2}-\sin ^{-1} x=2 \sin ^{-1} x$<br/>$\Rightarrow \frac{\pi}{2}=3 \sin ^{-1} \mathrm{x}$<br/>$\Rightarrow \sin ^{-1} x=\frac{\pi}{6}$<br/>So, $x=\sin \frac{\pi}{6}=\frac{1}{2}$</div>


title: what-is-the-value-of-x-x-1-dx
slug: what-is-the-value-of-x-x-1-dx-28685
question: <div class="question">What is the value of $\int(\sqrt{x}+x)^{-1} \mathrm{dx}?$</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\operatorname{In}(x+\sqrt{x})+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \ln (1+\sqrt{x})+c$</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">C</span> <span class="option-data">$2 \ln (x+\sqrt{x})+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \ln (x-\sqrt{x})+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \ln (1+\sqrt{x})+c$</span> </div>

answer:
<div class="solution">$\int(x+\sqrt{x})^{-1} d x=\int \frac{1}{(x+\sqrt{x})} d x$<br/>$=\int \frac{1}{(\sqrt{x} \cdot \sqrt{x}+\sqrt{x})} \cdot d x$<br/>Let $\sqrt{x}+1=t$<br/>then, $\frac{1}{2 \sqrt{x}} d x=d t \Rightarrow \frac{1}{\sqrt{x}} d x=2 d t$<br/>$\therefore \int \frac{1}{\sqrt{x}(\sqrt{x}+1)} d x=\int \frac{2 d t}{t}=2 \log t+c$<br/>$=2 \log (1+\sqrt{x})+c$</div>


title: what-is-the-variance-of-the-firs-ilnatural-number
slug: what-is-the-variance-of-the-firs-ilnatural-number-90791
question: <div class="question">What is the variance of the firs<br/>ILnatural number:</div>
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
answer: <div class="solution">Variance of 11 natural numbers $=\frac{11^{2}-1}{12}=10$</div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">10</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">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">11</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">13</span> </li> </ul>


title: what-is-the-vector-equally-inclined-to-the-vectors-i-3-j-and-3-i-j
slug: what-is-the-vector-equally-inclined-to-the-vectors-i-3-j-and-3-i-j-77106
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the vector equally inclined to the vectors $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}$<br/>and $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\hat{\mathbf{i}}+\hat{\mathbf{j}}$</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">$2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \hat{\mathbf{i}}+\hat{\mathbf{j}}$</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">$\hat{\mathbf{i}}+\hat{\mathbf{j}}$</span> </div>

answer:
<div class="solution">Let the required vector be $\hat{i}+\hat{j}$<br/>Since the vector $\hat{i}+\hat{j}$ is equally inclined to the vectors<br/>$\hat{i}+3 \hat{j}$ and $3 \hat{i}+\hat{j}$ therefore<br/><br/>Angle $\mathrm{b} / \mathrm{w} \hat{i}+\hat{j}$ and $\hat{i}+3 \hat{j}=\theta_{1}$ is equal to angle<br/>between $\hat{i}+\hat{j}$ and $3 \hat{i}+\hat{j}=\theta_{2}$<br/>$\therefore \quad$ Angle between $\hat{i}+\hat{j}$ and $\hat{i}+3 \hat{j}$<br/>$=\cos ^{-1}\left[\frac{(1)(1)+(1)(3)}{\sqrt{(1)^{2}+(1)^{2}} \sqrt{(1)^{2}+(3)^{2}}}\right]$<br/>$=\cos ^{-1}\left[\frac{1+3}{\sqrt{2} \sqrt{10}}\right]=\cos ^{-1}\left[\frac{4}{\sqrt{2} \sqrt{10}}\right]$<br/>$=\cos ^{-1}\left[\frac{2}{\sqrt{5}}\right]$ and<br/>angle between $\hat{i}+\hat{j}$ and $(3 \hat{i}+\hat{j})$<br/>$=\cos ^{-1}\left|\frac{1+3}{\sqrt{10} \sqrt{2}}\right|$<br/>$=\cos ^{-1}\left(\frac{4}{\sqrt{2} \sqrt{10}}\right)=\cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)$<br/>Hence required vector is $\hat{i}+\hat{j}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P4.db
title:
what-is-the-vector-in-the-x-y-plane-through-origin-and-perpendicular-to-the-vector-r-a-i-b-j-and-of-the-same-length

slug:
what-is-the-vector-in-the-x-y-plane-through-origin-and-perpendicular-to-the-vector-r-a-i-b-j-and-of-the-same-length-47906

question:
<div class="question">What is the vector in the $x y$ -plane through origin and perpendicular to the vector $\vec{r}=a \hat{i}+b \hat{j}$ and of the same length?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-a \hat{i}-b \vec{b}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a \hat{i}-b \hat{j}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-a \hat{i}+b \hat{j}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\quad b \hat{i}-a \hat{j}$</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">$\quad b \hat{i}-a \hat{j}$</span> </div>

answer:
<div class="solution">Let $\vec{r}_{1}=b \hat{i}-a \hat{j}$ be the required vector<br/>Given, $\vec{r}=a \vec{i}+b \vec{j}$<br/>Now, $\vec{r}_{1} \cdot \vec{r}=(b \hat{i}-a \hat{j}) \cdot(a \hat{i}+b \hat{j})$<br/>$=a b-a b=0$<br/>Hence, option (d) is the correct answer</div>


title: what-is-the-vector-perpendicular-to-both-the-vectors-i-j-and-i
slug: what-is-the-vector-perpendicular-to-both-the-vectors-i-j-and-i-62483
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the vector perpendicular to both the vectors<br/>$\hat{i}-\hat{j}$ and $\hat{\mathrm{i}} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\hat{i}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\hat{j}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\hat{j}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\hat{\mathrm{k}}$</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">$\hat{\mathrm{k}}$</span> </div>

answer:
<div class="solution">$\quad(\hat{\mathrm{i}}-\hat{\mathrm{j}}) \times(\hat{\mathrm{j}})$<br/>$\quad=\hat{\mathrm{i}} \times \hat{\mathrm{i}}-\hat{\mathrm{j}} \times \hat{\mathrm{j}}=\hat{\mathrm{k}}$</div>


title: what-is-the-vector-whose-magnitude-is-3-and-is-perpendicular-to-i-j-and-j-k
slug: what-is-the-vector-whose-magnitude-is-3-and-is-perpendicular-to-i-j-and-j-k-80850
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the vector whose magnitude is 3 , and is perpendicular<br/>to $\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\hat{\mathrm{j}}+\hat{\mathrm{k}} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3(\overrightarrow{\mathrm{i}}+\overrightarrow{\mathrm{j}}-\overrightarrow{\mathrm{k}})$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{3}(\overrightarrow{\mathrm{i}}-\overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}})$</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">$\sqrt{3}(\overrightarrow{\mathrm{i}}+\overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3(\overrightarrow{\mathrm{i}}-\overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{3}(\overrightarrow{\mathrm{i}}-\overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}})$</span> </div>

answer:
<div class="solution">Let the vector be $\mathrm{r}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$. Since, $\overrightarrow{\mathrm{r}}$ and $\hat{\mathrm{i}}+\hat{\mathrm{j}}$<br/>are perpendicular to each other.<br/>Hence, $\overrightarrow{\mathrm{r}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})=0 \Rightarrow \mathrm{x}+\mathrm{y}=0$<br/>also $\overrightarrow{\mathrm{r}}$ and $\hat{\mathrm{j}}+\hat{\mathrm{k}}$ are perpendicular to each other. so,<br/>$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{j}}+\hat{\mathrm{k}})=0$<br/>So, $\mathrm{y}+\mathrm{z}=0$<br/>(ii) and $x^{2}+y^{2}+z^{2}=9$<br/>$\Rightarrow \quad\left(-y^{2}\right)+y^{2}+(-y)^{2}=9$<br/>$\Rightarrow \quad 3 y^{2}=9$<br/>$\Rightarrow \quad y=\pm \sqrt{3}$<br/>$\therefore \quad \mathrm{x}=\mp \sqrt{3} \quad[$ from $(\mathrm{i})$<br/>and $\mathrm{z}=\pm \sqrt{3}$<br/>[from (ii)]<br/>So, vector is $\sqrt{3}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$</div>


tags: ['Biology', 'Transport in Plants', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-water-potential-of-pure-water-at-standard-temperatures-when-it-is-not-under-any-pressure

slug:
what-is-the-water-potential-of-pure-water-at-standard-temperatures-when-it-is-not-under-any-pressure-53076

question:
<div class="question">What is the water potential of pure water at standard temperatures when it is not under any pressure?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">-10</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

answer:
<div class="solution">By convention, the water potential of pure water at standard temperatures, which is not under any pressure, is taken to be zero.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
title:
what-is-the-x-coordinate-of-the-point-on-the-curve-f-x-x-7-x-6-where-the-tangent-is-parallel-to-x-axis

slug:
what-is-the-x-coordinate-of-the-point-on-the-curve-f-x-x-7-x-6-where-the-tangent-is-parallel-to-x-axis-10591

question:
<div class="question">What is the $\mathrm{x}$ -coordinate of the point on the curve $\mathrm{f}(\mathrm{x})=$ $\sqrt{\mathrm{x}}(7 \mathrm{x}-6)$, where the tangent is parallel to $\mathrm{x}$ -axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{2}{7}$</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">$\frac{6}{7}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{7}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{f}(\mathrm{x}) &amp;=\sqrt{\mathrm{x}}(7 \mathrm{x}-6)=7 \mathrm{x}^{3 / 2}-6 \mathrm{x}^{1 / 2} \\ \mathrm{f}^{\prime}(\mathrm{x}) &amp;=7 \times \frac{3}{2} \mathrm{x}^{1 / 2}-6 \times \frac{1}{2} \mathrm{x}^{-1 / 2} \end{aligned}$<br/>When tangent is parallel to $\mathrm{x}$ axis $\mathrm{f}^{\prime}(\mathrm{x})=0$ or, $\frac{21}{2} \mathrm{x}^{1 / 2}-3 \mathrm{x}^{-1 / 2}=0$<br/>$\frac{21}{2} \sqrt{x}=\frac{3}{\sqrt{x}}$<br/>or, $7 x=2 \Rightarrow x=\frac{2}{7}$</div>


title: what-is-true-about-earthworm-scorpion-cockroach-and-silverfish
slug: what-is-true-about-earthworm-scorpion-cockroach-and-silverfish-72447
question: <div class="question">What is true about earthworm, scorpion, cockroach, and silverfish?</div>
tags: ['Biology', 'Body Fluids and Circulation', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">They all posses dorsal heart</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">None of them in aquatic</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">They all belong to the same phylum</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">They all have joined paired appendages</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">They all posses dorsal heart</span> </div>

answer:
<div class="solution">Earthworm belonging to phylum Annelida, scorpion, silverfish, and cockroach (belonging to phylum Arthropoda) all have dorsal heart.</div>


title: what-is-true-about-ecosystem
slug: what-is-true-about-ecosystem-15383
question: <div class="question">What is true about ecosystem?</div>
tags: ['Biology', 'Ecosystem', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Primary consumers are least dependent upon producers</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Primary consumers out-number producers</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Producers are more than primary consumers</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">Secondary consumers are the largest and most powerful</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Producers are more than primary consumers</span> </div>

answer:
<div class="solution">In an ecosystem producers (green plants) are always more than consumers (herbivores, carnivores etc.)</div>


title: what-is-true-about-nereis-scorpion-cockroach-and-silver-fish
slug: what-is-true-about-nereis-scorpion-cockroach-and-silver-fish-26934
question: <div class="question">What is true about Nereis, scorpion, cockroach and silver fish?</div>
tags: ['Biology', 'Animal Kingdom', 'NEET', 'NEET 2007']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">they all possess dorsal heart</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">none of them is aquatic</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">they all belong to the same phylum</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">they all have jointed paired appendages.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">they all possess dorsal heart</span> </div>

answer:
<div class="solution">(Ref to Table)<br/>$\begin{array}{|l|l|l|l|l|}
<br/>\hline &amp; \text{Nereis} &amp; \text{Scorpion} &amp; \text{Cockroach} &amp; \text{Silver fish} \\
<br/>\hline \text{Dorsal heart} &amp; \text{Present} &amp; \text{Present} &amp; \text{Present} &amp; \text{Present} \\
<br/>\hline \text{Habitat} &amp; \begin{array}{l} 
<br/>\text{Sand or rocks in intertidal and} \\
<br/>\text{shallow marine waters.}
<br/>\end{array} &amp; \text{under logs, rocks, boards and clutter.} &amp; \text{damp but warm places} &amp; \text{dark, damp areas} \\
<br/>\hline \text{Phylum} &amp; \text{Annelida} &amp; \text{Arthropoda} &amp; \text{Arthropoda} &amp; \text{Arthropoda} \\
<br/>\hline \text{Presence of jointed appendages} &amp; \text{Absent} &amp; \text{Present} &amp; \text{Present} &amp; \text{Present} \\
<br/>\hline
<br/>\end{array}$<br/>Related Theory<br/>Scorpions are found on every continent except Antarctica, in habitats ranging from tropical rainforests to grasslands and deserts. As aduits, most scorpions are nocturnal and solitary, usually staying in the same territory throughout their lives. Many live in burrows they dig or claim and defend from other wildlife.</div>


title: what-is-true-about-the-euro-ii-norms
slug: what-is-true-about-the-euro-ii-norms-59994
question: <div class="question">What is true about the Euro II norms?</div>
tags: ['Biology', 'Environmental Issues', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">It stipulates to control sulphur at 350 ppm in diesel and 150 ppm in petrol</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">It stipulates to reduce sulphur levels to 50 ppm in petrol and diesel</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">It stipulates to reduce sulphur levels to 200 ppm in diesel and petrol</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It stipulates to reduce sulphur levels to 200 ppm in diesel and 100 ppm in petrol</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It stipulates to control sulphur at 350 ppm in diesel and 150 ppm in petrol</span> </div>

answer:
<div class="solution">Standard norms to keep check on air pollution is Euro II that stipulates that vehicular sulphur emission should be at 350ppm for diesel and 150ppm for petrol. Aromatic hydrocarbons should contain at 42% of the concerned fuel whether diesel or petrol.</div>


title: what-is-water-potential
slug: what-is-water-potential-11837
question: <div class="question">What is water potential?</div>
tags: ['Biology', 'Transport in Plants', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The energy of water molecules in random motion</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The concentration of water in a system</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The movement of water from high energy to low energy system</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">A concept fundamental to understanding water movement</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">A concept fundamental to understanding water movement</span> </div>

answer:
<div class="solution">Water potential (Î¨w) is a concept fundamental to understanding water movement.</div>


title: what-is-x-1-2-x-e-x-d-x-equal-to
slug: what-is-x-1-2-x-e-x-d-x-equal-to-62721
question: <div class="question">What is $\int \frac{x e^{x} d x}{(x+1)^{2}}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(x+1)^{2} e^{x}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(x+1) e^{x}+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{e^{x}}{x+1}+c$</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">$\frac{e^{x}}{(x+1)^{2}}+c$<br/>where $c$ is the constant integration.</span> </li> </ul>

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

answer:
<div class="solution">$\int \frac{x \mathrm{e}^{x}}{(1+x)^{2}} d x$<br/>$=\int \mathrm{e}^{x}\left[\frac{1}{1+x}-\frac{1}{(1+x)^{2}}\right] \mathrm{d} x$<br/>$=\frac{e^{x}}{1+x}+C \quad\left\{\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)\right\}$</div>


title: what-is-x-2-1-2-5-x-d-x-equal-to
slug: what-is-x-2-1-2-5-x-d-x-equal-to-49913
question: <div class="question">What is $\int\left(x^{2}+1\right)^{\frac{5}{2}} x d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(x^{2}+1\right)^{\frac{7}{2}}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2}{7}\left(x^{2}+1\right)^{\frac{7}{2}}+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{7}\left(x^{2}+1\right)^{\frac{7}{2}}+c$</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">None of the above<br/>where $\mathrm{c}$ is a constant of integration.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{7}\left(x^{2}+1\right)^{\frac{7}{2}}+c$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int \mathrm{x}\left(\mathrm{x}^{2}+1\right)^{5 / 2} \mathrm{dx}$<br/>Put $\mathrm{x}^{2}+1=\mathrm{t}$<br/>$2 \mathrm{xdx}=\mathrm{dt}$<br/>$\mathrm{xdx}=\frac{\mathrm{dt}}{2}$<br/>$\therefore \mathrm{I}=\int(\mathrm{t})^{5 / 2} \frac{\mathrm{dt}}{2}=\frac{1}{2}\left(\frac{\mathrm{t}^{7 / 2}}{7 / 2}\right)+\mathrm{c}=\frac{1}{7}\left(\mathrm{x}^{2}+\mathrm{l}\right)^{7 / 2}+\mathrm{c}$</div>


title: what-is-x-2-1-x-4-1-d-x-equal-to
slug: what-is-x-2-1-x-4-1-d-x-equal-to-38175
question: <div class="question">What is $\int \frac{x^{4}+1}{x^{2}+1} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{x^{3}}{3}-x+4 \tan ^{-1} x+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{x^{3}}{3}+x+4 \tan ^{-1} x+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{x^{3}}{3}-x+2 \tan ^{-1} x+c$</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">$\frac{x^{3}}{3}-x-4 \tan ^{-1} x+c$<br/>Where 'c' is a constant of integration.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x^{3}}{3}-x+2 \tan ^{-1} x+c$</span> </div>

answer:
<div class="solution">$\int \frac{x^{4}+1}{x^{2}+1} d x=\int \frac{x^{4}+2-1}{x^{2}+1} d x=\int\left(\frac{x^{4}-1}{x^{2}+1}+\frac{2}{x^{2}+1}\right) d x$<br/>$=\int\left[\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x^{2}+1}+\frac{2}{x^{2}+1}\right] d x$<br/>$=\int\left(x^{2}-1+\frac{2}{x^{2}+1}\right) d x=\frac{x^{3}}{3}-x+2 \tan ^{-1} x+c$</div>


title: what-is-x-2-a-2-d-x-equal-to
slug: what-is-x-2-a-2-d-x-equal-to-82547
question: <div class="question">What is $\int \frac{d x}{\sqrt{x^{2}+a^{2}}}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\ln \left|\frac{x+\sqrt{x^{2}+a^{2}}}{a}\right|+c$</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">$\ln \left|\frac{x-\sqrt{x^{2}+a^{2}}}{a}\right|+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\ln \left|\frac{x^{2}+\sqrt{x^{2}+a^{2}}}{a}\right|+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these<br/>where $c$ is the constant of integration.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ln \left|\frac{x+\sqrt{x^{2}+a^{2}}}{a}\right|+c$</span> </div>

answer:
<div class="solution">$\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{2}+\mathrm{a}^{2}}}$<br/>Let $\mathrm{x}=\mathrm{a} \tan \mathrm{u}$<br/>$\mathrm{dx}=\mathrm{a} \sec ^{2} \mathrm{u} \mathrm{du}$<br/>$=\int \frac{\mathrm{a} \sec ^{2} \mathrm{u} \mathrm{du}}{\sqrt{\mathrm{a}^{2} \tan ^{2} \mathrm{u}+\mathrm{a}^{2}}}$<br/>$=\mathrm{a} \int \frac{\sec ^{2} \mathrm{u} \mathrm{du}}{\sqrt{\mathrm{a}^{2}\left(1+\tan ^{2} \mathrm{u}\right)}} \Rightarrow \frac{\mathrm{a}}{\mathrm{a}} \int \frac{\sec ^{2} \mathrm{u} \mathrm{du}}{\sec \mathrm{u}}$<br/>$=\int \frac{\sec ^{2} \mathrm{u} \mathrm{du}}{\sec \mathrm{u}}=\int \sec \mathrm{u} \mathrm{du}$<br/>$=\ln [\tan (\mathrm{u})+\sec (\mathrm{u})]+\mathrm{c} \quad\left[\because \int \sec \mathrm{x} \mathrm{d} \mathrm{x}=\tan \mathrm{x}\right]$<br/>$=\ln \left[\frac{\mathrm{x}}{\sqrt{\mathrm{a}^{2}}}+\sqrt{1+\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}}\right]+\mathrm{c}$<br/>$=\ln \left[\frac{x}{a}+\frac{\sqrt{a^{2}+x^{2}}}{a}\right]+c$<br/>$=\ln \left[\frac{x+\sqrt{x^{2}+a^{2}}}{a}\right]+c$<br/>$\therefore \quad$ Option (a) is correct.</div>


title: what-is-x-2-f-x-equal-to-consider-the-function-f-x-2-x-2-1-s-i-n-x-where-x-2-and-f-2
slug: what-is-x-2-f-x-equal-to-consider-the-function-f-x-2-x-2-1-s-i-n-x-where-x-2-and-f-2-92490
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $x \rightarrow \frac{\pi}{2} f(x)$ equal to ?<br/>Consider the function $f(x)=\frac{1-\sin x}{(\pi-2 x)^{2}}$<br/>Where $x \neq \frac{\pi}{2}$ and $f\left(\frac{\pi}{2}\right)=\lambda$</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">$1 / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 4$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 8$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 8$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow \frac{\pi}{2}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow \frac{\pi}{2} \frac{1-\sin \mathrm{x}}{(\pi-2 \mathrm{x})^{2}}}$<br/>$=\lim _{\mathrm{x} \rightarrow \frac{\pi}{2}} \frac{-\cos \mathrm{x}}{(\pi-2 \mathrm{x})(-2)}$<br/>$=\lim _{\mathrm{x} \rightarrow \frac{\pi}{2}} \frac{\cos \mathrm{x}}{4(\pi-2 \mathrm{x})}$<br/>and $=\lim _{\mathrm{x} \rightarrow \frac{\pi}{2}} \frac{-\sin \mathrm{x}}{4(-2)}=\lim _{\mathrm{x} \rightarrow \frac{\pi}{2}} \frac{\sin \mathrm{x}}{8}$<br/>$=\frac{1}{8} \cdot \sin \frac{\pi}{2}=\frac{1}{8} \times 1=\frac{1}{8}$</div>


title: what-is-x-2-x-4-x-2-1-x-4-1-d-x-equal-to
slug: what-is-x-2-x-4-x-2-1-x-4-1-d-x-equal-to-24446
question: <div class="question">What is $\int \frac{x^{4}-1}{x^{2} \sqrt{x^{4}+x^{2}+1}} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{\frac{x^{4}+x^{2}+1}{x}}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{x^{4}+2-\frac{1}{x^{2}}}+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{\mathrm{x}^{2}+\frac{1}{\mathrm{x}^{2}}+1}+\mathrm{c}$</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">$\sqrt{\frac{x^{4}-x^{2}+1}{x}}+c$</span> </li> </ul>

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

answer:
<div class="solution">Take option (a) <br/>$I_{1}=\sqrt{\frac{x^{4}+x^{3}+1}{x}}+C$<br/>$\frac{d I_{1}}{d x}=\frac{d}{d x}\left[\left(x^{3}+x^{2}+x^{-1}\right)^{1 / 2}+C\right]$<br/>$\frac{d I_{2}}{d x}=\frac{1}{2}\left(x^{3}+x^{2}+x^{-1}\right)^{-1 / 2}\left(3 x^{2}+2 x-x^{-2}\right)$<br/>$=\frac{1}{2}\left[\frac{3 x^{2}+2 x-\frac{1}{x^{2}}}{\sqrt{x^{3}+x^{2}+\frac{1}{x}}}\right]$<br/>$\frac{d I_{2}}{d x}=\frac{1}{2}\left[\frac{3 x^{4}+2 x^{3}-1}{\frac{x^{2}}{\sqrt{x}} \sqrt{x^{4}+x^{3}+1}}\right]$<br/>Take option (b):<br/>$I_{2}=\sqrt{x^{4}+2-\frac{1}{x^{2}}}+C$<br/>$\frac{d I_{2}}{d x}=\frac{1}{2}\left[x^{4}+2-x^{-2}\right]^{-1 / 2}\left[4 x^{3}+0+2 x^{-3}\right]$<br/>$\quad=\frac{1}{2}\left[\frac{4 x^{3}+\frac{2}{x^{3}}}{\sqrt{x^{4}+2-\frac{1}{x^{2}}}}\right]=\frac{2 x^{6}+1}{\frac{x^{3}}{x} \sqrt{x^{6}+2 x^{2}-1}}$<br/>$\frac{d I_{2}}{d x}=\frac{2 x^{6}+1}{x^{2} \sqrt{x^{6}+2 x-1}}$<br/>Take option (c):<br/>$I_{3}=\sqrt{x^{2}+x^{-2}+1}+C$<br/>$\frac{d I_{3}}{d x}=\frac{1}{2}\left[x^{2}+x^{-2}+1\right]^{-1 / 2}\left[2 x-2 x^{-3}+0\right]$<br/>$=\frac{1}{2}\left[\frac{2 x-\frac{2}{x^{3}}}{\sqrt{x^{2}+\frac{1}{x^{2}}+1}}\right]$<br/>$=\frac{1}{2}\left[\frac{2\left(x^{4}-1\right)}{x^{3} \sqrt{\frac{x^{4}+1+x^{2}}{x^{2}}}}\right]$<br/>$\frac{d I_{3}}{d x}=\frac{x^{4}-1}{x^{2} \sqrt{x^{4}+x^{2}+1}}$</div>


title: what-is-x-3-sin-x-dx-equal-to
slug: what-is-x-3-sin-x-dx-equal-to-72581
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $\int\left(\mathrm{x}^{3}+\sin \mathrm{x}\right) \mathrm{d} \mathrm{x}$ equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{a}$</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</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{I}=\int_{-\mathrm{a}}^{\mathrm{a}}\left(\mathrm{x}^{3}+\sin \mathrm{x}\right) \mathrm{d} \mathrm{x}$<br/>Let $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\sin \mathrm{x}$<br/>$\mathrm{f}(-\mathrm{x})=(-\mathrm{x})^{3}+\sin (-\mathrm{x})=-\mathrm{x}^{3}-\sin \mathrm{x}$<br/>$=-\left(\mathrm{x}^{3}+\sin \mathrm{x}\right)=-\mathrm{f}(\mathrm{x})$<br/>Since $\mathrm{f}(\mathrm{x})$ is an odd function<br/>$\therefore \quad \int_{-a}^{a} f(x) d x=0$<br/>Since $\mathrm{f}(\mathrm{x})$ is an odd function<br/>$\therefore \quad \int_{-a}^{a} f(x) d x=0$</div>


title: what-is-x-cos-x-sin-x-d-x-equal-to
slug: what-is-x-cos-x-sin-x-d-x-equal-to-50668
question: <div class="question">What is $\int(x \cos x+\sin x) d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x \sin x+c$</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">$x \cos x+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-x \sin x+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-x \cos x+c$<br/>Where $c$ is an arbitrary constant.</span> </li> </ul>

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

answer:
<div class="solution">$\int(\mathrm{x} \cos \mathrm{x}+\sin \mathrm{x}) \mathrm{dx}=\int \mathrm{x} \cos \mathrm{x} \mathrm{dx}+\int \sin \mathrm{x} \mathrm{d} \mathrm{x}$<br/>$=\mathrm{x} \sin \mathrm{x}-\int \sin \mathrm{x} \mathrm{dx}+\int \sin \mathrm{x} \mathrm{dx}=\mathrm{x} \sin \mathrm{x}+\mathrm{c}$</div>


title: what-is-x-dx-b-x-dx-equal-to-where-is-the-greatest-a-a-integer-function
slug: what-is-x-dx-b-x-dx-equal-to-where-is-the-greatest-a-a-integer-function-23157
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$a-b$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $\int[\mathrm{x}] \mathrm{dx}+\int^{\mathrm{b}}[-\mathrm{x}] \mathrm{dx}$ equal to, where $[.]$ is the greatest $\mathrm{a} \quad \mathrm{a}$<br/>integer function?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{b}-\mathrm{a}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a-b$</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">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2(\mathrm{~b}-\mathrm{a})$</span> </li> </ul>

answer:
<div class="solution">$\int_{a}^{b}[x] d x+\int_{a}^{b}[-x] d x$<br/>$\quad=\int_{a}^{b}([x]+[-x]) d x=\int_{a}^{b}(-1) d x=a-b$</div>


title: what-is-x-dx-equal-to
slug: what-is-x-dx-equal-to-68573
question: <div class="question">What is $\quad$ | $\mathrm{x}$ $\mathrm{dx}$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
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">1</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">$-1$</span> </li> </ul>

answer:
<div class="solution">$\begin{array}{l}\text { Let } \mathrm{I}=\int_{-1}^{1} \mathrm{x}|\mathrm{x}| \mathrm{dx}=\int_{-1}^{0} \mathrm{x}(-\mathrm{x}) \mathrm{d} \mathrm{x}+\int_{0}^{1} \mathrm{x}(\mathrm{x}) \mathrm{d} \mathrm{x} \\ \left(\because|\mathrm{x}|=\left\{\begin{array}{ccc}\mathrm{x} &amp; \text { if } &amp; \mathrm{x} \geq 0 \\ -\mathrm{x} &amp; \text { if } &amp; \mathrm{x}   &lt; 0\end{array}\right)\right. \\ = &amp; -\int_{-1}^{0} \mathrm{x}^{2} \mathrm{~d} \mathrm{x}+\int_{0}^{1} \mathrm{x}^{2} \mathrm{dx} \\ = &amp; \left.\frac{-\mathrm{x}^{3}}{3}\right|_{-1} ^{0}+\left.\frac{\mathrm{x}^{3}}{3}\right|_{0} ^{1}=-\left(0+\frac{1}{3}\right)+\frac{1}{3}=0\end{array}$</div>


title: what-is-x-e-e-x-x-e-1-e-x-1-dx-equal-to
slug: what-is-x-e-e-x-x-e-1-e-x-1-dx-equal-to-37950
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $\int \frac{\left(x^{\mathrm{e}-1}+\mathrm{e}^{\mathrm{x}-1}\right) \mathrm{dx}}{\mathrm{x}^{\mathrm{e}}+\mathrm{e}^{\mathrm{x}}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{2}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\operatorname{In}(x+e)+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\operatorname{In}\left(\mathrm{x}^{\mathrm{e}}+\mathrm{e}^{\mathrm{x}}\right)+\mathrm{c}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{\mathrm{e}} \operatorname{In}\left(\mathrm{x}^{\mathrm{e}}+\mathrm{e}^{\mathrm{x}}\right)+\mathrm{c}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\mathrm{e}} \operatorname{In}\left(\mathrm{x}^{\mathrm{e}}+\mathrm{e}^{\mathrm{x}}\right)+\mathrm{c}$</span> </div>

answer:
<div class="solution">$\int \frac{\left(x^{e-1}+e^{x-1}\right) d x}{x^{e}+e^{x}}$<br/>Put $x^{e}+e^{x}=t \Rightarrow e x^{e-1}+e^{x}=\frac{d t}{d x}$<br/>$\Rightarrow\left(e x^{e-1}+e^{x}\right) d x=d t$<br/>$\therefore \int \frac{\left(x^{e-1}+e^{x-1}\right) d x}{x^{e}+e^{x}}=\frac{1}{e} \int \frac{e x^{e-1}+e^{x}}{x^{e}+e^{x}} \cdot d x$<br/>$=\frac{1}{e} \int \frac{d t}{t}=\frac{1}{e} \cdot \ln t+c$<br/>$=\frac{1}{e} \cdot \ln \left(x^{e}+e^{x}\right)+c$</div>


title: what-is-x-e-x-d-x-equal-to
slug: what-is-x-e-x-d-x-equal-to-85995
question: <div class="question">What is $\int \sqrt{x} e^{\sqrt{x}} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 e^{\sqrt{x}}(x-2 \sqrt{x}+2)+c$</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">$2 e^{\sqrt{x}}(x+2 \sqrt{x}+2)+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 e^{\sqrt{x}}(x+2 \sqrt{x}-2)+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 e^{\sqrt{x}}(x-2 \sqrt{x}-2)+c$<br/>Where ' $\mathrm{c}$ ' is a constant of integration.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 e^{\sqrt{x}}(x-2 \sqrt{x}+2)+c$</span> </div>

answer:
<div class="solution">Let $I=\int \sqrt{x} e^{\sqrt{x}} d x$<br/>Put, $\sqrt{x}=t$<br/>Differentiate both side w.r.t (t)<br/>$\Rightarrow \quad \frac{1}{2 \sqrt{x}} d x=d t \Rightarrow d x=2 t d t$<br/>$I=\int t e^{t} 2 t d t=2 \int t^{2} e^{t} d t$<br/>By parts, let first function $=t^{2}$ and second function $=e^{t .}$<br/>$=2\left[t^{2} e^{t}-\int 2 t e^{t} d t\right]=2\left[t^{2} e^{t}-2\left\{t e^{t}-\int e^{t} d t\right\}\right]$<br/>$=2\left[t^{2} e^{t}-2 t e^{t}+2 e^{t}\right]+c$<br/>where $c$ is a constant of integration.<br/>$=2\left[x e^{\sqrt{x}}-2 \sqrt{x} e^{\sqrt{x}}+2 e^{\sqrt{x}}\right]+c$<br/>$=2 e^{\sqrt{x}}(x-2 \sqrt{x}+2)+c$</div>


title: what-is-x-l-n-x-dx-equal-to
slug: what-is-x-l-n-x-dx-equal-to-44871
question: <div class="question">What is $\int \frac{\mathrm{dx}}{\mathrm{x} \ln \mathrm{x}}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\ln (\ln x)+c$</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">$\ln x+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(\ln x)^{2}+c$</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">$\ln (\ln x)+c$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int \frac{\mathrm{dx}}{\mathrm{x} \ell \mathrm{n} \mathrm{x}}$<br/>Put $\ell \mathrm{n} \mathrm{x}=\mathrm{t} \Rightarrow \frac{1}{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$<br/>$\therefore \quad \mathrm{I}=\int \frac{1}{\mathrm{t}} \mathrm{dt}=\ell \mathrm{nt}+\mathrm{c}=\ell \mathrm{n}(\ln \mathrm{x})+\mathrm{c}$<br/>where $\mathrm{c}$ is the constant of integration.</div>


title: what-is-x-nx-dx-equal-to
slug: what-is-x-nx-dx-equal-to-15813
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $\int \frac{\ell \mathrm{n} \mathrm{x}}{\mathrm{x}} \mathrm{d} \mathrm{x}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{(\ell \mathrm{n} \mathrm{x})^{2}}{2}+\mathrm{c}$</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">$\frac{(\ell \mathrm{n} \mathrm{x})}{2}+\mathrm{c}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(\ln x)^{2}+\mathrm{c}$</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">$\frac{(\ell \mathrm{n} \mathrm{x})^{2}}{2}+\mathrm{c}$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int \frac{\ell \mathrm{nx}}{\mathrm{x}} \mathrm{dx}$<br/>Put $\ell \mathrm{n} \mathrm{x}=\mathrm{t}$<br/>$\frac{1}{x} \mathrm{dx}=\mathrm{dt}$<br/>$\therefore \mathrm{I}=\int \mathrm{t} \mathrm{dt}=\frac{\mathrm{t}^{2}}{2}+\mathrm{c}$<br/>where $\mathrm{c}$ is the constant of integration. <br/>$=\frac{(\ln \mathrm{x})^{2}}{2}+\mathrm{c}$</div>


title: what-is-x-s-i-n-x-d-x-equal-to
slug: what-is-x-s-i-n-x-d-x-equal-to-20914
question: <div class="question">What is $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\cos \sqrt{x}}{2}+C$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \cos \sqrt{x}+C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-\cos \sqrt{x}}{2}+C$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-2 \cos \sqrt{x}+C$</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> </ul>

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

answer:
<div class="solution">Let $I=\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$<br/>Put $\sqrt{x}=t \Rightarrow \frac{1}{2 \sqrt{x}} d x=d t \Rightarrow \frac{1}{\sqrt{x}} d x=2 d t$<br/>$\therefore \quad I=2 \int \sin t d t=-2 \cos t+C=-2 \cos \sqrt{x}+C$<br/>where ' $C^{\prime}$ is a constant of integration.</div>


title: what-is-x-x-7-1-dx-equal-to
slug: what-is-x-x-7-1-dx-equal-to-10933
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $\int \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}^{7}+1\right)}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \ln \left|\frac{x^{7}-1}{x^{7}+1}\right|+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{7} \ln \left|\frac{x^{7}+1}{x^{7}}\right|+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\ln \left|\frac{x^{7}-1}{7 x}\right|+c$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{7} \ln \left|\frac{x^{7}}{x^{7}+1}\right|+c$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{7} \ln \left|\frac{x^{7}}{x^{7}+1}\right|+c$</span> </div>

answer:
<div class="solution">$\int \frac{\mathrm{d} x}{x\left(x^{7}+1\right)}=\int \frac{x^{6}}{x^{7}\left(x^{7}+1\right)} \cdot \mathrm{dx}$<br/>Let $x^{7}=t$<br/>$\Rightarrow 7 x^{6} \cdot \mathrm{dx}=\mathrm{dt} \Rightarrow x^{6} \mathrm{dx}=\frac{\mathrm{dt}}{7}$<br/>Then $\int \frac{x^{6} \mathrm{~d} x}{x^{7}\left(x^{7}+1\right)}=\frac{1}{7} \int \frac{d t}{t(t+1)}$<br/>$=\frac{1}{7}\left[\int_{t}^{1} \frac{d t}{t+1} d t\right]$<br/>$=\frac{1}{7}[\ell n|t|-\ell n|t+1|]+c$<br/>$=\frac{1}{7} \ell n\left|\frac{t}{t+1}\right|+c$<br/>$=\frac{1}{7} \ell n\left|\frac{x^{7}}{x^{7}+1}\right|+c$</div>


title: what-is-x-y-z-a-h-g-h-b-f-g-f-c-equal-to
slug: what-is-x-y-z-a-h-g-h-b-f-g-f-c-equal-to-30250
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is $\left[\begin{array}{lll}\mathrm{x} &amp; \mathrm{y} &amp; \mathrm{z}\end{array}\right]\left[\begin{array}{lll}\mathrm{a} &amp; \mathrm{h} &amp; \mathrm{g} \\ \mathrm{h} &amp; \mathrm{b} &amp; \mathrm{f} \\ \mathrm{g} &amp; \mathrm{f} &amp; \mathrm{c}\end{array}\right]$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[a x+h y+g z h+b+f \quad g+f+c]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{lll}a &amp; h &amp; g \\ h x &amp; b y &amp; f z \\ g &amp; f &amp; c\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{l}a x+h y+g z \\ h x+b y+f z \\ g x+f y+c z\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$[a x+h y+g z \quad h x+b y+f z \quad g x+f y+c z]$</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">$[a x+h y+g z \quad h x+b y+f z \quad g x+f y+c z]$</span> </div>

answer:
<div class="solution">$\left[\begin{array}{lll}x &amp; y &amp; z\end{array}\right]\left[\begin{array}{lll}a &amp; h &amp; g \\ h &amp; b &amp; f \\ g &amp; f &amp; c\end{array}\right]$<br/>$=[a x+h y+g z \quad h x+b y+f z \quad g x+f y+c z]$</div>


title: what-is-xe-x-dx-equal-to
slug: what-is-xe-x-dx-equal-to-38835
question: <div class="question">What is $\int \mathrm{xe}^{\mathrm{x}} \mathrm{dx}$ equal to</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
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">$\mathrm{e}$</span> </li> </ul>

answer:
<div class="solution">Let $I=\int_{0}^{1} x e^{x} d x=x e^{x}-\int_{0}^{1} 1 . e^{x} d x=\left[x e^{x}-e^{x}\right]_{0}^{1}$<br/>$=(e-e)-[0-1]=1$</div>


title: what-isare-the-functional-groups-of-ranitidine
slug: what-isare-the-functional-groups-of-ranitidine-32044
tags: ['Chemistry', 'Chemistry in Everyday Life', 'NEET']
answer: <div class="solution">Correct Option is : 4 <br/> All of the above</div>
source_db: MarksBatch2_P4.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/modules/dbq/cheas77dtry/Z8OgZ4uH2bUhHB0mc7Ew9Hn7VBw3BCqfpuXS8Ey8RkFg798zFh.png"/>What is/are the functional groups of ranitidine?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">primary amine</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">secondary amine</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">nitro</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">All 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>


title: what-isare-the-solutions-of-the-trigonometric-equation-cosec-x-cot-x-3-where-0-x-2
slug: what-isare-the-solutions-of-the-trigonometric-equation-cosec-x-cot-x-3-where-0-x-2-59538
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is/are the solutions of the trigonometric<br/>equation cosec $x+\cot x=\sqrt{3}$, where $0   &lt; x &lt; 2 \pi$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5 \pi}{3}$ only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{3}$ 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">$\pi$ only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi, \frac{\pi}{3}, \frac{5 \pi}{3}$</span> </li> </ul>

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

answer:
<div class="solution">Given equation:<br/>$\operatorname{cosec} x+\cot x=\sqrt{3}$<br/>$\operatorname{cosec} x-\cot x=\frac{1}{\sqrt{3}}$<br/>$\Rightarrow \quad 2 \operatorname{cosec} x=\sqrt{3}+\frac{1}{\sqrt{3}}=\frac{4}{\sqrt{3}}$<br/>$\Rightarrow \operatorname{cosec} x=\frac{2}{\sqrt{3}}$<br/>$\Rightarrow \sin x=\frac{\sqrt{3}}{2}$<br/>Possible values of $x=\frac{\pi}{3}, \frac{2 \pi}{3}$ $\therefore$ Option (b) is correct.</div>


title: what-ix-the-alea-enclused-between-the-cul-ves-y-2-12-x-and-the-lines-x-0-and-y-6
slug: what-ix-the-alea-enclused-between-the-cul-ves-y-2-12-x-and-the-lines-x-0-and-y-6-10465
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What ix the alea enclused between the cul ves $y^{2}-12 x$ and the lines $x=0$ and $y=6 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2 sq unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4 sq unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">6 sq unit</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">8 sq unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">6 sq unit</span> </div>

answer:
<div class="solution">Equation of given curve is $y^{2}=12 x$ At $y=6,36=12 x \Rightarrow \quad x=3$<br/>$\therefore$ Required area $=\int_{0}^{3}\left(y_{1}-y_{2}\right) d x$ where $y_{1}$ represents line and $y_{2}$ represents the curve.<br/>$=\int_{0}^{3}(6-\sqrt{12 x}) d x=[6 x]_{0}^{3}-\sqrt{12}\left[\frac{2 x^{3 / 2}}{3}\right]_{0}^{3}$<br/>$=[6 \times 3]-\frac{\sqrt{12} \times 2 \times \sqrt{27}}{3}=18-12=6$ sq unit</div>


title: what-measures-should-be-taken-for-public-hygiene-to-prevent-the-spread-of-infectious-diseases
slug: what-measures-should-be-taken-for-public-hygiene-to-prevent-the-spread-of-infectious-diseases-13405
tags: ['Biology', 'Human Health and Diseases', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What measures should be taken for public hygiene to prevent the spread of infectious diseases?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Avoid close contact with infected persons</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Consume clean food and water</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Proper disposal of waste and excreta</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">Spraying insecticides in ditches, drainage areas and swamps</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Proper disposal of waste and excreta</span> </div>

answer:
<div class="solution">Public hygiene measures include proper disposal of waste and excreta, periodic cleaning and disinfection of water reservoirs, pools, cesspools, and tanks, and observing standard practices of hygiene in public catering.</div>


tags: ['Chemistry', 'Surface Chemistry', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-optical-phenomenon-is-described-in-the-paragraph-as-the-scattering-of-light-in-all-directions-by-colloidal-particles

slug:
what-optical-phenomenon-is-described-in-the-paragraph-as-the-scattering-of-light-in-all-directions-by-colloidal-particles-13919

question:
<div class="question">What optical phenomenon is described in the paragraph as the scattering of light in all directions by colloidal particles?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Diffraction</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Dispersion</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Tyndall effect</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">Refraction</span> </li> </ul>

answer:
<div class="solution">The optical phenomenon described in the paragraph as the scattering of light in all directions by colloidal particles is called the Tyndall effect.</div>


title: what-organization-provides-recommendations-for-the-nomenclature-of-coordination-entities
slug: what-organization-provides-recommendations-for-the-nomenclature-of-coordination-entities-24583
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What organization provides recommendations for the nomenclature of coordination entities?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">United Nations</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">World Health Organization (WHO)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">International Union of Pure and Applied Chemistry (IUPA<br/>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">European Space Agency (ES<br/>A)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">International Union of Pure and Applied Chemistry (IUPA<br/>C)</span> </div>

answer:
<div class="solution">The recommendations for the nomenclature of coordination entities are based on the guidelines provided by the International Union of Pure and Applied Chemistry (IUPA</div>


title: what-practical-significance-does-the-concept-of-surroundings-hold-in-thermodynamics
slug: what-practical-significance-does-the-concept-of-surroundings-hold-in-thermodynamics-74032
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What practical significance does the concept of surroundings hold in thermodynamics?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">It defines the boundaries of the universe</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">It helps in understanding the behavior of the system</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">It isolates the system from external influences</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">It determines the speed of energy transformations</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It isolates the system from external influences</span> </div>

answer:
<div class="solution">The surroundings in thermodynamics serve the practical purpose of isolating the system from external influences.</div>


title: what-product-is-formed-when-secondary-alcohols-are-oxidised-using-chromic-anhydride-cro-3
slug: what-product-is-formed-when-secondary-alcohols-are-oxidised-using-chromic-anhydride-cro-3-45292
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Ketone</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What product is formed when secondary alcohols are oxidised using chromic anhydride (CrO<sub>3</sub>)?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Carboxylic acid</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Aldehyde</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Ketone</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">Alkene</span> </li> </ul>

answer:
<div class="solution">Secondary alcohols are oxidised to form ketones when using chromic anhydride (CrO<sub>3</sub>).</div>


title: what-property-of-alkali-metals-makes-them-strong-reducing-agents
slug: what-property-of-alkali-metals-makes-them-strong-reducing-agents-37410
question: <div class="question">What property of alkali metals makes them strong reducing agents?</div>
tags: ['Chemistry', 's Block Elements', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">High ionization enthalpy</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Low sublimation enthalpy</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">High hydration enthalpy</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">Low electronegativity</span> </li> </ul>

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

answer:
<div class="solution">The high reducing power of alkali metals is primarily attributed to their high hydration enthalpy. This is because the small size of alkali metal ions, such as lithium (Li+), results in strong electrostatic attraction with water molecules, leading to a highly negative standard electrode potential (E<sub>0</sub>) and strong reducing properties.</div>


tags: ['Chemistry', 'Amines', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-property-of-amines-is-used-for-the-distinction-of-primary-secondary-and-tertiary-amines-using-benzenesulfonyl-chloride

slug:
what-property-of-amines-is-used-for-the-distinction-of-primary-secondary-and-tertiary-amines-using-benzenesulfonyl-chloride-46913

question:
<div class="question">What property of amines is used for the distinction of primary, secondary, and tertiary amines using benzenesulfonyl chloride?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Solubility in water</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Boiling point</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Acidic character</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">Basicity</span> </li> </ul>

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

answer:
<div class="solution">The reaction of amines with benzene sulphonyl chloride helps distinguish their acidic character.</div>


title: what-recommendation-has-the-iupac-made-to-avoid-naming-disputes-for-new-elements
slug: what-recommendation-has-the-iupac-made-to-avoid-naming-disputes-for-new-elements-12488
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What recommendation has the IUPAC made to avoid naming disputes for new elements?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The IUPAC recommends that new elements should be named after famous scientists.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The IUPAC recommends that new elements should not be named until their discovery is proven.</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">The IUPAC recommends that new elements should be named by the first laboratory to synthesize them.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The IUPAC recommends that new elements should be named after the laboratory where they were discovered.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The IUPAC recommends that new elements should not be named until their discovery is proven.</span> </div>

answer:
<div class="solution">To avoid problems, the IUPAC has made a recommendation that until a new elementâs discovery is proved, and its name is officially recognised, a systematic nomenclature be derived directly from the atomic number of the element using the numerical roots for 0 and numbers 1-9.</div>


title: what-remains-intact-in-a-protein-during-denaturation
slug: what-remains-intact-in-a-protein-during-denaturation-37878
question: <div class="question">What remains intact in a protein during denaturation?</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Primary structure</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">Secondary structure</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Tertiary structure</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Quaternary structure</span> </li> </ul>

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

answer:
<div class="solution">During denaturation, secondary and tertiary structures are destroyed, but the primary structure remains intact.</div>


title: what-s-e-c-x-d-x-is-equal-to
slug: what-s-e-c-x-d-x-is-equal-to-10385
question: <div class="question">What $\int_{\sec x^{\circ}} d x$ is equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\log \left(\sec x^{\circ}+\tan x^{\circ}\right)+\mathrm{c}$</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">$\frac{\pi \log \tan \left(\frac{\pi}{4}+\frac{\pi}{2}\right)}{180^{\circ}}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{180^{\circ} \log \tan \left(\frac{\pi}{4}+\frac{x}{2}\right)}{\pi}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{180^{\circ} \log \tan \left(\frac{\pi}{4}+\frac{x}{360^{\circ}}\right)}{\pi}+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log \left(\sec x^{\circ}+\tan x^{\circ}\right)+\mathrm{c}$</span> </div>

answer:
<div class="solution">$\int \sec x^{\circ} \cdot d x=\int \frac{\sec x^{\circ} \cdot\left(\sec x^{\circ}+\tan x^{\circ}\right)}{\sec x^{\circ}+\tan x^{\circ}} \cdot d x$<br/>Let $u=\sec x^{\circ}+\tan x^{\circ}$<br/>$\Rightarrow \quad \frac{d u}{d x}=\sec x^{\circ}+\tan x^{\circ}+\sec ^{2} x^{\circ}$<br/>$\Rightarrow \quad d u=\left(\sec x^{\circ} \times \tan x^{\circ}+\sec ^{2} x^{\circ}\right) \cdot d x$<br/>$\therefore \quad \int \frac{\sec x^{\circ} \cdot\left(\sec x^{\circ}+\tan x^{\circ}\right)}{\sec x^{\circ}+\tan x^{\circ}} d x$<br/>$=\int \frac{\sec ^{2} x^{\circ}+\sec x^{\circ} \cdot \tan x^{\circ}}{\sec x^{\circ}+\tan x^{\circ}} \cdot d x$<br/>$\quad=\int \frac{d u}{u}=\log (u)+\mathrm{C}=\log \left(\sec x^{\circ}+\tan x^{\circ}\right)+\mathrm{C}$</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
title:
what-should-be-the-height-of-a-flag-where-a-20-feet-long-ladder-reaches-20-feet-below-the-flag-the-angle-of-elevation-of-the-top-of-the-flag-at-the-fo

slug:
what-should-be-the-height-of-a-flag-where-a-20-feet-long-ladder-reaches-20-feet-below-the-flag-the-angle-of-elevation-of-the-top-of-the-flag-at-the-fo-44242

question:
<div class="question">What should be the height of a flag where a 20 feet long ladder reaches 20 feet below the flag (The angle of elevation of the top of the flag at the foot of the ladder is $60^{\circ}$ )?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20 feet</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">30 feet</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">40 feet</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\begin{array}{l}\text { } 20 \sqrt{2} \text { feet } &amp; {\underline{}}\end{array}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">30 feet</span> </div>

answer:
<div class="solution">$\begin{array}{ll}\text { } &amp; \text {  } \operatorname{In} \Delta \mathrm{ABD} \text { , }\end{array}$<br/>$\tan 60^{\circ}=\frac{B D}{A B}$<br/>$\Rightarrow \sqrt{3}=\frac{\mathrm{h}}{\mathrm{AB}}$<br/>$\Rightarrow \mathrm{AB}=\frac{\mathrm{h}}{\sqrt{3}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/LBrs15-PYcqTBP1gFxPq9XGcTZmhC79Ev0aNX2EEvTw.original.fullsize.png"/><br/><br/>$\Rightarrow \mathrm{AB}=\frac{\mathrm{h}}{3} \sqrt{3}$<br/>Now, in $\triangle \mathrm{ABC}$<br/>$\mathrm{AC}^{2}=\mathrm{AB}^{2}+\mathrm{BC}^{2}$<br/>$\Rightarrow 20^{2}=\left(\frac{\mathrm{h}}{\sqrt{3}}\right)^{2}+(\mathrm{h}-20)^{2}$<br/>$\Rightarrow \mathrm{h}^{2}+3 \mathrm{~h}^{2}-120 \mathrm{~h}=0$<br/>$\Rightarrow 4 \mathrm{~h}^{2}-120 \mathrm{~h}=0$<br/>$\Rightarrow \mathrm{h}(\mathrm{h}-30)=0$<br/>$\mathrm{h}=0$ or 30<br/>$\mathrm{h}=0$ not possible $\Rightarrow \mathrm{h}=30 \mathrm{ft}$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (19 Oct Shift 1)']
source_db: MarksBatch2_P4.db
title:
what-should-be-the-length-of-a-closed-pipe-to-produce-resonance-with-sound-wave-of-wavelength-62-cm-in-fundamental-mode-neglect-end-correction

slug:
what-should-be-the-length-of-a-closed-pipe-to-produce-resonance-with-sound-wave-of-wavelength-62-cm-in-fundamental-mode-neglect-end-correction-71866

question:
<div class="question">What should be the length of a closed pipe to produce resonance with sound wave<br/>of wavelength $62 \mathrm{~cm}$, in fundamental mode?<br/>[Neglect end correction]</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$31 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$15 \cdot 5 \mathrm{~cm}$</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">$20 \cdot 6 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$46 \cdot 5 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$15 \cdot 5 \mathrm{~cm}$</span> </div>

answer:
<div class="solution">(C)<br/>In a closed pipe in fundamental mode the length of the pipe is $\frac{\lambda}{4}$. $\therefore \ell=\frac{62}{4}=15.5 \mathrm{~cm}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', '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_P4.db
title:
what-should-be-the-value-of-k-for-which-the-equation-3-x-2-3-y-2-k-1-z-2-x-y-z-0-represents-the-sphere

slug:
what-should-be-the-value-of-k-for-which-the-equation-3-x-2-3-y-2-k-1-z-2-x-y-z-0-represents-the-sphere-84624

question:
<div class="question">What should be the value of $\mathrm{k}$ for which the equation $3 x^{2}+3 y^{2}+(k+1) z^{2}+x-y+z=0$ represents the sphere?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-1$</span> </li> </ul>

answer:
<div class="solution">Given equation is $3 x^{2}+3 y^{2}+(k+1) z^{2}+x-y+z=0$<br/>which will represents a sphere if coeff of $x^{2}=$ coeff of $y^{2}=$ coeff of $z^{2}$ $\Rightarrow 3=\mathrm{k}+1$<br/>$\Rightarrow \mathrm{k}=2$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P4.db
title:
what-should-be-the-value-of-k-so-that-the-system-of-linear-equations-x-y-2-z-0-k-x-y-z-0-3-x-y-3-z-0-does-not-possess-a-unique-solution

slug:
what-should-be-the-value-of-k-so-that-the-system-of-linear-equations-x-y-2-z-0-k-x-y-z-0-3-x-y-3-z-0-does-not-possess-a-unique-solution-76649

question:
<div class="question">What should be the value of $\mathrm{k}$ so that the system of linear equations $x-y+2 z=0, k x-y+z=0,3 x+y-3 z=0$ does not possess a unique solution?</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">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">Equations are<br/>$\begin{array}{l}<br/>x-y+2 x=0 ...(i) \\ <br/>k x-y+z=0 ...(ii) \\<br/>3 x+y-3 x=0 ...(iii)<br/>\end{array}$<br/>System of equations posses a unique solution, if<br/>$|\mathrm{A}|=\left|\begin{array}{rrr}<br/>1 &amp; -1 &amp; 2 \\<br/>\mathrm{k} &amp; -1 &amp; 1 \\<br/>3 &amp; 1 &amp; -3<br/>\end{array}\right| \neq 0$<br/>$\text { Applying } \mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2} \text { and } \mathrm{C}_{2} \rightarrow \mathrm{C}_{2}+\frac{1}{2} \mathrm{C}_{3}$<br/>$|\mathbf{A}|=\left|\begin{array}{ccc}0 &amp; 0 &amp; 2 \\ \mathrm{k}-1 &amp; -\frac{1}{2} &amp; 1 \\ 4 &amp; -\frac{1}{2} &amp; -3\end{array}\right|=-\frac{1}{2}\left|\begin{array}{lrr}0 &amp; 0 &amp; 2 \\ \mathrm{k}-1 &amp; -1 &amp; 1 \\ 4 &amp; -1 &amp; -3\end{array}\right|$<br/>$=\frac{1}{2} \times 2[(\mathrm{k}-1)(-1)-(4)(-1)], 0$<br/>$\Rightarrow-(-k+1+4) \neq 0 \Rightarrow k-5 \neq 0 \Rightarrow k \neq 5$<br/>Thus, the system does not posses unique solution, if $\mathrm{k}=5$</div>


title: what-should-be-the-value-of-x-so-that-the-matrix-2-8-4-x-does-not-have-an-inverse
slug: what-should-be-the-value-of-x-so-that-the-matrix-2-8-4-x-does-not-have-an-inverse-89025
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-16$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What should be the value of $x$ so that the matrix $\left(\begin{array}{ll}2 &amp; 4 \\ -8 &amp; x\end{array}\right)$<br/>$\begin{array}{ll}\text { does not have an inverse? } &amp; \end{array}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">16</span> </li><li class="correct"> <span class="option-label">B</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><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">$-8$</span> </li> </ul>

answer:
<div class="solution">A matrix does not have an inverse if $|\mathrm{A}|=0$<br/>$\left|\begin{array}{cc}2 &amp; 4 \\ -8 &amp; x\end{array}\right|=0$<br/>$\Rightarrow x=-16$</div>


tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-term-describes-the-arrangement-of-ionsgroups-around-the-central-atomion-in-a-coordination-entity

slug:
what-term-describes-the-arrangement-of-ionsgroups-around-the-central-atomion-in-a-coordination-entity-82014

question:
<div class="question">What term describes the arrangement of ions/groups around the central atom/ion in a coordination entity?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Secondary valence</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Coordination number</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">Counter ions</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Lewis base</span> </li> </ul>

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

answer:
<div class="solution">The coordination number describes the arrangement of ions/groups around the central atom/ion in a coordination entity.</div>


title: what-term-is-used-for-ligands-with-one-donor-atom-per-molecule
slug: what-term-is-used-for-ligands-with-one-donor-atom-per-molecule-44413
question: <div class="question">What term is used for ligands with one donor atom per molecule?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Hexadentate</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Didendate</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Polydentate</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Unidentate</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">Unidentate</span> </div>

answer:
<div class="solution">Unidentate ligands have one donor atom per molecule and can bind through that atom.</div>


