# questions

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
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-87842
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
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="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{n}=\mathrm{m}\left(\frac{\pi}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{m}=\mathrm{n}=\frac{\pi}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Given function is<br/>$f(x)=\left\{\begin{array}{ll}m x+1, &amp; x \leq \frac{\pi}{2} \\ \sin x+n, &amp; x&gt;\frac{\pi}{2}\end{array}\right.$<br/>As given this function is continuous at $x=\frac{\pi}{2}$.<br/>So, limit of function when $\mathrm{x} \rightarrow \frac{\pi}{2}=\mathrm{f}\left(\frac{\pi}{2}\right)$<br/>$\Rightarrow \lim _{x \rightarrow \frac{\pi}{2}+}(\sin x+n)=f\left(\frac{\pi}{2}\right)$<br/>$\Rightarrow \lim _{h \rightarrow 0}\left(\sin \left(\frac{\pi}{2}+h\right)+n\right)=\frac{m \pi}{2}+1$<br/>$\Rightarrow \quad \sin \frac{\pi}{2}+n=\frac{m \pi}{2}+1$<br/>$\Rightarrow 1+\mathrm{n}=\frac{\mathrm{m} \pi}{2}+1$<br/>$\Rightarrow \mathrm{n}=\frac{\mathrm{m} \pi}{2}$</div>


title: if-f-x-sin-2-x-2-then-what-is-f-x-equal-to
slug: if-f-x-sin-2-x-2-then-what-is-f-x-equal-to-11202
question: <div class="question">If $f(x)=\sin ^{2} x^{2}$, then what is $f^{\prime}(x)$ equal to?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$4 x \sin \left(x^{2}\right) \cos \left(x^{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sin \left(x^{2}\right) \cos \left(x^{2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \sin \left(x^{2}\right) \sin ^{2} x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \mathrm{x} \cos ^{2}\left(\mathrm{x}^{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 x \sin \left(x^{2}\right) \cos \left(x^{2}\right)$</span> </div>

answer:
<div class="solution">Given $f(x)=\sin ^{2} x^{2}$<br/>$\therefore f^{\prime}(x)=2 \sin x^{2} \cos x^{2} \cdot 2 x$<br/>$=4 x \sin x^{2} \cos x^{2}$</div>


title: if-f-x-tan-x-e-2-x-7-x-3-then-what-is-the-value-of-f-0
slug: if-f-x-tan-x-e-2-x-7-x-3-then-what-is-the-value-of-f-0-65371
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(\mathrm{x})=\tan \mathrm{x}+\mathrm{e}^{-2 \mathrm{x}}-7 \mathrm{x}^{3}$, then what is the value of $f^{\prime}(0)$?</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">3</span> </li> </ul>

answer:
<div class="solution">Given $f(x)=\tan x+e^{-2 x}-7 x^{3}$<br/>On differentiating w.r.t. $\mathrm{x}$, we get<br/>$f^{\prime}(x)=\sec ^{2} x-2 e^{-2 x}-21 x^{2}$<br/>Put $x=0$<br/>$\Rightarrow f^{\prime}(0)=\sec ^{2} 0-2 e^{0}-21 \times 0=1-2=-1$</div>


title: if-f-x-x-1-cotx-is-continuous-at-x-0-then-what-is-f-0-equal-to
slug: if-f-x-x-1-cotx-is-continuous-at-x-0-then-what-is-f-0-equal-to-88247
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">e</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=(x+1)^{\text {cotx }}$ is continuous at $x=0$, then what is $f(0)$ 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">e</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\mathrm{e}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{e}^{2}$</span> </li> </ul>

answer:
<div class="solution">For a function to be continuous at a point the limit should exist and should be equal to the value of the function at that point.<br/>Here point is $\mathrm{x}=0$<br/>and $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}(x+1)^{\cot x}$<br/>$=\lim _{x \rightarrow 0}(1+x)^{\cot x}=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x} \cdot x \cot x}$<br/>$=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x} \lim _{x \rightarrow 0} \frac{x}{\tan x}}=e^{1}=e$<br/>Since limiting value of $f(x)=e$, when $x \rightarrow 0, f(0)$ should also be equal to e.</div>


title: if-f-x-x-1-x-then-what-is-f-a-1-f-a-equal-to
slug: if-f-x-x-1-x-then-what-is-f-a-1-f-a-equal-to-64641
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(\mathrm{x})=\frac{\mathrm{x}}{\mathrm{x}-1}$, then what is $\frac{f(\mathrm{a})}{f(\mathrm{a}+1)}$ equal to?</div>

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

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\mathrm{x}-1}$<br/>$\mathrm{f}(\mathrm{a})=\frac{\mathrm{a}}{\mathrm{a}-1}$<br/>$\mathrm{f}(\mathrm{a}+1)=\frac{\mathrm{a}+1}{\mathrm{a}+1-1}=\frac{\mathrm{a}+1}{\mathrm{a}}$<br/>$\therefore \quad \frac{\mathrm{f}(\mathrm{a})}{\mathrm{f}(\mathrm{a}+1)}=\frac{\frac{\mathrm{a}}{\mathrm{a}-1}}{\frac{\mathrm{a}+1}{\mathrm{a}}}=\frac{\mathrm{a}^{2}}{\mathrm{a}^{2}-1}$<br/>$\mathrm{f}\left(\mathrm{a}^{2}\right)=\frac{\mathrm{a}^{2}}{\mathrm{a}^{2}-1}$</div>


title: if-f-x-x-2-2-x-3-x-2-9-x-3-is-continuous-at-x-3-then-which-one-of-the-following-is-correct
slug: if-f-x-x-2-2-x-3-x-2-9-x-3-is-continuous-at-x-3-then-which-one-of-the-following-is-correct-95368
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^{2}-9}{\mathrm{x}^{2}-2 \mathrm{x}-3}, \mathrm{x} \neq 3$ is continuous at $\mathrm{x}=3$, then<br/>which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad f(3)=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}(3)=1.5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad f(3)=3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(3)=-1.5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}(3)=1.5$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^{2}-9}{\mathrm{x}^{2}-2 \mathrm{x}-3}$<br/>Given, the function is continuous at $\mathrm{x}=3$.<br/>$f(3)=\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-2 x-3}$<br/>$=\lim _{x \rightarrow 3} \frac{(x+3)(x-3)}{(x-3)(x+1)}$<br/>$=\frac{3+3}{3+1}=\frac{6}{4}=\frac{3}{2}=1.5$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
if-f-x-x-2-6-x-8-and-there-exists-a-point-c-in-the-interval-2-4-such-that-f-c-0-then-what-is-the-value-of-c

slug:
if-f-x-x-2-6-x-8-and-there-exists-a-point-c-in-the-interval-2-4-such-that-f-c-0-then-what-is-the-value-of-c-68775

question:
<div class="question">If $f(x)=x^{2}-6 x+8$ and there exists a point $c$ in the interval $[2,4]$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$, then what is the value of $\mathrm{c}$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2.5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2.8</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">$3.5$</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-6 \mathrm{x}+8$<br/>$\begin{aligned} \mathrm{f}^{\prime}(\mathrm{x}) &amp;=2 \mathrm{x}-6 \\ \mathrm{f}^{\prime}(\mathrm{c}) &amp;=2 \mathrm{c}-6 \\ \mathrm{f}^{\prime}(\mathrm{c}) &amp;=0 \\ \Rightarrow 2 \mathrm{c}-6 &amp;=0 \Rightarrow \mathrm{c}=3 \end{aligned}$</div>


title: if-f-x-x-2-x-2-x-2-then-what-is-f-1-x-equal-to
slug: if-f-x-x-2-x-2-x-2-then-what-is-f-1-x-equal-to-23457
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\frac{x-2}{x+2}, x \neq-2$, then what is $\mathrm{f}^{-1}(\mathrm{x})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{4(x+2)}{x-2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{x+2}{4(x-2)}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x+2}{x-2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{2(1+x)}{1-x}$</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> </ul>

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

answer:
<div class="solution">$f(x) \mathrm{y}=4 \frac{x-2}{x+2}, x \neq-2$<br/>$\frac{y}{1}=\frac{x-2}{x+2} \Rightarrow \frac{y+1}{y-1}=\frac{x-2+x+2}{x-2-x-2}$<br/>$\Rightarrow \frac{y+1}{y-1}=\frac{2 x}{-4}$<br/>$\Rightarrow \frac{y+1}{y-1}=\frac{-x}{2} \Rightarrow x=-2\left(\frac{y+1}{y-1}\right)$<br/>Now, $\mathrm{y}=\frac{-2(x+1)}{x-1}=\frac{2(x+1)}{1-x}$</div>


title: if-f-x-x-4-x-1-defines-a-function-of-r-then-what-is-its-domain
slug: if-f-x-x-4-x-1-defines-a-function-of-r-then-what-is-its-domain-49419
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\frac{\sqrt{\mathrm{x}-1}}{\mathrm{x}-4}$ defines a function of $\mathrm{R}$, then what is its domain?</div>

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

answer:
<div class="solution">$f(x)=\frac{\sqrt{x-1}}{x-4}$<br/>$f(x)$ is defined for $(x-1) \geq 0 \Rightarrow x \geq 1$ and $x-4 \neq 0$ $\Rightarrow x \neq 4$<br/>$\therefore \quad$ Domain of $f(x)=1 \leq x   &lt; \infty-\{4\}$<br/>or, $\quad x=[1,4) \cup(4, \infty)$.</div>


title: if-f-x-x-and-g-x-x-then-what-is-f-g-x-equal-to
slug: if-f-x-x-and-g-x-x-then-what-is-f-g-x-equal-to-68762
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\mathrm{x}$ and $\mathrm{g}(\mathrm{x})=|\mathrm{x}|$, then what is $(\mathrm{f}+\mathrm{g})(\mathrm{x})$<br/>equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0 for all $\mathrm{x} \in \mathrm{R}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{x}$ for all $\mathrm{x} \in \mathrm{R}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left\{\begin{array}{l}2 \mathrm{x}, \text { for } \mathrm{x} \geq 0 \\ 0, \text { for } \mathrm{x}   &lt; 0\end{array}\right.$</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">D</span> <span class="option-data">$\left\{\begin{array}{l}0, \text { for } \mathrm{x} \geq 0 \\ 2 \mathrm{x}, \text { for } \mathrm{x}   &lt; 0\end{array}\right.$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left\{\begin{array}{l}2 \mathrm{x}, \text { for } \mathrm{x} \geq 0 \\ 0, \text { for } \mathrm{x}   &lt; 0\end{array}\right.$</span> </div>

answer:
<div class="solution">Given functions are : $\mathrm{f}(\mathrm{x})=\mathrm{x}$ and $\mathrm{g}(\mathrm{x})=|\mathrm{x}|$<br/>$\therefore \quad(\mathrm{f}+\mathrm{g})(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})=\mathrm{x}+|\mathrm{x}|$<br/>According to defintion of modulus function,<br/>$(f+g)(x)=\left\{\begin{array}{ll}x+x, &amp; x \geq 0 \\ x-x, &amp; x   &lt; 0\end{array}\right.$<br/>$=\left\{\begin{array}{cl}2 \mathrm{x}, &amp; \mathrm{x} \geq 0 \\ 0, &amp; \mathrm{x} &lt; 0\end{array}\right.$</div>


title: if-f-x-x-n-x-then-f-x-attains-minimum-value-at-which-one-of-the-following-points
slug: if-f-x-x-n-x-then-f-x-attains-minimum-value-at-which-one-of-the-following-points-58078
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=x \ell \operatorname{n} x$, then $f(x)$ attains minimum value at which one of the following points?</div>

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

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

answer:
<div class="solution">Let $f(x)=x \ln x$<br/>$f^{\prime}(x)=\frac{x}{x}+\ln x=1+\ln x$<br/>Put $f(x)=0 \Rightarrow 1+\ln x=0$<br/>$\Rightarrow \ln x=-1 \Rightarrow x=\mathrm{e}^{-1}$<br/>Now, $f^{\prime \prime}(x)=\frac{1}{x}$<br/>$\left.f^{\prime \prime}(x)\right|_{x=e^{-1}}=\frac{1}{e^{-1}}=e&gt;0$<br/>Hence, $f(x)$ attains minimum value at $x=e^{-1}$.</div>


title: if-f-x-x-x-0-x-and-x-is-continuous-at-x-x-0-then-what-is-f-x-0-equal-to
slug: if-f-x-x-x-0-x-and-x-is-continuous-at-x-x-0-then-what-is-f-x-0-equal-to-48515
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\left(x-x_{0}\right) \phi(x)$ and $\phi(x)$ is continuous at $x=x_{0}$, then what is $\mathrm{f}^{\prime}\left(\mathrm{x}_{0}\right)$ equal to ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\phi\left(\mathrm{x}_{0}\right)$</span> </div>

answer:
<div class="solution">$\quad$  Given that $\mathrm{f}(\mathrm{x})=\left(\mathrm{x}-\mathrm{x}_{0}\right) \phi(\mathrm{x})$<br/>Differentiating w. r. t. $\mathrm{x}$ $\mathrm{f}^{\prime}(\mathrm{x})=\left(\mathrm{x}-\mathrm{x}_{0}\right) \phi^{\prime}(\mathrm{x})+\phi(\mathrm{x})(1)$<br/>Putting $\mathrm{x}=\mathrm{x}_{0}$<br/>$\mathrm{f}^{\prime}\left(\mathrm{x}_{0}\right)=\phi\left(\mathrm{x}_{0}\right)$</div>


title: if-f-x-x-x-1-then-which-one-of-the-following-is-correct
slug: if-f-x-x-x-1-then-which-one-of-the-following-is-correct-42677
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=|\mathrm{x}|+|\mathrm{x}-1|$, 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">$\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$ and $\mathrm{x}=1$</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">$\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$ but not at $\mathrm{x}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=1$ but not at $\mathrm{x}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ is neither continuous at $\mathrm{x}=0$ nor at $\mathrm{x}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$ and $\mathrm{x}=1$</span> </div>

answer:
<div class="solution">Given function is continuous at $\mathrm{x}=0$ and 1 .<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/FyJCnc24kdBBUeVYGxWh_VOcv3e89zNYko71c1Y6jqg.original.fullsize.png"/></div>


title: if-f-x-x-x-x-1-then-f-x-is
slug: if-f-x-x-x-x-1-then-f-x-is-53494
question: <div class="question">If $f(x)=x(\sqrt{x}-\sqrt{x+1})$, then $f(x)$ is</div>
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">continuous but not differentiable at $x=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">differentiable at $\mathrm{x}=0$</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">not continuous at $\mathrm{x}=0$</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">differentiable at $\mathrm{x}=0$</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{f}(\mathrm{x}) &amp;=\mathrm{x}(\sqrt{\mathrm{x}}-\sqrt{\mathrm{x}+1}) \\ \mathrm{f}(\mathrm{x}) &amp;=\frac{\mathrm{x}(\sqrt{\mathrm{x}}-\sqrt{\mathrm{x}+1})(\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}+1})}{(\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}+1})} \\ &amp;=\frac{\mathrm{x}(\mathrm{x}-\mathrm{x}-1)}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}+1}}=\frac{-\mathrm{x}}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}+1}} \end{aligned}$<br/>Hence, $\mathrm{f}(\mathrm{x})$ is continuous as well as differentiable at $\mathrm{x}=0$.</div>


title: if-f-x-x-x-x-then-what-is-f-x-equal-to
slug: if-f-x-x-x-x-then-what-is-f-x-equal-to-24370
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\ldots . \infty},}}}$ then what is $\mathrm{f}^{\prime}(\mathrm{x})$ equal<br/>to?</div>

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

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

answer:
<div class="solution">Given function is:<br/>$f(x)=\sqrt{x+\sqrt{x+\sqrt{x \sqrt{\infty}}}}$<br/>$\Rightarrow \quad f(x)=\sqrt{x+f(x)} \Rightarrow(f(x))^{2}=x+f(x)$<br/>On differentiating both sides wrt. $\mathrm{x}$, we get $2 \mathrm{f}(\mathrm{x}) \mathrm{f}^{\prime}(\mathrm{x})=1+\mathrm{f}^{\prime}(\mathrm{x})$<br/>$\mathrm{f}^{\prime}(\mathrm{x})\{2 \mathrm{f}(\mathrm{x})-1\}=1$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{2 \mathrm{f}(\mathrm{x})-1}$</div>


title: if-f-x-y-f-x-f-y-then-f-t-may-be-of-the-form-where-k-c-are-constants
slug: if-f-x-y-f-x-f-y-then-f-t-may-be-of-the-form-where-k-c-are-constants-32312
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f[x y]=f[x] f[y]$, then $f[t]$ may be of the form:<br/>where $\mathrm{k}, \mathrm{c}$ are constants</div>

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

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

answer:
<div class="solution">By observing the options Let $f(t)=t^{k}$<br/>Suppose $\mathrm{t}=\mathrm{xy}$<br/>$\therefore \quad \mathrm{f}(\mathrm{xy})=(\mathrm{xy})^{\mathrm{k}}=\mathrm{x}^{\mathrm{k}} \cdot \mathrm{y}^{\mathrm{k}}=\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})$<br/>Hence, $\mathrm{f}(\mathrm{t})=\mathrm{t}^{\mathrm{k}}$ where $\mathrm{k}$ ' is a constant.</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
title:
if-for-a-distribution-x-5-3-x-5-2-43-and-the-total-number-of-items-is-18-find-the-mean-and-standard-deviation

slug:
if-for-a-distribution-x-5-3-x-5-2-43-and-the-total-number-of-items-is-18-find-the-mean-and-standard-deviation-65215

question:
<div class="question">If for a distribution $\Sigma(x-5)=3, \Sigma(x-5)^2=43$ and the total number of items is 18 , find the mean and standard deviation.</div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp;\text { We have, } \Sigma(x-5)=\Sigma x-\Sigma 5=\Sigma x-5(18)=\Sigma x-90\\<br/>&amp;\therefore \quad \Sigma \mathbf{x}-90=3[\text { Given })]\\<br/>&amp;\Rightarrow \quad \Sigma \mathrm{x}=93\\<br/>&amp;\therefore \quad \overline{\mathrm{x}}=\frac{\Sigma \mathrm{x}}{18}=\frac{93}{18}=5.167\\<br/>&amp;\text { Also } 43=\Sigma(x-5)^2=\Sigma\left(x^2-10 x+25\right)\\<br/>&amp;\Rightarrow \quad 43=\Sigma x^2-10 \Sigma x+25(18)\\<br/>&amp;\Rightarrow \quad 43=\Sigma x^2-10(93)+450\\<br/>&amp;\Rightarrow \quad \Sigma x^2=523<br/>\end{aligned}<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>\therefore \text { S.D. } &amp;=\sqrt{\frac{523}{18}-(5.167)^2} \\<br/>&amp;=\sqrt{2.357}=1.535=1.53<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-for-positive-real-numbers-x-y-z-the-numbers-x-y-2-y-and-y-z-are-in-harmonic-progression-then-which-one-of-the-following-is-correct

slug:
if-for-positive-real-numbers-x-y-z-the-numbers-x-y-2-y-and-y-z-are-in-harmonic-progression-then-which-one-of-the-following-is-correct-66749

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in geometric progression</span> </div>

answer:
<div class="solution">As given:<br/>$\mathrm{x}+\mathrm{y}, 2 \mathrm{y}$ and $\mathrm{y}+\mathrm{z}$ are in harmonic progression. $2 y=\frac{(x+y)(y+z)}{x+y+y+z}$<br/>$\Rightarrow \quad \frac{1}{2 y}=\frac{1}{x+y}+\frac{1}{y+z}$<br/>$\Rightarrow \quad y(x+2 y+z)=\left(x y+x z+y^{2}+y z\right)$<br/>$\Rightarrow x y+2 y^{2}+y z=x y+x z+y^{2}+y z$<br/>$\Rightarrow \quad y^{2}=x z$<br/>$\Rightarrow \mathrm{x}, \mathrm{y}, \mathrm{z}$ are in geometric progession.</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-for-the-given-reaction-taking-place-at-25-c-in-a-cell-is-1-2-cu-s-1-2-cl-2-g-1-2-cu-2-cl-having-standard-emf-of-102-v-o-l-t-s-calculate-standard-fr

slug:
if-for-the-given-reaction-taking-place-at-25-c-in-a-cell-is-1-2-cu-s-1-2-cl-2-g-1-2-cu-2-cl-having-standard-emf-of-102-v-o-l-t-s-calculate-standard-fr-62695

question:
<div class="question">If for the given reaction taking place at <math><mn>25</mn><mo>Â°</mo><mi mathvariant="normal">C</mi></math> in a cell is,<br/> <math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mtext>Cu</mtext><mfenced> <mtext>s</mtext> </mfenced><mo>+</mo><mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mrow> <mtext>Cl</mtext></mrow> <mn>2</mn> </msub> <mfenced> <mtext>g</mtext> </mfenced><mo>â</mo><mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mtext>Cu</mtext></mrow> <mrow> <mn>2</mn><mo>+</mo></mrow> </msup> <mo>+</mo><msup> <mrow> <mtext>Cl</mtext></mrow> <mo>â</mo> </msup> </mrow> </math><br/>
having standard e.m.f. of <math><mn>1.02 </mn><mi mathvariant="normal"></mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">s</mi></math> . Calculate standard free energy change.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mo>-</mo><mn>98430</mn></math> <math><mi mathvariant="normal">J</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>98430</mn></math> <math><mi mathvariant="normal">J</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>96500</mn></math> <math><mi mathvariant="normal">J</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mo>-</mo><mn>96500</mn></math> <math><mi mathvariant="normal">J</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mo>-</mo><mn>98430</mn></math> <math><mi mathvariant="normal">J</mi></math></span> </div>

answer:
<div class="solution">The relation between free energy change and standard e.m.f. is given by <math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">G</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">n</mi><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mo>Â°</mo></math> <br/><math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">G</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>Ã</mo><mn>96500</mn><mo>Ã</mo><mn>1.02</mn></math> ; <br/><math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">G</mi><mo>=</mo><mo>-</mo><mn>98430</mn></math> .</div>


tags: ['Physics', 'Units and Dimensions', 'NEET', 'NEET 2021']
source_db: MarksBatch2_P1.db
title:
if-force-f-acceleration-a-and-time-t-are-chosen-as-the-fundamental-physical-quantities-find-the-dimensions-of-energy

slug:
if-force-f-acceleration-a-and-time-t-are-chosen-as-the-fundamental-physical-quantities-find-the-dimensions-of-energy-90776

question:
<div class="question">If force <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>F</mi><mo>]</mo></math>, acceleration <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>A</mi><mo>]</mo></math> and time <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>T</mi><mo>]</mo></math> are chosen as the fundamental physical quantities. Find the dimensions of energy.</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"><mo>[</mo><mi mathvariant="normal">F</mi><mo>]</mo><mo>[</mo><mi mathvariant="normal">A</mi><mo>]</mo><mo>[</mo><mi mathvariant="normal">T</mi><mo>]</mo></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi mathvariant="normal">F</mi><mo>]</mo><mo>[</mo><mi mathvariant="normal">A</mi><mo>]</mo><mfenced close="]" open="["><msup><mi mathvariant="normal">T</mi><mn>2</mn></msup></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">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi mathvariant="normal">F</mi><mo>]</mo><mo>[</mo><mi mathvariant="normal">A</mi><mo>]</mo><mfenced close="]" open="["><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi mathvariant="normal">F</mi><mo>]</mo><mfenced close="]" open="["><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo>[</mo><mi mathvariant="normal">T</mi><mo>]</mo></math></span> </li> </ul>

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

answer:
<div class="solution"><p>Let consider,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>Energy</mi><mo>]</mo><mo>=</mo><mo>[</mo><mi mathvariant="normal">F</mi><msup><mo>]</mo><mi mathvariant="normal">a</mi></msup><mo>[</mo><mi mathvariant="normal">A</mi><msup><mo>]</mo><mi mathvariant="normal">b</mi></msup><mo>[</mo><mi mathvariant="normal">T</mi><msup><mo>]</mo><mi mathvariant="normal">c</mi></msup></math></p><p>Now put the dimensional formula of the given quantities,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="["><msup><mi>MLT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">a</mi></msup><msup><mfenced close="]" open="["><msup><mi>LT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mi mathvariant="normal">b</mi></msup><mo>[</mo><mi mathvariant="normal">T</mi><msup><mo>]</mo><mi mathvariant="normal">c</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mi mathvariant="normal">a</mi></msup><msup><mi mathvariant="normal">L</mi><mrow><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><mn>2</mn><mi mathvariant="normal">b</mi><mo>+</mo><mi mathvariant="normal">c</mi></mrow></msup></mrow></mfenced></math></p><p>On comparing powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>M</mi><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn></math></p><p>On comparing powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>L</mi><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn><mo>+</mo><mi>b</mi><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>b</mi><mo>=</mo><mn>1</mn></math></p><p>On comparing powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>T</mi><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo> </mo><mi>a</mi><mo>-</mo><mn>2</mn><mo> </mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>-</mo><mn>2</mn><mo>-</mo><mn>2</mn><mo>+</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>c</mi><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mo> </mo><mi>E</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>g</mi><mi>y</mi><mo> </mo><mo>]</mo><mo>=</mo><mo>[</mo><mi>F</mi><mo>]</mo><mo>[</mo><mi>A</mi><mo>]</mo><mfenced close="]" open="["><msup><mi>T</mi><mn>2</mn></msup></mfenced></math></p></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-force-f-length-l-and-time-t-are-assumed-to-be-fundamental-units-then-the-dimensional-formula-of-the-mass-will-be-1

slug:
if-force-f-length-l-and-time-t-are-assumed-to-be-fundamental-units-then-the-dimensional-formula-of-the-mass-will-be-1-81836

question:
<div class="question">If force $(F)$, length $(L)$ and time $(T)$ are assumed to be fundamental units, then the dimensional formula of the mass will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{FL}^{2} \mathrm{~T}^{2}\right]$</span> </div>

answer:
<div class="solution">Let $\mathrm{m}=\mathrm{KF}^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}}$
<br/> <br/>Substituting the dimensions of
<br/> <br/>$[\mathrm{F}]=\left[\mathrm{MLT}^{-2}\right],[\mathrm{L}]=[\mathrm{L}]$ and $[\mathrm{T}]=[\mathrm{T}]$ and comparing both side, we get $\mathrm{m}=\mathrm{FL}^{-1} \mathrm{~T}^{2}$</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
title:
if-force-f-length-l-and-time-t-are-assumed-to-be-fundamental-units-then-the-dimensional-formula-of-the-mass-will-be

slug:
if-force-f-length-l-and-time-t-are-assumed-to-be-fundamental-units-then-the-dimensional-formula-of-the-mass-will-be-70276

question:
<div class="question">If force $(F)$, length $(L)$ and time $(T)$ are assumed to be fundamental units, then the dimensional formula of the mass will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{FL}^{2} \mathrm{~T}^{2}\right]$</span> </div>

answer:
<div class="solution">Let $\mathrm{m}=\mathrm{KF}^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}}$
<br/> <br/>Substituting the dimensions of
<br/> <br/>$[\mathrm{F}]=\left[\mathrm{MLT}^{-2}\right],[\mathrm{L}]=[\mathrm{L}]$ and $[\mathrm{T}]=[\mathrm{T}]$ and comparing both side, we get $\mathrm{m}=\mathrm{FL}^{-1} \mathrm{~T}^{2}$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2021 (27 Aug Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-force-f-length-l-and-time-t-are-taken-as-the-fundamental-quantities-then-what-will-be-the-dimension-of-density

slug:
if-force-f-length-l-and-time-t-are-taken-as-the-fundamental-quantities-then-what-will-be-the-dimension-of-density-69420

question:
<div class="question">If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density:</div>

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

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

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">d</mi><mo>∝</mo><msup><mi>F</mi><mi mathvariant="normal">a</mi></msup><msup><mi mathvariant="normal">L</mi><mi mathvariant="normal">b</mi></msup><msup><mi mathvariant="normal">T</mi><mi mathvariant="normal">c</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mi>M</mi><msup><mi>L</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfenced><mo>=</mo><msup><mfenced close="]" open="["><mrow><mi>M</mi><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mi>a</mi></msup><mfenced close="]" open="["><msup><mi>L</mi><mi>b</mi></msup></mfenced><mfenced close="]" open="["><mrow><mi>T</mi><msup><mo> </mo><mi>c</mi></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi>M</mi><mi>a</mi></msup><msup><mi>L</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi></mrow></msup></mrow></mfenced></math></p><p>comparing :</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn></math> ......(i)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> ....(ii)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo> </mo><mi>a</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math> .......(iii)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">c</mi><mo>=</mo><mn>2</mn><mo>  </mo><mo>;</mo><mo>  </mo><mi mathvariant="normal">b</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>  </mo><mo>;</mo><mo>  </mo><mo>∴</mo><mo>  </mo><mo>[</mo><mi mathvariant="normal">d</mi><mo>]</mo><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi>FL</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mrow><mo> </mo><mi mathvariant="normal">T</mi></mrow><mn>2</mn></msup></mrow></mfenced></math></p></div>


title: if-force-f-length-l-and-time-t-be-considered-fundamental-units-then-the-units-of-mass-will-be-1
tags: ['Physics', 'Units and Dimensions', 'NEET']
source_db: MarksBatch2_P1.db
slug:
if-force-f-length-l-and-time-t-be-considered-fundamental-units-then-the-units-of-mass-will-be-1-44758

question:
<div class="question">If force <math><mfenced separators="|"><mrow><mi>F</mi></mrow></mfenced><mo>,</mo></math> length <math><mfenced separators="|"><mrow><mi>L</mi></mrow></mfenced></math> and time <math><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math> be considered fundamental units, then the units of mass will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal"> </mi><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mn>2</mn></msup><mo>]</mo></math></span> </div>

answer:
<div class="solution">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">M</mi><mo>∝</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mi mathvariant="normal">F</mi><mi mathvariant="normal">a</mi></msup><msup><mi mathvariant="normal">L</mi><mi mathvariant="normal">b</mi></msup><msup><mi mathvariant="normal">T</mi><mi mathvariant="normal">c</mi></msup></mrow></mfenced></math><br/><br/>Writing dimensions on both sides and using the principle of homogeneity of dimensions we have,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><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><mn>0</mn></msup></mrow></mfenced><mo>=</mo><mi mathvariant="normal">k</mi><msup><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><mtext> </mtext><mi mathvariant="normal">L</mi><mtext> </mtext><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="|"><mi mathvariant="normal">L</mi></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/><br/>On comparing the power on both sides we get <math><mi>a</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math> and <math><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math><br/><br/>On solving we have <math><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn></math><br/><br/><math><mo>∴</mo></math> units of mass is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">F</mi><mtext> </mtext><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mn>2</mn></msup></mrow></mfenced></math></div>


title: if-force-f-length-l-and-time-t-be-considered-fundamental-units-then-the-units-of-mass-will-be
slug: if-force-f-length-l-and-time-t-be-considered-fundamental-units-then-the-units-of-mass-will-be-16778
tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If force <math><mfenced separators="|"><mrow><mi>F</mi></mrow></mfenced><mo>,</mo></math> length <math><mfenced separators="|"><mrow><mi>L</mi></mrow></mfenced></math> and time <math><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math> be considered fundamental units, then the units of mass will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mo>[</mo><mi>F</mi><mi>Â </mi><mi>L</mi><mi>Â </mi><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>]</mo></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mo>[</mo><mi>F</mi><mi>Â </mi><msup><mrow><mi>L</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mo>[</mo><mi>F</mi><mi>Â </mi><msup><mrow><mi>L</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>]</mo></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><mo>[</mo><msup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>L</mi><mi>Â </mi><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>]</mo></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mo>[</mo><mi>F</mi><mi>Â </mi><msup><mrow><mi>L</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>]</mo></math></span> </div>

answer:
<div class="solution">Let  <math><mi>M</mi><mo>â</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>F</mi></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msup></mrow></mfenced></math> <br/><br/>        Writing dimensions on both sides and using the principle of homogeneity of dimensions we have, <br/><br/> <math><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></mfenced><mo>=</mo><mi>k</mi><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>M</mi><mtext>â</mtext><mi>L</mi><mtext>â</mtext><msup><mrow><mi>T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>L</mi></mrow></mfenced></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>T</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msup></math> <br/><br/>        On comparing the power on both sides we get  <math><mi>a</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math>  and  <math><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math> <br/><br/>        On solving we have  <math><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn></math> <br/><br/> <math><mo>â´</mo></math>  units of mass is  <math><mfenced close="]" open="[" separators="|"><mrow><mi>F</mi><mtext>â</mtext><msup><mrow><mi>L</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2023 (11 Apr Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-force-f-velocity-v-and-time-t-are-considered-as-fundamental-physical-quantity-then-dimensional-formula-of-density-will-be

slug:
if-force-f-velocity-v-and-time-t-are-considered-as-fundamental-physical-quantity-then-dimensional-formula-of-density-will-be-80949

question:
<div class="question">If force <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>F</mi><mo>)</mo></math>, velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>V</mi><mo>)</mo></math> and time <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>T</mi><mo>)</mo></math> are considered as fundamental physical quantity, then dimensional formula of density will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>F</mi><mo> </mo><msup><mi>V</mi><mn>4</mn></msup><mo> </mo><msup><mi>T</mi><mrow><mo>-</mo><mn>6</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo> </mo><msup><mi>V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>F</mi><mn>2</mn></msup><mo mathvariant="italic"> </mo><msup><mi>V</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo> </mo><msup><mi>T</mi><mn>6</mn></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo> </mo><msup><mi>V</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo> </mo><msup><mi>T</mi><mn>2</mn></msup></math></span> </li> </ul>

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

answer:
<div class="solution"><p>The dimensional formula of force is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mi>M</mi><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p>The dimensional formula of velocity is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>The formula for density is </p><p> Density <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>M</mi><msup><mi>L</mi><mn>3</mn></msup></mfrac><mo>=</mo><mi>M</mi><msup><mi>L</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math></p><p>Comparing the density dimensions with the assumed fundamental quantities in the question,<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mi>L</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mfenced close="]" open="["><mi>F</mi></mfenced><mi>x</mi></msup><msup><mfenced close="]" open="["><mi>V</mi></mfenced><mi>y</mi></msup><msup><mfenced close="]" open="["><mi>T</mi></mfenced><mi>z</mi></msup><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>M</mi><msup><mi>L</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mfenced close="]" open="["><mrow><mi>M</mi><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mi>x</mi></msup><msup><mfenced close="]" open="["><mrow><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mi>y</mi></msup><mo>[</mo><mi>T</mi><msup><mo>]</mo><mi>z</mi></msup><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>M</mi><msup><mi>L</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mi>M</mi><mi>x</mi></msup><msup><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow></msup></math></p><p>Comparing the exponents,<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>i</mi><mo>)</mo><mspace linebreak="newline"></mspace><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo><mspace linebreak="newline"></mspace><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>i</mi><mi>i</mi><mi>i</mi><mo>)</mo></math></p><p>Solving equations (i), (ii) &amp; (iii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn></math>.</p><p>The dimensions of density <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>F</mi><msup><mi>V</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>.</p></div>


title: if-force-f-velocity-v-and-time-t-are-taken-as-fundamental-units-the-dimensions-of-mass-are
slug: if-force-f-velocity-v-and-time-t-are-taken-as-fundamental-units-the-dimensions-of-mass-are-55998
tags: ['Physics', 'Units and Dimensions', 'NEET', 'NEET 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">If force <math><mfenced separators="|"><mrow><mi>F</mi></mrow></mfenced><mo>,</mo><mi> </mi></math>velocity <math><mfenced separators="|"><mrow><mi>V</mi></mrow></mfenced></math> and time <math><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math> are taken as fundamental units, the dimensions of mass are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfenced close="]" open="[" separators="|"><mrow><mi>F</mi><msup><mrow><mi>V</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>T</mi></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><blockquote><p>The force is equal to the mass and the acceleration of the body.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mi>t</mi></mfrac></math></p><p>It can be dimensionally represented as</p><p><math><mi>F</mi><mo>=</mo><mi>M</mi><mo>×</mo><mfrac><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>M</mi><mi>L</mi></mrow><mrow><mi>T</mi><mi>T</mi></mrow></mfrac></math><br/><math><mi>F</mi><mo>=</mo><mfrac><mrow><mi>M</mi><mi>V</mi></mrow><mrow><mi>T</mi></mrow></mfrac></math></p><p>So, the dimension of the mass is<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>M</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><mi>F</mi><mi>T</mi><msup><mi>V</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p></blockquote></div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-four-dice-are-thrown-together-then-what-is-the-probability-that-the-sum-of-the-numbers-appearing-on-them-is-25

slug:
if-four-dice-are-thrown-together-then-what-is-the-probability-that-the-sum-of-the-numbers-appearing-on-them-is-25-28146

question:
<div class="question">If four dice are thrown together, then what is the probability that the sum of the numbers appearing on them is 25 ?</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">$1 / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 1296$</span> </li> </ul>

answer:
<div class="solution">Maximum sum of numbers appearing on four dice together $=24$ $\therefore$ Required probability $=0$</div>


tags: ['Mathematics', 'Functions', 'AP EAMCET', 'AP EAMCET 2019 (20 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-frac3x1leftx2x1rightfrac1x1-beginaligned-fracx2x2x1f1xf2x-text-and-fracx1x12leftx2x1righta-f1xleftbfracdx1right-f2xfraccx12-abcd-endaligned

slug:
if-frac3x1leftx2x1rightfrac1x1-beginaligned-fracx2x2x1f1xf2x-text-and-fracx1x12leftx2x1righta-f1xleftbfracdx1right-f2xfraccx12-abcd-endaligned-10492

question:
<div class="question">If \(\frac{3}{(x-1)\left(x^2+x+1\right)}=\frac{1}{x-1}\)<br/>\(\begin{aligned} &amp; -\frac{x+2}{x^2+x+1}=f_1(x)-f_2(x) \text { and } \frac{x+1}{(x-1)^2\left(x^2+x+1\right)}=A f_1(x)+\left(B+\frac{D}{x-1}\right) \\ &amp; f_2(x)+\frac{C}{(x-1)^2}, A+B+C+D= \end{aligned}\)</div>

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

answer:
<div class="solution">It is given that<br/>\(\begin{aligned}<br/>&amp; \frac{3}{(x-1)\left(x^2+x+1\right)}=\frac{1}{x-1}-\frac{x+2}{x^2+x+1} \\<br/>&amp; =f_1(x)-f_2(x) \\<br/>&amp; \text { and } \frac{x+1}{(x-1)^2\left(x^2+x+1\right)}=A f_1(x)+\left(B+\frac{D}{x-1}\right) \\<br/>&amp; f_2(x)+\frac{C}{(x-1)^2}<br/>\end{aligned}\)<br/>From above informations, it is necessary, that<br/>\(\begin{aligned}<br/>f_1(x) &amp; =\frac{1}{x-1} \\<br/>\text { and } \quad f_2(x) &amp; =\frac{x+2}{x^2+x+1}<br/>\end{aligned}\)<br/>So,<br/>\(\begin{array}{r}<br/>\frac{x+1}{(x-1)^2\left(x^2+x+1\right)}=\frac{A}{x-1}+\frac{(B x-B+D)(x+2)}{\left(x^2+x+1\right)(x-1)} \\<br/>+\frac{C}{(x-1)^2}<br/>\end{array}\)<br/>\(\begin{aligned}<br/>\Rightarrow x+1 &amp; =A(x-1)\left(x^2+x+1\right)+B(x-1) \\<br/>&amp; (x+2)(x-1)+D(x+2)(x-1)+C\left(x^2+x+1\right)<br/>\end{aligned}\)<br/>On comparing the coefficient, we get coefficient of \(x^3=0 \Rightarrow A+B=0\) coefficient of \(x^2=0 \Rightarrow A-A+2 B-2 B+D+C=0\)<br/>\(\begin{array}{llll} <br/>\Rightarrow &amp; D+C  =0 \\<br/>\therefore &amp; A+B+C+D &amp; =0<br/>\end{array}\)<br/>Hence, option (3) is correct.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Median</span> </div>
source_db: MarksBatch2_P1.db
title:
if-from-the-point-of-intersection-of-two-ogives-a-perpendicular-is-drawn-on-the-x-axis-what-does-the-x-coordinate-give

slug:
if-from-the-point-of-intersection-of-two-ogives-a-perpendicular-is-drawn-on-the-x-axis-what-does-the-x-coordinate-give-61731

question:
<div class="question">If from the point of intersection of two ogives, a perpendicular is drawn on the $x$ -axis, what does the $x-$ coordinate give?</div>

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

answer:
<div class="solution">The x-coordinate of the point of intersection of two ogives, gives median.</div>


title: if-fundamental-frequency-is-50-and-next-successive-frequencies-are-150-and-250-then-it-is
slug: if-fundamental-frequency-is-50-and-next-successive-frequencies-are-150-and-250-then-it-is-84234
tags: ['Physics', 'Waves and Sound', 'AIIMS', 'AIIMS 2001']
source_db: MarksBatch2_P1.db
question:
<div class="question">If fundamental frequency is 50 and next successive frequencies are 150 and 250 then it is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a pipe closed at one end</span> </div>

answer:
<div class="solution">For a pipe closed at one end, the frequencies are odd multiples of the fundamental frequencies.</div>


title: if-fx-sincosx-then-fx-is-equal-to
slug: if-fx-sincosx-then-fx-is-equal-to-62180
question: <div class="question">If f(x) = sin(cosx) , then f'(x) is equal to ?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos (\cos x)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sin (-\sin x)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(\sin x) \cos (\cos x)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(-\sin x) \cos (\cos x)$</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> </ul>

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

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


tags: ['Mathematics', 'Sets and Relations', 'KCET', 'KCET 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3/4</span> </div>
source_db: MarksBatch2_P1.db
title:
if-fxleftbeginaligned-k-x3-text-if-x-leq-2-3-text-if-x2-endalignedright-is-continuous-at-x2-then-the-value-of-k-is

slug:
if-fxleftbeginaligned-k-x3-text-if-x-leq-2-3-text-if-x2-endalignedright-is-continuous-at-x2-then-the-value-of-k-is-35403

question:
<div class="question">If \( f(x)=\left\{\begin{aligned} K x^{3} ; &amp; \text { if } x \leq 2 \\ 3 ; &amp; \text { if } x&gt;2 \end{aligned}\right. \) is continuous at \( x=2 \), then the value of \( K \) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( 13 \)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\( 04 \)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3/4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( 4 / 3 \)</span> </li> </ul>

answer:
<div class="solution">Given that \( f(x)=\left\{\begin{array}{cc}K \chi^{2} &amp; x \leq 2 \\ 3 &amp; x&gt;2\end{array}\right. \)<br/>Since, \( f(x) \) is continuous at \( x=2 \). Then, L.H.L = R.H.L<br/>So, \( \lim _{x \rightarrow 2} K x^{2}=\lim _{x \rightarrow 2} 3 \) \( \Rightarrow K \cdot 2^{2}=3 \Rightarrow K=\frac{3}{4} \)</div>


title: if-fxleftbeginarraycc3x-1-leq-x-leq-1-4x-1-x-4endarrayright-then-at-x1-fx-will-be
slug: if-fxleftbeginarraycc3x-1-leq-x-leq-1-4x-1-x-4endarrayright-then-at-x1-fx-will-be-13425
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \( f(x)=\left\{\begin{array}{cc}3^{x}, &amp; -1 \leq x \leq 1 \\ 4-x, &amp; 1 &lt; x &lt; 4\end{array}\right. \), then at \( x=1, f(x) \) will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Continuous but not differentiable</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Neither continuous nor differentiable</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Continuous and differentiable</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Differentiable but not continuous</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Continuous but not differentiable</span> </div>

answer:
<div class="solution">Since <math> <mrow> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>-</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>→</mo> <mn>1</mn> </mrow> </munder> <msup> <mn>3</mn> <mtext>x</mtext> </msup> <mo>=</mo> <mn>3</mn> </mrow> </math><br/><math> <mrow> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>→</mo> <mn>1</mn> </mrow> </munder> <mfenced close=")" open="(" separators=","> <mrow> <mn>4</mn> <mo>-</mo> <mtext>x</mtext> </mrow> </mfenced> <mo>=</mo> <mn>3</mn> </mrow> </math><br/>and  f(1) = 3<sup>1</sup> = 3<br/><math> <mrow> <mo>∴   </mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>-</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </math><br/><math> <mrow> <mo>⇒  </mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>x</mtext> </mrow> </mfenced> </mrow> </math> is continuous at x = 1<br/>Again  <math> <mrow> <msup> <mtext>f</mtext> <mo>′</mo> </msup> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>→</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> </mrow> </munder> <mfrac> <mrow> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>x</mtext> </mrow> </mfenced> <mo>-</mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> <mrow> <mtext>x</mtext> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>→</mo> <mn>1</mn> </mrow> </munder> <mfrac> <mrow> <msup> <mn>3</mn> <mtext>x</mtext> </msup> <mo>-</mo> <mn>3</mn> </mrow> <mrow> <mtext>x</mtext> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>h</mtext> <mo>→</mo> <mn>0</mn> </mrow> </munder> <mfrac> <mrow> <msup> <mn>3</mn> <mrow> <mn>1</mn> <mo>+</mo> <mtext>h</mtext> </mrow> </msup> <mo>-</mo> <mn>3</mn> </mrow> <mrow> <mtext>h</mtext> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <munder> <mtext>lim</mtext> <mrow> <mtext>h</mtext> <mo>→</mo> <mn>0</mn> </mrow> </munder> <mfrac> <mrow> <msup> <mn>3</mn> <mtext>h</mtext> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mtext>h</mtext> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <mtext>log </mtext> <mn>3</mn> </mrow> </math><br/>and  <math> <mrow> <msup> <mtext>f</mtext> <mo>′</mo> </msup> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>→</mo> <mn>1</mn> </mrow> </munder> <mfrac> <mrow> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>x</mtext> </mrow> </mfenced> <mo>-</mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> <mrow> <mtext>x</mtext> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>→</mo> <mn>1</mn> </mrow> </munder> <mfrac> <mrow> <mn>4</mn> <mo>-</mo> <mtext>x</mtext> <mo>-</mo> <mn>3</mn> </mrow> <mrow> <mtext>x</mtext> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math><br/><math> <mrow> <mo>∴   </mo> <msup> <mtext>f</mtext> <mo>′</mo> </msup> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mrow> </mfenced> <mo>≠</mo> <msup> <mtext>f</mtext> <mo>′</mo> </msup> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>-</mo> <mn>0</mn> </mrow> </mfenced> </mrow> </math><br/><math> <mrow> <mo>⇒</mo> </mrow> </math>   f(x) is not differentiable at x = 1</div>


tags: ['Mathematics', 'Determinants', 'AP EAMCET', 'AP EAMCET 2020 (17 Sep Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-fxleftbeginarraycccx3-2-x218-3-x381-x5-2-x250-4-x3500-1-2-3endarrayright-then-fmathrml-f3f3-f5f5-fmathrml-is-equal-to

slug:
if-fxleftbeginarraycccx3-2-x218-3-x381-x5-2-x250-4-x3500-1-2-3endarrayright-then-fmathrml-f3f3-f5f5-fmathrml-is-equal-to-92187

question:
<div class="question">If \(f(x)=\left|\begin{array}{ccc}x-3 &amp; 2 x^2-18 &amp; 3 x^3-81 \\ x-5 &amp; 2 x^2-50 &amp; 4 x^3-500 \\ 1 &amp; 2 &amp; 3\end{array}\right|\), then \(f(\mathrm{l}) f(3)+f(3) f(5)+f(5) f(\mathrm{l})\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(f(1)\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(f(3)\)</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">\(f(1)+f(3)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(f(1)+f(5)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(f(3)\)</span> </div>

answer:
<div class="solution">\(f(x)=(x-3)(x-5)\left|\begin{array}{ccc}1 &amp; 2(x+3) &amp; 3\left(x^2+9+3 x\right) \\ 1 &amp; 2(x+5) &amp; 4\left(x^2+25+5 x\right) \\ 1 &amp; 2 &amp; 3\end{array}\right|\)<br/>\(\Rightarrow \quad f(3)=f(5)=0\)<br/>So, \(f(1) f(3)+f(3) \cdot f(5)+f(5) f(1)=0=f(3)\)</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'KCET', 'KCET 2018']
source_db: MarksBatch2_P1.db
title:
if-fxleftbeginarrayccfraclog-e-xx1-x-neq-1-k-x1endarrayright-is-continuous-at-x1-then-the-value-of-k-is

slug:
if-fxleftbeginarrayccfraclog-e-xx1-x-neq-1-k-x1endarrayright-is-continuous-at-x1-then-the-value-of-k-is-80210

question:
<div class="question">If \( f(x)=\left\{\begin{array}{cc}\frac{\log _{e} x}{x-1} &amp; ; x \neq 1 \\ k &amp; ; x=1\end{array}\right. \) is continuous at \( x=1 \), then the value of \( k \) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">e</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\( 11 \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\( 00 \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( 11 \)</span> </div>

answer:
<div class="solution">Given that, \( f(x)=\left\{\begin{array}{cc}\frac{\log _{e} x}{x-1} &amp; x \neq 1 \\ k &amp; x=1\end{array}\right. \)<br/>For continuous at \( x=1 \), we have<br/>\[<br/>\begin{array}{l}<br/>\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x) \\<br/>k=\lim _{x \rightarrow 1} \frac{\log _{e} x}{x-1}=\lim _{x \rightarrow 1} \frac{1 / x}{1}=1 \\<br/>\text { So, } k=1<br/>\end{array}<br/>\]</div>


title: if-fxleftbeginarrayccfracsin-3-xe2-x1-x-neq-0-k2-x0endarrayright-is-continuous-at-x0-then-k
slug: if-fxleftbeginarrayccfracsin-3-xe2-x1-x-neq-0-k2-x0endarrayright-is-continuous-at-x0-then-k-26894
tags: ['Mathematics', 'Continuity and Differentiability', 'KCET', 'KCET 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">none</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If \( f(x)=\left\{\begin{array}{cc}\frac{\sin 3 x}{e^{2 x}-1} &amp; x \neq 0 \\ k-2 &amp; x=0\end{array}\right. \) is continuous at \( x=0 \), then \( k= \)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( \frac{9}{5} \)</span> </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">\( \frac{3}{2} \)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">none</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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 Options are not matching<br/>\( f(x)=\left\{\begin{array}{c}\frac{\sin 3 x}{e^{2 x}-1} x \neq 0 \\ k-2 \quad x=0\end{array}\right. \) Since \( f \) is continuous at \( x=0 \) \( \Rightarrow \operatorname{lt}_{x \rightarrow 0} f(x)=f(0) \) \( \lim _{x \rightarrow 0} \frac{\sin 3 x}{2 x-1}=k-2 \) \( \Rightarrow \lim _{x \rightarrow 0} \frac{\frac{\sin 3 x}{x}}{\frac{e^{2 x}-1}{x}}=k-2 \) \( \Rightarrow \frac{\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}}{\text { lt } \frac{e^{2 x}-1}{x}}=k-2 \) \( \Rightarrow \frac{3}{2}=k-2 \Rightarrow k=\frac{3}{2}+2=\frac{7}{2} \) \( \therefore k=\frac{7}{2} \)<br/>\( \Rightarrow \operatorname{lt}_{x \rightarrow 0} f(x)=f(0) \) \( \lim _{x \rightarrow 0} \frac{\sin 3 x}{2 x-1}=k-2 \)<br/>\( \Rightarrow \lim _{x \rightarrow 0} \frac{\frac{\sin 3 x}{x}}{\frac{e^{2 x}-1}{x}}=k-2 \)<br/>\( \rightarrow \frac{\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}}{\operatorname{lt}_{x \rightarrow 0} \frac{e^{2 x}-1}{x}}=k-2 \)<br/>\( \Rightarrow \frac{3}{2}=k-2 \Rightarrow k=\frac{3}{2}+2=\frac{7}{2} \)<br/>\( \therefore \mathrm{k}=\frac{7}{2} \)</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'KCET', 'KCET 2018']
source_db: MarksBatch2_P1.db
title:
if-fxleftbeginarraycl-fracsqrt1mathrmkxsqrt1mathrmkxmathrmx-text-if-1-leq-mathrmx-leq-0-frac2-mathrmx1mathrmx1-text-if-0-leq-mathrmx-leq-1-endarrayrig

slug:
if-fxleftbeginarraycl-fracsqrt1mathrmkxsqrt1mathrmkxmathrmx-text-if-1-leq-mathrmx-leq-0-frac2-mathrmx1mathrmx1-text-if-0-leq-mathrmx-leq-1-endarrayrig-31699

question:
<div class="question">If<br/>\[<br/>f(x)=\left\{\begin{array}{cl}<br/>\frac{\sqrt{1+\mathrm{kx}}-\sqrt{1-\mathrm{kx}}}{\mathrm{x}} ; &amp; \text { if }-1 \leq \mathrm{x} \leq 0 \\<br/>\frac{2 \mathrm{x}+1}{\mathrm{x}-1} ; &amp; \text { if } 0 \leq \mathrm{x} \leq 1<br/>\end{array}\right.<br/>\]<br/>is continuous at \( x=0 \), then the value of \( k \) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( k=1 \)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\( k=-1 \)</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">\( k=0 \)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( k=2 \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( k=-1 \)</span> </div>

answer:
<div class="solution">Given that<br/>\[<br/>f(x)=\mid \begin{array}{cc}<br/>\frac{\sqrt{1+k x}-\sqrt{1-k x}}{x} &amp; ;-1 \leq x &lt; 0 \\<br/>\frac{2 x+1}{x-1} &amp; ; 0 \leq x \leq 1<br/>\end{array}<br/>\]<br/>For continues at \( x=0 \), we have<br/>\[<br/>\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)<br/>\]<br/>\[<br/>\begin{array}{l}<br/>\text { So, } \lim _{x \rightarrow 0^{-}} \frac{\sqrt{1+k x}-\sqrt{1-k x}}{x} \\<br/>=\lim _{x \rightarrow 0^{-}} \frac{2 k(\sqrt{1-k x}+\sqrt{1+k x})}{x(\sqrt{x})}=k \\<br/>\text { and } \lim _{x \rightarrow 0^{+}} \frac{2 x+1}{x-1}=-1=k \\<br/>\text { So, } k=-1<br/>\end{array}<br/>\]</div>


title: if-fxleftbeginarrayclfracsin-p1-xsin-xx-x-0endarrayright
slug: if-fxleftbeginarrayclfracsin-p1-xsin-xx-x-0endarrayright-45599
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \( f(x)=\left\{\begin{array}{cl}\frac{\sin (p+1) x+\sin x}{x} &amp; , x &lt; 0 \\ q &amp; , x=0 \text { is continuous at } x=0, \text { then the ordered pair }(p, q) \text { is equal to } \\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}} &amp; , x&gt;0\end{array}\right. \)</div>

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

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

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mi> </mi><mi> </mi><mi> </mi><mi>f</mi><mfenced separators="|"><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>x</mi><mi> </mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi></mrow><mtable><mtr><mtd><mi>x</mi></mtd></mtr></mtable></mfrac></mtd></mtr><mtr><mtd><mi>q</mi></mtd></mtr><mtr><mtd><mfrac><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mo>-</mo><msqrt><mi>x</mi></msqrt></mrow><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mtd></mtr></mtable><mi> </mi><mi mathvariant="normal"> </mi><mtable><mtr><mtd><mo>:</mo><mi>x</mi><mo>&lt;</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>:</mo><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>:</mo><mi>x</mi><mo>&gt;</mo><mn>0</mn></mtd></mtr></mtable></mrow></mfenced><mi> </mi></math></p><p><math><mo>∵</mo><mi>f</mi></math> is continuous at <math><mi>x</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>.</mo><mi>H</mi><mo>.</mo><mi>L</mi><mo>=</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mi> </mi><mfrac><mrow><mi>sin</mi><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced separators="|"><mrow><mo>-</mo><mi>h</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><mo>-</mo><mi>h</mi><mo>)</mo></mrow><mrow><mo>-</mo><mi>h</mi></mrow></mfrac><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>h</mi><mo>+</mo><mi>sin</mi><mi>h</mi></mrow><mi>h</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>h</mi></mrow><mrow><mi>h</mi><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mfenced separators="|"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mi>h</mi></mrow><mi>h</mi></mfrac><mo>=</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math></p><p><math><mo>=</mo><mi>p</mi><mo>+</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>.</mo><mi>H</mi><mo>.</mo><mi>L</mi><mo>=</mo><mi> </mi><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mi> </mi><mfrac><mrow><msqrt><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mi>h</mi><mo>-</mo></msqrt><mi> </mi><msqrt><mi>h</mi></msqrt></mrow><msup><mi>h</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></mfrac><mo>×</mo><mi> </mi><mfrac><mrow><msqrt><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mi>h</mi></msqrt><mo>+</mo><msqrt><mi>h</mi></msqrt></mrow><mrow><msqrt><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mi>h</mi></msqrt><mo>+</mo><msqrt><mi>h</mi></msqrt></mrow></mfrac><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mi>h</mi><mo>-</mo><mi>h</mi></mrow><mrow><msup><mi>h</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mi> </mi><msup><mi>h</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced separators="|"><mrow><msqrt><mi>h</mi><mo>+</mo><mn>1</mn></msqrt><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p><p><math><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>1</mn><mo>+</mo><mi>h</mi></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>L</mi><mo mathvariant="italic">.</mo><mi>H</mi><mo mathvariant="italic">.</mo><mi>L</mi><mo mathvariant="italic">.</mo><mo>=</mo><mi>R</mi><mo>.</mo><mi>H</mi><mo>.</mo><mi>L</mi><mo>.</mo><mo>=</mo><mi>f</mi><mfenced separators="|"><mn>0</mn></mfenced><mo>⇒</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mi>q</mi></math><br/><math><mo>=</mo><mtable><mtr><mtd><mi>p</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mo>&amp;</mo><mi> </mi><mi> </mi><mi>q</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mtd></mtr></mtable><mi> </mi></math></p></div>


tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-fxleftbeginarraylx-text-when-x-text-is-rational-1x-text-when-x-text-is-irrational-endarrayright-then-fof-x-is-given-as

slug:
if-fxleftbeginarraylx-text-when-x-text-is-rational-1x-text-when-x-text-is-irrational-endarrayright-then-fof-x-is-given-as-69548

question:
<div class="question">If \(f(x)=\left\{\begin{array}{l}x ; \text { when } x \text { is rational } \\ 1-x ; \text { when } x \text { is irrational }\end{array}\right.\), then fof \((x)\) is given as</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">\(\mathrm{x}\)</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">C</span> <span class="option-data">\(1+\mathrm{x}\)</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">\(\mathrm{x}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; f o f(x)=\left\{\begin{array}{l}<br/>f(x) ; \text { when } f(x) \text { is rational } \\<br/>1-f(x) \text {; when } f(x) \text { is irrational }<br/>\end{array}\right. \\<br/>&amp; =\left\{\begin{array}{l}<br/>\mathrm{x} ; \quad \text { when } \mathrm{x} \text { is rational } \\<br/>1-(1-\mathrm{x}) ; \text { when } \mathrm{x} \text { is irrational }<br/>\end{array}=\mathrm{x}\right. \\<br/>&amp;<br/>\end{aligned}\)</div>


title: if-fxx24-x5-then-fa-where-aleftbeginarraylll1-2-2-2-1-2-2-2-1endarrayright-equals
slug: if-fxx24-x5-then-fa-where-aleftbeginarraylll1-2-2-2-1-2-2-2-1endarrayright-equals-64421
tags: ['Mathematics', 'Matrices', 'VITEEE', 'VITEEE 2023']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(f(x)=x^2-4 x-5\), then \(f(A)\), where \(A=\left[\begin{array}{lll}1 &amp; 2 &amp; 2 \\ 2 &amp; 1 &amp; 2 \\ 2 &amp; 2 &amp; 1\end{array}\right]\) equals</div>

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

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

answer:
<div class="solution">Given \(f(x)=x^2-4 x-5\)<br/>\(\therefore f(A)=A^2-4 A-5 I\)<br/>Now, \(A^2=\left[\begin{array}{lll}1 &amp; 2 &amp; 2 \\ 2 &amp; 1 &amp; 2 \\ 2 &amp; 2 &amp; 1\end{array}\right]\left[\begin{array}{lll}1 &amp; 2 &amp; 2 \\ 2 &amp; 1 &amp; 2 \\ 2 &amp; 2 &amp; 1\end{array}\right]=\left[\begin{array}{lll}9 &amp; 8 &amp; 8 \\ 8 &amp; 9 &amp; 8 \\ 8 &amp; 8 &amp; 9\end{array}\right]\)<br/>\(\begin{aligned}<br/>\therefore \mathrm{A}^2-4 \mathrm{~A}-5 \mathrm{I} &amp; =\left[\begin{array}{lll}<br/>9 &amp; 8 &amp; 8 \\<br/>8 &amp; 9 &amp; 8 \\<br/>8 &amp; 8 &amp; 9<br/>\end{array}\right]   -4\left[\begin{array}{lll}<br/>1 &amp; 2 &amp; 2 \\<br/>2 &amp; 1 &amp; 2 \\<br/>2 &amp; 2 &amp; 1<br/>\end{array}\right]-5\left[\begin{array}{lll}<br/>1 &amp; 0 &amp; 0 \\<br/>0 &amp; 1 &amp; 0 \\<br/>0 &amp; 0 &amp; 1<br/>\end{array}\right]<br/>\end{aligned}\)<br/>\(=\left[\begin{array}{lll}<br/>0 &amp; 0 &amp; 0 \\<br/>0 &amp; 0 &amp; 0 \\<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right]=\mathrm{O}\)</div>


title: if-g-x-25-x-2-then-what-is-lim-x-1-x-1-g-x-g-1-equal-to
slug: if-g-x-25-x-2-then-what-is-lim-x-1-x-1-g-x-g-1-equal-to-28258
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $G(x)=\sqrt{25-x^{2}}$, then what is $\lim _{x \rightarrow 1} \frac{G(x)-G(1)}{x-1}$ equal to?</div>

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

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

answer:
<div class="solution">$\mathrm{G}(\mathrm{x})=\sqrt{25-\mathrm{x}^{2}}$<br/>Now,<br/>$\lim _{x \rightarrow 1} \frac{G(x)-G(1)}{x-1}$<br/>$\lim _{x \rightarrow 1} \frac{\sqrt{25-x^{2}}-\sqrt{24}}{x-1}$<br/>Applying L'Hospital rule,<br/>$\lim _{x \rightarrow 1} \frac{(-2 x)}{2 \sqrt{25-x^{2}}}=\frac{-2}{2 \sqrt{25-x^{2}}}$<br/>$=\frac{-2}{2 \sqrt{24}}=\frac{-1}{\sqrt{24}}=\frac{-1}{2 \sqrt{6}}$<br/>Option (a) is correct.</div>


title: if-g-x-f-x-1-and-f-x-x-x-0-then-which-one-of-the-following-is-correct-2015-ii
slug: if-g-x-f-x-1-and-f-x-x-x-0-then-which-one-of-the-following-is-correct-2015-ii-33413
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{g}(\mathrm{x})=\frac{1}{\mathrm{f}(\mathrm{x})}$ and $\mathrm{f}(\mathrm{x})=\mathrm{x}, \mathrm{x} \neq 0$, then which one of the<br/>following is correct $\quad[2015-I I]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \operatorname{f}(\mathrm{f}(\mathrm{f}(\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x})))))$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{f}(\mathrm{f}(\mathrm{g}(3(\mathrm{~g}(\mathrm{f}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))$</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">C</span> <span class="option-data">$\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x}))))))=\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x})))))$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \mathrm{f}(\mathrm{f}(\mathrm{g}(3(\mathrm{~g}(\mathrm{f}(\mathrm{x}))))))=\mathrm{g}(\mathrm{g}(\mathrm{f}(\mathrm{g}(\mathrm{f}(\mathrm{x}))))))$</span> </div>

answer:
<div class="solution">$f(x)=x$ $\mathrm{g}(\mathrm{x})=1 / \mathrm{x}$<br/>Putting these values in the options, only (b) is correct.</div>


title: if-g-x-sin-x-x-r-and-f-x-s-i-n-x-1-x-0-2-what-is-gof-x-equal-to
slug: if-g-x-sin-x-x-r-and-f-x-s-i-n-x-1-x-0-2-what-is-gof-x-equal-to-83081
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}, \mathrm{x} \in \mathrm{R}$ and $\mathrm{f}(\mathrm{x})=\frac{1}{\sin \mathrm{x}}, \mathrm{x} \in\left(0, \frac{\pi}{2}\right)$ what is (gof) (x) equal to?</div>

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

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

answer:
<div class="solution">Given function are :<br/>$g(x)=\sin x$ and $f(x)=\frac{1}{\sin x}$<br/>$($ gof $)(x)=g[f(x)]$<br/>$=\sin \mathrm{f}(\mathrm{x})$<br/>$=\sin \left(\frac{1}{\sin \mathrm{x}}\right)$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-gravitational-constant-g-planks-constant-h-and-speed-of-light-c-considered-as-fundamental-physical-quantities-then-find-dimensional-formula-of-time

slug:
if-gravitational-constant-g-planks-constant-h-and-speed-of-light-c-considered-as-fundamental-physical-quantities-then-find-dimensional-formula-of-time-39988

question:
<div class="question">If gravitational constant <math><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>,</mo></math> planks constant <math><mfenced separators="|"><mrow><mi>h</mi></mrow></mfenced></math> and speed of light <math><mfenced separators="|"><mrow><mi>c</mi></mrow></mfenced></math> considered as fundamental physical quantities then find dimensional formula of time</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mfenced close="]" open="[" separators="|"><mrow><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>Â </mi><mi>h</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><mfrac><mrow><mo>-</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></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">B</span> <span class="option-data"><math><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msup><mi>Â </mi><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>Â </mi><msup><mrow><mi>c</mi></mrow><mrow><mfrac><mrow><mo>-</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfenced close="]" open="[" separators="|"><mrow><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>h</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><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>Â </mi><msup><mrow><mi>h</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><mo>-</mo><mn>2</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><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>Â </mi><mi>h</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><mfrac><mrow><mo>-</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Let <br/><br/> <math><mfenced close="]" open="[" separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>G</mi></mrow></mfenced></mrow><mrow><mi>Î±</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>L</mi></mrow></mfenced></mrow><mrow><mi>Î²</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>C</mi></mrow></mfenced></mrow><mrow><mi>Î³</mi></mrow></msup></math> <br/><br/> <math><mo>=</mo><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><mn>9</mn></mrow></msup><mo>⋅</mo><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>Î²</mi></mrow></msup><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>Î³</mi></mrow></msup></math> <br/><br/> <math><mo>â</mo><mi>Â </mi><mo>-</mo><mi>Î±</mi><mo>+</mo><mi>Î²</mi><mo>=</mo><mn>0</mn></math> <br/><br/> <math><mn>3</mn><mo>â</mo><mo>+</mo><mn>2</mn><mi>Î²</mi><mo>+</mo><mi>Î³</mi><mo>=</mo><mn>0</mn></math> <br/><br/> <math><mo>-</mo><mn>2</mn><mi>Î±</mi><mo>-</mo><mi>Î²</mi><mo>-</mo><mi>Î³</mi><mo>=</mo><mn>1</mn></math> <br/><br/>On solving we get<br/><br/> <math><mi>Î±</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi>Â </mi><mi>Î²</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi>Â </mi><mi>Î³</mi><mo>=</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
title:
if-i-1-5-x-i-6-5-and-i-1-5-x-i-6-2-25-then-the-standard-deviation-of-observations-3-x-1-2-3-x-2-2-3-x-3-2-3-x-4-2-and-3-x-5-2-is-equal-to

slug:
if-i-1-5-x-i-6-5-and-i-1-5-x-i-6-2-25-then-the-standard-deviation-of-observations-3-x-1-2-3-x-2-2-3-x-3-2-3-x-4-2-and-3-x-5-2-is-equal-to-66388

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></munderover><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>25</mn></math>, then the standard deviation of observations <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>2,</mn><mo> </mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>2,</mn><mo> </mo><mn>3</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>2</mn><mo>,</mo><mo> </mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>x</mi><mn>4</mn></msub><mo>+</mo><mn>2</mn></math> and<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>3</mn><msub><mi>x</mi><mn>5</mn></msub><mo>+</mo><mn>2</mn></math> is equal to</div>

answer:
<div class="solution">Variance of <math><mfenced separators="|"><mrow><mn>3</mn><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>9</mn><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math><br/><math><mo>=</mo><mn>9</mn><mfenced separators="|"><mrow><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mn>5</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></math><br/><math><mo>=</mo><mn>9</mn><mfenced separators="|"><mrow><mn>5</mn><mo>-</mo><mn>1</mn></mrow></mfenced></math><br/><math><mo>=</mo><mn>36</mn></math><br/><math><mo>â</mo><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>=</mo><mn>6</mn></math><br/></div>


title: if-i-1-9-x-i-5-9-and-i-1-9-x-i-5-2-45-then-the-standard-deviation-of-the-9-items-x-1-x-2-x-9-is
tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2018 (08 Apr)']
source_db: MarksBatch2_P1.db
slug:
if-i-1-9-x-i-5-9-and-i-1-9-x-i-5-2-45-then-the-standard-deviation-of-the-9-items-x-1-x-2-x-9-is-95886

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>9</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>45</mn></math>, then the standard deviation of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> items <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi><mn>9</mn></msub></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Given <math> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mn>9</mn> </munderover> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mstyle><mo>−</mo><mn>45</mn><mo>=</mo><mn>9</mn></mrow> </math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>54</mn></math></p><p>Also <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><mfenced><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mn>10</mn><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>25</mn></mrow></mfenced><mo>=</mo><mn>45</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mn>10</mn><mo>×</mo><mn>54</mn><mo>+</mo><mn>25</mn><mo>×</mo><mn>9</mn><mo>=</mo><mn>45</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>45</mn><mo>+</mo><mn>540</mn><mo>−</mo><mn>225</mn></math></p><p><math> <mrow> <mo>=</mo><mn>360</mn></mrow> </math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mstyle displaystyle="true"><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mstyle><mi>N</mi></mfrac><mo>−</mo><msup><mfenced><mfrac><mstyle displaystyle="true"><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mstyle><mi>N</mi></mfrac></mfenced><mn>2</mn></msup></math></p><p><math> <mrow> <mo>=</mo><mfrac> <mrow> <mn>360</mn></mrow> <mn>9</mn> </mfrac> <mo>−</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mfrac> <mrow> <mn>54</mn></mrow> <mn>9</mn> </mfrac> </mrow> <mo>)</mo></mrow></mrow> <mn>2</mn> </msup> </mrow> </math></p><p>So, variance <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>40</mn><mo>−</mo><mn>36</mn><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>S</mi><mi>D</mi><mo>=</mo><mn>2</mn></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2020 (06 Sep Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-i-1-n-x-i-a-n-and-i-1-n-x-i-a-2-n-a-n-a-1-then-the-standard-deviation-of-n-observations-x-1-x-2-x-n-is

slug:
if-i-1-n-x-i-a-n-and-i-1-n-x-i-a-2-n-a-n-a-1-then-the-standard-deviation-of-n-observations-x-1-x-2-x-n-is-41702

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo largeop="true">Σ</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo largeop="true">Σ</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mi>n</mi><mi>a</mi><mo>,</mo><mo> </mo><mfenced><mrow><mi>n</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>&gt;</mo><mn>1</mn></mrow></mfenced></math>, then the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> 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><msub><mi>x</mi><mi>n</mi></msub></math> is<br/> </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>a</mi><mo>-</mo><mn>1</mn></math><br/> </span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msqrt><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo></msqrt></math><br/> </span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo></msqrt></math><br/> </span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo></msqrt></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"><msqrt><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo></msqrt></math></span> </div>

answer:
<div class="solution"><p>SD, <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><mi>n</mi></mfrac><mo>-</mo><msup><mfenced><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mfenced><mn>2</mn></msup></msqrt></math></p><p>SD is independent of change of origin, so, for given data<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><msqrt><mfrac><mrow><mi>Σ</mi><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup></mrow><mi>n</mi></mfrac><mo>-</mo><msup><mfenced><mfrac><mrow><mi>Σ</mi><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mi>n</mi></mfrac></mfenced><mn>2</mn></msup></msqrt></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mi>n</mi><mi>a</mi></mrow><mi>n</mi></mfrac><mo>-</mo><msup><mfenced><mfrac><mi>n</mi><mi>n</mi></mfrac></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mi>a</mi><mo>-</mo><mn>1</mn></msqrt></math></p></div>


title: if-i-n-0-4-tan-n-x-d-x-then-what-is-i-n-i-n-2-equal-to
slug: if-i-n-0-4-tan-n-x-d-x-then-what-is-i-n-i-n-2-equal-to-88278
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $I_{n}=\int_{0}^{\frac{\pi}{4}} \tan ^{n} x d x$ then what is $I_{n}+I_{n-2}$ equal to?</div>

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

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

answer:
<div class="solution">Let $I_{n}=\int_{0}^{\pi / 4} \tan ^{n} x d x$<br/>Consider,<br/>$I_{n}+I_{n-2}=\int_{0}^{\pi / 4} \tan ^{n} x d x+\int_{0}^{\pi / 4} \tan ^{n-2} x d x$<br/>$=\int_{0}^{\pi / 4} \tan ^{n-2} x\left(\tan ^{2} x+1\right) d x$<br/>$=\int_{0}^{\pi / 4} \sec ^{2} x \tan ^{n-2} x d x$<br/>Put $\tan x=t$<br/>$\sec ^{2} x d x=d t$<br/>when $x=0$ then $t=0$ and when $x=\frac{\pi}{4}, t=1$<br/>$\therefore I_{n}+I_{n-2}=\int_{0}^{1} t^{n-2} d t$<br/>$=\left.\frac{t^{n-2+1}}{n-2+1}\right|_{0} ^{1}=\left.\frac{t^{n-1}}{n-1}\right|_{0} ^{1}=\frac{1}{n-1}[1-0]=\frac{1}{n-1}$</div>


title: if-in-a-a-bc-cos-b-sin-a-2-sin-c-then-the-triangle-is
slug: if-in-a-a-bc-cos-b-sin-a-2-sin-c-then-the-triangle-is-66754
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If in a $\Delta A B C, \cos B=(\sin A) /(2 \sin C)$, then the triangle is</div>

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

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

answer:
<div class="solution">Consider $\cos B=\frac{\sin A}{2 \sin C}$<br/>$\Rightarrow \frac{c^{2}+a^{2}-b^{2}}{2 a c}=\frac{a}{2 c}$<br/>$\quad\left(\because \frac{a}{\sin A}=\frac{c}{\sin C} \Rightarrow \frac{\sin A}{\sin C}=\frac{a}{c}\right)$<br/>$\Rightarrow c^{2}+a^{2}-b^{2}=\frac{2 a^{2} c}{2 c}$<br/>$\Rightarrow c^{2}+a^{2}-b^{2}=a^{2}$<br/>$\Rightarrow c^{2}-b^{2}=0$<br/>$\Rightarrow c=b$<br/>Hence, $\Delta \mathrm{ABC}$ is isosceles triangle.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$36.1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-in-a-frequency-distribution-table-with-12-classes-the-width-of-each-class-is-25-and-the-lowest-class-boundary-is-61-then-what-is-the-upper-class-bo

slug:
if-in-a-frequency-distribution-table-with-12-classes-the-width-of-each-class-is-25-and-the-lowest-class-boundary-is-61-then-what-is-the-upper-class-bo-88245

question:
<div class="question">If in a frequency distribution table with 12 classes, the width of each class is $2.5$ and the lowest class boundary is $6.1$. then what is the upper class boundary of the highest class?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">30 .</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$27.6$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30.6$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$36.1$</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">Given : lowest class boundary $=6.1$ Class width $=2.5$, Number of elasses $=12$<br/>$\Rightarrow$ Upper class boundary of the highest class $=6.1+(2.5 \times 12)=6.1+30=36.1$</div>


title: if-in-a-rightangled-triangle-abc-hypotenuse-ac-p-then-what-is-a-b-a-c-bc-b-a-c-a-cb-equal-to
slug: if-in-a-rightangled-triangle-abc-hypotenuse-ac-p-then-what-is-a-b-a-c-bc-b-a-c-a-cb-equal-to-43434
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If in a right-angled triangle $\mathrm{ABC}$, hypotenuse $\mathrm{AC}=\mathrm{p}$, then what is $\overrightarrow{A B} \cdot \overrightarrow{A C}+\overrightarrow{B C} \cdot \overrightarrow{B A}+\overrightarrow{C A} \cdot \overrightarrow{C B}$ equal to ?</div>

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

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

answer:
<div class="solution">Hypotenuse, $\mathrm{AC}=\mathrm{P}$ $\mathrm{BC}$ is perpendicular to $\mathrm{AB}$.<br/>$\therefore \overrightarrow{B C} \cdot \overrightarrow{B A}=0$<br/>$\therefore \overrightarrow{A B} \cdot \overrightarrow{A C}+\overrightarrow{B C} \cdot \overrightarrow{B A}+\overrightarrow{C A} \cdot \overrightarrow{C B}$<br/>$=\overrightarrow{A B} \cdot \overrightarrow{A C}+0+\overrightarrow{A C} \cdot \overrightarrow{B C}=\overrightarrow{A C}(\overrightarrow{A B}+\overrightarrow{B C})$<br/>$=\overrightarrow{A C} \cdot \overrightarrow{A C}=\overrightarrow{A C}^{2}=\mathrm{P}^{2}$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-in-general-the-value-of-sin-a-is-known-but-the-value-of-a-is-not-known-then-how-many-values-of-tan-2-a-can-be-calculated

slug:
if-in-general-the-value-of-sin-a-is-known-but-the-value-of-a-is-not-known-then-how-many-values-of-tan-2-a-can-be-calculated-60943

question:
<div class="question">If in general, the value of $\sin A$ is known, but the value of $\mathrm{A}$<br/>is not known, then how many values of $\tan \left(\frac{A}{2}\right)$ can be calculated?</div>

answer:
<div class="solution">We know, $\sin A=\frac{2 \tan \frac{A}{2}}{1+\tan ^{2} \frac{A}{2}}$<br/>If $\sin A$ is known then equation (1) becomes<br/>quadratic equation in $\tan \left(\frac{A}{2}\right)$. This mean 2 values of<br/>$\tan \left(\frac{A}{2}\right)$ can be calculated.</div>


title: if-instantaneous-current-is-given-by-i-4-cos-t-then-the-rms-value-of-current-is
slug: if-instantaneous-current-is-given-by-i-4-cos-t-then-the-rms-value-of-current-is-97367
tags: ['Physics', 'Alternating Current', 'COMEDK', 'COMEDK 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If instantaneous current is given by $i=4 \cos (\omega t+\phi)$, then the rms value of current is</div>

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

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

answer:
<div class="solution">Instantaneous current,<br/>$$<br/>i=4 \cos (\omega t+\phi)<br/>$$<br/>Comparing with standard equation, $i=i_{0} \cos (\omega t+\phi)$, we get<br/>$\therefore$ Peak current, $i_{0}=4 \mathrm{~A}$<br/>$$<br/>\text { Hence, } \begin{aligned}<br/>i_{\mathrm{rms}} &amp;=\frac{i_{0}}{\sqrt{2}} \\<br/>&amp;=\frac{4}{\sqrt{2}}=2 \sqrt{2} \mathrm{~A}<br/>\end{aligned}<br/>$$</div>


title: if-is-a-complex-cube-root-of-unity-and-x-2-2-then-what-is-the-value-of-x-2-4-x-7
slug: if-is-a-complex-cube-root-of-unity-and-x-2-2-then-what-is-the-value-of-x-2-4-x-7-86597
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is a complex cube root of unity and $x=\omega^{2}-\omega-2$, then what is the value of $x^{2}+4 x+7 ?$</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="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 $x=\omega^{2}-\omega-2$<br/>$\Rightarrow x+2=\omega^{2}-\omega$<br/>On squaring both sides, we get<br/>$(x+2)^{2}=\left(\omega^{2}-\omega\right)^{2}$<br/>$\Rightarrow x^{2}+4 x+4=\omega^{4}+\omega^{2}-2 \omega^{3}$<br/>Add 3 on both side<br/>$\Rightarrow x^{2}+4 x+4+3=\omega+\omega^{2}-2+3\left(\because \omega^{3}=1\right)$<br/>$\Rightarrow x^{2}+4 x+7=1+\omega+\omega^{2}$<br/>$\Rightarrow \quad x^{2}+4 x+7=0\left(\because 1+\omega+\omega^{2}=0\right)$</div>


title: if-is-a-complex-cube-root-of-unity-then-what-is-10-10-equal-to
slug: if-is-a-complex-cube-root-of-unity-then-what-is-10-10-equal-to-23389
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is a complex cube root of unity, then what is $\omega^{10}+\omega^{-10}$ 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">-2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Consider $\omega^{10}+\omega^{-10}=\omega^{10}+\frac{1}{\omega^{10}}$<br/>$=\left(\omega^{3}\right)^{3} \cdot \omega+\frac{1}{\left(\omega^{3}\right)^{3} \cdot \omega}=\omega+\frac{1}{\omega}=-1$</div>


title: if-is-a-complex-cube-root-of-unity-then-what-is-the-value-of-1-1-1-1-2-1
slug: if-is-a-complex-cube-root-of-unity-then-what-is-the-value-of-1-1-1-1-2-1-24622
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is a complex cube root of unity, then what is the value<br/>of $1-\frac{1}{(1+\omega)}-\frac{1}{\left(1+\omega^{2}\right)} ?$</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">$\omega$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\omega^{2}$</span> </li> </ul>

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


title: if-is-a-complex-nonreal-cube-root-of-unity-then-satisfies-which-one-of-the-following-equations
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-is-a-complex-nonreal-cube-root-of-unity-then-satisfies-which-one-of-the-following-equations-53077

question:
<div class="question">If $\omega$ is a complex non-real cube root of unity, then $\omega$ satisfies which one of the following equations?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+x+1=0$</span> </div>

answer:
<div class="solution">If $\omega$ is cube root of unity then $\omega^{2}+\omega+1=0$ So, \omega satisfies the equaion $x^{2}+x+1=0$</div>


title: if-is-a-complex-number-such-that-2-1-0-then-what-is-31-equal-to
slug: if-is-a-complex-number-such-that-2-1-0-then-what-is-31-equal-to-42029
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ is a complex number such that $\alpha^{2}+\alpha+1=0$, then what is $\alpha^{31}$ equal to?</div>

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

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

answer:
<div class="solution">Since, $\alpha$ is a complex root therefore $\alpha^{2}+\alpha+1=0 \Rightarrow \alpha=\omega$ or $\omega^{2}$<br/>consider $\alpha^{31}=(\omega)^{31}$<br/>$=\left(\omega^{3}\right)^{10} \cdot \omega$<br/>$=\omega \quad\left(\because \omega^{3}=1\right)$<br/>$=\alpha$</div>


title: if-is-a-null-set-then-which-one-of-the-following-is-correct
slug: if-is-a-null-set-then-which-one-of-the-following-is-correct-65882
question: <div class="question">If $\phi$ is a null set, then which one of the following is correct?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 2)']
answer: <div class="solution">Since $\phi$ is null set therefore $\phi=$</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\phi=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\phi=\{0\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\phi=\{\phi$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\phi=\{\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\phi=\{\}$</span> </div>


title: if-is-perpendicular-to-both-and-where-k-and-2-i-3-j-4-k-then-what-is-equal-to
slug: if-is-perpendicular-to-both-and-where-k-and-2-i-3-j-4-k-then-what-is-equal-to-97924
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{\beta}$ is perpendicular to both $\vec{\alpha}$ and $\vec{\gamma}$ where $\vec{\alpha}=\overrightarrow{\mathrm{k}}$ and<br/>$\vec{\gamma}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$, then what is $\vec{\beta}$ equal to?</div>

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

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

answer:
<div class="solution">Let $\vec{\beta}=a \hat{i}+b \hat{j}+c \hat{k}$<br/>since $\vec{\beta} \cdot \vec{\alpha}=0$ and $\vec{\beta} \cdot \vec{\gamma}=0$<br/>$\therefore \quad c=0$ and $2 a+3 b=0 \Rightarrow a=-\frac{3 b}{2}$<br/>Hence, $\vec{\beta}=-\frac{3 b}{2} \hat{i}+b \hat{j}=0 \Rightarrow-3 \hat{i}+2 \hat{j}=0$</div>


title: if-is-the-acute-angle-between-the-diagonals-of-a-cube-then-which-one-of-the-following-is-correct-1
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-is-the-acute-angle-between-the-diagonals-of-a-cube-then-which-one-of-the-following-is-correct-1-77569

question:
<div class="question">If $\theta$ is the acute angle between the diagonals of a cube, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \theta&gt;60^{\circ} \quad$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/gZdnATvrvbZt-VWbi5KJuUNEhS99b1tH3yppkIUpL7g.original.fullsize.png"/><br/><br/>Let there be cube of side 'a'. Co-ordinates of its vertices $\mathrm{O}, \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}$ have bemarked in the figure. Diagonals<br/>are $\mathrm{OE}, \mathrm{FC}, \mathrm{GB}$ and $\mathrm{AD}$. Direction ratios $\left(\mathrm{dr}_{3}\right.$ ) of these diagonals are : $\mathrm{OE}\langle(\mathrm{a}-0),(\mathrm{a}-0),(\mathrm{a}-0)\rangle=(\mathrm{a}, \mathrm{a}, \mathrm{a})$<br/>$F C\langle(-a, a,-a)\rangle ; G B\langle(-a, a, a)\rangle$ and $A D\langle(a, a,-a)\rangle$<br/>Their des are :<br/>OE, $\left\langle\frac{a}{\sqrt{a^{2}+a^{2}+a^{2}}}, \frac{a}{\sqrt{a^{2}+a^{2}+a^{2}}}, \frac{a}{\sqrt{a^{2}+a^{2}+a^{2}}}\right\rangle$<br/>$=\left\langle\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right\rangle$<br/>$\mathrm{AD},\left\langle\frac{\mathrm{a}}{\sqrt{\Sigma \mathrm{a}^{2}}}, \frac{\mathrm{a}}{\sqrt{\Sigma \mathrm{a}^{2}}}, \frac{-\mathrm{a}}{\sqrt{\Sigma \mathrm{a}^{2}}}\right\rangle=\left\langle\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}\right\rangle$<br/>Angle, $\theta$, between $\mathrm{AD}$ and $\mathrm{OE}$ is given by<br/>$\cos \theta=\pm \frac{\frac{1}{\sqrt{3}} \times \frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}} \times \frac{1}{\sqrt{3}}-\frac{1}{\sqrt{3}} \times \frac{1}{\sqrt{3}}}{\sqrt{\left\{\left(\frac{1}{\sqrt{3}}\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}\right\}\left\{\left(\frac{1}{\sqrt{3}}\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}+\left(-\frac{1}{\sqrt{3}}\right)^{2}\right\}}}$<br/>$=\frac{\frac{1}{3}}{1 \times 1}=\pm \frac{1}{3}$<br/>Since the cube is in positive octant, we take $+\frac{1}{3}$.<br/>So, $\cos \theta=\frac{1}{3} \Rightarrow \theta&gt;60$ '<br/>[Since value of $\cos \theta$ decreases as $\theta$ increases in 0 to $90^{\circ} \cdot \cos \theta=1$ when $\theta=0^{\circ}$ and $\cos \theta=0$ when $\theta=90^{\circ}$ ]</div>


title: if-is-the-acute-angle-between-the-diagonals-of-a-cube-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-is-the-acute-angle-between-the-diagonals-of-a-cube-then-which-one-of-the-following-is-correct-29866

question:
<div class="question">If $\theta$ is the acute angle between the diagonals of a cube, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \theta=30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \theta=45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \cos \theta=1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3 \cos \theta=1$</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">$3 \cos \theta=1$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/uiyuUAIurIWGn5Snd1A6F6jJzp6wfRjg0WZmT5VNtUQ.original.fullsize.png"/><br/><br/>diagonals are OP and AD and Acute anngle $=\theta$<br/>$\cos \theta=\left|\frac{\mathrm{a}_{1} \mathrm{a}_{2}+\mathrm{b}_{1} \mathrm{~b}_{2}+\mathrm{c}_{1} \mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}} \sqrt{\mathrm{a}_{2}^{2}+\mathrm{b}_{2}^{2}+\mathrm{c}_{2}^{2}}}\right|$<br/>$=\left|\frac{\mathrm{a}(-\mathrm{a})+(\mathrm{a})(\mathrm{a})+(\mathrm{a})(\mathrm{a})}{\sqrt{\mathrm{a}^{2}+\mathrm{a}^{2}+\mathrm{a}^{2}} \sqrt{\mathrm{a}^{2}+\mathrm{a}^{2}+\mathrm{a}^{2}}}\right|$<br/>$=\left|\frac{-\mathrm{a}^{2}+\mathrm{a}^{2}+\mathrm{a}^{2}}{\sqrt{3 \mathrm{a}^{2}} \sqrt{3 \mathrm{a}^{2}}}\right|=\left|\frac{\mathrm{a}^{2}}{3 \mathrm{a}^{2}}\right|=\frac{1}{3}$<br/>$\Rightarrow 3 \cos \theta=1$</div>


title: if-is-the-angle-between-the-vectors-is-4-i-k-and-i-j-k-then-what-is-sin-cos-equal-to
slug: if-is-the-angle-between-the-vectors-is-4-i-k-and-i-j-k-then-what-is-sin-cos-equal-to-37618
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\theta$ is the angle between the vectors is $4(\hat{i}-\hat{k})$ and $\hat{i}+\hat{j}+\hat{k}$, then what is $(\sin \theta+\cos \theta)$ equal to ?</div>

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

answer:
<div class="solution">Angle $\mathrm{b} / \mathrm{w}$ the vectors is $\cos \theta=\frac{4(1)-4(1)}{\sqrt{32} \sqrt{3}}=0$<br/>$\Rightarrow \theta=\frac{\pi}{2}$<br/>Hence, $\cos \theta+\sin \theta=\cos \frac{\pi}{2}+\sin \frac{\pi}{2}=0+1=1$.</div>


title: if-is-the-cube-root-of-unity-then-what-is-one-root-of-the-equation-x-2-2-0-2-x-2-2-1-0
slug: if-is-the-cube-root-of-unity-then-what-is-one-root-of-the-equation-x-2-2-0-2-x-2-2-1-0-12790
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-2$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is the cube root of unity, then what is one root of the equation<br/>$\left|\begin{array}{ccc}x^{2} &amp; -2 x &amp; -2 \omega^{2} \\ 2 &amp; \omega &amp; -\omega \\ 0 &amp; \omega &amp; 1\end{array}\right|=0 ?$</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">$-2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\omega$</span> </li> </ul>

answer:
<div class="solution">Given matrix is :<br/>$\left|\begin{array}{ccc}x^{2} &amp; -2 x &amp; -2 \omega^{2} \\ 2 &amp; \omega &amp; -\omega \\ 0 &amp; \omega &amp; 1\end{array}\right|=0$<br/>$\mathrm{By} \mathrm{C}_{2} \rightarrow \mathrm{C}_{2}+\mathrm{C}_{3}$, we get<br/>$\Rightarrow\left|\begin{array}{ccc}\mathrm{x}^{2} &amp; -2 \mathrm{x}-2 \omega^{2} &amp; -2 \omega^{2} \\ 2 &amp; 0 &amp; -\omega \\ 0 &amp; 1+\omega &amp; 1\end{array}\right|=0$<br/>$\Rightarrow\left|\begin{array}{ccc}\mathrm{x}^{2} &amp; -2 \mathrm{x}-2 \omega^{2} &amp; -2 \omega^{2} \\ 2 &amp; 0 &amp; -\omega \\ 0 &amp; -\omega^{2} &amp; 1\end{array}\right|=0$<br/>$\quad\left[\because \quad 1+\omega=-\omega^{2}\right]$<br/>$\Rightarrow \omega^{2}\left|\begin{array}{cc}x^{2} &amp; -2 \omega^{2} \\ 2 &amp; -\omega\end{array}\right|+1\left|\begin{array}{cc}x^{2} &amp; -2 x-2 \omega^{2} \\ 2 &amp; -0\end{array}\right|=0$<br/>$\Rightarrow \omega^{2}\left(-\omega x^{2}+4 \omega^{2}\right)-\left(-4 x-4 \omega^{2}\right)=0$<br/>$\Rightarrow-x^{2}+4 \omega+4 x+4 \omega^{2}=0$<br/>$\Rightarrow-x^{2}+4 \omega-4 x-4-4 \omega=0 \Rightarrow-x^{2}-4 x-4=0$<br/>$\Rightarrow(x+2)^{2}=0 \Rightarrow x=-2$</div>


title: if-is-the-cube-root-of-unity-then-what-is-the-conjugate-of-2-2-3-i
slug: if-is-the-cube-root-of-unity-then-what-is-the-conjugate-of-2-2-3-i-25736
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is the cube root of unity, then what is the conjugate of $2 \omega^{2}+3 i ?$</div>

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

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

answer:
<div class="solution">Let $z=2 \omega^{2}+3 i$<br/>Since, $\omega$ is the cube root of unity<br/>$\therefore \omega=\frac{-1+\sqrt{3} i}{2}$ and $\omega^{2}=\frac{-1-\sqrt{3} i}{2}$<br/>$\therefore z=2 \omega^{2}+3 i$<br/>$\quad=2\left[\frac{-1-\sqrt{3} i}{2}\right]+3 i$<br/>$\quad=-1-\sqrt{3} i+3 i=-1+(3-\sqrt{3}) i$<br/>$\bar{z}=-1-(3-\sqrt{3}) i=-1+\sqrt{3} i-3 i=2\left(\frac{-1+\sqrt{3} i}{2}\right)-3 i$<br/>$=2 \omega-3 i$</div>


title: if-is-the-imaginary-cube-root-of-unity-then-what-is-2-2-2-27-equal-to
slug: if-is-the-imaginary-cube-root-of-unity-then-what-is-2-2-2-27-equal-to-22352
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is the imaginary cube root of unity, then what is $\left(2-\omega+2 \omega^{2}\right)^{27}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3^{27} \omega$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-3^{27} \omega^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3^{27}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-3^{27}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$-3^{27}$</span> </div>

answer:
<div class="solution">Consider, $\left(2-\omega+2 \omega^{2}\right)^{27}=\left[2\left(1+\omega^{2}\right)-\omega\right]^{27}$<br/>$=(-2 \omega-\omega)^{27}\left[\because 1+\omega+\omega^{2}=0\right]$<br/>$=(-3 \omega)^{27}=-3^{27} \cdot \omega^{27}$<br/>$=(-3)^{27} \cdot\left(\omega^{3}\right)^{9}=(-3)^{27}$</div>


title: if-is-the-universal-set-and-p-is-a-subset-of-then-what-is-p-p-p-equal-to
slug: if-is-the-universal-set-and-p-is-a-subset-of-then-what-is-p-p-p-equal-to-54815
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\phi$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mu$ is the universal set and $\mathrm{P}$ is a subset of $\mu$, then what is $\mathrm{P} \cap(\mathrm{P}-\mu) \cup(\mu-\mathrm{P})\}$ equal to ?</div>

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

answer:
<div class="solution">Since $\mu$ is universal set and $\mathrm{P} \subseteq \mu, \mathrm{P}-\mu=\phi$ and $\mu-\mathrm{P}=\mathrm{P}$<br/>So, $(P-\mu) \cup(\mu-P)=\phi \cup P=P^{\prime}$<br/>Now, $\operatorname{P} \cap\{P-\mu) \cup(\mu-P)\}=\operatorname{P} \cap P=\phi$</div>


title: if-k-sin-18-sin-18-5-sin-18-7-then-what-is-the-value-of-k
slug: if-k-sin-18-sin-18-5-sin-18-7-then-what-is-the-value-of-k-29522
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{K}=\sin \left(\frac{\pi}{18}\right) \sin \left(\frac{5 \pi}{18}\right) \sin \left(\frac{7 \pi}{18}\right)$, then what is the value<br/>of K?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{16}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{K}=\sin \left(\frac{\pi}{18}\right) \sin \left(\frac{5 \pi}{18}\right) \sin \left(\frac{7 \pi}{18}\right)$<br/>We know, $2 \sin A \sin B=\cos (A-B)-\cos (A+B)$<br/>$K=\frac{1}{2} \cdot \sin \frac{\pi}{18}\left[2 \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18}\right]$<br/>$=\frac{1}{2} \cdot \sin \frac{\pi}{18}\left[\cos \frac{2 \pi}{18}-\cos \frac{2}{\frac{\not / \pi} \pi}{3^{6}}\right]$<br/>$=\frac{1}{2} \cdot \frac{1}{2}\left[2 \sin \frac{\pi}{18} \cos \frac{2 \pi}{18}-2 \sin \frac{\pi}{18} \cos \frac{2 \pi}{3}\right]$<br/>$=\frac{1}{4}\left[\sin \left(\frac{3 \pi}{18}\right)+\sin \left(\frac{-\pi}{18}\right)-2 \sin \frac{\pi}{18} \cos \left(\pi-\frac{\pi}{3}\right)\right]$<br/>$=\frac{1}{4}\left[\sin \frac{\pi}{6}-\sin \frac{\pi}{18}+2 \sin \frac{\pi}{18} \cdot \cos \frac{\pi}{3}\right]$<br/>$=\frac{1}{4}\left[\sin \frac{\pi}{6}-\sin \frac{\pi}{18}+\not 2 \sin \frac{\pi}{18} \cdot \frac{1}{\not}\right]$<br/>$=\frac{1}{4} \sin \frac{\pi}{6}=\frac{1}{4} \times \frac{1}{2}=\frac{1}{8}$</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-l-1-dx-d-e-s-i-n-x-2017-ii-l-2-lim-h-0-h-e-s-i-n-x-h-e-s-i-n-x-l-3-e-s-i-n-x-cos-xdx-then-which-one-of-the-following-is-correct

slug:
if-l-1-dx-d-e-s-i-n-x-2017-ii-l-2-lim-h-0-h-e-s-i-n-x-h-e-s-i-n-x-l-3-e-s-i-n-x-cos-xdx-then-which-one-of-the-following-is-correct-47263

question:
<div class="question">If $l_{1}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\sin \mathrm{x}}\right) \quad[2017-\mathrm{II}]$<br/>$l_{2}=\lim _{h \rightarrow 0} \frac{e^{\sin (x+h)}-e^{\sin x}}{h}$<br/>$l_{3}=\int \mathrm{e}^{\sin \mathrm{x}} \cos \mathrm{x} \mathrm{d} \mathrm{x}$<br/>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">$l_{1} \neq l_{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{d}}{\mathrm{dx}}\left(l_{3}\right)=l_{2}$</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">$\int l_{3} \mathrm{dx}=l_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$l_{2}=l_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{d}}{\mathrm{dx}}\left(l_{3}\right)=l_{2}$</span> </div>

answer:
<div class="solution">$l_{1}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\sin \mathrm{x}}\right)$<br/>$l_{2}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\sin (\mathrm{x}+\mathrm{h})}-\mathrm{e}^{\sin \mathrm{x}}}{\mathrm{h}}$<br/>$l_{3}=\int \mathrm{e}^{\sin \mathrm{x}} \cdot \cos \mathrm{x} \mathrm{d} \mathrm{x}$<br/>$l_{2}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\sin (\mathrm{x}+\mathrm{h})}-\mathrm{e}^{\sin \mathrm{x}}}{\mathrm{h}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\sin \mathrm{x}}\right)=l_{1}$<br/>$l_{3}=\int \mathrm{e}^{\sin \mathrm{x}} \cdot \cos \mathrm{x} \mathrm{d} \mathrm{x}$<br/>Let $\sin \mathrm{x}=\mathrm{t} \Rightarrow \cos \mathrm{x} \mathrm{dx}=\mathrm{dt}$<br/>$l_{3}=\int \mathrm{e}^{\mathrm{t}} \cdot \mathrm{dt}=\mathrm{e}^{\mathrm{t}}+\mathrm{c}=\mathrm{e}^{\sin \mathrm{x}}+\mathrm{c}$<br/>$\frac{\mathrm{d}}{\mathrm{dx}}\left(l_{3}\right)=\frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}\left(\mathrm{e}^{\sin \mathrm{x}}+\mathrm{c}\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\sin \mathrm{x}}\right)=l_{1}=l_{2}$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2023 (14 May Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-l-is-the-length-of-the-open-pipe-r-is-the-internal-radius-of-the-pipe-and-v-is-the-velocity-of-sound-in-air-then-fundamental-frequency-of-open-pipe

slug:
if-l-is-the-length-of-the-open-pipe-r-is-the-internal-radius-of-the-pipe-and-v-is-the-velocity-of-sound-in-air-then-fundamental-frequency-of-open-pipe-22346

question:
<div class="question">If ' $l$ ' is the length of the open pipe, ' $r$ ' is the internal radius of the pipe and ' $V$ ' is the velocity of sound in air then fundamental frequency of open pipe is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{V}}{(l+0.3 \mathrm{r})}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{V}}{(l+1.2 \mathrm{r})}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{V}}{(l+0.6 \mathrm{r})}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{V}}{2(l+1.2 \mathrm{r})}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{V}}{2(l+1.2 \mathrm{r})}$</span> </div>

answer:
<div class="solution">For an open organ pipe, the length of the pipe with end correction is given as:<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{L}=l+2 \mathrm{e}=l+2 \times 0.6 \mathrm{r} \\<br/>&amp; \mathrm{L}=l+1.2 \mathrm{r}<br/>\end{aligned}<br/>$$<br/>$\therefore \quad$ The fundamental frequency of open pipe is:<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{f}=\frac{\mathrm{v}}{2 \mathrm{~L}} \\<br/>&amp; \mathrm{f}=\frac{\mathrm{v}}{2(l+1.2 \mathrm{r})}<br/>\end{aligned}<br/>$$</div>


title: if-l-m-n-0-then-the-system-of-equations-2-x-y-z-l-x-2-y-z-m-x-y-2-z-n-has
slug: if-l-m-n-0-then-the-system-of-equations-2-x-y-z-l-x-2-y-z-m-x-y-2-z-n-has-35415
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $l+m+\mathrm{n}=0$, then the system of equations  $-2 x+y+z=l$<br/>$x-2 y+z=m$<br/>$x+y-2 z=n$<br/>has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a trivial solution</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">no solution</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a unique solution</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">infinitely many solutions</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">infinitely many solutions</span> </div>

answer:
<div class="solution">$\begin{aligned} \text { Here, } A &amp;=\left[\begin{array}{ccc}-2 &amp; 1 &amp; 1 \\ 1 &amp; -2 &amp; 1 \\ 1 &amp; 1 &amp; -2\end{array}\right] \\ B &amp;=\left[\begin{array}{c}1 \\ m \\ n\end{array}\right] \text { and } \quad X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \end{aligned}$<br/>$\begin{aligned} \therefore|A| &amp;=-2\left|\begin{array}{cc}-2 &amp; 1 \\ 1 &amp; -2\end{array}\right|-1\left|\begin{array}{cc}1 &amp; 1 \\ 1 &amp; -2\end{array}\right|+1\left|\begin{array}{cc}1 &amp; 1 \\ -2 &amp; 1\end{array}\right| \\ &amp;=-2(4-1)-1(-2-1)+(1+2) \\ &amp;=-6+3+3=0 \end{aligned}$<br/>Now, adj $A=\left[\begin{array}{lll}3 &amp; 3 &amp; 3 \\ 3 &amp; 3 &amp; 3 \\ 3 &amp; 3 &amp; 3\end{array}\right]$<br/>$\therefore(\operatorname{adj} A) \cdot B=0$<br/>So, the given system of equations has an infinitely many solutions.</div>


title: if-l-n-2-x-e-x-1-1-dx-ln-2-3-then-what-is-the-value-of-x
slug: if-l-n-2-x-e-x-1-1-dx-ln-2-3-then-what-is-the-value-of-x-28078
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int_{\ln 2}^{\mathrm{x}}\left(\mathrm{e}^{\mathrm{x}}-1\right)^{-1} \mathrm{dx}=\ln \frac{3}{2}$, then what is the value of $\mathrm{x}$ ?</div>

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

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

answer:
<div class="solution">Let $\mathrm{I}=\int_{\ell \mathrm{n} 2}^{\mathrm{x}}\left(\mathrm{e}^{\mathrm{x}}-1\right)^{-1} \mathrm{dx}$<br/>$=\int_{\ell^{n} 2}^{x} \frac{1}{e^{x}-1} d x$<br/>Put $\mathrm{e}^{x}-1=\mathrm{t} \Rightarrow \mathrm{e}^{\mathrm{x}}=\mathrm{t}+1$<br/>$\mathrm{e}^{x} \mathrm{dx}=\mathrm{dt} \Rightarrow \mathrm{dx}=\frac{\mathrm{dt}}{\mathrm{e}^{\mathrm{x}}}$<br/>or, $\mathrm{dx}=\frac{\mathrm{dt}}{\mathrm{t}+1}$<br/>when $x=\ell \mathrm{n} 2, \mathrm{t}=\mathrm{e}^{\ln 2}-1=2-1=1$<br/>and $\mathrm{I}=\int_{1}^{\mathrm{t}} \frac{1}{\mathrm{t}(\mathrm{t}+1)} \mathrm{dt}$<br/>breaking into partial fractions.<br/>$\frac{1}{t(t+1)}=\frac{1}{t}-\frac{1}{t+1}$<br/>and $\mathrm{I}=\int_{1}^{\mathrm{t}}\left(\frac{1}{\mathrm{t}}-\frac{1}{\mathrm{t}+1}\right) \mathrm{dt}=\left[\log _{\mathrm{e}}^{\mathrm{t}}-\log _{\mathrm{e}}(\mathrm{t}+1)\right]_{1}^{\mathrm{t}}$<br/>or $\mathrm{I}=\left[\log _{\mathrm{e}} \frac{\mathrm{t}}{\mathrm{t}+1}\right]_{1}^{\mathrm{t}}=\log _{\mathrm{e}} \frac{\mathrm{t}}{\mathrm{t}+1}-\log _{\mathrm{e}} \frac{1}{2}$<br/>$-\log _{\mathrm{e}} \frac{2 \mathrm{t}}{\mathrm{t}+1}=\log _{\mathrm{e}} \frac{3}{2}$<br/>$\left[\right.$ Since, $\left.\int_{\ell \mathrm{n} 2}^{\mathrm{x}}\left(\mathrm{e}^{\mathrm{x}}-1\right)^{-1} \mathrm{dx}=\log _{\mathrm{e}} \frac{3}{2}\right]$<br/>So, $\frac{2 t}{t+1}=\frac{3}{2}$<br/>or, $4 \mathrm{t}=3 \mathrm{t}+3 \Rightarrow \mathrm{t}=3$<br/>So, $\mathrm{e}^{\mathrm{x}}-1=3, \mathrm{e}^{\mathrm{x}}=4 \Rightarrow \mathrm{x}=\ln 4$</div>


tags: ['Mathematics', 'Matrices', 'KCET', 'KCET 2018']
source_db: MarksBatch2_P1.db
title:
if-leftbeginarraycc1-1-1-1endarrayrightleftbeginarraylmathrmx-mathrmyendarrayrightleftbeginarrayc2-4endarrayright-then-the-values-of-mathrmx-and-mathr

slug:
if-leftbeginarraycc1-1-1-1endarrayrightleftbeginarraylmathrmx-mathrmyendarrayrightleftbeginarrayc2-4endarrayright-then-the-values-of-mathrmx-and-mathr-61639

question:
<div class="question">If \( \left[\begin{array}{cc}1 &amp; 1 \\ -1 &amp; 1\end{array}\right]\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}2 \\ 4\end{array}\right] \), then the values of \( \mathrm{x} \) and \( \mathrm{y} \) respectively are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( -1,3 \)</span> </div>

answer:
<div class="solution">Given that, \( \left[\begin{array}{cc}1 &amp; 1 \\ -1 &amp; 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right] \)<br/>\( \left[\begin{array}{c}x+y \\ -x+y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right] \)<br/>So, \( x+y=2 \rightarrow \) (1)<br/>and \( -x+y=4 \rightarrow(2) \)<br/>From Eqs. (1) and (2), we have \( x=-1, y=3 \)</div>


title: if-leftbeginarrayccc1-1-x-1-x-1-x-1-1endarrayright-has-no-inverse-then-the-real-value-of-x-is
slug: if-leftbeginarrayccc1-1-x-1-x-1-x-1-1endarrayright-has-no-inverse-then-the-real-value-of-x-is-33684
tags: ['Mathematics', 'Matrices', 'AP EAMCET', 'AP EAMCET 2020 (21 Sep Shift 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 \(\left[\begin{array}{ccc}1 &amp; -1 &amp; x \\ 1 &amp; x &amp; 1 \\ x &amp; -1 &amp; 1\end{array}\right]\) has no inverse, then the real value of \(x\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">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">It is given that matrix \(A=\left[\begin{array}{ccc}1 &amp; -1 &amp; x \\ 1 &amp; x &amp; 1 \\ x &amp; -1 &amp; 1\end{array}\right]\) have no inverse, so<br/>\(\begin{array}{rlrl} <br/>&amp; |A|=0 \Rightarrow\left|\begin{array}{ccc}<br/>1 &amp; -1 &amp; x \\<br/>1 &amp; x &amp; 1 \\<br/>x &amp; -1 &amp; 1<br/>\end{array}\right| &amp; =0 \\<br/>\Rightarrow \quad 1(x+1)+1(1-x)+x\left(-1-x^2\right) &amp; =0 \\<br/>\Rightarrow \quad x+1+1-x-x-x^3 &amp; =0 \\<br/>\Rightarrow x^3+x-2=0 \Rightarrow(x-1)\left(x^2+x+2\right) &amp; =0<br/>\end{array}\)<br/>Either \(x=1\) or \(x^2+x+2=0\)<br/>But discriminant of quadratic equation \(x^2+x+2=0\) is negative so no real roots.<br/>\(\therefore|A|=0 \Rightarrow x=1\)<br/>Hence, option (d) is correct</div>


tags: ['Mathematics', 'Basic of Mathematics', 'KCET', 'KCET 2015']
source_db: MarksBatch2_P1.db
title:
if-leftbeginarrayccc2-a-x1-y1-2-b-x2-y2-2-c-x3-y3endarrayrightfraca-b-c2-neq-0-then-the-area-of-the-triangle-whose-vertices-are-leftfracx1a-fracy1arig

slug:
if-leftbeginarrayccc2-a-x1-y1-2-b-x2-y2-2-c-x3-y3endarrayrightfraca-b-c2-neq-0-then-the-area-of-the-triangle-whose-vertices-are-leftfracx1a-fracy1arig-27792

question:
<div class="question">If \( \left|\begin{array}{ccc}2 a &amp; x_{1} &amp; y_{1} \\ 2 b &amp; x_{2} &amp; y_{2} \\ 2 c &amp; x_{3} &amp; y_{3}\end{array}\right|=\frac{a b c}{2} \neq 0 \), then the area of the triangle whose vertices are<br/>\[<br/>\left(\frac{x_{1}}{a}, \frac{y_{1}}{a}\right),\left(\frac{x_{2}}{b}, \frac{y_{2}}{b}\right),\left(\frac{x_{3}}{c}, \frac{y_{3}}{c}\right) \text { is }<br/>\]</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( \frac{1}{8} \)</span> </div>

answer:
<div class="solution">Given vertices of triangle \( \mathrm{ABC} \) are<br/>\( A\left(\frac{x_{1}}{a}, \frac{y_{1}}{a}\right), B\left(\frac{x_{2}}{b}, \frac{y_{2}}{b}\right), C\left(\frac{x_{3}}{c}, \frac{y_{3}}{c}\right) \)<br/>Now, \( \left|\begin{array}{ccc}2 a &amp; x_{1} &amp; y_{1} \\ 2 b &amp; x_{2} &amp; y_{2} \\ 2 c &amp; x_{3} &amp; y_{3}\end{array}\right|=\frac{a b c}{2} \rightarrow(1) \)<br/>Taking L.H.S. of the Eq. (1) we have<br/>\[<br/>\left|\begin{array}{lll}<br/>2 a &amp; x_{1} &amp; y_{1} \\<br/>2 h &amp; x_{2} &amp; y_{2} \\<br/>2 c &amp; x_{3} &amp; y_{3}<br/>\end{array}\right|<br/>\]<br/>Now \( R_{1} \rightarrow \frac{R_{1}}{a}, R_{2} \rightarrow \frac{R_{2}}{a}, R_{3} \rightarrow \frac{R_{3}}{a} \) then \( C_{1} \rightarrow \frac{C_{1}}{2} \) we have<br/>\( \left|\begin{array}{ccc}2 &amp; \frac{x_{1}}{a} &amp; \frac{y_{1}}{a} \\ 2 &amp; \frac{x_{2}}{b} &amp; \frac{y_{2}}{b} \\ 2 &amp; \frac{x_{3}}{c} &amp; \frac{y_{3}}{c}\end{array}\right| \begin{array}{ccc}1 &amp; \frac{x_{1}}{a} &amp; \frac{y_{1}}{a} \\ 1 &amp; \frac{x_{2}}{b} &amp; \frac{y_{2}}{b} \\ 1 &amp; \frac{x_{3}}{c} &amp; \frac{y_{3}}{c}\end{array} \mid \)<br/>We know that area od triangle is given by<br/>\( 1 \frac{x_{1}}{a} \frac{y_{1}}{a} \)</div>


title: if-leftbeginarrayccc32k-42-323k-42k-52-424k-52k-62-525kendarrayright0-then-the-value-of-k-is
slug: if-leftbeginarrayccc32k-42-323k-42k-52-424k-52k-62-525kendarrayright0-then-the-value-of-k-is-78655
tags: ['Mathematics', 'Determinants', 'VITEEE', 'VITEEE 2023']
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 \(\left|\begin{array}{ccc}3^2+k &amp; 4^2 &amp; 3^2+3+k \\ 4^2+k &amp; 5^2 &amp; 4^2+4+k \\ 5^2+k &amp; 6^2 &amp; 5^2+5+k\end{array}\right|=0\), then the value of \(k\) is</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">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">Breaking the given determinant into two determinants, we get<br/>\(\begin{aligned}<br/>&amp; \left|\begin{array}{lll}<br/>3^2+\mathrm{k} &amp; 4^2 &amp; 3^2+\mathrm{k} \\<br/>4^2+\mathrm{k} &amp; 5^2 &amp; 4^2+\mathrm{k} \\<br/>5^2+\mathrm{k} &amp; 6^2 &amp; 5^2+\mathrm{k}<br/>\end{array}\right|+\left|\begin{array}{lll}<br/>3^2+\mathrm{k} &amp; 4^2 &amp; 3 \\<br/>4^2+\mathrm{k} &amp; 5^2 &amp; 4 \\<br/>5^2+\mathrm{k} &amp; 6^2 &amp; 5<br/>\end{array}\right|=0 \\<br/>&amp; \Rightarrow 0+\left|\begin{array}{ccc}<br/>9+\mathrm{k} &amp; 16 &amp; 3 \\<br/>7 &amp; 9 &amp; 1 \\<br/>9 &amp; 11 &amp; 1<br/>\end{array}\right|=0<br/>\end{aligned}\)<br/>[Applying \(R_3-R_2\) and \(R_2-R_1\) in second det.]<br/>\(\begin{aligned}<br/>&amp; \Rightarrow\left|\begin{array}{ccc}<br/>9+\mathrm{k} &amp; 16 &amp; 3 \\<br/>7 &amp; 9 &amp; 1 \\<br/>2 &amp; 2 &amp; 0<br/>\end{array}\right|=0\left[\text { Applying } \mathrm{R}_3-\mathrm{R}_2\right] \\<br/>&amp; \left.\Rightarrow\left|\begin{array}{ccc}<br/>9+\mathrm{k} &amp; 7-\mathrm{k} &amp; 3 \\<br/>7 &amp; 2 &amp; 1 \\<br/>2 &amp; 0 &amp; 0<br/>\end{array}\right|=0 \text { [Applying } \mathrm{C}_2-\mathrm{C}_1\right] \\<br/>&amp; \Rightarrow 2(7-\mathrm{k}-6)=0 \Rightarrow \mathrm{k}=1<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Determinants', 'AP EAMCET', 'AP EAMCET 2017 (24 Apr Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch2_P1.db
title:
if-leftbeginarraycccx2x-x1-x-2-2-x23-x1-3-x-3-x3-x-2-2-x-3-2-x1-2-x1endarrayrightx-ab-where-a-and-b-are-determinants-of-order-3-not-involving-x-then-a

slug:
if-leftbeginarraycccx2x-x1-x-2-2-x23-x1-3-x-3-x3-x-2-2-x-3-2-x1-2-x1endarrayrightx-ab-where-a-and-b-are-determinants-of-order-3-not-involving-x-then-a-33160

question:
<div class="question">If $\left|\begin{array}{ccc}x^2+x $x+1 $x-2$ 2 x^2+3 x-1 $3 x $3 x-3$ x^2+2 x+3 $2 x-1 $2 x-1\end{array}\right|=x A+B$, where $A$ and $B$ are determinants of order 3 not involving $x$, then $|A|=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">27</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">C</span> <span class="option-data">19</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-8$</span> </li> </ul>


title: if-leftbeginarrayll3-x-x-1endarrayrightleftbeginarrayll3-2-4-1endarrayright-then-x-is-equal-to
tags: ['Mathematics', 'Probability', 'KCET', 'KCET 2017']
source_db: MarksBatch2_P1.db
slug:
if-leftbeginarrayll3-x-x-1endarrayrightleftbeginarrayll3-2-4-1endarrayright-then-x-is-equal-to-37701

question:
<div class="question">If \( \left|\begin{array}{ll}3 &amp; x \\ x &amp; 1\end{array}\right|=\left|\begin{array}{ll}3 &amp; 2 \\ 4 &amp; 1\end{array}\right| \) then \( x \) 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">\( 04 \)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\( 08 \)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\( \pm 2 \sqrt{2} \)</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> </ul>

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

answer:
<div class="solution">Given that, \( \left|\begin{array}{cc}3 &amp; x \\ x &amp; 1\end{array}\right|=\left|\begin{array}{ll}3 &amp; 2 \\ 4 &amp; 1\end{array}\right| \)<br/>\( 3-x^{2}=3-8 \)<br/>\( \Rightarrow x^{2}=8 \Rightarrow x=\pm 2 \sqrt{2} \)</div>


title: if-lim-x-0-x-a-2-where-a-0-then-what-is-lim-x-0-a-x-equal-to
slug: if-lim-x-0-x-a-2-where-a-0-then-what-is-lim-x-0-a-x-equal-to-94201
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow 0} \phi(x)=a^{2}$, where $a \neq 0$, then what is $\lim _{x \rightarrow 0} \phi\left(\frac{x}{a}\right)$ equal to?</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \lim _{x \rightarrow 0} \phi(x)=a^{2} \\ &amp; \Rightarrow \lim _{x \rightarrow 0} \phi\left(\frac{x}{a}\right)=a^{2} \end{aligned}$<br/>$a \neq 0$<br/>[because function value is constant]</div>


title: if-lim-x-2-x-s-i-n-x-and-lim-x-x-c-o-s-x-m-then-which-one-of-the-following-is-correct
slug: if-lim-x-2-x-s-i-n-x-and-lim-x-x-c-o-s-x-m-then-which-one-of-the-following-is-correct-79317
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x}{x}=$ and $\lim _{x \rightarrow \infty} \frac{\cos x}{x}=m$, then which one of the<br/>following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad l=1, \mathrm{~m}=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$l=\frac{2}{\pi}, \mathrm{m}=\infty$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$l=\frac{2}{\pi}, \mathrm{m}=0$</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">$l=1, \mathrm{~m}=\infty$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$l=\frac{2}{\pi}, \mathrm{m}=0$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x}{x}=l$ and $\lim _{x \rightarrow \infty} \frac{\cos x}{x}=m$<br/>$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x}{x}=\frac{1}{\frac{\pi}{2}}=\frac{2}{\pi} ; \lim _{x \rightarrow \infty} \frac{\cos x}{x}=0$<br/>$\therefore l=\frac{2}{\pi}, \mathrm{m}=0$</div>


title: if-lim-x-a-g-x-f-x-exists-then-which-one-of-the-following-correct
slug: if-lim-x-a-g-x-f-x-exists-then-which-one-of-the-following-correct-71513
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\lim _{\mathrm{x} \rightarrow \mathrm{a}}\left[\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}\right]$ exists, then which one of the following correct $?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ must exist</span> </div>

answer:
<div class="solution">For $\lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]$ to exist, then both $\lim _{x \rightarrow a} f(x)$ and<br/>$\lim _{x \rightarrow a} g(x)$ must exist.</div>


title: if-lines-represented-by-equation-p-x-2-q-y-2-0-are-distinct-then
slug: if-lines-represented-by-equation-p-x-2-q-y-2-0-are-distinct-then-74640
tags: ['Mathematics', 'Pair of Lines', 'MHT CET', 'MHT CET 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If lines represented by equation <math><mi>p</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>q</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn></math> are distinct then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mi>p</mi><mi>q</mi><mo>&gt;</mo><mn>0</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">B</span> <span class="option-data"><math><mi>p</mi><mi>q</mi><mo>&lt;</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mi>p</mi><mi>q</mi><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mn>0</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mi>p</mi><mi>q</mi><mo>&gt;</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><math><mi>p</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>q</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn></math><br/>Comparing with standard homogeneous equation we get, <br/><math><mi>a</mi><mo>=</mo><mi>p</mi><mo>,</mo><mi> </mi><mi>b</mi><mo>=</mo><mo>-</mo><mi>q</mi><mo>,</mo><mi> </mi><mi>h</mi><mo>=</mo><mn>0</mn></math><br/>Lines are real and distinct if <math><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>a</mi><mi>b</mi><mo>&gt;</mo><mn>0</mn></math><br/><math><mo>â</mo><mi> </mi><mi> </mi><mn>0</mn><mo>+</mo><mi>p</mi><mi>q</mi><mo>&gt;</mo><mn>0</mn></math><br/><math><mi>p</mi><mi>q</mi><mo>&gt;</mo><mn>0</mn></math></div>


title: if-lo-g-3-lo-g-3-lo-g-3-x-lo-g-3-3-then-what-is-the-value-of-x
slug: if-lo-g-3-lo-g-3-lo-g-3-x-lo-g-3-3-then-what-is-the-value-of-x-17258
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\log _{3}\left[\log _{3}\left[\log _{3} x\right]\right]=\log _{3} 3$, then what is the value of $x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3^{9}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">27</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3^{27}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$3^{27}$</span> </div>

answer:
<div class="solution">Consider $\log _{3}\left[\log _{3}\left[\log _{3} x\right]\right]=\log _{3} 3$<br/>$\Rightarrow \log _{3}\left[\log _{3} x\right]$<br/>$\Rightarrow \log _{3} x=$</div>


title: if-lo-g-3-x-2-lo-g-3-x-2-then-which-one-of-the-following-is-correct
slug: if-lo-g-3-x-2-lo-g-3-x-2-then-which-one-of-the-following-is-correct-64777
tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">$\mathrm{If}\left(\log _{3} \mathrm{x}\right)^{2}+\log _{3} \mathrm{x}   &lt; 2$, then which one of the following is<br/>correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0   &lt; \mathrm{x} &lt; \frac{1}{9}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{9}   &lt; \mathrm{x} &lt; 3$</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><li class=""> <span class="option-label">C</span> <span class="option-data">$3   &lt; x &lt; \infty$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{9} \leq x \leq 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{9}   &lt; \mathrm{x} &lt; 3$</span> </div>

answer:
<div class="solution">Given equation is $\left(\log _{3} x\right)^{2}+\log _{3} x   &lt; 2$<br/>$\Rightarrow\left(\log _{3} x\right)^{2}+\left(\log _{3} x\right)-2 &lt; 0$<br/>$\Rightarrow\left(\log _{3} x+2\right)\left(\log _{3} x-1\right) &lt; 0$<br/>$-2 &lt; \log _{3} x &lt; $<br/>$\Rightarrow \log _{3} 3^{-2} &lt; \log _{3} x &lt; \log _{3} 3$<br/>$\Rightarrow \frac{1}{9} &lt; \mathrm{x} &lt; 3$</div>


title: if-lo-g-3-x-lo-g-x-2-x-lo-g-2-x-y-lo-g-x-x-2-then-what-is-y-equal-to
slug: if-lo-g-3-x-lo-g-x-2-x-lo-g-2-x-y-lo-g-x-x-2-then-what-is-y-equal-to-22960
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 2)']
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 $\left(\log _{3} x\right)\left(\log _{x} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$, then what is $y$ equal<br/>to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.5$</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">27</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">18</span> </li> </ul>

answer:
<div class="solution">$\left(\log _{3} x\right)\left(\log _{x} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$<br/>$\frac{\log x}{\log 3} \times \frac{\log 2 x}{\log x} \times \frac{\log y}{\log 2 x}=\frac{\log x^{2}}{\log x}$<br/>$\frac{\log y}{\log 3}=\frac{2 \log x}{\log x}$<br/>$\log y=2 \log 3$<br/>$\log y=\log 9$<br/>$y=9$</div>


title: if-lo-g-8-m-lo-g-8-6-1-3-2-then-m-is-equal-to
slug: if-lo-g-8-m-lo-g-8-6-1-3-2-then-m-is-equal-to-42032
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\log _{8} \mathrm{~m}+\log _{8} \frac{1}{6}=\frac{2}{3}$, then $\mathrm{m}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">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">18</span> </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">4</span> </li> </ul>

answer:
<div class="solution">$\log _{8} \mathrm{~m}+\log _{8} \frac{1}{6}=\frac{2}{3}$<br/>$\Rightarrow \quad \log _{8}\left(\mathrm{~m}, \frac{1}{6}\right)=\frac{2}{3}$<br/>$\Rightarrow(8)^{\frac{2}{3}}=\mathrm{m} \cdot \frac{1}{6}$<br/>$\Rightarrow \mathrm{m}=24$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-lo-g-10-2-lo-g-10-2-x-1-and-lo-g-10-2-x-3-are-three-consecutive-terms-of-an-ap-then-the-value-of-x

slug:
if-lo-g-10-2-lo-g-10-2-x-1-and-lo-g-10-2-x-3-are-three-consecutive-terms-of-an-ap-then-the-value-of-x-30797

question:
<div class="question">If $\log _{10} 2, \log _{10}\left(2^{x}-1\right)$ and $\log _{10}\left(2^{x}+3\right)$ are three<br/>consecutive terms of an A.P, then the value of $x$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\log _{5} 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\log _{2}$</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">$\log _{10} 5$</span> </li> </ul>

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

answer:
<div class="solution">$\log _{10} 2, \log _{10}\left(2^{x}-1\right)$ and $\log _{10}\left(2^{x}+3\right)$ are in A.P.<br/>$\log _{10}\left(2^{x}-1\right)-\log _{10} 2=\log \left(2^{x}+3\right)-\log _{10}\left(2^{x}-1\right)$<br/>$\log _{10}\left(\frac{2^{\mathrm{x}}-1}{2}\right)=\log _{10}\left(\frac{2^{\mathrm{x}}+3}{2^{\mathrm{x}}-1}\right)$<br/>$\Rightarrow \frac{2^{x}-1}{2}=\frac{2^{x}+3}{2^{x}-1}$<br/>Let $2^{\mathrm{x}}=\mathrm{y}$, then<br/>$y^{2}-4 y-5=0$<br/>$y^{2}-5 y+y-5=0$<br/>$y(y-5)+1(y-5)=0$<br/>$y=-1, y=5$<br/>Therefore, $2^{\mathrm{x}}=5$ $\mathrm{x}=\log _{2} 5$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-lo-g-10-2-lo-g-10-2-x-1-lo-g-10-2-x-3-are-three-consecutive-terms-of-an-ap-then-which-one-of-the-following-is-correct

slug:
if-lo-g-10-2-lo-g-10-2-x-1-lo-g-10-2-x-3-are-three-consecutive-terms-of-an-ap-then-which-one-of-the-following-is-correct-77518

question:
<div class="question">If $\log _{10} 2, \log _{10}\left(2^{x}-1\right), \log _{10}\left(2^{x}+3\right)$ are three<br/>consecutive terms of an AP, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x=\log _{2} 5$</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">D</span> <span class="option-data">$x=\log _{5} 2$</span> </li> </ul>

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

answer:
<div class="solution">Let $\log _{10} 2, \log _{10}\left(2^{x}-1\right)$ and $\log _{10}\left(2^{x}+3\right)$ are<br/>in A.P<br/>$\therefore 2 \log _{10}\left(2^{x}-1\right)=\log _{10} 2+\log _{10}\left(2^{x}+3\right)$<br/>$\Rightarrow \log _{10}\left(2^{x}-1\right)^{2}=\log _{10} 2\left(2^{x}+3\right)$<br/>$\Rightarrow 2^{2 x}+1-2^{x+1}=2.2^{x}+6$<br/>$\Rightarrow a^{2}+1-2 a=2 a+6$ where $a=2^{x} .$<br/>$\Rightarrow a^{2}-4 a-5=0$<br/>$\Rightarrow a=5$ or $a=-1 \quad 2^{x}=5 \Rightarrow x \log _{2}=\log 5$<br/>$\Rightarrow x=\frac{\log 5}{\log 2} \Rightarrow x=\log _{2} 5$</div>


title: if-lo-g-10-x-1-lo-g-10-5-3-then-what-is-the-value-of-x
slug: if-lo-g-10-x-1-lo-g-10-5-3-then-what-is-the-value-of-x-27862
question: <div class="question">If $\log _{10}(x+1)+\log _{10} 5=3$, then what is the value of $x$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">199</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">199</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">200</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">290</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">300</span> </li> </ul>

answer:
<div class="solution">Let $\log _{10}(x+1)+\log _{10} 5=3$<br/>$\Rightarrow \log _{10} 5(x+1)=3 \quad(\because \log m+\log n=\log m n)$<br/>ì´ $5(x+1)=10$<br/>$\Rightarrow(x+1)=\frac{1000}{5}=200$<br/>$\Rightarrow x=200-1=199$</div>


title: if-lo-g-a-ab-x-then-what-is-lo-g-h-ab-equal-to
slug: if-lo-g-a-ab-x-then-what-is-lo-g-h-ab-equal-to-25571
question: <div class="question">If $\log _{a}(a b)=x$, then what is $\log _{h}(a b)$ equal to?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{x}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{x}{x+1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x}{1-x}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{x}{x-1}$</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">$\frac{x}{x-1}$</span> </div>

answer:
<div class="solution">$\log _{a} a b=x$<br/>$\log _{a} a+\log _{a} b=x$<br/>$\frac{1}{\log _{b} a}=x-1$<br/>$\log _{b} a=\frac{1}{x-1}$...(i)<br/>$\log _{b} a b=\log _{b} a+\log _{b} b$<br/>$\quad=\frac{1}{x-1}+1 \quad($ From $(1))$<br/>$=\frac{1+x-1}{x-1}$<br/>$\log _{b} a b=\frac{x}{x-1}$</div>


title: if-lo-g-k-x-lo-g-5-k-3-then-what-is-x-equal-to
slug: if-lo-g-k-x-lo-g-5-k-3-then-what-is-x-equal-to-54510
question: <div class="question">If $\log _{k} x \log _{5} k=3$, then what is $x$ equal to?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">125</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$k^{5}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 k^{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$24^{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">125</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\Rightarrow \log y=\log 3^{2} \Rightarrow y=3^{2} \Rightarrow y=9$<br/>Given, $\log _{5} k \log _{k} x=3$<br/>$\frac{\log k}{\log 5} \cdot \frac{\log x}{\log k}=3 \Rightarrow \frac{\log x}{\log 5}=3$<br/>$\Rightarrow \log x=3 \log 5 \Rightarrow \log x=\log 5^{3}$<br/>$\Rightarrow x=5^{3} \Rightarrow x=125$</div>


title: if-lo-g-x-x-lo-g-3-2-x-lo-g-2-x-y-lo-g-x-x-2-then-what-is-the-value-of-y
slug: if-lo-g-x-x-lo-g-3-2-x-lo-g-2-x-y-lo-g-x-x-2-then-what-is-the-value-of-y-97256
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
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 $\left(\log _{x} x\right)\left(\log _{3} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$, then what is the value<br/>of $y$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$9 / 2$</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">27</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">18</span> </li> </ul>

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


title: if-m-1-0-0-1-and-n-0-1-1-0-then-what-is-the-value-of-the-determinant-of-m-cos-n-sin
slug: if-m-1-0-0-1-and-n-0-1-1-0-then-what-is-the-value-of-the-determinant-of-m-cos-n-sin-79683
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2016 (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 $\mathrm{m}=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$ and $\mathrm{n}=\left[\begin{array}{ll}0 &amp; 1 \\ -1 &amp; 0\end{array}\right]$, then what is the value of the determinant of $m \cos \theta-n \sin \theta ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="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">$\mathrm{m} \cos \theta-\mathrm{n} \sin \theta=\left[\begin{array}{cc}\cos \theta &amp; 0 \\ 0 &amp; \cos \theta\end{array}\right]-\left[\begin{array}{cc}0 &amp; \sin \theta \\ -\sin \theta &amp; 0\end{array}\right]$<br/>$=\left[\begin{array}{cc}\cos \theta &amp; -\sin \theta \\ \sin \theta &amp; \cos \theta\end{array}\right]$<br/>det. $(\mathrm{m} \cos \theta-\mathrm{n} \sin \theta)=\cos ^{2} \theta-\left(-\sin ^{2} \theta\right)$<br/>$=\cos ^{2} \theta+\sin ^{2} \theta=1$</div>


title: if-m-and-n-are-integers-then-what-is-the-value-of-0-sin-m-x-sin-n-x-d-x-if-m-n
slug: if-m-and-n-are-integers-then-what-is-the-value-of-0-sin-m-x-sin-n-x-d-x-if-m-n-74856
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">If $\mathrm{m}$ and $\mathrm{n}$ are integers, then what is the value of<br/>$\int_{0}^{\pi} \sin m x \sin n x d x$, if $m \neq n ?$</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">$\frac{1}{m+n}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\mathrm{~m}-\mathrm{n}} \quad$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{m} \quad\mathrm{}$</span> </li> </ul>

answer:
<div class="solution">The given integral<br/>$\int_{0}^{\pi} \sin m x \cdot \sin n x d x$<br/>$=\int_{0}^{\pi} \sin m(\pi-x) \cdot \sin n(\pi-x) d x$<br/>$=\int_{0}^{\pi} \sin m x \cdot \sin n x d x$<br/>So, $\int_{0}^{\pi} \sin m x \cdot \sin n x d x$<br/>$=2 \int_{0}^{\pi / 2} \sin m x \cdot \sin n x d x$<br/>$=2 \int_{0}^{\pi / 2} \frac{1}{2}[\cos (m x-n x)-\cos (m x+n x)] d x$<br/>$=\int_{0}^{\pi / 2}[\cos (m-n) x-\cos (m+n) x] d x$<br/>$=\left[\frac{\sin (m-n) x}{m-n}-\frac{\sin (m+n) x}{m+n}\right]_{0}^{\pi / 2}=0$</div>


title: if-m-and-n-are-the-roots-of-the-equation-x-p-x-q-k-0-then-the-roots-of-the-equation-x-m-x-n-k-0-are
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-m-and-n-are-the-roots-of-the-equation-x-p-x-q-k-0-then-the-roots-of-the-equation-x-m-x-n-k-0-are-22480

question:
<div class="question">If $m$ and $n$ are the roots of the equation $(x+p)(x+q)-k=0$, then the roots of the equation $(x-m)(x-n)+k=0$ are</div>

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

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

answer:
<div class="solution">Here $\mathrm{m}$ and $\mathrm{n}$ are the roots of equation. $(\mathrm{x}+\mathrm{p})(\mathrm{x}+\mathrm{q})-\mathrm{k}=0$<br/>$\mathrm{x}^{2}+\mathrm{x}(\mathrm{p}+\mathrm{q})+\mathrm{p} \mathrm{q}-\mathrm{k}=0$...(i)<br/>If $\mathrm{m}$ and $\mathrm{n}$ are the roots of equation, then $(\mathrm{x}-\mathrm{m})(\mathrm{x}-\mathrm{n})=0$<br/>$\mathrm{x}^{2}-(\mathrm{m}+\mathrm{n}) \mathrm{x}+\mathrm{mn}=0$...(ii)<br/>Now equation (i) should be equal to equation (ii), $(m+n)=-(p+q) a n d m n=p q-k$<br/>Now, we have to find roots of $(x-m)(x-n)+k=0$ $x^{2}-(m+n) x+m n+k=0$<br/>$x^{2}+(p+q) x+(p q-k)+k=0$<br/>$\mathrm{x}^{2}+(\mathrm{p}+\mathrm{q}) \mathrm{x}+\mathrm{pq}=0$<br/>$\mathrm{x}^{2}+\mathrm{px}+\mathrm{qx}+\mathrm{pq}=0$<br/>$x(x+p)+q(x+p)=0$<br/>$x+q=0$ or $x+p=0$<br/>$\mathrm{x}=-\mathrm{q}$ and $\mathrm{x}=-\mathrm{p}$<br/>Option $(\mathrm{c})$ is correct</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2016 (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-m-is-the-geometric-mean-of-z-y-l-o-g-yz-x-z-l-o-g-z-x-and-y-x-l-o-g-x-y-then-what-is-the-value-of-m

slug:
if-m-is-the-geometric-mean-of-z-y-l-o-g-yz-x-z-l-o-g-z-x-and-y-x-l-o-g-x-y-then-what-is-the-value-of-m-83602

question:
<div class="question">If $\mathrm{m}$ is the geometric mean of <br/>$\left(\frac{y}{z}\right)^{\log (y z)},\left(\frac{z}{x}\right)^{\log (z x)}$ and $\left(\frac{x}{y}\right)^{\log (x y)}$<br/>then what is the value of $\mathrm{m}$ ?</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">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9</span> </li> </ul>

answer:
<div class="solution">Three terms are<br/>$G_{1}=\left(\frac{y}{z}\right)^{\log y z} G_{2}=\left(\frac{z}{x}\right)^{\log z x} G_{3}=\left(\frac{x}{y}\right)^{\log x y}$<br/>Geometric mean of three terms is<br/>$m=\sqrt[3]{G_{1} G_{2} G_{3}}$ ...(i)<br/>$\begin{aligned} G_{1} G_{2} G_{3}=&amp;\left(\frac{y}{z}\right)^{\log y z} \cdot\left(\frac{z}{x}\right)^{\log z x} \cdot\left(\frac{x}{y}\right)^{\log x y} \\ &amp;=\frac{y^{\log y} \cdot y^{\log z}}{z^{\log y} \cdot z^{\log z}} \times \frac{z^{\log z} \cdot z^{\log x}}{x^{\log z} \cdot x^{\log x}} \times \frac{x^{\log x} \cdot x^{\log y}}{y^{\log x} \cdot y^{\log y}} \\ &amp;=\left(\frac{y}{x}\right)^{\log z} \cdot\left(\frac{z}{y}\right)^{\log x} \cdot\left(\frac{x}{z}\right)^{\log y} \end{aligned}$<br/>Taking log both sides $\begin{aligned} \log G_{1} G_{2} G_{3}=&amp; \log \left[\left(\frac{y}{x}\right)^{\log z}\right]+\log \left[\left(\frac{z}{y}\right)^{\log x}\right] \\ &amp;+\log \left[\left(\frac{x}{z}\right)^{\log y}\right] \\=&amp; \log z \log y-\log z \log x+\log x \log z \\ &amp;-\log x \log y+\log y \log x-\log y \log z \end{aligned}$<br/>$\log G_{1} G_{2} G_{3}=0$<br/>$G_{1} G_{2} G_{3}=\mathrm{e}^{0}=1$<br/>Hence $m=\sqrt[3]{G_{1} G_{2} G_{3}}=(1)^{\frac{1}{3}}$<br/>$m=1$</div>


title: if-m-n-r-are-three-vectors-is-the-angle-between-the-vectors-m-and-n-what-is-mnr-cos-equal-to
slug: if-m-n-r-are-three-vectors-is-the-angle-between-the-vectors-m-and-n-what-is-mnr-cos-equal-to-92652
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{m}, \mathrm{n}, \mathrm{r}$ are three vectors, $\theta$ is the angle between the<br/>vectors $\overrightarrow{\mathrm{m}}$ and $\overrightarrow{\mathrm{n}}$, what is $\mathrm{mnr} \cos \theta$ equal to ?</div>

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

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

answer:
<div class="solution">Given that $\overrightarrow{\mathrm{m}}, \overrightarrow{\mathrm{n}}$ and $\overrightarrow{\mathrm{r}}$ are three vectors and $\theta$ is the<br/>angle between $\overrightarrow{\mathrm{m}}$ and $\overrightarrow{\mathrm{n}}$,<br/>We get, $\cos \theta=\frac{\overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{n}}}{|\overrightarrow{\mathrm{m}} \| \overrightarrow{\mathrm{n}}|}$<br/>$\Rightarrow m n \cos \theta=\overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{n}}$<br/>[where, $|\overrightarrow{\mathrm{m}}|=\mathrm{m}$ and $|\overrightarrow{\mathrm{n}}|=\mathrm{n}]$<br/>$\Rightarrow \mathrm{mn} \cos \theta=(\overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{n}}) \frac{\overrightarrow{\mathrm{r}}}{|\overrightarrow{\mathrm{r}}|}$<br/>$\Rightarrow \mathrm{mn} \cos \theta=(\overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{n}}) \frac{\overrightarrow{\mathrm{r}}}{\mathrm{r}}$<br/>[where $|\overrightarrow{\mathrm{r}}|=\mathrm{r}]$<br/>$\Rightarrow \operatorname{mn} r \cos \theta=(\overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{n}}) \overrightarrow{\mathrm{r}}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-malphaleftbeginarrayccccos-alpha-sin-alpha-0-sin-alpha-cos-alpha-0-0-0-1endarrayright-mbetaleftbeginarrayccccos-beta-0-sin-beta-0-1-0-sin-beta-0-co

slug:
if-malphaleftbeginarrayccccos-alpha-sin-alpha-0-sin-alpha-cos-alpha-0-0-0-1endarrayright-mbetaleftbeginarrayccccos-beta-0-sin-beta-0-1-0-sin-beta-0-co-33688

question:
<div class="question">If \(M(\alpha)=\left[\begin{array}{ccc}\cos \alpha &amp; -\sin \alpha &amp; 0 \\ \sin \alpha &amp; \cos \alpha &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]\); \(M(\beta)=\left[\begin{array}{ccc}\cos \beta &amp; 0 &amp; \sin \beta \\ 0 &amp; 1 &amp; 0 \\ -\sin \beta &amp; 0 &amp; \cos \beta\end{array}\right]\) then \([M(\alpha) M(\beta)]^{-1}\) is equal to -</div>

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

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \left.[\mathrm{M}(\alpha) \mathrm{M}(\beta)]^{-1}=\mathrm{M}(\beta)^{-1} \mathrm{M}(\alpha)^{-1}\right] \\<br/>&amp; \operatorname{Now~M(\alpha )^{-1}}=\left[\begin{array}{ccc}<br/>\cos \alpha &amp; \sin \alpha &amp; 0 \\<br/>-\sin \alpha &amp; \cos \alpha &amp; 0 \\<br/>0 &amp; 0 &amp; 1<br/>\end{array}\right] \\<br/>&amp; =\left[\begin{array}{ccc}<br/>\cos (-\alpha) &amp; -\sin (-\alpha) &amp; 0 \\<br/>\sin (-\alpha) &amp; \cos (-\alpha) &amp; 0 \\<br/>0 &amp; 0 &amp; 1<br/>\end{array}\right]=\mathrm{M}(-\alpha) \\<br/>&amp; M(\beta)^{-1}=\left[\begin{array}{ccc}<br/>\cos \beta &amp; 0 &amp; -\sin \beta \\<br/>0 &amp; 1 &amp; 0 \\<br/>\sin \beta &amp; 0 &amp; \cos \beta<br/>\end{array}\right] \\<br/>&amp; =\left[\begin{array}{ccc}<br/>\cos (-\beta) &amp; 0 &amp; \sin (-\beta) \\<br/>0 &amp; 1 &amp; 0 \\<br/>-\sin (-\beta) &amp; 0 &amp; \cos (-\beta)<br/>\end{array}\right]=\mathrm{M}(-\beta) \\<br/>&amp; {[\mathrm{M}(\alpha) \mathrm{M}(\beta)]^{-1}=\mathrm{M}(-\beta) \mathrm{M}(-\alpha)} \\<br/>&amp;<br/>\end{aligned}\)</div>


title: if-mammalian-ovum-fails-to-get-fertilized-which-one-of-the-following-is-unlikely
slug: if-mammalian-ovum-fails-to-get-fertilized-which-one-of-the-following-is-unlikely-41329
tags: ['Biology', 'Human Reproduction', 'NEET', 'NEET 2005']
source_db: MarksBatch2_P1.db
question:
<div class="question">If mammalian ovum fails to get fertilized, which one of the following is unlikely?</div>

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

answer:
<div class="solution">A) Estrogen secretion further decreases- Estrogen is secreted by the corpus luteum. Since it disintegrates in the absence of fertilization, estrogen levels fall.
<br/>B) Progesterone secretion starts declining- If pregnancy occurs, human chorionic gonadotropin is released maintaining the corpus luteum allowing it to maintain levels of progesterone. In the other way round, (i.e., in case of fertilization does not occur), progesterone levels fall due to the disintegration of the corpus luteum.
<br/>C) Corpus luteum will disintegrate- If the egg is not fertilized, the corpus luteum stops secreting progesterone and decays (after approximately 14 days in humans). It then degenerates into a corpus Albicans, which is a mass of fibrous scar tissue.
<br/>D) Primary follicle starts developing- Primary follicle is a precursor to the corpus luteum and thus this is most unlikely to happen. At the start of the menstrual cycle, some 12-20 primary follicles begin to develop under the influence of elevated FSH to form secondary follicles. The primary follicles have formed from primordial follicles, which developed in the ovary at around 10-30 weeks after conception.
<br/>So, the correct answer is option B.</div>


title: if-mass-m-distance-l-and-time-t-are-fundamental-quantities-then-find-dimension-of-torque
slug: if-mass-m-distance-l-and-time-t-are-fundamental-quantities-then-find-dimension-of-torque-26822
tags: ['Physics', 'Units and Dimensions', 'JIPMER', 'JIPMER 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If mass $(\mathrm{M})$, distance $(\mathrm{L})$ and time $(\mathrm{T})$ are fundamental quantities, then find dimension of torque.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$</span> </div>

answer:
<div class="solution">Dimension of torque<br/>$\begin{aligned}<br/>\tau &amp; =\mathrm{F} \times \mathrm{r}=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}] \\<br/>&amp; =\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Matrices', 'KCET', 'KCET 2016']
source_db: MarksBatch2_P1.db
title:
if-mathrmafrac1pileftbeginarrayllsin-1mathrmx-pi-tan-1leftfracmathrmxpiright-sin-1leftfracmathrmxpiright-cot-1pi-mathrmxendarrayright-mathrmbleftbegin

slug:
if-mathrmafrac1pileftbeginarrayllsin-1mathrmx-pi-tan-1leftfracmathrmxpiright-sin-1leftfracmathrmxpiright-cot-1pi-mathrmxendarrayright-mathrmbleftbegin-20179

question:
<div class="question">If \(\mathrm{A}=\frac{1}{\pi}\left[\begin{array}{ll}\sin ^{-1}(\mathrm{x} \pi) &amp; \tan ^{-1}\left(\frac{\mathrm{x}}{\pi}\right) \\ \sin ^{-1}\left(\frac{\mathrm{x}}{\pi}\right) &amp; \cot ^{-1}(\pi \mathrm{x})\end{array}\right] \mathrm{B}=\left[\begin{array}{cc}-\cos ^{-1}(\mathrm{x} \pi) &amp; \tan ^{-1}\left(\frac{\mathrm{x}}{\pi}\right) \\ \sin ^{-1}\left(\frac{\mathrm{x}}{\pi}\right) &amp; -\tan ^{-1}(\pi x)\end{array}\right]\) then \(\mathrm{A}-\mathrm{B}\) equal to</div>

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

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

answer:
<div class="solution">We have, \(A=\left[\begin{array}{ll}\frac{1}{\pi} \sin ^{-1}(x \pi) &amp; \frac{1}{\pi} \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \frac{1}{\pi} \sin ^{-1}\left(\frac{x}{\pi}\right) &amp; \frac{1}{\pi} \cot ^{-1}(\pi x)\end{array}\right] B=\left[\begin{array}{cc}-\frac{1}{\pi} \cos ^{-1}(x \pi) &amp; \frac{1}{\pi} \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \frac{1}{\pi} \sin ^{-1}\left(\frac{x}{\pi}\right) &amp; -\frac{1}{\pi} \tan ^{-1}(\pi x)\end{array}\right]\)<br/>\(\therefore A-B=\left[\begin{array}{ll}<br/>\frac{1}{\pi}\left(\sin ^{-1} x \pi+\cos ^{-1} x \pi\right) &amp; \frac{1}{\pi}\left(\tan ^{-1} \frac{x}{\pi}-\tan ^{-1} \frac{x}{\pi}\right) \\<br/>\frac{1}{\pi}\left(\sin ^{-1} \frac{x}{\pi}-\sin ^{-1} \frac{x}{\pi}\right) &amp; \frac{1}{\pi}\left(\cot ^{-1} \pi x\right)+\tan ^{-1} \pi x<br/>\end{array}\right]\)<br/>\(\left[\begin{array}{cc}<br/>\frac{1}{\pi} \cdot \frac{\pi}{2} &amp; 0 \\<br/>0 &amp; \frac{1}{\pi} \cdot \frac{\pi}{2}<br/>\end{array}\right]\left[\begin{array}{l}<br/>\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2} \\<br/>\text { and } \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}<br/>\end{array}\right]\)<br/>\(=\frac{1}{2}\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right]=\frac{1}{2} \mathrm{I}\)</div>


title: if-mathrmaleftbeginarraycccos-alpha-sin-alpha-sin-alpha-cos-alphaendarrayright-then-a-aprime
slug: if-mathrmaleftbeginarraycccos-alpha-sin-alpha-sin-alpha-cos-alphaendarrayright-then-a-aprime-14725
tags: ['Mathematics', 'Matrices', 'KCET', 'KCET 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \( \mathrm{A}=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right] \), then \( A A^{\prime}= \)</div>

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

answer:
<div class="solution">Given that \( A=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right] \)<br/>So, \( A=\left[\begin{array}{cc}\cos \alpha &amp; -\sin \alpha \\ \sin \alpha &amp; \cos \alpha\end{array}\right] \)<br/>Then,<br/>\[<br/>\begin{array}{l}<br/>A \cdot A=\left[\begin{array}{cc}<br/>\cos \alpha &amp; \sin \alpha \\<br/>-\sin \alpha &amp; \cos \alpha<br/>\end{array}\right]\left[\begin{array}{cc}<br/>\cos \alpha &amp; -\sin \alpha \\<br/>\sin \alpha &amp; \cos \alpha<br/>\end{array}\right] \\<br/>=\left[\begin{array}{cc}<br/>\cos ^{2} \alpha+\sin ^{2} \alpha &amp; -\cos \alpha \sin \alpha+\cos \alpha \sin \alpha \\<br/>-\cos \alpha \sin \alpha+\cos \alpha \sin \alpha &amp; \cos ^{2} \alpha+\sin ^{2} \alpha<br/>\end{array}\right] \\<br/>=\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right]=I<br/>\end{array}<br/>\]</div>


title: if-mathrmaleftbeginarrayllalpha-2-2-alphaendarrayright-and-leftmathrma3right27-then-alpha
slug: if-mathrmaleftbeginarrayllalpha-2-2-alphaendarrayright-and-leftmathrma3right27-then-alpha-38150
tags: ['Mathematics', 'Continuity and Differentiability', 'KCET', 'KCET 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \( \mathrm{A}=\left[\begin{array}{ll}\alpha &amp; 2 \\ 2 &amp; \alpha\end{array}\right] \) and \( \left|\mathrm{A}^{3}\right|=27 \), then \( \alpha= \)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( \pm \sqrt{7} \)</span> </div>

answer:
<div class="solution">Given that $A=\left[\begin{array}{ll}\alpha &amp; 2 \\ 2 &amp; \alpha\end{array}\right]$<br/>And $\left|A^{3}\right|=27=|A|^{3}$<br/>$\Rightarrow|A|=3$<br/>So, $|A|=\alpha^{2}-4=3$<br/>$\Rightarrow \alpha^{2}=7 \Rightarrow \alpha=\pm \sqrt{7}$</div>


tags: ['Mathematics', 'Matrices', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-mathrmaleftbeginarraylll0-1-2-1-2-3-3-mathrma-1endarrayright-mathrma1leftbeginarraylll1-2-1-2-1-2-4-3-mathrmc-5-2-3-2-1-2endarrayright-then-values-

slug:
if-mathrmaleftbeginarraylll0-1-2-1-2-3-3-mathrma-1endarrayright-mathrma1leftbeginarraylll1-2-1-2-1-2-4-3-mathrmc-5-2-3-2-1-2endarrayright-then-values-99220

question:
<div class="question">If \( \mathrm{A}=\left[\begin{array}{lll}0 &amp; 1 &amp; 2 \\ 1 &amp; 2 &amp; 3 \\ 3 &amp; \mathrm{a} &amp; 1\end{array}\right] \&amp; \mathrm{~A}^{-1}=\left[\begin{array}{lll}1 / 2 &amp; -1 / 2 &amp; 1 / 2 \\ -4 &amp; 3 &amp; \mathrm{C} \\ 5 / 2 &amp; -3 / 2 &amp; 1 / 2\end{array}\right] \), then values of a \&amp; \( C \) are</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">\( 1,2 \)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\( 1,-1 \)</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">\( -2,-1 \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( 1,-1 \)</span> </div>

answer:
<div class="solution"><p>We have, <math><mi mathvariant="normal">A</mi><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mi mathvariant="normal">a</mi></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mrow></mfenced></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mfenced close="]" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>3</mn></mtd><mtd><mi mathvariant="normal">C</mi></mtd></mtr><mtr><mtd><mn>5</mn><mo>/</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>3</mn><mo>/</mo><mn>2</mn></mtd><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> </p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi><mo>.</mo><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>1</mn><mi mathvariant="normal"> </mi></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">I</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi mathvariant="normal">A</mi><mo>.</mo><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced close="]" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mi mathvariant="normal">a</mi></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mi mathvariant="normal"> </mi><mfenced close="]" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>3</mn></mtd><mtd><mi mathvariant="normal">C</mi></mtd></mtr><mtr><mtd><mn>5</mn><mo>/</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>3</mn><mo>/</mo><mn>2</mn></mtd><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="]" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi mathvariant="normal">C</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">C</mi></mtd></mtr><mtr><mtd><mn>4</mn><mo>-</mo><mn>4</mn><mi mathvariant="normal">a</mi></mtd><mtd><mn>3</mn><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mtd><mtd><mn>2</mn><mo>+</mo><mi>ac</mi></mtd></mtr></mtable></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfenced close="]" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi mathvariant="normal">C</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">C</mi></mtd></mtr><mtr><mtd><mn>4</mn><mo>-</mo><mn>4</mn><mi mathvariant="normal">a</mi></mtd><mtd><mn>3</mn><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mtd><mtd><mn>2</mn><mo>+</mo><mi>ac</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mn>4</mn><mi>a</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi mathvariant="normal">a</mi><mo>=</mo><mn>1</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>C</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p></div>


tags: ['Mathematics', 'Matrices', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">128B</span> </div>
source_db: MarksBatch2_P1.db
title:
if-mathrmaleftbeginarrayllmathrmi-mathrmi-mathrmi-mathrmiendarrayright-mathrmbleftbeginarrayll1-1-1-1endarrayright-then-mathrma8-equals

slug:
if-mathrmaleftbeginarrayllmathrmi-mathrmi-mathrmi-mathrmiendarrayright-mathrmbleftbeginarrayll1-1-1-1endarrayright-then-mathrma8-equals-98050

question:
<div class="question">If \( \mathrm{A}=\left[\begin{array}{ll}\mathrm{i} &amp; -\mathrm{i} \\ -\mathrm{i} &amp; \mathrm{i}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 &amp; -1 \\ -1 &amp; 1\end{array}\right] \), then \( \mathrm{A}^{8} \) equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( 4 \mathrm{~B} \)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">128B</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\( -128 \mathrm{~B} \)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( -64 \mathrm{~B} \)</span> </li> </ul>

answer:
<div class="solution">We have <math><mi mathvariant="normal">A</mi><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">B</mi></math><br/><math><mo>⇒</mo><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">B</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><msup><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mo>-</mo><msup><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mrow></mfenced><mi mathvariant="normal"> </mi><mo>=</mo><mo>-</mo><mn>2</mn><mi mathvariant="normal">B</mi></math><br/><math><mo>⇒</mo><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mn>2</mn><mi mathvariant="normal">B</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mn>4</mn><msup><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mn>4</mn><mfenced separators="|"><mrow><mn>2</mn><mi mathvariant="normal">B</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal"> </mi><mn>8</mn><mi mathvariant="normal">B</mi></math><br/><math><mo>⇒</mo><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn>8</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><msup><mrow><mfenced separators="|"><mrow><mn>8</mn><mi mathvariant="normal">B</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mn>64</mn><msup><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi mathvariant="normal"> </mi><mn>128</mn><mi mathvariant="normal">B</mi></math></div>


tags: ['Mathematics', 'Differentiation', 'KCET', 'KCET 2015']
source_db: MarksBatch2_P1.db
title:
if-mathrmpleftbeginarrayllx-1-1-xendarrayright-and-mathrmqleftbeginarraylllx-1-1-1-x-1-1-1-xendarrayright-then-fracd-qd-x

slug:
if-mathrmpleftbeginarrayllx-1-1-xendarrayright-and-mathrmqleftbeginarraylllx-1-1-1-x-1-1-1-xendarrayright-then-fracd-qd-x-69487

question:
<div class="question">If \( \mathrm{P}=\left|\begin{array}{ll}x &amp; 1 \\ 1 &amp; x\end{array}\right| \) and \( \mathrm{Q}=\left|\begin{array}{lll}x &amp; 1 &amp; 1 \\ 1 &amp; x &amp; 1 \\ 1 &amp; 1 &amp; x\end{array}\right| \),then \( \frac{d Q}{d x}= \)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( 3 \mathrm{P} \)</span> </div>

answer:
<div class="solution">Given that<br/>\( P=\left|\begin{array}{ll}x &amp; 1 \\ 1 &amp; x\end{array}\right|, Q=\left|\begin{array}{lll}x &amp; 1 &amp; 1 \\ 1 &amp; x &amp; 1 \\ 1 &amp; 1 &amp; x\end{array}\right| \)<br/>So, \( P=x^{2}-1 \)<br/>And \( Q=x\left(x^{2}-1\right)-(x-1)+(1-x) \)<br/>\( =x^{3}-3 x+2 \)<br/>Now, \( \frac{d Q}{d x}=3 x^{2}-3 \)<br/>\( =3\left(x^{2}-1\right)=3 P \)</div>


title: if-md-is-12-the-value-of-sd-will-be-where-md-is-mean-deviation-and-sd-is-standard-deviation
slug: if-md-is-12-the-value-of-sd-will-be-where-md-is-mean-deviation-and-sd-is-standard-deviation-22897
tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$15$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If M.D. is 12, the value of S.D. will be (where M.D. is mean deviation and S.D. is standard deviation)</div>

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

answer:
<div class="solution">We know that Q.D. $=\frac{5}{6} \times$ M.D. $=\frac{5}{6} \times 12=10$<br/>$\therefore$ S.D. $=\frac{3}{2} \times$ Q.D. $=\frac{3}{2} \times 10 \Rightarrow$ S.D. $=15$.<br/>where Q.D. is Quartile Deviation, S.D. is Standard deviation and M.D. is Mean deviation.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-mean-and-variance-of-a-binomial-variate-x-are-2-and-1-respectively-then-the-probability-that-x-takes-a-value-greater-than-1-is

slug:
if-mean-and-variance-of-a-binomial-variate-x-are-2-and-1-respectively-then-the-probability-that-x-takes-a-value-greater-than-1-is-39681

question:
<div class="question">If mean and variance of a Binomial variate $\mathrm{X}$ are 2 and 1 respectively, then the probability that $\mathrm{X}$ takes a value greater than 1 is</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">$\frac{4}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{8}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{11}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{11}{16}$</span> </div>

answer:
<div class="solution">We have; $\mathrm{np}=2=$ mean $\mathrm{npq}=1=$ variance<br/>$\Rightarrow \mathrm{p}=\frac{1}{2} ; \mathrm{q}=\frac{1}{2} \&amp; \mathrm{n}=4$<br/>$\begin{aligned} \text { Required probability } &amp;=\mathrm{P}(\mathrm{x}&gt;1) \\ &amp;=1-\mathrm{P}(\mathrm{x} \leq 1) \\ &amp;=1-[\mathrm{P}(\mathrm{x}=0)+\mathrm{P}(\mathrm{x}=1)] \\ &amp;=1-\left[{ }^{4} \mathrm{C}_{0} \mathrm{q}^{4}+{ }^{4} \mathrm{C}_{1} \mathrm{q}^{3} \mathrm{p}^{1}\right] \\ &amp;=1-\frac{5}{16}=\frac{11}{16} \end{aligned}$</div>


title: if-median-of-the-abc-through-a-is-perpendicular-to-bc-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-median-of-the-abc-through-a-is-perpendicular-to-bc-then-which-one-of-the-following-is-correct-95550

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan B-\tan C=0$</span> </div>

answer:
<div class="solution">In the given $\Delta \mathrm{ABC}$ let $\mathrm{BC}=\mathrm{a}$<br/>$\therefore \quad \mathrm{BD}=\mathrm{CD}=\frac{\mathrm{a}}{2}$ <br/>In $\Delta \mathrm{ADB}$,<br/>$\tan B=\frac{A D}{B D}=\frac{A D}{a / 2}$<br/>$\Rightarrow \quad \tan B=\frac{2 \mathrm{AD}}{\mathrm{a}}$ ...(1)<br/>In $\Delta \mathrm{ADC}$<br/>$\tan \mathrm{C}=\frac{\mathrm{AD}}{\mathrm{CD}}=\frac{\mathrm{AD}}{\mathrm{a} / 2}$<br/>$\tan C=\frac{2 A D}{a}$ ...(2)<br/>From eqs. (1) and (2), we get <br/>$\tan B=\tan C$<br/>$\Rightarrow \tan B-\tan C=0$</div>


tags: ['Physics', 'Units and Dimensions']
source_db: MarksBatch2_P1.db
title:
if-momentum-p-area-a-and-time-t-are-taken-to-be-fundamental-quantities-then-energy-has-the-dimensional-formula

slug:
if-momentum-p-area-a-and-time-t-are-taken-to-be-fundamental-quantities-then-energy-has-the-dimensional-formula-81370

question:
<div class="question">If momentum $(p)$, area $(A)$ and time $(T)$ are taken to be fundamental quantities, then energy has the dimensional formula</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\left[p \mathrm{~A}^{1 / 2} \mathrm{~T}\right]$</span> </div>

answer:
<div class="solution">Given that fundamental quantities are momentum $(p)$, area $(A)$ and time $(T)$.<br/>Let us consider the dimensional formula for<br/>$$<br/>\begin{aligned}<br/>&amp;E \propto\left[p^a A^b T^c\right] \\<br/>&amp;\left.E=k p^a A^b T^c\right]<br/>\end{aligned}<br/>$$<br/>where $k$ is dimensionless constant of proportionality.<br/>Dimensions of energy $[E]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$ and Dimension of momentum $p=m v=\left[\mathrm{MLT}^{-1}\right]$<br/>Dimension of Area $[A]=\left[\mathrm{L}^2\right]$<br/>Dimension of Time $[T]=[\mathrm{T}]$<br/>Dimension of energy $[E]=[K][p]^a[A]^b[T]^c$<br/>Putting all the dimensions, value<br/>$\mathrm{ML}^2 \mathrm{~T}^{-2}=\left[\mathrm{MLT}^{-1}\right]^a\left[\mathrm{~L}^2\right]^b\left[\mathrm{~T}^c\right.$<br/>$=\mathrm{M}^a \mathrm{~L}^{2 b+a} \mathrm{~T}^{-a+c}$<br/>By principle of homogeneity of dimensions,<br/>$$<br/>\begin{aligned}<br/>&amp;a=1,2 b+a=2 \Rightarrow 2 b+1=2 \Rightarrow b=\frac{1}{2} \\<br/>&amp;-a+c=-2 \\<br/>&amp;c=-2+a=-2+1=-1<br/>\end{aligned}<br/>$$<br/>So, Dimensional formula (of energy)<br/>$$<br/>E=\left[p A^{1 / 2} T^{-1}\right]<br/>$$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2020 (02 Sep Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-momentum-p-area-a-and-time-t-are-taken-to-be-the-fundamental-quantities-then-the-dimensional-formula-for-energy-is

slug:
if-momentum-p-area-a-and-time-t-are-taken-to-be-the-fundamental-quantities-then-the-dimensional-formula-for-energy-is-50449

question:
<div class="question">If momentum<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">P</mi><mo>)</mo></math>, area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">A</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 taken to be the fundamental quantities then the dimensional formula for energy is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">P</mi><mn>2</mn></msup><mo> </mo><msup><mi>AT</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>PA</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>PA</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</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 xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>P</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>AT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>PA</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Let dimension formula of energy will be</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msup><mi>A</mi><mi>a</mi></msup><msup><mi>T</mi><mi>b</mi></msup><msup><mi>P</mi><mi>c</mi></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><msup><mi>M</mi><mn>1</mn></msup><msup><mi>L</mi><mn>1</mn></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msup><mi>L</mi><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><msup><mi>T</mi><mn>1</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>=</mo><msup><mi mathvariant="normal">M</mi><mi mathvariant="normal">C</mi></msup><msup><mi mathvariant="normal">L</mi><mrow><mn>2</mn><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">c</mi></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mi mathvariant="normal">b</mi><mo>-</mo><mi mathvariant="normal">c</mi></mrow></msup></math><br/>by comparison <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>1</mn></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>2</mn></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>2</mn></mfenced></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>-</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>2</mn></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>3</mn></mfenced></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>  </mo><mi>a</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>P</mi><mn>1</mn></msup></math></p></div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-n-1-and-n-2-are-the-sizes-g-1-and-g-2-the-geometric-means-of-two-series-respectively-then-which-one-of-the-following-expresses-the-geometric-mean-g

slug:
if-n-1-and-n-2-are-the-sizes-g-1-and-g-2-the-geometric-means-of-two-series-respectively-then-which-one-of-the-following-expresses-the-geometric-mean-g-88441

question:
<div class="question">If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$ are the sizes, $\mathrm{G}_{1}$ and $\mathrm{G}_{2}$ the geometric means of two series respectively, then which one of the following expresses the geometric mean (G) of the combined series?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\log \mathrm{G}=\frac{\mathrm{n}_{1} \mathrm{G}_{1}+\mathrm{n}_{2} \mathrm{G}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\log \mathrm{G}=\frac{\mathrm{n}_{2} \log \mathrm{G}_{1}+\mathrm{n}_{1} \log \mathrm{G}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$</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">$\mathrm{G}=\frac{\mathrm{n}_{1} \log \mathrm{G}_{1}+\mathrm{n}_{2} \log \mathrm{G}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$</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">$\log \mathrm{G}=\frac{\mathrm{n}_{2} \log \mathrm{G}_{1}+\mathrm{n}_{1} \log \mathrm{G}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$</span> </div>

answer:
<div class="solution">Geometric Mean of combined series is given by the expression<br/>$\log \mathrm{G}=\frac{\mathrm{n}_{2} \log \mathrm{G}_{1}+\mathrm{n}_{1} \log \mathrm{G}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$</div>


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

slug:
if-n-1-n-2-a-n-d-n-3-are-the-fundamental-frequencies-of-three-segments-into-which-a-string-is-divided-then-the-original-fundamental-frequency-n-of-the-37805

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

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

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

answer:
<div class="solution">In stretched string for transverse wave, Frequency is inversely proportional to lengthAs <math><mi>n</mi><mo>â</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>l</mi></mrow></mfrac><mi> </mi><mi>a</mi><mi>n</mi><mi>d</mi><mi> </mi><mi>l</mi><mo>=</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub></math><br/><br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mfrac></math></div>


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

slug:
if-n-1-n-2-and-n-3-are-the-fundamental-frequencies-of-three-segments-into-which-a-string-is-divided-then-the-original-fundamental-frequency-n-of-the-s-1-55870

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

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

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

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


