# questions

title: what-is-lim-x-0-e-x-2-1-equal-to
slug: what-is-lim-x-0-e-x-2-1-equal-to-46371
question: <div class="question">What is $\lim _{x \rightarrow 0} e^{-\frac{1}{x^{2}}}$ equal to?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
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">Limit does not exist</span> </li> </ul>

answer:
<div class="solution">$\lim _{x \rightarrow 0} e^{-\frac{1}{x^{2}}}=e^{\frac{-1}{0}}=e^{-\infty}=0$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-lim-x-0-f-x-equal-to-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-lim-x-0-f-x-equal-to-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-30906

question:
<div class="question">What is $\lim _{x \rightarrow 0} f(x)$ equal to?<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=""> <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">$\frac{3}{\pi}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{6}{\pi}$</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">$\frac{6}{\pi}$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow 0} f(x)=\frac{\alpha \cos x}{\pi-2 x}$<br/>$=\frac{\alpha(\cos 0)}{\pi-2(0)}=\frac{6}{\pi}$<br/>$\therefore \quad$ Option $(\mathrm{d})$ is correct.</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-lim-x-0-f-x-equal-to-consider-the-function-f-x-x-x-a-x-x-1-where-denotes-the-greatest-integer-function-1

slug:
what-is-lim-x-0-f-x-equal-to-consider-the-function-f-x-x-x-a-x-x-1-where-denotes-the-greatest-integer-function-1-37817

question:
<div class="question">What is $\lim _{x \rightarrow 0^{-}} f(x)$ equal to?<br/>Consider the function $\mathrm{f}(\mathrm{x})=\frac{\mathrm{a}^{[\mathrm{x}]+\mathrm{x}}-1}{[\mathrm{x}]+\mathrm{x}}$ where $[\cdot]$ denotes the<br/>greatest integer function.</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">$\ln$ a</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1-\mathrm{a}^{-1}$</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">D</span> <span class="option-data">Limit does not exist</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} \lim _{x \rightarrow 0^{-}} f(x) &amp;=\lim _{h \rightarrow 0}\left[\frac{a^{[0-h]+(0-h)}-1}{[0-h]+(0-h)}\right] \\ &amp;=\lim _{h \rightarrow 0} \frac{a^{[-h]-h}-1}{[-h]+(-h)} \\ &amp;=\lim _{h \rightarrow 0} \frac{a^{-1-h}-1}{-1-h} \\ &amp;=\frac{a^{-1-0}-1}{-1-0}=\frac{a^{-1}-1}{-1} \\ &amp;=\lim _{h \rightarrow 0^{-}} f(x)=\left(1-a^{-1}\right) \end{aligned}$</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-lim-x-0-f-x-equal-to-consider-the-function-f-x-x-x-a-x-x-1-where-denotes-the-greatest-integer-function

slug:
what-is-lim-x-0-f-x-equal-to-consider-the-function-f-x-x-x-a-x-x-1-where-denotes-the-greatest-integer-function-29337

question:
<div class="question">What is $\lim _{x \rightarrow 0^{+}} f(x)$ equal to?<br/>Consider the function $\mathrm{f}(\mathrm{x})=\frac{\mathrm{a}^{[\mathrm{x}]+\mathrm{x}}-1}{[\mathrm{x}]+\mathrm{x}}$ where $[\cdot]$ denotes the<br/>greatest integer function.</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">$\ln$ a</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 1-\mathrm{a}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Limit does not exist</span> </li> </ul>

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

answer:
<div class="solution">Given $f(x)=\frac{a^{[x]+x}-1}{[x]+x}$<br/>$\therefore \lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} \frac{a^{[0+h]+(0+h)}-1}{[0+h]+(0+h)}$<br/>$=\lim _{h \rightarrow 0} \frac{a^{[h]+(h)}-1}{[h]+h}$<br/>$=\lim _{h \rightarrow 0} \frac{\left(a^{h}-1\right)}{h}$<br/>$=\log _{e} a$</div>


title: what-is-lim-x-0-s-i-n-2-x-t-a-n-x-equal-to
slug: what-is-lim-x-0-s-i-n-2-x-t-a-n-x-equal-to-21962
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{\tan x}{\sin 2 x}$ equal to?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Limit does not exist</span> </li> </ul>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{\tan x}{\sin 2 x}$<br/>$\lim _{x \rightarrow 0} \frac{\frac{\tan x}{x}}{2\left(\frac{\sin 2 x}{2 x}\right)}=\frac{1}{2} .$</div>


title: what-is-lim-x-0-x-1-1-x-equal-to
slug: what-is-lim-x-0-x-1-1-x-equal-to-34763
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{1-\sqrt{1+x}}{x}$ equal to?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1/2</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1/2</span> </li><li class="correct"> <span class="option-label">B</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">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">Consider $\lim _{x \rightarrow 0} \frac{1-\sqrt{1+x}}{x}$<br/>$=\lim _{x \rightarrow 0} \frac{1-(1+x)}{x(1+\sqrt{1+x})}($ ByRationalizing $)$<br/>$=\lim _{x \rightarrow 0} \frac{-x}{x(1+\sqrt{1+x})}=\frac{-1}{2}$</div>


title: what-is-lim-x-0-x-1-c-o-s-x-equal-to
slug: what-is-lim-x-0-x-1-c-o-s-x-equal-to-48866
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}$ equal to ?</div>
tags: ['Mathematics', 'Limits', '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_P3.db
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">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \lim _{x \rightarrow 0} \frac{1-\cos x}{x}=\lim _{x \rightarrow 0} \frac{2 \sin ^{2} \frac{x}{2}}{x} \\ &amp;=2\left[\lim _{x \rightarrow 0} \frac{\sin \frac{x}{2}}{\frac{x}{2} \times 2} \cdot \lim _{x \rightarrow 0} \sin \frac{x}{2}\right] \\ &amp;=2\left[\frac{1}{2} \times 0\right]=0 \end{aligned}$</div>


title: what-is-lim-x-0-x-1-x-1-equal-to
slug: what-is-lim-x-0-x-1-x-1-equal-to-68584
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
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">$\frac{1}{2}$</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">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">$\begin{aligned} &amp; \lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x} \\ &amp;=\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x} \times \frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}=\lim _{x \rightarrow 0} \frac{1+x-1}{x[\sqrt{1+x}+1]} \\ &amp;=\lim _{x \rightarrow 0} \frac{1}{\sqrt{1+x}+1}=\frac{1}{\sqrt{1+0}+1}=\frac{1}{2} \end{aligned}$</div>


title: what-is-lim-x-0-x-1-x-n-1-equal-to
slug: what-is-lim-x-0-x-1-x-n-1-equal-to-16485
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{(1+x)^{n}-1}{x}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">n</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">n</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">n-1</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \lim _{x \rightarrow 0} \frac{(1+x)^{n}-1}{x} \\ &amp;=\lim _{x \rightarrow 0} \frac{{ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots .+{ }^{n} C_{n} x^{n}-1}{x} \\ &amp;=\lim _{x \rightarrow 0} \frac{x\left({ }^{n} C_{1}+{ }^{n} C_{2} x+\ldots .+{ }^{n} C_{n} x^{n-1}\right)}{x} \\ &amp;=\lim _{x \rightarrow 0}{ }^{n} C_{1}+{ }^{n} C_{2} x+\ldots .{ }^{n} C_{n} x^{n-1} \\ &amp; \text { Put } x=0 \Rightarrow{ }^{n} C_{1}=n \end{aligned}$</div>


title: what-is-lim-x-0-x-2-2-1-c-o-s-x-equal-to
slug: what-is-lim-x-0-x-2-2-1-c-o-s-x-equal-to-32015
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow 0} \frac{2(1-\cos \mathrm{x})}{\mathrm{x}^{2}}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 2$</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</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">$\begin{aligned} \lim _{x \rightarrow 0} \frac{2(1-\cos x)}{x^{2}}=\lim _{x \rightarrow 0} \frac{2.2 \sin ^{2} \frac{x}{2}}{x^{2}} \\=&amp; 4 \lim _{x \rightarrow 0} \frac{\sin \frac{x}{2}}{\frac{x}{2} \times 2} \cdot \lim _{x \rightarrow 0} \frac{\sin \frac{x}{2}}{\frac{x}{2} \times 2} \\=&amp; \lim _{x \rightarrow 0} \frac{\sin \frac{x}{2}}{\frac{x}{2}} \cdot \lim _{x \rightarrow 0} \frac{\sin \frac{x}{2}}{\frac{x}{2}}=1 \times 1=1 \end{aligned}$</div>


title: what-is-lim-x-0-x-2-e-x-1-x-equal-to
slug: what-is-lim-x-0-x-2-e-x-1-x-equal-to-48607
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{\mathrm{x}}-(1+\mathrm{x})}{\mathrm{x}^{2}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$</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">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">$\lim _{x \rightarrow 0} \frac{e^{x}-(1+x)}{x^{2}}$<br/>If we keep $\mathrm{x}=0$, it is $\frac{0}{0}$.<br/>So, applying L'Hospital role.<br/>$\lim _{x \rightarrow 0} \frac{e^{x}-(1+x)}{x^{2}}=\frac{1}{2} \lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=\frac{1}{2} \times 1=\frac{1}{2}$</div>


title: what-is-lim-x-0-x-2-sin-x-1-equal-to
slug: what-is-lim-x-0-x-2-sin-x-1-equal-to-37856
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow 0} x^{2} \sin \left(\frac{1}{x}\right)$ equal to?</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">$1 / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Limit does not exist</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} \lim _{\mathrm{x} \rightarrow 0}\left(\mathrm{x}^{2}\right)\left[\sin \left(\frac{1}{\mathrm{x}}\right)\right] &amp;=0 \times \lim _{\mathrm{x} \rightarrow 0} \sin \left(\frac{1}{\mathrm{x}}\right) \\ &amp;=0 \times \text { finite quantity }=0 \end{aligned}$</div>


title: what-is-lim-x-0-x-5-x-1-equal-to
slug: what-is-lim-x-0-x-5-x-1-equal-to-37579
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{5^{x}-1}{x}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\log _{\mathrm{e}} 5$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$\log _{5} \mathrm{e}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">5</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">$\log _{\mathrm{e}} 5$</span> </div>

answer:
<div class="solution">$\because \lim _{x \rightarrow 0} \frac{\mathrm{a}^{\mathrm{x}}-1}{\mathrm{x}}=\operatorname{loe}_{\mathrm{e}} \mathrm{a}$<br/>$\therefore \lim _{x \rightarrow 0} \frac{5^{x}-1}{x}=\log _{e} 5$<br/>Hence, option (a) is correct</div>


title: what-is-lim-x-0-x-a-x-b-x-equal-to
slug: what-is-lim-x-0-x-a-x-b-x-equal-to-15656
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\operatorname{Lim}_{x \rightarrow 0} \frac{a^{x}-b^{x}}{x}$ equal to?</div>

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

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

answer:
<div class="solution">Consider<br/>$\lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{x}=\lim _{x \rightarrow 0} \frac{a^{x} \log a-b^{x} \log b}{1}$<br/>(Using L' Hospital's rule)<br/>$=\log \mathrm{a}-\log \mathrm{b}=\log \frac{\mathrm{a}}{\mathrm{b}}$</div>


title: what-is-lim-x-0-x-a-x-b-x
slug: what-is-lim-x-0-x-a-x-b-x-41391
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow 0} \frac{\mathrm{a}^{\mathrm{X}}-\mathrm{b}^{\mathrm{X}}}{\mathrm{x}}$ ?</div>

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

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

answer:
<div class="solution">Let $\mathrm{L}=\lim _{\mathrm{x} \rightarrow 0} \frac{\mathrm{a}^{\mathrm{x}}-\mathrm{b}^{\mathrm{x}}}{\mathrm{x}}$<br/>This is of $\frac{0}{0}$ form so, $L$ 'hospital's rule is applicable.<br/>$\Rightarrow \mathrm{L}=\lim _{\mathrm{x} \rightarrow 0} \frac{\mathrm{a}^{\mathrm{x}} \log \mathrm{a}-\mathrm{b}^{\mathrm{x}} \log \mathrm{b}}{1}$ (byL'Hospital's rule)<br/>$=\log a-\log b=\log \frac{a}{b}$</div>


title: what-is-lim-x-0-x-c-o-s-x-equal-to
slug: what-is-lim-x-0-x-c-o-s-x-equal-to-32125
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pi$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\pi}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}=\frac{\cos 0}{\pi-0}=\frac{1}{\pi}$</div>


title: what-is-lim-x-0-x-l-o-g-5-1-x-equal-to
slug: what-is-lim-x-0-x-l-o-g-5-1-x-equal-to-59610
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{\log _{5}(1+x)}{x}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.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">$\log _{5} \mathrm{e}$</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">$\log _{e} 5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{5} \mathrm{e}$</span> </div>

answer:
<div class="solution">Given equation $\lim _{x \rightarrow 0} \frac{\log _{5}(1+x)}{x}$<br/>$=\lim _{x \rightarrow 0} \frac{\log _{\mathrm{e}}(1+\mathrm{x})}{\mathrm{x} \log _{\mathrm{e}} 5} \quad\left[\because \log _{\mathrm{x}} \mathrm{y}=\frac{\log _{\mathrm{e}} \mathrm{y}}{\log _{\mathrm{e}} \mathrm{x}}\right]$<br/>$=\frac{1}{\log _{\mathrm{e}} 5} \lim _{\mathrm{x} \rightarrow 0} \frac{\log _{\mathrm{e}}(1+\mathrm{x})}{\mathrm{x}}=\log _{5} \mathrm{e}$<br/>$\left[\because \lim _{\mathrm{x} \rightarrow 0} \frac{\log _{\mathrm{e}}(1+\mathrm{x})}{\mathrm{x}}=1, \log _{\mathrm{x}} \mathrm{y}=\frac{1}{\log _{\mathrm{y}} \mathrm{x}}\right]$</div>


title: what-is-lim-x-2-x-2-4-x-2-equal-to
slug: what-is-lim-x-2-x-2-4-x-2-equal-to-50413
question: <div class="question">What is $\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}$ equal to?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
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">$\frac{1}{4}$</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">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Let $\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}=\lim _{x \rightarrow 2} \frac{x-2}{(x-2)(x+2)}$<br/>$=\lim _{x \rightarrow 2} \frac{1}{x+2}=\frac{1}{2+2}=\frac{1}{4}$</div>


title: what-is-lim-x-2-x-3-8-2-x-equal-to
slug: what-is-lim-x-2-x-3-8-2-x-equal-to-88139
question: <div class="question">What is $\lim _{x \rightarrow 2} \frac{2-x}{x^{3}-8}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{8}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{8}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{12}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow 2} \frac{2-x}{x^{3}-8}=\lim _{x \rightarrow 2} \frac{2-x}{(x-2)\left(x^{2}+2 x+4\right)}$<br/>$=\lim _{x \rightarrow 2} \frac{-1}{x^{2}+2 x+4}$<br/>Putting $\mathrm{x}=2$, we get<br/>$\frac{-1}{2^{2}+4+4}=-\frac{1}{12}$</div>


title: what-is-lim-x-2-x-3-8-x-2-equal-to
slug: what-is-lim-x-2-x-3-8-x-2-equal-to-66870
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{\mathrm{x} \rightarrow-2}\left(\frac{\mathrm{x}+2}{\mathrm{x}^{3}+8}\right)$ equal to?</div>

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

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

answer:
<div class="solution">$\lim _{x \rightarrow-2}\left(\frac{x+2}{x^{3}+8}\right)=\lim _{x \rightarrow-2} \frac{x+2}{(x)^{3}+(2)^{3}}$<br/>$=\lim _{x \rightarrow-2} \frac{x+2}{(x+2)\left(x^{2}-2 x+4\right)}$<br/>$=\frac{1}{(-2)^{2}-2(-2)+4}=\frac{1}{12}$</div>


title: what-is-lim-x-3-f-x-equal-to-consider-the-function-f-x-x-2-5-x-13-x-3-x-3-2014-ii
slug: what-is-lim-x-3-f-x-equal-to-consider-the-function-f-x-x-2-5-x-13-x-3-x-3-2014-ii-94425
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow 3} f(x)$ equal to ?<br/>Consider the function $f(x)=\left\{\begin{array}{ll}x^{2}-5 &amp; x \leq 3 \\ \sqrt{x+13} &amp; x&gt;3\end{array} \quad[2014-I I]\right.$</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">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">5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">13</span> </li> </ul>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\mathrm{x}^{2}-5 &amp; , \quad \mathrm{x} \leq 3 \\ \sqrt{\mathrm{x}+13} &amp; , \quad \mathrm{x}&gt;3\end{array}\right.$<br/>To find $\lim _{x \rightarrow 3} f(x)$<br/>$\mathrm{LHL}=\lim _{\mathrm{x} \rightarrow 3^{-}} \mathrm{f}(\mathrm{x})$<br/>$=\lim _{\mathrm{x} \rightarrow 3^{-}}\left(\mathrm{x}^{2}-5\right)=\lim _{\mathrm{x} \rightarrow(3-\mathrm{h})}\left[(3-\mathrm{h})^{2}-5\right]$<br/>$=\lim _{\mathrm{h} \rightarrow 0}\left(9-6 \mathrm{~h}+\mathrm{h}^{2}-5\right)=4$<br/>$\mathrm{RH}=\lim _{\mathrm{x} \rightarrow 3^{+}}(\sqrt{\mathrm{x}+13})$<br/>$=\lim _{\mathrm{x} \rightarrow(3+\mathrm{h})}(\sqrt{3+\mathrm{h}+13})=\lim _{\mathrm{h} \rightarrow 0}(\sqrt{16+\mathrm{h}})=4$<br/>$\because \lim _{\mathrm{x} \rightarrow 3^{-}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 3^{+}} \mathrm{f}(\mathrm{x})=4$<br/>$\therefore \lim _{\mathrm{x} \rightarrow 3} \mathrm{f}(\mathrm{x})=4$</div>


title: what-is-lim-x-3-x-x-3-x-equal-to
slug: what-is-lim-x-3-x-x-3-x-equal-to-83708
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow \infty}\left(\frac{x}{3+x}\right)^{3 x}$ equal to?</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">$e^{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$e^{-9}$</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">$e^{9}$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\lim _{x \rightarrow \infty}\left(\frac{x}{3+x}\right)^{3 x}=\lim _{x \rightarrow \infty}\left(\frac{1}{1+\frac{3}{x}}\right)^{3 x}$<br/>$=\lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right)^{-3 x}=\lim _{x \rightarrow \infty}\left[\left(1+\frac{3}{x}\right)^{x}\right]^{-3}$<br/>$=\left[e^{3}\right]^{-3}=e^{-9} \quad\left(\because \lim _{n \rightarrow \infty}\left(1+\frac{\lambda}{n}\right)^{n}=e^{\lambda}\right)$</div>


title: what-is-lim-x-6-2-s-i-n-2-x-3-s-i-n-x-1-2-s-i-n-2-x-s-i-n-x-1-to
slug: what-is-lim-x-6-2-s-i-n-2-x-3-s-i-n-x-1-2-s-i-n-2-x-s-i-n-x-1-to-89753
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-3$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \sin ^{2} x+\sin x-1}{2 \sin ^{2} x-3 \sin x+1}$ to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{3}$</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">$-3$</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">$\quad \lim _{x \rightarrow \frac{\pi}{6}} \frac{(\sin x+1)(2 \sin x-1)}{(\sin <br/>x-1)(2 \sin x-1)}=\frac{\frac{1}{2}+1}{\frac{1}{2}-1}=-3$</div>


title: what-is-lim-x-a-2-x-2-a-x-1-a-2-x-2-1-equal-to
slug: what-is-lim-x-a-2-x-2-a-x-1-a-2-x-2-1-equal-to-85614
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow \infty}\left(\sqrt{a^{2} x^{2}+a x+1}-\sqrt{a^{2} x^{2}+1}\right)$ equal to?</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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">Consider, $\lim _{x \rightarrow \infty}\left[\sqrt{a^{2} x^{2}+a x+1}-\sqrt{a^{2} x^{2}+1}\right]$<br/>$=\lim _{x \rightarrow \infty} \frac{\left(\sqrt{a^{2} x^{2}+a x+1}-\sqrt{a^{2} x^{2}+1}\right)\left(\sqrt{a^{2} x^{2}+a x+1}+\sqrt{a^{2} x^{2}+1}\right)}{\left(\sqrt{a^{2} x^{2}+a x+1}+\sqrt{a^{2} x^{2}+1}\right)}$<br/>$=\lim _{x \rightarrow \infty}\left[\frac{a^{2} x^{2}+a x+1-a^{2} x^{2}-1}{\sqrt{a^{2} x^{2}+a x+1}+\sqrt{a^{2} x^{2}+1}}\right]$<br/>$=\lim _{x \rightarrow \infty} \frac{a}{\sqrt{a^{2}+\frac{a}{x}+\frac{1}{x^{2}}}+\sqrt{a^{2}+\frac{1}{x^{2}}}}$<br/>$=\frac{a}{\sqrt{a^{2}}+\sqrt{a^{2}}}=\frac{a}{2 a}=\frac{1}{2}$</div>


title: what-is-lim-x-o-b-x-s-i-n-2-a-x-a-b-are-constants-equal-to
slug: what-is-lim-x-o-b-x-s-i-n-2-a-x-a-b-are-constants-equal-to-97527
tags: ['Mathematics', 'Limits', '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_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow o} \frac{\sin ^{2} a x}{b x}(\mathrm{a}, \mathrm{b}$ are constants) equal to?</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</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{a} / \mathrm{b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Does not exist</span> </li> </ul>

answer:
<div class="solution">Consider $\lim _{x \rightarrow 0} \frac{\sin ^{2} a x}{b x}$<br/>Multiply and divide by $a^{2} x^{2}$<br/>$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin ^{2} a x}{b x}=\lim _{x \rightarrow 0}\left[\frac{\sin ^{2} a x}{b x} \times \frac{a^{2} x^{2}}{a^{2} x^{2}}\right]$<br/>$=\lim _{x \rightarrow 0}\left[\frac{\sin ^{2} a x}{a^{2} x^{2}} \times \frac{a^{2} x^{2}}{b x}\right]$<br/>$=\lim _{x \rightarrow 0}\left[\left(\frac{\sin a x}{a x}\right)^{2} \times \frac{a^{2} x}{b}\right]$<br/>$=\lim _{x \rightarrow 0}\left(\frac{\sin a x}{a x}\right)^{2} \cdot \lim _{x \rightarrow 0} \frac{a^{2} x}{b}$<br/>$=1 \times 0=0 \quad\left(\because \lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right)$</div>


title: what-is-ln-x-2-d-x-equal-to
slug: what-is-ln-x-2-d-x-equal-to-34978
question: <div class="question">What is $\int \ln \left(x^{2}\right) d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 x \ln (x)-2 x+c$</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">$\frac{2}{x}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x \ln (x)+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2 \ln (x)}{x}-2 x+c$</span> </li> </ul>

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

answer:
<div class="solution">$\int \ln \left(x^{2}\right) \cdot d x=2 \cdot \int \ln x \cdot d x$<br/>$=2 \int 1 \cdot \ln x d x=2\left[\ln x_{-} x-\int \frac{1}{x} x \cdot d x\right.$<br/>$=2(x, \ln x-x)+c=2 x \ln x-2 x+c$</div>


title: what-is-lo-g-81-243-equal-to
slug: what-is-lo-g-81-243-equal-to-72548
question: <div class="question">What is $\log _{81} 243$ equal to?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1.25$</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.75$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.25$</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">$1.5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">$\log _{81} 243=\log _{3} 53^{6}=\frac{\log 3^{6}}{\log 3^{5}}=\frac{6}{5}=1.25$</div>


title: what-is-lo-g-a-a-2-1-lo-g-a-a-2-1-1-equal-to
slug: what-is-lo-g-a-a-2-1-lo-g-a-a-2-1-1-equal-to-88380
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_P3.db
question:
<div class="question">What is<br/>$\log \left(a+\sqrt{a^{2}+1}\right)+\log \left(\frac{1}{a+\sqrt{a^{2}+1}}\right)$ equal to?</div>

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

answer:
<div class="solution">Let $\log \left(a+\sqrt{a^{2}+1}\right)+\log \left(\frac{1}{a+\sqrt{a^{2}+1}}\right)$<br/>$=\log \left(a+\sqrt{a^{2}+1}\right)+\log 1-\log \left(a+\sqrt{a^{2}+1}\right)$<br/>$=\log \left(a+\sqrt{a^{2}+1}\right)-\log \left(a+\sqrt{a^{2}+1}\right)$<br/>$=0$</div>


title: what-is-lo-g-x-1-d-x-is-equal-to
slug: what-is-lo-g-x-1-d-x-is-equal-to-73951
question: <div class="question">What is $\int \log (x+1) d x$ is equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x \log (x+1)-x+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(x+1) \log (x+1)-x+c$</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">C</span> <span class="option-data">$\frac{1}{x+1}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\log (x+1)}{x+1}+c$</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{I}=\int \log (\mathrm{x}+1) \mathrm{d} \mathrm{x}$<br/>Let $x+1=t$ $\Rightarrow \mathrm{dx}=\mathrm{dt}$<br/>$\Rightarrow \quad \mathrm{I}=\int 1 . \log \mathrm{tdt}$<br/>Integrating by parts, taking log tas first function $\Rightarrow \mathrm{I}=\mathrm{t} \log \mathrm{t}-\int \frac{1}{\mathrm{t}} \cdot \mathrm{tdt}+\mathrm{c}_{1}$<br/>$=\mathrm{t} \log \mathrm{t}-\int 1 \mathrm{dt}+\mathrm{c}_{1}=\mathrm{t} \log \mathrm{t}-\mathrm{t}+\mathrm{c} 1$<br/>$=(x+1) \log (x+1)-x-1+c_{1}$<br/>$=(x+1) \log (x+1)-x+c \quad\left[\because c=c_{1}-1\right]$</div>


title: what-is-maximum-value-of-sin-2-x
slug: what-is-maximum-value-of-sin-2-x-93924
question: <div class="question">What is maximum value of $\sin ^{2} \mathrm{x} ?$</div>
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_P3.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">0</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">Infinity</span> </li> </ul>

answer:
<div class="solution">$-1   &lt; \sin x &lt; 1 \Rightarrow 1 &lt; \sin ^{2} x &lt; 1$<br/>Hence, maximum value of $\sin ^{2} x=1$</div>


title: what-is-meant-by-tertiary-structure-of-proteins
slug: what-is-meant-by-tertiary-structure-of-proteins-61448
question: <div class="question">What is meant by tertiary structure of proteins?</div>
tags: ['Biology', 'Biomolecules']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">Tertiary structure of protein : When the individual peptide chains of secondary structure of protein are further extensively coiled and folded into sphere like shapes with the hydrogen bonds between the amino and carboxyl group and various other kinds of bonds cross linking on chain to another they form tertiary structure.<br/>The ability of proteins to carry out specific reactions is the result of their primary, secondary and tertiary structure.<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/q-fpiunBwZt9AtOVu-FP5mT1yCQWaOjUiRR5eeUtxkk.original.fullsize.png"/><br/></div>


title: what-is-meant-by-the-term-tyndall-effect
slug: what-is-meant-by-the-term-tyndall-effect-29558
question: <div class="question">What is meant by the term "Tyndall effect"?</div>
tags: ['Chemistry', 'Surface Chemistry', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The scattering of light by colloidal particles, making the beam visible.</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">The irreversible aggregation of colloidal particles leading to the formation of a precipitate.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The process of removing small dissolved molecules from a colloid by diffusion through a semipermeable membrane.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The migration of charged colloidal particles under the influence of an electric field.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The scattering of light by colloidal particles, making the beam visible.</span> </div>

answer:
<div class="solution">Correct Option is : (A)<br/>The scattering of light by colloidal particles, making the beam visible.</div>


title: what-is-modulus-of-1-3-i-1-1-3-i-1
slug: what-is-modulus-of-1-3-i-1-1-3-i-1-70420
question: <div class="question">What is modulus of $\frac{1}{1+3 i}-\frac{1}{1-3 i} ?$</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{25}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{25}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5}{3}$</span> </li> </ul>

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

answer:
<div class="solution">Let $z=\frac{1}{1+3 i}-\frac{1}{1-3 i}$<br/>$=\frac{(1-3 i)-(1+3 i)}{(1+3 i)(1-3 i)}=\frac{-6 i}{(1)^{2}-(3 i)^{2}}=-\frac{6 i}{10}=-\frac{3}{5} i$<br/>$\therefore \quad|z|=\sqrt{(0)^{2}+\left(-\frac{3}{5}\right)^{2}}=\sqrt{\frac{9}{25}}=\frac{3}{5}$</div>


title: what-is-n-1-n-1-n-2-n-1-n-1-equal-to
slug: what-is-n-1-n-1-n-2-n-1-n-1-equal-to-57365
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{(\mathrm{n}+2) !+(\mathrm{n}+1)(\mathrm{n}-1) !}{(\mathrm{n}+1)(\mathrm{n}-1) !}$ equal to ?</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">Always an odd integer</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">A perfect square</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

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

answer:
<div class="solution">Given expression is:<br/>$\frac{(\mathrm{n}+2) !+(\mathrm{n}+1) !(\mathrm{n}-1) !}{(\mathrm{n}+1) !(\mathrm{n}-1) !}=\mathrm{x}$ (let)<br/>$\Rightarrow \quad x=\frac{(n+2)(n+1) n(n-1) !+(n+1)(n-1) !}{(n+1)(n-1) !}$<br/>$=(n+2) n+1=n^{2}+2 n+1=(n+1)^{2}$<br/>Which is a perfect square.</div>


title: what-is-o-0-1-1-x-2-t-a-n-1-dx-equal-to
slug: what-is-o-0-1-1-x-2-t-a-n-1-dx-equal-to-49561
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\mathrm{O}_{0}^{1} \frac{\tan ^{-1}}{1+\mathrm{x}^{2}} \mathrm{dx}$ equal to?</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">$\frac{\pi}{8}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi^{2}}{8}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi^{2}}{32}$</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">$\frac{\pi^{2}}{32}$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\dot{\mathrm{O}}_{0}^{1} \frac{\tan ^{-1}}{1+\mathrm{x}^{2}} \mathrm{dx}$<br/>$\operatorname{Put} \tan ^{-1} \mathrm{x}=\mathrm{t}$<br/>$\frac{1}{1+\mathrm{x}^{2}} \mathrm{dx}=\mathrm{dt}$<br/>$\mathrm{x}=0, \Rightarrow \mathrm{t}=0$<br/>$\mathrm{x}=1, \Rightarrow \mathrm{t}=\pi / 4$<br/>$\therefore \mathrm{I}=\int_{0}^{\pi / 4} \mathrm{tdt}=\left.\frac{\mathrm{t}^{2}}{2}\right|_{0} ^{\pi / 4}=\frac{\pi^{2}}{32}$</div>


title: what-is-one-of-the-square-roots-of-3-4-i-where-i-1
slug: what-is-one-of-the-square-roots-of-3-4-i-where-i-1-11230
question: <div class="question">What is one of the square roots of $3+4 i$, where $i=\sqrt{-1}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2+i$</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2+i$</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">B</span> <span class="option-data">$2-i$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-2+i$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3-i$</span> </li> </ul>

answer:
<div class="solution">$3+4 \mathrm{i}=2^{2}+\mathrm{i}^{2}+2 \times 2 \times \mathrm{i}=(2+\mathrm{i})^{2}$<br/>$\sqrt{3+4 i}=2+i$</div>


title: what-is-one-of-the-values-of-i-i
slug: what-is-one-of-the-values-of-i-i-59478
question: <div class="question">What is one of the values of $\sqrt{\mathrm{i}}+\sqrt{-\mathrm{i}}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{2}$</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">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pm \frac{1+\mathrm{i}}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pm \frac{1-\mathrm{i}}{\sqrt{2}}$</span> </li> </ul>

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

answer:
<div class="solution">$\sqrt{i}=\sqrt{\frac{2 i}{2}}=\sqrt{\frac{1+2 i-1}{2}}$<br/>$=\sqrt{\frac{1+2 i+i^{2}}{2}}=\sqrt{\frac{(1+i)^{2}}{2}}=\frac{1+i}{\sqrt{2}}$<br/>$\sqrt{-i}=\sqrt{-\frac{2 i}{2}}$<br/>$=\sqrt{\frac{1-2 i-1}{2}}=\sqrt{\frac{1-2 i+i^{2}}{2}}=\sqrt{\frac{(1-i)^{2}}{2}}=\frac{(1-i)}{\sqrt{2}}$<br/>So, value of $\sqrt{i}+\sqrt{-i}$<br/>$=\frac{(1+i)+(1-i)}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$</div>


title: what-is-oogenesis-give-a-brief-account-of-oogenesis
slug: what-is-oogenesis-give-a-brief-account-of-oogenesis-90966
question: <div class="question">What is oogenesis? Give a brief account of oogenesis.</div>
tags: ['Biology', 'Human Reproduction']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">The process of formation of a mature female gamete (ovum) is called oogenesis. It occurs in the ovaries of female reproductive system. Oogenesis is a discontinuous process it begins before birth, stops in midprocess \&amp; only resumes after menarch. It occurs in three phases:<br/>Multiplicative phase (formation of oogonia mitotically from the primary germ cells), Growth phase (growth of oogonia into primary oocyte) \&amp; Maturation phase (formation of mature ova from primary oocyte through meiosis). Maturation phase produces two haploid cells Larger one called secondary oocyte \&amp; the smaller one called polar bodies (1st polar body). Meiosis II of secondary oocyte results in the formation of functional egg or ovum<br/>and a second polar body. The first polar body may also divide to form two polar bodies of equal sizes which do not take part in reproduction \&amp; ultimately degenerates. First maturation division may be completed in the ovaries just prior to ovulation but second one (Final) is completed outside the ovary after fertilization. Secondary oocyte is female gamete in which the 1st meiotic division is completed \&amp; second meiotic division (Metaphase stage) has begin. The egg is released at secondary oocyte stage under the effect of LH.<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/b-82PecGwdMfyV0sqJnqC9BsrulpIWCvTzK4rIxJhBM.original.fullsize.png"/><br/></div>


title: what-is-periderm-how-does-periderm-formation-take-place-in-the-dicot-stems
slug: what-is-periderm-how-does-periderm-formation-take-place-in-the-dicot-stems-39459
tags: ['Biology', 'Anatomy of Flowering Plants']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is periderm? How does periderm formation take place in the dicot stems?</div>

answer:
<div class="solution">As the stem continues to increase in girth due to activity of vascular cambium, the outer cortical and epidermis layers get broken and need to be replaced to provide new protective cell layers. Hence, sooner or later, another meristematic tissue called cork cambium or phellogen develops, usually in the cortex region. Phellogen is made of thinwalled rectangular cells. Its cuts off cells on both side. The outer cell differentiate into cork or phellem while inner cells differentiate into secondary cortex or phelloderm. Phelloderm, phellem and phelloderm are collectively called as periderm.</div>


title: what-is-primary-productivity-give-brief-description-of-factors-that-affect-primary-productivity
tags: ['Biology', 'Ecosystem']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
slug:
what-is-primary-productivity-give-brief-description-of-factors-that-affect-primary-productivity-30813

question:
<div class="question">What is primary productivity? Give brief description of factors that affect primary productivity.</div>

answer:
<div class="solution">Primary productivity of an ecosystem is the amount of energy fixed or biomass synthesized by primary producers or green plants per unit area per unit time during photosynthesis.<br/>Factors affecting primary productivity are -<br/>- Plant species inhabiting a particular area<br/>- Sunlight<br/>- Temperature<br/>- Soil water<br/>- Nutrients<br/>In deserts, sunlight is abundant but water is scarce or nutrients are lacking. Therefore, in such areas, water \&amp; nutrients supply become the limiting factors.</div>


title: what-is-r-0-1-n-r-c-n-equal-to
slug: what-is-r-0-1-n-r-c-n-equal-to-14000
question: <div class="question">What is $\sum_{r=0}^{1}{ }^{n+r} C_{n}$ equal to?</div>
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$n+3 C_{n+1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$n+3 C_{n}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$n+2 C_{n}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$n+2 C_{n+1}$</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> </ul>

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

answer:
<div class="solution">$\sum_{\mathrm{r}=0}^{1} \sum^{\mathrm{n}+\mathrm{r}} \mathrm{C}_{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{n}}$<br/>$=1+\frac{(\mathrm{n}+1) !}{(\mathrm{n}+1-\mathrm{n}) ! \mathrm{n} !}=1+\frac{(\mathrm{n}+1)(\mathrm{n} !)}{\mathrm{n} !}$<br/>$=1+\mathrm{n}+1=\mathrm{n}+2$<br/>${ }^{n+2} C_{n+1}=\frac{(n+2) !}{(n+2-n-1) !(n+1) !}$<br/>$=\frac{(\mathrm{n}+2)(\mathrm{n}+1) !}{(\mathrm{n}+1) !}=\mathrm{n}+2$<br/>$\mathrm{OR}$<br/>$\sum_{\mathrm{r}=0}^{1}{ }^{\mathrm{n}+\mathrm{r}} \mathrm{C}_{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{n}}$<br/>$=1+(\mathrm{n}+1)=\mathrm{n}+2$<br/>Now,<br/>${ }^{n+2} C_{1}=\frac{(n+2) !}{1 !(n+2-1) !}=\frac{(n+2)(n+1) !}{(n+1) !}=(n+2)$<br/>$\therefore$ Option (d) is correct.</div>


title: what-is-r-0-n-c-n-r-equal-to
slug: what-is-r-0-n-c-n-r-equal-to-88210
question: <div class="question">What is $\sum_{r=0}^{n} C(n, r)$ equal to ?</div>
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{n}-1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$n$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$n l$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2^{n}$</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, $(1+\mathrm{x})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{0}+{ }^{\mathrm{n}} \mathrm{C}_{1}^{\mathrm{x}}+{ }^{\mathrm{n}} \mathrm{C}_{2}^{\mathrm{x}^{2}}+\ldots .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}}$<br/>For $\mathrm{x}=1,(1+1)^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{0}+{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}+\ldots .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}$<br/>$\therefore \sum_{\mathrm{r}=0}^{\mathrm{n}} \mathrm{c}(\mathrm{n}, \mathrm{r})=2^{\mathrm{n}}$</div>


title: what-is-red-data-book
slug: what-is-red-data-book-75238
question: <div class="question">What is Red Data Book?</div>
tags: ['Biology', 'Biodiversity and its Conservation']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">Red Data Book is a compilation of data or records of species on the verge of extinction or species that are endangered. That as he book is maintained by IUCN.</div>


title: what-is-referred-to-as-satellite-chromosome
slug: what-is-referred-to-as-satellite-chromosome-11785
question: <div class="question">What is referred to as satellite chromosome?</div>
tags: ['Biology', 'Cell - The Unit of Life']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">Additional constriction or secondary constriction at the chromosomal ends as distal part of the arm formed by chromatin thread are known satellite chromosomees. These constriction gives appearance of an outgrowth or a small fragment.<br/>These are also known as (sat) chromosomes or marker chromosomes. Chromosomes $13,14,15,16,21$ and 21 satellite chromosomes.<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/-Lso1YNYUdtxBt5JMmqzZv9ypAbKzGGz8RamZ-x24T0.original.fullsize.png"/><br/></div>


title: what-is-s-e-c-14-4-s-e-c-1-8-cosec-14-4-cosec-1-8-equal-to
slug: what-is-s-e-c-14-4-s-e-c-1-8-cosec-14-4-cosec-1-8-equal-to-35023
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\left(\frac{\sec 18^{\circ}}{\sec 144^{\circ}}+\frac{\operatorname{cosec} 18^{\circ}}{\operatorname{cosec} 144^{\circ}}\right)$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sec 18^{\circ}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \text { Consider }\left(\frac{\sec 18^{\circ}}{\sec 144^{\circ}}+\frac{\cos e c 18^{\circ}}{\cos e c 144^{\circ}}\right) \\=&amp; \frac{\sec 18^{\circ}}{\sec \left(180^{\circ}-36^{\circ}\right)}+\frac{\cos e c 18^{\circ}}{\cos e c\left(180^{\circ}-36^{\circ}\right)} \\=&amp;-\frac{\sec 18^{\circ}}{\sec 36^{\circ}}+\frac{\cos e c 18^{\circ}}{\cos e c 36^{\circ}}(\because \sin , \operatorname{cosec} \operatorname{are}+\mathrm{ve}\\ &amp; \left.\text { in } 2^{\text {nd }} \text { quadrant and sec is }-\text { ve in } 2^{\text {nd }} \text { quadrant }\right) \\=&amp; \frac{\sin 36^{\circ}}{\sin 18^{\circ}}-\frac{\cos 36^{\circ}}{\cos 18^{\circ}}=\frac{\sin 36^{\circ} \cos 18^{\circ}-\cos 36^{\circ} \sin 18^{\circ}}{\sin 18^{\circ} \cos 18^{\circ}} \\=&amp; \frac{\sin \left(36^{\circ}-18^{\circ}\right)}{\sin 18^{\circ} \cos 18^{\circ}}=\frac{\sin 18^{\circ}}{\sin 18^{\circ} \cos 18^{\circ}}=\sec 18^{\circ} \\ &amp;\left(\because \frac{1}{\cos x}=\sec x\right) \end{aligned}$</div>


title: what-is-s-i-n-1-0-1-c-o-s-1-0-3-equal-to
slug: what-is-s-i-n-1-0-1-c-o-s-1-0-3-equal-to-68513
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</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">$\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}=\frac{\cos 10^{\circ}-\sqrt{3} \sin 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}$<br/>$=\frac{2 \times 2\left[\frac{1}{2} \cos 10^{\circ}-\frac{\sqrt{3}}{2} \sin 10^{\circ}\right]}{2 \sin 10^{\circ} \cos 10^{\circ}}$<br/>$=\frac{4\left(\cos 60^{\circ} \cos 10^{\circ}-\sin 60^{\circ} \sin 10^{\circ}\right)}{\sin 20^{\circ}}$<br/>$=\frac{4 \cdot \cos \left(60^{\circ}+10^{\circ}\right)}{\sin 20^{\circ}}=4 \cdot \frac{\cos 70^{\circ}}{\sin 20^{\circ}}=4 \cdot \frac{\sin 20^{\circ}}{\sin 20^{\circ}}=4$</div>


title: what-is-s-i-n-2-2-c-o-s-2-1-d-equal-to
slug: what-is-s-i-n-2-2-c-o-s-2-1-d-equal-to-21155
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int \frac{d \theta}{\sin ^{2} \theta+2 \cos ^{2} \theta-1}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\tan \theta+c$</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">$\cot \theta+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2} \tan \theta+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2} \cot \theta+c$</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int \frac{d \theta}{\sin ^{2} \theta+2 \cos ^{2} \theta-1}$<br/>$=\int \frac{d \theta}{1-\cos ^{2} \theta+2 \cos ^{2} \theta-1}=\int \frac{d \theta}{\cos ^{2} \theta}$<br/>$=\int \sec ^{2} \theta d \theta=\tan \theta+c$</div>


title: what-is-s-i-n-2-x-c-o-s-2-x-d-x-equal-to
slug: what-is-s-i-n-2-x-c-o-s-2-x-d-x-equal-to-60602
question: <div class="question">What is $\int \frac{d x}{\sin ^{2} x \cos ^{2} x}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan x+\cot x+\mathrm{c}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\tan x-\cot x+\mathrm{c}$</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">C</span> <span class="option-data">$(\tan x+\cot x)^{2}+\mathrm{c}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(\tan x-\cot x)^{2}+c$<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">$\tan x-\cot x+\mathrm{c}$</span> </div>

answer:
<div class="solution">Let $I=\int \frac{d x}{\sin ^{2} x \cos ^{2} x}=\int \frac{\left(\sin ^{2} x+\cos ^{2} x\right) d x}{\sin ^{2} x \cos ^{2} x}$<br/>$=\int\left[\frac{\sin ^{2} x}{\sin ^{2} x \cos ^{2} x}+\frac{\cos ^{2} x}{\sin ^{2} x \cos ^{2} x}\right] d x$<br/>$=\int\left[\frac{1}{\cos ^{2} x}+\frac{1}{\sin ^{2} x}\right] d x$<br/>$=\int\left(\sec ^{2} x+\operatorname{cosec}^{2} x\right) d x$<br/>$=\tan x-\cot x+c$ where $c$ is constant of Integration.</div>


title: what-is-s-i-n-6-i-1-c-o-s-6-s-i-n-6-i-1-c-o-s-6-3-where-i-1-equal-to
slug: what-is-s-i-n-6-i-1-c-o-s-6-s-i-n-6-i-1-c-o-s-6-3-where-i-1-equal-to-81808
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">i</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\left[\frac{\sin \frac{\pi}{6}+i\left(1-\cos \frac{\pi}{6}\right)}{\sin \frac{\pi}{6}-i\left(1-\cos \frac{\pi}{6}\right)}\right]^{3}$<br/>where $i=\sqrt{-1}$, equal to?</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">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">-i</span> </li> </ul>

answer:
<div class="solution">$\left[\frac{\sin \frac{\pi}{6}+i\left(1-\cos \frac{\pi}{6}\right)}{\sin \frac{\pi}{6}-i\left(1-\cos \frac{\pi}{6}\right)}\right]^{3}$<br/>$=\left[\frac{2 \sin \frac{\pi}{12} \cos \frac{\pi}{12}+i\left(2 \sin ^{2} \frac{\pi}{12}\right)}{2 \sin \frac{\pi}{12} \cos \frac{\pi}{12}-i\left(2 \sin ^{2} \frac{\pi}{12}\right)}\right]^{3}$<br/>$=\left[\frac{\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}}{\cos \frac{\pi}{12}-i \sin \frac{\pi}{12}}\right]^{3}=\left(\frac{e^{i \frac{\pi}{12}}}{e^{-i \frac{\pi}{12}}}\right)^{3}$<br/>$=\left(e^{i \frac{\pi}{6}}\right)^{3}=e^{i \times 3 \times \frac{\pi}{6}}=e^{i \frac{\pi}{2}}=\cos \frac{\pi}{2}+\mathrm{i} \sin \frac{\pi}{2}=\mathrm{i}$<br/>$\therefore$ Option (c) is correct.</div>


title: what-is-s-i-n-7-x-2-s-i-n-5-x-s-i-n-3-x-c-o-s-7-x-c-o-s-3-x-equal-to
slug: what-is-s-i-n-7-x-2-s-i-n-5-x-s-i-n-3-x-c-o-s-7-x-c-o-s-3-x-equal-to-63023
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{\cos 7 x-\cos 3 x}{\sin 7 x-2 \sin 5 x+\sin 3 x}$ equal to ?</div>

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

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

answer:
<div class="solution">$\frac{\cos 7 x-\cos 3 x}{\sin 7 x-2 \sin 5 x+\sin 3 x}$<br/>$=\frac{-2 \sin \frac{7 x+3 x}{2} \cdot \sin \frac{7 x-3 x}{2}}{2 \sin \frac{7 x+3 x}{2} \cos \frac{7 x-3 x}{2}-2 \sin 5 x}$<br/>$\quad\left(\because \sin C+\sin D=2 \sin \left(\frac{C+D}{2}\right) \cdot \cos \left(\frac{C-D}{2}\right)\right.$<br/>and $\left.\cos C-\cos D=-2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)\right)$<br/>$=\frac{-2 \sin 5 x \cdot \sin 2 x}{2 \sin 5 x \cos 2 x-2 \sin 5 x}$<br/>$=\frac{-2 \sin 5 x \cdot \sin 2 x}{-2 \sin 5 x[1-\cos 2 x]}$<br/>$=\frac{\sin 2 x}{1-1+2 \sin ^{2} x} \quad\left(\because \cos 2 x=1-2 \sin ^{2} x\right)$<br/>$=\frac{2 \sin x \cos x}{2 \sin ^{2} x}=\cot x$</div>


title: what-is-sec-n-x-tan-x-d-x-equal-to
slug: what-is-sec-n-x-tan-x-d-x-equal-to-17975
question: <div class="question">What is $\int \sec ^{n} x \tan x d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sec ^{n} x}{n}+c$</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">$\frac{\sec ^{n-1} x}{n-1}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\tan ^{n} x}{n}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\tan ^{n-1} x}{n-1}+c$<br/>Where $^{\prime} \mathrm{c}$ ' is a constant of integration.</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{I}=\int \sec ^{n} x \tan x d x$<br/>Put, sec $x=t \Rightarrow \sec x \tan x d x=d t \Rightarrow d x=\frac{d t}{t}$<br/>$\therefore \quad I=\int t^{n} \cdot \frac{d t}{t}=\int t^{n-1} d t=\frac{t^{n}}{n}+c=\frac{\sec ^{n} x}{n}+c$<br/>where ' $\mathrm{c}$ ' is a constant of integration.</div>


title: what-is-secondary-growth-in-plants
slug: what-is-secondary-growth-in-plants-79803
question: <div class="question">What is secondary growth in plants?</div>
tags: ['Biology', 'Plant - Growth and Development', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Increase in cell number</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Increase in length</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Increase in size of the cell</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Increase in girth of the organs</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">Increase in girth of the organs</span> </div>

answer:
<div class="solution">Secondary growth in plants is caused by lateral meristems, vascular cambium, and cork-cambium. It results in an increase in girth of the organs in which they are active.</div>


title: what-is-secondary-metabolism
slug: what-is-secondary-metabolism-72146
question: <div class="question">What is secondary metabolism?</div>
tags: ['Biology', 'Human Health and Diseases']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">Secondary metabolism is also known as specialised metabolism. It is a term for pathways and small molecule metabolites produced by metabolism that are not absolutely essential for the organism's survical. Metabolites aid in the growth and development in case of plants. They also facilitie the primary metabolism in plants.</div>


title: what-is-secondary-productivity-in-an-ecosystem
slug: what-is-secondary-productivity-in-an-ecosystem-42263
question: <div class="question">What is secondary productivity in an ecosystem?</div>
tags: ['Biology', 'Ecosystem', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The rate of production of organic matter during photosynthesis</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The rate of formation of new organic matter by 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">C</span> <span class="option-data">The rate of biomass production</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither A) nor B)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The rate of formation of new organic matter by consumers</span> </div>

answer:
<div class="solution">Secondary productivity is defined as the rate of formation of new organic matter by consumers.</div>


title: what-is-sin-1-5-3-sin-1-5-4-equal-to
slug: what-is-sin-1-5-3-sin-1-5-4-equal-to-39710
question: <div class="question">What is $\sin ^{-1} \frac{3}{5}-\sin ^{-1} \frac{4}{5}$ equal to ?</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\pi / 2$</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">B</span> <span class="option-data">$\pi / 3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi / 6$</span> </li> </ul>

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

answer:
<div class="solution">$\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{4}{5}=\sin ^{-1} \frac{3}{5}+\cos ^{-1} \frac{3}{5}$<br/>$\left(\right.$ let $\left.\sin ^{-1} \frac{4}{5}=\theta \Rightarrow \sin \theta=\frac{4}{5} \Rightarrow \cos \theta=\frac{3}{5}\right)$<br/>$=\frac{\pi}{2}$</div>


title: what-is-sin-1-cos-x-dx-equal-to
slug: what-is-sin-1-cos-x-dx-equal-to-78160
question: <div class="question">What is $\int \sin ^{-1}(\cos x) \mathrm{dx}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x \pi}{2}-\frac{x^{2}}{2}+K$</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">B</span> <span class="option-data">$\frac{\pi}{2}+\frac{x^{2}}{2}+K$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{x \pi}{2}-\frac{x^{2}}{2}+K$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2}-\frac{x^{2}}{2}+K$<br/>Where $\mathrm{K}$ is a constant of integration.</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int \sin ^{-1}(\cos x) d x$<br/>$=\int \sin ^{-1}\left[\sin \left(\frac{\pi}{2}-x\right)\right] d x=\int\left(\frac{\pi}{2}-x\right) d x$<br/>$=\frac{x \pi}{2}-\frac{x^{2}}{2}+K$<br/>where $K$ is a constant of integration.</div>


title: what-is-sin-1-sin-5-3-equal-to
slug: what-is-sin-1-sin-5-3-equal-to-40052
question: <div class="question">What is $\sin ^{-1} \sin \frac{3 \pi}{5}$ equal to ?</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3 \pi}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{2 \pi}{5}$</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">$\frac{\pi}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

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

answer:
<div class="solution">$\sin ^{-1} \sin \frac{3 \pi}{5}=\sin ^{-1} \sin \left(\pi-\frac{2 \pi}{5}\right)$<br/>$=\sin ^{-1} \sin \frac{2 \pi}{5}=\frac{2 \pi}{5}$</div>


title: what-is-sin-2-2-0-sin-2-7-0-equal-to
slug: what-is-sin-2-2-0-sin-2-7-0-equal-to-28931
question: <div class="question">What is $\sin ^{2} 20^{\circ}+\sin ^{2} 70^{\circ}$ equal to ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', '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_P3.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">$-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

answer:
<div class="solution">$\sin ^{2} 20^{\circ}+\sin ^{2} 70^{\circ}=\sin ^{2} 20^{\circ}+\sin ^{2}\left(90^{\circ}-20^{\circ}\right)$<br/>$=\sin ^{2} 20^{\circ}+\cos ^{2} 20^{\circ}=1$</div>


title: what-is-sin-2-3-cos-2-4-tan-2-5-equal-to
slug: what-is-sin-2-3-cos-2-4-tan-2-5-equal-to-39901
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\sin ^{2}(3 \pi)+\cos ^{2}(4 \pi)+\tan ^{2}(5 \pi)$ equal to $?</div>

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

answer:
<div class="solution">$\sin ^{2}(3 \pi)+\cos ^{2}(4 \pi)+\tan ^{2}(5 \pi)$<br/>$=\sin ^{2}(3 \pi)+\cos ^{2}(\pi+3 \pi)+\tan ^{2}(5 \pi)$<br/>$=\left(\sin ^{2}(3 \pi)+\cos ^{2}(3 \pi)\right)+\tan ^{2}(2 \times 2 \pi+\pi)$<br/>$=1+\tan ^{2} \pi=\sec ^{2} \pi=1$</div>


title: what-is-sin-2-66-2-1-sin-2-23-2-1-equal-to
slug: what-is-sin-2-66-2-1-sin-2-23-2-1-equal-to-12791
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\sin ^{2} 66 \frac{1^{\circ}}{2}-\sin ^{2} 23 \frac{1}{2}^{\circ}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin 47^{\circ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\cos 47^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \sin 47^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \cos 47^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\cos 47^{\circ}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \sin ^{2} 66 \frac{1^{\circ}}{2}-\sin ^{2} 23 \frac{1^{\circ}}{2} \\=&amp;\left[\sin \left(90^{\circ}-23 \frac{1^{\circ}}{2}\right)\right]^{2}-\sin ^{2} 23 \frac{1^{\circ}}{2} \\=&amp; \cos ^{2} 23 \frac{1^{\circ}}{2}-\sin ^{2} 23 \frac{1^{\circ}}{2} \\=&amp; \cos 2\left(23 \frac{1^{\circ}}{2}\right)=\cos 47^{\circ} \\\left(\because \cos 2 \mathrm{~A}=\cos ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~A}\right) \\=\cos \left[2 \times\left(\frac{47}{2}\right)\right]=\cos 47^{\circ} \end{aligned}$</div>


title: what-is-sin-2-a-sin-2-b-equal-to
slug: what-is-sin-2-a-sin-2-b-equal-to-11265
question: <div class="question">What is $\sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
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">$1 / 2$</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">2</span> </li> </ul>

answer:
<div class="solution">$\sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\sin ^{2} 60^{\circ}-\sin ^{2} 30^{\circ}=\frac{3}{4}-\frac{1}{4}=\frac{1}{2}$</div>


title: what-is-sin-2-x-d-x-cos-2-x-d-x-equal-to
slug: what-is-sin-2-x-d-x-cos-2-x-d-x-equal-to-91790
question: <div class="question">What is $\int \sin ^{2} x d x+\int \cos ^{2} x d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x+c$</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x+c$</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">$\frac{x^{2}}{2}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these <br/>Where $c$ is an arbitary constant.</span> </li> </ul>

answer:
<div class="solution">$\int \sin ^{2} x d x+\int \cos ^{2} x d x=\int\left(\sin ^{2} x+\left(\cos ^{2} x\right) d x\right.$<br/>$\int \mathrm{dx}=\mathrm{x}+\mathrm{c}$</div>


title: what-is-sin-2-x-n-cot-x-d-x-equal-to
slug: what-is-sin-2-x-n-cot-x-d-x-equal-to-69111
question: <div class="question">What is $\int \sin 2 x \ell n(\cot x) d x$ 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_P3.db
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">$\pi \ln 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\pi \ln 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi \ell \mathrm{n} 2}{2}$</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\cot \mathrm{x}) \mathrm{d} \mathrm{x}$<br/>$=\left(\because \cot x=\frac{\cos x}{\sin x}\right)$<br/>$=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ell \mathrm{n}(\cos \mathrm{x}) \mathrm{d} \mathrm{x}-\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\sin \mathrm{x}) \mathrm{dx}$<br/>$=\int_{0}^{\frac{\pi}{2}} \sin \left[2\left(\frac{\pi}{2}+\mathrm{x}\right)\right] \ln \cos \left(\frac{\pi}{2}+\mathrm{x}\right) \mathrm{d} \mathrm{x}$<br/>$=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\sin \mathrm{x}) \mathrm{dx}$<br/>$=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\sin \mathrm{x}) \mathrm{d} \mathrm{x}-\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\sin \mathrm{x}) \mathrm{dx}=0$</div>


title: what-is-sin-3-x-cos-x-d-x-equal-to
slug: what-is-sin-3-x-cos-x-d-x-equal-to-21133
question: <div class="question">What is $\int \sin ^{3} x \cos x d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos ^{4} x+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sin ^{4} x+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\left(1-\sin ^{2} x\right)^{2}}{4}+c$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\left(1-\cos ^{2} x\right)^{2}}{4}+c$<br/>Where $c$ is the constant of integration.</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">$\frac{\left(1-\cos ^{2} x\right)^{2}}{4}+c$<br/>Where $c$ is the constant of integration.</span> </div>

answer:
<div class="solution">Let $t=\sin x \Rightarrow d t=\cos x d x$<br/>$\int \sin ^{3} x \cos x d x=\int t^{3} d t=\frac{t^{4}}{4}+c=\frac{\sin ^{4} x}{4}+c$<br/>$=\frac{\left(1-\cos ^{2} x\right)^{2}}{4}+c$</div>


title: what-is-sin-10-5-cos-10-5-equal-to
slug: what-is-sin-10-5-cos-10-5-equal-to-94175
question: <div class="question">What is $\sin 105^{\circ}+\cos 105^{\circ}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin 50^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos 50^{\circ}$</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">0</span> </li> </ul>

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

answer:
<div class="solution">\sin 105^{\circ}+\cos 105^{\circ} \\<br/>=\frac{\sin \left(\frac{2 x}{2}\right)}{\cos \left(\frac{2 x}{2}\right)}=\frac{\sin x}{\cos x}=\tan x . \\<br/>=\left(\sin 60^{\circ} \cdot \sin 45^{\circ}+\cos 60^{\circ} \cdot \cos 45^{\circ}\right)+\left(\cos 60^{\circ} \cos 45^{\circ}\right. \\<br/>\left.-\sin 60^{\circ} \sin 45^{\circ}\right) \\<br/>=\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}}+\frac{1}{2} \cdot \frac{1}{\sqrt{2}}+\frac{1}{2} \cdot \frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} \\</div>


title: what-is-sin-cot-1-cos-tan-1-x-where-x-0-equal-to
slug: what-is-sin-cot-1-cos-tan-1-x-where-x-0-equal-to-30441
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\sin \left[\cot ^{-1}\left\{\cos \left(\tan ^{-1} x\right)\right]\right.$ where $x&gt;0$, equal to?</div>

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

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

answer:
<div class="solution">Let $\alpha=\tan ^{-1} \mathrm{x} \Rightarrow \tan \alpha=\mathrm{x}$<br/>then $\cos \alpha=\frac{1}{\sqrt{1+\tan ^{2} \alpha}}=\frac{1}{\sqrt{1+\mathrm{x}^{2}}}$<br/>$\Rightarrow \cos \left(\tan ^{-1} x\right)=\left\{\frac{1}{\sqrt{1+x^{2}}}\right\}$<br/>so, $\cot ^{-1}\left\{\cos \left(\tan ^{-1} x\right)\right\}=\cot ^{-1}\left\{\frac{1}{\sqrt{1+x^{2}}}\right\}$<br/>Let $\cot ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)=\beta$<br/>$\Rightarrow \cot \beta=\frac{1}{\sqrt{1+\mathrm{x}^{2}}}$<br/>and $\sin \beta=\frac{1}{\sqrt{1+\cot ^{2} \beta}}=\frac{\sqrt{1+\mathrm{x}^{2}}}{\sqrt{\mathrm{x}^{2}+1+1}}=\sqrt{\frac{\mathrm{x}^{2}+1}{\mathrm{x}^{2}+2}}$<br/>$\Rightarrow \sin \left[\cot ^{-1}\left\{\cos \left(\tan ^{-1}\right)\right\}\right]=\sqrt{\frac{x^{2}+1}{x^{2}+2}}$</div>


title: what-is-sin-sin-1-5-3-sin-1-5-4-equal-to
slug: what-is-sin-sin-1-5-3-sin-1-5-4-equal-to-18879
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\sin \left[\sin ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{4}{5}\right)\right]$ equal to ?</div>

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

answer:
<div class="solution">$\quad \sin \left[\sin ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{4}{5}\right)\right]$<br/>$=\sin \left[\sin ^{-1}\left\{\frac{3}{5} \sqrt{1-\frac{16}{25}}+\frac{4}{5} \sqrt{1-\frac{9}{25}}\right\}\right]$<br/>$=\sin \left[\sin ^{-1}\left\{\frac{3}{5} \times \frac{3}{5}+\frac{4}{5} \times \frac{4}{5}\right\}\right]$<br/>$=\sin \left[\sin ^{-1}\left\{\frac{9}{25}+\frac{16}{25}\right\}\right]=\sin \left[\sin ^{-1}(1)\right]$<br/>$=\sin \frac{\pi}{2}=1$</div>


title: what-is-sin-x-lo-g-tan-x-d-x-equal-to
slug: what-is-sin-x-lo-g-tan-x-d-x-equal-to-77271
question: <div class="question">What is $\int \sin x \log (\tan x) d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos x \log \tan x+\log \tan (x / 2)+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\cos x \log \tan x+\log \tan (x / 2)+c$</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">C</span> <span class="option-data">$\cos x \log \tan x+\log \cot (x / 2)+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\cos x \log \tan x+\log \cot (x / 2)+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\cos x \log \tan x+\log \tan (x / 2)+c$</span> </div>

answer:
<div class="solution">$\int \sin x \log (\tan x) d x$<br/>$=-\cos x \log \tan x-\int(-\cos x) \frac{1}{\tan x} \cdot \sec ^{2} x d x$<br/>$=-\cos x \log \tan x+\int \frac{1}{\sin x} d x$<br/>$=-\cos x \log (\tan x)+\int \frac{1+\tan ^{2} \frac{x}{2}}{2 \tan \frac{x}{2}} d x$<br/>Let $t=\tan \frac{x}{2}$<br/>$\Rightarrow \frac{d x}{d t}=\frac{2}{1+t^{2}} \Rightarrow d x=\frac{2}{1+t^{2}} \cdot d t$<br/>So, $-\cos x \cdot \log (\tan x)+\int \frac{1+\tan ^{2} \frac{x}{2}}{2 \tan \frac{x}{2}} \cdot d x$<br/>$=-\cos x \cdot \log (\tan x)+\int \frac{1+t^{2}}{2 t} \cdot \frac{2}{1+t^{2}} \cdot d t$<br/>$=-\cos x \log \tan x+\int \frac{1}{t} \cdot d t$<br/>$=-\cos x \log \tan x+\log (t)+c$<br/>$=-\cos x \log \tan x+\log \tan \left(\frac{x}{2}\right)+c$</div>


title: what-is-spermatogenesis-briefly-describe-the-process-of-spermatogenesis
slug: what-is-spermatogenesis-briefly-describe-the-process-of-spermatogenesis-38605
tags: ['Biology', 'Human Reproduction']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is spermatogenesis? Briefly describe the process of spermatogenesis.</div>

answer:
<div class="solution">Spermatogenesis is the process of producing sperms with half the number of chromosomes (haploid) as somatic cells. It occurs in seminiferous tubules. Sperm production begins at puberty continues throughout life with several hundred million sperms being produced each day. Once sperm are formed they move into the epididymis, where they mature and are stored.<br/>During spermatogenesis one spermatogonium produces 4 sperms. Spermatogenesis completes through the following phases - multiplicative phase, growth phase, maturation phase \&amp; spermiogenesis. In multiplicative phase the sperm mother cells divide by mitosis \&amp; produce spermatogonia. The spermatogonia grow in size to form large primary spermatocytes by getting nourishment from sertoli cells in growth phase. Maturation phase involves meiosis I in which primary spermatocytes divide to produce secondary spermatocyte and meiosis II which produces spermatids. Thus each primary spermatocyte gives rise to four haploid spermatids. Spermiogenesis or spermateliosis is process of formation of flagellated spermatozoa from spermatids. Spermiogenesis begins in the seminiferous tubules but usually completed in epididymis.<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/q60Hlh-UOXvpJBu6rB_ITDp8p2ECvrVBv_z-HudKI4c.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Thermodynamics (C)', 'MHT CET', 'MHT CET 2020 (19 Oct Shift 2)']
source_db: MarksBatch2_P3.db
title:
what-is-standard-n-n-bond-enthalpy-from-following-reaction-n-2-g-3-h-2-g-2-nh-3-g-h-83-kj-h-h-h-435-kj-h-n-h-389-kj

slug:
what-is-standard-n-n-bond-enthalpy-from-following-reaction-n-2-g-3-h-2-g-2-nh-3-g-h-83-kj-h-h-h-435-kj-h-n-h-389-kj-17504

question:
<div class="question">What is standard $\mathrm{N} \equiv \mathrm{N}$ bond enthalpy from following reaction? $\mathrm{N}_{2_{(\mathrm{g})}}+3 \mathrm{H}_{2_{(\mathrm{g})}} \longrightarrow 2 \mathrm{NH}_{3_{(\mathrm{g})}} \Delta \mathrm{H}^{\circ}=-83 \mathrm{KJ}$<br/>$\left(\Delta \mathrm{H}^{\circ}{ }_{(\mathrm{H}-\mathrm{H})}=435 \mathrm{~kJ}, \Delta \mathrm{H}^{\circ}{ }_{(\mathrm{N}-\mathrm{H})}=389 \mathrm{~kJ}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">435 kJ</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2334 \mathrm{~kJ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$946 \mathrm{~kJ}$</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">$1305 \mathrm{~kJ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$946 \mathrm{~kJ}$</span> </div>

answer:
<div class="solution">(C)<br/>$\mathrm{N}_{2(8)}+3 \mathrm{H}_{2(8)} \longrightarrow 2 \mathrm{NH}_{3(8)} \quad \Delta \mathrm{H}^{\circ}=-83 \mathrm{~kJ}$<br/>$\Delta \mathrm{H}^{0}=\sum \Delta \mathrm{H}^{0}$ (reactant bonds) $-\sum \Delta \mathrm{H}^{0}$ (product bonds)<br/>$\left.\therefore \Delta H^{0}=\left[\Delta H^{0}{ }_{(N=N)}^{0}+3 \Delta H^{0}{ }^{0} H-H\right)\right]-\left[6 \Delta H^{0}{ }^{0}(N-H)\right]$<br/>$\therefore-83=\Delta H_{(N=N)}^{0}+3(435)-6(389)$<br/>$\therefore-83=\Delta H_{(N=N)}^{0}+1305-2334$<br/>$\therefore \Delta H^{0}(N \sim N)=-83-1305+2334=946 \mathrm{~kJ}$</div>


tags: ['Chemistry', 'Electrochemistry', 'MHT CET', 'MHT CET 2022 (05 Aug Shift 1)']
source_db: MarksBatch2_P3.db
title:
what-is-standard-reduction-potential-of-cu-2-cu-s-if-e-of-following-cell-is-046-cu-s-cu-aq-2-ag-aq-ag-s-e-ag-ag-080-v

slug:
what-is-standard-reduction-potential-of-cu-2-cu-s-if-e-of-following-cell-is-046-cu-s-cu-aq-2-ag-aq-ag-s-e-ag-ag-080-v-81514

question:
<div class="question">What is standard reduction potential of $\mathrm{Cu}^{2+} \mid \mathrm{Cu}_{(\mathrm{s})}$ if $\mathrm{E}^{\circ}$ of following cell is 0.46 ?<br/>$\mathrm{Cu}_{(\mathrm{s})}\left|\mathrm{cu}_{(\mathrm{aq})}^{2+} \| \mathrm{Ag}_{(\mathrm{aq})}^{+}\right| \mathrm{Ag}_{(\mathrm{s})} ; \mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^{\circ}=0.80 \mathrm{v} \mid$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.56 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.34 \mathrm{~V}$</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">$1.44 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.26 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.34 \mathrm{~V}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{E}_{\text {cell }}^{\circ}=\mathrm{E}_{\mathrm{Ag}}^{\circ}-\mathrm{E}_{\mathrm{cu}}^{\circ} \\ &amp; 0.46=0.80-\mathrm{E}_{\text {cell }}^{\circ} \\ &amp; \mathrm{E}_{\mathrm{cu}}^{\circ}=0.34 \mathrm{v}\end{aligned}$</div>


title: what-is-stomatal-apparatus-explain-the-structure-of-stomata-with-a-labelled-diagram
slug: what-is-stomatal-apparatus-explain-the-structure-of-stomata-with-a-labelled-diagram-43554
tags: ['Biology', 'Anatomy of Flowering Plants']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is stomatal apparatus? Explain the structure of stomata with a labelled diagram.<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/xA6RdMN7dDKJNkC_DlajKUlB-xevv3dBmfSd0_-1edA.original.fullsize.png"/><br/></div>

answer:
<div class="solution">Several minute openings or stomata are found on the epidermis of all the green aerial parts of plants, but are abundant on the lower surface on the leaves as they regulate the process of transpiration. A large number of stomata occur on the upper surface of leaves of aquatic plants. Each stomata is surrounded by two cells known as the guard cells. In the dicotyledons plants these are bean shaped, but in sedges and grasses these are dumb-bell shaped. The guard cell are living. Their outer walls are thin where as inner ones surrounding the aperture are highly thickened. Due to this variation in the thickening, the guard cell may becomes turgid and flaccid, depending upon the supply of water in them, which makes the opening and closing of stomata possible. Some times a few neighbouring epidermal cells in vicinity of guard cells become specialized in their shape and size and contents. These are known as subsidiary cells. The stomatal aperture, guard cells and the surrounding subsidiary cell are together called stomatal apparatus.</div>


title: what-is-sum-to-the-100-terms-of-the-series-9-99-999
slug: what-is-sum-to-the-100-terms-of-the-series-9-99-999-74641
question: <div class="question">What is sum to the 100 terms of the series $9+99+999+\ldots ?$</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{10}{9}\left(10^{100}-1\right)-100$</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">$\frac{10}{9}\left(10^{99}-1\right)-100$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$100\left(100^{10}-1\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9}{100}\left(10^{100}-1\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{10}{9}\left(10^{100}-1\right)-100$</span> </div>

answer:
<div class="solution">Let $\mathrm{S}=9+99+999+\ldots$<br/>$=\left(10^{1}-1\right)+\left(10^{2}-1\right)+\left(10^{3}-1\right)+\ldots$<br/>$=\left(10+10^{2}+10^{3}+\ldots\right)-(1+1+1+\ldots 100$ times $)$<br/>$=\frac{10\left(10^{100}-1\right)}{10-1}-100 \quad\left(\because 10,10^{2}, 10^{3} \ldots \ldots .\right.$ is GP. with<br/>$\left.\mathrm{a}=10, \mathrm{r}=10, \mathrm{~S}_{100}=\frac{a\left(r^{100}-1\right)}{r-1}\right)$<br/>$=\frac{10}{9}\left(10^{100}-1\right)-100$</div>


title: what-is-t-a-n-3-6-c-o-t-5-4-c-o-t-7-0-t-a-n-2-0-equal-to
slug: what-is-t-a-n-3-6-c-o-t-5-4-c-o-t-7-0-t-a-n-2-0-equal-to-52183
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}$ equal to ?</div>

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

answer:
<div class="solution">\quad &amp; \frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}=\frac{\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}}{\cos ^{2} \frac{x}{2}+\sin ^{2} \frac{x}{2}} \\ \text { 147. (a) } &amp; \sin ^{2} 20^{\circ}+\sin ^{2} 70^{\circ}=\sin ^{2} 20^{\circ}+\sin ^{2}\left(90^{\circ}-20^{\circ}\right) \\=\sin ^{\circ} 20^{\circ}+\cos ^{2} 20^{\circ}=&amp; 1 \\ \text { 148. (d) }\left(1-\sin ^{2} \theta\right)\left(1+\tan ^{2} \theta\right) &amp;=\cos ^{2} \theta \cdot \sec ^{2} \theta \\=\cos ^{2} \theta \cdot \frac{1}{\cos ^{2} \theta}=1 \end{aligned}$</div>


title: what-is-t-a-n-15-2-c-o-t-8-8-1-t-a-n-2-0-c-o-t-6-2-equal-to
slug: what-is-t-a-n-15-2-c-o-t-8-8-1-t-a-n-2-0-c-o-t-6-2-equal-to-79566
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{1-\tan 2^{0} \cot 62^{\circ}}{\tan 152^{\circ}-\cot 88^{\circ}}$ equal to?</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">$-\sqrt{3}$</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">$\sqrt{2}-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1-\sqrt{2}$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} L=\frac{1-\tan 2^{\circ} \cot 62^{\circ}}{\tan 152^{\circ}-\cot 88^{\circ}} &amp;=\frac{1-\tan 2^{\circ} \cot (90-28)^{\circ}}{\tan (180-28)^{\circ}-\cot (90-2)^{\circ}} \\ \Rightarrow L=\frac{1-\tan 2^{\circ} \tan 28^{\circ}}{-\tan 28^{\circ}-\tan 2^{\circ}}=-\left[\frac{1-\tan 2^{\circ} \tan 28^{\circ}}{\tan 2^{\circ}+\tan 28^{\circ}}\right] \\ \Rightarrow L=-\frac{1}{\tan (2+28)^{\circ}}=-\frac{1}{\tan 30^{\circ}}=-\sqrt{3} \\\left[\cdots \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}\right] \end{aligned}$</div>


title: what-is-tan-1-2-1-tan-1-3-1-equal-to
slug: what-is-tan-1-2-1-tan-1-3-1-equal-to-19610
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{3}\right)$ equal to ?</div>

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

answer:
<div class="solution">$\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{3}\right)=\tan ^{-1}\left(\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}}\right)$<br/>$=\tan ^{-1}\left(\frac{\frac{5}{6}}{\frac{5}{6}}\right)=\tan ^{-1}(1)=\frac{\pi}{4}$</div>


title: what-is-tan-1-4-1-tan-1-5-3-equal-to
slug: what-is-tan-1-4-1-tan-1-5-3-equal-to-50700
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{3}{5}\right)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4}$</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">C</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2}$</span> </li> </ul>

answer:
<div class="solution">$\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{3}{5}\right)$<br/>We know, $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$<br/>So, $\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{3}{5}\right)=\tan ^{-1}\left(\frac{\frac{1}{4}+\frac{3}{5}}{1-\left(\frac{1}{4}\right)\left(\frac{3}{5}\right)}\right)$<br/>$=\tan ^{-1}\left(\frac{\frac{5+12}{20}}{1-\frac{3}{20}}\right)$<br/>$=\tan ^{-1}\left(\frac{\frac{17}{20}}{\frac{17}{20}}\right)=\tan ^{-1}(1)=\frac{\pi}{4}$</div>


title: what-is-tan-1-5-equal-to
slug: what-is-tan-1-5-equal-to-42063
question: <div class="question">What is tan $15^{\circ}$ equal to ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2-\sqrt{3}$</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">B</span> <span class="option-data">$2+\sqrt{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 \sqrt{3}</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 i $\sqrt{3}$</span> </li> </ul>

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

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


title: what-is-tan-1-sec-x-tan-x-dx-equal-to
slug: what-is-tan-1-sec-x-tan-x-dx-equal-to-28531
question: <div class="question">What is $\int \tan ^{-1}(\sec x+\tan x) \mathrm{dx}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\pi x}{4}+\frac{x^{2}}{4}+c$</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">$\frac{\pi x}{2}+\frac{x^{2}}{4}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi x}{4}+\frac{\pi x^{2}}{4}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi x}{4}-\frac{x^{2}}{4}+c$</span> </li> </ul>

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

answer:
<div class="solution">$\int \tan ^{-1}(\sec x+\tan x) d x$<br/>$=\int \tan ^{-1}\left(\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\right) d x$ $\quad \because \sec x+\tan x=\frac{1+\sin x}{\cos x}$<br/>$=\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)=\int\left(\frac{\pi}{4}+\frac{x}{2}\right) \mathrm{d} x$<br/>$=\frac{\pi x}{4}+\frac{x^{2}}{4}+c .$</div>


title: what-is-tan-2-x-sec-4-x-d-x-equal-to
slug: what-is-tan-2-x-sec-4-x-d-x-equal-to-81107
question: <div class="question">What is $\int \tan ^{2} x \sec ^{4} x d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sec ^{5} x}{5}+\frac{\sec ^{3} x}{3}+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\tan ^{5} x}{5}+\frac{\tan ^{3} x}{3}+c$</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">$\frac{\tan ^{5} x}{5}+\frac{\sec ^{3} x}{3}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sec ^{5} x}{5}+\frac{\tan ^{3} x}{3}+c$</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int \tan ^{2} x \sec ^{4} x d x$<br/>Let $\tan x=t$<br/>$\Rightarrow \sec ^{2} x d x=d t$<br/>$\begin{aligned} \therefore \quad &amp; \mathrm{I}=\int \tan ^{2} x \cdot \sec ^{2} x \cdot \sec ^{2} x \cdot d x \\ &amp;=\int \tan ^{2} x\left(1+\tan ^{2} x\right) \sec ^{2} x \cdot d x \end{aligned}$<br/>$\begin{aligned} \therefore \quad I &amp;=\int \mathrm{t}^{2}\left(1+t^{2}\right) d t=\int\left(t^{2}+t^{4}\right) d t \\ &amp;=\frac{t^{5}}{5}+\frac{t^{3}}{3}+c=\frac{\tan ^{5} x}{5}+\frac{\tan ^{3} x}{3}+c \end{aligned}$</div>


title: what-is-tan-4-a-sec-4-a-tan-2-a-sec-2-a-equal-to
slug: what-is-tan-4-a-sec-4-a-tan-2-a-sec-2-a-equal-to-12061
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\tan ^{4} \mathrm{~A}-\sec ^{4} \mathrm{~A}+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\overline{0}$</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 $[2013-I]$</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>

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

answer:
<div class="solution">Consider $\tan ^{4} \mathrm{~A}-\sec ^{4} \mathrm{~A}+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}$<br/>$=\left(\tan ^{2} \mathrm{~A}\right)^{2}-\left(\sec ^{2} \mathrm{~A}\right)^{2}+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A} .$<br/>$=\left(\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}\right)\left(\tan ^{2} \mathrm{~A}-\sec ^{2} \mathrm{~A}\right)$<br/>$+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}$<br/>$=-\left(\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}\right)+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}=0$</div>


title: what-is-tan-7-2-1-equal-to
slug: what-is-tan-7-2-1-equal-to-65210
question: <div class="question">What is tan $\left(7 \frac{1}{2}\right)^{\circ}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{6}+\sqrt{3}-\sqrt{2}+2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{6}+\sqrt{3}+\sqrt{2}+2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{6}-\sqrt{3}+\sqrt{2}-2$</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">D</span> <span class="option-data">$\sqrt{6}+\sqrt{3}+\sqrt{2}-2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{6}-\sqrt{3}+\sqrt{2}-2$</span> </div>

answer:
<div class="solution">$\tan \left(7 \frac{1}{2}\right)^{-}=\frac{\sin \left(7 \frac{1}{2}\right)^{\circ}}{\cos \left(7 \frac{1}{2}\right)}^{\prime}$<br/>Multiply and divide by $2 \sin \left(7 \frac{1}{2}\right)$; we get<br/>$\frac{2 \sin ^{2}\left(7 \frac{1}{2}\right)^{\circ}}{2 \sin \left(7 \frac{1}{2}\right)^{\circ} \cos \left(7 \frac{1}{2}\right)^{\circ}}=\frac{2 \sin ^{2}\left(\frac{15}{2}\right)^{\circ}}{2 \sin \left(\frac{15}{2}\right)^{\circ} \cos \left(\frac{15}{2}\right)^{\circ}}$<br/>$\left(\because \cos 2 \theta=1-2 \sin ^{2} \theta\right.$ and $\left.\sin 2 \theta=2 \sin \theta \cos \theta\right)$<br/>$=\frac{1-\cos \left(2 \times \frac{15}{2}\right)^{\circ}}{\sin \left(2 \times \frac{15}{2}\right)^{\circ}}$<br/>$=\frac{1-\cos 15^{\circ}}{\sin 15^{\circ}}=\frac{1-\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)}{\frac{\sqrt{3}-1}{2 \sqrt{2}}}=\sqrt{2}(\sqrt{3}+1)-(2+\sqrt{3})$<br/>$=\sqrt{6}+\sqrt{2}-2-\sqrt{3}=\sqrt{6}-\sqrt{3}+\sqrt{2}-2$</div>


title: what-is-tan-12-equal-to
slug: what-is-tan-12-equal-to-75426
question: <div class="question">What is $\tan \left(\frac{\pi}{12}\right)$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2-\sqrt{3}$</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">B</span> <span class="option-data">$2+\sqrt{3}$</span> </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">$\sqrt{3}-\sqrt{2}$</span> </li> </ul>

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

answer:
<div class="solution">$\quad \tan \frac{\pi}{12}=\tan \left(\frac{\pi}{3}-\frac{\pi}{4}\right)=\frac{\tan \pi / 3-\tan \pi / 4}{1+\tan \pi / 3 \tan \pi / 4}$<br/>$=\frac{\sqrt{3}-1-3+\sqrt{3}}{(1)^{2}-(\sqrt{3})^{2}}=\frac{2(\sqrt{3}-2)}{-2}=2-\sqrt{3}$</div>


title: what-is-tan-a-2-b-tan-2-a-b-equal-to
slug: what-is-tan-a-2-b-tan-2-a-b-equal-to-99588
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_P3.db
question:
<div class="question">What is tan $(\mathrm{A}+2 \mathrm{~B}) \cdot \tan (2 \mathrm{~A}+\mathrm{B})$ equal to?</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">$\underline{0}$</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">$\tan (\mathrm{A}+2 \mathrm{~B}) \cdot \tan (2 \mathrm{~A}+\mathrm{B})$<br/>Put $\mathrm{A}=60$ and $\mathrm{B}=30$ in above expression We get $\tan \left(120^{\circ}\right) \cdot \tan \left(150^{\circ}\right)$<br/>$=\tan \left(90^{\circ}+30^{\circ}\right) \tan \left(90^{\circ}+60^{\circ}\right)$<br/>$=\cot 30^{\circ} \cdot \cot 60^{\circ}=\sqrt{3} \cdot \frac{1}{\sqrt{3}}=1$</div>


title: what-is-tan-cos-1-x-equal-to
slug: what-is-tan-cos-1-x-equal-to-10216
question: <div class="question">What is tan $\left(\cos ^{-1} x\right)$ equal to?</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{1-x^{2}}}{x}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{x}}{1+\mathrm{x}^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{1+x^{2}}}{x}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{1-\mathrm{x}^{2}}$</span> </li> </ul>

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

answer:
<div class="solution">Let $\cos ^{-1} \mathrm{x}=\theta$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/wrEwsyZs4CRdbuo32fD-qIzn3tiujrUJWguotceiRJ8.original.fullsize.png"/><br/><br/>$\Rightarrow \cos \theta=\mathrm{x} \Rightarrow \sin \theta=\sqrt{1-\mathrm{x}^{2}}$<br/>$\Rightarrow \tan \theta=\frac{\sqrt{1-x^{2}}}{x}$ and $\theta=\cos ^{-1} x$<br/>This can be represented by a triangle with hypotenuous $=1$ and sides $\mathrm{x}$ and $\sqrt{1-\mathrm{x}^{2}}$.<br/>$\Rightarrow \tan \left(\cos ^{-1} x\right)=\frac{\sqrt{1-x^{2}}}{x}$</div>


title: what-is-the-1-5-th-term-of-the-series-3-7-13-21-31-43
slug: what-is-the-1-5-th-term-of-the-series-3-7-13-21-31-43-52161
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">241</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the $15^{\text {th }}$ term of the series $3,7,13,21,31,43, \ldots . ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">205</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">225</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">238</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">241</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<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/jOmuc3nP-rihXGl1JBkiH-5XH1Q5R2gExHygI-D5x-U.original.fullsize.png"/><br/><br/>$\Rightarrow \mathrm{a}_{\mathrm{n}}=3+[4+6+8+10+12+\ldots(\mathrm{n}-1)]$<br/>$=3+\frac{(n-1)}{2}[8+\{(n-1)-1\} \times 2]$<br/>$=3+\frac{(n-1)}{2}[8+2 n-4]$<br/>$=3+\frac{(n-1)}{2}(2 n+4)$<br/>$=3+(\mathrm{n}-1)(\mathrm{n}+2)$<br/>$\therefore 15$ th term $=\mathrm{a}_{15}=3+(15-1)(15+2)$<br/>$=3+14 \times 17=241$</div>


title: what-is-the-10-th-common-term-between-the-series-2-6-10-and-1-6-11
slug: what-is-the-10-th-common-term-between-the-series-2-6-10-and-1-6-11-70244
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">186</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the 10 th common term between the series $2+6+10+\ldots .$ and $1+6+11+\ldots . ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">180</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">186</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">196</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">206</span> </li> </ul>

answer:
<div class="solution">Given two series are $2+6+10+14+18+22+26+30+\ldots$<br/>and $1+6+11+16+21+26+31+\ldots$<br/>So, series which are made to be common terms of both series is $6,26,-\mathrm{u}$<br/>This is an A. P where $a=6, d=20$ $\begin{aligned} \therefore \mathrm{a}_{10} &amp;=\mathrm{a}+(\mathrm{n}-1) \mathrm{d} \\ &amp;=6+(10-1) 20=6+9 \times 20 \\ &amp;=180+6=186 \end{aligned}$</div>


title: what-is-the-abundance-of-boron-in-the-earths-crust-by-mass
slug: what-is-the-abundance-of-boron-in-the-earths-crust-by-mass-37340
question: <div class="question">What is the abundance of boron in the Earth's crust by mass?</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.001%</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.01%</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.1%</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0.0001%</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">0.0001%</span> </div>

answer:
<div class="solution">The abundance of boron in the Earth's crust is less than 0.0001% by mass.</div>


title: what-is-the-acid-soluble-pool-representative-of
slug: what-is-the-acid-soluble-pool-representative-of-22345
question: <div class="question">What is the acid soluble pool representative of?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
answer: <div class="solution">The acid soluble pool represents roughly the cytoplasmic composition.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The entire chemical composition of living tissues or organisms</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The chemical composition of living tissue from abundance point of view</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The cytoplasmic composition</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The macromolecules from cytoplasm and organelles</span> </li> </ul>

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


title: what-is-the-acute-angle-between-the-lines-ax-by-a-b-and-a-x-y-b-x-y-2-b
slug: what-is-the-acute-angle-between-the-lines-ax-by-a-b-and-a-x-y-b-x-y-2-b-14918
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the acute angle between the lines $\mathrm{Ax}+\mathrm{By}=\mathrm{A}+\mathrm{B}$ and $A(x-y)+B(x+y)=2 B ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$45^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\tan ^{-1}\left(\frac{A}{\sqrt{A^{2}+B^{2}}}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$60^{\circ}$</span> </li> </ul>

answer:
<div class="solution">Lines are $L_{1} \equiv A x+B y=A+B$ and $L_{2} \equiv A(x-y)+B(x+y)=2 B$<br/>Slope of $L_{1}$ is $-\frac{A}{B}$<br/>$\mathrm{m}_{1}=-\frac{\mathrm{A}}{\mathrm{B}} \quad\left[\mathrm{m}_{1}\right.$ is the side of line $\left.\mathrm{L}_{1}\right]$<br/>For line $\mathrm{L}_{2}$ :<br/>$\mathrm{Ax}-\mathrm{Ay}+\mathrm{Bx}+\mathrm{By}=2 \mathrm{~B}$<br/>$(\mathrm{A}+\mathrm{B}) \mathrm{x}-(\mathrm{A}-\mathrm{B}) \mathrm{y}=2 \mathrm{~B}$<br/>Slope of line $L_{2}$ in $\frac{(A+B)}{A-B}$<br/>$\mathrm{m}_{2}=\frac{(\mathrm{A}+\mathrm{B})}{(\mathrm{A}-\mathrm{B})} \quad\left[\mathrm{m}_{2}\right.$ is the slope of line $\left.\mathrm{L}_{2}\right]$<br/>If angle between line $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ is $\theta$ then<br/>$\tan \theta=\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}$<br/>$=\frac{-\frac{A}{B}-\frac{A+B}{A-B}}{1+\left(-\frac{A}{B}\right) \times\left(\frac{A+B}{A-B}\right)}=\frac{\frac{-A(A-B)-B(A+B)}{B(A-B)}}{\frac{B(A-B)-A(A+B)}{B(A-B)}}$<br/>$=\frac{-A^{2}+A B-A B-B^{2}}{A B-B^{2}-A^{2}-A B}=\frac{-B^{2}-A^{2}}{-B^{2}-A^{2}}=1$<br/>so, $\theta=\frac{\pi}{4}$</div>


title: what-is-the-acute-angle-between-the-lines-represented-by-the-equations-y-3-x-5-0-and-3-y-x-6-0
tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-acute-angle-between-the-lines-represented-by-the-equations-y-3-x-5-0-and-3-y-x-6-0-56199

question:
<div class="question">What is the acute angle between the lines represented by the equations $y-\sqrt{3} x-5=0$ and $\sqrt{3} y-x+6=0$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$60^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$75^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$30^{\circ}$</span> </div>

answer:
<div class="solution">$y-\sqrt{3} x-5=0$<br/>(i)<br/>$\sqrt{3} y-x+6=0$<br/>$y=m x+c$<br/>From (i) and from (ii)<br/>$y=\sqrt{3} x+5 \quad y=\frac{x}{\sqrt{3}}-\frac{6}{\sqrt{3}}$<br/>$m_{1}=\sqrt{3}$<br/>$m_{2}=\frac{1}{\sqrt{3}}$<br/>Angle between two lines, $\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|=\left|\frac{\sqrt{3}-\frac{1}{\sqrt{3}}}{1+\sqrt{3} \frac{1}{\sqrt{3}}}\right|=\frac{1}{\sqrt{3}}$<br/>$=\tan 30^{\circ}$<br/>$\theta=30^{\circ}$</div>


title: what-is-the-acute-angle-between-the-pair-of-straight-lines-2-x-3-y-1-and-3-x-2-y-2
slug: what-is-the-acute-angle-between-the-pair-of-straight-lines-2-x-3-y-1-and-3-x-2-y-2-43056
tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the acute angle between the pair of straight lines<br/>$\sqrt{2} \mathrm{x}+\sqrt{3} \mathrm{y}=1$ and $\sqrt{3} \mathrm{x}+\sqrt{2} \mathrm{y}=2$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\tan ^{-1}\left(\frac{1}{2 \sqrt{6}}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan ^{-1}(3)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1}\left(\frac{1}{2 \sqrt{6}}\right)$</span> </div>

answer:
<div class="solution">Given lines, $\sqrt{2} \mathrm{x}+\sqrt{3} \mathrm{y}=1$ and $\sqrt{3} \mathrm{x}+\sqrt{2} \mathrm{y}=2$<br/>We know, Slope $(\mathrm{m})=\frac{-\mathrm{a}}{\mathrm{b}}$<br/>$\therefore$ Slope of $\sqrt{2} \mathrm{x}+\sqrt{3} \mathrm{y}=1$ is $\mathrm{m}_{1}=\frac{-\sqrt{2}}{\sqrt{3}}$<br/>Slope of $\sqrt{3} x+\sqrt{2} y=2$ is $m_{2}=\frac{-\sqrt{3}}{\sqrt{2}}$<br/>$\tan \theta=\left|\frac{\mathrm{m}_{1}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\right|=\left|\frac{\frac{-\sqrt{3}}{\sqrt{2}}+\frac{\sqrt{2}}{\sqrt{3}}}{1+\left(\frac{-\sqrt{2}}{\sqrt{3}}\right)\left(\frac{-\sqrt{3}}{\sqrt{2}}\right)}\right|$<br/>$=\left|\frac{\frac{-3+2}{\sqrt{6}}}{2}\right|=\left|\frac{-1}{2 \sqrt{6}}\right|=\frac{1}{2 \sqrt{6}}$<br/>$\therefore \theta=\tan ^{-1}\left(\frac{1}{2 \sqrt{6}}\right)$</div>


title: what-is-the-acute-angle-between-the-planes-x-y-2-z-3-and-2-x-y-z-11
slug: what-is-the-acute-angle-between-the-planes-x-y-2-z-3-and-2-x-y-z-11-56719
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the acute angle between the planes $\mathrm{x}+\mathrm{y}+2 \mathrm{z}=3$ and $-2 x+y-z=11 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi / 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pi / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi / 6$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\pi / 3$</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>

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

answer:
<div class="solution">The given equation of the planes are $x+y+2 z=3$ and $-2 x+y-z=11$<br/>We know that, the angle between the planes $a_{1} x+b_{1} y+c_{1} z+d_{1}=0$<br/>and $a_{2} x+b_{2} y+c_{2} z+d_{2}=0$ is given by<br/>$\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\right|$<br/>Here, $a_{1}=1, b_{1}=1, c_{1}=2, a_{2}=-2, b_{2}=1, c_{2}=-1$<br/>$\therefore \quad \cos \theta=\left|\frac{1 \times(-2)+1 \times 1+2 \times(-1)}{\sqrt{1+1+4} \sqrt{4+1+1}}\right|$<br/>$=\left|\frac{-2+1-2}{\sqrt{6} \sqrt{6}}\right|=\left|\frac{3}{6}\right|=\frac{1}{2}=\cos \frac{\pi}{3} \Rightarrow \theta=\frac{\pi}{3}$</div>


title: what-is-the-adjoint-of-the-matrix-cos-sin-sin-cos
slug: what-is-the-adjoint-of-the-matrix-cos-sin-sin-cos-66520
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the adjoint of the matrix<br/>$\left(\begin{array}{l}\cos (-\theta)-\sin (-\theta) \\ -\sin (-\theta) \cos (-\theta)\end{array}\right) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\begin{array}{l}\cos \theta-\sin \theta \\ -\sin \theta \cos \theta\end{array}\right)$</span> </div>

answer:
<div class="solution">$\mathrm{A}=\left[\begin{array}{ll}\cos \theta &amp; \sin \theta \\ \sin \theta &amp; \cos \theta\end{array}\right]$<br/>$(\because \cos (-\theta)=\cos \theta$ and $\sin (-\theta)=-\sin \theta)$<br/>$\operatorname{adj} \mathrm{A}=\left[\begin{array}{cc}\cos \theta &amp; -\sin \theta \\ -\sin \theta &amp; \cos \theta\end{array}\right]$</div>


tags: ['Chemistry', 'Kinetic Theory of Gases', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">374 J</span> </div>
answer: <div class="solution">Correct Option is : (C)<br/>374 J</div>
source_db: MarksBatch2_P3.db
title:
what-is-the-amount-of-heat-needed-to-raise-the-temperature-of-the-gas-in-a-448liter-cylinder-containing-helium-gas-at-standard-temperature-and-pressur

slug:
what-is-the-amount-of-heat-needed-to-raise-the-temperature-of-the-gas-in-a-448liter-cylinder-containing-helium-gas-at-standard-temperature-and-pressur-43831

question:
<div class="question">What is the amount of heat needed to raise the temperature of the gas in a 44.8-liter cylinder containing helium gas at standard temperature and pressure by 15.0 Â°C, assuming the gas constant (R) is 8.31 J molâ»Â¹ Kâ»Â¹?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">449.76 J</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">66.45 J</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">374 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">D</span> <span class="option-data">977 J</span> </li> </ul>


tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-the-amount-of-heat-required-to-vaporize-one-mole-of-a-liquid-at-constant-temperature-and-under-standard-pressure-1-bar-called

slug:
what-is-the-amount-of-heat-required-to-vaporize-one-mole-of-a-liquid-at-constant-temperature-and-under-standard-pressure-1-bar-called-29316

question:
<div class="question">What is the amount of heat required to vaporize one mole of a liquid at constant temperature and under standard pressure (1 bar) called?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Standard enthalpy of sublimation</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Standard enthalpy of fusion</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Standard enthalpy of vaporization or molar enthalpy of vaporization</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">Standard enthalpy of condensation</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Standard enthalpy of vaporization or molar enthalpy of vaporization</span> </div>

answer:
<div class="solution">The amount of heat  required to vaporize one mole of a liquid at constant temperature and under  standard pressure (1 bar) is called its standard enthalpy of vaporization or  molar enthalpy of vaporization, â<sub>vap</sub>H.</div>


title: what-is-the-angle-between-the-lines-1-x-2-2-y-1-1-z-2-and-1-x-1-3-2-y-3-2-z-5
slug: what-is-the-angle-between-the-lines-1-x-2-2-y-1-1-z-2-and-1-x-1-3-2-y-3-2-z-5-27047
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between the lines $\frac{x-2}{1}=\frac{y+1}{-2}=\frac{z+2}{1}$ and $\frac{x-1}{1}=\frac{2 y+3}{3}=\frac{z+5}{2} ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{6}$</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">The given lines are:-<br/>$\frac{\mathrm{x}-2}{1}=\frac{\mathrm{y}-(-1)}{-2}=\frac{\mathrm{z}-(-2)}{1}$ and<br/>$\frac{x-1}{1}=\frac{y-\left(-\frac{3}{2}\right)}{\frac{3}{2}}=\frac{z-(-5)}{2}$<br/>dr's of Ist line are:-<br/>$a_{1}=1, b_{1}=-2, c_{1}=1$<br/>dr's of IInd line are:$\mathrm{a}_{2}=2, \mathrm{~b}_{2}=3, \mathrm{c}_{2}=4$<br/>Let $^{\prime} \theta^{\prime}$ be the angle $\mathrm{b} / \mathrm{w}$ two lines, then,<br/>$\cos \theta=\frac{\left|\mathrm{a}_{1} \mathrm{a}_{2}+\mathrm{b}_{1} \mathrm{~b}_{2}+\mathrm{c}_{1} \mathrm{c}_{2}\right|}{\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}} \cdot \sqrt{\mathrm{a}_{2}^{2}+\mathrm{b}_{2}^{2}+\mathrm{c}_{2}^{2}}}$<br/>$\cos \theta=0$<br/>$\Rightarrow \quad \theta=\frac{\pi}{2}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-angle-between-the-lines-whose-direction-cosines-are-proportional-to-2-3-4-and-1-2-1-respectively

slug:
what-is-the-angle-between-the-lines-whose-direction-cosines-are-proportional-to-2-3-4-and-1-2-1-respectively-37941

question:
<div class="question">What is the angle between the lines whose direction cosines are proportional to $(2,3,4)$ and $(1,-2,1)$ respectively?</div>

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

answer:
<div class="solution">Since, direction cosines are proportional to $(2,3,4)$ and $(1,-2,1)$ respectively $2 \times 1+3 \times(-2)+4 \times 1=0$<br/>Angle between the lines is $90^{\circ}$.</div>


title: what-is-the-angle-between-the-lines-x-y-1-and-x-y-1
slug: what-is-the-angle-between-the-lines-x-y-1-and-x-y-1-47681
question: <div class="question">What is the angle between the lines $x+y=1$ and $x-y=1$ ?</div>
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Slope of $\mathrm{x}+\mathrm{y}=1$ is $-1$ Slope of $\mathrm{x}-\mathrm{y}=1$ is 1 Let $\tan A=-1, \tan B=1$<br/>$\mathrm{A}=\frac{3 \pi}{4}, \mathrm{~B}=\frac{\pi}{4}$<br/>$\mathrm{~A}-\mathrm{B}=\frac{\pi}{2}$</div>


title: what-is-the-angle-between-the-lines-x-z-0-y-0-and-20-x-15-v-12-z
slug: what-is-the-angle-between-the-lines-x-z-0-y-0-and-20-x-15-v-12-z-34728
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between the lines $x+z=0, y=0$ and $20 x$ $=15 v=12 z ? \quad$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\cos ^{-1}(1 / 5)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos ^{-1}(1 / 7)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\cos ^{-1} \frac{45}{7 \sqrt{61}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sin ^{-1}(1 / 7)$</span> </li> </ul>

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

answer:
<div class="solution">Given, $x+z=0, y=0$ and $20 x=15 y=12 z$<br/>$\Rightarrow \frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ and $\frac{x}{3}=\frac{y}{4}=\frac{z}{5}$<br/>Let $\theta$ be angle between two lines. $\cos \theta=\left|\frac{(1)(3)+(0)(4)+(-1)(5)}{\sqrt{1+0+1} \sqrt{9+16+25}}\right|=\left|\frac{3+0-5}{\sqrt{2} \sqrt{50}}\right|$<br/>$=\frac{2}{\sqrt{2} \cdot 5 \sqrt{2}}=\frac{1}{5}$<br/>$\theta=\cos ^{-1}\left(\frac{1}{5}\right)$</div>


title: what-is-the-angle-between-the-planes-2-x-y-2-z-1-0-and-3-x-4-y-5-z-3-0
slug: what-is-the-angle-between-the-planes-2-x-y-2-z-1-0-and-3-x-4-y-5-z-3-0-72362
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between the planes $2 \mathrm{x}-\mathrm{y}-2 \mathrm{z}+1=0$ and $3 x-4 y+5 z-3=0 ? \quad$</div>

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

answer:
<div class="solution">Given equation of plane are $2 x-y-2 z+1=0$<br/>$\Rightarrow \mathrm{a}_{1}=2, \mathrm{~b}_{1}=-1, \mathrm{c}_{1}=-2, \mathrm{~d}_{1}=1$<br/>and $3 x-4 y+5 z-3=0$<br/>$\Rightarrow a_{2}=3, b_{2}=-4, c_{2}=5, d_{2}=-3$<br/>$\therefore$ Required angle is<br/>$\cos \theta=\pm \frac{\mathrm{a}_{1} \mathrm{a}_{2}+\mathrm{b}_{1} \mathrm{~b}_{2}+\mathrm{c}_{1} \mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}} \sqrt{\mathrm{a}_{2}^{2}+\mathrm{b}_{2}^{2}+\mathrm{c}_{2}^{2}}}$<br/>$\Rightarrow \cos \theta=\frac{2(3)+(-1)(-4)+5(-2)}{\sqrt{2^{2}+1^{2}+2^{2}} \sqrt{3^{2}+4^{2}+5^{2}}}=0$<br/>$\Rightarrow \theta=\frac{\pi}{2}$</div>


title: what-is-the-angle-between-the-planes-2-x-y-z-6-and-x-y-2-z-3
slug: what-is-the-angle-between-the-planes-2-x-y-z-6-and-x-y-2-z-3-76578
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between the planes $2 x-y+z=6$ and $x+y+2 z=3 ? \quad$</div>

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

answer:
<div class="solution">We know, if<br/>$$<br/>\begin{array}{l}<br/>a_{1} x+b_{1} y+c_{1} z=d_{1} \text { and } \\<br/>a_{2} x+b_{2} y+c_{2} z=d_{2}<br/>\end{array}$$ are two planes then angle between them is $$<br/>\cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} $$<br/>Let $\theta$ be the angle between given planes Here, $a_{1}=2, b_{1}=-1, c_{1}=1$ $$ \begin{aligned}<br/>&amp; a_{2}=1, b_{2}=1, c_{2}=2 \\<br/>\therefore \quad &amp; \cos \theta=\frac{2 \times 1+1 \times(-1)+1 \times 2}{\sqrt{4+1+1} \sqrt{1+1+4}} \\<br/>=&amp; \frac{3}{6}=\frac{1}{2}=\cos \frac{\pi}{3} \\<br/>\Rightarrow \quad \theta=\frac{\pi}{3}<br/>\end{aligned} $$ $\Rightarrow \theta=\frac{\pi}{3}$</div>


tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-angle-between-the-straight-lines-m-2-mn-y-mn-n-2-x-n-3-and-mn-m-2-y-mn-n-2-x-m-3-where-m-n

slug:
what-is-the-angle-between-the-straight-lines-m-2-mn-y-mn-n-2-x-n-3-and-mn-m-2-y-mn-n-2-x-m-3-where-m-n-55179

question:
<div class="question">What is the angle between the straight lines $\left(\mathrm{m}^{2}-\mathrm{mn}\right)$ $y=\left(m n+n^{2}\right) x+n^{3}$ and $\left(m n+m^{2}\right) y=\left(m n-n^{2}\right) x+m^{3}$<br/>where $\mathrm{m}&gt;\mathrm{n} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan ^{-1}\left(\frac{2 \mathrm{mn}}{\mathrm{m}^{2}+\mathrm{n}^{2}}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\tan ^{-1}\left(\frac{4 \mathrm{~m}^{2} \mathrm{n}^{2}}{\mathrm{~m}^{4}-\mathrm{n}^{4}}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan ^{-1}\left(\frac{4 \mathrm{~m}^{2} \mathrm{n}^{2}}{\mathrm{~m}^{4}+\mathrm{n}^{4}}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$45^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1}\left(\frac{4 \mathrm{~m}^{2} \mathrm{n}^{2}}{\mathrm{~m}^{4}-\mathrm{n}^{4}}\right)$</span> </div>

answer:
<div class="solution">Given straight lines are <br/>$\left(m^{2}-m n\right) y=\left(m n+n^{2}\right) x+n^{3}$ ...(1)<br/>$\left(m n+m^{2}\right) y=\left(m n-n^{2}\right) x+m^{3}$ ...(2)<br/>(1) $\Rightarrow \mathrm{y}=\frac{\left(\mathrm{mn}+\mathrm{n}^{2}\right)}{\mathrm{m}^{2}-\mathrm{mn}} \mathrm{x}+\frac{\mathrm{n}^{3}}{\mathrm{~m}^{2}-\mathrm{mn}}$<br/>So, slope of $(1), \mathrm{m}_{1}=\frac{\mathrm{mn}+\mathrm{n}^{2}}{\mathrm{~m}^{2}-\mathrm{mn}}$<br/>(2) $\Rightarrow \mathrm{y}=\frac{\mathrm{mn}-\mathrm{n}^{2}}{\mathrm{mn}+\mathrm{m}^{2}} \mathrm{x}+\frac{\mathrm{m}^{3}}{\mathrm{mn}+\mathrm{m}^{2}}$<br/>So, slope of $(2), \mathrm{m}_{2}=\frac{\mathrm{mn}-\mathrm{n}^{2}}{\mathrm{mn}+\mathrm{m}^{2}}$<br/>If $\alpha$ is the angle between lines $(1),(2)$, then<br/>$\tan \alpha=\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}$<br/>$=\frac{\frac{m n+n^{2}}{m^{2}-m n}-\frac{m n-n^{2}}{m n+m^{2}}}{1+\left(\frac{m n+n^{2}}{m^{2}-m n}\right)\left(\frac{m n-n^{2}}{m n+m^{2}}\right)}$<br/>$=\frac{\left(m n+n^{2}\right)\left(m n+m^{2}\right)-\left(m n-n^{2}\right)\left(m^{2}-m n\right)}{\left(m^{2}-m n\right)\left(m n+m^{2}\right)+\left(m n+n^{2}\right)\left(m n-n^{2}\right)}$<br/>$=\frac{\mathrm{m}^{2} \mathrm{n}^{2}+\mathrm{m}^{3} \mathrm{n}+\mathrm{mn}^{3}+\mathrm{m}^{2} \mathrm{n}^{2}-\mathrm{m}^{3} \mathrm{n}+\mathrm{m}^{2} \mathrm{n}^{2}+\mathrm{m}^{2} \mathrm{n}^{2}-\mathrm{mn}^{3}}{\mathrm{~m}^{3} \mathrm{n}+\mathrm{m}^{4}-\mathrm{m}^{2} \mathrm{n}^{2}-\mathrm{m}^{3} \mathrm{n}+\mathrm{m}^{2} \mathrm{n}^{2}-\mathrm{mn}^{3}+\mathrm{mn}^{3}-\mathrm{n}^{4}}$<br/>$\tan \alpha=\frac{4 \mathrm{~m}^{2} \mathrm{n}^{2}}{\mathrm{~m}^{4}-\mathrm{n}^{4}} \Rightarrow \alpha=\tan ^{-1}\left(\frac{4 \mathrm{~m}^{2} \mathrm{n}^{2}}{\mathrm{~m}^{4}-\mathrm{n}^{4}}\right)$</div>


title: what-is-the-angle-between-the-two-lines-whose-direction-numbers-are-3-1-3-1-4-and-3-1-3-1-4
slug: what-is-the-angle-between-the-two-lines-whose-direction-numbers-are-3-1-3-1-4-and-3-1-3-1-4-50315
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between the two lines whose direction numbers are $(\sqrt{3}-1,-\sqrt{3}-1,4)$ and $(-\sqrt{3}-1, \sqrt{3}-1,4) ?$</div>

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

answer:
<div class="solution">Let position vectors of two points be<br/>$\overrightarrow{\mathrm{OA}}=(\sqrt{3}-1) \hat{\mathrm{i}}-(\sqrt{3}+1) \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and<br/>$\overrightarrow{\mathrm{OB}}=-(\sqrt{3}+1) \hat{\mathrm{i}}+(\sqrt{3}-1) \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$<br/>$|\overrightarrow{\mathrm{OA}}|=\sqrt{(\sqrt{3}-1)^{2}+(\sqrt{3}+1)^{2}+4^{2}}$<br/>$=\sqrt{3+1-2 \sqrt{3}+3+1+2 \sqrt{3}+16}=\sqrt{24}$<br/>Also, $|\overrightarrow{\mathrm{OB}}|=\sqrt{(\sqrt{3}+1)^{2}+(\sqrt{3}-1)^{2}+4^{2}}=\sqrt{24}$<br/>$\overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}=|\overrightarrow{\mathrm{OA}}| \cdot|\overrightarrow{\mathrm{OB}}| \cdot \cos \theta$, where $\theta$ is the angle<br/>between $\overrightarrow{\mathrm{OA}} \&amp; \overrightarrow{\mathrm{OB}}$<br/>$\overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}=-(\sqrt{3}-1)(\sqrt{3}+1)-(\sqrt{3}+1)(\sqrt{3}-1)+16$<br/>$=-3+1-3+1+16=12$<br/>So, $\cos \theta=\frac{\overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}}{|\overrightarrow{\mathrm{OA}}| \overrightarrow{\mid \mathrm{OB}} \mid}=\frac{12}{24}=\frac{1}{2}$<br/>$\Rightarrow \theta=\frac{\pi}{3}$</div>


title: what-is-the-angle-between-the-two-straight-lines-y-2-3-x-5-and-y-2-3-x-7
slug: what-is-the-angle-between-the-two-straight-lines-y-2-3-x-5-and-y-2-3-x-7-93503
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between the two straight lines<br/>$\mathrm{y}=(2-\sqrt{3}) \mathrm{x}+5$ and $\mathrm{y}=(2+\sqrt{3}) \mathrm{x}-7 ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$60^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$15^{\circ}$</span> </li> </ul>

answer:
<div class="solution">The given lines are $y=(2-\sqrt{3}) x+5$ and $y=(2+\sqrt{3}) x-7$<br/>Therefore, slope of first line $\mathrm{m}_{1}=2-\sqrt{3}$ and slope of second line $\mathrm{m}_{2}=2+\sqrt{3}$ $\tan \theta=\left|\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\right|=\left|\frac{2+\sqrt{3}-2+\sqrt{3}}{1+(4-3)}\right|$<br/>$=\left|\frac{2 \sqrt{3}}{2}\right|=\sqrt{3}=\tan \frac{\pi}{3} \Rightarrow \theta=\frac{\pi}{3}=60^{\circ}$</div>


title: what-is-the-angle-between-two-planes-2-x-y-z-4-and-x-y-2-z-6
slug: what-is-the-angle-between-two-planes-2-x-y-z-4-and-x-y-2-z-6-61969
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the angle between two planes $2 x-y+z=4$ and $x+y+2 z=6 ? \quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{2}$</span> </li><li class="correct"> <span class="option-label">B</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><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">Given equations of two planes are $2 x-y+z=4$ and $x+y+2 z=6$<br/>So, angle between them is<br/>$\cos \theta=\frac{2(1)+(-1)(1)+(1)(2)}{\sqrt{4+1+1} \sqrt{4+1+1}}$<br/>$=\frac{2-1+2}{\sqrt{6} \sqrt{6}}=\frac{3}{6}=\frac{1}{2}$<br/>$\Rightarrow \theta=\frac{\pi}{3}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-angle-in-circular-measure-between-the-hour-hand-and-the-minute-hand-of-a-clock-when-thetime-is-half-past-4-m

slug:
what-is-the-angle-in-circular-measure-between-the-hour-hand-and-the-minute-hand-of-a-clock-when-thetime-is-half-past-4-m-21964

question:
<div class="question">What is the angle (in circular measure) between the hour hand and the minute hand of a clock when thetime is half past $4 \mathrm{~m}$</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">$\frac{\pi}{4}$</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">$\frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Angle traced by the hour hand in 12 hours $=360^{\circ}$ Angle traced by it in $4 \mathrm{hr} 30 \mathrm{~min}\left(4 \mathrm{~h}+\frac{30}{60} \mathrm{hr}\right)=\frac{9}{2} \mathrm{hr}$<br/>$=\frac{9}{2} \times \frac{360}{12}=13$<br/>Angle traced by minute hand is $60^{\circ} \mathrm{min}=360^{\circ}$<br/>Angle traced by it in $30 \mathrm{~min}=\frac{30}{60} \times 360=180^{\circ}$<br/>RequiredAngle $=180^{\circ}-135^{\circ}=45^{\circ} \Rightarrow 45 \times \frac{\pi}{100}=\frac{\pi}{4}$ radian</div>


title: what-is-the-angle-subtended-by-1-m-pole-at-a-distance-1-km-on-the-ground-in-sexagesimal-measure
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-angle-subtended-by-1-m-pole-at-a-distance-1-km-on-the-ground-in-sexagesimal-measure-44798

question:
<div class="question">What is the angle subtended by $1 \mathrm{~m}$ pole at a distance $1 \mathrm{~km}$ on the ground in sexagesimal measure?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{9}{50 \pi}$ degrec</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{5 \pi}$ degrec</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.4$ minute</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.5$ minute</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{9}{50 \pi}$ degrec</span> </div>

answer:
<div class="solution">Let $\mathrm{AB}$ be the pole of $1 \mathrm{~m}$. $B C=1 \mathrm{~km}=1000 \mathrm{~m}$<br/>Let ' $\theta$ ' be the required angle.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/CmKQj9zNO_TDCw7Dn-oCNTNQJGuKc0KogwSsaxUcprM.original.fullsize.png"/><br/><br/>Now, $\tan \theta=\left(\frac{1}{1000}\right)^{\circ}$<br/>Since ${ }^{\circ} \theta$ ' is very small<br/>$\therefore \tan \theta=\theta=\left(\frac{1}{1000}\right)^{\circ}$<br/>Now, consider option (a)<br/>$\left(\frac{9}{50 \pi}\right)^{\circ}=\left(\frac{9}{50 \times 180}\right)^{\circ}=\left(\frac{9}{9000}\right)^{\circ}=\left(\frac{1}{1000}\right)^{\circ}$<br/>Hence, required angle $=\left(\frac{9}{50 \pi}\right)^{\circ}$</div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-212.3</span> </div>
source_db: MarksBatch2_P3.db
title:
what-is-the-approximate-standard-free-energy-change-per-mole-of-zn-in-kj-mol-1-for-a-daniel-cell-at-298-k-1

slug:
what-is-the-approximate-standard-free-energy-change-per-mole-of-zn-in-kj-mol-1-for-a-daniel-cell-at-298-k-1-18757

question:
<div class="question">What is the approximate standard free energy change per mole of $\mathrm{Zn}$ (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) for a Daniel cell at $298 \mathrm{~K}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-212.3</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">230.0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1.10</span> </li> </ul>

answer:
<div class="solution">For Daniell cell we have<br/>$$<br/>\mathrm{Zn}(s)+\mathrm{Cu}^{2+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{Cu}(s)<br/>$$<br/>Gibbs free energy $=-n F E$<br/>$n=$ number of electrons $=2$<br/>$F=$ Faraday's constant $=96500 \mathrm{C}$<br/>$E=\operatorname{emf}$ of cell $=1.1 \mathrm{~V}$<br/>$\therefore$ Gibbs free energy $=-2 \times 96500 \times 1.1 \mathrm{~J}=-212.3 \mathrm{~kJ}$</div>


tags: ['Chemistry', 'Electrochemistry', 'TS EAMCET', 'TS EAMCET 2018 (07 May Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-212.3</span> </div>
source_db: MarksBatch2_P3.db
title:
what-is-the-approximate-standard-free-energy-change-per-mole-of-zn-in-kj-mol-1-for-a-daniel-cell-at-298-k

slug:
what-is-the-approximate-standard-free-energy-change-per-mole-of-zn-in-kj-mol-1-for-a-daniel-cell-at-298-k-18538

question:
<div class="question">What is the approximate standard free energy change per mole of $\mathrm{Zn}$ (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) for a Daniel cell at $298 \mathrm{~K}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-212.3</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">230.0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1.10</span> </li> </ul>

answer:
<div class="solution">For Daniell cell we have<br/>$$<br/>\mathrm{Zn}(s)+\mathrm{Cu}^{2+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{Cu}(s)<br/>$$<br/>Gibbs free energy $=-n F E$<br/>$n=$ number of electrons $=2$<br/>$F=$ Faraday's constant $=96500 \mathrm{C}$<br/>$E=\operatorname{emf}$ of cell $=1.1 \mathrm{~V}$<br/>$\therefore$ Gibbs free energy $=-2 \times 96500 \times 1.1 \mathrm{~J}=-212.3 \mathrm{~kJ}$</div>


title: what-is-the-approximate-value-of-102-8
slug: what-is-the-approximate-value-of-102-8-26446
question: <div class="question">What is the approximate value of $(1.02)^{8}$ ?</div>
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.171</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1.171</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">1.175</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.177</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.179</span> </li> </ul>

answer:
<div class="solution">$(1.02)^{8}=(1+0.02)^{8}$<br/>$(1+x)^{\mathrm{n}}=1+\mathrm{nx}+\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{3 !} \mathrm{x}^{3}+\ldots .$<br/>$\mathrm{n}=8, \mathrm{x}=0.02$<br/>$(1+0.02)^{8}$<br/>$=1+8 \times 0.02+\frac{8 \times 7}{2 !} \cdot(0.02)^{2}+\frac{8.7 .6}{3 !}(.02)^{3}$<br/>Neglecting higher terms $=1+0.16+28 \times 0.0004+56 \times 0.000008$<br/>$\simeq 1+0.16+0.0112=1.171$</div>


title: what-is-the-area-between-the-curve-y-cos-3-x-0-x-6-and-the-coordinate-axes
slug: what-is-the-area-between-the-curve-y-cos-3-x-0-x-6-and-the-coordinate-axes-19895
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area between the curve<br/>$y=\cos 3 x, 0 \leq x \leq \frac{\pi}{6}$ and the co-ordinate axes?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 squale unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$ squaue unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{3}$ square 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">$\frac{1}{4}$ square unit</span> </li> </ul>

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

answer:
<div class="solution">Required Area $=\int_{0}^{\pi / 6} \cos 3 x d x$<br/>$=\left.\frac{\sin 3 x}{3}\right|_{0} ^{\pi / 6}=\frac{\sin 3(\pi / 6)}{3}-\sin 0$<br/>$=\frac{1}{3} \sin \pi / 2-0=\frac{1}{3}(1)=\frac{1}{3}$ sq. unit.</div>


title: what-is-the-area-bounded-by-the-curve-y-4-x-x-2-3-and-the-x-axis
slug: what-is-the-area-bounded-by-the-curve-y-4-x-x-2-3-and-the-x-axis-44022
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area bounded by the curve $y=4 x-x^{2}-3$ and the $x$ -axis? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 / 3$ sq unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4 / 3$ 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">C</span> <span class="option-data">$5 / 3 \mathrm{sq}$ unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 / 5 \mathrm{sq}$ unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 / 3$ sq unit</span> </div>

answer:
<div class="solution">Given curve is $y=4 x-x^{2}-3$ Since, area bounded by $x$ -axis $\therefore y=0$<br/>$\Rightarrow 4 x-x^{2}-3=0 \Rightarrow x^{2}-4 x+3=0$<br/>$\Rightarrow x^{2}-3 x-x+3=0 \Rightarrow(x-3)(x-1)=0$<br/>$\Rightarrow x=1,3$<br/>$\therefore$ Required area $=\int_{1}^{3}\left(4 x-x^{2}-3\right) d x$<br/>$=\frac{4 x^{2}}{2}-\frac{x^{3}}{3}-\left.3 x\right|_{1} ^{3}=\left(\frac{36}{2}-\frac{27}{3}-9\right)-\left(\frac{4}{2}-\frac{1}{3}-3\right)$<br/>$=(18-9-9)-\left(2-\frac{10}{3}\right)=0-\left(\frac{-4}{3}\right)=\frac{4}{3}$ sq. unit.</div>


title: what-is-the-area-bounded-by-the-curve-y-x-2-and-the-line-y-16
slug: what-is-the-area-bounded-by-the-curve-y-x-2-and-the-line-y-16-11187
question: <div class="question">What is the area bounded by the curve $y=x^{2}$ and the line $y=16 ?$</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$32 /$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$64 / 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$256 / 3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$128 /$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$256 / 3$</span> </div>

answer:
<div class="solution">Given, equations of curves are $y=x^{2} \quad \ldots$ (i)<br/>and $y=16$ $\ldots$ (ii) On solving Eqs. (i) and (ii), we get $x^{2}=16 \Rightarrow x=4,-4$<br/>$\therefore \quad$ Points of intersection are $(4,16)$, and $(-4,16)$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/EWY4sr32QnfaJZUJUtkWOEn5j7naeJFEfgo_wILi3Sw.original.fullsize.png"/><br/><br/>Required area $=\int_{-4}^{4}\left(16-x^{2}\right) d x=2 \int_{0}^{4}\left(16-x^{2}\right) d x$<br/>$=2\left[16 x-\frac{x^{3}}{3}\right]_{0}^{4}=2\left[64-\frac{64}{3}\right]=2 \times 64 \times \frac{2}{3}$<br/>$=\frac{256}{3}$ sq unit</div>


tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-area-bounded-by-the-curves-consider-the-curves-f-x-x-x-1-and-g-x-2-3-x-2-x-x-0-x-0-2016-i

slug:
what-is-the-area-bounded-by-the-curves-consider-the-curves-f-x-x-x-1-and-g-x-2-3-x-2-x-x-0-x-0-2016-i-49232

question:
<div class="question">What is the area bounded by the curves?<br/>Consider the curves $\mathrm{f}(\mathrm{x})=\mathrm{x}|\mathrm{x}|-1$ and $\mathrm{g}(\mathrm{x})=\left\{\begin{array}{ll}\frac{3 \mathrm{x}}{2}, &amp; \mathrm{x}&gt;0 \\ 2 \mathrm{x}, &amp; \mathrm{x} \leq 0\end{array} \quad[2016-\mathrm{I}]\right.$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{17}{6}$ square units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{8}{3}$ square units</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 square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{3}$ square unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{8}{3}$ square units</span> </div>

answer:
<div class="solution">$\begin{aligned} \text { Area } &amp;=\int_{-1}^{2}[g(x)-f(x)] d x \\ &amp;=\int_{-1}^{0}[g(x)-f(x)] d x+\int_{0}^{2}[g(x)-f(x)] d x \end{aligned}$<br/>$\begin{aligned} &amp;=\int_{-1}^{0}\left(2 x+x^{2}+1\right) d x+\int_{0}^{2}\left[\frac{3 x}{2}-x^{2}+1\right] d x \\ &amp;=I_{1}+I_{2} \\ I_{1}=&amp;\left(\frac{2 x^{2}}{2}+\frac{1}{3} x^{3}+x\right)_{-1}^{0} \\=&amp;(0+0+0)-\left((-1)^{2}+\frac{1}{3}(-1)^{3}+(-1)\right) \\=&amp;-\left(1-\frac{1}{3}-1\right)=\frac{1}{3} \\ I_{2}=&amp; \int_{0}^{2}\left[\frac{3 x}{2}-x^{2}+1\right] d x \\=&amp;\left[\frac{3}{2} \frac{x^{2}}{2}-\frac{1}{3} x^{3}+x\right]_{0}^{2} \\=&amp;\left(\frac{3}{4} \times 4-\frac{1}{3} \times 8+2\right)-(0) \\=&amp; 5-\frac{8}{3}=\frac{7}{3} \\ \text { Area } &amp;=I_{1}+I_{2}=\frac{1}{3}+\frac{7}{3} \end{aligned}$<br/>Area $=\frac{8}{3}$ square units.</div>


title: what-is-the-area-bounded-by-the-curves-y-1-x-2
slug: what-is-the-area-bounded-by-the-curves-y-1-x-2-65757
question: <div class="question">What is the area bounded by the curves $|y|=1-x^{2}$ ?</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{4}{3}$ square units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{8}{3}$ square units</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 square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{16}{3}$ square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{8}{3}$ square units</span> </div>

answer:
<div class="solution">Since $|y|=\left\{\begin{array}{cc}y &amp; y&gt;0 \\ -y &amp; y   &lt; 0 \\ 0 &amp; y=0\end{array}\right.$<br/>For $y&gt;0 \Rightarrow y=1-x^{2}$<br/>For $y &lt; 0 \Rightarrow y=x^{2}-1$<br/>For $y=0 \Rightarrow x=\pm 1$<br/>So area under the curve $=4 \times$ Area under the region OABO (symmetry)<br/>$\begin{aligned}=&amp; 4 \times \int_{0}^{1} 1-x^{2} \mathrm{~d} \mathrm{x} \\ &amp; \left.=4 \times\left[x-\frac{x^{3}}{3}\right]_{0}^{1}\right] \end{aligned}$<img src="https://cdn-question-pool.getmarks.app/pyq/nda/5RXQ88h19fkRdRGlXPPof0ELFTExOeuwZiWNJPB5t50.original.fullsize.png"/><br/><br/>$\quad=4\left(1-\frac{1}{3}\right)=4 \times \frac{2}{3}=\frac{8}{3}$ sq. units</div>


title: what-is-the-area-bounded-by-the-curves-y-e-x-y-e-x-and-the-straight-line-x-1
slug: what-is-the-area-bounded-by-the-curves-y-e-x-y-e-x-and-the-straight-line-x-1-33011
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area bounded by the curves $y=e^{x}$, $y=e^{-x}$ and the straight line $x=1 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(e+\frac{1}{e}\right)$ sq unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(e-\frac{1}{e}\right)$ sq unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(e+\frac{1}{e}-2\right)$ 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">$\left(e-\frac{1}{e}-2\right)$ sq unit</span> </li> </ul>

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

answer:
<div class="solution">Given equations of curves are $y=e^{x}$ and $y=e^{-x}$. $\Rightarrow e^{x}=\frac{1}{e^{x}} \quad \Rightarrow e^{2 x}=e^{0} \Rightarrow x=0$<br/>Also, equation of straight line gives $x=1$<br/>$\begin{aligned} \text { Required area } &amp;=\int_{0}^{1}\left(e^{x}-e^{-x}\right) d x \\ &amp;=\left[e^{x}+e^{-x}\right]_{0}^{1}=e+e^{-1}-\left(e^{0}+e^{-0}\right) \\ &amp;=\left(e+\frac{1}{e}-2\right) \text { sq unit } \end{aligned}$</div>


title: what-is-the-area-bounded-by-the-lines-x-0-y-0-and-x-2-0
slug: what-is-the-area-bounded-by-the-lines-x-0-y-0-and-x-2-0-57075
question: <div class="question">What is the area bounded by the lines $x=0, y=0$ and $x+$ ] $2=0 ?$</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$ square unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 square unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2 square units</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 square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 square units</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/2yI8tHUhv7mGOszQlvD_ZGyfCfRV_LkJx9WNktIkYl4.original.fullsize.png"/><br/><br/>Area of $\Delta \mathrm{OAB}=\frac{1}{2} \times 2 \times 2=2$ square units</div>


title: what-is-the-area-bounded-by-y-tan-x-y-0-and-x-4
slug: what-is-the-area-bounded-by-y-tan-x-y-0-and-x-4-79297
question: <div class="question">What is the area bounded by $y=\tan x, y=0$ and $x=\frac{\pi}{4}$ ?</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ln 2$ square units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\ln 2}{2}$ square units</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 ( en 2 )square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Nonc of these</span> </li> </ul>

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

answer:
<div class="solution">Required area $=\int_{0}^{\frac{\pi}{4}} \tan x \mathrm{dx}$<br/>$\quad=\ln |\sec x|_{0}^{\frac{\pi}{4}}=\ln \sqrt{2}=\frac{\ln 2}{2}$</div>


title: what-is-the-area-bounded-hy-the-curve-x-y-a-x-y-0-and-the-coordinate-axes
slug: what-is-the-area-bounded-hy-the-curve-x-y-a-x-y-0-and-the-coordinate-axes-30553
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area bounded hy the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$<br/>$(x, y \geq 0)$ and the coordinate axes?</div>

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

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

answer:
<div class="solution">Area bounded by curve $\sqrt{x}+\sqrt{y}=\sqrt{a}(\mathrm{x}, \mathrm{y} \geq 0$ ) and coordinate axes is<br/>$=\int_{0}^{a} y d x=\int_{0}^{a} a+x-2 \sqrt{a} \sqrt{x} d x$<br/>$\quad(\because \sqrt{y}=\sqrt{a}-\sqrt{x} \Rightarrow y=a+x-2 \sqrt{a} \sqrt{x})$<br/>$=a x+\frac{x^{2}}{2}=\left.\frac{2 \sqrt{a} x^{3 / 2}}{3 / 2}\right|_{0} ^{a}$<br/>$=a^{2}+\frac{a^{2}}{2}-\frac{4}{3} a^{2}=\frac{3 a^{2}}{2}-\frac{4}{3} a^{2}$<br/>$=\frac{9 a^{2}-8 a^{2}}{6}=\frac{a^{2}}{6}$ sq. unit</div>


title: what-is-the-area-enclosed-by-the-curve-2-x-2-y-2-1
slug: what-is-the-area-enclosed-by-the-curve-2-x-2-y-2-1-40520
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area enclosed by the curve $2 \mathrm{x}^{2}+\mathrm{y}^{2}=1$ ?</div>

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

answer:
<div class="solution">Given equation of curve $2 x^{2}+y^{2}=1$ is an ellipse which can be written as $\frac{x^{2}}{y^{2}}+\frac{y^{2}}{1}=1$<br/>Area of ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is $A=\pi$ ab sq unit<br/>Here, $\mathrm{a}=\frac{1}{\sqrt{2}}, \mathrm{~b}=1$.<br/>: Required area $=\pi \cdot \frac{1}{\sqrt{2}} \cdot 1=\frac{\pi}{\sqrt{2}}$ sq unit</div>


title: what-is-the-area-enclosed-by-the-equation-x-2-y-2-2
slug: what-is-the-area-enclosed-by-the-equation-x-2-y-2-2-95800
question: <div class="question">What is the area enclosed by the equation<br/>$x^{2}+y^{2}=2 ?$</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 \pi$ square units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \pi$ square units</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">$4 \pi^{2}$ square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4 square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \pi$ square units</span> </div>

answer:
<div class="solution">Given equation of circle is $x^{2}+y^{2}=2$ $\Rightarrow y=\sqrt{2-x^{2}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/sV96pqa2TUxtQVbG6-XVOt-H5D_-2rIj9FE5S6MKoRg.original.fullsize.png"/><br/><br/>Required area $=4 \times$ Area of shaded portion $-4 \int_{0}^{\sqrt{2}} \sqrt{2-x^{2}} d x$<br/>$\sqrt{2-x^{2}} d x$<br/>Let $x=\sqrt{2} \sin t \Rightarrow t=\sin ^{-1}\left(\frac{x}{\sqrt{2}}\right)$<br/>$d x=\sqrt{2} \cos t \cdot d t$<br/>$\therefore \int \sqrt{2-x^{2}} \cdot d x=\int \sqrt{2-2 \sin ^{2} t} \cdot \sqrt{2} \cos t \cdot d t$<br/>$=\int \sqrt{2 \cos ^{2} t} \cdot \sqrt{2} \cos t \cdot d t$<br/>$=2 \int \cos ^{2} t \cdot d t$<br/>We know,<br/>$\int \cos ^{n} x \cdot d x=\frac{n-1}{n} \int \cos ^{n-2}(x) d x+\frac{\cos ^{n-1} x \cdot \sin x}{n}$<br/>$=2\left[\frac{\cos t \cdot \sin t}{2}+\frac{1}{2} \int 1 \cdot d t\right]=2\left[\frac{\cos t \cdot \sin t}{2}+\frac{t}{2}\right]$<br/>$=\cos t \sin t+t$<br/>$=\cos \left(\sin ^{-1} \frac{x}{\sqrt{2}}\right) \times \sin \left(\sin ^{-1} \frac{x}{\sqrt{2}}\right)+\sin ^{-1} \frac{x}{\sqrt{2}}$<br/>$=\sqrt{1-\frac{x^{2}}{2}} \cdot \frac{x}{\sqrt{2}}+\sin ^{-1} \frac{x}{\sqrt{2}}$<br/>$=\frac{x}{2} \sqrt{2-x^{2}}+\sin ^{-1} \frac{x}{\sqrt{2}}$<br/>$\therefore 4 \cdot \int_{0}^{\sqrt{2}} \sqrt{2-x^{2}} \cdot d x=4\left[\frac{x}{2} \sqrt{2-x^{2}}+\sin ^{-1} \frac{x}{\sqrt{2}}\right]_{0}^{\sqrt{2}}$<br/>$=4\left[0+\sin ^{-1} \frac{\sqrt{2}}{\sqrt{2}}-0-\sin ^{-1} \frac{0}{\sqrt{2}}\right]$<br/>$=4\left[\sin ^{-1} 1-\sin ^{-1} 0\right]=4\left(\frac{\pi}{2}-0\right)=2 \pi$ sq. units.</div>


title: what-is-the-area-of-a-triangle-whose-vertices-are-at-3-1-2-1-1-3-and-4-3-1
slug: what-is-the-area-of-a-triangle-whose-vertices-are-at-3-1-2-1-1-3-and-4-3-1-85601
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of a triangle whose vertices are at $(3,-1,2)$, $(1,-1,-3)$ and $(4,-3,1)$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{165}}{2}$</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">$\frac{\sqrt{135}}{2}$</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">2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{165}}{2}$</span> </div>

answer:
<div class="solution">Let the vertices of the $\Delta \mathrm{ABC}$ are $\mathrm{A}(3,-1,2)$, $\mathrm{B}(1,-1,-3)$ and $\mathrm{C}(4,-3,1)$<br/>Let $\overrightarrow{\mathrm{OA}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and<br/>$$<br/>\overrightarrow{\mathrm{OC}}=4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\mathrm{k}<br/>$$<br/>Area of $\Delta \mathrm{ABC}=\frac{1}{2}|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}|$<br/>Now, $\overrightarrow{\mathrm{AB}}=\overline{\mathrm{OA}}-\overrightarrow{\mathrm{OB}}=2 \hat{\mathrm{i}}+5 \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OC}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\therefore \quad$ Required Area $=\frac{1}{2}\left\|\begin{array}{|ccc}\hat{\mathrm{i}} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ 2 &amp; 0 &amp; 5 \\ -1 &amp; 2 &amp; 1\end{array}\right\|$<br/>$=\frac{1}{2}|\hat{i}(-10)-\hat{j}(2+5)+\hat{k}(4)|$<br/>$=\frac{1}{2}|-10 \hat{i}-7 \hat{j}+4 \hat{k}|$<br/>$=\frac{1}{2} \sqrt{100+49+16}=\frac{1}{2} \sqrt{165}$ squnit</div>


title: what-is-the-area-of-one-of-the-loops-between-the-curve-y-c-sin-x-and-x-axis-1
slug: what-is-the-area-of-one-of-the-loops-between-the-curve-y-c-sin-x-and-x-axis-1-71442
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of one of the loops between the curve $y=$ $\mathrm{c} \sin \mathrm{x}$ and $\mathrm{x}$ -axis ?</div>

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

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

answer:
<div class="solution">Area of one of loop between $\mathrm{y}=\mathrm{c} \sin \mathrm{x}$ and $\mathrm{x}$ -axis<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Gk1hHwG_OD4N1gTmq6Ph9OwYz1TjX_itcCmPl4Upzdo.original.fullsize.png"/><br/><br/>$=\mathrm{c}+\mathrm{c}=2 \mathrm{c}$</div>


title: what-is-the-area-of-the-ellipse-4-x-2-9-y-2-1
slug: what-is-the-area-of-the-ellipse-4-x-2-9-y-2-1-78506
question: <div class="question">What is the area of the ellipse $4 x^{2}+9 y^{2}=1 \quad$</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 \pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{36}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{6}$</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">D</span> <span class="option-data">$\frac{\pi}{\sqrt{6}}$</span> </li> </ul>

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

answer:
<div class="solution">Given $4 x^{2}+9 y^{2}=1$<br/>$\Rightarrow \frac{x^{2}}{\left(\frac{1}{2}\right)^{2}}+\frac{y^{2}}{\left(\frac{1}{3}\right)^{2}}=1$<br/>$\therefore \quad a=\frac{1}{2}$ and $b=\frac{1}{3}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/m_c9GHtQN3joALesrwYiwkizbjmGlhydwcqsy8jbtBo.original.fullsize.png"/><br/><br/>Now, area of ellipse $=4 \int_{0}^{1 / 2} y d x$<br/>$=4 \int_{0}^{1 / 2} \sqrt{\frac{1-4 x^{2}}{9}} d x=\frac{4}{3} \int_{0}^{1 / 2} \sqrt{1-(2 x)^{2}} d x$<br/>$=\frac{2}{3} \int_{0}^{1} \sqrt{1-t^{2}} d t=\frac{2}{3}\left[\frac{1}{2} t \sqrt{1-t^{2}}+\frac{1}{2} \sin ^{-1}(t)\right]_{0}^{1}$<br/>$=\frac{2}{3}\left[0+\frac{1}{2} \sin ^{-1}(1)-\frac{1}{2} \sin ^{-1}(0)\right]$<br/>$=\frac{2}{3}\left[\frac{1}{2} \times \frac{\pi}{2}-\frac{1}{2} \times 0\right]=\frac{\pi}{6}$</div>


title: what-is-the-area-of-the-largest-rectangular-field-which-can-be-enclosed-with-200-m-of-fencing
slug: what-is-the-area-of-the-largest-rectangular-field-which-can-be-enclosed-with-200-m-of-fencing-67676
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the largest rectangular field which can be enclosed with $200 \mathrm{~m}$ of fencing ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1600 \mathrm{~m}^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2100 \mathrm{~m}^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2400 \mathrm{~m}^{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\begin{array}{ll}\text {  } 2500 \mathrm{~m}^{2} &amp; {}\end{array}$</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">$\begin{array}{ll}\text {  } 2500 \mathrm{~m}^{2} &amp; {}\end{array}$</span> </div>

answer:
<div class="solution">Area of largest rectangular field for a given perimeter for is possible if length and breadth of rectangular field are equal i.e. it is a square<br/>$\Rightarrow 4 \mathrm{x}=200 \Rightarrow \mathrm{x}=\frac{200}{4}=50 \mathrm{~m}$<br/>$\therefore$ Area of largest rectangular field $=50 \times 50=2500 \mathrm{~m}^{2}$ Aliter:<br/>Let length and breadth of rectangular field be $\mathrm{x}$ and $\mathrm{y}$ respectively $\therefore \quad 2(\mathrm{x}+\mathrm{y})=200$<br/>$\Rightarrow \mathrm{y}=100-\mathrm{x}$<br/>and area, $\mathrm{A}=\mathrm{xy}$<br/>$=x(100-x)=100 x-x^{2}$<br/>$\because \frac{\mathrm{d} \mathrm{A}}{\mathrm{dx}}=100-2 \mathrm{x}$<br/>Put $\frac{\mathrm{dA}}{\mathrm{dx}}=0$ for maxima or minima<br/>$100-2 x=0$<br/>$\Rightarrow \mathrm{x}=50 \Rightarrow \mathrm{y}=50$<br/>Now, $\frac{\mathrm{d}^{2} \mathrm{~A}}{\mathrm{dx}^{2}}=-2   &lt; 0$, which shows maximum, indepenent of values of $\mathrm{x}$ and $\mathrm{y}$, but only when they are equal.<br/>$\therefore \quad$ Ais maximumat $x=50$.<br/>Hence, required area $=50(100-50)=50 \times 50=2500 \mathrm{~m}^{2}$</div>


title: what-is-the-area-of-the-parabola-x-2-y-bounded-by-the-line-y-1
slug: what-is-the-area-of-the-parabola-x-2-y-bounded-by-the-line-y-1-57697
question: <div class="question">What is the area of the parabola $x^{2}=y$ bounded by the line $y=1$</div>
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$ square unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2}{3}$ square unÄ±t</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{4}{3}$ square units</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 square units</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/1gQu5aXC9gjdlgxCKEXFKA2nF8uM4O-tIMrxZj5d9tY.original.fullsize.png"/><br/><br/>Area of $\mathrm{ABCD}=2 \times 1=2 \mathrm{sq}$. units<br/>Area of $\mathrm{AOD}=\int_{-1}^{1} \mathrm{x}^{2} \mathrm{dx}=\left.\frac{\mathrm{x}^{3}}{3}\right|_{-1} ^{1}=\frac{2}{3} \mathrm{sq} .$ units<br/>Required area $=2-\frac{2}{3}=\frac{4}{3}$ sq. units</div>


title: what-is-the-area-of-the-parabola-y-2-x-bounded-by-its-latus-rectum-1
slug: what-is-the-area-of-the-parabola-y-2-x-bounded-by-its-latus-rectum-1-82688
question: <div class="question">What is the area of the parabola $y^{2}=x$ bounded by its latus rectum?</div>
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{12}$ square unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{6}$ square 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">C</span> <span class="option-data">$\frac{1}{3}$ square unit</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">$\frac{1}{6}$ square unit</span> </div>

answer:
<div class="solution">Required Area $=2 \int_{0}^{\frac{1}{4}} \sqrt{\mathrm{x}} \mathrm{dx}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/3fihbIQEBpMMPyVFPBs88rjj72pQnmu16M_nZrmkuL4.original.fullsize.png"/><br/><br/>$=2 \cdot \frac{2}{3}\left[\mathrm{x}^{3 / 2}\right]_{0}^{1 / 4}=\frac{4}{3}\left[\frac{1}{8}-0\right]=\frac{1}{6} \mathrm{sq} .$ unit.</div>


title: what-is-the-area-of-the-parabola-y-4-b-x-bounded-by-its-latus-rectum
slug: what-is-the-area-of-the-parabola-y-4-b-x-bounded-by-its-latus-rectum-72574
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the parabola $y^{\prime}-4 b x$ bounded by its latus rectum?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 b^{2} / 3$ square unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 b^{2} / 3$ square unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$b^{2}$ square unit</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$8 b^{2} / 3$ square uni</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">$8 b^{2} / 3$ square uni</span> </div>

answer:
<div class="solution">Given equation $y^{2}=4 b x=2 \int_{0}^{b} \sqrt{4 b x} d x$<br/>$=4 \sqrt{\mathrm{b}} \times \frac{2}{3}\left[\mathrm{x}^{\frac{3}{2}}\right]_{0}^{\mathrm{b}}=\frac{8 \sqrt{\mathrm{b}}}{3}\left[\mathrm{~b}^{\frac{3}{2}}-0\right]$<br/>$\therefore$ area of parabola bounded by its latus rectum<br/>$=\frac{8 \mathrm{~b}^{2}}{3}$ sq. units</div>


title: what-is-the-area-of-the-parallelogram-having-diagonals-3-i-j-2-k-and-i-3-j-4-k
slug: what-is-the-area-of-the-parallelogram-having-diagonals-3-i-j-2-k-and-i-3-j-4-k-73546
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the parallelogram having diagonals $3 \hat{i}+\hat{j}-2 \hat{k}$ and $\hat{i}-3 \hat{j}+4 \hat{k} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \sqrt{5}$ square units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \sqrt{5}$ square units</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$5 \sqrt{3}$ square units</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">$15 \sqrt{2}$ square units</span> </li> </ul>

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

answer:
<div class="solution">Diagonal $d_{1}, \overline{A C}=3 i+j-2 k$<br/>Diagonal $d_{2}, \overrightarrow{B D}=i-3 j+4 k$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/znXdNs59KRK-NdZjxx-jUR79bmRb9hf9k0UHktDjcgE.original.fullsize.png"/><br/><br/>Area of parallelogram is $\frac{1}{2}\left|\overrightarrow{d_{1}} \times \overrightarrow{d_{2}}\right|$<br/>$\begin{aligned} \text { Hence area } &amp;=\frac{1}{2}\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ 3 &amp; 1 &amp; -2 \\ 1 &amp; -3 &amp; 4\end{array}\right| \\ &amp;=\frac{1}{2}|[\hat{i}(4-6)-\hat{j}(12+2)+\hat{k}(-9-1)]| \\ &amp;=\frac{1}{2}|-2 \hat{i}-14 \hat{j}-10 \hat{k}| \\ &amp;=\frac{1}{2} \sqrt{4+196+100} \\ &amp;=\frac{10 \sqrt{3}}{2}=5 \sqrt{3} \text { square units } \end{aligned}$</div>


title: what-is-the-area-of-the-portion-of-the-curve-y-sin-x-lying-between-x-0-y-0-and-x-2
slug: what-is-the-area-of-the-portion-of-the-curve-y-sin-x-lying-between-x-0-y-0-and-x-2-88007
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the portion of the curve $y=\sin x$, lying between $\mathrm{x}=0, \mathrm{y}=0$ and $\mathrm{x}=2 \pi$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 square unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 square units</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 square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8 square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 square units</span> </div>

answer:
<div class="solution">$\begin{aligned} \text { Required area }=\int_{0}^{2 \pi} \sin x \mathrm{dx} \\=&amp;-\left.\cos \mathrm{x}\right|_{0} ^{2 \pi}=-\cos 2 \pi-(-\cos 0) \\=-\cos (\pi+\pi)+1=-[-\cos \pi]+1 \\=+\cos \left(\frac{\pi}{2}+\frac{\pi}{2}\right)+1=\sin \frac{\pi}{2}+1=1+1=2 \mathrm{sq} . \text { units. } \end{aligned}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-area-of-the-rectangle-having-vertices-a-b-c-and-d-with-positive-vectors-i-2-1-j-4-k-i-2-1-j-4-k-i-2-1-j-4-k-and-i-2-1-j-4-k

slug:
what-is-the-area-of-the-rectangle-having-vertices-a-b-c-and-d-with-positive-vectors-i-2-1-j-4-k-i-2-1-j-4-k-i-2-1-j-4-k-and-i-2-1-j-4-k-72569

question:
<div class="question">What is the area of the rectangle having vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and<br/>$\mathrm{D}$ with positive vectors $-\hat{\mathrm{i}}+\frac{1}{2} \hat{\mathrm{j}}+4 \hat{\mathrm{k}}, \hat{\mathrm{i}}+\frac{1}{2} \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$<br/>$\hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $-\hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{j}}+4 \hat{\mathrm{k}} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 square unit</span> </div>

answer:
<div class="solution">From the given vectors we can conclude that<br/>$\mathrm{A}\left(-1, \frac{1}{2}, 4\right), \mathrm{B}\left(1, \frac{1}{2}, 4\right), \mathrm{C}\left(1, \frac{-1}{2}, 4\right), \mathrm{D}\left(-1,-\frac{1}{2}, 4\right)$<br/>Length $=\mathrm{AB}=2, \mathrm{BC}=1$<br/>Area $=\mathrm{AB} \times \mathrm{BC}=2$</div>


title: what-is-the-area-of-the-rectangle-of-which-r-a-i-b-j-is-a-semidiagonal
slug: what-is-the-area-of-the-rectangle-of-which-r-a-i-b-j-is-a-semidiagonal-29231
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the rectangle of which $\overrightarrow{\mathrm{r}}=\mathrm{a} \overrightarrow{\mathrm{i}}+\mathrm{b} \overrightarrow{\mathrm{j}}$ is a semidiagonal?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a^{2}+b^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2\left(a^{2}+b^{2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4\left(a^{2}+b^{2}\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{ab}$</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">$4 \mathrm{ab}$</span> </div>

answer:
<div class="solution">As given Semidiagonal is $\overrightarrow{\mathrm{r}}=\mathrm{a} \overrightarrow{\mathrm{i}}+\overrightarrow{\mathrm{b}}$<br/>So diagonal is $2 \overrightarrow{\mathrm{r}}=2 \mathrm{a} \overrightarrow{\mathrm{i}}+2 \mathrm{~b} \overrightarrow{\mathrm{j}}$<br/>$\Rightarrow$ Sides of rectangle are $2 \mathrm{a}$ and $2 \mathrm{~b}$ Hence, area of rectangle $=2 \mathrm{a} \times 2 \mathrm{~b}=4 \mathrm{ab}$.</div>


title: what-is-the-area-of-the-region-bounded-by-the-curve-f-x-1-4-x-2-x-2-2-and-the-x-axis
slug: what-is-the-area-of-the-region-bounded-by-the-curve-f-x-1-4-x-2-x-2-2-and-the-x-axis-80312
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the region bounded by the curve<br/>$f(x)=1-\frac{x^{2}}{4}, x \in[-2,2]$, and the $x$ -axis? $\quad$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{8}{3} \mathrm{sq}$ 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">B</span> <span class="option-data">$\frac{4}{3} \mathrm{sq}$ unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3} \mathrm{sq}$ uni</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{3}$ sq unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{8}{3} \mathrm{sq}$ unit</span> </div>

answer:
<div class="solution">Required area $=\int_{-2}^{2}\left(1-\frac{x^{2}}{4}\right) \mathrm{dx}$<br/>Since, $\left(1-\frac{\mathrm{x}^{2}}{4}\right)$ is an even function therefore<br/>$\int_{-2}^{2}\left(1-\frac{x^{2}}{4}\right) d x=2 \int_{0}^{2}\left(1-\frac{x^{2}}{4}\right) d x$<br/>$=2\left[\mathrm{x}-\frac{\mathrm{x}^{3}}{12}\right]_{0}^{2}=2\left[2-\frac{2^{3}}{12}\right]$<br/>$=2\left(2-\frac{2}{3}\right)=\frac{8}{3} \mathrm{sq}$ unit</div>


title: what-is-the-area-of-the-region-bounded-by-the-line-3-x-5-y-15-x-1-x-3-and-x-axis-in-sq-unit
slug: what-is-the-area-of-the-region-bounded-by-the-line-3-x-5-y-15-x-1-x-3-and-x-axis-in-sq-unit-56944
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the region bounded by the line $3 \mathrm{x}-5 \mathrm{y}=15$, $\mathrm{x}=1, \mathrm{x}=3$ and $\mathrm{x}$ -axis in sq unit?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{36}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{18}{5}$</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">$\frac{9}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{18}{5}$</span> </div>

answer:
<div class="solution">The given equation of line can be rewritten as<br/>$\frac{x}{5}-\frac{y}{3}=1$ and $y=\frac{3 x-15}{5}$<br/>$\begin{array}{ll}\therefore \quad &amp; \text { Required area }=\int_{1}^{3} \mathrm{ydx} \\ &amp; =\int_{1}^{3}\left(\frac{3 \mathrm{x}-15}{5}\right) \mathrm{d} \mathrm{x}=\frac{1}{5} \int_{1}^{3}(3 \mathrm{x}-15) \mathrm{d} \mathrm{x} \\ &amp; =\frac{1}{5}\left[\frac{3 \mathrm{x}^{2}}{2}-15 \mathrm{x}\right]_{1}^{3}=\frac{1}{5}\left[\frac{27}{2}-45-\frac{3}{2}+15\right]\end{array}$<br/>$=\frac{1}{5}\left[\frac{24}{3}-30\right]=\frac{1}{5}[12-30]$<br/>$=\frac{-18}{5}=\frac{18}{5}$ sq. unit (neglecting -ve sign)</div>


title: what-is-the-area-of-the-region-bounded-by-the-lines-y-y-0-and-x-4
slug: what-is-the-area-of-the-region-bounded-by-the-lines-y-y-0-and-x-4-36786
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the region bounded by the lines $y=$ $\mathrm{y}=0$ and $\mathrm{x}=4 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4 square units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">8 square units</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">12 squate units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16 syuare urtib</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/_npUc7yJGB6vEwm9HEeGg86m312CNksOHaEWW4G7ExM.original.fullsize.png"/><br/><br/>Area of shaded region $=\frac{1}{2} \times \mathrm{AB} \times \mathrm{BC}$<br/>$=\frac{1}{2} \times 4 \times 4=8$ sq. units.</div>


title: what-is-the-area-of-the-region-bounded-by-the-parabolas-y-2-6-x-1-and-y-2-3-x
slug: what-is-the-area-of-the-region-bounded-by-the-parabolas-y-2-6-x-1-and-y-2-3-x-54436
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the region bounded by the parabolas $y^{2}=6(x-1)$ and $y^{2}=3 x ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4 \sqrt{6}}{3}$</span> </div>

answer:
<div class="solution">Given parabolas are $y^{2}=6(x-1)$....(1) and $y^{2}=3 x$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/bG66zK-A-2fPsO6cUVi0ARYYw9kfT-89EHRz3jE_5HE.original.fullsize.png"/><br/><br/>$y^{2}=6(x-1)$<br/>$\Rightarrow \quad 6 x-6=y^{2}$<br/>$\Rightarrow x=\frac{y^{2}+6}{6}=\frac{y^{2}}{6}+1$<br/>Also, $\mathrm{y}^{2}=3 \mathrm{x} \Rightarrow \mathrm{x}=\frac{\mathrm{y}^{2}}{3}$.<br/>Solving $(1),(2), 3 x=6(x-1)$<br/>$\Rightarrow 3 x=6 x-6$<br/>$\Rightarrow 3 x=6 \Rightarrow x=2 .$<br/>$y^{2}=6 \Rightarrow y=\pm \sqrt{6}$<br/>$\therefore$ Area $=$<br/>$\int_{-\sqrt{6}}^{\sqrt{6}}\left(\frac{y^{2}}{6}+1-\frac{y^{2}}{3}\right) \cdot d y=2 \int_{0}^{\sqrt{6}}\left(1-\frac{y^{2}}{6}\right) d y=2\left[y-\frac{y^{3}}{18}\right]_{0}^{\sqrt{6}}$<br/>$=2\left[\sqrt{6}-\frac{(\sqrt{6})^{3}}{18}\right]=2 \times \frac{2 \sqrt{6}}{3}=\frac{4 \sqrt{6}}{3}$</div>


tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-area-of-the-region-bounded-by-x-axis-the-curve-y-f-x-and-the-two-ordinates-x-1-and-x-2-3-consider-the-function-f-x-x-1-x-2-where-x-r

slug:
what-is-the-area-of-the-region-bounded-by-x-axis-the-curve-y-f-x-and-the-two-ordinates-x-1-and-x-2-3-consider-the-function-f-x-x-1-x-2-where-x-r-51459

question:
<div class="question">What is the area of the region bounded by $x$ -axis, the curve<br/>$\mathrm{y}=\mathrm{f}(\mathrm{x})$ and the two ordinates $\mathrm{x}=1$ and $\mathrm{x}=\frac{3}{2}$ ?<br/>Consider the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|+\mathrm{x}^{2}$, where $\mathrm{x} \in \mathbf{R}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{12}$ square unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7}{12}$ square unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$ square unit</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{11}{12}$ square 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{11}{12}$ square unit</span> </div>

answer:
<div class="solution">Area required for given region is<br/>$\begin{aligned} A_{2} &amp;=\int_{1}^{3 / 2} f(x) d x \\ &amp;=\int_{1}^{3 / 2}\left[x^{2}+x-1\right] d x \\ &amp;=\left[\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-x\right]_{1}^{3 / 2} \\ A_{2} &amp;=\left(\frac{1}{3}\left(\frac{27}{8}\right)+\frac{1}{2}\left(\frac{9}{4}\right)-\frac{3}{2}\right)-\left(\frac{1}{3}(1)+\frac{1}{2}(1)-1\right) \\ A_{2} &amp;=\frac{11}{12} \quad \text { square unit } \end{aligned}$</div>


tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-area-of-the-region-bounded-by-xaxis-the-curve-y-f-x-and-the-two-ordinates-x-2-1-and-x-1-consider-the-function-f-x-x-1-x-2-where-x-r

slug:
what-is-the-area-of-the-region-bounded-by-xaxis-the-curve-y-f-x-and-the-two-ordinates-x-2-1-and-x-1-consider-the-function-f-x-x-1-x-2-where-x-r-16006

question:
<div class="question">What is the area of the region bounded by x-axis, the curve<br/>$\mathrm{y}=\mathrm{f}(\mathrm{x})$ and the two ordinates $\mathrm{x}=\frac{1}{2}$ and $\mathrm{x}=1 ?$<br/>Consider the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|+\mathrm{x}^{2}$, where $\mathrm{x} \in \mathbf{R}$.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{5}{12}$ square 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">B</span> <span class="option-data">$\frac{5}{6}$ square unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{-}$ square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2 square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{12}$ square unit</span> </div>

answer:
<div class="solution">$\quad f(x)=\left\{\begin{array}{ll}x^{2}+x-1 &amp; x \geq 1 \quad \forall x \in R \\ x^{2}-x+1 &amp; x   &lt; 1\end{array}\right.$<br/>Hence area required for given region is<br/>$\begin{aligned} A_{1} &amp;=\int_{1 / 2}^{1} f(x) d x \\ &amp;=\int_{1 / 2}^{1}\left(x^{2}-x+1\right) d x \\ &amp;=\left[\frac{1}{3} x^{3}-\frac{x^{2}}{2}+x\right]_{1 / 2}^{1} \\ &amp;=\left(\frac{1}{3} \times 1-\frac{1}{2}+1\right)-\left(\frac{1}{3} \times \frac{1}{8}-\frac{1}{2} \times \frac{1}{4}+\frac{1}{2}\right) \\ A_{1}=\frac{5}{12} &amp; \text { square unit. } \end{aligned}$</div>


title: what-is-the-area-of-the-region-cnclosed-by-y-2-x-and-y-4
slug: what-is-the-area-of-the-region-cnclosed-by-y-2-x-and-y-4-63886
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the region cnclosed by $\mathrm{y}-2|\mathrm{x}|$ and $\mathrm{y}=4 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2 square unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4 square unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">8 square 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">16 squareunit</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/ZmWOHIsXe2fFKeO6r4MjvF7DixYXpr0EcaOQAnqN11I.original.fullsize.png"/><br/><br/>Required Area $=2 \int_{y=0}^{4} \frac{y}{2} d y=\left.\frac{y^{2}}{2}\right|_{0} ^{4}=\frac{16}{2}=8$ sq. unit.<br/>or Area $=2 \times 4=8$ sq. unit.</div>


title: what-is-the-area-of-the-triangle-bounded-by-the-side-x-0-y-0-and-x-y-2
slug: what-is-the-area-of-the-triangle-bounded-by-the-side-x-0-y-0-and-x-y-2-31821
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle bounded by the side $\mathrm{x}=0$ $\mathrm{y}=0$, and $\mathrm{x}+\mathrm{y}=2 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 square unit</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Z-Hpnsao_y9kZZ67AbHvASvflp2i1ze7oZAoK0eThsI.original.fullsize.png"/><br/><br/>Area of $\Delta \mathrm{OAB}=\frac{1}{2} \times \mathrm{OB} \times \mathrm{OA}$<br/>$=\frac{1}{2} \times 2 \times 2=2$ square unit.</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
what-is-the-area-of-the-triangle-formed-by-the-lines-joining-the-vertex-of-the-parabola-x-2-12-y-to-the-ends-of-the-latus-rectum-1

slug:
what-is-the-area-of-the-triangle-formed-by-the-lines-joining-the-vertex-of-the-parabola-x-2-12-y-to-the-ends-of-the-latus-rectum-1-35330

question:
<div class="question">What is the area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12 y$ to the ends of the latus rectum?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9 square units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12 square units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">14 square units</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">18 square units</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">18 square units</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/xCYJK4UfmzdX0rUNq7Kg4NK9nMGg3L0Qth-dnxyRwjA.original.fullsize.png"/><br/><br/>Given parabola is $\mathrm{x}^{2}=12 y$ which is of the form $\mathrm{x}^{2}=4 a y$.<br/>$\Rightarrow 4 a=12 \Rightarrow a=3$<br/>Now, $L M$ is the latus rectum whose length $=4 a=4 \times 3$<br/>=12<br/>So, area of $\Delta L M V=\frac{1}{2} \times L M \times V F$.<br/>$=\left(\frac{1}{2} \times 12 \times 3\right)$ sq. unit<br/>$=18$ square unit</div>


tags: ['Mathematics', 'Parabola', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-area-of-the-triangle-formed-by-the-lines-joining-the-vertex-of-the-parabola-x-2-12-y-to-the-ends-of-the-latus-rectum

slug:
what-is-the-area-of-the-triangle-formed-by-the-lines-joining-the-vertex-of-the-parabola-x-2-12-y-to-the-ends-of-the-latus-rectum-41872

question:
<div class="question">What is the area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12 y$ to the ends of the latus rectum?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9 square units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12 square units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">14 square units</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">18 square units</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">18 square units</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/xCYJK4UfmzdX0rUNq7Kg4NK9nMGg3L0Qth-dnxyRwjA.original.fullsize.png"/><br/><br/>Given parabola is $\mathrm{x}^{2}=12 y$ which is of the form $\mathrm{x}^{2}=4 a y$.<br/>$\Rightarrow 4 a=12 \Rightarrow a=3$<br/>Now, $L M$ is the latus rectum whose length $=4 a=4 \times 3$<br/>=12<br/>So, area of $\Delta L M V=\frac{1}{2} \times L M \times V F$.<br/>$=\left(\frac{1}{2} \times 12 \times 3\right)$ sq. unit<br/>$=18$ square unit</div>


title: what-is-the-area-of-the-triangle-formed-by-the-lines-y-x-0-y-x-0-x-c-1
slug: what-is-the-area-of-the-triangle-formed-by-the-lines-y-x-0-y-x-0-x-c-1-43174
tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle formed by the lines $y-x=0, y+x=0, x=c ?$</div>

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

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

answer:
<div class="solution">The shaded portion shows the required area. Since the area is symmetric about the $x$ -axis therefore Required area $=2$ area $(\Delta A O D)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/MmJ0uktr-Ms0vHKTNvpsq0OYBDG2yoBEJYqw8yGe3JI.original.fullsize.png"/><br/><br/>$$<br/>\begin{array}{l}<br/>=2 \times \frac{1}{2} \times \text { base } \times \text { height } \\<br/>=2 \times \frac{1}{2} \times O D \times A D \\<br/>=c \times c \\<br/>=c^{2}<br/>\end{array}<br/>$$</div>


title: what-is-the-area-of-the-triangle-oab-where-o-is-the-origin-oa-3-i-j-k-and-ob-2-i-j-3-k
slug: what-is-the-area-of-the-triangle-oab-where-o-is-the-origin-oa-3-i-j-k-and-ob-2-i-j-3-k-57685
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle OAB where $\mathrm{O}$ is the origin, $\overrightarrow{\mathrm{OA}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OB}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \sqrt{6}$ square unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{5 \sqrt{6}}{2}$ square 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">C</span> <span class="option-data">$\sqrt{6}$ square unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{30}$ square unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5 \sqrt{6}}{2}$ square unit</span> </div>

answer:
<div class="solution">Area of $\Delta \mathrm{OAB}=\frac{1}{2}|\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}|$<br/>$\therefore \overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}=\left|\begin{array}{ccc}\hat{i} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ 3 &amp; -1 &amp; 1 \\ 2 &amp; 1 &amp; -3\end{array}\right|$<br/>$=\hat{\mathrm{i}}[3-1]-\hat{\mathrm{j}}[-9-2]+\hat{\mathrm{k}}[3+2]$<br/>$=2 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$<br/>$\therefore|\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}|=\sqrt{2^{2}+11^{2}+5^{2}}=\sqrt{150}=5 \sqrt{6}$<br/>$\therefore$ Required area $=\frac{1}{2} \times 5 \sqrt{6}=\frac{5 \sqrt{6}}{2} \mathrm{sq} .$ units</div>


