# questions

tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-are-the-roots-of-the-equation-x-2-mx-m-0-m-0-then-which-one-of-the-following-statements-is-correct

slug:
if-are-the-roots-of-the-equation-x-2-mx-m-0-m-0-then-which-one-of-the-following-statements-is-correct-36390

question:
<div class="question">If $\alpha, \beta$ are the roots of the equation $\ell \mathrm{x}^{2}-\mathrm{mx}+\mathrm{m}=0$, $\ell \neq \mathrm{m}, \ell \neq 0$, then which one of the following statements is<br/>correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$</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">$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{\mathrm{m}}{\ell}}=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{\frac{\alpha+\beta}{\alpha \beta}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The arithmetic mean of $\alpha$ and $\beta$ is the same as their geometric mean</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$</span> </div>

answer:
<div class="solution">As given : $\alpha$ and $\beta$ are the roots of the quadratio equation $\ell \mathrm{x}^{2}-\mathrm{mx}+\mathrm{m}=0$<br/>So, sum of roots,<br/>$\alpha+\beta=\frac{\mathrm{m}}{\ell}$ and product of roots, $\alpha \beta=\frac{\mathrm{m}}{\ell}$<br/>$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}=\frac{\alpha+\beta}{\sqrt{\alpha \beta}}=\frac{\mathrm{m} / \ell}{\sqrt{\mathrm{m} / \ell}}$<br/>$\Rightarrow \sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}=\sqrt{\frac{\mathrm{m}}{\ell}}=0 \Rightarrow \sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$</div>


title: if-are-the-roots-of-the-equation-x-2-x-2-0-then-what-is-10-10-10-10-equal-to
slug: if-are-the-roots-of-the-equation-x-2-x-2-0-then-what-is-10-10-10-10-equal-to-31534
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1024</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha, \beta$ are the roots of the equation $x^{2}+x+2=0$, then<br/>what is $\frac{\alpha^{10}+\beta^{10}}{\alpha^{-10}+\beta^{-10}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4096</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2048</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1024</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">512</span> </li> </ul>

answer:
<div class="solution">Here, $\alpha$ and $\beta$ are the roots of the equation $\alpha+\beta=-1$<br/>$\alpha \beta=2$<br/>$\frac{\alpha^{10}+\beta^{10}}{\alpha^{-10}+\beta^{-10}}=\frac{\alpha^{10}+\beta^{10}}{\frac{1}{\alpha^{10}}+\frac{1}{\beta^{10}}}$<br/>$=\frac{\left(\alpha^{10}+\beta^{10}\right)(\alpha \beta)^{10}}{\left(\alpha^{10}+\beta^{10}\right)}=(\alpha \beta)^{10}$<br/>$\therefore(\alpha \beta)^{10}=2^{10}=1024$</div>


title: if-are-the-roots-of-the-quadratic-equation-x-2-x-1-0-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-are-the-roots-of-the-quadratic-equation-x-2-x-1-0-then-which-one-of-the-following-is-correct-69543

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

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\alpha^{6}-\beta^{6}\right)=0$</span> </div>

answer:
<div class="solution">Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-x+1=0$.<br/>$\alpha+\beta=-(-1)=1$...(i)<br/>$\alpha \beta=1$<br/>Now, $\alpha-\beta=\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}$<br/>$=\sqrt{(1)^{2}-4(1)}=\sqrt{-3}=\sqrt{3} i$...(ii)<br/>On solving(i), (ii) we get<br/>$\alpha=\frac{1+i \sqrt{3}}{2}$ and $\beta=\frac{1-i \sqrt{3}}{2}$<br/>$\Rightarrow \alpha=\frac{1}{2}+i \cdot \frac{\sqrt{3}}{2}$ and $\beta=\frac{1}{2}-i \cdot \frac{\sqrt{3}}{2}$<br/>$\Rightarrow \quad \alpha=\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}$ and $\beta=\cos \frac{\pi}{3}-i \sin \frac{\pi}{3}$<br/>(a) $\alpha^{4}-\beta^{4}=\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}-\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}$<br/>$=2 i \sin \frac{4 \pi}{3}$<br/>(By Demoiver's thm)<br/>$\Rightarrow \alpha^{4}-\beta^{4}$ is not real.<br/>(b) $2\left(\alpha^{5}+\beta^{5}\right)$<br/>$=2\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}+\cos \frac{5 \pi}{3}-i \sin \frac{5 \pi}{3}\right)$<br/>$=2.2 \cos \frac{5 \pi}{3}=4 \cdot \frac{1}{2}=2$<br/>Now, $(\alpha \beta)^{5}=1$<br/>$\Rightarrow 2\left(\alpha^{5}+\beta^{5}\right) \neq(\alpha \beta)^{5}$<br/>(c) $\alpha^{6}-\beta^{6}=\cos \frac{6 \pi}{3}+i \sin \frac{6 \pi}{3}-\cos \frac{6 \pi}{3}+i \sin \frac{6 \pi}{3}$<br/>$=2 i \sin 2 \pi=0$<br/>Hence, option (c) is correct.</div>


title: if-are-the-roots-of-x-2-p-x-q-0-and-are-the-roots-of-x-2-px-r-0-then-what-is-equal-to
slug: if-are-the-roots-of-x-2-p-x-q-0-and-are-the-roots-of-x-2-px-r-0-then-what-is-equal-to-55949
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha, \beta$ are the roots of $x^{2}+p x-q=0$ and $\gamma, \delta$ are the roots of $\mathrm{x}^{2}-\mathrm{px}+\mathrm{r}=0$ then what is $(\beta+\gamma)(\beta+\delta)$ equal to $?$</div>

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

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

answer:
<div class="solution">Since, $\alpha \&amp; \beta$ are the roots of $x^{2}+p x-q=0$, then, $\alpha+\beta=-\mathrm{p}$...(1)<br/>and $\alpha \beta=-\mathrm{q}$...(2)<br/>(2) put the value of $\alpha$ from $(2)$ in $(1)$<br/>$-\frac{q}{\beta}+\beta=-p \Rightarrow-q+\beta^{2}=-p \beta$<br/>$\Rightarrow \beta^{2}=q-p \beta$...(3)<br/>Since $\gamma \&amp; \delta$ are the roots of $x^{2}-p x+r=0$, then, $\gamma+\delta=\mathrm{p}, \gamma \delta=\mathrm{r}$<br/>Now, $(\beta+\gamma)(\beta+\delta)=\beta^{2}+\beta \delta+\beta \gamma+\gamma \delta$.<br/>$=\beta^{2}+\beta[\gamma+\delta]+\gamma \delta .=\mathrm{q}-\mathrm{p} \beta+\mathrm{p} \beta+\mathrm{r}=\mathrm{q}-\mathrm{r}$</div>


title: if-are-the-rootsofthe-equations-x-2-2-x-1-0-then-what-is-the-value-of-2-2-2-2
slug: if-are-the-rootsofthe-equations-x-2-2-x-1-0-then-what-is-the-value-of-2-2-2-2-25099
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha, \beta$ are the rootsofthe equations $x^{2}-2 x-1=0$, then what is the value of $\alpha^{2} \beta^{-2}+\alpha^{-2} \beta^{2}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">30</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">Since, $\alpha$ and $\beta$ are the roots of the equation $\mathrm{x}^{2}-2 \mathrm{x}-1=0$, then<br/>Sum of roots, $\alpha+\beta=2$ and<br/>product of the roots $\alpha \beta=-1$ Since, $(\alpha+\beta)=\alpha^{2}+\beta^{2}+2 \alpha \beta$<br/>$\Rightarrow 4=\alpha^{2}+\beta^{2}-2$<br/>$\Rightarrow \alpha^{2}+\beta^{2}=6$<br/>Now, $\alpha^{2} \beta^{-2}+\alpha^{-2} \beta^{2}=\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{\alpha^{4}+\beta^{4}}{(\alpha \beta)^{2}}$<br/>$\Rightarrow\left(\alpha^{2}+\beta^{2}\right)^{2}=6^{2}$<br/>$\Rightarrow \alpha^{4}+\beta^{4}+2 \alpha^{2} \beta^{2}=36$<br/>$\begin{aligned} \Rightarrow \alpha^{4}+\beta^{4}+2 &amp;=36 &amp;(\because \alpha \beta=-1) \end{aligned}$<br/>$\Rightarrow \alpha^{4}+\beta^{4}=34$...(i)<br/>$\Rightarrow \frac{\alpha^{4}+\beta^{4}}{(\alpha \beta)^{2}}=\frac{34}{(-1)^{2}}=34$<br/>[Putting value of $\alpha^{4}+\beta^{4}=34$ from Equation (i)]</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-arethe-roots-of-a-x-2-2-b-x-c-0-and-are-the-roots-of-ax-2-2-bx-c-0-then-what-is-b-2-ac-b-2-ac-equal-to

slug:
if-arethe-roots-of-a-x-2-2-b-x-c-0-and-are-the-roots-of-ax-2-2-bx-c-0-then-what-is-b-2-ac-b-2-ac-equal-to-94567

question:
<div class="question">If $\alpha, \beta$ arethe roots of $a x^{2}+2 b x+c=0$ and $\alpha+\delta, \beta+\delta$ are the roots of $\mathrm{Ax}^{2}+2 \mathrm{Bx}+\mathrm{C}=0$, then what is $\left(\mathrm{b}^{2}-\mathrm{ac}\right) / \mathrm{B}^{2}-\mathrm{AC}$<br/>equal to?</div>

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

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

answer:
<div class="solution">Since, $\alpha$ and $\beta$ are the roots of $a x^{2}+2 b x+c=0$<br/>so, $k+\beta=-\frac{2 b}{a}$ and $\alpha \beta=\frac{c}{a}$<br/>$\delta$ and $\beta+\delta$ are theroots of $A x^{2}+2 B x+C=0$<br/>so, sum of the roots $=\alpha+\beta+2 \delta=-\frac{2 \mathrm{~B}}{\mathrm{~A}}$ and product<br/>ofthe roots $(\alpha+\delta)(\beta+\delta)=\frac{\mathrm{C}}{\mathrm{A}}$<br/>$\Rightarrow-\frac{2 b}{a}+2 \delta=-\frac{2 B}{A}$<br/>$\Rightarrow \delta=\frac{b}{a}-\frac{B}{A}$.. (i)<br/>and $(\alpha+\delta)(\beta+\delta)=\frac{\mathrm{C}}{\mathrm{A}}$<br/>$\Rightarrow \alpha \beta+(\alpha+\beta) \delta+\delta^{2}=\frac{\mathrm{C}}{\mathrm{A}}$...(ii)<br/>Putting value of $\delta$ from equation (i) in equation (ii),<br/>$\frac{\mathrm{c}}{\mathrm{a}}-\frac{2 \mathrm{~b}}{\mathrm{a}}\left(\frac{\mathrm{b}}{\mathrm{a}}-\frac{\mathrm{B}}{\mathrm{A}}\right)+\left(\frac{\mathrm{b}}{\mathrm{a}}-\frac{\mathrm{B}}{\mathrm{A}}\right)^{2}=\frac{\mathrm{C}}{\mathrm{A}}$<br/>$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}}-\frac{2 \mathrm{~b}^{2}}{\mathrm{a}^{2}}+\frac{2 \mathrm{bB}}{\mathrm{aA}}+\left(\frac{\mathrm{b}}{\mathrm{a}}\right)^{2}+\left(\frac{\mathrm{B}}{\mathrm{A}}\right)^{2}-\frac{2 \mathrm{bB}}{\mathrm{aA}}=\frac{\mathrm{C}}{\mathrm{A}}$<br/>$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}}-\left(\frac{\mathrm{b}}{\mathrm{a}}\right)^{2}+\left(\frac{\mathrm{B}}{\mathrm{A}}\right)^{2}=\frac{\mathrm{C}}{\mathrm{A}}$<br/>$\Rightarrow \frac{\mathrm{B}^{2}}{\mathrm{~A}^{2}}-\frac{\mathrm{C}}{\mathrm{A}}=\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{c}}{\mathrm{a}} \Rightarrow \frac{\mathrm{B}^{2}-\mathrm{AC}}{\mathrm{A}^{2}}=\frac{\mathrm{b}^{2}-\mathrm{ac}}{\mathrm{a}^{2}}$<br/>$\Rightarrow \frac{b^{2}-a c}{B^{2}-A C}=\left(\frac{a}{A}\right)^{2}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-at-any-instant-t-for-a-sphere-r-denotes-the-radius-s-denotes-the-surface-area-and-v-denotes-the-volume-then-what-is-d-t-d-v-equal-to

slug:
if-at-any-instant-t-for-a-sphere-r-denotes-the-radius-s-denotes-the-surface-area-and-v-denotes-the-volume-then-what-is-d-t-d-v-equal-to-74392

question:
<div class="question">If at any instant $t$, for a sphere, $r$ denotes the radius, $S$ denotes the surface area and $V$ denotes the volume, then<br/>what is $\frac{d V}{d t}$ equal to?</div>

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

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

answer:
<div class="solution">Surface area of sphere $\mathrm{S}=4 \pi r^{2}$ Differentiate both sides w.r.t. $\mathrm{t}^{\prime}$<br/>$\Rightarrow \frac{\mathrm{dS}}{\mathrm{dt}}=\frac{8 \pi \mathrm{rdr}}{\mathrm{dt}} \Rightarrow \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}=\frac{1}{8 \pi \mathrm{r}} \cdot \frac{\mathrm{d} \mathrm{s}}{\mathrm{dt}}$<br/>and Volume $=\mathrm{V}=\frac{4}{3} \pi \mathrm{r}^{3}$<br/>$\Rightarrow \quad \frac{\mathrm{dV}}{\mathrm{dt}}=\frac{4}{3} \pi \cdot 3 \mathrm{r}^{2} \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}=4 \pi \mathrm{r}^{2} \frac{\mathrm{d} \mathrm{r}}{\mathrm{dt}}$<br/>$\quad=\frac{4 \pi \mathrm{r}^{2}}{8 \pi \mathrm{r}} \cdot \frac{\mathrm{d} \mathrm{S}}{\mathrm{dt}}=\frac{1}{2} \mathrm{r} \frac{\mathrm{dS}}{\mathrm{dt}}$</div>


tags: ['Mathematics', 'Matrices', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-athetaleftbeginarraylr1-tan-theta-tan-theta-1endarrayright-and-a-bi-then-leftsec-2-thetaright-b-is-equal-to

slug:
if-athetaleftbeginarraylr1-tan-theta-tan-theta-1endarrayright-and-a-bi-then-leftsec-2-thetaright-b-is-equal-to-61274

question:
<div class="question">If \( A(\theta)=\left[\begin{array}{lr}1 &amp; \tan \theta \\ -\tan \theta &amp; 1\end{array}\right] \) and \( A B=I \), then \( \left(\sec ^{2} \theta\right) B \) is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( A(-\theta) \)</span> </div>

answer:
<div class="solution"><math><mi>A</mi><mi>B</mi><mo>=</mo><mi>I</mi><mo>â</mo><mi>Â </mi><mi>Â </mi><mi>B</mi><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mrow><msup><mrow><mi mathvariant="normal">tan</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi><mi mathvariant="normal">Â </mi></mrow></mrow></mrow></mfrac><mfenced close="]" open="[" separators="|"><mrow><mtable><mtr><mtd><mrow><maligngroup></maligngroup><mn>1</mn><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mo>-</mo><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup></maligngroup><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi></mrow></mrow><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mrow><msup><mrow><mi mathvariant="normal">sec</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi><mi mathvariant="normal">Â </mi></mrow></mrow></mrow></mfrac><mfenced close="]" open="[" separators="|"><mrow><mtable><mtr><mtd><mrow><maligngroup></maligngroup><mn>1</mn><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mo>-</mo><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup></maligngroup><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi></mrow></mrow><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math><br/><math><mo>â</mo><mi mathvariant="normal">Â </mi><mi mathvariant="normal">Â </mi><mi mathvariant="normal">Â </mi><mo>(</mo><mrow><mrow><msup><mrow><mi mathvariant="normal">sec</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi><mo>)</mo><mi mathvariant="normal">Â </mi><mi mathvariant="normal">B</mi><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mtable><mtr><mtd><mrow><maligngroup></maligngroup><mn>1</mn><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mo>-</mo><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup></maligngroup><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi mathvariant="normal">Î¸</mi></mrow></mrow><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mi>Â </mi><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mi mathvariant="normal">A</mi><mo>(</mo><mo>-</mo><mi mathvariant="normal">Î¸</mi><mo>)</mo></mrow></mrow></math></div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-b-1-b-2-b-3-are-three-consecutive-terms-of-an-arithmetic-progression-with-common-difference-d-0-then-what-is-the-value-of-d-for-which-b-3-2-b-2-b-3

slug:
if-b-1-b-2-b-3-are-three-consecutive-terms-of-an-arithmetic-progression-with-common-difference-d-0-then-what-is-the-value-of-d-for-which-b-3-2-b-2-b-3-60655

question:
<div class="question">If $\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}$ are three consecutive terms of an arithmetic progression with common difference $\mathrm{d}&gt;0$, then what is the<br/>value of $\mathrm{d}$ for which $\mathrm{b}_{3}^{2}=\mathrm{b}_{2} \mathrm{~b}_{3}+\mathrm{b}_{1} \mathrm{~d}+2 ?$</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">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">$b_{1}, b_{2}, b_{3}$ are in $A P$ with common difference $d$, so $b_{2}=b_{1}+d$ and $b_{3}=b_{1}+2 d$<br/>As given, $\mathrm{b}_{3}^{2}=\mathrm{b}_{2} \mathrm{~b}_{3}+\mathrm{b}_{1} \mathrm{~d}+2$<br/>$\Rightarrow\left(b_{1}+2 d\right)^{2}=\left(b_{1}+d\right)\left(b_{1}+2 d\right)+b_{1} d+2$<br/>$\Rightarrow \mathrm{b}_{1}^{2}+4 \mathrm{~d}^{2}+4 \mathrm{~b}_{1} \mathrm{~d}=\mathrm{b}_{1}^{2}+2 \mathrm{~b}_{1} \mathrm{~d}+\mathrm{b}_{1} \mathrm{~d}+2 \mathrm{~d}^{2}+\mathrm{b}_{1} \mathrm{~d}+2$<br/>$\Rightarrow 2 \mathrm{~d}^{2}=2$<br/>$\Rightarrow \mathrm{d}^{2}=1$<br/>$\Rightarrow \mathrm{d}=\pm 1$<br/>i.e. $\mathrm{d}=1$ or $-1$<br/>Since, $\mathrm{d}&gt;0,-1$ is discarded and $\mathrm{d}=1$</div>


title: if-b-3-2-1-2-4-1-0-0-0-then-what-is-adjoint-of-b-equal-to
slug: if-b-3-2-1-2-4-1-0-0-0-then-what-is-adjoint-of-b-equal-to-15490
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{B}=\left[\begin{array}{lll}3 &amp; 2 &amp; 0 \\ 2 &amp; 4 &amp; 0 \\ 1 &amp; 1 &amp; 0\end{array}\right]$, then what is adjoint of B equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{ccc}0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ -2 &amp; -1 &amp; 8\end{array}\right]$</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">$\left[\begin{array}{ccc}0 &amp; 0 &amp; -2 \\ 0 &amp; 0 &amp; -1 \\ 0 &amp; 0 &amp; 8\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{lll}0 &amp; 0 &amp; 2 \\ 0 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 0\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It does not exist</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{ccc}0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ -2 &amp; -1 &amp; 8\end{array}\right]$</span> </div>

answer:
<div class="solution">$B=\left[\begin{array}{lll}3 &amp; 2 &amp; 0 \\ 2 &amp; 4 &amp; 0 \\ 1 &amp; 1 &amp; 0\end{array}\right]$<br/>C factor matrix of<br/>$B=\left[\begin{array}{ccc}(4)(0)-(0)(1) &amp; -[(2)(0)-(0)(1)] &amp; (2)(1)-(4)(1) \\ -[2(0)-(0)(1)] &amp; 3(0)-(0)(1) &amp; -[(3)(1)-(2)(1)] \\ 2(0)-0(4) &amp; -[3(0)-2(0)] &amp; 3(4)-2(2)\end{array}\right]$<br/>$=\left[\begin{array}{ccc}0 &amp; 0 &amp; -2 \\ 0 &amp; 0 &amp; -1 \\ 0 &amp; 0 &amp; 8\end{array}\right]$<br/>Adjoint $B=\left[\begin{array}{ccc}0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ -2 &amp; -1 &amp; 8\end{array}\right]$</div>


title: if-b-a-1-b-c-1-a-1-c-1-then-a-b-c-are-in
slug: if-b-a-1-b-c-1-a-1-c-1-then-a-b-c-are-in-84839
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$, then $a, b, c$ are in</div>

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

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

answer:
<div class="solution">Suppose a, $\mathrm{b}$ and $\mathrm{c}$ are in $\mathrm{HP}$.<br/>$b=\frac{2 a c}{a+c}$<br/>Now, consider<br/>$\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{\frac{2 a c}{a+c}-a}+\frac{1}{\frac{2 a c}{a+c}-c}$<br/>$=\frac{a+c}{a(c-a)}+\frac{a+c}{c(a-c)}=\left(\frac{a+c}{c-a}\right)\left(\frac{1}{a}-\frac{1}{c}\right)$<br/>$=\frac{a+c}{c-a} \times \frac{c-a}{c a}=\frac{a+c}{c a}=\frac{1}{a}+\frac{1}{c}$<br/>Thus, our supposition is correct. Hence, $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are in $\mathrm{HP}$.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-b-and-c-are-the-position-vectors-of-the-points-b-and-c-respectively-then-the-position-vector-of-the-point-d-such-that-bd-4-bc-is

slug:
if-b-and-c-are-the-position-vectors-of-the-points-b-and-c-respectively-then-the-position-vector-of-the-point-d-such-that-bd-4-bc-is-49647

question:
<div class="question">If $\vec{b}$ and $\vec{c}$ are the position vectors of the points $B$ and $C$ respectively, then the position vector of the point D such<br/>that $\overrightarrow{\mathrm{BD}}=4 \overrightarrow{\mathrm{BC}}$ is\</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 4 \overrightarrow{\mathrm{c}}-3 \overrightarrow{\mathrm{b}}$</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}}$<br/>$\overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{c}}$<br/>$\overrightarrow{\mathrm{BD}}=4 \overrightarrow{\mathrm{BC}}$<br/>$\overrightarrow{\mathrm{BO}}+\overrightarrow{\mathrm{OD}}=4(\overrightarrow{\mathrm{BO}}+\overrightarrow{\mathrm{OC}})$<br/>$\overrightarrow{\mathrm{OD}}=3 \overrightarrow{\mathrm{BO}}+4 \overrightarrow{\mathrm{OC}}$<br/>$\overrightarrow{\mathrm{OD}}=4 \overrightarrow{\mathrm{OC}}-3 \overrightarrow{\mathrm{OB}}$<br/>$\overrightarrow{\mathrm{OD}}=\overrightarrow{4 \mathrm{c}}-3 \overrightarrow{\mathrm{b}}$</div>


title: if-b-c-x-a-c-a-x-b-c-b-x-c-3-then-what-is-the-value-of-x
slug: if-b-c-x-a-c-a-x-b-c-b-x-c-3-then-what-is-the-value-of-x-65245
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">a+b+c</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{x-a}{b+c}+\frac{x-b}{c+a}+\frac{x-c}{c+b}=3$, then what is the value of $x$ ?</div>

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

answer:
<div class="solution">$\frac{x-a}{b+c}+\frac{x-b}{c+a}+\frac{x-c}{a+b}=3$<br/>Such an equation is possible only, if<br/>$\frac{x-a}{b+c}=\frac{x-b}{c+a}=\frac{x-c}{a+b}=1$<br/>$\Rightarrow x=a+b+c$</div>


title: if-b-is-a-nonsingular-matrix-and-a-is-a-square-matrix-then-the-value-of-det-b-1-ab-is-equal-to
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-b-is-a-nonsingular-matrix-and-a-is-a-square-matrix-then-the-value-of-det-b-1-ab-is-equal-to-92124

question:
<div class="question">If $\mathrm{B}$ is a non-singular matrix and $\mathrm{A}$ is a square matrix, then the value of det $\left(\mathrm{B}^{-1} \mathrm{AB}\right)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\operatorname{det}(\mathrm{B})$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\operatorname{det}(\mathrm{A})$</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">$\operatorname{det}\left(\mathrm{B}^{-1}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\operatorname{det}\left(\mathrm{A}^{-1}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\operatorname{det}(\mathrm{A})$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp;\left|\mathrm{B}^{-1} \mathrm{AB}\right|=\left|\mathrm{B}^{-1}\right||\mathrm{A}||\mathrm{B}| \\ &amp;=\frac{1}{|\not{B} \mid}|\mathrm{A}||\not{B}| \quad \because \quad\left|\mathrm{B}^{-1}\right|=\frac{1}{|\mathrm{~B}|} \\ &amp;=|\mathrm{A}| \end{aligned}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-be-angles-which-the-vector-r-i-2-j-k-makes-with-the-coordinate-axes-then-what-is-the-value-of-sin-2-sin-2-sin-2

slug:
if-be-angles-which-the-vector-r-i-2-j-k-makes-with-the-coordinate-axes-then-what-is-the-value-of-sin-2-sin-2-sin-2-30584

question:
<div class="question">If $\alpha, \beta, \gamma$ be angles which the vector $\overrightarrow{\mathrm{r}}=\lambda \overrightarrow{\mathrm{i}}+2 \overrightarrow{\mathrm{j}}-\overrightarrow{\mathrm{k}}$ makes<br/>with the coordinate axes, then what is the value of $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma$ ?</div>

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

answer:
<div class="solution">We know that for any line that makes $\alpha, \beta$ and $\gamma$ angle With axis $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$<br/>$\Rightarrow \quad 1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$<br/>$\Rightarrow \quad \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">377</span> </div>
source_db: MarksBatch2_P1.db
title:
if-both-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-mean-of-x-1-5-2-x-2-5-2-x-50-5-2-is-1

slug:
if-both-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-mean-of-x-1-5-2-x-2-5-2-x-50-5-2-is-1-79110

question:
<div class="question">If both mean and the standard deviation of 50 observations $x_1, x_2, \ldots ., x_{50}$ are equal to 16 , then mean of $\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots \ldots .,\left(x_{50}-5\right)^2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">378</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">377</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">357</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">397</span> </li> </ul>

answer:
<div class="solution">$\vec{x}=16 \text { and S.D. }=16$<br/>if we subtract 5 from each observations mean will become $16-5=11$ and S.D. remains unchanged i.e., 16<br/>$\begin{aligned}<br/>&amp; \Rightarrow \sqrt{\frac{\sum\left(x_i-5\right)^2}{50}-11^2}=16 \\<br/>&amp; \Rightarrow \frac{\sum\left(x_i-5\right)^2}{50}=256+121=377<br/>\end{aligned}$<br/>hence, the required mean is 377</div>


tags: ['Mathematics', 'Statistics', 'MHT CET', 'MHT CET 2022 (08 Aug Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">377</span> </div>
source_db: MarksBatch2_P1.db
title:
if-both-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-mean-of-x-1-5-2-x-2-5-2-x-50-5-2-is

slug:
if-both-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-mean-of-x-1-5-2-x-2-5-2-x-50-5-2-is-70066

question:
<div class="question">If both mean and the standard deviation of 50 observations $x_1, x_2, \ldots ., x_{50}$ are equal to 16 , then mean of $\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots \ldots .,\left(x_{50}-5\right)^2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">378</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">377</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">357</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">397</span> </li> </ul>

answer:
<div class="solution">$\vec{x}=16 \text { and S.D. }=16$<br/>if we subtract 5 from each observations mean will become $16-5=11$ and S.D. remains unchanged i.e., 16<br/>$\begin{aligned}<br/>&amp; \Rightarrow \sqrt{\frac{\sum\left(x_i-5\right)^2}{50}-11^2}=16 \\<br/>&amp; \Rightarrow \frac{\sum\left(x_i-5\right)^2}{50}=256+121=377<br/>\end{aligned}$<br/>hence, the required mean is 377</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2019 (10 Apr Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-both-the-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-the-mean-of-x-1-4-2-x-2-4-2-x-50-4-2-is-1

slug:
if-both-the-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-the-mean-of-x-1-4-2-x-2-4-2-x-50-4-2-is-1-80573

question:
<div class="question">If both the mean and the standard deviation of <math><mn>50</mn></math> observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal"> </mi><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal"> </mi><mo>…</mo><mo>,</mo><mi mathvariant="normal"> </mi><msub><mi>x</mi><mn>50</mn></msub></math> are equal to <math><mn>16</mn></math>, then the mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>,</mo><mi mathvariant="normal"> </mi><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>,</mo><mi mathvariant="normal"> </mi><mo>…</mo><mo>,</mo><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mn>50</mn></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>525</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>480</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mn>400</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>380</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>400</mn></math></span> </div>

answer:
<div class="solution"><p>For observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mn>50</mn></msub></math></p><p>We have, mean, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mi> </mi></msubsup></mrow><mn>50</mn></mfrac><mo>=</mo><mn>16</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>i</mi></mfenced></math></p><p>And, standard deviation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><msqrt><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>-</mo><msup><mfenced><mover><mi>x</mi><mo>¯</mo></mover></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><mn>16</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>=</mo><msup><mn>16</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mover><mi>x</mi><mo>¯</mo></mover></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>=</mo><msup><mn>16</mn><mn>2</mn></msup><mo>+</mo><msup><mn>16</mn><mn>2</mn></msup><mo>=</mo><mn>512</mn><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mrow><mi>i</mi><mi>i</mi></mrow></mfenced></math></p><p>Now, the mean value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>,</mo><mo> </mo><msup><mfenced><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mo> </mo><msup><mfenced><mrow><msub><mi>x</mi><mn>50</mn></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></math> will be</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></mrow><mn>50</mn></mfrac><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mn>8</mn><mo> </mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>16</mn><mo>×</mo><mn>50</mn></mrow><mn>50</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>-</mo><mn>8</mn><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>50</mn></mfrac><mo>+</mo><mn>16</mn></math></p><p> Using <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>i</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>i</mi><mi>i</mi></mrow></mfenced><mo>,</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>512</mn><mo>-</mo><mn>8</mn><mo>×</mo><mn>16</mn><mo>+</mo><mn>16</mn><mo>=</mo><mn>400</mn><mo>.</mo></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-both-the-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-the-mean-of-x-1-4-2-x-2-4-2-x-50-4-2-is-2

slug:
if-both-the-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-the-mean-of-x-1-4-2-x-2-4-2-x-50-4-2-is-2-55238

question:
<div class="question">If both the mean and the standard deviation of <math><mn>50</mn></math> observations <math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi mathvariant="normal"> </mi><mo>â¦</mo><mo>,</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>50</mn></mrow></msub></math> are equal to <math><mn>16</mn></math>, then the mean of <math><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi mathvariant="normal"> </mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi mathvariant="normal"> </mi><mo>â¦</mo><mo>,</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>50</mn></mrow></msub><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>525</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>480</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mn>400</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>380</mn></math></span> </li> </ul>

answer:
<div class="solution">For observations <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo> </mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>50</mn></mrow></msub></math><br/>Mean, <math><mover><mrow><mi>x</mi></mrow><mrow><mo>-</mo></mrow></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi> </mi></mrow></msubsup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>=</mo><mn>16</mn></math> . . . . . (i)<br/>Variance, <math><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>-</mo><msup><mrow><mo>(</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>-</mo></mrow></mover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>⇒</mo><mfrac><mrow><mo>∑</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>=</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>-</mo></mrow></mover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>512</mn></math> . . . . . (ii)<br/>So, the mean value of <math><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mo> </mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>50</mn></mrow></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> will be<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msup><mrow><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mn>2</mn></msup></mrow><mn>50</mn></mfrac><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mn>8</mn><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>16</mn><mo>×</mo><mn>50</mn></mrow><mn>50</mn></mfrac></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>-</mo><mn>8</mn><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>50</mn></mfrac><mo>+</mo><mn>16</mn><mo>=</mo><mn>512</mn><mo>-</mo><mn>8</mn><mo>×</mo><mn>16</mn><mo>+</mo><mn>16</mn><mo>=</mo><mn>400</mn></math> (using (i) and (ii))</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-both-the-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-the-mean-of-x-1-4-2-x-2-4-2-x-50-4-2-is

slug:
if-both-the-mean-and-the-standard-deviation-of-50-observations-x-1-x-2-x-50-are-equal-to-16-then-the-mean-of-x-1-4-2-x-2-4-2-x-50-4-2-is-44797

question:
<div class="question">If both the mean and the standard deviation of <math><mn>50</mn></math> observations <math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi mathvariant="normal"> </mi><mo>â¦</mo><mo>,</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>50</mn></mrow></msub></math> are equal to <math><mn>16</mn></math>, then the mean of <math><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi mathvariant="normal"> </mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi mathvariant="normal"> </mi><mo>â¦</mo><mo>,</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>50</mn></mrow></msub><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>525</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>480</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mn>400</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>380</mn></math></span> </li> </ul>

answer:
<div class="solution">For observations <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo> </mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>50</mn></mrow></msub></math><br/>Mean, <math><mover><mrow><mi>x</mi></mrow><mrow><mo>-</mo></mrow></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi> </mi></mrow></msubsup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>=</mo><mn>16</mn></math> . . . . . (i)<br/>Variance, <math><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>-</mo><msup><mrow><mo>(</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>-</mo></mrow></mover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>⇒</mo><mfrac><mrow><mo>∑</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>=</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>-</mo></mrow></mover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mn>16</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>512</mn></math> . . . . . (ii)<br/>So, the mean value of <math><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mo> </mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>50</mn></mrow></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> will be<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msup><mrow><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>4</mn><mo>)</mo></mrow><mn>2</mn></msup></mrow><mn>50</mn></mfrac><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mn>8</mn><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>16</mn><mo>×</mo><mn>50</mn></mrow><mn>50</mn></mfrac></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>-</mo><mn>8</mn><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>50</mn></mfrac><mo>+</mo><mn>16</mn><mo>=</mo><mn>512</mn><mo>-</mo><mn>8</mn><mo>×</mo><mn>16</mn><mo>+</mo><mn>16</mn><mo>=</mo><mn>400</mn></math> (using (i) and (ii))</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-both-the-roots-of-the-equation-x-2-2-k-x-k-2-4-0-lie-between-3-and-5-then-which-one-of-the-following-is-correct

slug:
if-both-the-roots-of-the-equation-x-2-2-k-x-k-2-4-0-lie-between-3-and-5-then-which-one-of-the-following-is-correct-76137

question:
<div class="question">If both the roots of the equation $x^{2}-2 k x+k^{2}-4=0$ lie between $-3$ and 5 , then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2   &lt; \mathrm{k} &lt; 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-5   &lt; \mathrm{k} &lt; 3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-3   &lt; \mathrm{k} &lt; 5$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-1   &lt; \mathrm{k} &lt; 3$</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">$-1   &lt; \mathrm{k} &lt; 3$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; x^{2}-2 k x+k^{2}-4=0 \\ &amp; \Rightarrow(x-k)^{2}-2^{2}=0 \\ &amp; \Rightarrow(x-k-2)(x-k+2)=0 \\ &amp; \Rightarrow x=k+2, k-2 \\ &amp; \Rightarrow k+2   &lt; 5 \&amp; k-2&gt;-3 \\ &amp; \Rightarrow k &lt; 3 \&amp; k&gt;-1 \\ &amp; \Rightarrow-1 &lt; k &lt; 3 \end{aligned}$</div>


title: if-c-0-and-4-a-c-2-b-then-ax-2-bx-c-0-has-a-root-in-which-one-of-the-following-intervals
slug: if-c-0-and-4-a-c-2-b-then-ax-2-bx-c-0-has-a-root-in-which-one-of-the-following-intervals-11365
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{c}&gt;0$ and $4 \mathrm{a}+\mathrm{c}   &lt; 2 \mathrm{~b}$, then $\mathrm{ax}^{2}-\mathrm{bx}+\mathrm{c}=0$ has a root in<br/>which one of the following intervals?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(0,2)$</span> </div>

answer:
<div class="solution">Let $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}-\mathrm{bx}+\mathrm{c}$<br/>$\mathrm{f}(2)=4 \mathrm{a}-2 \mathrm{~b}+\mathrm{c}   &lt; 0 \quad$ (given)<br/>$f(0)=c&gt;0 \quad$ (given)<br/>So, we can see that sign of $f(x)$ changes, when $x$ changes from 0 to 2, so it has a root in the interval $(0,2) .$</div>


title: if-c-20-n-2-c-20-n-2-then-what-is-n-equal-to
slug: if-c-20-n-2-c-20-n-2-then-what-is-n-equal-to-21492
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{C}(20, \mathrm{n}+2)=\mathrm{C}(20, \mathrm{n}-2)$, then what is $\mathrm{n}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">Given, $\mathrm{C}(20, \mathrm{n}+2)=\mathrm{C}(20, \mathrm{n}-2)$<br/>$\Rightarrow{ }^{20} \mathrm{C}_{\mathrm{n}+2}={ }^{20} \mathrm{C}_{\mathrm{n}-2}$<br/>$\Rightarrow 20=\mathrm{n}+2+\mathrm{n}-2 \quad\left(\because{ }^{n} c_{r}={ }^{n} c_{s} \Rightarrow n=r+s\right)$<br/>$\Rightarrow 20=2 \mathrm{n}$<br/>$\Rightarrow \mathrm{n}=10$</div>


title: if-c-28-2-r-c-28-2-r-4-then-what-is-r-equal-to
slug: if-c-28-2-r-c-28-2-r-4-then-what-is-r-equal-to-26534
question: <div class="question">If $C(28,2 r)=C(28,2 r-4)$, then what is $r$ equal to ?</div>
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">$\mathrm{C}(28,2 \mathrm{r})=\mathrm{C}(28,2 \mathrm{r}-4)$<br/>${ }^{28} \mathrm{C}_{2 \mathrm{r}}={ }^{28} \mathrm{C}_{2 \mathrm{r}-4}$<br/>$\Rightarrow 2 \mathrm{r}+2 \mathrm{r}-4=28$<br/>$\Rightarrow 4 \mathrm{r}=32$<br/>$\Rightarrow \mathrm{r}=8$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-c-is-the-unit-vector-perpendicular-to-both-the-vectors-a-and-b-then-what-is-another-unit-vector-perpendicular-to-both-the-vectors-a-and-b

slug:
if-c-is-the-unit-vector-perpendicular-to-both-the-vectors-a-and-b-then-what-is-another-unit-vector-perpendicular-to-both-the-vectors-a-and-b-57314

question:
<div class="question">If $\vec{c}$ is the unit vector perpendicular to both the vectors<br/>$\overrightarrow{\text { a and }} \overrightarrow{\mathrm{b}}$, then what is another unit vector perpendicular<br/>to both the vectors a and $\overrightarrow{\mathrm{b}}$ ?</div>

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

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

answer:
<div class="solution">Let $\vec{c}$ is the unit vector perpendicular to both the<br/>vectors $\vec{a}$ and $\vec{b}$.<br/><br/>So, A unit vector which is perpendicular to both the<br/>vectors $\vec{a}$ and $\vec{b}$ is $\frac{(\vec{a} \times \vec{b})}{|\vec{a} \times \vec{b}|}$</div>


title: if-c-n-12-c-n-8-then-what-is-the-value-of-c-22-n
slug: if-c-n-12-c-n-8-then-what-is-the-value-of-c-22-n-16294
question: <div class="question">If $C(n, 12)=C(n, 8)$, then what is the value of $C(22, n)$ ?</div>
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">231</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">131</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">231</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">256</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">292</span> </li> </ul>

answer:
<div class="solution">Given $C(n, 12)=C(n, 8)$ $\begin{array}{l}\Rightarrow{ }^{n} C_{12}={ }^{n} C_{8} \\ \Rightarrow \frac{n !}{(n-12) ! 12 !}=\frac{n !}{(n-8) ! 8 !} \\ \Rightarrow \frac{1}{(n-12) !(12 \times 11 \times 10 \times 9 \times 8 !)} \\ = &amp; \frac{1}{(n-8)(n-9)(n-10)(n-11)(n-12) ! 8 !} \\ \Rightarrow \frac{1}{12 \times 11 \times 10 \times 9}=\frac{1}{(\mathrm{n}-8)(\mathrm{n}-9)(\mathrm{n}-10)(\mathrm{n}-11)} \\ \Rightarrow(n-8)(n-9)(n-10)(n-11) \\ =12 \times 11 \times 10 \times 9 \\ \Rightarrow n-8=12, n-9=11, n-10=10 \text { and } n-11=9 \\ \Rightarrow \mathrm{n}=20 \\ \Rightarrow C(22, n)={ }^{22} C_{20} \\ \quad=\frac{22 !}{2 ! 20 !}=\frac{22 \times 21}{2}=231\end{array}$</div>


tags: ['Physics', 'Units and Dimensions', 'AP EAMCET', 'AP EAMCET 2014']
source_db: MarksBatch2_P1.db
title:
if-c-the-velocity-of-light-h-plancks-constant-and-g-gravitational-constant-are-taken-as-fundamental-quantities-then-the-dimensional-formula-of-mass-is-1

slug:
if-c-the-velocity-of-light-h-plancks-constant-and-g-gravitational-constant-are-taken-as-fundamental-quantities-then-the-dimensional-formula-of-mass-is-1-96869

question:
<div class="question">If $C$ the velocity of light, $h$ Planck's constant and $G$ gravitational constant are taken as fundamental quantities, then the dimensional formula of mass is</div>

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

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

answer:
<div class="solution">Let,<br/>$\begin{aligned}<br/>M &amp; =C^a h^b G^c \\<br/>\mathrm{ML}^0 \mathrm{~T}^0 &amp; =\left[\mathrm{LT}^{-1}\right]^a\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^{\mathrm{b}}\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^{\mathrm{a}}<br/>\end{aligned}$<br/>where, $h=\frac{\text { Energy }}{\text { Frequency }}$<br/>$\begin{aligned}<br/>&amp; =\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right] \\<br/>C &amp; =\frac{\text { Metre }}{\text { Second }}=\left[\mathrm{LT}^{-1}\right] \\<br/>G &amp; =\frac{\text { Force } \times \text { (distance) }}{\text { (mass) }} \\<br/>&amp; =\frac{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^2\right]}{\left[\mathrm{M}^2\right]}=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]<br/>\end{aligned}$<br/>Comparing the coefficients $M, L, T$, of both sides we get<br/>$\begin{array}{l}<br/>b-c=1 \\<br/>a+2 b+3 c=0 \\<br/>-(a+b+2 c)=0<br/>\end{array}$<br/>Solve the Eqs. (ii), (iii) and (iv), we get<br/>$a=\frac{1}{2}, b=\frac{1}{2}, c=-\frac{1}{2}$<br/>So, $M=h^{1 / 2} C^{1 / 2} G^{-1 / 2}$</div>


tags: ['Physics', 'Units and Dimensions', 'TS EAMCET', 'TS EAMCET 2014']
source_db: MarksBatch2_P1.db
title:
if-c-the-velocity-of-light-h-plancks-constant-and-g-gravitational-constant-are-taken-as-fundamental-quantities-then-the-dimensional-formula-of-mass-is

slug:
if-c-the-velocity-of-light-h-plancks-constant-and-g-gravitational-constant-are-taken-as-fundamental-quantities-then-the-dimensional-formula-of-mass-is-84682

question:
<div class="question">If $C$ the velocity of light, $h$ Planck's constant and $G$ gravitational constant are taken as fundamental quantities, then the dimensional formula of mass is</div>

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

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

answer:
<div class="solution">Let,<br/>$\begin{aligned}<br/>M &amp; =C^a h^b G^c \\<br/>\mathrm{ML}^0 \mathrm{~T}^0 &amp; =\left[\mathrm{LT}^{-1}\right]^a\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^{\mathrm{b}}\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^{\mathrm{a}}<br/>\end{aligned}$<br/>where, $h=\frac{\text { Energy }}{\text { Frequency }}$<br/>$\begin{aligned}<br/>&amp; =\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right] \\<br/>C &amp; =\frac{\text { Metre }}{\text { Second }}=\left[\mathrm{LT}^{-1}\right] \\<br/>G &amp; =\frac{\text { Force } \times \text { (distance) }}{\text { (mass) }} \\<br/>&amp; =\frac{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^2\right]}{\left[\mathrm{M}^2\right]}=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]<br/>\end{aligned}$<br/>Comparing the coefficients $M, L, T$, of both sides we get<br/>$\begin{array}{l}<br/>b-c=1 \\<br/>a+2 b+3 c=0 \\<br/>-(a+b+2 c)=0<br/>\end{array}$<br/>Solve the Eqs. (ii), (iii) and (iv), we get<br/>$a=\frac{1}{2}, b=\frac{1}{2}, c=-\frac{1}{2}$<br/>So, $M=h^{1 / 2} C^{1 / 2} G^{-1 / 2}$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-c-velocity-of-light-h-plancks-constant-and-g-universal-gravitational-constant-are-taken-as-fundamental-quantities-then-the-dimensional-formula-of-m

slug:
if-c-velocity-of-light-h-plancks-constant-and-g-universal-gravitational-constant-are-taken-as-fundamental-quantities-then-the-dimensional-formula-of-m-76139

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> (velocity of light), <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> (Planck's constant) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> (Universal gravitational constant) are taken as fundamental quantities, then the dimensional formula of mass is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi> </mi><msup><mrow><mi>C</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi> </mi><msup><mrow><mi>G</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> </div>

answer:
<div class="solution">Let, <math><mi>M</mi><mo>=</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msup></math><br/><br/><math><mi>M</mi><msup><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>L</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>a</mi></mrow></msup><mi> </mi><mi> </mi><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>M</mi><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>b</mi></mrow></msup><mi> </mi><mi> </mi><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>M</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msup></math> ......(i)<br/><br/>Where, <math><mi>h</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">y</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">y</mi></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfrac><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>M</mi><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mi> </mi><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mi>M</mi><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math><br/><br/><math><mi>C</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow></mfrac><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mi>L</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math><br/><br/><math><mi>G</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi><mo>Ã</mo><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfrac><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>M</mi><mi>L</mi><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mi> </mi><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>M</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math><br/><br/>Comparing the coefficients M, L, T, of both sides we get<br/><br/><math><mi>b</mi><mo>-</mo><mi>c</mi><mo>=</mo><mn>1</mn></math> ......(ii)<br/><br/><math><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo>+</mo><mn>3</mn><mi>c</mi><mo>=</mo><mn>0</mn></math> ......(iii)<br/><br/><math><mo>-</mo><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> .....(iv)<br/><br/>Solve the equations (ii), (iii) and (iv), we get<br/><br/><math><mi>a</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi> </mi><mi>b</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi> </mi><mi>c</mi><mo>=</mo><mi> </mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math><br/><br/>So, <math><mi>M</mi><mo>=</mo><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi> </mi><msup><mrow><mi>C</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi> </mi><msup><mrow><mi>G</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></div>


title: if-coefficient-of-variation-is-60-and-standarddeviation-is-24-then-arithmetic-mean-is
slug: if-coefficient-of-variation-is-60-and-standarddeviation-is-24-then-arithmetic-mean-is-88790
tags: ['Mathematics', 'Definite Integration', 'KCET', 'KCET 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If coefficient of variation is \( 60 \) and standarddeviation is \( 24 \), then Arithmetic mean is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\( 40 \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\( \frac{7}{20} \)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\( \frac{20}{7} \)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( \frac{1}{40} \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( 40 \)</span> </div>

answer:
<div class="solution">Coefficient of variation is given by<br/>\[<br/>C V=\frac{\sigma}{x} \times 100<br/>\]<br/>Here, \( \mathrm{CV}=60 \) and \( \sigma=24 \). Then,</div>


title: if-cos-1-5-1-then-what-is-the-value-of-cosec-1-5
slug: if-cos-1-5-1-then-what-is-the-value-of-cosec-1-5-19182
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)=\theta$, then what is the value of $\operatorname{cosec}^{-1}(\sqrt{5)}$ ?</div>

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

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

answer:
<div class="solution">Let, $\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)=\theta \Rightarrow \cos \theta=\frac{1}{\sqrt{5}}$<br/>$\Rightarrow \quad \sec \theta=\sqrt{5} \Rightarrow \sec ^{-1}(\sqrt{5})=\theta$<br/>$\Rightarrow \frac{\pi}{2}-\operatorname{cosec}^{-1}(\sqrt{5})=\theta \quad\left(\because \sec ^{-1} x+\operatorname{cosec}^{-1} x=\frac{\pi}{2}\right)$<br/>$\Rightarrow \quad \operatorname{cosec}^{-1}(\sqrt{5})=\frac{\pi}{2}-\theta$</div>


title: if-cos-3-a-2-1-then-how-many-values-can-sin-a-assume-0-a-36-0
slug: if-cos-3-a-2-1-then-how-many-values-can-sin-a-assume-0-a-36-0-70050
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos 3 \mathrm{~A}=\frac{1}{2}$, then how many values can $\sin \mathrm{A}$ assume? $\left(0   &lt; \mathrm{A} &lt; 360^{\circ}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">6</span> </li> </ul>

answer:
<div class="solution">Given, $\cos 3 \mathrm{~A}=\frac{1}{2}$<br/>$\Rightarrow \cos 3 \mathrm{~A}=\cos \left(\frac{\pi}{3}\right)$ or $\left(\frac{5 \pi}{3}\right)$<br/>Since, $\mathrm{O}   &lt; \mathrm{A} &lt; 360^{\circ}$<br/>A can take the values, $\frac{\pi}{9}, \frac{5 \pi}{9}, \frac{7 \pi}{9}, \frac{11 \pi}{9}, \frac{13 \pi}{9}$ and<br/>$\frac{17 \pi}{9}$<br/>So, $\sin$ A can assume 6 values.</div>


title: if-cos-a-4-3-then-what-is-the-value-of-sin-2-a-sin-2-3-a
slug: if-cos-a-4-3-then-what-is-the-value-of-sin-2-a-sin-2-3-a-45903
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos A=\frac{3}{4}$, then what is the value of $\sin \left(\frac{A}{2}\right) \sin \left(\frac{3 A}{2}\right)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{8}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{5}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{5}{24}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{7}{32}$</span> </li> </ul>

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

answer:
<div class="solution">$\cos A=\frac{3}{4}$<br/>$\sin \left(\frac{A}{2}\right) \cdot \sin \left(\frac{3 A}{2}\right)=\frac{1}{2}\left[2 \sin \frac{A}{2} \sin \frac{3 A}{2}\right]$<br/>$=\frac{1}{2}\left[\cos \left(\frac{3 A}{2}-\frac{A}{2}\right)-\cos \left(\frac{3 A}{2}+\frac{A}{2}\right)\right]$<br/>$=\frac{1}{2}[\cos A-\cos 2 A]=\frac{1}{2}\left[\cos A-\left(2 \cos ^{2} A-1\right)\right]$<br/>$=\frac{1}{2}\left[\cos A-2 \cos ^{2} A+1\right]$<br/>$=\frac{1}{2}\left[\left(\frac{3}{4}\right)-2\left(\frac{3}{4}\right)^{2}+1\right]$<br/>$=\frac{1}{2}\left[\frac{3}{4}-\frac{18}{16}+1\right]=\frac{1}{2}\left[\frac{12-18+16}{16}\right]$<br/>$=\frac{1}{2}\left[\frac{10}{16}\right]=\frac{5}{16}$</div>


title: if-cos-a-cos-b-m-and-sin-a-sin-b-n-where-m-n-0-then-what-is-sin-a-b-equal-to
slug: if-cos-a-cos-b-m-and-sin-a-sin-b-n-where-m-n-0-then-what-is-sin-a-b-equal-to-21625
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos A+\cos B=m$ and $\sin A+\sin B=n$, where $\mathrm{m}, n \neq 0$,<br/>then what is $\sin (A+B)$ equal to ?</div>

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

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

answer:
<div class="solution">Let $\cos A+\cos B=m$<br/>and $\sin A+\sin B=n$<br/>Consider $\sin (A+B)=\frac{\left(m^{2}+n^{2}\right) \sin (A+B)}{m^{2}+n^{2}}$<br/>$=\frac{[2+2 \cos (A-B)] \sin (A+B)}{2+2 \cos (A-B)}$<br/>(from i and ii)<br/>$=\frac{2 \sin (\mathrm{A}+\mathrm{B})+2 \sin (\mathrm{A}+\mathrm{B}) \cos (\mathrm{A}-\mathrm{B})}{1+1+2 \cos (\mathrm{A}-\mathrm{B})}$<br/>$=\frac{2 \sin (A+B)+\sin (A+B+A-B)+\sin (A+B-A+B)}{1+1+2 \cos (A-B)}$<br/>$=\frac{2 \mathrm{mn}}{\mathrm{m}^{2}+\mathrm{n}^{2}}($ from (i) and (ii))<br/>Hence, $\sin (\mathrm{A}+\mathrm{B})=\frac{2 \mathrm{mn}}{\mathrm{m}^{2}+\mathrm{n}^{2}}$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-cos-and-cos-0-are-the-roots-of-the-quadratic-equation-4-x-2-3-0-then-what-is-the-value-of-sec-sec

slug:
if-cos-and-cos-0-are-the-roots-of-the-quadratic-equation-4-x-2-3-0-then-what-is-the-value-of-sec-sec-92423

question:
<div class="question">If $\cos \alpha$ and $\cos \beta(0   &lt; \alpha &lt; \beta &lt; \pi)$ are the roots of the quadratic equation $4 \mathrm{x}^{2}-3=0$, then what is the value of $\sec \alpha \times \sec \beta$ ?</div>

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

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

answer:
<div class="solution">Product of roots $=\frac{\mathrm{c}}{\mathrm{a}}$<br/>$\Rightarrow \cos \alpha \cdot \cos \beta=-\frac{3}{4}$<br/>$\Rightarrow \frac{1}{\cos \alpha \cdot \cos \beta}=\sec \alpha \cdot \sec \beta=-\frac{4}{3}$</div>


title: if-cos-cos-cos-0-where-0-2-0-2-0-2-then-what-is-the-value-of-sin-sin-sin
slug: if-cos-cos-cos-0-where-0-2-0-2-0-2-then-what-is-the-value-of-sin-sin-sin-68077
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos \alpha+\cos \beta+\cos \gamma=0$, where $0   &lt; \alpha \leq \frac{\pi}{2}, 0 &lt; \beta \leq \frac{\pi}{2}$,<br/>$0 &lt; \gamma \leq \frac{\pi}{2}$, then what is the value of $\sin \alpha+\sin \beta+\sin \gamma ?$</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">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">$\frac{5 \sqrt{2}}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3 \sqrt{2}}{2}$</span> </li> </ul>

answer:
<div class="solution">$\cos \alpha+\cos \beta+\cos \gamma=0$<br/>Given, $0   &lt; \alpha \leq \frac{\pi}{2}, 0 &lt; \beta \leq \frac{\pi}{2}, 0 &lt; \gamma \leq \frac{\pi}{2}$.<br/>(1) is satisfied when $\alpha=\frac{\pi}{2}, \beta=\frac{\pi}{2}$ and $\gamma=\frac{\pi}{2}$.<br/>$\therefore \sin \alpha+\sin \beta+\sin \gamma=\sin \frac{\pi}{2}+\sin \frac{\pi}{2}+\sin \frac{\pi}{2} .$<br/>$=1+1+1=3 .$</div>


title: if-cos-sin-and-lies-in-the-first-quadrant-then-which-one-of-the-following-is-correct
slug: if-cos-sin-and-lies-in-the-first-quadrant-then-which-one-of-the-following-is-correct-23159
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos \theta   &lt; \sin \theta$ and $\theta$ lies in the first quadrant, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pi / 4   &lt; \theta &lt; \pi / 2$</span> </div>

answer:
<div class="solution">We know that, for $\frac{\pi}{4}   &lt; \theta &lt; \frac{\pi}{2}$<br/>$\cos \theta &lt; \sin \theta$</div>


title: if-cos-x-1-then-what-is-1-c-o-s-x-s-i-n-x-equal-to
slug: if-cos-x-1-then-what-is-1-c-o-s-x-s-i-n-x-equal-to-45543
question: <div class="question">If $\cos x \neq-1$, then what is $\frac{\sin x}{1+\cos x}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\cot \frac{x}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cot \frac{x}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\tan \frac{x}{2}$</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">$-\tan \frac{x}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\frac{\sin x}{1+\cos x}=\frac{2 \sin x / 2 \cos x / 2}{1+2 \cos ^{2}(x / 2)-1}$<br/>$\left(\because \sin 2 x=2 \sin x \cos x\right.$ and $\left.\cos 2 x=2 \cos ^{2} x-1\right)$<br/>$=\frac{2 \sin x / 2 \cos x / 2}{2 \cos ^{2} x / 2}=\frac{\sin x / 2}{\cos x / 2}=\tan x / 2$</div>


title: if-cos-x-3-1-then-what-is-sin-x-cot-x-cosecx-tan-x-equal-to
slug: if-cos-x-3-1-then-what-is-sin-x-cot-x-cosecx-tan-x-equal-to-94566
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_P1.db
question:
<div class="question">If $\cos x=\frac{1}{3}$, then what is $\sin x \cdot \cot x$. cosecx $\cdot \tan x$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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">$$<br/>\begin{aligned} \sin x &amp; \cdot \cot x \cdot \operatorname{cosec} x \cdot \tan x \\ &amp;=(\sin x \cdot \operatorname{cosec} x) \cdot(\cot x \cdot \tan x) \\ &amp;=1 \times 1=1\end{aligned}<br/>$$</div>


title: if-cosec-cot-3-1-where-0-then-what-is-the-value-of-cos
slug: if-cosec-cot-3-1-where-0-then-what-is-the-value-of-cos-93530
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\operatorname{cosec} \theta-\cot \theta=\frac{1}{\sqrt{3}}$ where $\theta \neq 0$, then what is the value<br/>of $\cos \theta ?$</div>

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

answer:
<div class="solution">Consider, $\operatorname{cosec} \theta-\cot \theta=\frac{1}{\sqrt{3}}$<br/>$\Rightarrow \frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}=\frac{1}{\sqrt{3}} \Rightarrow \frac{1-\cos \theta}{\sin \theta}=\frac{1}{\sqrt{3}}$<br/>$\Rightarrow \frac{1-\left(1-2 \sin ^{2} \theta / 2\right)}{2 \sin \theta / 2 \cos \theta / 2}=\frac{1}{\sqrt{3}} \Rightarrow \frac{2 \sin ^{2} \theta / 2}{2 \sin \theta / 2 \cos \theta / 2}=\frac{1}{\sqrt{3}}$<br/>$\Rightarrow \frac{\sin \theta / 2}{\cos \theta / 2}=\frac{1}{\sqrt{3}} \Rightarrow \tan \frac{\theta}{2}=\tan 30^{\circ}$<br/>$\Rightarrow \theta=60^{\circ}$<br/>$\therefore \cos \theta=\cos 60^{\circ}=\frac{1}{2}$</div>


title: if-cosec-cot-c-then-what-is-cos-equal-to
slug: if-cosec-cot-c-then-what-is-cos-equal-to-72053
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\operatorname{cosec} \theta+\cot \theta=\mathrm{c}$, then what is $\cos \theta$ equal to $?</div>

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

answer:
<div class="solution">Let $\operatorname{cosec} \theta+\cot \theta=\mathrm{c}$<br/>$\Rightarrow \frac{1}{\sin \theta}+\frac{\cos \theta}{\sin \theta}=\mathrm{c} \Rightarrow \frac{1+\cos \theta}{\sin \theta}=\mathrm{c}$<br/>$\Rightarrow \frac{1+\left(2 \cos ^{2} \frac{\theta}{2}-1\right)}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}=\mathrm{c} \Rightarrow \frac{2 \cos ^{2} \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}=\mathrm{c}$<br/>$\Rightarrow \cot \frac{\theta}{2}=\mathrm{c} \Rightarrow \cos \theta=\frac{1-\frac{1}{\mathrm{c}^{2}}}{1+\frac{1}{\mathrm{c}^{2}}}=\frac{\mathrm{c}^{2}-1}{\mathrm{c}^{2}+1}$<br/>$\left(\because \cos \theta=\frac{1-\tan ^{2}\left(\frac{\theta}{2}\right)}{1+\tan ^{2}\left(\frac{\theta}{2}\right)}\right)$<br/>$\Rightarrow \tan \frac{\theta}{2}=\frac{1}{\mathrm{c}}$</div>


title: if-cot-2-cos-where-2-then-what-is-the-value-of
slug: if-cot-2-cos-where-2-then-what-is-the-value-of-47062
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If cot $\theta=2 \cos \theta$, where $(\pi / 2)   &lt; \theta &lt; \pi$, then what is the value of $\theta ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$5 \pi 6$</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">B</span> <span class="option-data">$2 \pi / 3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 \pi 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$11 \pi / 12$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 \pi 6$</span> </div>

answer:
<div class="solution">Given $\cot \theta=2 \cos \theta, \frac{\pi}{2}   &lt; \theta &lt; \pi$<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \frac{\cos \theta}{\sin \theta}=2 \cos \theta \Rightarrow \frac{\cos \theta}{2 \cos \theta}=\sin \theta \\<br/>\Rightarrow \frac{1}{2}=\sin \theta \Rightarrow \sin \frac{\pi}{6}=\sin \theta \Rightarrow \theta=\frac{\pi}{6}<br/>\end{array}<br/>$$<br/>But $\frac{\pi}{2} &lt; \theta &lt; \pi \therefore \theta=\pi-\frac{\pi}{6}=\frac{5 \pi}{6}$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-cot-512-and-lies-in-the-third-quadrant-then-what-is-beginarrayll2-sin-theta3-cos-theta-text-equal-to

slug:
if-cot-512-and-lies-in-the-third-quadrant-then-what-is-beginarrayll2-sin-theta3-cos-theta-text-equal-to-94824

question:
<div class="question">If $\cot \theta=5 / 12$ and $\theta$ lies in the third quadrant, then what is $\begin{array}{ll}(2 \sin \theta+3 \cos \theta) \text { equal to? }</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-p^{2}$ for some odd prime $p$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-q / p)$ where $p$ is an odd prime and $q$ a positive integer with $(q / p)$ not an integer</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-p$ for some odd prime $p$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-p$ for some odd prime $p$</span> </div>

answer:
<div class="solution">Let $\cot \theta=\frac{5}{12}$<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \tan \theta=\frac{12}{5}=\frac{\text { perpendicular }(\mathrm{p})}{\text { base }(\mathrm{b})} \\<br/>\therefore \text { Hypotenuse }(\mathrm{H})=\sqrt{p^{2}+b^{2}} \\<br/>=\sqrt{(12)^{2}+(5)^{2}}=\sqrt{144+25}=\sqrt{169}=13<br/>\end{array}<br/>$$<br/>Consider $2 \sin \theta+3 \cos \theta$<br/>$=2\left(\frac{p}{H}\right)+3\left(\frac{b}{H}\right) \quad(\mathrm{H}-\mathrm{Height})$<br/>But $\theta$ lies in $3^{\text {rd }}$ quadrant and $\sin \theta, \cos \theta$ both are negative in $3^{\text {rd }}$ quadrant<br/>$\begin{aligned} \therefore 2 \sin \theta+3 \cos \theta &amp;=2\left(\frac{-p}{H}\right)+3\left(\frac{-b}{H}\right) \\ &amp;=2\left(\frac{-12}{13}\right)+3\left(\frac{-5}{13}\right) \\ &amp;=\frac{-24-15}{13}=\frac{-39}{13}=-3 \end{aligned}$<br/>which is an odd prime.</div>


title: if-cot-a-2-and-cot-b-3-then-what-is-the-value-of-a-b
slug: if-cot-a-2-and-cot-b-3-then-what-is-the-value-of-a-b-94396
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cot \mathrm{A}=2$ and $\cot \mathrm{B}=3$, then what is the value of $\mathrm{A}+\mathrm{B}$ ?</div>

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

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

answer:
<div class="solution">\cot \mathrm{A}=2 \text { and } \cot \mathrm{B}=3\\<br/>\begin{aligned}<br/>&amp; \cot (A+B)=\frac{\cot A \cot B-1}{\cot A+\cot B}=\frac{6-1}{2+3}=\frac{5}{5}=1 \\<br/>&amp; \Rightarrow \cot (A+B)=\cot \left(\frac{\pi}{4}\right) \Rightarrow A+B=\frac{\pi}{4} \\<br/>&amp;=\left[\sin \left(90^{\circ}-23 \frac{1^{\circ}}{2}\right)\right]^{2}-\sin ^{2} 23 \frac{1^{\circ}}{2} \\<br/>=&amp; \cos ^{2} 23 \frac{1^{\circ}}{2}-\sin ^{2} 23 \frac{1^{\circ}}{2} \\<br/>=&amp; \cos 2\left(23 \frac{1^{\circ}}{2}\right)=\cos 47^{\circ} \\<br/>\left(\because \cos 2 \mathrm{~A}=\cos ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~A}\right) \\</div>


title: if-cot-and-cot-are-the-roots-of-the-equation-x-2-b-x-c-0-with-b-0-then-the-value-of-cot
slug: if-cot-and-cot-are-the-roots-of-the-equation-x-2-b-x-c-0-with-b-0-then-the-value-of-cot-23621
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cot \alpha$ and $\cot \beta$ are the roots of the equation $x^{2}+b x+c=0$ with $b \neq 0$, then the value of $\cot (\alpha+\beta)$</div>

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

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

answer:
<div class="solution">Given equation, $\mathrm{x}^{2}+\mathrm{bx}+\mathrm{c}=0$<br/>Roots are $\cot \alpha, \cot \beta$<br/>Sum of roots $=\cot \alpha+\cot \beta=-b$<br/>Product of roots $=\cot \alpha, \cot \beta=c$<br/>$\cot (\alpha+\beta) \frac{\cot \alpha \cdot \cot \beta-1}{\cot \beta+\cot \alpha}=\frac{\mathrm{c}-1}{-\mathrm{b}}=\frac{1-\mathrm{c}}{\mathrm{b}}$</div>


title: if-cot-x-y-1-3-cot-x-y-3-then-what-are-the-smallest-positive-values-of-x-and-y-respectively
slug: if-cot-x-y-1-3-cot-x-y-3-then-what-are-the-smallest-positive-values-of-x-and-y-respectively-89978
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If cot $(x+y)=1 / \sqrt{3}, \cot (x-y)=\sqrt{3}$ then what are the smallest positive values of $x$ and $y$ respectively?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$45^{\circ}, 30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$30^{\circ}, 45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15^{\circ}, 60^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$45^{\circ}, 15^{\circ}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$45^{\circ}, 15^{\circ}$</span> </div>

answer:
<div class="solution">Since $\quad \cot (x+y)=\frac{1}{\sqrt{3}}=\cot 60^{\circ} \quad\left[\cot 60^{\circ}=\frac{1}{\sqrt{3}}\right]$<br/>$\Rightarrow x+y=60^{\circ}$<br/>and $\cot (x-y)=\sqrt{3}=\cot 30^{\circ}$<br/>$\Rightarrow x-y=30^{\circ}$<br/>From equations (i) and (ii), we get $x=45^{\circ}$ and $y=15^{\circ}$</div>


title: if-cota-cot-b-2-then-what-is-the-value-of-cos-a-b-sec-a-b
slug: if-cota-cot-b-2-then-what-is-the-value-of-cos-a-b-sec-a-b-53754
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If cotA $\cot B=2$, then what is the value of $\cos (\mathrm{A}+\mathrm{B}) \sec (\mathrm{A}-\mathrm{B}) ?$</div>

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

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

answer:
<div class="solution">Consider $\cos (\mathrm{A}+\mathrm{B}) \cdot \sec (\mathrm{A}-\mathrm{B})$<br/>$=\frac{\cos (\mathrm{A}+\mathrm{B})}{\cos (\mathrm{A}-\mathrm{B})}=\frac{\cos \mathrm{A} \cos \mathrm{B}-\sin \mathrm{A} \sin \mathrm{B}}{\cos \mathrm{A} \cos \mathrm{B}+\sin \mathrm{A} \sin \mathrm{B}}$<br/>Divide $\mathrm{Nr}$ and $\mathrm{Dr}$ by $\sin \mathrm{A} \sin \mathrm{B}$,<br/>$\frac{\cot \mathrm{A} \cot \mathrm{B}-1}{\cot \mathrm{A} \cot \mathrm{B}+1}=\frac{2-1}{2+1}=\frac{1}{3}$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-d-is-determinant-of-order-3-and-d-is-the-determinant-obtained-by-replacing-the-elements-of-d-by-their-cofactors-then-which-one-of-the-following-is-

slug:
if-d-is-determinant-of-order-3-and-d-is-the-determinant-obtained-by-replacing-the-elements-of-d-by-their-cofactors-then-which-one-of-the-following-is-29087

question:
<div class="question">If $\mathrm{D}$ is determinant of order 3 and $\mathrm{D}^{\prime}$ is the determinant obtained by replacing the elements of $D$ by their cofactors, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">$\mathrm{D}^{\prime}=$ cofactor $\mathrm{D}$ $\Rightarrow\left|\mathrm{D}^{\prime}\right|=\mid$ cofactor $\mathrm{D} \mid$<br/>$\Rightarrow\left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{3-1}$<br/>$\Rightarrow\left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{2}$<br/>So $\mathrm{D}^{\prime}=\mathrm{D}^{2}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-d-is-the-number-of-degrees-contained-in-an-angle-m-is-the-number-of-minutes-and-s-is-the-number-of-seconds-then-the-value-of-s-m-m-d-is-1

slug:
if-d-is-the-number-of-degrees-contained-in-an-angle-m-is-the-number-of-minutes-and-s-is-the-number-of-seconds-then-the-value-of-s-m-m-d-is-1-74481

question:
<div class="question">If d is the number of degrees contained in an angle, $\mathrm{m}$ is the number of minutes and s is the number of seconds, then the value of $(\mathrm{s}-\mathrm{m}) /(\mathrm{m}-\mathrm{d})$ is:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{60}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/vdzwto9n9bVC0Kc4NGRZiaXJYD0qnVksKR5_kZ3e12E.original.fullsize.png"/><br/><br/>$40 \%$ of total population $=0.4 \times 75 \times 10^{6}=3 \times 10^{7}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-d-is-the-number-of-degrees-contained-in-an-angle-m-is-the-number-of-minutes-and-s-is-the-number-of-seconds-then-the-value-of-s-m-m-d-is

slug:
if-d-is-the-number-of-degrees-contained-in-an-angle-m-is-the-number-of-minutes-and-s-is-the-number-of-seconds-then-the-value-of-s-m-m-d-is-93303

question:
<div class="question">If d is the number of degrees contained in an angle, $\mathrm{m}$ is the number of minutes and s is the number of seconds, then the value of $(\mathrm{s}-\mathrm{m}) /(\mathrm{m}-\mathrm{d})$ is:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{60}$</span> </div>

answer:
<div class="solution">$\mathrm{d}=60 \times 60 \mathrm{~s}$<br/>$\mathrm{~m}=60 \mathrm{~s}$<br/>$\frac{s-m}{m-d}=\frac{s-60 s}{60 s-60 \times 60 \mathrm{~s}}=\frac{-59}{60 \times(-59)}=\frac{1}{60}$</div>


title: if-d-x-d-y-1-x-y-x-y-and-y-1-0-then-what-is-y-x-equal-to
slug: if-d-x-d-y-1-x-y-x-y-and-y-1-0-then-what-is-y-x-equal-to-40133
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{d y}{d x}=1+x+y+x y$ and $y(-1)=0$, then what is $y(x)$ equal to?</div>

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

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

answer:
<div class="solution">Let $\frac{d y}{d x}=1+x+y+x y$<br/>$\Rightarrow \frac{d y}{d x}=(1+x)(1+y)$<br/>$\Rightarrow \frac{d y}{1+y}=d x(1+x)$<br/>$\Rightarrow \int \frac{1}{1+y} d y=\int(1+x) d x$<br/>$\Rightarrow \log (1+y)=x+\frac{x^{2}}{2}+c$<br/>Given At $x=-1, y=0$<br/>$\Rightarrow \log 1=-1+\frac{1}{2}+c$<br/>$\Rightarrow c=\frac{1}{2}$<br/>$\therefore \quad \log (1+y)=x+\frac{x^{2}}{2}+\frac{1}{2}=\frac{(1+x)^{2}}{2}$<br/>$\Rightarrow 1+y=e^{\frac{(1+x)^{2}}{2}}$<br/>$\Rightarrow y=e^{\frac{(1+x)^{2}}{2}}-1$</div>


title: if-denotes-the-cube-root-of-unity-then-what-is-the-real-root-of-the-equation-x-3-27-0
slug: if-denotes-the-cube-root-of-unity-then-what-is-the-real-root-of-the-equation-x-3-27-0-91126
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ denotes the cube root of unity, then what is the real root of the equation $x^{3}-27=0$ ?</div>

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

answer:
<div class="solution">The given equation is $x^{3}-27=0$ $\Rightarrow \mathrm{x}^{3}-3^{3}=0 \Rightarrow \mathrm{x}=3,3 \omega, 3 \omega^{2}$<br/>Thus, real root of given equation is $3 \omega^{3}$, since $\omega^{3}=1$</div>


tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-different-words-are-formed-with-all-the-letters-of-the-word-again-and-are-arrangedalphabeticallyamong-themselves-as-in-a-dictionary-the-word-at-the

slug:
if-different-words-are-formed-with-all-the-letters-of-the-word-again-and-are-arrangedalphabeticallyamong-themselves-as-in-a-dictionary-the-word-at-the-65470

question:
<div class="question">If different words are formed with all the letters of the word 'AGAIN' and are arrangedalphabeticallyamong themselves as in a dictionary, the word at the 50 th place will be</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/CCFAKEkvGcrsTxQuqU2HFR4q8pLN0SbeRpffmgROlRI.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/BoflYH4DenMto4K_wq9Qn9HXvSoBf8PBCh5rIx2NeUc.original.fullsize.png"/></div>


tags: ['Physics', 'Motion In One Dimension', 'TS EAMCET', 'TS EAMCET 2020 (09 Sep Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-e-and-e-0-represent-the-energies-t-and-t-0-represent-the-times-in-which-of-the-following-is-dimensionally-correct-relation

slug:
if-e-and-e-0-represent-the-energies-t-and-t-0-represent-the-times-in-which-of-the-following-is-dimensionally-correct-relation-35068

question:
<div class="question"><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub></math> represent the energies, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>0</mn></msub></math> represent the times, in which of the following is dimensionally correct relation?</p></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msub><mi>E</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi><mo>/</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>The dimensional formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msub><mi>E</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi><mo>/</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></msup></math> as  <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi><mo>/</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></msup></math> is a dimensionless quantity.</p><p>In the question, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub></math> represents the energies. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> is the time-averaged energy of the oscillator which is dimensionally represented as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mi>L</mi><mn>2</mn></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub></math> is the total energy of an undamped oscillator which is represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>A</mi><mn>2</mn></msup><msubsup><mi>w</mi><mn>0</mn><mn>2</mn></msubsup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the mass, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is the amplitude, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> is the angular velocity.</p><p>The dimensional representation of the total energy of an undamped oscillator is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>=</mo><mi>M</mi><msup><mfenced><mi>L</mi></mfenced><mn>2</mn></msup><msup><mfenced><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mn>2</mn></msup><mo>=</mo><mi>M</mi><msup><mi>L</mi><mn>2</mn></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p>Thus, the dimensional formula is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msub><mi>E</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi><mo>/</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></msup></math>.</p></div>


title: if-e-cos-sin-sin-cos-then-e-e-is-equal-to
slug: if-e-cos-sin-sin-cos-then-e-e-is-equal-to-47511
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $E(\theta)=\left[\begin{array}{cc}\cos \theta &amp; \sin \theta \\ -\sin \theta &amp; \cos \theta\end{array}\right]$, then $E(\alpha) E(\beta)$ is equal to</div>

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

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

answer:
<div class="solution">$\mathrm{E}(\theta)=\left[\begin{array}{cc}\cos \theta &amp; \sin \theta \\ -\sin \theta &amp; \cos \theta\end{array}\right]$<br/>Now $\mathrm{E}(\alpha)=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right]$<br/>and $\mathrm{E}(\beta)=\left[\begin{array}{cc}\cos \beta &amp; \sin \beta \\ -\sin \beta &amp; \cos \beta\end{array}\right]$<br/>Now $\mathrm{E}(\alpha) \mathrm{E}(\beta)=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \beta &amp; \sin \beta \\ -\sin \beta &amp; \cos \beta\end{array}\right]$<br/>$=\left[\begin{array}{lr}\cos \alpha \cdot \cos \beta &amp; \cos \alpha \sin \beta \\ -\sin \alpha \sin \beta &amp; +\sin \alpha \cos \beta \\ -\sin \alpha \cos \beta &amp; -\sin \alpha \sin \beta \\ -\cos \alpha \sin \beta &amp; +\cos \alpha \cos \beta\end{array}\right]$<br/>$=\left[\begin{array}{cc}\cos (\alpha+\beta) &amp; \sin (\alpha+\beta) \\ -\sin (\alpha+\beta) &amp; \cos (\alpha+\beta)\end{array}\right]$<br/>$=\mathrm{E}(\alpha+\beta)$<br/>$\therefore$ Option $(\mathrm{c})$ is correct.</div>


title: if-e-is-the-universal-set-and-a-b-c-then-the-set-e-e-e-e-e-a-is-same-as-the-set
slug: if-e-is-the-universal-set-and-a-b-c-then-the-set-e-e-e-e-e-a-is-same-as-the-set-41567
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If E is the universal set and $\mathrm{A}=\mathrm{B} \cup \mathrm{C}$, then the set $\mathrm{E}-(\mathrm{E}-$ $(\mathrm{E}-(\mathrm{E}-(\mathrm{E}-\mathrm{A}))))$ is same as the set</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{B}^{\prime} \cap \mathrm{C}$</span> </div>

answer:
<div class="solution">$\mathrm{E}$ is the universal set and $\mathrm{A}=\mathrm{B} \cup \mathrm{C}$.<br/>Since, $\mathrm{E}$ is the universal set, $\mathrm{E}-\mathrm{A}=\mathrm{A}^{\prime}$<br/>$\therefore \mathrm{E}-(\mathrm{E}-(\mathrm{E}-(\mathrm{E}-(\mathrm{E}-\mathrm{A}))))$<br/>$=\mathrm{E}-\left(\mathrm{E}-\left(\mathrm{E}-\left(\mathrm{E}-\mathrm{A}^{\prime}\right)\right)\right)$<br/>$=\mathrm{E}-(\mathrm{E}-(\mathrm{E}-\mathrm{A}))$<br/>$=\mathrm{E}-(\mathrm{E}-\mathrm{A})^{\prime}$<br/>$=\mathrm{E}-\mathrm{A}$<br/>$=\mathrm{A}^{\prime}$<br/>$=(\mathrm{B} \cup \mathrm{C})^{\prime}$<br/>$=\mathrm{B}^{1} \cap \mathrm{C}^{\prime}$</div>


title: if-e-y-x-y-e-then-what-is-the-value-of-d-x-2-d-2-y-at-x-0
slug: if-e-y-x-y-e-then-what-is-the-value-of-d-x-2-d-2-y-at-x-0-66057
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $e^{y}+x y=e$, then what is the value of $\frac{d^{2} y}{d x^{2}}$ at $x=0$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$e^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$e^{-2}$</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">e</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">$e^{-2}$</span> </div>

answer:
<div class="solution">Given $e^{y}+x y=e$ On differentiating w.r.t. $x$, we get<br/>$e^{y} \frac{d y}{d x}+y+x \frac{d y}{d x}=0$ ...(i)<br/>At $x=0$ we get $e^{y}+0 . y=e \Rightarrow e^{y}=e \Rightarrow y=1$<br/>$\therefore$ By putting $y=1$ in equation (i)<br/>we get<br/>$e \frac{d y}{d x}+1+0=0$<br/>$\Rightarrow \frac{d y}{d x}=-\frac{1}{e}$<br/>Again differentiating Eq. (i), we get<br/>$e^{y} \frac{d^{2} y}{d x^{2}}+e^{y}\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}+x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=0$<br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}\left(e^{y}+x\right)+e^{y}\left(\frac{d y}{d x}\right)^{2}+\frac{2 d y}{d x}=0$<br/>Now, At $x=0, y=1$<br/>$\frac{d^{2} y}{d x^{2}}(e+0)+e\left(-\frac{1}{e}\right)^{2}+2\left(-\frac{1}{e}\right)=0$<br/>$\Rightarrow e \frac{d^{2} y}{d x^{2}}-\frac{1}{e}=0$<br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{1}{e^{2}}=e^{-2}$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-each-element-in-a-row-of-a-determinant-is-multiplied-by-the-same-factor-r-then-the-value-of-the-determinant

slug:
if-each-element-in-a-row-of-a-determinant-is-multiplied-by-the-same-factor-r-then-the-value-of-the-determinant-95437

question:
<div class="question">If each element in a row of a determinant is multiplied by the same factor $\mathrm{r}$, then the value of the determinant:</div>

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

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

answer:
<div class="solution">As we know that if each element of any row (or column) of a determinant is multiplied by the same number, then the value of determinant is multiplied by that number.</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2015 (11 Apr Online)']
source_db: MarksBatch2_P1.db
title:
if-electronic-charge-e-electron-mass-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-the-permeability-of-vacuu-1

slug:
if-electronic-charge-e-electron-mass-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-the-permeability-of-vacuu-1-91930

question:
<div class="question">If electronic charge <math><mi>e</mi></math>, electron mass <math><mi>m</mi></math>, speed of light in vacuum <math><mi>c</mi></math> and Planck's constant <math><mi>h</mi></math> are taken as fundamental quantities, the permeability of vacuum <math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub></math> can be expressed in units of:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfenced separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Magnetic potential energy inside a solenoid is<br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>L</mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math> Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo></math>self-inductor<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>U</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced separators="|"><mrow><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>n</mi><mn>2</mn></msup><mi>A</mi><mi> </mi><mi>l</mi></mrow></mfenced><msup><mi>I</mi><mn>2</mn></msup></math><br/><br/><math><mo>∵</mo><mi>n</mi><mo>=</mo></math> number of turns per unit length<br/><br/><math><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></mfrac></math><br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><mi>l</mi><mi> </mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mfrac><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mi>A</mi><mi>l</mi><mi> </mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup><mi> </mi></math><br/><br/><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>U</mi><mi>l</mi></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/>Dimensionally, <math><mfenced close="]" open="[" separators="|"><mrow><mi>U</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>e</mi></mrow><mrow><mi>T</mi></mrow></mfrac></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>A</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>l</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>l</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mi>′</mi></mrow></msup></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mi> </mi><mfrac><mrow><mo>[</mo><mi>l</mi><mo>]</mo></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mfrac><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>e</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mfrac><mrow><mi>t</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math><br/><br/><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-electronic-charge-e-electron-mass-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-the-permeability-of-vacuu-2

slug:
if-electronic-charge-e-electron-mass-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-the-permeability-of-vacuu-2-33908

question:
<div class="question">If electronic charge <math><mi>e</mi></math>, electron mass <math><mi>m</mi></math>, speed of light in vacuum <math><mi>c</mi></math> and Planck's constant <math><mi>h</mi></math> are taken as fundamental quantities, the permeability of vacuum <math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub></math> can be expressed in units of:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfenced separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Magnetic potential energy inside a solenoid is<br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>L</mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math> Where L = self-inductor<br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><mi> </mi><mi>l</mi></mrow></mfenced><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mo>âµ</mo><mi>n</mi><mo>=</mo></math> no. of turns per unit length<br/><br/><math><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></mfrac></math><br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><mi>l</mi><mi> </mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><mfrac><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mi>A</mi><mi>l</mi><mi> </mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup><mi> </mi></math><br/><br/><math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>U</mi><mi>l</mi></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/>Dimensionally, <math><mfenced close="]" open="[" separators="|"><mrow><mi>U</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mrow><mi>Î»</mi></mrow></mfrac></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>e</mi></mrow><mrow><mi>T</mi></mrow></mfrac></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>A</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>l</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>l</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mi>â²</mi></mrow></msup></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mrow><mi>Î»</mi></mrow></mfrac></mrow></mfenced><mi> </mi><mfrac><mrow><mo>[</mo><mi>l</mi><mo>]</mo></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mfrac><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>e</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>Î»</mi></mrow></mfrac><mfrac><mrow><mi>t</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math><br/><br/><math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-electronic-charge-e-electron-mass-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-the-permeability-of-vacuu

slug:
if-electronic-charge-e-electron-mass-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-the-permeability-of-vacuu-22840

question:
<div class="question">If electronic charge <math><mi>e</mi></math>, electron mass <math><mi>m</mi></math>, speed of light in vacuum <math><mi>c</mi></math> and Planck's constant <math><mi>h</mi></math> are taken as fundamental quantities, the permeability of vacuum <math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub></math> can be expressed in units of:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfenced separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Magnetic potential energy inside a solenoid is<br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>L</mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math> Where L = self-inductor<br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><mi> </mi><mi>l</mi></mrow></mfenced><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mo>âµ</mo><mi>n</mi><mo>=</mo></math> no. of turns per unit length<br/><br/><math><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></mfrac></math><br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><mi>l</mi><mi> </mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mi>U</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><mfrac><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mi>A</mi><mi>l</mi><mi> </mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup><mi> </mi></math><br/><br/><math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>U</mi><mi>l</mi></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>A</mi><msup><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/>Dimensionally, <math><mfenced close="]" open="[" separators="|"><mrow><mi>U</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mrow><mi>Î»</mi></mrow></mfrac></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>I</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>e</mi></mrow><mrow><mi>T</mi></mrow></mfrac></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>A</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>l</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><mi>l</mi></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mi>â²</mi></mrow></msup></mrow></mfenced></math><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mrow><mi>Î»</mi></mrow></mfrac></mrow></mfenced><mi> </mi><mfrac><mrow><mo>[</mo><mi>l</mi><mo>]</mo></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mfrac><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>e</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>Î»</mi></mrow></mfrac><mfrac><mrow><mi>t</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math><br/><br/><math><msub><mrow><mi>Î¼</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>h</mi></mrow><mrow><mi>c</mi><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></div>


tags: ['Mathematics', 'Permutation Combination', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-eleven-members-of-a-committee-sit-at-a-round-table-so-that-the-president-and-secretary-always-sit-together-then-the-number-of-arrangements-is

slug:
if-eleven-members-of-a-committee-sit-at-a-round-table-so-that-the-president-and-secretary-always-sit-together-then-the-number-of-arrangements-is-15136

question:
<div class="question">If eleven members of a committee sit at a round table so that the President and Secretary always sit together, then the number of arrangements is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10!\times 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10!$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$9!\times 2$</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">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9!\times 2$</span> </div>

answer:
<div class="solution">Required number of ways $9!^* 2$.<br/>\{By fundamental property of circular permutation\}</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-energy-e-velocity-v-and-time-t-are-chosen-as-the-fundamental-quantities-the-dimensional-formula-of-surface-tension-will-befor-surface-tension-force-1

slug:
if-energy-e-velocity-v-and-time-t-are-chosen-as-the-fundamental-quantities-the-dimensional-formula-of-surface-tension-will-befor-surface-tension-force-1-51574

question:
<div class="question">If energy <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">E</mi><mo>)</mo></math>, velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">V</mi><mo>)</mo></math> and time <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">T</mi><mo>)</mo></math> are chosen as the fundamental quantities, the dimensional formula of surface tension will be___(For surface tension, Force=Surface tension<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo></math>length)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal"> </mi><msup><mrow><mi mathvariant="normal">V</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><mi mathvariant="normal"> </mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Dimensional formula of surface tension, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>S</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>0</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math><br/>Dimensional formula of energy, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>E</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math><br/>Dimensional formula of velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>V</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>0</mn></msup><msup><mi mathvariant="normal">L</mi><mn>1</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math><br/><br/>By applying dimensional analysis,<br/><math><mi mathvariant="normal">S</mi><mo>=</mo><msup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msup><msup><mrow><mi mathvariant="normal">V</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>0</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mi mathvariant="normal">a</mi></msup><msup><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">M</mi><mn>0</mn></msup><msup><mi mathvariant="normal">L</mi><mn>1</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mi mathvariant="normal">b</mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">T</mi></mfenced><mi mathvariant="normal">c</mi></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>0</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mi mathvariant="normal">M</mi><mi mathvariant="normal">a</mi></msup><msup><mi mathvariant="normal">L</mi><mrow><mn>2</mn><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mi mathvariant="normal">b</mi><mo>+</mo><mi mathvariant="normal">c</mi></mrow></msup></math><br/><br/>On comparing,<br/><math><mi mathvariant="normal">a</mi><mo>=</mo><mn>1</mn></math><br/><math><mn>2</mn><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi mathvariant="normal">b</mi><mo>=</mo><mi mathvariant="normal"> </mi><mo>-</mo><mn>2</mn></math><br/><math><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mi mathvariant="normal">b</mi><mo>+</mo><mi mathvariant="normal">c</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>⇒</mo><mi mathvariant="normal">c</mi><mo>=</mo><mi mathvariant="normal"> </mi><mo>-</mo><mn>2</mn></math><br/><math><mo>∴</mo><mi mathvariant="normal">S</mi><mo>=</mo><msup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">V</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p></div>


tags: ['Physics', 'Units and Dimensions', 'NEET', 'NEET 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-energy-e-velocity-v-and-time-t-are-chosen-as-the-fundamental-quantities-the-dimensional-formula-of-surface-tension-will-befor-surface-tension-force

slug:
if-energy-e-velocity-v-and-time-t-are-chosen-as-the-fundamental-quantities-the-dimensional-formula-of-surface-tension-will-befor-surface-tension-force-19380

question:
<div class="question">If energy <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">E</mi><mo>)</mo></math>, velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">V</mi><mo>)</mo></math> and time <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">T</mi><mo>)</mo></math> are chosen as the fundamental quantities, the dimensional formula of surface tension will be___(For surface tension, Force=Surface tension<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo></math>length)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal"> </mi><msup><mrow><mi mathvariant="normal">V</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><mi mathvariant="normal"> </mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Dimensional formula of surface tension, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>S</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>0</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math><br/>Dimensional formula of energy, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>E</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math><br/>Dimensional formula of velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>V</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>0</mn></msup><msup><mi mathvariant="normal">L</mi><mn>1</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math><br/><br/>By applying dimensional analysis,<br/><math><mi mathvariant="normal">S</mi><mo>=</mo><msup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msup><msup><mrow><mi mathvariant="normal">V</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>0</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mi mathvariant="normal">a</mi></msup><msup><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">M</mi><mn>0</mn></msup><msup><mi mathvariant="normal">L</mi><mn>1</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mi mathvariant="normal">b</mi></msup><msup><mfenced close="]" open="[" separators="|"><mi mathvariant="normal">T</mi></mfenced><mi mathvariant="normal">c</mi></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>0</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mi mathvariant="normal">M</mi><mi mathvariant="normal">a</mi></msup><msup><mi mathvariant="normal">L</mi><mrow><mn>2</mn><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mi mathvariant="normal">b</mi><mo>+</mo><mi mathvariant="normal">c</mi></mrow></msup></math><br/><br/>On comparing,<br/><math><mi mathvariant="normal">a</mi><mo>=</mo><mn>1</mn></math><br/><math><mn>2</mn><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi mathvariant="normal">b</mi><mo>=</mo><mi mathvariant="normal"> </mi><mo>-</mo><mn>2</mn></math><br/><math><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mi mathvariant="normal">b</mi><mo>+</mo><mi mathvariant="normal">c</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>⇒</mo><mi mathvariant="normal">c</mi><mo>=</mo><mi mathvariant="normal"> </mi><mo>-</mo><mn>2</mn></math><br/><math><mo>∴</mo><mi mathvariant="normal">S</mi><mo>=</mo><msup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">V</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p></div>


title: if-entropy-of-a-solid-is-greater-than-zero-at-t-0-it-is-called
slug: if-entropy-of-a-solid-is-greater-than-zero-at-t-0-it-is-called-82240
tags: ['Chemistry', 'Thermodynamics (C)', 'MHT CET', 'MHT CET 2020 (16 Oct Shift 2)']
answer: <div class="solution">(C)</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If entropy of a solid is greater than zero, at $\mathrm{T}=0$, it is called</div>

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

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


title: if-equation-of-sound-wave-is-y-00015-sin-628-x-316-t-then-its-wavelength-will-be
slug: if-equation-of-sound-wave-is-y-00015-sin-628-x-316-t-then-its-wavelength-will-be-63201
tags: ['Physics', 'Waves and Sound', 'AIIMS', 'AIIMS 2002']
source_db: MarksBatch2_P1.db
question:
<div class="question">If equation of sound wave is $y=0.0015 \sin (62.8 x+316 t)$, then its wavelength will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.2 unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.3 unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0.1 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">2 unit.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">0.1 unit</span> </div>

answer:
<div class="solution">Equation of sound wave is<br/>$$<br/>y=0.0015 \sin (62.4 x+316 t) \text {. }<br/>$$<br/>Standard equation of a sound wave is<br/>$$<br/>y=A \sin 2 \pi\left[\frac{x}{\lambda}+\frac{t}{T}\right]<br/>$$<br/>Comparing the given equation with the standard equation, we get<br/>$$<br/>2 \pi / \lambda=62.8 \text { or, } \lambda=\frac{2 \pi}{62.8}=0.1 \text { unit. }<br/>$$<br/>(where $\lambda=$ wavelength of the wave)</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'TS EAMCET', 'TS EAMCET 2018 (05 May Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-equilibrium-constant-of-a-process-is-38-1-0-3-at-2-5-c-standard-free-energy-change-of-the-process-is-r-8314-j-mol-1-k-1-lo-g-00038-242

slug:
if-equilibrium-constant-of-a-process-is-38-1-0-3-at-2-5-c-standard-free-energy-change-of-the-process-is-r-8314-j-mol-1-k-1-lo-g-00038-242-99280

question:
<div class="question">If equilibrium constant of a process is $3.8 \times 10^{-3}$ at $25^{\circ} \mathrm{C}$, standard free energy change of the process is<br/>$$<br/>\left(R=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \log 0.0038=-2.42\right)<br/>$$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5.7 \mathrm{~kJ} \mathrm{~mol}^1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$9.9 \mathrm{~kJ} \mathrm{~mol}^1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$13.8 \mathrm{~kJ} \mathrm{~mol}^1$</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">$15.6 \mathrm{~kJ} \mathrm{~mol}^1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$13.8 \mathrm{~kJ} \mathrm{~mol}^1$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\Delta G^{\circ} &amp; =-R T \ln K \\<br/>\Delta G^{\circ} &amp; =-2.303 R T \log K \\<br/>\Delta G^{\circ} &amp; =-2.303 \times 8.314 \mathrm{~mol}^{-1} \times 298 \log 3.8 \times 10^{-3} \\<br/>\Delta G^{\circ} &amp; =13.8 \times 10^3 \mathrm{~J} \mathrm{~mol}^{-1} \\<br/>\Delta G^{\circ} &amp; =13.8 \mathrm{~kJ} \mathrm{~mol}^{-1}<br/>\end{aligned}<br/>$$</div>


title: if-f-9-9-and-f-9-4-then-what-is-lim-x-9-x-3-f-x-3-equal-to
slug: if-f-9-9-and-f-9-4-then-what-is-lim-x-9-x-3-f-x-3-equal-to-75917
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(9)=9$ and $f^{\prime}(9)=4$ then what is $\lim _{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">36</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$\lim _{\mathrm{x} \rightarrow 0} \frac{\sqrt{\mathrm{f}(\mathrm{x})}-3}{\sqrt{\mathrm{x}}-3}=\lim _{\mathrm{x} \rightarrow 0} \frac{\frac{1}{2 \sqrt{\mathrm{f}(\mathrm{x})}} \mathrm{f}^{\prime}(\mathrm{x})}{\frac{1}{2 \sqrt{\mathrm{x}}} .1}$<br/>(ByL' Hospital rule)<br/>$=\lim _{x \rightarrow 0} \frac{\mathrm{f}^{\prime}(\mathrm{x}) \times \sqrt{\mathrm{x}}}{\sqrt{\mathrm{f}(\mathrm{x})}}=\frac{\mathrm{f}^{\prime}(9) \times \sqrt{9}}{\sqrt{\mathrm{f}(9)}}$<br/>$=\frac{4 \times 3}{\sqrt{9}}=\frac{4 \times 3}{3}=4$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-f-and-g-are-two-increasing-functions-such-that-f-o-g-is-defined-then-which-one-of-the-following-is-correct

slug:
if-f-and-g-are-two-increasing-functions-such-that-f-o-g-is-defined-then-which-one-of-the-following-is-correct-36487

question:
<div class="question">If $f$ and $g$ are two increasing functions such that $f o g$ is defined, then which one of the following is correct?</div>

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

answer:
<div class="solution">Product of two increasing function is always an increasing function. $\therefore f o g$ is always an increasing function</div>


title: if-f-ir-ir-g-ir-ir-be-two-functions-given-by-f-x-2-x-3-and-g-x-x-3-5-then-fog-1-x-is-equal-to
slug: if-f-ir-ir-g-ir-ir-be-two-functions-given-by-f-x-2-x-3-and-g-x-x-3-5-then-fog-1-x-is-equal-to-92566
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}: \mathrm{IR} \rightarrow \mathrm{IR}, \mathrm{g}: \mathrm{IR} \rightarrow \mathrm{IR}$ be two functions given by<br/>$\mathrm{f}(\mathrm{x})=2 \mathrm{x}-3$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}^{3}+5$, then $(\mathrm{fog})^{-1}(\mathrm{x})$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\frac{x+7}{2}\right)^{\frac{1}{3}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\frac{x-7}{2}\right)^{\frac{1}{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(x-\frac{7}{2}\right)^{\frac{1}{3}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(x+\frac{7}{2}\right)^{\frac{1}{3}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{x-7}{2}\right)^{\frac{1}{3}}$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=2 \mathrm{x}-3$<br/>$\mathrm{~g}(\mathrm{x})=\mathrm{x}^{3}+5$<br/>$\operatorname{fog}(\mathrm{x})=\mathrm{f}[\mathrm{g}(\mathrm{x})]=\mathrm{f}\left(\mathrm{x}^{3}+5\right)$<br/>$=2\left(\mathrm{x}^{3}+5\right)-3$<br/>$\operatorname{fog}(\mathrm{x})=2 \mathrm{x}^{3}+7=\mathrm{y}(\mathrm{say})$<br/>$\Rightarrow 2 \mathrm{x}^{3}=\mathrm{y}-7$<br/>$\Rightarrow \mathrm{x}=\left[\frac{\mathrm{y}-7}{2}\right]^{\frac{1}{3}}$<br/>$\Rightarrow \operatorname{fog}^{-1}(\mathrm{y})=\left[\frac{\mathrm{y}-7}{2}\right]^{\frac{1}{3}}$<br/>$\Rightarrow \mathrm{fog}^{-1}(\mathrm{x})=\left[\frac{\mathrm{x}-7}{2}\right]^{\frac{1}{3}}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'AP EAMCET', 'AP EAMCET 2017 (25 Apr Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1$</span> </div>
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch2_P1.db
title:
if-f-is-defined-by-fxleftbeginarrayccfrac1cos-a-xx-sin-x-x-neq-0-frac12-quadx0endarrayright-and-f-is-continuous-at-x-0-then-a-2

slug:
if-f-is-defined-by-fxleftbeginarrayccfrac1cos-a-xx-sin-x-x-neq-0-frac12-quadx0endarrayright-and-f-is-continuous-at-x-0-then-a-2-49311

question:
<div class="question">If $f$ is defined by $f(x)=\left\{\begin{array}{cc}\frac{1-\cos a x}{x \sin x}, $(x \neq 0)$ \frac{1}{2} \quad,(x=0)\end{array}\right.$ and $f$ is continuous at $x=0$, then $a^2=$</div>

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


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
if-f-n-denotes-the-set-of-all-divisors-of-n-except-1-what-is-the-least-value-of-y-satisfying-f-20-f-16-f-y

slug:
if-f-n-denotes-the-set-of-all-divisors-of-n-except-1-what-is-the-least-value-of-y-satisfying-f-20-f-16-f-y-29434

question:
<div class="question">If $F(n)$ denotes the set of all divisors of $n$ except 1, what is the least value of y satisfying $[F(20) \cap F(16)] \subseteq F(y$</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">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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Given that $\mathrm{F}(\mathrm{n})=$ set of all divisors of $\mathrm{n}$ except $\mathrm{F}(20)=\{2,4,5,10,20\}$<br/>and $\mathrm{F}(16)=\{2,4,8,16\}$<br/>$\mathrm{F}(20) \cap \mathrm{F}(16)=\{2,4,5,10,20\} \cap\{2,4,8,16\}$<br/>$=\{2,4\}$<br/>Also, $\{\mathrm{F}(20) \cap \mathrm{F}(16)\} \subseteq \mathrm{F}(\mathrm{y})$<br/>So, least value of $y=2$</div>


title: if-f-r-r-g-r-r-and-g-x-x-3-and-f-g-x-x-3-2-then-what-is-the-value-of-f-3
slug: if-f-r-r-g-r-r-and-g-x-x-3-and-f-g-x-x-3-2-then-what-is-the-value-of-f-3-41995
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f: R \rightarrow R, g: R \rightarrow R$ and $g(x)=x+3$ and $(f \circ g)(x)=(x+3)^{2}$, then what is the value of $f(-3)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">3</span> </li> </ul>

answer:
<div class="solution">Given $f \circ g(x)=(x+3)^{2}$ and $g(x)=x+3$ $f o g(x)=(x+3)^{2} \Rightarrow f[g(x)]=(x+3)^{2}$<br/>$\Rightarrow \quad f[x+3]=(x+3)^{2}(\because g(x)=x+3)$<br/>$\Rightarrow f(x)=x^{2}$<br/>Hence, $f(-3)=(-3)^{2}=9$</div>


title: if-f-r-s-defined-by-f-x-4-sin-x-3-cos-x-1-is-onto-then-what-is-s-equal-to
slug: if-f-r-s-defined-by-f-x-4-sin-x-3-cos-x-1-is-onto-then-what-is-s-equal-to-44861
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}: \mathbb{R} \rightarrow \mathrm{S}$ defined by $\mathrm{f}(\mathrm{x})=4 \sin \mathrm{x}-3 \cos \mathrm{x}+1$ is onto,<br/>then what is S equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[-5,5]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-5,5)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-4,6)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$[-4,6]$</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">$[-4,6]$</span> </div>

answer:
<div class="solution">$f(x)=4 \sin x-3 \cos x+1$<br/>To find range of this function, we have to find maximum and minimum values.<br/>We know, for a $\sin \theta+\mathrm{b} \cos \theta+\mathrm{c}, \max$ value is<br/>$\sqrt{a^{2}+b^{2}}+c$ and minimum value is $-\sqrt{a^{2}+b^{2}}+c$. $\therefore$ Maximum value<br/>$\begin{array}{ll} &amp; =\sqrt{(4)^{2}+(-3)^{2}}+1=\sqrt{16+9}+1=5+1=6 \\ &amp; \text { Minimum value }=-\sqrt{16+9}+1=-5+1=-4 \\ \therefore \quad &amp; \text { Range, } \mathrm{S} \in[-4,6]\end{array}$</div>


title: if-f-x-1-0-x-is-a-rational-number-x-is-an-irrational-number-what-isare-the-values-of-f-of-3
slug: if-f-x-1-0-x-is-a-rational-number-x-is-an-irrational-number-what-isare-the-values-of-f-of-3-47510
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}1 &amp; \mathrm{x} \text { is a rational number } \\ 0, &amp; \mathrm{x} \text { is an irrational number, }\end{array}\right.$ what is/are the value(s) of (f of) $(\sqrt{3)}$ ?</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">Both 0 and 1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">From the given direction of function<br/>$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}1, &amp; \mathrm{x} \text { is a rational number } \\ 0, &amp; \mathrm{x} \text { is an irrational number }\end{array}\right.$<br/>$($ f 0 f $) \sqrt{3}=\mathrm{f}\{\mathrm{f}(\sqrt{3})\}$<br/>$=\mathrm{f}(0) \quad(\because \sqrt{3}$ is an irrational number $)$ $=1$<br/>$(\because 0$ is a rational number)</div>


title: if-f-x-1-4-x-3-4-x-x-4-and-g-x-in-1-x-1-x-then-what-is-the-value-of-f-g-e-1-e-1-equal-to
slug: if-f-x-1-4-x-3-4-x-x-4-and-g-x-in-1-x-1-x-then-what-is-the-value-of-f-g-e-1-e-1-equal-to-97927
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\frac{4 x+x^{4}}{1+4 x^{3}}$ and $g(x)=\operatorname{In}\left(\frac{1+x}{1-x}\right)$, then what is the<br/>value of $f^{\circ} \mathrm{g}\left(\frac{\mathrm{e}-1}{\mathrm{e}+1}\right)$ equal to?</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">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">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\frac{4 \mathrm{x}+\mathrm{x}^{4}}{1+4 \mathrm{x}^{3}}, \mathrm{~g}(\mathrm{x})=\ln \left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right)$<br/>$g\left(\frac{e-1}{e+1}\right)=\ell n\left(\frac{1+\left(\frac{e-1}{e+1}\right)}{1-\left(\frac{e-1}{e+1}\right)}\right)=\ell n\left(\frac{e+1+e-1}{e+1-e+1}\right)$<br/>$=\ln \left(\frac{2 \ell}{2}\right)=\ell$ n $\mathrm{e}=1$<br/>fo $\mathrm{g}\left(\frac{\mathrm{e}-1}{\mathrm{e}+1}\right)=\mathrm{f}(1)=\frac{4(1)+(1)^{4}}{1+4(1)^{3}}=\frac{4+1}{1+4}=\frac{5}{5}=1$</div>


title: if-f-x-1-f-x-2-f-1-x-1-x-2-x-1-x-2-for-x-1-x-2-1-1-then-what-is-f-x-equal-to
slug: if-f-x-1-f-x-2-f-1-x-1-x-2-x-1-x-2-for-x-1-x-2-1-1-then-what-is-f-x-equal-to-83858
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}\left(\mathrm{x}_{1}\right)-\mathrm{f}\left(\mathrm{x}_{2}\right)=\mathrm{f}\left(\frac{\mathrm{x}_{1}-\mathrm{x}_{2}}{1-\mathrm{x}_{1} \mathrm{x}_{2}}\right)$ for $\mathrm{x}_{1}, \mathrm{x}_{2} \in(-1,1)$, then what<br/>is $\mathrm{f}(\mathrm{x})$ equal to?</div>

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

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

answer:
<div class="solution">$f\left(x_{1}\right)-f\left(x_{2}\right)=f\left(\frac{x_{1}-x_{2}}{1-x_{1} x_{2}}\right)$<br/>$x_{1}, x_{2} \in(-1,1)$<br/>then $f(x)=\log \frac{(1-x)}{(1+x)}$<br/>$\begin{aligned} f\left(x_{1}\right)=\log \frac{1-x_{1}}{1+x_{1}} &amp; f\left(x_{2}\right)=\log \frac{1-x_{2}}{1+x_{2}} \\ f\left(x_{1}\right)-f\left(x_{2}\right) &amp;=\log \frac{1-x_{1}}{1+x_{1}}-\log \frac{1-x_{2}}{1+x_{2}} \\=&amp; \log \frac{\left(1-x_{1}\right)}{\left(1+x_{1}\right)} \times \frac{\left(1+x_{2}\right)}{\left(1-x_{2}\right)} \\=&amp; \log \frac{\left(1-x_{1}+x_{2}-x_{1} x_{2}\right)}{\left(1+x_{1}-x_{2}-x_{1} x_{2}\right)} \\=&amp; \log \frac{\left(1-x_{1} x_{2}\right)-\left(x_{1}-x_{2}\right)}{\left(1-x_{1} x_{2}\right)+\left(x_{1}-x_{2}\right)} \\ f\left(x_{1}\right)-f\left(x_{2}\right)=f\left(\frac{x_{1}-x_{2}}{1-x_{1} x_{2}}\right) \end{aligned}$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
title:
if-f-x-1-sin-2-x-sin-2-x-sin-2-x-cos-2-x-1-cos-2-x-cos-2-x-4-sin-2-x-4-sin-2-x-1-4-sin-2-x-what-is-the-maximum-value-of-f-x

slug:
if-f-x-1-sin-2-x-sin-2-x-sin-2-x-cos-2-x-1-cos-2-x-cos-2-x-4-sin-2-x-4-sin-2-x-1-4-sin-2-x-what-is-the-maximum-value-of-f-x-29809

question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc}1+\sin ^{2} \mathrm{x} &amp; \cos ^{2} \mathrm{x} &amp; 4 \sin 2 \mathrm{x} \\ \sin ^{2} \mathrm{x} &amp; 1+\cos ^{2} \mathrm{x} &amp; 4 \sin 2 \mathrm{x} \\ \sin ^{2} \mathrm{x} &amp; \cos ^{2} \mathrm{x} &amp; 1+4 \sin 2 \mathrm{x}\end{array}\right|$<br/>What is the maximum value of $\mathrm{f}(\mathrm{x})$?</div>

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc}1+\sin ^{2} \mathrm{x} &amp; \cos ^{2} x &amp; 4 \sin 2 \mathrm{x} \\ \sin ^{2} \mathrm{x} &amp; 1+\cos ^{2} \mathrm{x} &amp; 4 \sin 2 \mathrm{x} \\ \sin ^{2} \mathrm{x} &amp; \cos ^{2} \mathrm{x} &amp; 1+4 \sin 2 \mathrm{x}\end{array}\right|$<br/>Applying $\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}$<br/>$=\left|\begin{array}{ccc}2 &amp; \cos ^{2} x &amp; 4 \sin 2 x \\ 2 &amp; 1+\cos ^{2} x &amp; 4 \sin 2 x \\ 1 &amp; \cos ^{2} x &amp; 1+4 \sin 2 x\end{array}\right|$<br/>(Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ and $\left.\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\right)$<br/>$=\left|\begin{array}{ccc}2 &amp; \cos ^{2} x &amp; 4 \sin 2 x \\ 0 &amp; 1 &amp; 0 \\ -1 &amp; 0 &amp; 1\end{array}\right|$<br/>$\mathrm{f}(\mathrm{x})=2+4 \sin 2 \mathrm{x}$<br/>$\therefore \quad-1 \leq \sin 2 x \leq 1$, maximum value of $\sin 2 x=1$<br/>Thus, maximum value of $f(x)=2+4=6$</div>


title: if-f-x-1-x-5-x-is-continuous-at-x-0-then-what-is-the-value-of-f-0
slug: if-f-x-1-x-5-x-is-continuous-at-x-0-then-what-is-the-value-of-f-0-69594
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=(1+\mathrm{x})^{5 / \mathrm{x}}$ is continuous at $\mathrm{x}=0$, then what is the value of $\mathrm{f}(0)$ ?</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">$\infty$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{e}^{5}$</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> </ul>

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

answer:
<div class="solution">Given that $\mathrm{f}(\mathrm{x})=(1+\mathrm{x})^{5 / \mathrm{x}}$ and $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$. Value of function at $\mathrm{x}=0$ is same as limit of the function at $\mathrm{x}=0$.<br/>$\mathrm{f}(0)=\lim _{\mathrm{x} \rightarrow 0}(1+\mathrm{x})^{5 / \mathrm{x}}=\left\{\lim _{\mathrm{x} \rightarrow 0}(1+\mathrm{x})^{1 / \mathrm{x}}\right\}^{5}=\mathrm{e}^{5}$</div>


title: if-f-x-2-s-i-n-x-then-what-is-the-derivative-of-f-x
slug: if-f-x-2-s-i-n-x-then-what-is-the-derivative-of-f-x-19571
question: <div class="question">If $f(x)=2^{\sin x}$, then what is the derivative of $f(x)$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{\sin x} \ln 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(\sin x) 2^{2 \sin x-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(\cos x) 2^{2 \sin x-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of the above</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $f(x)=2^{\sin x}$. <br/>$\mathrm{f}^{\prime}(\mathrm{x})=2^{\sin \mathrm{x}} . \ln 2 \cos \mathrm{x}$</div>


title: if-f-x-2-x-1-then-on-the-interval-0-which-one-of-the-following-is-correct
slug: if-f-x-2-x-1-then-on-the-interval-0-which-one-of-the-following-is-correct-37756
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(\mathrm{x})=\frac{\mathrm{x}}{2}-1$, then on the interval $[0, \pi]$ which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan [f(\mathrm{x})]$, where $[\cdot]$ is the greatest integer function, and $\frac{1}{f(\mathrm{x})}$ are both discontinuous.</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}-1, \quad[0, \pi] \\ &amp; \tan \cdot \mathrm{f}(\mathrm{x})=\tan \left(\frac{\mathrm{x}}{2}-1\right) \end{aligned}$<br/>$\frac{1}{\mathrm{f}(\mathrm{x})}=\frac{1}{\frac{\mathrm{x}}{2}-1}$ is discontinuous at $\mathrm{x}=2$<br/>tan. $\mathrm{f}(\mathrm{x})$ is discontinuous for $\mathrm{x}=2$ in $[0, \pi]$</div>


title: if-f-x-2-x-2-3-x-5-then-what-is-f-0-3-f-1-equal-to
slug: if-f-x-2-x-2-3-x-5-then-what-is-f-0-3-f-1-equal-to-87736
tags: ['Mathematics', 'Application of Derivatives', '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_P1.db
question:
<div class="question">If $f(x)=2 x^{2}+3 x-5$, then what is $f^{\prime}(0)+3 f^{\prime}(-1)$ equal to?</div>

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

answer:
<div class="solution">$\mathrm{F}(\mathrm{x})=2 \mathrm{x}^{2}+3 \mathrm{x}-5$<br/>$\mathrm{~F}^{\prime}(\mathrm{x})=4 \mathrm{x}+3$<br/>$\mathrm{~F}^{\prime}(0)+3 \mathrm{~F}^{\prime}(-1)=3+3(-4+3)=0$</div>


title: if-f-x-2-x-7-and-g-x-x-2-7-x-r-then-what-are-values-of-x-for-which-fog-x-25
slug: if-f-x-2-x-7-and-g-x-x-2-7-x-r-then-what-are-values-of-x-for-which-fog-x-25-29210
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=2 x+7$ and $g(x)=x^{2}+7, x \in R$, then what are values of $x$ for which $\operatorname{fog}(x)=25 ?$</div>

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

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

answer:
<div class="solution">Let, $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+7$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+7, \mathrm{x} \in \mathrm{R}$<br/>Now fog $(x)=f[g(x)]=f\left(x^{2}+7\right)$<br/>$=2\left(\mathrm{x}^{2}+7\right)+7=2 \mathrm{x}^{2}+14+7$<br/>But fog $(x)=25$ $\Rightarrow \quad 2 x^{2}+21=25$<br/>$\Rightarrow \quad x^{2}=2$<br/>$\Rightarrow \quad \mathrm{x}=\pm \sqrt{2}$</div>


title: if-f-x-2-x-then-what-is-f-x-equal-to
slug: if-f-x-2-x-then-what-is-f-x-equal-to-82242
question: <div class="question">If $f(x)=2^{x}$, then what is $f^{\prime \prime}(x)$ equal to?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2^{x}(\ln 2)^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x(x-1) 2^{x-2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{x+1}(\ln 2)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2^{x}\left(\log _{10} 2\right)^{2}$</span> </li> </ul>

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

answer:
<div class="solution">Let $f(x)=2^{x}$<br/>On differentiating w.r.t. $x$, we get<br/>$f^{\prime}(x)=2^{x}(\ln 2)$<br/>On again differentiating w.r.t. $x$, we get <br/>$f^{\prime \prime}(x)=2^{x}(\ln 2)^{2}$</div>


title: if-f-x-3-1-x-then-f-x-f-y-f-z-is-equal-to
slug: if-f-x-3-1-x-then-f-x-f-y-f-z-is-equal-to-76266
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=3^{1+\mathrm{x}}$, then $\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y}) \mathrm{f}(\mathrm{z})$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x+y+z+2)$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=3^{1+\mathrm{x}}$<br/>$\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y}) \cdot \mathrm{f}(\mathrm{z})=3^{1+\mathrm{x}} \cdot 3^{1+\mathrm{y}} \cdot 3^{1+\mathrm{z}}$<br/>$=3^{1+\mathrm{x}+1+\mathrm{y}+1+\mathrm{z}}=3^{3+\mathrm{x}+\mathrm{y}+\mathrm{z}}=3^{1+\mathrm{x}+\mathrm{y}+\mathrm{z}+2}$<br/>$=\mathrm{f}(\mathrm{x}+\mathrm{y}+\mathrm{z}+2)$</div>


title: if-f-x-3-2-x-2-3-x-r-2010ii-then-what-is-f-1-x-equal-to
slug: if-f-x-3-2-x-2-3-x-r-2010ii-then-what-is-f-1-x-equal-to-30437
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\frac{2}{3} x+\frac{3}{2}, x \in R, \quad$ [2010-II]<br/>then what is $f^{-1}(x)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{2} x+\frac{2}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{3}{2} x-\frac{9}{4}$</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">$\frac{2}{3} x-\frac{4}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3} x-\frac{2}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{2} x-\frac{9}{4}$</span> </div>

answer:
<div class="solution">Let $f(x)=\frac{2}{3} x+\frac{3}{2}=y($ say $)=\frac{4 x+9}{6}=y$<br/>$\Rightarrow \quad 4 x+9=6 y$<br/>$\Rightarrow \quad x=\frac{6 y-9}{4}$<br/>$\quad x=f^{-1}(y)$<br/>$\Rightarrow \quad f^{-1}(x)=\frac{6 x-9}{4}=\frac{3 x}{2}-\frac{9}{4}$</div>


title: if-f-x-3-x-2-6-x-9-then
slug: if-f-x-3-x-2-6-x-9-then-66240
question: <div class="question">If $f(x)=3 x^{2}+6 x-9$, then</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)$ is increasing in $(-1,3)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(x)$ is decreasing in $(3, \infty)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(x)$ is increasing in $(-\infty,-1)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$f(x)$ is decreasing in $(-\infty,-1)$</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">$f(x)$ is decreasing in $(-\infty,-1)$</span> </div>

answer:
<div class="solution">Given $f(x)=3 x^{2}+6 x-9$ On differentiating w.r.t. $x$, we get $f^{\prime}(x)=6 x+6$<br/>$\begin{aligned} &amp; f^{\prime}(x)   &lt; 0 \\ \Rightarrow &amp; 6 x+6 &lt; 0 \end{aligned}$<br/>$\Rightarrow 6 \mathrm{x} &lt; -6$<br/>$\Rightarrow \mathrm{x} &lt; -1$<br/>Hence $f(x)$ is decreasing in $(-\infty,-1)$</div>


title: if-f-x-3-x-4-0-x-2-2-x-2-x-3-is-continuous-at-x-2-then-what-is-the-value-of
slug: if-f-x-3-x-4-0-x-2-2-x-2-x-3-is-continuous-at-x-2-then-what-is-the-value-of-45247
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\left\{\begin{array}{ll}3 x-4,0 \leq x \leq 2 \\ 2 x+\lambda, 2   &lt; x \leq 3\end{array} \right.$<br/>is continuous at $x=2$, then what is the value of $\lambda$ ?</div>

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

answer:
<div class="solution">Given $f(x)=\left\{\begin{array}{l}3 x-4,0 \leq x \leq 2 \\ 2 x+\lambda, 2   &lt; x \leq 3\end{array}\right.$<br/>Also $f(x)$ is continuous at $x=2$<br/>$\therefore \lim _{x \rightarrow 2} f(x)=f(2)$<br/>Now, $f(2)=3 \times 2-4=6-4=2$<br/>$\Rightarrow \lim _{x \rightarrow 2}(2 x+\lambda)=2$<br/>$\Rightarrow 4+\lambda=2$<br/>$\Rightarrow \quad \lambda=-2$</div>


title: if-f-x-9-x-2-then-what-is-lim-x-1-x-1-f-x-f-1-equal-to
slug: if-f-x-9-x-2-then-what-is-lim-x-1-x-1-f-x-f-1-equal-to-30962
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{F}(\mathrm{x})=\sqrt{9-\mathrm{x}^{2}}$, then what is $\lim _{\mathrm{x} \rightarrow 1} \frac{\mathrm{F}(\mathrm{x})-\mathrm{F}(1)}{\mathrm{x}-1}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{4 \sqrt{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{8}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{1}{2 \sqrt{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2 \sqrt{2}}$</span> </li> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow 1} \frac{F(x)-F(1)}{x-1}$<br/>$\quad$ We know, $_{x \rightarrow a}{\lim } \frac{F(n)-F(a)}{x-a}=F^{\prime}(a)$<br/>$\therefore F^{\prime}(x)=\frac{d}{d x}\left(\sqrt{9-x^{2}}\right)$<br/>$=\frac{1(0-2 x)}{2 \sqrt{9-x^{2}}}=\frac{-x}{\sqrt{9-x^{2}}}$<br/>$\quad F^{\prime}(1)=\frac{-1}{\sqrt{9-1}}=\frac{-1}{\sqrt{8}}=\frac{-1}{2 \sqrt{2}}$</div>


title: if-f-x-25-x-2-then-what-is-lim-x-1-x-1-f-x-f-1-equal-to-1
slug: if-f-x-25-x-2-then-what-is-lim-x-1-x-1-f-x-f-1-equal-to-1-55760
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\sqrt{25-\mathrm{x}^{2}}$, then what is $\lim _{\mathrm{x} \rightarrow 1} \frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(1)}{\mathrm{x}-1}$ equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{1}{\sqrt{24}}$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow 1} \frac{f(x)-f(1)}{x-1}=f^{\prime}(x)$<br/>Now, $f^{\prime}(x)=\frac{-2 x}{2 \sqrt{25-x^{2}}}=\frac{-x}{\sqrt{25-x^{2}}}$<br/>$f^{\prime}(1)=\frac{-1}{\sqrt{25-1}}=\frac{-1}{\sqrt{24}}$</div>


title: if-f-x-25-x-2-then-what-is-lim-x-1-x-1-f-x-f-1-equal-to
slug: if-f-x-25-x-2-then-what-is-lim-x-1-x-1-f-x-f-1-equal-to-70796
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\sqrt{25-\mathrm{x}^{2}}$, then what is $\operatorname{Lim}_{\mathrm{x} \rightarrow 1} \frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(1)}{\mathrm{x}-1}$ equal to ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{1}{\sqrt{24}}$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\sqrt{25-\mathrm{x}^{2}}$<br/>$\mathrm{f}(1)=\sqrt{24}$<br/>$\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(1)}{\mathrm{x}-1}$<br/>$\therefore$ It is $\frac{0}{0}($ undefined condition $)$ so using L'hospital's<br/>rule<br/>$\Rightarrow \lim _{x \rightarrow 1} \frac{\mathrm{f}^{\prime}(\mathrm{x})-0}{1}=\lim _{\mathrm{x} \rightarrow 1}\left(\sqrt{25-\mathrm{x}^{2}}\right)^{\prime}$<br/>$\Rightarrow \lim _{x \rightarrow 1} \frac{1}{2} \times \frac{1}{\sqrt{25-x^{2}}}(-2 x)$<br/>$=\frac{1}{2} \times \frac{1}{\sqrt{25-(1)^{2}}} \times(-2)$<br/>$=-\frac{1}{\sqrt{24}}$</div>


title: if-f-x-a-b-x-c-x-2-then-what-is-0-1-f-x-d-x-equal-to
slug: if-f-x-a-b-x-c-x-2-then-what-is-0-1-f-x-d-x-equal-to-88667
question: <div class="question">If $f(x)=a+b x+c x^{2}$, then what is $\int_{0}^{1} f(x) d x$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$[f(0)+4 f(1 / 2)+f(1)] / 6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$[f(0)+4 f(1 / 2)+f(1)] / 3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[f(0)+4 f(1 / 2)+f(1)]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$[f(0)+2 f(1 / 2)+f(1)] / 6$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$[f(0)+4 f(1 / 2)+f(1)] / 6$</span> </div>

answer:
<div class="solution">Given, $f(x)=a+b x+c x^{2}$<br/>$\therefore \int_{0}^{1} f(x) d x=\int_{0}^{1}\left(a+b x+c x^{2}\right) d x$<br/>$=\left[a x+\frac{b x^{2}}{2}+\frac{c x^{3}}{3}\right]_{0}^{1}$<br/>$=a+\frac{b}{2}+\frac{c}{3}$<br/>Here, $f(0)=a, f\left(\frac{1}{2}\right)=a+\frac{b}{2}+\frac{c}{4}$<br/>and $f(1)=a+b+c$<br/>Now, $\frac{f(0)+4 f\left(\frac{1}{2}\right)+f(1)}{6}$<br/>$=\frac{a+4\left(a+\frac{b}{2}+\frac{c}{4}\right)+a+b+c}{6}$<br/>$=\frac{a+4\left(\frac{4 a+2 b+c}{4}\right)+a+b+c}{6}$<br/>$=\frac{a+4 a+2 b+c+a+b+c}{6}=\frac{6 a+3 b+2 c}{6}$<br/>$=a+\frac{b}{2}+\frac{c}{3}$<br/>$\therefore$ From Eqs. (i) and (ii), we get<br/>$\int_{0}^{1} f(x) d x=\frac{f(0)+4 f\left(\frac{1}{2}\right)+f(1)}{6}$</div>


title: if-f-x-a-sin-2-x-b-and-f-2-1-2-and-0-1-f-x-dx-2-a-then-what-is-the-value-of-b
slug: if-f-x-a-sin-2-x-b-and-f-2-1-2-and-0-1-f-x-dx-2-a-then-what-is-the-value-of-b-89886
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">$\quad$ If $f(x)=A \sin \left(\frac{\pi x}{2}\right)+B$ and $f^{\prime}\left(\frac{1}{2}\right)=\sqrt{2}$ and<br/>$\int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{2 \mathrm{~A}}{\pi}$, then what is the value of $\mathrm{B}$ ?</div>

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

answer:
<div class="solution">Given function $\mathrm{f}(\mathrm{x})=\mathrm{A} \sin \left(\frac{\pi \mathrm{x}}{2}\right)+\mathrm{B}$<br/>Differentiating w. r. t. x $f^{\prime}(x)=A \cos \left(\frac{\pi x}{2}\right) \cdot \frac{\pi}{2}$<br/>$\mathrm{f}^{\prime}\left(\frac{1}{2}\right)=\sqrt{2}=\mathrm{A}\left(\cos \frac{\pi}{4}\right) \frac{\pi}{2}=\mathrm{A} \cdot \frac{1}{\sqrt{2}} \cdot \frac{\pi}{2}$<br/>$\Rightarrow \quad A=\frac{(\sqrt{2} \times \sqrt{2}) \times 2}{\pi}=\frac{4}{\pi}$<br/>Now, $\int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\frac{2 \mathrm{~A}}{\pi}$<br/>$\Rightarrow \quad \int_{0}^{1}\left\{\mathrm{~A} \sin \left(\frac{\pi \mathrm{x}}{2}\right)+\mathrm{B}\right\} \mathrm{d} \mathrm{x}=\frac{2 \times 4}{\pi^{2}}$<br/>$\Rightarrow\left[-\mathrm{A} \cos \frac{\pi \mathrm{x}}{2} \cdot \frac{2}{\pi}+\mathrm{Bx}\right]_{0}^{1}=\frac{8}{\pi^{2}}$<br/>$\Rightarrow \quad-\frac{4}{\pi} \cdot \frac{2}{\pi} \cos \frac{\pi}{2}+\mathrm{B}+\frac{4}{\pi} \cdot \frac{2}{\pi} \cos 0=\frac{8}{\pi^{2}}$<br/>$\Rightarrow \quad B+\frac{8}{\pi^{2}}=\frac{8}{\pi^{2}} \Rightarrow B=0$</div>


title: if-f-x-a-x-2-b-x-1-3-x-2-if-2-x-3-2-if-2-1-x-2-5-is-continuous-on-its-domain-then-a-b-has-the-value
tags: ['Mathematics', 'Continuity and Differentiability', 'MHT CET', 'MHT CET 2022 (06 Aug Shift 2)']
source_db: MarksBatch2_P1.db
slug:
if-f-x-a-x-2-b-x-1-3-x-2-if-2-x-3-2-if-2-1-x-2-5-is-continuous-on-its-domain-then-a-b-has-the-value-88085

question:
<div class="question">If $f(x)= \begin{cases}a x^2+b x+1 &amp; \text { if }|2 x-3| \geq 2 \\ 3 x+2 &amp; \text {; if } \frac{1}{2}<x continuous="" domain="" has="" is="" its="" on="" the="" then="" value=""> <div class="tags" data-pagefind-ignore=""> <span class="tag">Mathematics</span><span class="tag">Continuity and Differentiability</span><span class="tag">MHT CET</span><span class="tag">MHT CET 2022 (06 Aug Shift 2)</span> </div> <span class="label">Options:</span>
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{23}{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{1}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{13}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{31}{5}$</span> </li> </ul> <!-- /Question --> <!-- Solution --> <section class="solution-container" id="accepted-answer"> <div class="solution-header"> <span class="label">Solution:</span> <div class="badges-container" data-pagefind-ignore=""> <span class="badge upvotes-count"> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.625 5.0075C14.4842 4.84795 14.3111 4.72019 14.1171 4.63269C13.9231 4.54519 13.7128 4.49996 13.5 4.5H10V3.5C10 2.83696 9.73661 2.20107 9.26777 1.73223C8.79893 1.26339 8.16304 1 7.5 1C7.40711 0.999934 7.31604 1.02574 7.237 1.07454C7.15795 1.12333 7.09407 1.19318 7.0525 1.27625L4.69125 6H2C1.73478 6 1.48043 6.10536 1.29289 6.29289C1.10536 6.48043 1 6.73478 1 7V12.5C1 12.7652 1.10536 13.0196 1.29289 13.2071C1.48043 13.3946 1.73478 13.5 2 13.5H12.75C13.1154 13.5001 13.4684 13.3668 13.7425 13.1252C14.0166 12.8835 14.1931 12.5501 14.2388 12.1875L14.9888 6.1875C15.0153 5.97626 14.9966 5.76179 14.9339 5.55833C14.8712 5.35488 14.7659 5.16711 14.625 5.0075ZM2 7H4.5V12.5H2V7Z" fill="#014A82"></path> </svg> 1892 Upvotes
</span> <span class="badge expert-verification-badge" data-pagefind-ignore=""> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.1163 6.42625C13.8806 6.18 13.6369 5.92625 13.545 5.70312C13.46 5.49875 13.455 5.16 13.45 4.83187C13.4406 4.22187 13.4306 3.53062 12.95 3.05C12.4694 2.56937 11.7781 2.55937 11.1681 2.55C10.84 2.545 10.5012 2.54 10.2969 2.455C10.0744 2.36312 9.82 2.11937 9.57375 1.88375C9.1425 1.46937 8.6525 1 8 1C7.3475 1 6.85812 1.46937 6.42625 1.88375C6.18 2.11937 5.92625 2.36312 5.70312 2.455C5.5 2.54 5.16 2.545 4.83187 2.55C4.22187 2.55937 3.53062 2.56937 3.05 3.05C2.56937 3.53062 2.5625 4.22187 2.55 4.83187C2.545 5.16 2.54 5.49875 2.455 5.70312C2.36312 5.92562 2.11937 6.18 1.88375 6.42625C1.46937 6.8575 1 7.3475 1 8C1 8.6525 1.46937 9.14187 1.88375 9.57375C2.11937 9.82 2.36312 10.0738 2.455 10.2969C2.54 10.5012 2.545 10.84 2.55 11.1681C2.55937 11.7781 2.56937 12.4694 3.05 12.95C3.53062 13.4306 4.22187 13.4406 4.83187 13.45C5.16 13.455 5.49875 13.46 5.70312 13.545C5.92562 13.6369 6.18 13.8806 6.42625 14.1163C6.8575 14.5306 7.3475 15 8 15C8.6525 15 9.14187 14.5306 9.57375 14.1163C9.82 13.8806 10.0738 13.6369 10.2969 13.545C10.5012 13.46 10.84 13.455 11.1681 13.45C11.7781 13.4406 12.4694 13.4306 12.95 12.95C13.4306 12.4694 13.4406 11.7781 13.45 11.1681C13.455 10.84 13.46 10.5012 13.545 10.2969C13.6369 10.0744 13.8806 9.82 14.1163 9.57375C14.5306 9.1425 15 8.6525 15 8C15 7.3475 14.5306 6.85812 14.1163 6.42625ZM10.8538 6.85375L7.35375 10.3538C7.30731 10.4002 7.25217 10.4371 7.19147 10.4623C7.13077 10.4874 7.06571 10.5004 7 10.5004C6.93429 10.5004 6.86923 10.4874 6.80853 10.4623C6.74783 10.4371 6.69269 10.4002 6.64625 10.3538L5.14625 8.85375C5.05243 8.75993 4.99972 8.63268 4.99972 8.5C4.99972 8.36732 5.05243 8.24007 5.14625 8.14625C5.24007 8.05243 5.36732 7.99972 5.5 7.99972C5.63268 7.99972 5.75993 8.05243 5.85375 8.14625L7 9.29313L10.1462 6.14625C10.1927 6.09979 10.2479 6.06294 10.3086 6.0378C10.3692 6.01266 10.4343 5.99972 10.5 5.99972C10.5657 5.99972 10.6308 6.01266 10.6914 6.0378C10.7521 6.06294 10.8073 6.09979 10.8538 6.14625C10.9002 6.1927 10.9371 6.24786 10.9622 6.30855C10.9873 6.36925 11.0003 6.4343 11.0003 6.5C11.0003 6.5657 10.9873 6.63075 10.9622 6.69145C10.9371 6.75214 10.9002 6.8073 10.8538 6.85375Z" fill="#00672E"></path> </svg>
Verified Answer
</span> </div> </div> <div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{23}{5}$</span> </div> <div class="solution">$\begin{aligned} &amp; f(x)=\left\{\begin{array}{ccc}a x^2+b x+1 &amp; ; &amp; |2 x-3| \geq 2 \\ 3 x+2 &amp; ; &amp; \frac{1}{2}<x a="" x=""></x>for continuity at $x=\frac{1}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/B7jw2n1orE52k3R183BlwbMF_y5zSlpUa3NEqXT8-SY.original.fullsize.png"/><br/>for continuity at $x=\frac{5}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/hUBSqhM293JDBdKv-vpL8SqOFYqHanhJjHLwOXvBBFI.original.fullsize.png"/><br/>from (1) and (2)<br/>$a=-\frac{4}{5}$ and $b=\frac{27}{5}$<br/>$a+b=\frac{-4+27}{5}=\frac{23}{5}$</div> </section> <!-- /Solution --> <!-- Call to Action --> <section class="call-to-action" data-pagefind-ignore=""> <div class="call-to-action-header"> <svg fill="none" height="24" viewbox="0 0 25 24" width="25" xmlns="http://www.w3.org/2000/svg"> <path d="M19.9999 13.5C20.0018 13.8058 19.9089 14.1047 19.734 14.3555C19.559 14.6063 19.3107 14.7968 19.023 14.9006L14.1874 16.6875L12.4062 21.5269C12.3007 21.8134 12.1098 22.0608 11.8594 22.2355C11.6089 22.4101 11.3109 22.5038 11.0055 22.5038C10.7002 22.5038 10.4022 22.4101 10.1517 22.2355C9.90124 22.0608 9.71039 21.8134 9.60491 21.5269L7.8124 16.6875L2.97303 14.9062C2.68646 14.8008 2.43915 14.6099 2.26445 14.3595C2.08976 14.109 1.99609 13.811 1.99609 13.5056C1.99609 13.2003 2.08976 12.9022 2.26445 12.6518C2.43915 12.4013 2.68646 12.2105 2.97303 12.105L7.8124 10.3125L9.59366 5.47313C9.69914 5.18656 9.88998 4.93924 10.1404 4.76455C10.3909 4.58985 10.6889 4.49619 10.9943 4.49619C11.2996 4.49619 11.5977 4.58985 11.8481 4.76455C12.0986 4.93924 12.2894 5.18656 12.3949 5.47313L14.1874 10.3125L19.0268 12.0938C19.3146 12.1986 19.5628 12.3901 19.7371 12.642C19.9114 12.8939 20.0033 13.1937 19.9999 13.5ZM14.7499 4.5H16.2499V6C16.2499 6.19891 16.3289 6.38968 16.4696 6.53033C16.6102 6.67098 16.801 6.75 16.9999 6.75C17.1988 6.75 17.3896 6.67098 17.5302 6.53033C17.6709 6.38968 17.7499 6.19891 17.7499 6V4.5H19.2499C19.4488 4.5 19.6396 4.42098 19.7802 4.28033C19.9209 4.13968 19.9999 3.94891 19.9999 3.75C19.9999 3.55109 19.9209 3.36032 19.7802 3.21967C19.6396 3.07902 19.4488 3 19.2499 3H17.7499V1.5C17.7499 1.30109 17.6709 1.11032 17.5302 0.96967C17.3896 0.829018 17.1988 0.75 16.9999 0.75C16.801 0.75 16.6102 0.829018 16.4696 0.96967C16.3289 1.11032 16.2499 1.30109 16.2499 1.5V3H14.7499C14.551 3 14.3602 3.07902 14.2196 3.21967C14.0789 3.36032 13.9999 3.55109 13.9999 3.75C13.9999 3.94891 14.0789 4.13968 14.2196 4.28033C14.3602 4.42098 14.551 4.5 14.7499 4.5ZM22.9999 7.5H22.2499V6.75C22.2499 6.55109 22.1709 6.36032 22.0302 6.21967C21.8896 6.07902 21.6988 6 21.4999 6C21.301 6 21.1102 6.07902 20.9696 6.21967C20.8289 6.36032 20.7499 6.55109 20.7499 6.75V7.5H19.9999C19.801 7.5 19.6102 7.57902 19.4696 7.71967C19.3289 7.86032 19.2499 8.05109 19.2499 8.25C19.2499 8.44891 19.3289 8.63968 19.4696 8.78033C19.6102 8.92098 19.801 9 19.9999 9H20.7499V9.75C20.7499 9.94891 20.8289 10.1397 20.9696 10.2803C21.1102 10.421 21.301 10.5 21.4999 10.5C21.6988 10.5 21.8896 10.421 22.0302 10.2803C22.1709 10.1397 22.2499 9.94891 22.2499 9.75V9H22.9999C23.1988 9 23.3896 8.92098 23.5302 8.78033C23.6709 8.63968 23.7499 8.44891 23.7499 8.25C23.7499 8.05109 23.6709 7.86032 23.5302 7.71967C23.3896 7.57902 23.1988 7.5 22.9999 7.5Z" fill="#1589EE"></path> </svg> <h3>Looking for more such questions to practice?</h3> </div> <p>
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options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{23}{5}$</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">$\frac{1}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{13}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{31}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{23}{5}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; f(x)=\left\{\begin{array}{ccc}a x^2+b x+1 &amp; ; &amp; |2 x-3| \geq 2 \\ 3 x+2 &amp; ; &amp; \frac{1}{2}<x a="" x=""></x>for continuity at $x=\frac{1}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/B7jw2n1orE52k3R183BlwbMF_y5zSlpUa3NEqXT8-SY.original.fullsize.png"/><br/>for continuity at $x=\frac{5}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/hUBSqhM293JDBdKv-vpL8SqOFYqHanhJjHLwOXvBBFI.original.fullsize.png"/><br/>from (1) and (2)<br/>$a=-\frac{4}{5}$ and $b=\frac{27}{5}$<br/>$a+b=\frac{-4+27}{5}=\frac{23}{5}$</div>


tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-f-x-and-g-x-are-continuous-functions-satisfying-f-x-f-a-x-and-g-x-g-a-x-2-then-what-is-0-a-f-x-g-x-d-x-equal-to

slug:
if-f-x-and-g-x-are-continuous-functions-satisfying-f-x-f-a-x-and-g-x-g-a-x-2-then-what-is-0-a-f-x-g-x-d-x-equal-to-87047

question:
<div class="question">If $f(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are continuous functions satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then what is<br/>$\int_{0}^{a} f(x) g(x) d x$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\int_{0}^{a} g(x) d x$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\int_{0}^{a} f(x) d x$</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">$2 \int_{0}^{a} f(x) d x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\int_{0}^{a} f(x) d x$</span> </div>

answer:
<div class="solution">$\begin{aligned} I &amp;=\int_{0}^{a} f(x) \cdot g(x) d x \\ &amp; I=\int_{0}^{a} f(a-x) \cdot g(a-x) d x \\ &amp; I=\int_{0}^{a} f(x) \cdot[2-g(x)] d x \end{aligned}$<br/>$\therefore f(a-x)=f(x)$ and $g(a-x)=2 g(x)$<br/>$\mathrm{I}-\int_{0}^{\mathrm{a}} 2 . \mathrm{f}(\mathrm{x}) \cdot \mathrm{d} \mathrm{x}-\int_{0}^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{s}) \cdot \mathrm{d} \mathrm{x}$<br/>$\mathrm{I}=2 \cdot \int_{0}^{\mathrm{a}} \mathrm{f}(\mathrm{x})-\mathrm{I}$<br/>$\Rightarrow \not z^{\prime} I=\not^{\prime} \int_{0}^{a} f(x) d x$<br/>$\therefore I=\int_{0}^{a} f(x) d x$</div>


title: if-f-x-cos-x-g-x-lo-g-x-and-y-gof-x-then-what-is-the-value-of-d-x-d-y-at-x-0
slug: if-f-x-cos-x-g-x-lo-g-x-and-y-gof-x-then-what-is-the-value-of-d-x-d-y-at-x-0-34088
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\cos x, g(x)=\log x$ and $y=($ gof $)(x)$, then what is the value of $\frac{d y}{d x}$ at $x=0 ?$</div>

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

answer:
<div class="solution">Given $f(x)=\cos x$ and $g(x)=\log x$<br/>Consider $\mathrm{y}=\operatorname{gof}(x)$<br/>$\quad=g(f(x)\}$<br/>$\quad=\log (f(x))$<br/>$\quad=\log (\cos x)$<br/>$\therefore \frac{d y}{d x}=\frac{1}{\cos x}(-\sin x)=-\tan x$<br/>$\Rightarrow\left(\frac{d y}{d x}\right)_{x=0}=-\tan 0=0$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-f-x-cos-x-sin-x-0-sin-x-cos-x-0-0-0-1-then-which-of-the-following-are-correct-1-f-f-f-2-the-value-of-the-determinant-of-the-matrix-f-f-is-1-3-the-d

slug:
if-f-x-cos-x-sin-x-0-sin-x-cos-x-0-0-0-1-then-which-of-the-following-are-correct-1-f-f-f-2-the-value-of-the-determinant-of-the-matrix-f-f-is-1-3-the-d-76029

question:
<div class="question">If $\mathrm{f}\{\mathrm{x}\}=\left[\begin{array}{lcl}\cos \mathrm{x} &amp; -\sin x &amp; 0 \\ \sin \mathrm{x} &amp; \cos \mathrm{x} &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$, then which of the following are<br/>correct ?<br/>$1.$ $\quad \mathrm{f}(\theta) \times \mathrm{f}(\phi)=\mathrm{f}(\theta+\phi)$<br/>$2.$ The value of the determinant of the matrix $\mathrm{f}(\theta) \times \mathrm{f}(\phi)$ is 1 .<br/>$3.$ The determinant of $f(x)$ is an even function. <br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">$\quad f(\theta)=\left[\begin{array}{ccc}\cos \theta &amp; -\sin \theta &amp; 0 \\ \sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right] \&amp; f(\phi)\left[\begin{array}{ccc}\cos \phi &amp; -\sin \phi &amp; 0 \\ \sin \phi &amp; \cos \phi &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>$\mathbf{f}(\theta) \times \mathrm{f}(\phi)=\left[\begin{array}{ccc}\cos \theta \cos \phi-\sin \theta \sin \phi &amp; -\cos \theta \sin \phi-\sin \theta \cos \phi &amp; 0 \\ \sin \theta \cos \phi+\cos \theta \sin \phi &amp; -\sin \theta \sin \phi+\cos \theta \cos \phi &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>(using Trigonometric Identities)<br/>$\Rightarrow \quad \mathrm{f}(\theta) \times \mathrm{f}(\phi)=\mathrm{f}(\theta+\phi)$<br/>Also, det, $[\mathrm{f}(\theta) \times \mathrm{f}(\phi)]=1\left[\cos ^{2}(\theta+\phi)-\left(-\sin ^{2}(\theta+\phi)\right)\right]$<br/>$=\cos ^{2}(\theta+\phi)+\sin ^{2}(\theta+\phi)=1 .$<br/>$\&amp; \operatorname{det}(\mathrm{f}(x))=\left(\cos ^{2} x-\left(-\sin ^{2} x\right)=\cos ^{2} x+\sin ^{2} x=1 .\right.$<br/>For $x=-x$<br/>det. $(\mathrm{f}(-x))=\cos ^{2}(-x)+\operatorname{Sin}^{2}(-x)=1$<br/>Hence, det. $(\mathrm{f}(-x))=\operatorname{det}(\mathrm{f}(x))$<br/>Hence, $\operatorname{det} .(\mathrm{f}(\mathrm{x}))$ is even function.</div>


title: if-f-x-dx-lo-g-f-x-2-c-then-what-is-f-x-equal-to
slug: if-f-x-dx-lo-g-f-x-2-c-then-what-is-f-x-equal-to-81029
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int \frac{\mathrm{dx}}{\mathrm{f}(\mathrm{x})}=\log \{\mathrm{f}(\mathrm{x})\}^{2}+\mathrm{c}$, then what is $\mathrm{f}(\mathrm{x})$ equal to?</div>

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

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

answer:
<div class="solution">We check from the given options one by one. Options<br/>(a) and (b) do not satisfy. We check option (c).<br/>Let $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}+\alpha$<br/>$\therefore \int \frac{\mathrm{dx}}{\frac{\mathrm{x}}{2}+\alpha}=\int \frac{2 \mathrm{dx}}{(\mathrm{x}+2 \alpha)}$<br/>$=2 \log (x+2 \alpha)+c_{1}=\log (x+2 \alpha)^{2}+c_{1}$<br/>$=\log \left(\frac{x}{2}+\alpha\right)^{2}+\log 2^{2}+c_{1}=\log \left(\frac{x}{2}+\alpha\right)^{2}+c$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-f-x-e-s-i-n-l-o-g-c-o-s-x-and-g-x-lo-g-cos-x-then-what-is-the-derivative-of-f-x-with-respect-to-g-x

slug:
if-f-x-e-s-i-n-l-o-g-c-o-s-x-and-g-x-lo-g-cos-x-then-what-is-the-derivative-of-f-x-with-respect-to-g-x-88160

question:
<div class="question">If $f(x)=e^{\sin (\log \cos x)}$ and $g(x)=\log \cos x$, then what is the derivative of $\mathrm{f}(\mathrm{x})$ with respect to $\mathrm{g}(\mathrm{x})$?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{f}(\mathrm{x}) \cos [\mathrm{g}(\mathrm{x})]$</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">$\mathrm{f}(\mathrm{x}) \sin [\mathrm{g}(\mathrm{x})]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$g(x) \cos [f(x)]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$g(x) \sin [\mathrm{f}(\mathrm{x})]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}(\mathrm{x}) \cos [\mathrm{g}(\mathrm{x})]$</span> </div>

answer:
<div class="solution">Given funcion is: $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\sin (\log \cos \mathrm{x})}$<br/>Differentiating w.r.t. $\mathrm{x}$<br/>$f^{\prime}(x)=e^{\sin (\log \cos x)} \cdot \cos (\log \cos x) \cdot \frac{1}{\cos x}(-\sin x)$<br/>$=-\mathrm{e}^{\sin (\log \cos x)} \cdot \cos (\log \cos x) \cdot \tan x$<br/>and $\mathrm{g}(\mathrm{x})=\log \cos \mathrm{x}$<br/>$\therefore g^{\prime}(x)=\frac{1}{\cos x}(-\sin x)=-\tan x$<br/>Hence,<br/>$\frac{f^{\prime}(x)}{g^{\prime}(x)}=\frac{-e^{\sin (\log \cos x)} \cdot \cos (\log \cos x) \cdot \tan x}{-\tan x}$<br/>$=\mathrm{e}^{\sin (\log \cos x)} \cdot \cos (\log \cos x)$<br/>$=\mathrm{f}(\mathrm{x}) \cdot \cos [\mathrm{g}(\mathrm{x})]$</div>


title: if-f-x-e-x-and-g-x-lo-g-x-then-what-is-the-value-of-gof-x
slug: if-f-x-e-x-and-g-x-lo-g-x-then-what-is-the-value-of-gof-x-78039
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=e^{x}$ and $g(x)=\log x$, then what is the value of (gof)' $(x)$?</div>

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

answer:
<div class="solution">Let $f(x)=e^{x}, g(x)=\log x$<br/>Consider $($ g of $)(x)=g[f(x)]$<br/>$=\log f(x) \quad($ By defn of $g(x))$<br/>$=\log \left(e^{x}\right) \quad\left(\because f(x)=e^{x}\right)$<br/>$=x \quad(\because \log e=1)$</div>


title: if-f-x-is-an-even-function-then-what-is-0-f-cos-x-d-x-equal
slug: if-f-x-is-an-even-function-then-what-is-0-f-cos-x-d-x-equal-50118
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)$ is an even function, then what is $\int_{0}^{\pi} f(\cos x) d x$ equal</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">$\int_{0}^{\frac{\pi}{2}} f(\cos x) d x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \int_{0}^{\frac{\pi}{2}} f(\cos x) d x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \int_{0}^{\frac{\pi}{2}} f(\cos x) d x$</span> </div>

answer:
<div class="solution">Since $f(x)$ is an even function therefore<br/>$$<br/>\begin{array}{l}<br/>\int_{0}^{\pi} f(x) d x=2 \int_{0}^{\pi / 2} f(x) d x \\<br/>\text { Hence, } \int_{0}^{\pi} f(\cos x) d x=2 \int_{0}^{\pi / 2} f(\cos x) d x<br/>\end{array}<br/>$$</div>


title: if-f-x-is-an-even-function-where-f-x-0-then-which-one-of-the-following-is-correct
slug: if-f-x-is-an-even-function-where-f-x-0-then-which-one-of-the-following-is-correct-63712
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})$ is an even function, where $\mathrm{f}(\mathrm{x}) \neq 0$, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}^{\prime}(\mathrm{x})$ is an odd function</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})$ is an even function. Let's see some examples<br/>(1) If $\mathrm{f}(\mathrm{x})=\cos \mathrm{x}$, which is even function, $\mathrm{f}^{\prime}(\mathrm{x})=-\sin \mathrm{x}$, which is odd function.<br/>(2) If $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$, which is even function, $\mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}$, which is odd function.</div>


title: if-f-x-is-differentiable-everywherethen-which-one-of-the-following-is-correct
slug: if-f-x-is-differentiable-everywherethen-which-one-of-the-following-is-correct-62621
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})$ is differentiable everywhere,then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$|\mathrm{f}|$ is differentiable everywhere</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|\mathrm{f}|^{2}$ is differentiable everywhere</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}|\mathrm{f}|$ is not differentiable at some points</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\text { None of the above }$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}|\mathrm{f}|$ is not differentiable at some points</span> </div>

answer:
<div class="solution">If $\mathrm{f}(\mathrm{x})$ is differential everywhere then $\mid \mathrm{f}$ | is not differentiable at some point, so, $\mathrm{f}|\mathrm{f}|$ is not differentiable at some point. [Example: $\mathrm{f}(\mathrm{x})=\mathrm{x}$ is differentiable everywhere but $|\mathrm{f}(\mathrm{x})|=|\mathrm{x}|$ is not differentiable at $\mathrm{x}=0]$</div>


tags: ['Mathematics', 'Functions', 'JEE Main']
answer: <div class="solution">It is a fundamental concept</div>
source_db: MarksBatch2_P1.db
title:
if-f-x-is-periodic-function-with-period-t-then-the-function-f-a-x-b-where-a-0-is-periodic-with-period

slug:
if-f-x-is-periodic-function-with-period-t-then-the-function-f-a-x-b-where-a-0-is-periodic-with-period-91262

question:
<div class="question">If $f(x)$ is periodic function with period $T$ then the function $f(a x+b)$ where $a\gt0$, is periodic with period</div>

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

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


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-f-x-k-x-3-9-x-2-9-x-3-is-monotonically-increasing-in-every-interval-then-which-one-of-the-following-is-correct

slug:
if-f-x-k-x-3-9-x-2-9-x-3-is-monotonically-increasing-in-every-interval-then-which-one-of-the-following-is-correct-85430

question:
<div class="question">If $f(x)=k x^{3}-9 x^{2}+9 x+3$ is monotonically increasing in every interval, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$k   &lt; 3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$k \leq 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$k&gt;3$</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">$k \geq 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$k&gt;3$</span> </div>

answer:
<div class="solution">Given $f(x)=k x^{3}-9 x^{2}+9 x+3$ On differentiating w.r.t. $x$, we get $f^{\prime}(x)=3 k x^{2}-18 x+9$<br/>For a function to be monotonically increasing. $b^{2}-4 a c   &lt; 0$<br/>Here, $a=3 k, b=-18, c=9$<br/>$\therefore^{2}-4 a c=(-18)^{2}-4(3 k)(9)=(-18)(-18)-(3 k) 18 \times 2$<br/>$\Rightarrow 36-12 k &lt; 0$<br/>$\Rightarrow k&gt;3$</div>


title: if-f-x-l-n-x-1-s-i-n-e-x-2-1-then-lim-x-2-f-x-is-equal-to
slug: if-f-x-l-n-x-1-s-i-n-e-x-2-1-then-lim-x-2-f-x-is-equal-to-82767
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\frac{\sin \left(e^{x-2}-1\right)}{\ln (x-1)}$, then $\lim _{x \rightarrow 2} f(x)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</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">$\mathrm{f}(\mathrm{x})=\frac{\sin \left(\mathrm{e}^{\mathrm{x}-2}-1\right)}{\ln (\mathrm{x}-1)}$<br/>$\lim _{\mathrm{x} \rightarrow 2} \frac{\sin \left(\mathrm{e}^{\mathrm{x}-2}-1\right)}{\ln (\mathrm{x}-1)}=\mathrm{L}$<br/>It is $\frac{0}{0}$ (undefined) condition so using L'hospital's rule<br/>$\Rightarrow \mathrm{L}=\lim _{\mathrm{x} \rightarrow 2}\left[\frac{\left\{\sin \left(\mathrm{e}^{\mathrm{x}-2}-1\right)\right\}^{-}}{\{\ln (\mathrm{x}-1)\}^{-}}\right]$<br/>$\Rightarrow \mathrm{L}=\lim _{\mathrm{x} \rightarrow 2} \frac{\cos \left(\mathrm{e}^{\mathrm{x}-2}-1\right) \cdot \mathrm{e}^{(\mathrm{x}-2)}}{1 /(\mathrm{x}-1)}$<br/>$\Rightarrow \mathrm{L}=\lim _{\mathrm{x} \rightarrow 2} \cos \left(\mathrm{e}^{2-2}-1\right) \mathrm{e}^{2-2} \cdot(2-1)$<br/>$\Rightarrow \mathrm{L}=\cos (0) \mathrm{e}^{0} \cdot 1$<br/>$\Rightarrow \mathrm{L}=1$</div>


title: if-f-x-ln-x-1-x-2-then-what-is-f-x-d-x-equal-to
slug: if-f-x-ln-x-1-x-2-then-what-is-f-x-d-x-equal-to-76410
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\ln \left(x-\sqrt{1+x^{2}}\right)$, then what is $\int f^{\prime \prime}(x) d x$ equal to?</div>

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

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

answer:
<div class="solution">Given that $f(x)=\ln \left(x-\sqrt{1+x^{2}}\right)$<br/>$\int \mathrm{f}^{\prime \prime}(\mathrm{x}) \mathrm{dx}=\mathrm{f}^{\prime}(\mathrm{x})+\mathrm{c}$ where $\mathrm{c}$ is a constant<br/>$=\frac{1}{\left(x-\sqrt{1+x^{2}}\right)} \cdot\left(1-\frac{2 x}{2 \sqrt{1+x^{2}}}\right)+c$<br/>$=\frac{-\left(x-\sqrt{1+x^{2}}\right)}{\left(\sqrt{1+x^{2}}\right)\left(x-\sqrt{1+x^{2}}\right)}+c=-\frac{1}{\sqrt{1+x^{2}}}+c$</div>


title: if-f-x-lo-g-1-x-1-x-then-what-is-f-1-x-2-2-x-equal-to
slug: if-f-x-lo-g-1-x-1-x-then-what-is-f-1-x-2-2-x-equal-to-23360
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\log \left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right)$, then what is $\mathrm{f}\left(\frac{2 \mathrm{x}}{1-\mathrm{x}^{2}}\right)$ equal to ?</div>

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

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

answer:
<div class="solution">Given that $\mathrm{f}(\mathrm{x})=\log \left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right)$<br/>So, $f\left(\frac{2 x}{1+x^{2}}\right)=\log \left(\frac{1+\frac{2 x}{1+x^{2}}}{1-\frac{2 x}{1+x^{2}}}\right)$<br/>$=\log \left(\frac{1+x^{2}+2 x}{1+x^{2}-2 x}\right)=\log \left(\frac{(1+x)^{2}}{(1-x)^{2}}\right)$<br/>$=\log \left(\frac{1+x}{1-x}\right)^{2}=2 \log \left(\frac{1+x}{1-x}\right)$<br/>$=2 \mathrm{f}(\mathrm{x})$<br/>$\left[\right.$ since $\left.f(x)=\log \left(\frac{1+x}{1-x}\right)\right]$</div>


title: if-f-x-lo-g-10-1-x-then-what-is-4-f-4-5-f-1-lo-g-10-2-equal-to
slug: if-f-x-lo-g-10-1-x-then-what-is-4-f-4-5-f-1-lo-g-10-2-equal-to-48573
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\log _{10}(1+x)$, then what is $4 \mathrm{f}(4)+5 f(1)-\log _{10} 2$ equal<br/>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">$\mathrm{f}(\mathrm{x})=\log _{10}(1+\mathrm{x})$<br/>$4 \cdot \mathrm{f}(4)+5 \cdot \mathrm{f}(1)-\log _{10} 2$<br/>$=4 \cdot \log _{10}(1+4)+5 \cdot \log _{10}(1+1)-\log _{10} 2$<br/>$=4 \log _{10} 5+5 \log _{10} 2-\log _{10} 2$<br/>$=4 \log _{10} 5+4 \log _{10} 2$<br/>$=4\left(\log _{10} 5+\log _{10} 2\right)$<br/>$=4\left(\log _{10} 10\right)=$</div>


title: if-f-x-lo-g-e-1-x-1-x-g-x-1-3-x-2-3-x-x-3-and-g-of-t-g-f-t-then-what-is-g-f-e-1-e-1-equal-to
slug: if-f-x-lo-g-e-1-x-1-x-g-x-1-3-x-2-3-x-x-3-and-g-of-t-g-f-t-then-what-is-g-f-e-1-e-1-equal-to-35096
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\log _{e}\left(\frac{1+x}{1-x}\right), g(x)=\frac{3 x+x^{3}}{1+3 x^{2}}$ and g of $(\mathrm{t})=\mathrm{g}(\mathrm{f}(\mathrm{t}))$,<br/>then what is $g \circ f\left(\frac{e-1}{e+1}\right)$ equal to?</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">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">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>


title: if-f-x-lo-g-e-lo-g-e-x-then-what-is-f-e-equal-to
slug: if-f-x-lo-g-e-lo-g-e-x-then-what-is-f-e-equal-to-25256
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\log _{\mathrm{e}}\left[\log _{\mathrm{e}} \mathrm{x}\right]$, then what is $\mathrm{f}^{\prime}(\mathrm{e})$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{e}^{-1}$</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">e</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

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

answer:
<div class="solution">The given funcion is: $\mathrm{f}(\mathrm{x})=\log _{\mathrm{e}}\left[\log _{\mathrm{e}} \mathrm{x}\right]$<br/>Differentiating w.r.t. $\mathrm{x}$, we get<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\log _{\mathrm{e}} \mathrm{x}} \cdot \frac{1}{\mathrm{x}} \Rightarrow \mathrm{f}^{\prime}(\mathrm{e})=\frac{1}{\log _{\mathrm{e}} \mathrm{e}} \cdot \frac{1}{\mathrm{e}}=\frac{1}{\mathrm{c}}=\mathrm{e}^{-1}$</div>


title: if-f-x-lo-g-x-x-0-then-what-is-f-x-equal-to
slug: if-f-x-lo-g-x-x-0-then-what-is-f-x-equal-to-95742
question: <div class="question">If $f(x)=\log |x|, x \neq 0$, then what is $f^{\prime}(x)$ equal to?</div>
tags: ['Mathematics', 'Differentiation', 'NDA', 'NDA 2008 (Phase 2)']
answer: <div class="solution">$f(x)=\log |x|$<br/>$f^{\prime}(x)=\frac{1}{|x|}$</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{|x|}$</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">$\frac{1}{x}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-1}{x}$</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">$\frac{1}{|x|}$</span> </div>


title: if-f-x-m-x-1-sin-x-n-x-2-x-2-is-continuous-at-x-2-then-which-one-of-the-following-is-correct-1
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-f-x-m-x-1-sin-x-n-x-2-x-2-is-continuous-at-x-2-then-which-one-of-the-following-is-correct-1-80437

question:
<div class="question">If $f(x)=\left\{\begin{array}{ll}m x+1 &amp; x \leq \frac{\pi}{2} \\ \sin x+n &amp; x&gt;\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\frac{\pi}{2}$, then<br/>which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{m}=1, \mathrm{n}=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{m}=\frac{\mathrm{n} \pi}{2}+1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{n}=\mathrm{m}\left(\frac{\pi}{2}\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{m}=\mathrm{n}=\frac{\pi}{2}$</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>

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ and $\mathrm{g}(\mathrm{x})=\mathrm{cx}+\mathrm{d}$<br/>$\mathrm{f}[\mathrm{g}(\mathrm{x}))]=\mathrm{a}(\mathrm{cx}+\mathrm{d})+\mathrm{b}$<br/>$=\mathrm{acx}+\mathrm{ad}+\mathrm{b}$<br/>and $\mathrm{g}[\mathrm{f}(\mathrm{x})]=\mathrm{c}(\mathrm{ax}+\mathrm{b})+\mathrm{d}$<br/>$=\mathrm{acx}+\mathrm{dc}+\mathrm{d}$<br/>Now from $\mathrm{f}[(\mathrm{gx})]=\mathrm{g}[\mathrm{f}(\mathrm{d})]$<br/>$\Rightarrow \mathrm{ad}+\mathrm{b}=\mathrm{bc}+\mathrm{d}$<br/>$\Rightarrow \mathrm{f}(\mathrm{d})=\mathrm{g}(\mathrm{b})$</div>


