# questions

title: let-f-x-x-x-1-2-what-is-f-x-equal-to-when-0-x-1
slug: let-f-x-x-x-1-2-what-is-f-x-equal-to-when-0-x-1-31110
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=[|\mathrm{x}|-\mid \mathrm{x}-1]]^{2}$<br/>What is $\mathrm{f}^{\prime}(\mathrm{x})$ equal to when $0   &lt; \mathrm{x} &lt; 1 ?$</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">$2 x-1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 x-2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$8 x-4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 x-4$</span> </div>

answer:
<div class="solution">Given $f(x)=(|x|-|x+1|)^{2}$<br/>$f(x)=\left\{\begin{array}{cc}1 &amp; x \leq 0 \\ (2 x-1)^{2} &amp; 0   &lt; x &lt; 1 \\ 1 &amp; x \geq 1\end{array}\right.$<br/>When $0 &lt; x &lt; 1$<br/>$f(x)=(2 x-1)^{2}$<br/>$f^{\prime}(x)=2(2 x-1) \cdot 2=4(2 x-1)$<br/>$f^{\prime}(x)=8 x-4$</div>


title: let-f-x-x-x-1-2-what-is-f-x-equal-to-when-x-1
slug: let-f-x-x-x-1-2-what-is-f-x-equal-to-when-x-1-47202
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (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">Let $\mathrm{f}(\mathrm{x})=[|\mathrm{x}|-\mid \mathrm{x}-1]]^{2}$<br/>What is $\mathrm{f}^{\prime}(\mathrm{x})$ equal to when $\mathrm{x}&gt;1$?</div>

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

answer:
<div class="solution">Given $f(x)=(|x|-|x+1|)^{2}$<br/>$f(x)=\left\{\begin{array}{cc}1 &amp; x \leq 0 \\ (2 x-1)^{2} &amp; 0   &lt; x &lt; 1 \\ 1 &amp; x \geq 1\end{array}\right.$<br/>When $x&gt;1$ $f(x)=1$<br/>$f^{\prime}(x)=0$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-x-x-x-1-2-which-of-the-following-equations-isare-correct-1-f-2-f-5-2-f-2-f-05-f-3-4-select-the-correct-answer-using-the-code-given-below

slug:
let-f-x-x-x-1-2-which-of-the-following-equations-isare-correct-1-f-2-f-5-2-f-2-f-05-f-3-4-select-the-correct-answer-using-the-code-given-below-42046

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=[|\mathrm{x}|-\mid \mathrm{x}-1]]^{2}$<br/>Which of the following equations is/are correct?<br/> $1 \quad \mathrm{f}(-2)=\mathrm{f}(5)$<br/> $2 \quad f^{\prime \prime}(-2)+f^{\prime \prime}(0.5)+f^{\prime \prime}(3)=4$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">Given $f(x)=(|x|-|x+1|)^{2}$<br/>$f(x)=\left\{\begin{array}{cc}1 &amp; x \leq 0 \\ (2 x-1)^{2} &amp; 0   &lt; x &lt; 1 \\ 1 &amp; x \geq 1\end{array}\right.$<br/>For $x=-2$<br/>$f(x)=1$ so $f(-2)=1$<br/>For $x=5$ $f(x)=1 \Rightarrow f(5)=1$<br/>Hence $f(-2)=f(5)$<br/>Now, for $x=-1$<br/>$f^{\prime \prime}(x)=0$<br/>$f^{\prime \prime}(-2)=0$<br/>For $x=0.5$<br/>$f^{\prime \prime}(x)=8 \Rightarrow f^{\prime \prime}(0.5)=8$<br/>For $x=3$<br/>$f^{\prime \prime}(x)=0 \Rightarrow f^{\prime \prime}(3)=0$<br/>$\Rightarrow f^{\prime \prime}(-2)+f^{\prime \prime}(0.5)+f^{\prime \prime}(3)=8 \neq 4$<br/>Only statement 1 is correct.</div>


title: let-f-x-x-x-1-when-x-0-1-then-which-one-of-the-following-is-correct
slug: let-f-x-x-x-1-when-x-0-1-then-which-one-of-the-following-is-correct-57108
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(\mathrm{x})=\mathrm{x}+\frac{1}{\mathrm{x}}$, when $\mathrm{x} \in(0,1)$. 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">$f(\mathrm{x})$ fluctuates in the interval</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(\mathrm{x})$ increases in the interval</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f(\mathrm{x})$ decreases in the interval</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

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

answer:
<div class="solution">$f(x)=x+\frac{1}{x}$<br/>$f^{\prime}(x)=1-\frac{1}{x^{2}}=\frac{x^{2}-1}{x^{2}}=\frac{(x-1)(x+1)}{x^{2}}$<br/>For $x \in(0,1), f^{\prime}(x)   &lt; 0$<br/>$\Rightarrow \mathrm{f}(\mathrm{x})$ decreases</div>


title: let-f-x-y-f-x-f-y-and-f-1-2-for-all-x-y-r-where-f-x-is-continuous-function-what-is-f-1-equal-to
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
let-f-x-y-f-x-f-y-and-f-1-2-for-all-x-y-r-where-f-x-is-continuous-function-what-is-f-1-equal-to-85934

question:
<div class="question">$\quad$ Let $f(x+y)=f(x) \cdot f(y)$ and $f(1)=2$ for all $x, y \in R$, where $f(x)$ is continuous function. What is $\mathrm{f}^{\prime}(1)$ equal to ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \ln 2$</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">$\ln 2$</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">0</span> </li> </ul>

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

answer:
<div class="solution">Given that $\mathrm{f}(1)=2$ and $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})$<br/>These are value for all values of $x$ and $y$ Puting $\mathrm{x}=1$ and $\mathrm{y}=1$, we get $\mathrm{f}(2)=\mathrm{f}(1) \cdot \mathrm{f}(1)=2.2=2^{2}$<br/>Similarly, $\mathrm{f}(3)=\mathrm{f}(1) \cdot \mathrm{f}(2)=2 \cdot 2^{2}=2^{3}$<br/>$\Rightarrow \quad \mathrm{f}(\mathrm{x})=2^{\mathrm{x}}$<br/>$\Rightarrow \quad \mathrm{f}^{\prime}(\mathrm{x})=2^{\mathrm{x}} \log _{\mathrm{e}} 2$<br/>Hence, $\mathrm{f}^{\prime}(1)=2 \log _{\mathrm{e}} 2$</div>


title: let-f-x-y-f-x-f-y-and-f-x-1-xg-x-x-where-lim-x-0-g-x-a-and-lim-x-0-x-b-then-what-is-f-x-equal-to
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
let-f-x-y-f-x-f-y-and-f-x-1-xg-x-x-where-lim-x-0-g-x-a-and-lim-x-0-x-b-then-what-is-f-x-equal-to-98617

question:
<div class="question">Let $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})$ and $\mathrm{f}(\mathrm{x})=1+\mathrm{xg}(\mathrm{x}) \phi(\mathrm{x})$,<br/>where $\lim _{x \rightarrow 0} g(x)=a$ and $\lim _{x \rightarrow 0} \phi(x)=b$. Then what is $f^{\prime}(x)$<br/>equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+\mathrm{ab} \mathrm{f}(\mathrm{x})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1+\mathrm{ab}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">ab</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\operatorname{abf}(\mathrm{x})$</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> </ul>

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

answer:
<div class="solution">$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{f(x) f(h)-f(x)}{h}$<br/>$=\lim _{h \rightarrow 0} \frac{f(x)[f(h)-1]}{h}=\lim _{h \rightarrow 0} \frac{f(x)[h g(h) \phi(h)]}{h}$<br/>$=a b f(x)$</div>


title: let-f-x-y-f-x-f-y-for-all-x-and-y-then-what-is-f-5-equal-to-where-f-x-is-the-derivative-of-f-x
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
let-f-x-y-f-x-f-y-for-all-x-and-y-then-what-is-f-5-equal-to-where-f-x-is-the-derivative-of-f-x-68230

question:
<div class="question">Let $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y})$ for all $\mathrm{x}$ and $\mathrm{y}$. Then what is $f^{\prime}(5)$<br/>equal to [where $f^{\prime}(x)$ is the derivative of $\left.f(x)\right] ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(5) f^{\prime} 0$</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">$f(5)-f^{\prime}(0)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(5) f(0)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(5)+f^{\prime}(0)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(5) f^{\prime} 0$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})$<br/>$\operatorname{Let} \mathrm{f}(\mathrm{x})=\mathrm{a}^{\mathrm{x}}$<br/>$\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{a}^{\mathrm{x}+\mathrm{y}}=\mathrm{a}^{\mathrm{x}} \cdot \mathrm{a}^{\mathrm{y}}=\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y}) .$<br/>$\mathrm{f}(5)=\mathrm{a}^{5}$<br/>$\mathrm{f}^{\prime}(5)=\mathrm{a}^{5} \cdot \log \mathrm{a}$<br/>$=\mathrm{f}(5) \cdot \mathrm{f}(0)$<br/>$\mathrm{f}^{\mathrm{l}}(0)=\mathrm{a}^{0} \cdot \log \mathrm{a}$<br/>$=\log \mathrm{a}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
let-fxleftbeginarrayccfrac1cos-xpix2-cdot-fracsin-2-xlog-left1pi22-pi-xx2right-quad-x-neq-pi-k-quad-xpiendarrayright-if-fx-is-continuous-functions-at-

slug:
let-fxleftbeginarrayccfrac1cos-xpix2-cdot-fracsin-2-xlog-left1pi22-pi-xx2right-quad-x-neq-pi-k-quad-xpiendarrayright-if-fx-is-continuous-functions-at-93671

question:
<div class="question">Let \( f(x)=\left\{\begin{array}{cc}\frac{1+\cos x}{(\pi-x)^{2}} \cdot \frac{\sin ^{2} x}{\log \left(1+\pi^{2}-2 \pi x+x^{2}\right)} &amp; , \quad x \neq \pi \\ k &amp; , \quad x=\pi\end{array}\right. \). If \( f(x) \) is continuous functions at \( x=\pi \), then \( k \) is equal to</div>

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

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

answer:
<div class="solution">The given function is<br/><math> <mrow> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>x</mtext> </mrow> </mfenced> <mo>=</mo> <mo>{</mo> <mtable> <mtr> <mtd> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mtext>cos x</mtext> </mrow> <mrow> <msup> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <mtext>Ï</mtext> <mo>-</mo> <mtext>x</mtext> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>Â·</mo> <mfrac> <mrow> <msup> <mtext>sin</mtext> <mn>2</mn> </msup> <mtext>x</mtext> </mrow> <mrow> <mtext>log</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mtext>Ï</mtext> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mtext>Ï</mtext> <mtext>x</mtext> <mo>+</mo> <msup> <mtext>x</mtext> <mn>2</mn> </msup> </mrow> </mfenced> </mrow> </mfrac> </mtd> <mtd> <mtext>,</mtext> </mtd> <mtd> <mtext>x</mtext> <mo>â </mo> <mtext>Ï</mtext> </mtd> </mtr> <mtr> <mtd> <mtext>k</mtext> </mtd> <mtd> <mtext>,</mtext> </mtd> <mtd> <mtext>x</mtext> <mo>=</mo> <mtext>Ï</mtext> </mtd> </mtr> </mtable> </mrow> </math><br/>Since, f(x) continuous at x = <mtext>Ï</mtext>.<br/><math> <mrow> <mo>â´                     </mo> <munder> <mtext>lim</mtext> <mrow> <mtext>x</mtext> <mo>â</mo> <mtext>Ï</mtext> </mrow> </munder> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>x</mtext> </mrow> </mfenced> <mo>=</mo> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>Ï</mtext> </mrow> </mfenced> </mrow> </math><br/><math> <mrow> <mo>â              </mo> <munder> <mtext>lim</mtext> <mrow> <mtext>h</mtext> <mo>â</mo> <mn>0</mn> </mrow> </munder> <mtext>f</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>Ï</mtext> <mo>+</mo> <mtext>h</mtext> </mrow> </mfenced> <mo>=</mo> <mtext>k</mtext> <mo>â</mo> <munder> <mtext>lim</mtext> <mrow> <mtext>h</mtext> <mo>â</mo> <mn>0</mn> </mrow> </munder> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mtext>cos</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>Ï</mtext> <mo>+</mo> <mtext>h</mtext> </mrow> </mfenced> </mrow> <mrow> <msup> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <mtext>Ï</mtext> <mo>-</mo> <mtext>Ï</mtext> <mo>-</mo> <mtext>h</mtext> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>Â·</mo> </mrow> </math><br/>                                     <math> <mrow> <mfrac> <mrow> <msup> <mtext>sin</mtext> <mn>2</mn> </msup> <mfenced close=")" open="(" separators=","> <mrow> <mi>Ï</mi> <mo>+</mo> <mtext>h</mtext> </mrow> </mfenced> </mrow> <mrow> <mtext>log</mtext> <msup> <mrow> <mfenced close="]" open="[" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>Ï</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mi>Ï</mi> <mfenced close=")" open="(" separators=","> <mrow> <mi>Ï</mi> <mo>+</mo> <mtext>h</mtext> </mrow> </mfenced> <mo>+</mo> <mfenced close=")" open="(" separators=","> <mrow> <mi>Ï</mi> <mo>+</mo> <mtext>h</mtext> </mrow> </mfenced> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>=</mo> <mtext>k</mtext> </mrow> </math><br/><math> <mrow> <mo>â                 </mo> <munder> <mtext>lim</mtext> <mrow> <mtext>h</mtext> <mo>â</mo> <mn>0</mn> </mrow> </munder> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <mtext>cos h</mtext> </mrow> <mrow> <msup> <mi>Ï</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>Ã</mo> <mfrac> <mrow> <msup> <mtext>sin</mtext> <mn>2</mn> </msup> <mtext>h</mtext> </mrow> <mrow> <mtext>log</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mtext>h</mtext> <mn>2</mn> </msup> </mrow> </mfenced> </mrow> </mfrac> <mo>=</mo> <mtext>k</mtext> </mrow> </math><br/><math> <mrow> <mo>â            </mo> <munder> <mtext>lim</mtext> <mrow> <mtext>h</mtext> <mo>â</mo> <mn>0</mn> </mrow> </munder> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <msup> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <mfrac> <mrow> <mtext>sin h/2</mtext> </mrow> <mrow> <mtext>h/2</mtext> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> <mo>Â·</mo> <mfrac> <mrow> <msup> <mtext>h</mtext> <mn>2</mn> </msup> </mrow> <mrow> <mtext>log</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mtext>h</mtext> <mn>2</mn> </msup> </mrow> </mfenced> </mrow> </mfrac> <mo>Â·</mo> <msup> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <mfrac> <mrow> <mtext>sin h</mtext> </mrow> <mrow> <mtext>h</mtext> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mtext>k</mtext></mrow> </math><br/><math> <mrow> <mo>â                                 </mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mtext>k</mtext> </mrow> </math></div>


title: let-fxleftbeginarrayccx1-1-leq-x-leq-0-x-0-leq-x-leq-1endarrayright
slug: let-fxleftbeginarrayccx1-1-leq-x-leq-0-x-0-leq-x-leq-1endarrayright-15165
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2023']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let \(f(x)=\left\{\begin{array}{cc}x+1, &amp; -1 \leq x \leq 0 \\ -x, &amp; 0 \leq x \leq 1\end{array}\right.\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(f(x)\) is discontinuous in \([-1,1]\) and so has no maximum value or minimum value in \([-1,1]\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(f(x)\) is continuous in \([-1,1]\) and so has maximum value and minimum value</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(f(x)\) is discontinuous in \([-1,1]\) but still has the maximum and minimum value</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(f(x)\) is bounded in \([-1,1]\) and does not attain maximum or minimum value</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(f(x)\) is discontinuous in \([-1,1]\) but still has the maximum and minimum value</span> </div>

answer:
<div class="solution">Hint : \(f(x)= \begin{cases}x+1 &amp; -1 \leq x \leq 0 \\ -x &amp; 0  &lt; x \leq 1\end{cases}\)<br/>\(\left.\begin{array}{l}f\left(-1^{-}\right)=0 \\ f(-1)=0 \\ f\left(-1^{+}\right)=0\end{array}\right]\) continuous at \(x=-1\)<br/>\(\left.\begin{array}{l}\mathrm{f}(0)=1 \\ \mathrm{f}\left(0^{-}\right)=1 \\ \mathrm{f}\left(0^{+}\right)=0\end{array}\right]\) discontinuous at \(\mathrm{x}=0\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fCmf9DcaEDChL3cuKJBO1iTpYEmuRZAxNIZ_yy47gQ0.original.fullsize.png"/><br/><br/>obviously \(f(x)\) has local maximum at \(x=0\)</div>


title: let-g-be-the-greatest-integer-function-then-the-function-f-x-g-x-2-g-x-is-discontinuous-at
slug: let-g-be-the-greatest-integer-function-then-the-function-f-x-g-x-2-g-x-is-discontinuous-at-79259
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{g}$ be the greatest integer function. Then the function $f(\mathrm{x})=(\mathrm{g}(\mathrm{x}))^{2}-\mathrm{g}(\mathrm{x})$ is discontinuous at</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">all integers</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">all integers except 0 and 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">all integers except 0</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">all integers except 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">all integers except 1</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=(\mathrm{g}(\mathrm{x}))^{2}-\mathrm{g}(\mathrm{x})$<br/>Given, $\mathrm{g}(\mathrm{x})$ is greatest integer function. So, $\mathrm{g}(\mathrm{x})=[\mathrm{x}]$. $\therefore \mathrm{f}(\mathrm{x})=[\mathrm{x}]^{2}-[\mathrm{x}]$<br/>$\mathrm{f}(\mathrm{x})$ is discontinuous at every integer except $\mathrm{x}=1$</div>


title: let-g-r-r-be-a-function-such-that-g-x-2-x-5-then-what-is-g-1-x-equal-to
slug: let-g-r-r-be-a-function-such-that-g-x-2-x-5-then-what-is-g-1-x-equal-to-21424
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2008 (Phase 2)']
answer: <div class="solution">Let $y=2 x+5$<br/>$\Rightarrow y-5=2 x$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $g: R \rightarrow R$ be a function such that, $g(x)=2 x+5$. Then, what is $g^{-1}(x)$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x-5}{2}$</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">$2 x-5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x-\frac{5}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x}{2}+\frac{5}{2}$</span> </li> </ul>

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


tags: ['Physics', 'Mathematics in Physics', 'TS EAMCET', 'TS EAMCET 2020 (11 Sep Shift 1)']
source_db: MarksBatch2_P1.db
title:
let-g-w-e-and-s-be-relative-strength-of-gravitational-weaknuclear-electromagnetic-and-strongnuclear-forces-respectively-which-of-the-correct-statement

slug:
let-g-w-e-and-s-be-relative-strength-of-gravitational-weaknuclear-electromagnetic-and-strongnuclear-forces-respectively-which-of-the-correct-statement-13887

question:
<div class="question">Let $G, W, E$ and $S$ be relative strength of gravitational, weak-nuclear, electromagnetic and strong-nuclear forces, respectively. Which of the correct statement?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$S&gt;E&gt;W&gt;G$</span> </div>

answer:
<div class="solution">We know that, among all inverse forces based on inverse square law's gravitational force is the weakest and nuclear force is the strongest.<br/>Hence, $F_{\text {Strongnuclear }}&gt;E_{\text {Electromagnetic }}&gt;F_{\text {Gravitational }}$ $\ldots(\mathrm{i})$<br/>Weak nuclear force is a fundamental force in nature that underlies some forms of radioactivity, governs the decay of unstable subatomic particles such as meson and initiates the nuclear fusion reaction that fuels the sun.<br/>These forces are considered weaker than electromagnetic forces and stronger than gravitational forces, i.e.<br/>$F_{\text {Electromagnetic }}&gt;F_{\text {Weak nuclear }}&gt;F_{\text {Gravitational }}$ $\ldots(\mathrm{ii})$<br/>Given, $\quad F_{\text {Electromagnetic }}=E \Rightarrow F_{\text {Strongnuclear }}=S$<br/>$F_{\text {Weak nuclear }}=W$ and $F_{\text {Gravitational }}=G$<br/>Using Eqs. (i) and (ii), we get<br/>$S&gt;E&gt;W&gt;G$</div>


title: let-g-x-x-3-4-x-6-if-f-x-g-x-and-f-1-2-then-what-is-f-x-equal-to
slug: let-g-x-x-3-4-x-6-if-f-x-g-x-and-f-1-2-then-what-is-f-x-equal-to-38504
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $g(x)=x^{3}-4 x+6$. If $f^{\prime}(x)=g^{\prime}(x)$ and $f(1)=2$, then what is $f(x)$ equal to?</div>

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

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

answer:
<div class="solution">Now, (gof)' $(x)=1$ Given, $g(x)=x^{3}-4 x+6$<br/>$\operatorname{But} f^{\prime}(x)=g^{\prime}(x)$<br/>$\Rightarrow \int f^{\prime}(x) d x=\int g^{\prime}(x) d x$<br/>$\Rightarrow f(x)=g(x)+c$<br/>$\therefore f(x)=x^{3}-4 x+6+c$ where ' $\mathrm{c}$ ' is a constant. Now, $f(1)=2$ $\Rightarrow 2=f(1)=(1)^{3}-4(1)+6+c$<br/>$\Rightarrow 2=1-4+6+c$<br/>$\Rightarrow c=-1$<br/>$\therefore f(x)=x^{3}-4 x+6-1=x^{3}-4 x+5$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-i-2-j-k-2-i-j-3-k-and-2-i-j-6-k-be-three-vectors-if-and-are-both-perpendicular-to-the-vector-and-10-then-what-is-the-magnitude-of

slug:
let-i-2-j-k-2-i-j-3-k-and-2-i-j-6-k-be-three-vectors-if-and-are-both-perpendicular-to-the-vector-and-10-then-what-is-the-magnitude-of-73879

question:
<div class="question">Let $\vec{\alpha}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \vec{\beta}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\vec{\gamma}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+6 \hat{\mathrm{k}}$ be<br/>three vectors. If $\vec{\alpha}$ and $\vec{\beta}$ are both perpendicular to the<br/>vector $\vec{\delta}$ and $\vec{\delta} \cdot \vec{\gamma}=10$, then what is the magnitude of $\vec{\delta}$ ?</div>

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

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

answer:
<div class="solution">$\begin{aligned} \vec{\alpha} &amp;=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}} \\ \vec{\beta} &amp;=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\ \vec{\gamma} &amp;=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+6 \hat{\mathrm{k}} \end{aligned}$<br/>Let $\vec{\delta}=a \hat{i}+b \hat{j}+c \hat{k}$<br/>Since $\vec{\alpha}$ and $\vec{\delta}$ are perpendicular, $\vec{\alpha} \cdot \vec{\delta}=0$<br/>$\Rightarrow \mathrm{a}+2 \mathrm{~b}-\mathrm{c}=0$<br/>$\vec{\beta}$ and $\vec{\delta}$ are perpendicular, $\vec{\beta} \cdot \vec{\delta}=0$<br/>$\Rightarrow 2 \mathrm{a}-\mathrm{b}+3 \mathrm{c}=0$<br/>from $(1),(2), \frac{a}{5}=\frac{b}{-5}=\frac{c}{-5} \Rightarrow \frac{a}{1}=\frac{b}{-1}=\frac{c}{-1}=x($ say $)$<br/>So, $a=x, b=-x, c=-x$<br/>Also, it is given $\vec{\gamma} \cdot \vec{\delta}=10$<br/>$\Rightarrow 2 \mathrm{a}+\mathrm{b}+6 \mathrm{c}=10$<br/>$\Rightarrow 2 x-x-6 x=10$<br/>$\Rightarrow-5 x=10$<br/>$\Rightarrow x=-2$<br/>$\therefore \vec{\delta}=-2 \hat{i}+2 \hat{j}+2 \hat{k} .$<br/>$\therefore|\vec{\delta}|=\sqrt{4+4+4}=\sqrt{12}=2 \sqrt{3}$</div>


tags: ['Mathematics', 'Indefinite Integration', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
let-i-d-x-1-3-s-i-n-2-x-1-2-t-a-n-1-2-f-x-c-where-c-is-the-constant-of-integration-if-f-4-1-then-the-fundamental-period-of-y-f-x-is

slug:
let-i-d-x-1-3-s-i-n-2-x-1-2-t-a-n-1-2-f-x-c-where-c-is-the-constant-of-integration-if-f-4-1-then-the-fundamental-period-of-y-f-x-is-15121

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mo>â«</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mi mathvariant="normal">n</mi><mn>2</mn></msup><mi>x</mi></mrow></mfrac><mo>=</mo></math><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced separators="|"><mrow><mn>2</mn><mi>f</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mi>C</mi></math> (where, <math><mi>C</mi></math> is the constant of integration). If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced separators="|"><mfrac><mi>Ï</mi><mn>4</mn></mfrac></mfenced><mo>=</mo><mn>1</mn><mo>,</mo></math> then the fundamental period of <math><mi mathvariant="normal"></mi><mi>y</mi><mo>=</mo><mi>f</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mi>Ï</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mi mathvariant="normal"></mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mi>Ï</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">C</span> <span class="option-data"><math><mn>2</mn><mi>Ï</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mi>Ï</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mi mathvariant="normal"></mi></math></span> </li> </ul>

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

answer:
<div class="solution">Dividing the numerator and denominator by <math><msup><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi><mo>,</mo><mi></mi></math> we get<br/><math><mi>I</mi><mo>=</mo><mo>â«</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi><mo>+</mo><mn>3</mn><mi></mi><msup><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></mrow></mfrac><mi>d</mi><mi>x</mi></math><br/><math><mo>=</mo><mo>â«</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>4</mn><mi mathvariant="normal"></mi><msup><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></mrow></mfrac><mi>d</mi><mi>x</mi></math><br/>Let, <math><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><mi>x</mi><mi></mi></mrow></mrow><mo>=</mo><mi></mi><mi>t</mi></math><br/><math><mo>â</mo><msup><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi><mi>d</mi><mi>x</mi><mo>=</mo><mi>d</mi><mi>t</mi></math><br/><math><mo>â´</mo><mi>I</mi><mo>=</mo><mrow><mo stretchy="false">â«</mo><mrow><mfrac><mrow><mi>d</mi><mi>t</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>4</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mrow><mi mathvariant="normal"></mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mi mathvariant="normal">n</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced separators="|"><mrow><mn>2</mn><mi>t</mi></mrow></mfenced><mo>+</mo><mi>C</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mi mathvariant="normal">n</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced separators="|"><mrow><mn>2</mn><mi>tan</mi><mo>â¡</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mi>C</mi><mspace linebreak="newline"></mspace><mo>â</mo><mi>f</mi><mfenced separators="|"><mi>x</mi></mfenced><mo>=</mo><mi>tan</mi><mo>â¡</mo><mi>x</mi></math><br/>Hence, the fundamental period is <math><mi>â</mi><mi>Ï</mi><mi>â</mi></math></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2021 (18 Mar Shift 2)']
source_db: MarksBatch2_P1.db
title:
let-in-a-series-of-2-n-observations-half-of-them-are-equal-to-a-and-remaining-half-are-equal-to-a-also-by-adding-a-constant-b-in-each-of-these-observa

slug:
let-in-a-series-of-2-n-observations-half-of-them-are-equal-to-a-and-remaining-half-are-equal-to-a-also-by-adding-a-constant-b-in-each-of-these-observa-22160

question:
<div class="question">Let in a series of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math> observations, half of them are equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and remaining half are equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>a</mi></math>. Also by adding a constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> in each of these observations, the mean and standard deviation of new set become <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math>, respectively. Then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> is equal to :</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"><mn>425</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 xmlns="http://www.w3.org/1998/Math/MathML"><mn>650</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>250</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>925</mn></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"><mn>425</mn></math></span> </div>

answer:
<div class="solution"><p>Let observations are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>i</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mn>2</mn><mi>n</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>a</mi><mo>+</mo><mo>…</mo><mo>+</mo><mi>a</mi><mo>)</mo><mo>-</mo><mo>(</mo><mi>a</mi><mo>+</mo><mi>a</mi><mo>+</mo><mo>…</mo><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mn>0</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mfenced><mover><mi>x</mi><mo>¯</mo></mover></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mo>-</mo><mn>0</mn><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>σ</mi><mi mathvariant="normal">x</mi></msub><mo>=</mo><mi mathvariant="normal">a</mi></math></p><p>Now, adding a constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>¯</mo></mover><mo>=</mo><mover><mi>x</mi><mo>¯</mo></mover><mo>+</mo><mi>b</mi><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>b</mi><mo>=</mo><mn>5</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>y</mi></msub><mo>=</mo><msub><mi>σ</mi><mi>x</mi></msub></math> (No change in S.D.)  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>20</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>425</mn></math></p></div>


title: let-m-be-the-set-of-men-and-r-is-a-relation-is-son-of-defined-on-m-then-r-is
slug: let-m-be-the-set-of-men-and-r-is-a-relation-is-son-of-defined-on-m-then-r-is-23835
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $M$ be the set of men and $R$ is a relation 'is son of' defined on $M$. Then, $R$ is</div>

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

answer:
<div class="solution">Let $M=$ Set of men and $R$ is arelation'is son of defined on $M$. Reflexive : aRa (\because" $a$ can not be a son of $a$ ) Symmetric : $a R b \Rightarrow b R a$<br/>which is not also possible (\because: If $a$ is a son of $b$ then $b$ can not be a son of $a$ )<br/>Transitive : $a R b, b R c \Rightarrow a R c$<br/>which is not possible.</div>


tags: ['Mathematics', 'Binomial Theorem', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">45</span> </div>
source_db: MarksBatch2_P1.db
title:
let-m-n-in-n-and-g-c-d2-n1-if-30leftbeginarrayc30-0endarrayright29leftbeginarrayc30-1endarrayrightldots-ldots2leftbeginarrayc30-28endarrayright1leftbe

slug:
let-m-n-in-n-and-g-c-d2-n1-if-30leftbeginarrayc30-0endarrayright29leftbeginarrayc30-1endarrayrightldots-ldots2leftbeginarrayc30-28endarrayright1leftbe-97841

question:
<div class="question">Let \( m, n \in N \) and \( g c d(2, n)=1 . \) If \( 30\left(\begin{array}{c}30 \\ 0\end{array}\right)+29\left(\begin{array}{c}30 \\ 1\end{array}\right)+\ldots \ldots+2\left(\begin{array}{c}30 \\ 28\end{array}\right)+1\left(\begin{array}{c}30 \\ 29\end{array}\right)=n .2^{m} \), then \( n+m \) is equal to<br/>\( \left(\right. \) Here \( \left.\left(\begin{array}{l}n \\ k\end{array}\right)={ }^{n} C_{k}\right) \)</div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mfenced><mmultiscripts><mi>C</mi><mn>0</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced><mo>+</mo><mn>29</mn><mfenced><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced><mo>+</mo><mo>…</mo><mo>+</mo><mn>2</mn><mfenced><mmultiscripts><mi>C</mi><mn>28</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced><mo>+</mo><mn>1</mn><mfenced><mmultiscripts><mi>C</mi><mn>29</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced></math><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>30</mn><mfenced><mmultiscripts><mi>C</mi><mn>30</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced><mo>+</mo><mn>29</mn><mfenced><mmultiscripts><mi>C</mi><mn>29</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><mn>2</mn><mfenced><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced><mo>+</mo><mn>1</mn><mfenced><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>30</mn></munderover><mi>r</mi><mfenced><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>30</mn></munderover><mi>r</mi><mfenced><mfrac><mn>30</mn><mi>r</mi></mfrac></mfenced><mfenced><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mn>29</mn></mmultiscripts></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>30</mn><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>30</mn></munderover><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mn>29</mn></mmultiscripts></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>30</mn><mfenced><mrow><mmultiscripts><mi>C</mi><mn>0</mn><none></none><mprescripts></mprescripts><none></none><mn>29</mn></mmultiscripts><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>29</mn></mmultiscripts><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>29</mn></mmultiscripts><mo>+</mo><mo>…</mo><mo>+</mo><mmultiscripts><mi>C</mi><mn>29</mn><none></none><mprescripts></mprescripts><none></none><mn>29</mn></mmultiscripts></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>30</mn><mfenced><msup><mn>2</mn><mn>29</mn></msup></mfenced><mo>=</mo><mn>15</mn><msup><mfenced><mn>2</mn></mfenced><mn>30</mn></msup><mo>=</mo><mi>n</mi><msup><mfenced><mn>2</mn></mfenced><mi>m</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>n</mi><mo>=</mo><mn>15</mn><mo>,</mo><mi>m</mi><mo>=</mo><mn>30</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mi>m</mi><mo>=</mo><mn>45</mn></math></p></div>


title: let-mathrmfmathrmx-leftbeginarrayla-x21-x-1-xa-x-leq-1endarrayright-then-fx-is-derivable-at-x1-if
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
slug:
let-mathrmfmathrmx-leftbeginarrayla-x21-x-1-xa-x-leq-1endarrayright-then-fx-is-derivable-at-x1-if-21298

question:
<div class="question">Let \(\mathrm{f}(\mathrm{x})=\) \(\left\{\begin{array}{l}a x^2+1, x &gt; 1 \\ x+a, x \leq 1\end{array}\right.\). Then \(f(x)\) is derivable at \(x=1\), if</div>

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

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \operatorname{Lf}^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{-h} \\<br/>&amp; =\lim _{h \rightarrow 0} \frac{(1-h+a)-(1+a)}{-h}=\lim _{h \rightarrow 0} \frac{-h}{-h}=1 \\<br/>&amp; \operatorname{Rf}^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h} \\<br/>&amp; =\lim _{h \rightarrow 0} \frac{\left[a(1+h)^2+1\right]-(1+a)}{h} \\<br/>&amp; =\lim _{h \rightarrow 0}(a h+2 a)=2 a<br/>\end{aligned}\)<br/>Since f' (1) exists, \(\therefore L f^{\prime}(1)=R f^{\prime}(1) \Rightarrow a=1 / 2\)</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-matrix-b-be-the-adjoint-of-a-square-matrix-a-be-the-identitymatrix-of-same-order-as-a-if-k-0-is-the-determinant-of-the-matrix-a-then-what-is-ab-eq

slug:
let-matrix-b-be-the-adjoint-of-a-square-matrix-a-be-the-identitymatrix-of-same-order-as-a-if-k-0-is-the-determinant-of-the-matrix-a-then-what-is-ab-eq-18644

question:
<div class="question">Let matrix $\mathrm{B}$ be the adjoint of a square matrix $\mathrm{A}, \ell$ be the identitymatrix of same order as A. If $k(\neq 0)$ is the determinant of the matrix $\mathrm{A}$, then what is $\mathrm{AB}$ equal to?</div>

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

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

answer:
<div class="solution">$\mathrm{B}=\mathrm{adj} \mathrm{A}, \mathrm{I}=$ Identity matrix, $|\mathrm{A}|=k$<br/>$\mathrm{AB}=\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}| \mathrm{I}=k \mathrm{I}$</div>


title: let-mleftbeginarrayll1-1-0-1endarrayright-and-nleftbeginarrayll1-0-0-2endarrayright-then-n-m10-n1
tags: ['Mathematics', 'Determinants', 'AP EAMCET', 'AP EAMCET 2020 (18 Sep Shift 2)']
source_db: MarksBatch2_P1.db
slug:
let-mleftbeginarrayll1-1-0-1endarrayright-and-nleftbeginarrayll1-0-0-2endarrayright-then-n-m10-n1-30054

question:
<div class="question">Let \(M=\left[\begin{array}{ll}1 &amp; 1 \\ 0 &amp; 1\end{array}\right]\) and \(N=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 2\end{array}\right]\). Then \(N M^{10} N^{-1}=\)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left[\begin{array}{ll}1 &amp; 5 \\ 0 &amp; 1\end{array}\right]\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>M &amp; =\left[\begin{array}{ll}<br/>1 &amp; 1 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>M^2 &amp; =\left[\begin{array}{ll}<br/>1 &amp; 1 \\<br/>0 &amp; 1<br/>\end{array}\right]\left[\begin{array}{ll}<br/>1 &amp; 1 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>M^2 &amp; =\left[\begin{array}{ll}<br/>1 &amp; 2 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>M^3 &amp; =\left[\begin{array}{ll}<br/>1 &amp; 3 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>&amp; \vdots \\<br/>M^{10} &amp; =\left[\begin{array}{ll}<br/>1 &amp; 10 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>N &amp; =\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 2<br/>\end{array}\right] \\<br/>|N| &amp; =2-0=2 \\<br/>\operatorname{Adj}^N &amp; =\left[\begin{array}{ll}<br/>2 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>N^{-1} &amp; =\frac{1}{|N|} \text { Adj A } \\<br/>N^{-1} &amp; =\frac{1}{2}\left[\begin{array}{ll}<br/>2 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right] \\<br/>N^{-1} &amp; =\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; \frac{1}{2}<br/>\end{array}\right]<br/>\end{aligned}\)<br/>Consider,<br/>\(\begin{aligned}<br/>\mathbf{N} \cdot \mathbf{M}^{10} \cdot \mathbf{N}^{-1} &amp; =\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 2<br/>\end{array}\right]\left[\begin{array}{cc}<br/>1 &amp; 10 \\<br/>0 &amp; 1<br/>\end{array}\right]\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; \frac{1}{2}<br/>\end{array}\right] \\<br/>&amp; =\left[\begin{array}{cc}<br/>1 &amp; 10 \\<br/>0 &amp; 2<br/>\end{array}\right]\left[\begin{array}{cc}<br/>1 &amp; 0 \\<br/>0 &amp; \frac{1}{2}<br/>\end{array}\right]=\left[\begin{array}{ll}<br/>1 &amp; 5 \\<br/>0 &amp; 1<br/>\end{array}\right]<br/>\end{aligned}\)<br/>Hence, option (a) is correct.</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
let-n-3-a-list-of-numbers-0-x-1-x-2-x-n-has-mean-and-standard-deviatiion-a-new-list-of-numbers-is-made-as-follows-y-1-0-y-2-x-2-y-n-1-x-n-1-y-n-x-1-x--1

slug:
let-n-3-a-list-of-numbers-0-x-1-x-2-x-n-has-mean-and-standard-deviatiion-a-new-list-of-numbers-is-made-as-follows-y-1-0-y-2-x-2-y-n-1-x-n-1-y-n-x-1-x-1-88392

question:
<div class="question">Let $n \geq 3$. A list of numbers $0 &lt; x_{1} &lt; x_{2} &lt; \ldots &lt; x_{n}$ has mean $\mu$ and standard deviatiion $\sigma$. A new list of numbers is made as follows : $y_{1}=0, y_{2}=x_{2}, \ldots, y_{n-1}=x_{n-1}, y_{n}=x_{1}+x_{n}$. The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Which of the following is necessarily true?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mu=\hat{\mu}, \sigma \leq \hat{\sigma}$</span> </div>

answer:
<div class="solution">$\hat{\mu}=\frac{0+x_{2}+x_{3} \ldots \ldots x_{n-1}+\left(x_{1}+x_{n}\right)}{n}=\mu$<br/>$\hat{\sigma}^{2}=\frac{1}{n} \Sigma y_{i}^{2}-\hat{\mu}^{2}$<br/>$=\frac{1}{n}\left[0^{2}+x_{2}^{2}+x_{3}^{2} \ldots\left(x_{n-1}\right)^{2}+\left(x_{1}+x_{n}\right)^{2}\right]-\mu^{2}$<br/>$\hat{\sigma}^{2}-\sigma^{2}=\frac{2 x_{1} x_{n}}{n}&gt;0$<br/>so $\hat{\sigma}&gt;\sigma$</div>


tags: ['Mathematics', 'Statistics', 'KVPY', 'KVPY 2013 (SB/SX)']
source_db: MarksBatch2_P1.db
title:
let-n-3-a-list-of-numbers-0-x-1-x-2-x-n-has-mean-and-standard-deviatiion-a-new-list-of-numbers-is-made-as-follows-y-1-0-y-2-x-2-y-n-1-x-n-1-y-n-x-1-x-

slug:
let-n-3-a-list-of-numbers-0-x-1-x-2-x-n-has-mean-and-standard-deviatiion-a-new-list-of-numbers-is-made-as-follows-y-1-0-y-2-x-2-y-n-1-x-n-1-y-n-x-1-x-43281

question:
<div class="question">Let $n \geq 3$. A list of numbers $0 &lt; x_{1} &lt; x_{2} &lt; \ldots &lt; x_{n}$ has mean $\mu$ and standard deviatiion $\sigma$. A new list of numbers is made as follows : $y_{1}=0, y_{2}=x_{2}, \ldots, y_{n-1}=x_{n-1}, y_{n}=x_{1}+x_{n}$. The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Which of the following is necessarily true?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mu=\hat{\mu}, \sigma \leq \hat{\sigma}$</span> </div>

answer:
<div class="solution">$\hat{\mu}=\frac{0+x_{2}+x_{3} \ldots \ldots x_{n-1}+\left(x_{1}+x_{n}\right)}{n}=\mu$<br/>$\hat{\sigma}^{2}=\frac{1}{n} \Sigma y_{i}^{2}-\hat{\mu}^{2}$<br/>$=\frac{1}{n}\left[0^{2}+x_{2}^{2}+x_{3}^{2} \ldots\left(x_{n-1}\right)^{2}+\left(x_{1}+x_{n}\right)^{2}\right]-\mu^{2}$<br/>$\hat{\sigma}^{2}-\sigma^{2}=\frac{2 x_{1} x_{n}}{n}&gt;0$<br/>so $\hat{\sigma}&gt;\sigma$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
let-n-3-a-list-of-numbers-x-1-x-2-x-n-has-mean-and-standard-deviation-a-new-list-of-numbers-y-1-y-2-y-n-is-made-as-follows-y-1-2-x-1-x-2-y-2-2-x-1-x-2-1

slug:
let-n-3-a-list-of-numbers-x-1-x-2-x-n-has-mean-and-standard-deviation-a-new-list-of-numbers-y-1-y-2-y-n-is-made-as-follows-y-1-2-x-1-x-2-y-2-2-x-1-x-2-1-86281

question:
<div class="question">Let $n \geq 3$.A list of numbers $x_{1}, x_{2}, \ldots \ldots \ldots x_{n}$ has mean $\mu$ and standard deviation $\sigma .$ A new list of numbers $y_{1}, y_{2}, \ldots \ldots \ldots . y_{n}$ is made as follows: $y_{1}=\frac{x_{1}+x_{2}}{2}, y 2=\frac{x_{1}+x_{2}}{2}$ and $y_{j}=x$ for $j=3,4, \ldots \ldots . n$. The mean and the standard deviation of the list are $\hat{\mu}$ and $\hat{\sigma}$. Then which of the following is necessarily true?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mu=\hat{\mu}$ and $\sigma \geq \hat{\sigma}$</span> </div>

answer:
<div class="solution">$\mu=\frac{\sum x_{i}}{n} \quad n \geq 3$<br/>$\sigma=\sqrt{\frac{\sum x_{i}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}}$<br/>$\hat{\mu}=\frac{y_{1}+y_{2}+y_{3}+\ldots \ldots+y_{n}}{n}$<br/>$=\frac{\frac{x_{1}+x_{2}}{2}+\frac{x_{1}+x_{2}}{2}+x_{3}+x_{4}+\ldots \ldots \ldots+x_{n}}{n}$<br/>$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots \ldots . .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}=\mu$<br/>$\hat{\sigma}=\sqrt{\frac{\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}}$<br/>$\begin{array}{l}=\sqrt{\frac{\frac{1}{4}\left(2 x_{1}^{2}+2 x_{2}^{2}+4 x_{1} x_{2}\right)+x_{3}^{2}+\ldots \ldots . .+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}} \\ =\sqrt{\frac{\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)+x_{1} x_{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \leq \sqrt{\frac{x_{1}^{2}+x_{2}^{2}+\ldots \ldots+x_{2}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \hat{\sigma} \leq \sigma &amp; \ldots \ldots \ldots(2) \\ \text { by }(1) \text { and }(2)\end{array}$</div>


tags: ['Mathematics', 'Statistics', 'KVPY', 'KVPY 2014 (SB/SX)']
source_db: MarksBatch2_P1.db
title:
let-n-3-a-list-of-numbers-x-1-x-2-x-n-has-mean-and-standard-deviation-a-new-list-of-numbers-y-1-y-2-y-n-is-made-as-follows-y-1-2-x-1-x-2-y-2-2-x-1-x-2

slug:
let-n-3-a-list-of-numbers-x-1-x-2-x-n-has-mean-and-standard-deviation-a-new-list-of-numbers-y-1-y-2-y-n-is-made-as-follows-y-1-2-x-1-x-2-y-2-2-x-1-x-2-18330

question:
<div class="question">Let $n \geq 3$.A list of numbers $x_{1}, x_{2}, \ldots \ldots \ldots x_{n}$ has mean $\mu$ and standard deviation $\sigma .$ A new list of numbers $y_{1}, y_{2}, \ldots \ldots \ldots . y_{n}$ is made as follows: $y_{1}=\frac{x_{1}+x_{2}}{2}, y 2=\frac{x_{1}+x_{2}}{2}$ and $y_{j}=x$ for $j=3,4, \ldots \ldots . n$. The mean and the standard deviation of the list are $\hat{\mu}$ and $\hat{\sigma}$. Then which of the following is necessarily true?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mu=\hat{\mu}$ and $\sigma \geq \hat{\sigma}$</span> </div>

answer:
<div class="solution">$\mu=\frac{\sum x_{i}}{n} \quad n \geq 3$<br/>$\sigma=\sqrt{\frac{\sum x_{i}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}}$<br/>$\hat{\mu}=\frac{y_{1}+y_{2}+y_{3}+\ldots \ldots+y_{n}}{n}$<br/>$=\frac{\frac{x_{1}+x_{2}}{2}+\frac{x_{1}+x_{2}}{2}+x_{3}+x_{4}+\ldots \ldots \ldots+x_{n}}{n}$<br/>$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots \ldots . .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}=\mu$<br/>$\hat{\sigma}=\sqrt{\frac{\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}}$<br/>$\begin{array}{l}=\sqrt{\frac{\frac{1}{4}\left(2 x_{1}^{2}+2 x_{2}^{2}+4 x_{1} x_{2}\right)+x_{3}^{2}+\ldots \ldots . .+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}} \\ =\sqrt{\frac{\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)+x_{1} x_{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \leq \sqrt{\frac{x_{1}^{2}+x_{2}^{2}+\ldots \ldots+x_{2}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \hat{\sigma} \leq \sigma &amp; \ldots \ldots \ldots(2) \\ \text { by }(1) \text { and }(2)\end{array}$</div>


title: let-n-be-a-positive-integer-and-1-x-n-a-0-a-1-x-a-2-x-2-a-n-x-n-what-is-a-0-a-1-a-2-a-n-equal-to
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
let-n-be-a-positive-integer-and-1-x-n-a-0-a-1-x-a-2-x-2-a-n-x-n-what-is-a-0-a-1-a-2-a-n-equal-to-91148

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

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

answer:
<div class="solution">$(1+\mathrm{x})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{0}+{ }^{\mathrm{n}} \mathrm{C}_{1} \mathrm{x}+{ }^{\mathrm{n}} \mathrm{C}_{2} \mathrm{x}^{2}+\ldots .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}}$<br/>Putting $\mathrm{x}=1$ $2^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{0}+{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}+\ldots \ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}$<br/>$\begin{aligned} \Rightarrow \mathrm{a}_{0}+\mathrm{a}_{1}+\ldots \mathrm{a}_{\mathrm{n}} &amp;=2^{\mathrm{n}} \\ <br/>&amp;\left[\text { here }{ }^{\mathrm{n}} \mathrm{C}_{0}=\mathrm{a}_{0},{ }^{\mathrm{n}} \mathrm{C}_{1}=\mathrm{a}_{1}, \ldots .,{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}}\right] \end{aligned}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-n-be-the-set-of-integers-a-relation-r-on-n-is-defined-as-r-x-y-xy-0-x-y-n-then-which-one-of-the-following-is-correct

slug:
let-n-be-the-set-of-integers-a-relation-r-on-n-is-defined-as-r-x-y-xy-0-x-y-n-then-which-one-of-the-following-is-correct-21126

question:
<div class="question">Let $\mathrm{N}$ be the set of integers. A relation $\mathrm{R}$ on $\mathrm{N}$ is defined as $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{xy}&gt;0, \mathrm{x}, \mathrm{y}, \in \mathrm{N}\}$. Then, which one of the following is correct?</div>

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

answer:
<div class="solution">A relation is equivalent if it is <br/>(i) Reflexive<br/>(ii)Symmetric and<br/>(iii) Transitive<br/>We check for the same, one by -one $\mathrm{x}, \mathrm{y} \in \mathrm{N} \Rightarrow \mathrm{x}&gt;0, \mathrm{y}&gt;0$<br/>$\mathrm{R}=\{(\mathrm{x}, \mathrm{y}) \mid \mathrm{xy}&gt;0, \mathrm{x}, \mathrm{y}, \in \mathrm{N}\}$<br/>(i) Reflexive $\mathrm{x}, \mathrm{y} \in \mathrm{N}$<br/>$\therefore \quad x, x, \in N \Rightarrow x^{2}&gt;0$<br/>$\therefore \quad \mathrm{R}$ is reflexive<br/>(ii) Symmetric $\mathrm{x}, \mathrm{y} \in \mathrm{N}$<br/>and $x y&gt;0 \Rightarrow y x&gt;0$<br/>$\mathrm{R}$ is also symmetric<br/>(iii) Transitive<br/>$\therefore \quad x, y, z \in N$<br/>$\Rightarrow x y&gt;0, y z&gt;0 \quad \Rightarrow x z&gt;0$<br/>$\therefore \quad \mathrm{R}$ is also transtive. Conclusion: $\mathrm{R}$ is an equivalence relation.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-n-be-the-set-of-natural-numbers-and-f-n-n-be-a-function-given-by-f-x-x-1-x-n-which-one-of-the-following-is-correct

slug:
let-n-be-the-set-of-natural-numbers-and-f-n-n-be-a-function-given-by-f-x-x-1-x-n-which-one-of-the-following-is-correct-94140

question:
<div class="question">Let $\mathrm{N}$ be the set of natural numbers and $\mathrm{f}: \mathrm{N} \rightarrow \mathrm{N}$, be a function given by $\mathrm{f}(\mathrm{x})=\mathrm{x}+1, \mathrm{x} \in \mathrm{N}$. <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">$\mathrm{f}$ is one-one and onto</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}$ is one-one but not onto</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{f}$ is only onto</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}$ is neither one-one nor onto</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}$ is one-one but not onto</span> </div>

answer:
<div class="solution">To show $\mathrm{f}$ is one-one. Let $f(x)=f(y)$ (To show: $x=y)$ $\Rightarrow \quad x+1=y+1$<br/>$\Rightarrow \quad x=y$<br/>Hence, $\mathrm{f}$ is one-one, Now, ' $\mathrm{f}$ ' is not onto because every element of codomain does not have it's pre-image in domain.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/8tNKsb8OjhTXV63tiBLYRvE8943NjzGSrCAyuRoIMhU.original.fullsize.png"/></div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-n-denote-the-set-of-all-nonnegative-integers-and-z-denote-the-set-of-all-integers-the-function-f-z-n-given-by-f-x-x-is

slug:
let-n-denote-the-set-of-all-nonnegative-integers-and-z-denote-the-set-of-all-integers-the-function-f-z-n-given-by-f-x-x-is-45583

question:
<div class="question">Let $N$ denote the set of all non-negative integers and $Z$ denote the set of all integers. The function $f: Z \rightarrow N$ given by<br/>$f(x)=|x|$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">One-one but not onto</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Onto but not one-one</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both one-one and onto</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither one-one nor onto</span> </li> </ul>

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

answer:
<div class="solution">$f: \mathrm{Z} \rightarrow \mathrm{N}$ and $f(\mathrm{x})=|\mathrm{x}|$<br/>When we draw a parallel line to $\mathrm{x}$ -axis. It cuts the curve into more than one point. Therefore, $f(\mathrm{x})=|\mathrm{x}|$ is not one-one. but onto<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/M0HhSrdul5ekVMmVaXcWjk0gRV7-avuDQ2Ta1AWUqbg.original.fullsize.png"/></div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-n-denote-the-setof-natural-numbers-and-a-n-2-n-n-and-b-n-3-n-n-which-one-of-the-following-is-incorrect

slug:
let-n-denote-the-setof-natural-numbers-and-a-n-2-n-n-and-b-n-3-n-n-which-one-of-the-following-is-incorrect-64669

question:
<div class="question">Let $N$ denote the setof natural numbers and $A=\left\{\mathrm{n}^{2}: \mathrm{n} \in N\right\}$ and $\mathrm{B}=\left\{\mathrm{n}^{3}: \mathrm{n} \in N\right\} .$ Which one of the  following is incorrect?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A \cup B=N$</span> </div>

answer:
<div class="solution">Let $\mathrm{A}=\left\{n^{2}: n \in \mathrm{N}\right\}$ and $\mathrm{B}=\left\{n^{3}: n \in \mathrm{N}\right\}$<br/>$\mathrm{A}=\{1,4,9,16, \ldots . .\}$<br/>and $\mathrm{B}=\{1,8,27,64, \ldots \ldots\}$<br/>Now, $\mathrm{A} \cap \mathrm{B}=\{1\}$ which is a finite set.<br/>Also, $\mathrm{A} \cup \mathrm{B}=\{1,4,8,9,27, \ldots . .\}$<br/>So, complement of $\mathrm{A} \cup \mathrm{B}$ is infinite set. Hence, $\mathrm{A} \cup \mathrm{B} \neq \mathrm{N}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15</span> </div>
source_db: MarksBatch2_P1.db
title:
let-o-0-0-0-p-3-4-5-q-m-n-r-and-r-1-1-1-be-the-vertices-of-a-parallelogram-taken-in-order-what-is-the-value-of-m-n-r

slug:
let-o-0-0-0-p-3-4-5-q-m-n-r-and-r-1-1-1-be-the-vertices-of-a-parallelogram-taken-in-order-what-is-the-value-of-m-n-r-91403

question:
<div class="question">Let $O(0,0,0), P(3,4,5), Q(m, n, r)$ and $R(1,1,1)$ be the vertices of a parallelogram taken in order. What is the value of $m+n+r ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">More than 15</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/pO3WwB4k-vVJQFyGVFyUQ8TSa5fxAsoEjEHcxBmTHwU.original.fullsize.png"/><br/><br/>OPQR is a parallelogram and $\mathrm{OQ}, \mathrm{PR}$ are the diagonals of parallelogram. We know that in a parallelogram, diagonals bisect each other.<br/>$\therefore\left(\frac{0+m}{2}, \frac{0+n}{2}, \frac{0+r}{2}\right) \equiv\left(\frac{1+3}{2}, \frac{1+4}{2}, \frac{1+5}{2}\right)$<br/>$\therefore \quad \frac{m}{2}=\frac{4}{2}, \frac{n}{2}=\frac{5}{2}, \frac{r}{2}=\frac{6}{2}$<br/>$\Rightarrow \quad m=4, n=5, r=6$<br/>Hence, $m+n+r=4+5+6=15$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-o-be-the-origin-and-p-q-r-be-the-points-such-that-po-oq-qo-or-then-which-one-of-the-following-is-correct

slug:
let-o-be-the-origin-and-p-q-r-be-the-points-such-that-po-oq-qo-or-then-which-one-of-the-following-is-correct-11091

question:
<div class="question">Let $\mathrm{O}$ be the origin and $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ be the points such<br/>that $\overrightarrow{\mathrm{PO}}+\overrightarrow{\mathrm{OQ}}=\overrightarrow{\mathrm{QO}}+\overrightarrow{\mathrm{OR}}$. Then which one of the following<br/>is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}, \mathrm{Q}, \mathrm{R}$ are collinear</span> </div>

answer:
<div class="solution">Consider $\overrightarrow{\mathrm{PO}}+\overrightarrow{\mathrm{OQ}}=\overrightarrow{\mathrm{QO}}+\overrightarrow{\mathrm{OR}}$<br/>$\Rightarrow \overrightarrow{\mathrm{OQ}}-\overrightarrow{\mathrm{QO}}=-\overrightarrow{\mathrm{PO}}+\overrightarrow{\mathrm{OR}}$<br/>$\Rightarrow \overrightarrow{\mathrm{OQ}}+\overrightarrow{\mathrm{OQ}}=\overrightarrow{\mathrm{OR}}+\overrightarrow{\mathrm{OP}}$<br/>$\Rightarrow \overrightarrow{\mathrm{OQ}}=\frac{1}{2}(\overrightarrow{\mathrm{OP}}+\overrightarrow{\mathrm{OR}})$<br/>Hence Q is the mid-point of P and $\mathrm{R}$. $\therefore \mathrm{P}, \mathrm{Q}, \mathrm{R}$ are collinear.</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">i</span> </div>
source_db: MarksBatch2_P1.db
title:
let-o-be-the-origin-and-point-a-be-represented-byz-if-oa-is-rotated-through-an-angle-2-in-the-anticlockwise-direction-keeping-the-length-of-oa-same-th

slug:
let-o-be-the-origin-and-point-a-be-represented-byz-if-oa-is-rotated-through-an-angle-2-in-the-anticlockwise-direction-keeping-the-length-of-oa-same-th-53756

question:
<div class="question">Let $\mathrm{O}$ be the origin and point $\mathrm{A}$ be represented byz. If OA is rotated through an angle $\pi / 2$ in the anticlockwise direction keeping the length of OA same, then what represents the new point?</div>

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

answer:
<div class="solution">Let $z=\cos \theta+i \sin \theta$<br/>Now, on rotating through an angle $\frac{\pi}{2}, z$ becomes<br/>$\mathrm{Z}=\cos \left(\frac{\pi}{2}+\theta\right)+\mathrm{i} \sin \left(\frac{\pi}{2}+\theta\right)$<br/>$=-\sin \theta+\mathrm{i} \cos \theta=\mathrm{i}^{2} \sin \theta+\mathrm{i} \cos \theta$<br/>$=\mathrm{i}(\cos \theta+\mathrm{i} \sin \theta)=\mathrm{i} \mathrm{z}$</div>


title: let-p-1-2-3-and-a-relation-on-set-p-is-given-by-the-set-r-1-2-1-3-2-1-1-1-2-2-3-3-2-3-then-r-is
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P2.db
slug:
let-p-1-2-3-and-a-relation-on-set-p-is-given-by-the-set-r-1-2-1-3-2-1-1-1-2-2-3-3-2-3-then-r-is-14966

question:
<div class="question">Let $\mathrm{P}=\{1,2,3\}$ and a relation on set $\mathrm{P}$ is given by the set $\mathrm{R}=\{(1,2),(1,3),(2,1),(1,1),(2,2),(3,3),(2,3)\}$. Then $\mathrm{R}$ is</div>

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

answer:
<div class="solution">Given relation $\mathrm{R}=\{(1,2),(1,3),(2,1),(1,1),(2,2),(3,3),(2,3)\}$ is<br/>reflexive and transitive only but not symmetric $\because(3,1)$ and $(3,2) \notin \mathrm{R}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-p-and-q-be-the-position-vectors-of-the-points-p-and-q-respectively-with-respect-to-origin-o-the-points-r-and-s-divide-pq-internally-and-externally

slug:
let-p-and-q-be-the-position-vectors-of-the-points-p-and-q-respectively-with-respect-to-origin-o-the-points-r-and-s-divide-pq-internally-and-externally-82370

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9 p^{2}=4 q^{2}$</span> </div>

answer:
<div class="solution">R divides PQ internally in ratio $2: 3$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/tovtg_pLUptZWeNhtVDZhv6WqYAHKu-38gWoB3AUzdk.original.fullsize.png"/><br/><br/>$\therefore \overrightarrow{\mathrm{OR}}=\frac{2 \overrightarrow{\mathrm{q}}+3 \overrightarrow{\mathrm{p}}}{5}$ ...(1)<br/>S divides $\mathrm{PQ}$ externally in ratio $2: 3$<br/>$\overrightarrow{\mathrm{OS}}=\frac{2 \overrightarrow{\mathrm{q}}-3 \overrightarrow{\mathrm{p}}}{2-3}=3 \overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}}$ ...(2)<br/>Given, $\overrightarrow{\mathrm{OR}}$ and $\overrightarrow{\mathrm{OS}}$ are perpendicular.<br/>$\therefore\left(\frac{3 \overrightarrow{\mathrm{p}}+2 \overrightarrow{\mathrm{q}}}{5}\right)(3 \overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}})=0$<br/>$\Rightarrow 9 p^{2}-4 q^{2}=0 \Rightarrow 9 p^{2}=4 q^{2}$</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
let-p-be-a-point-on-the-parabola-y-2-12-x-and-n-be-the-foot-of-the-perpendicular-drawn-from-p-on-the-axis-of-the-parabola-a-line-is-now-drawn-through--1

slug:
let-p-be-a-point-on-the-parabola-y-2-12-x-and-n-be-the-foot-of-the-perpendicular-drawn-from-p-on-the-axis-of-the-parabola-a-line-is-now-drawn-through-1-97725

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi></math> be a point on the parabola, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math> be the foot of the perpendicular drawn from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> , on the axis of the parabola. A line is now drawn through the mid-point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>N</mi></math>, parallel to its axis which meets the parabola at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math>. If the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo></math>intercept of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>NQ</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>,</mo></math> then :</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>P</mi><mi>N</mi><mo>=</mo><mn>4</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"><mi>M</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></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"><mi>M</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></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">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>N</mi><mo>=</mo><mn>3</mn></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>M</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Comparing the equation of the parabola, with standard equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>a</mi><mi>x</mi></math>, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn></math>.</p><p>Figure is drawn on the basis of the given information.</p><p><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_main/8436dc67-a9c2-48a5-ba9c-13ab46ac02f3-Q6JEEMAINSEP3SFT1(2).png" style="width: 307px; height: 264px;"/></p><p>Parametric form of parabola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn><mi>a</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>P</mi><mo>=</mo><mfenced><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mn>6</mn><mi>t</mi></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfenced><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mn>0</mn></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mfenced><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mn>3</mn><mi>t</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>Q</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>,</mo><mn>3</mn><mi>t</mi></mrow></mfenced></math></p><p>Equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mi>Q</mi></math>:  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>0</mn><mo>=</mo><mfenced><mfrac><mrow><mn>3</mn><mi>t</mi><mo>-</mo><mn>0</mn></mrow><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mstyle></mfrac></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></math></p><p>i.e., <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn></mrow><mrow><mn>3</mn><mi>t</mi></mrow></mfrac><mo>·</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo>⇒</mo><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn></mrow><mrow><mn>3</mn><mi>t</mi></mrow></mfrac><mo>·</mo><mi>x</mi><mo>+</mo><mn>4</mn><mi>t</mi><mspace linebreak="newline"></mspace></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>4</mn><mi>t</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>⇒</mo><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>N</mi><mo>=</mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>Q</mi><mo>=</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mfrac><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mfrac><mrow><mn>9</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p><p> </p></div>


tags: ['Mathematics', 'Parabola', 'JEE Main', 'JEE Main 2020 (03 Sep Shift 1)']
source_db: MarksBatch2_P2.db
title:
let-p-be-a-point-on-the-parabola-y-2-12-x-and-n-be-the-foot-of-the-perpendicular-drawn-from-p-on-the-axis-of-the-parabola-a-line-is-now-drawn-through-

slug:
let-p-be-a-point-on-the-parabola-y-2-12-x-and-n-be-the-foot-of-the-perpendicular-drawn-from-p-on-the-axis-of-the-parabola-a-line-is-now-drawn-through-64069

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi></math> be a point on the parabola, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math> be the foot of the perpendicular drawn from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> , on the axis of the parabola. A line is now drawn through the mid-point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>N</mi></math>, parallel to its axis which meets the parabola at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math>. If the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo></math>intercept of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>NQ</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>,</mo></math> then :</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>P</mi><mi>N</mi><mo>=</mo><mn>4</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"><mi>M</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></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"><mi>M</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></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">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>N</mi><mo>=</mo><mn>3</mn></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>M</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Comparing the equation of the parabola, with standard equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>a</mi><mi>x</mi></math>, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn></math>.</p><p>Figure is drawn on the basis of the given information.</p><p><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_main/8436dc67-a9c2-48a5-ba9c-13ab46ac02f3-Q6JEEMAINSEP3SFT1(2).png" style="width: 307px; height: 264px;"/></p><p>Parametric form of parabola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn><mi>a</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>P</mi><mo>=</mo><mfenced><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mn>6</mn><mi>t</mi></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfenced><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mn>0</mn></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mfenced><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mn>3</mn><mi>t</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>Q</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>,</mo><mn>3</mn><mi>t</mi></mrow></mfenced></math></p><p>Equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mi>Q</mi></math>:  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>0</mn><mo>=</mo><mfenced><mfrac><mrow><mn>3</mn><mi>t</mi><mo>-</mo><mn>0</mn></mrow><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mstyle></mfrac></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></math></p><p>i.e., <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn></mrow><mrow><mn>3</mn><mi>t</mi></mrow></mfrac><mo>·</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo>⇒</mo><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn></mrow><mrow><mn>3</mn><mi>t</mi></mrow></mfrac><mo>·</mo><mi>x</mi><mo>+</mo><mn>4</mn><mi>t</mi><mspace linebreak="newline"></mspace></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>4</mn><mi>t</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>⇒</mo><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>N</mi><mo>=</mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>Q</mi><mo>=</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mfrac><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mfrac><mrow><mn>9</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p><p> </p></div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P2.db
title:
let-p-p-1-p-2-p-3-p-4-q-q-1-q-2-q-3-q-4-and-if-r-r-1-r-2-r-3-r-4-10-p-i-q-f-r-k-i-j-k-10-how-many-elements-does-s-10-have

slug:
let-p-p-1-p-2-p-3-p-4-q-q-1-q-2-q-3-q-4-and-if-r-r-1-r-2-r-3-r-4-10-p-i-q-f-r-k-i-j-k-10-how-many-elements-does-s-10-have-11655

question:
<div class="question">Let $\mathrm{P}=\left\{\mathrm{p}_{1}, \mathrm{p}_{2}, \mathrm{p}_{3}, \mathrm{p}_{4}\right\}$<br/>$Q=\left\{q_{1}, q_{2}, q_{3}, q_{4}\right\}$ and<br/>If $\begin{aligned} \mathrm{R}=\left\{\mathrm{r}_{1}, \mathrm{r}_{2}, \mathrm{r}_{3}, \mathrm{r}_{4}\right\} \\ 10 &amp;=\left\{\left(\mathrm{p}_{i}, \mathrm{q}_{\mathrm{f}}, \mathrm{r}_{\mathrm{k}}\right): \mathrm{i}+\mathrm{j}+\mathrm{k}=10\right\} \end{aligned}$<br/>how many elements does $\mathrm{S}_{10}$ have?</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">6</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Given that $\mathrm{P}=\left\{\mathrm{p}_{1}, \mathrm{p}_{2}, \mathrm{p}_{3}, \mathrm{p}_{4}\right\}$<br/>$Q=\left\{q_{1}, q_{2}, q_{3}, q_{4}\right\}$<br/>and $\mathrm{R}=\left\{\mathrm{r}_{1}, \mathrm{r}_{2}, \mathrm{r}_{3}, \mathrm{r}_{4}\right\}$<br/>$\mathrm{S}_{10}=\left\{\mathrm{p}_{2}, \mathrm{q}_{4}, \mathrm{r}_{4}\right),\left(\mathrm{p}_{3}, \mathrm{q}_{3}, \mathrm{r}_{4}\right),\left(\mathrm{p}_{3}, \mathrm{q}_{4}, \mathrm{r}_{3}\right)$<br/>$\left(\mathrm{p}_{4}, \mathrm{q}_{2}, \mathrm{r}_{4}\right)$<br/>$\left.\left(\mathrm{p}_{4}, \mathrm{q}_{3}, \mathrm{r}_{3}\right),\left(\mathrm{p}_{4}, \mathrm{q}_{4}, \mathrm{r}_{2}\right)\right\}$<br/>Total number of elements in $\mathrm{S}_{10}$ are 6 .</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-p-q-r-and-s-be-respectively-the-magnitudes-of-the-vectors-3-i-2-j-2-i-2-j-k-4-i-j-k-2-i-2-j-3-k-which-one-of-the-following-is-correct

slug:
let-p-q-r-and-s-be-respectively-the-magnitudes-of-the-vectors-3-i-2-j-2-i-2-j-k-4-i-j-k-2-i-2-j-3-k-which-one-of-the-following-is-correct-65467

question:
<div class="question">Let $p, q, r$ and $s$ be respectively the magnitudes of the vectors<br/>$3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}, 2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$. Which<br/>one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad r&gt;s&gt;q&gt;p$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$s&gt;r&gt;p&gt;q$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad r&gt;s&gt;p&gt;q$</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">$\quad s&gt;r&gt;q&gt;p$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad r&gt;s&gt;p&gt;q$</span> </div>

answer:
<div class="solution">Let $p=$ Magnitude of $3 \hat{i}-2 \hat{j}=\sqrt{9+4}=\sqrt{13}$<br/>$q=$ Magnitude of $2 \hat{i}+2 \hat{j}+\hat{k}=\sqrt{4+4+1}=3$<br/>$r=$ Magnitude of $4 \hat{i}-\hat{j}+\hat{k}=\sqrt{16+1+1}=\sqrt{18}=3 \sqrt{2}$<br/>$s=$ Magnitude of $2 \hat{i}+2 \hat{j}+3 \hat{k}=\sqrt{4+4+9}=\sqrt{17}$<br/>$\therefore \quad r&gt;s&gt;p&gt;q$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-p-q-r-s-be-the-distances-from-origin-of-the-points-2-6-3-4-4-5-and-2-5-respectively-which-one-of-the-following-is-a-whole-number

slug:
let-p-q-r-s-be-the-distances-from-origin-of-the-points-2-6-3-4-4-5-and-2-5-respectively-which-one-of-the-following-is-a-whole-number-65817

question:
<div class="question">Let $\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}$ be the distances from origin of the points $(2,6)$, $(3,4),(4,5)$ and $(-2,5)$ respectively. Which one of the following is a whole number?</div>

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

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

answer:
<div class="solution">Let $\mathrm{A}(2,6) ; \mathrm{B}(3,4) ; \mathrm{C}(4,5)$ and $\mathrm{D}(-2,5)$ are the given points. Let $\mathrm{O}$ be the origin, i.e. $\mathrm{O}(0,0)$<br/>$\mathrm{OA}=\sqrt{(2-0)^{2}+(6-0)^{2}}=\sqrt{40}=2 \sqrt{10}$ units<br/>$O B=\sqrt{(3-0)^{2}+(4-0)^{2}}=\sqrt{9+16}=5$ units<br/>$\mathrm{OC}=\sqrt{(4-0)^{2}+(5-0)^{2}}=\sqrt{16+25}=\sqrt{41}$ units<br/>$O D=\sqrt{(-2-0)^{2}+(5-0)^{2}}=\sqrt{4+25}=\sqrt{29}$ units<br/>So, $\mathrm{q}=\mathrm{OB}=5$ units is the correct answer.</div>


tags: ['Mathematics', 'Mathematical Reasoning', 'JEE Main', 'JEE Main 2022 (28 Jul Shift 2)']
source_db: MarksBatch2_P2.db
title:
let-p-ramesh-listens-to-music-q-ramesh-is-out-of-his-village-r-it-is-sunday-s-it-is-saturday-then-the-statement-ramesh-listens-to-music-only-if-he-is-

slug:
let-p-ramesh-listens-to-music-q-ramesh-is-out-of-his-village-r-it-is-sunday-s-it-is-saturday-then-the-statement-ramesh-listens-to-music-only-if-he-is-90061

question:
<div class="question"><p>Let<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi><mo>:</mo></math> Ramesh listens to music.<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>:</mo></math> Ramesh is out of his village<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>:</mo></math> It is Sunday<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">s</mi><mo mathvariant="bold">:</mo></math> It is Saturday</p><p>Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as</p></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>⇒</mo><mfenced><mrow><mo>(</mo><mo>~</mo><mi>q</mi><mo>)</mo><mo>∧</mo><mfenced><mrow><mi>r</mi><mo>∨</mo><mi>s</mi></mrow></mfenced></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><blockquote><p>Given,<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi><mo>:</mo></math> Ramesh listens to music.<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>:</mo></math> Ramesh is out of his village<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>:</mo></math> It is Sunday<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">s</mi><mo mathvariant="bold">:</mo></math> It is Saturday</p><p>Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as</p><p>Now on breaking each statement and combining them we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≡</mo></math> Ramesh listens to music</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>~</mo><mi>q</mi><mo>≡</mo></math> He is in village.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∨</mo><mi>s</mi><mo>≡</mo></math> Saturday or Sunday</p><p>So, on combining all statement we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>⇒</mo><mfenced><mrow><mfenced><mrow><mo>~</mo><mi>q</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>r</mi><mo>∨</mo><mi>s</mi></mrow></mfenced></mrow></mfenced></math></p></blockquote></div>


tags: ['Mathematics', 'Determinants', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">36</span> </div>
source_db: MarksBatch2_P2.db
title:
let-pleftbeginarrayccc30-20-56-90-140-112-120-60-14endarrayright-and-aleftbeginarrayccc2-7-omega2-1-omega-1-0-omega-omega1endarrayright-where-omegafra

slug:
let-pleftbeginarrayccc30-20-56-90-140-112-120-60-14endarrayright-and-aleftbeginarrayccc2-7-omega2-1-omega-1-0-omega-omega1endarrayright-where-omegafra-41516

question:
<div class="question">Let \( P=\left[\begin{array}{ccc}-30 &amp; 20 &amp; 56 \\ 90 &amp; 140 &amp; 112 \\ 120 &amp; 60 &amp; 14\end{array}\right] \) and \( A=\left[\begin{array}{ccc}2 &amp; 7 &amp; \omega^{2} \\ -1 &amp; -\omega &amp; 1 \\ 0 &amp; -\omega &amp; -\omega+1\end{array}\right] \) where \( \omega=\frac{-1+i \sqrt{3}}{2} \), and \( I_{3} \) be the identity matrix of order \( 3 \). If the determinant of the matrix \( \left(P^{-1} A P-I_{3}\right)^{2} \) is \( \alpha \omega^{2} \), then the value of \( \alpha \) is equal to</div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><msup><mfenced><mrow><msup><mi>P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>P</mi><mo>-</mo><mi>I</mi></mrow></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><msup><mi>P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>P</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>P</mi><mo>+</mo><mi>I</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mn>2</mn></msup><mi>P</mi><mo>-</mo><mn>2</mn><msup><mi>P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>P</mi><mo>+</mo><mi>I</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>M</mi><mo>=</mo><msup><mi>A</mi><mn>2</mn></msup><mi>P</mi><mo>-</mo><mn>2</mn><mi>A</mi><mi>P</mi><mo>+</mo><mi>P</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>A</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo> </mo><mi>A</mi><mo>.</mo><mi>I</mi><mo>+</mo><msup><mi>I</mi><mn>2</mn></msup></mrow></mfenced><mi>P</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>Det</mo><mfenced><mrow><mi>P</mi><mi>M</mi></mrow></mfenced><mo>=</mo><mo>Det</mo><mfenced><mrow><msup><mfenced><mrow><mi>A</mi><mo>-</mo><mi>I</mi></mrow></mfenced><mn>2</mn></msup><mo>×</mo><mi>P</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>Det</mo><mfenced><mrow><mi>P</mi><mi>M</mi></mrow></mfenced><mo>=</mo><mo>=</mo><mo>Det</mo><msup><mfenced><mrow><mi>A</mi><mo>-</mo><mi>I</mi></mrow></mfenced><mn>2</mn></msup><mo>×</mo><mo>Det</mo><mfenced><mi>P</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>Det</mi><mo> </mo><mi>M</mi><mo>=</mo><mo>=</mo><msup><mfenced><mrow><mi>Det</mi><mfenced><mrow><mi>A</mi><mo>-</mo><mi>I</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p><p>Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>-</mo><mi>I</mi><mo>=</mo><mfenced close="]" open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>7</mn></mtd><mtd><msup><mi>w</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mi>w</mi><mo>-</mo><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mi>w</mi></mtd><mtd><mo>-</mo><mi>w</mi></mtd></mtr></mtable></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>Det</mo><mfenced><mrow><mi>A</mi><mo>-</mo><mi>I</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><msup><mi>w</mi><mn>2</mn></msup><mo>+</mo><mi>w</mi><mo>+</mo><mi>w</mi></mrow></mfenced><mo>+</mo><mn>7</mn><mfenced><mrow><mo>-</mo><mi>w</mi></mrow></mfenced><mo>+</mo><msup><mi>w</mi><mn>3</mn></msup><mo>=</mo><mo>-</mo><mn>6</mn><mi>w</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>Det</mo><msup><mfenced><mfenced><mrow><mi>A</mi><mo>-</mo><mi>I</mi></mrow></mfenced></mfenced><mn>2</mn></msup><mo>=</mo><mn>36</mn><msup><mi>w</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>α</mi><mo>=</mo><mn>36</mn></math></p></div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-q-be-the-image-of-the-point-p-2-1-5-in-the-plane-3-x-2-y-2-z-1-0-consider-the-following-1-the-coordinates-of-q-are-4-3-1-2-pq-is-of-length-more-th

slug:
let-q-be-the-image-of-the-point-p-2-1-5-in-the-plane-3-x-2-y-2-z-1-0-consider-the-following-1-the-coordinates-of-q-are-4-3-1-2-pq-is-of-length-more-th-87020

question:
<div class="question">Let $Q$ be the image of the point $P(-2,1,-5)$ in the plane $3 x-2 y+2 z+1=0 .$ Consider the following: <br/> 1 . The coordinates of $\mathrm{Q}$ are $(4,-3,-1)$.<br/> 2 . PQ is of length more than 8 units.<br/> 3 . The point $(1,-1,-3)$ is the mid-point of the line segment $\mathrm{PQ}$ and lies on the given plane. Which of the above statements are correct?</div>

answer:
<div class="solution">Let $Q\left(x_{1}, y_{1}, z_{1}\right)$ be the image of the point $P$. The direction ratios of $P Q$ are $3,-2,2$<br/>The Equation of line $P Q$ is $\frac{x+2}{3}=\frac{y-1}{-2}=\frac{z+5}{2}=r$ Coordinates of any point on the line $P Q$ is $3 r-2,-2 r+$ I and $2 r-5$.<br/>Let $Q(3 r-2,-2 r+1,2 r-5)$ be such a point.<br/>et $L$ be the mid point of $P Q, L=\left(\frac{3 r}{2}-2,1-r, r-5\right)$<br/>Since $L$ lies on the plane $3 x-2 y+2 z+1=0$<br/>So, $3\left(\frac{3 r}{2}-2\right)-2(1-r)+2(r-5)+1=0$<br/>$\Rightarrow \frac{17}{2} r-17=0 \Rightarrow r=2$<br/>So, coordinates of $Q$ are $(3 \times 2-2,-2 \times 2+1,2 \times 2-5)$<br/>$=(4,-3,-1)$<br/>Also the mid point of $P Q$ is $L=\left(\frac{3 \times 2}{2}-2,1-2,2-5\right)$ $\therefore P Q=\sqrt{(-2-4)^{2}+(1+3)^{2}+(-5+1)^{2}}=\sqrt{68}$<br/>$\Rightarrow P Q=2 \sqrt{17}&gt;8$<br/>Option (d) is correct.</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-q-be-the-image-of-the-point-p-2-1-5-in-the-plane-3-x-2-y-2-z-1-0-consider-the-following-1-the-direction-ratios-of-the-line-segment-pq-are-3-2-2-2-

slug:
let-q-be-the-image-of-the-point-p-2-1-5-in-the-plane-3-x-2-y-2-z-1-0-consider-the-following-1-the-direction-ratios-of-the-line-segment-pq-are-3-2-2-2-88974

question:
<div class="question">Let $Q$ be the image of the point $P(-2,1,-5)$ in the plane $3 x-2 y+2 z+1=0 .$ Consider the following:<br/> 1 . The direction ratios of the line segment $\mathrm{PQ}$ are $   &lt; 3,-2,2&gt;$.<br/> 2 . The sum of the squares of direction cosines of the line segment $\mathrm{PQ}$ is unity. Which of the above statements is/are correct?</div>

answer:
<div class="solution">From (i) above, 1 is correct. We know that, Sum of direction cosines of the line segment $\mathrm{PQ}=1$.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-r-be-a-relation-on-the-set-n-of-natural-numbers-defined-by-nrm-n-is-a-factor-of-m-then-which-one-of-the-following-is-correct

slug:
let-r-be-a-relation-on-the-set-n-of-natural-numbers-defined-by-nrm-n-is-a-factor-of-m-then-which-one-of-the-following-is-correct-86621

question:
<div class="question">Let $\mathrm{R}$ be a relation on the set $\mathrm{N}$ of natural numbers defined<br/>by 'nRM $\Leftrightarrow \mathrm{n}$ is a factor of $\mathrm{m}$ '. Then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{R}$ is reflexive, transitive but not symmetric</span> </div>

answer:
<div class="solution">$n R m \Leftrightarrow n$ is a factor of $m$ $\Rightarrow m$ is divisible by $n$. Reflexivity We know that<br/>$n$ is divisible by $n \forall n \in N$<br/>$(n, n) \in R \forall n \in N$<br/>$R$ is reflexive. Symmetric<br/>$n, m \in N$<br/>Let $n=2, m=6$<br/>$m$ is divisible by $n$ but $n$ is not divisible by $m$. Hence $R$ is not symmetric. Transitivity<br/>Let $(n, m) \in R$ and $(m, p) \in R$ then $(n, m) \in R$ and<br/>$(m, p) \in R \Rightarrow(\mathrm{n}, \mathrm{p}) \in \mathrm{R}$<br/>or If $m$ is divisible by $n$ and $p$ is divisible by $m$. Hence $p$ is divisible by $n$.<br/>$(n, p) \in R \forall n, p \in N$<br/>$R$ is transitive relation on $N$. Hence $R$ is reflexive, transitive but not symmetric.<br/>$\int_{0}^{2 a} f(x) d x=2 \int_{0}^{a} f(x) d x$ if $f(2 a-x)=f x$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-r-be-arelation-from-a-1-2-3-4-to-b-1-3-5-suchthat-r-a-b-a-b-where-a-a-an-db-b-what-is-ror-1-equal-to

slug:
let-r-be-arelation-from-a-1-2-3-4-to-b-1-3-5-suchthat-r-a-b-a-b-where-a-a-an-db-b-what-is-ror-1-equal-to-75025

question:
<div class="question">Let $\mathrm{R}$ be arelation from $\mathrm{A}=\{1,2,3,4\}$ to $\mathrm{B}=\{1,3,5\}$ suchthat $\mathrm{R}=[(\mathrm{a}, \mathrm{b}): \mathrm{a}   &lt; \mathrm{b}$, where $\mathrm{a} \in \mathrm{A}$ an $\mathrm{d} \mathrm{b} \in \mathrm{B}]$. What is $\mathrm{RoR}^{-1}$<br/>equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)\}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)\}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\{(3,3),(3,5),(5,3),(5,5)$</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">$\{(3,3),(3,4),(4,5)\}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{R}=\{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)\}$<br/>$\Rightarrow \mathrm{R}^{-1}=\{(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)\} .$<br/>$\Rightarrow \operatorname{RoR}^{-1}=\{(3,3),(3,5),(5,3),(5,5)\}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-r-be-the-relation-defined-on-the-set-of-natural-number-n-as-a-rb-a-b-n-if-a-divides-b-then-which-one-of-the-following-is-correct

slug:
let-r-be-the-relation-defined-on-the-set-of-natural-number-n-as-a-rb-a-b-n-if-a-divides-b-then-which-one-of-the-following-is-correct-30778

question:
<div class="question">Let $R$ be the relation defined on the set of natural number $\mathrm{N}$ as $a \mathrm{Rb} ; \mathrm{a}, \mathrm{b} \in \mathrm{N}$, if a divides $\mathrm{b}$. Then, which one of the following is correct?</div>

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

answer:
<div class="solution">For reflexive: (d) For reflexive:<br/>$\mathrm{aRa} \Rightarrow$ a divides a R is reflexive. For symmetric :<br/>$\mathrm{aRb} \Rightarrow$ a divides $\mathrm{b}$<br/>$\mathrm{bRa} \Rightarrow \mathrm{b}$ divides a<br/>which may not be true $\Rightarrow \quad \mathrm{R}$ is not symmetric For transitive $\mathrm{aRB} \Rightarrow$ a divides $\mathrm{b} \Rightarrow \mathrm{b}=\mathrm{ka}$<br/>$\mathrm{bRc} \Rightarrow \mathrm{b}$ divides $\mathrm{c} \Rightarrow \mathrm{c}=\mathrm{lb}$<br/>Now, $\mathrm{c}=1 \mathrm{ka}$<br/>$\Rightarrow$ a divides $\mathrm{c}$<br/>$\Rightarrow$ a $R c$<br/>$\Rightarrow \quad \mathrm{aRb}, \mathrm{bRc} \Rightarrow \mathrm{c} \mathrm{Ra}$<br/>$\Rightarrow \mathrm{R}$ is transitive.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-r-be-the-set-of-real-numbers-and-let-f-r-r-be-a-function-such-that-f-x-1-x-2-x-2-what-is-the-range-of-f

slug:
let-r-be-the-set-of-real-numbers-and-let-f-r-r-be-a-function-such-that-f-x-1-x-2-x-2-what-is-the-range-of-f-17927

question:
<div class="question">Let $\mathrm{R}$ be the set of real numbers and let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a<br/>function such that $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{2}}$. What is the range of $\mathrm{f}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{R}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{R}-\{1\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[0,1]$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$[0,1)$</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">$[0,1)$</span> </div>

answer:
<div class="solution">$\because \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{2}}$<br/>Since, numerator $   &lt; $ denominator $\mathrm{f}(\mathrm{x}) &lt; 1$ for all values of $\mathrm{x}$ (negative or positive) and $\mathrm{f}(\mathrm{x})=0$ for $\mathrm{x}=0$<br/>So, range of fis $[0,1)$.</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">36</span> </div>
source_db: MarksBatch2_P2.db
title:
let-r-x-x-n-x-is-a-multiple-of-3-and-x-100-s-x-x-n-x-is-a-multiple-of-5-and-x-100-what-is-the-number-of-elements-in-r-s-s-r-2

slug:
let-r-x-x-n-x-is-a-multiple-of-3-and-x-100-s-x-x-n-x-is-a-multiple-of-5-and-x-100-what-is-the-number-of-elements-in-r-s-s-r-2-75089

question:
<div class="question">Let $R=\{x \mid x \in N, x$ is a multiple of 3 and $x \leq 100\}$ $S=\{x \mid x \in N, x$ is a multiple of 5 and $x \leq 100\}$<br/>What is the number of elements in $(R \times S) \cap(S \times R)^{2}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">36</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">20</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">33</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">Let $R=\{x: x \in N, x$ is a multiple of 3 and $x \leq 100\}$<br/>and $S=\{x: x \in N, x$ is a multiple of 5 and $x \leq 100\}$ $R=\{3,6,9,12,15, \ldots, 99\}$<br/>and $S=\{5,10,15, \ldots, 95,100\}$<br/>Now, $(R \times S) \cap(S \times R)=(R \cap S) \times(S \cap R)$<br/>$=(15,30,45,60,75,90) \times(15,30,45,60,75,90)$<br/>Number of elements in $(R \times S) \cap(S \times R)$<br/>$=6 \times 6=36$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">23</span> </div>
source_db: MarksBatch2_P2.db
title:
let-s-be-a-set-of-all-distinct-numbers-of-the-form-q-p-where-p-q-1-2-3-4-5-6-what-is-the-cardinality-of-the-set-s

slug:
let-s-be-a-set-of-all-distinct-numbers-of-the-form-q-p-where-p-q-1-2-3-4-5-6-what-is-the-cardinality-of-the-set-s-53385

question:
<div class="question">Let $S$ be a set of all distinct numbers of the form $\frac{\mathrm{P}}{\mathrm{Q}}$, where<br/>$\mathrm{p}, \mathrm{q} \in\{1,2,3,4,5,6\}$. What is the cardinality of the set $\mathrm{S}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">21</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">23</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">32</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">36</span> </li> </ul>

answer:
<div class="solution">Given that p, q: $(1,2,3,4,5,6$, )<br/>For $\frac{\mathrm{p}}{\mathrm{q}}$ form, when $\mathrm{p}=1, \mathrm{q}=1,2,3,4,5,6$<br/>thus, $\frac{\mathrm{p}}{\mathrm{q}}=1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ and $\frac{1}{6}$<br/>$\mathrm{n}=\left(\frac{\mathrm{p}}{\mathrm{q}}\right)=6$<br/>When $p=2, q=1,3,5$<br/>thus $\frac{\mathrm{p}}{\mathrm{q}}=2, \frac{2}{3}, \frac{2}{5}$ and $\mathrm{n}\left(\frac{\mathrm{p}}{\mathrm{q}}\right)=3$<br/>When $\mathrm{p}=3, \mathrm{q}=1,2,4,5$<br/>thus $\frac{\mathrm{p}}{\mathrm{q}}=3, \frac{3}{2}, \frac{3}{4}, \frac{3}{5}$ and $\mathrm{n}\left(\frac{\mathrm{p}}{\mathrm{q}}\right)=4$<br/>When $\mathrm{p}=4, \mathrm{q}=1,3,5$<br/>thus $\frac{\mathrm{p}}{\mathrm{q}}=4, \frac{4}{3}, \frac{4}{5}$ and $\mathrm{n}\left(\frac{\mathrm{p}}{\mathrm{q}}\right)=3$<br/>When $p=5, q=1,2,3,4,6$<br/>thus $\left(\frac{p}{q}\right)=5, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}$ and $\frac{5}{6}$ and $\mathrm{n}\left(\frac{\mathrm{p}}{q}\right)=5$<br/>When $\mathrm{P}=6, \mathrm{q}=1,5$<br/>thus $\left(\frac{p}{q}\right)=6, \frac{6}{5}$ and $n\left(\frac{p}{q}\right)=2$<br/>Hence, cardinality of the set (s) $=6+3+4+3+5+2=23 .$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-s-be-the-set-of-all-persons-living-in-delhi-we-say-that-x-y-in-s-are-related-if-they-were-born-in-delhi-on-the-same-day-which-one-of-the-following

slug:
let-s-be-the-set-of-all-persons-living-in-delhi-we-say-that-x-y-in-s-are-related-if-they-were-born-in-delhi-on-the-same-day-which-one-of-the-following-35858

question:
<div class="question">Let $S$ be the set of all persons living in Delhi. We say that $x$, y in $\mathrm{S}$ are related if they were born in Delhi on the same day. 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">The relation is an equivalent relation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The relation is not reflexive but it is symmetric and transitive</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The relation is not symmetric but it is reflexive and transitive</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The relation is not transitive but it is reflexive and symmetric</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{S}=\{$ All persons living in Delhi\}<br/>A relationship is said to be equivalence relation if it is reflexive, symmetric and transitive<br/>$\Rightarrow$ Here, $(\mathrm{x}, \mathrm{y}) \in \mathrm{R} \Rightarrow$ reflexive relation<br/>$\Rightarrow$ Since, $x \varepsilon y$ are born on the same day, $x R y \Rightarrow y R x$<br/>So, it is symmeitric relation.<br/>$\Rightarrow \mathrm{x} \mathrm{R} \mathrm{y}, \mathrm{y} \mathrm{R} \mathrm{Z} \Rightarrow \mathrm{xR} \mathrm{Z}$ (Date of births are same)<br/>So, it is transitive relation. So, the given relation is an equivalent relation.</div>


title: let-s-denote-set-of-all-integers-define-a-relation-r-a-r-b-if-ab-0-where-a-b-s-then-r-is
slug: let-s-denote-set-of-all-integers-define-a-relation-r-a-r-b-if-ab-0-where-a-b-s-then-r-is-65745
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $S$ denote set of all integers. Define a relation $R^{\circ}$<br/>$' a R b$ if $a b \geq 0$ where $a, b \in S^{\prime}$. Then $R$ is:</div>

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

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

answer:
<div class="solution">$\mathrm{S}=$ Set of all integers and $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}), \mathrm{a}, \mathrm{b} \in \mathrm{S}$ and $\mathrm{ab} \geq 0\}$<br/>For reflexive $: \mathrm{aRa} \Rightarrow \mathrm{a}, \mathrm{a}=\mathrm{a}^{2} \geq 0$<br/>for all integers a. $\mathrm{a} \geq 0$ For symmetric $: \mathrm{aRb} \Rightarrow \mathrm{ab} \geq 0 \forall \mathrm{a}, \mathrm{b} \in \mathrm{S}$<br/>If $a b \geq 0$, then $b a \geq 0 \Rightarrow b R a$<br/>For transitive :<br/>If $a b \geq 0, b c \geq 0$, then also ac $\geq 0$ Relation $\mathrm{R}$ is reflexive, symmetric and transitive</div>


title: let-s-n-denote-the-sum-of-the-n-terms-of-an-ap-and-3-s-n-s-2-n-what-is-s-3-n-s-2-n-equal-to
slug: let-s-n-denote-the-sum-of-the-n-terms-of-an-ap-and-3-s-n-s-2-n-what-is-s-3-n-s-2-n-equal-to-36756
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $S_{n}$ denote the sum of the $n$ terms of an $\mathrm{AP}$ and $3 \mathrm{~S}_{\mathrm{n}}=\mathrm{S}_{2 \mathrm{n}}$.<br/>What is $\mathrm{S}_{3 \mathrm{n}}=\mathrm{S}_{2 \mathrm{n}}$ equal to?</div>

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

answer:
<div class="solution">From explanation $\frac{\mathrm{S}_{3 \mathrm{n}}}{\mathrm{S}_{2 \mathrm{n}}}=\frac{6 \mathrm{n}^{2} \mathrm{~d}}{3 \mathrm{n}^{2} \mathrm{~d}}=\frac{2}{1}=2: 1$</div>


title: let-s-n-denote-the-sum-of-the-n-terms-of-an-ap-and-3-s-n-s-2-n-what-is-s-3-n-s-n-equal-to
slug: let-s-n-denote-the-sum-of-the-n-terms-of-an-ap-and-3-s-n-s-2-n-what-is-s-3-n-s-n-equal-to-50324
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$6: 1$</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $S_{n}$ denote the sum of the $n$ terms of an $\mathrm{AP}$ and $3 \mathrm{~S}_{\mathrm{n}}=\mathrm{S}_{2 \mathrm{n}}$.<br/>What is $\mathrm{S}_{3 \mathrm{n}}=\mathrm{S}_{\mathrm{n}}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4: 1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$6: 1$</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">$8: 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10: 1$</span> </li> </ul>

answer:
<div class="solution">Given, $\mathrm{S}_{\mathrm{n}}=$ Sum of first n terms of an AP.<br/>$\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{h}}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}]$ or $\mathrm{S}_{2 \mathrm{n}}=\frac{2 \mathrm{n}}{2}[2 \mathrm{a}+(2 \mathrm{n}-1) \mathrm{d}]$<br/>Similarly, $\quad S_{3 n}=\frac{3 n}{2}[3 a+(3 n-1) d]$<br/>According to direction, $3 \mathrm{~S}_{\mathrm{n}}=2 \mathrm{~S}_{2 \mathrm{n}}$<br/>Putting the value of $\mathrm{S}_{\mathrm{n}}$ and $\mathrm{S}_{2 \mathrm{n}}$ in above equation. $3\left(\frac{\mathrm{n}}{2}\right)[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}]=2\left(\frac{\mathrm{n}}{2}\right)[2 \mathrm{a}+(2 \mathrm{n}-1) \mathrm{d}]$<br/>$6 a+3(n-1) d=4 a+2(n-1) d$<br/>$2 \mathrm{a}=\mathrm{d}(\mathrm{n}+1)$<br/>$\therefore \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[\mathrm{~d}(\mathrm{n}+1)+\mathrm{d}(\mathrm{n}-1)]$<br/>$=\frac{\mathrm{n}}{2}[\mathrm{dn}+\mathrm{d}+\mathrm{dn}-\mathrm{d}]$<br/>$=\frac{\mathrm{n}}{2}(2 \mathrm{dn})=\mathrm{n}^{2} \mathrm{~d}$<br/>Now, $\mathrm{S}_{2 \mathrm{n}}=\mathrm{n}[\mathrm{d}(\mathrm{n}+1)+(2 \mathrm{n}-1) \mathrm{d}]=3 \mathrm{n}^{2} \mathrm{~d}$<br/>$\mathrm{S}_{3 \mathrm{n}}=\frac{3 \mathrm{n}}{2}[\mathrm{~d}(\mathrm{n}+1+3 \mathrm{n}-1)]=6 \mathrm{n}^{2} \mathrm{~d}$<br/>Hence, $\frac{\mathrm{S}_{3 \mathrm{n}}}{\mathrm{S}_{\mathrm{n}}}=\frac{6 \mathrm{n}^{2} \mathrm{~d}}{\mathrm{n}^{2} \mathrm{~d}}=\frac{6}{1}=6: 1$</div>


title: let-sin-a-b-1-and-sin-a-b-2-1-where-a-b-0-2-what-is-the-value-of-a
slug: let-sin-a-b-1-and-sin-a-b-2-1-where-a-b-0-2-what-is-the-value-of-a-96550
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $\sin (\mathrm{A}+\mathrm{B})=1$ and $\sin (\mathrm{A}-\mathrm{B})=\frac{1}{2}$ where $\mathrm{A}, \mathrm{B} \in\left[0, \frac{\pi}{2}\right]$.<br/>What is the value of A?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{6}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{8}$</span> </li> </ul>

answer:
<div class="solution">$\begin{array}{l}\sin (\mathrm{A}+\mathrm{B})=1 \\ \Rightarrow \sin (\mathrm{A}+\mathrm{B})=\sin 90^{\circ} \\ \Rightarrow \mathrm{A}+\mathrm{B}=90^{\circ} \\ &amp; \text { Given } \sin (\mathrm{A}-\mathrm{B})=\frac{1}{2}=\sin 30^{\circ} \\ \Rightarrow \mathrm{A}-\mathrm{B}=30^{\circ}\end{array}$<br/>On solving $(1)$ and $(2)$, we get $\mathrm{A}=60$<br/>$\mathrm{B}=30$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P2.db
title:
let-t-r-be-the-r-th-term-of-an-ap-for-r-1-2-3-if-for-some-distinct-positive-integers-m-and-n-we-have-t-m-1-n-and-t-n-1-m-then-what-is-t-mn-equal-to

slug:
let-t-r-be-the-r-th-term-of-an-ap-for-r-1-2-3-if-for-some-distinct-positive-integers-m-and-n-we-have-t-m-1-n-and-t-n-1-m-then-what-is-t-mn-equal-to-75847

question:
<div class="question">Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an AP for $\mathrm{r}=1,2,3, \ldots$. If for some distinct positive integers $\mathrm{m}$ and $\mathrm{n}$ we have $\mathrm{T}_{\mathrm{m}}=1 / \mathrm{n}$ and $\mathrm{T}_{\mathrm{n}}=1 / \mathrm{m}$, then what is $\mathrm{T}_{\mathrm{mn}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(\mathrm{mn})^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{m}^{-1}+\mathrm{nn}^{-1}$</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">0</span> </li> </ul>

answer:
<div class="solution">$\quad \mathrm{T}_{\mathrm{n}}=\frac{1}{m}, \mathrm{~T}_{\mathrm{m}}=\frac{1}{n}$<br/>$\Rightarrow \quad 1^{\text {st }}$ term $=\mathrm{c.d}=\frac{1}{m n}$<br/>$\Rightarrow \quad \mathrm{T}_{m n}=\frac{1}{m n}+\frac{m n-1}{m n}=1$</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-the-coefficient-of-the-middle-term-of-the-binomial-expansion-of-1-x-2-n-be-and-those-of-two-middle-terms-of-the-binomial-expansion-of-1-x-2-n-1-be

slug:
let-the-coefficient-of-the-middle-term-of-the-binomial-expansion-of-1-x-2-n-be-and-those-of-two-middle-terms-of-the-binomial-expansion-of-1-x-2-n-1-be-39902

question:
<div class="question">Let the coefficient of the middle term of the binomial expansion of $(1+x)^{2 n}$ be $\alpha$ and those of two middle terms of the binomial expansion of $(1+\mathrm{x})^{2 n-1}$ be $\beta$ and $\gamma$. Which one of the $\begin{array}{ll}\text { following relations is correct? } &amp; \end{array}$</div>

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

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

answer:
<div class="solution">$\alpha={ }^{2 n} \mathrm{C}_{n}$<br/>$\beta^{-2 n-1} \mathrm{C}_{n}$<br/>$\gamma=^{2 n-1} \mathrm{C}_{n-1}$<br/>$\beta+\gamma={ }^{2 n-1} \mathrm{C}_{n}+{ }^{2 n-1} \mathrm{C}_{n-1}={ }^{2 n} \mathrm{C}_{n}=\alpha$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-the-coordinates-of-the-points-a-b-c-be-1-8-4-0-11-4-and-2-3-1-respectively-what-are-the-coordinates-of-the-point-d-which-is-the-foot-of-the-perpen

slug:
let-the-coordinates-of-the-points-a-b-c-be-1-8-4-0-11-4-and-2-3-1-respectively-what-are-the-coordinates-of-the-point-d-which-is-the-foot-of-the-perpen-48087

question:
<div class="question">Let the coordinates of the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be $(1,8,4)$, $(0,-11,4)$ and $(2,-3,1)$ respectively. What are the coordinates of the point $D$ which is the foot of the perpendicular from $\mathrm{A}$ on $\mathrm{BC}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(4,5,-2)$</span> </div>

answer:
<div class="solution">$\mathrm{A}(1,8,4), \mathrm{B}(0,-11,4), \mathrm{C}(2,-3,1)$<br/>Let $\mathrm{D}=(\alpha, \beta, \gamma)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/mShug5MpO-XdrlyzP4WD09mgLuv3fWv2oi9hlwQOim8.original.fullsize.png"/><br/><br/>Direction ratios of $\mathrm{BC}=\left(\mathrm{x}_{2}-\mathrm{x}_{1}, \mathrm{y}_{2}-\mathrm{y}_{1}, \mathrm{z}_{2}-\mathrm{z}_{1}\right)$<br/>Let, $(\mathrm{a}, \mathrm{b}, \mathrm{c})=(2,8,-3)$<br/>Direction 1 Let (a'. Since, $\mathrm{AD}$ is perpendicular to $\mathrm{BC}$,<br/>e find option (c) is correct.<br/>know, the equation of plane passing through 3</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.6$</span> </div>
source_db: MarksBatch2_P2.db
title:
let-the-correlation-coefficient-between-x-and-y-be-06-random-variables-z-and-w-are-defined-as-z-x-5-and-w-3-y-what-is-the-correlation-coefficient-betw

slug:
let-the-correlation-coefficient-between-x-and-y-be-06-random-variables-z-and-w-are-defined-as-z-x-5-and-w-3-y-what-is-the-correlation-coefficient-betw-71781

question:
<div class="question">Let the correlation coefficient between $\mathrm{X}$ and $\mathrm{Y}$ be $0.6$. Random variables $\mathrm{Z}$ and $\mathrm{W}$ are defined as $\mathrm{Z}=\mathrm{X}+5$ and $\mathrm{W}$<br/>$=\frac{Y}{3}$. What is the correlation coefficient between $\mathrm{Z}$ and $\mathrm{W}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\underline{0.1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.36$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, $\mathrm{r}_{x y}=0.6$<br/>$\mathrm{z}=\mathrm{x}+5 ; w=\frac{y}{3}$<br/>$\Rightarrow \mathrm{b}_{\mathrm{z}}=1 \Rightarrow b_{w y}=\frac{1}{3}$<br/>$b_{x x} b_{v y}=(1)\left(\frac{1}{3}\right)=\frac{1}{3}$<br/>$\Rightarrow \frac{r_{z w}}{r_{x y}}=\frac{1}{3} \Rightarrow r_{z w}=\frac{r_{x y}}{3}=\frac{0.6}{3}=0.2$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2024 (05 Apr Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P2.db
title:
let-the-mean-and-the-standard-deviation-of-the-probability-distribution-x-p-x-3-1-1-k-0-6-1-3-4-1-be-and-respectively-if-2-then-is-equal-to

slug:
let-the-mean-and-the-standard-deviation-of-the-probability-distribution-x-p-x-3-1-1-k-0-6-1-3-4-1-be-and-respectively-if-2-then-is-equal-to-14048

question:
<div class="question">Let the mean and the standard deviation of the probability distribution
<br/>$\begin{array}{|c|c|c|c|c|}
<br/>\hline \mathrm{X}  &amp; \mathrm{\alpha} &amp; \mathrm{1} &amp; 0 &amp; -3 \\
<br/>\hline \mathrm{P}(\mathrm{X}) &amp; \frac{1}{3} &amp; \mathrm{~K} &amp; \frac{1}{6} &amp; \frac{1}{4} \\
<br/>\hline
<br/>\end{array}$
<br/>be $\mu$ and $\sigma$, respectively. If $\sigma-\mu=2$, then $\sigma+\mu$ is equal to________</div>

answer:
<div class="solution">$\begin{aligned} &amp; \frac{1}{3}+\mathrm{k}+\frac{1}{6}+\frac{1}{4}=1 \quad \Rightarrow \mathrm{k}=\frac{1}{4} \\ &amp; \mu=\frac{\alpha}{3}+\frac{1}{4}-\frac{3}{4} \\ &amp; \mu=\frac{\alpha}{3}-\frac{1}{2} \\ &amp; \sigma=\sqrt{\left(\alpha^2 \frac{1}{3}+\frac{1}{4}+9 \frac{1}{4}\right)-\left(\frac{\alpha}{3}-\frac{1}{2}\right)^2} \\ &amp; \sigma=\sqrt{\frac{2 \alpha^2}{9}+\frac{\alpha}{3}+\frac{9}{4}} \\ &amp; \sigma=\mu+2 \\ &amp; \sigma^2=(\mu+2)^2 \Rightarrow \frac{2 \alpha^2}{9}+\frac{\alpha}{3}+\frac{9}{4}=\frac{\alpha^2}{9}+\frac{9}{4}+\alpha \\ &amp; \frac{\alpha^2}{9}-\frac{2 \alpha}{3}=0 \\ &amp; \alpha=0,(\text { reject }) \text { or } \alpha=6 \\ &amp; (\because \mathrm{x}=0 \text { is already given) } \\ &amp; \Rightarrow \sigma+\mu=2 \mu+2 \\ &amp; =5\end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
let-the-mean-of-50-observations-is-15-and-the-standard-deviation-is-2-however-one-observation-was-wrongly-recorded-the-sum-of-the-correct-and-incorrec-1

slug:
let-the-mean-of-50-observations-is-15-and-the-standard-deviation-is-2-however-one-observation-was-wrongly-recorded-the-sum-of-the-correct-and-incorrec-1-64372

question:
<div class="question">Let the mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> and the standard deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>70</mn></math>. If the mean of the correct set of observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math>, then the variance of the correct set is equal to</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>10</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>36</mn></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"><mn>43</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></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"><mn>43</mn></math></span> </div>

answer:
<div class="solution"><p>By using the formula of mean we get, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>50</mn></mfrac><mo>=</mo><mn>15</mn><mo>⇒</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>750</mn></math></p><p>Now variance<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>-</mo><msup><mn>15</mn><mn>2</mn></msup><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup><mo>⇒</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>11450</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mfenced><mi>new</mi></mfenced></munder><mo>=</mo><mn>50</mn><mo>×</mo><mn>16</mn><mo>=</mo><mn>800</mn></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mfenced><mi>new</mi></mfenced></munder><mo>-</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>800</mn><mo>-</mo><mn>750</mn><mo>=</mo><mn>50</mn></math></p><p>Hence, if wrong observation was <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> then corrected one is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>50</mn></math>.<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow></mfenced><mo>=</mo><mn>70</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>10</mn></math></p><p>Therefore, the correct observation <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>60</mn></math>.</p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo>∑</mo><msup><msub><mi>x</mi><mi>i</mi></msub><mn>2</mn></msup></mrow><mfenced><mi>new</mi></mfenced></munder><mo>=</mo><mn>11450</mn><mo>-</mo><msup><mfenced><mn>10</mn></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mn>60</mn></mfenced><mn>2</mn></msup><mo>=</mo><mn>14950</mn></math></p><p>Therefore, variance of the correct set <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>14950</mn><mn>50</mn></mfrac><mo>-</mo><msup><mfenced><mn>16</mn></mfenced><mn>2</mn></msup><mo>=</mo><mn>299</mn><mo>-</mo><mn>256</mn><mo>=</mo><mn>43</mn></math>.</p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2022 (26 Jun Shift 2)']
source_db: MarksBatch2_P2.db
title:
let-the-mean-of-50-observations-is-15-and-the-standard-deviation-is-2-however-one-observation-was-wrongly-recorded-the-sum-of-the-correct-and-incorrec

slug:
let-the-mean-of-50-observations-is-15-and-the-standard-deviation-is-2-however-one-observation-was-wrongly-recorded-the-sum-of-the-correct-and-incorrec-44554

question:
<div class="question">Let the mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> and the standard deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>70</mn></math>. If the mean of the correct set of observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math>, then the variance of the correct set is equal to</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>10</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>36</mn></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"><mn>43</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></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"><mn>43</mn></math></span> </div>

answer:
<div class="solution"><p>By using the formula of mean we get, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>50</mn></mfrac><mo>=</mo><mn>15</mn><mo>⇒</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>750</mn></math></p><p>Now variance<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>50</mn></mfrac><mo>-</mo><msup><mn>15</mn><mn>2</mn></msup><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup><mo>⇒</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>11450</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mfenced><mi>new</mi></mfenced></munder><mo>=</mo><mn>50</mn><mo>×</mo><mn>16</mn><mo>=</mo><mn>800</mn></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mfenced><mi>new</mi></mfenced></munder><mo>-</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>800</mn><mo>-</mo><mn>750</mn><mo>=</mo><mn>50</mn></math></p><p>Hence, if wrong observation was <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> then corrected one is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>50</mn></math>.<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow></mfenced><mo>=</mo><mn>70</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>10</mn></math></p><p>Therefore, the correct observation <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>60</mn></math>.</p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo>∑</mo><msup><msub><mi>x</mi><mi>i</mi></msub><mn>2</mn></msup></mrow><mfenced><mi>new</mi></mfenced></munder><mo>=</mo><mn>11450</mn><mo>-</mo><msup><mfenced><mn>10</mn></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mn>60</mn></mfenced><mn>2</mn></msup><mo>=</mo><mn>14950</mn></math></p><p>Therefore, variance of the correct set <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>14950</mn><mn>50</mn></mfrac><mo>-</mo><msup><mfenced><mn>16</mn></mfenced><mn>2</mn></msup><mo>=</mo><mn>299</mn><mo>-</mo><mn>256</mn><mo>=</mo><mn>43</mn></math>.</p></div>


title: let-the-random-variable-x-follow-b-6-p-if-16-p-x-4-p-x-2-then-what-is-the-value-of-p
slug: let-the-random-variable-x-follow-b-6-p-if-16-p-x-4-p-x-2-then-what-is-the-value-of-p-97217
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let the random variable $\mathrm{X}$ follow $\mathrm{B}(6, \mathrm{p})$. If $16 \mathrm{P}(\mathrm{X}=4)=$ $\mathrm{P}(\mathrm{X}=2)$, then what is the value of $\mathrm{p}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$</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">$\frac{1}{5}$</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">$\frac{1}{6}$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp; \text { X follows } B(6, p)=16 P(x=4)=P(x=2) \\ &amp; \Rightarrow 16{ }^{6} C_{4} \cdot p^{4}(1-p)^{6-4}={ }^{6} C_{2} \cdot p^{2}(1-p)^{6-2} \\ &amp; \Rightarrow 16 \times \frac{6 !}{4 ! 2 !} p^{4}(1-p)^{2}=\frac{6 !}{2 ! 4 !} p^{2}(1-p)^{4} \\ &amp; \Rightarrow 16 p^{2}=(1-p)^{2} \Rightarrow 16 p^{2}=1+p^{2}-2 p \\ &amp; \Rightarrow 15 p^{2}+2 p-1=0 \Rightarrow 15 p^{2}+5 p-3 p-1=0 \\ &amp; \Rightarrow p=\frac{1}{5},-\frac{1}{3} \\ &amp; \text { As } P&gt;0 \Rightarrow P=\frac{1}{5} \end{aligned}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-the-sample-space-consist-of-nonnegative-integers-up-to-50-x-denote-the-numbers-which-are-multiples-of-3-and-y-denote-the-odd-numbers-which-of-the-

slug:
let-the-sample-space-consist-of-nonnegative-integers-up-to-50-x-denote-the-numbers-which-are-multiples-of-3-and-y-denote-the-odd-numbers-which-of-the-61009

question:
<div class="question">Let the sample space consist of non-negative integers up to $50, \mathrm{X}$ denote the numbers which are multiples of 3 and $\mathrm{Y}$ denote the odd numbers. Which of the following is/are correct?<br/>$1.$ $\mathrm{P}(\mathrm{X})=\frac{8}{25}$<br/>$2.$ $\mathrm{P}(\mathrm{Y})=\frac{1}{2}$<br/>Select the correct answer using the code given below.</div>

answer:
<div class="solution">Given $\mathrm{S}=51$ (includes 0 ) $x$ denotes the multiples of 3 upto 51 . $\mathrm{n}(\mathrm{x})=16$<br/>y denotes the odd numbers upto 51 . $\mathrm{n}(\mathrm{y})=25$<br/>$\therefore \mathrm{P}(\mathrm{x})=\frac{16}{51}, \mathrm{P}(\mathrm{y})=\frac{25}{51}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-the-set-of-all-triangles-p-the-set-of-all-isosceles-triangles-q-the-set-of-all-equilateral-triangles-r-the-set-of-all-rightangled-triangles-what-d

slug:
let-the-set-of-all-triangles-p-the-set-of-all-isosceles-triangles-q-the-set-of-all-equilateral-triangles-r-the-set-of-all-rightangled-triangles-what-d-22430

question:
<div class="question">let $\mu=$ the set of all triangles, $\mathrm{P}=$ the set of all isosceles triangles, $\mathrm{Q}=$ the set of all equilateral triangles, $\mathrm{R}=$ the set of all right-angled triangles. What do the sets $\mathrm{P} \cap \mathrm{Q}$ and $\mathrm{R}$ - P represents respectively?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The set of isosceles triangles; the set of non- isosceles right angled triangles</span> </div>

answer:
<div class="solution">As given : $\mu=$ the set of all triangles $\mathrm{P}=$ the set of all isosceles triangles $\mathrm{Q}=$ the set of all equilateral triangles $\mathrm{R}=$ the set of all right angled triangles<br/>$\therefore \quad \mathrm{P} \cap \mathrm{Qrepresents}$ the set of isosceles triangles and $\mathrm{R}-\mathrm{P}$ represents the set of non-isosceles right angled triangles.</div>


title: let-the-slope-of-the-curve-y-cos-1-sin-x-be-tan-then-the-value-of-in-the-interval-0-is
slug: let-the-slope-of-the-curve-y-cos-1-sin-x-be-tan-then-the-value-of-in-the-interval-0-is-68062
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let the slope of the curve $y=\cos ^{-1}(\sin x)$ be $\tan \theta$ : Then the value of $\theta$ in the interval $(0, \pi)$ is</div>

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

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

answer:
<div class="solution">$\begin{aligned} y &amp;=\cos ^{-1}(\sin x) \\ &amp;=\cos ^{-1} \cos \left(\frac{\pi}{2}-x\right) \\ &amp;=\frac{\pi}{2}-x \end{aligned}$<br/>From question, slope of the curve $m=\tan \theta$<br/>$\tan \theta=-1 \Rightarrow \theta=\frac{3 \pi}{4}$</div>


tags: ['Physics', 'Ray Optics', 'JEE Main', 'JEE Main 2011']
source_db: MarksBatch2_P2.db
title:
let-the-x-z-plane-be-the-boundary-between-two-transparent-media-medium-1-in-z-0-has-a-refractive-index-of-2-and-medium-2-with-z-0-has-a-refractive-ind

slug:
let-the-x-z-plane-be-the-boundary-between-two-transparent-media-medium-1-in-z-0-has-a-refractive-index-of-2-and-medium-2-with-z-0-has-a-refractive-ind-51246

question:
<div class="question">Let the $x-z$ plane be the boundary between two transparent media. Medium 1 in $z \geq 0$ has a refractive index of $\sqrt{2}$ and medium 2 with $z &lt; 0$ has a refractive index of $\sqrt{3}$. A ray of light in medium 1 given by the vector $\vec{A}=6 \sqrt{3} \hat{i}+8 \sqrt{3} \hat{j}-10 \hat{k}$ is incident on the plane of separation. The angle of refraction in medium 2 is</div>

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

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

answer:
<div class="solution">Normal to the plane is $z$-axis<br/>$$<br/>\begin{aligned}<br/>&amp; \cos \theta_1=\frac{A_z}{A}=\frac{10}{20}=\frac{1}{2}, \theta_1=60 \\<br/>&amp; \mu_1 \sin \theta_1=\mu_2 \sin \theta_2 \Rightarrow \sqrt{2} \times \frac{\sqrt{3}}{2}=\sqrt{3} \sin \theta_2 \Rightarrow \theta_2=45^{\circ}<br/>\end{aligned}<br/>$$</div>


title: let-u-i-j-v-2-i-5-j-w-4-i-3-j-and-p-u-v-w-which-one-of-the-following-is-correct
slug: let-u-i-j-v-2-i-5-j-w-4-i-3-j-and-p-u-v-w-which-one-of-the-following-is-correct-91904
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $\overrightarrow{\mathbf{u}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}, \overrightarrow{\mathbf{v}}=2 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}, \overrightarrow{\mathbf{w}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{u}}+\overrightarrow{\mathbf{v}}+\overrightarrow{\mathbf{w}}$.<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">$-3 \overrightarrow{\mathbf{u}}+2 \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{p}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \overrightarrow{\mathbf{u}}-2 \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{p}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3 \overrightarrow{\mathbf{u}}+2 \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{p}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3 \overrightarrow{\mathbf{u}}-2 \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{p}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \overrightarrow{\mathbf{u}}+2 \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{p}}$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\text { Since, } \overrightarrow{\mathrm{p}}=\overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{v}}+\overrightarrow{\mathrm{w}} \\<br/>\Rightarrow &amp; \overrightarrow{\mathrm{p}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})+(4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}})=7 \hat{\mathrm{i}}+7 \hat{\mathrm{j}} \\<br/>&amp; \text { Now, } 3 \overrightarrow{\mathrm{u}}+2 \overrightarrow{\mathrm{v}}=3(\hat{\mathrm{i}}-\hat{\mathrm{j}})+2(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}})=\\<br/>&amp; \hat{3}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}=7 \hat{\mathrm{i}}+7 \hat{\mathrm{j}} \\<br/>\Rightarrow &amp; 3 \overrightarrow{\mathrm{u}}+2 \overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{p}}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-u-x-n-1-x-10-be-the-universal-set-n-being-the-set-of-natural-numbers-if-a-1-2-3-4-and-b-2-3-6-10-then-what-is-the-complement-of-a-b

slug:
let-u-x-n-1-x-10-be-the-universal-set-n-being-the-set-of-natural-numbers-if-a-1-2-3-4-and-b-2-3-6-10-then-what-is-the-complement-of-a-b-97894

question:
<div class="question">Let $\mathrm{U}=\{\mathrm{x} \in \mathrm{N}: 1 \leq \mathrm{x} \leq 10\}$ be the universal set, $\mathrm{N}$ being the set of natural numbers. If $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{2,3,6$. $10\}$ then what is the complement of $(\mathrm{A}-\mathrm{B})$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{6,10$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{1,4\}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\{2,3,5,6,7,8,9,10\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\{5,6,7,8,9,10\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\{2,3,5,6,7,8,9,10\}$</span> </div>

answer:
<div class="solution">$\begin{array}{l}\text { A-B }=\{1,4\} \\ &amp; (\mathrm{A}-\mathrm{B})^{\mathrm{c}}=\mathrm{U}-(\mathrm{A}-\mathrm{B})=\{2,3,5,6,7,8,9,10\}\end{array}$</div>


tags: ['Mathematics', 'Statistics']
source_db: MarksBatch2_P2.db
title:
let-x-1-x-2-x-3-x-4-and-x-5-be-the-observations-with-mean-m-and-standard-deviations-then-standard-deviation-of-the-observations-kx-1-kx-2-kx-3-kx-4-an

slug:
let-x-1-x-2-x-3-x-4-and-x-5-be-the-observations-with-mean-m-and-standard-deviations-then-standard-deviation-of-the-observations-kx-1-kx-2-kx-3-kx-4-an-56566

question:
<div class="question">Let $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4$ and $\mathrm{x}_5$ be the observations with mean $\mathrm{m}$ and standard deviations. Then, standard deviation of the observations $\mathrm{kx}_1, \mathrm{kx}_2, \mathrm{kx}_3, \mathrm{kx}_4$ and $\mathrm{kx}_5$ is</div>

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

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

answer:
<div class="solution">Standard deviation is dependent on change of scale. Therefore, the standard deviation of $\mathrm{kx}_1, \mathrm{kx}_2, \mathrm{kx}_3, \mathrm{kx}_4, \mathrm{kx}_5$ is $\mathrm{ks}$.</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
let-x-1-x-2-x-3-x-k-be-k-observations-and-w-i-a-x-i-b-for-i-1-2-3-k-where-a-and-b-are-constants-if-mean-of-x-i-is-52-and-their-standard-deviation-is-1

slug:
let-x-1-x-2-x-3-x-k-be-k-observations-and-w-i-a-x-i-b-for-i-1-2-3-k-where-a-and-b-are-constants-if-mean-of-x-i-is-52-and-their-standard-deviation-is-1-79607

question:
<div class="question">Let <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><msub><mi>x</mi><mn>3</mn></msub><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mi>k</mi></msub></math> be <math><mi>k</mi></math> observations and <math><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>a</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mi>b</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1,</mn><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are constants. If mean of <math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is <math><mn>52</mn></math> and their standard deviation is <math><mn>12</mn></math> and mean of <math><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is <math><mn>60</mn></math> and their standard deviation is <math><mn>15</mn></math>, then the value of <math><mi>a</mi></math> and <math><mi>b</mi></math> should be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mi>a</mi><mo>=</mo><mn>1.25</mn><mo>,</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>5</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>a</mi><mo>=</mo><mo>-</mo><mn>1.25</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>5</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mi>a</mi><mo>=</mo><mn>2.5</mn><mo>,</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>5</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mi>a</mi><mo>=</mo><mn>2.5</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>5</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mi>a</mi><mo>=</mo><mn>1.25</mn><mo>,</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mi>i</mi></msub><mo>=</mo><mi>a</mi><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi></math><br/><math><mo>⇒</mo><mover accent="false"><mrow><mi>w</mi></mrow><mo>¯</mo></mover><mo>=</mo><mi>a</mi><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover><mo>+</mo><mi>b</mi></math><br/><math><mo>⇒</mo><mn>60</mn><mo>=</mo><mi>a</mi><mi mathvariant="normal"></mi><mfenced separators="|"><mrow><mn>52</mn></mrow></mfenced><mo>+</mo><mi>b</mi></math> …. (i)<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo> </mo><mi>of</mi><mo> </mo><mi>w</mi><mo>=</mo><mfenced close="|" open="|" separators="|"><mi>a</mi></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo> </mo><mi>of</mi><mo> </mo><msub><mi>x</mi><mi>i</mi></msub></mrow></mfenced></math><br/><math><mn>15</mn><mo>=</mo><mfenced close="|" open="|" separators="|"><mrow><mi>a</mi></mrow></mfenced><mn>1</mn><mn>2</mn></math><br/><math><mo>⇒</mo><mi>a</mi><mo>=</mo><mo>±</mo><mn>1.25</mn></math> …. (ii)</p><p>From (i) and (ii), we get,<br/>If <math><mi>a</mi><mo>=</mo><mn>1.25</mn><mo>⇒</mo><mn>60</mn><mo>=</mo><mn>65</mn><mo>+</mo><mi>b</mi><mo>⇒</mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>5</mn></math><br/>If <math><mi>a</mi><mo>=</mo><mo>-</mo><mn>1.25</mn><mo>⇒</mo><mn>60</mn><mo>=</mo><mo>-</mo><mn>65</mn><mo>+</mo><mi>b</mi><mo>⇒</mo><mi>b</mi><mo>=</mo><mn>125</mn></math></p></div>


tags: ['Mathematics', 'Statistics']
source_db: MarksBatch2_P2.db
title:
let-x-1-x-2-x-n-be-n-observations-let-w-i-l-x-i-k-for-i-1-2-n-where-l-and-k-are-constants-if-the-mean-of-x-i-sis-48-and-their-standard-deviation-is-12

slug:
let-x-1-x-2-x-n-be-n-observations-let-w-i-l-x-i-k-for-i-1-2-n-where-l-and-k-are-constants-if-the-mean-of-x-i-sis-48-and-their-standard-deviation-is-12-81855

question:
<div class="question">Let $x_1, x_2, \ldots x_n$ be $n$ observations. Let $w_i=l x_i+k$ for $i=1$, $2, \ldots, n$, where $l$ and $k$ are constants. If the mean of $x_i^{\prime} \operatorname{sis} 48$ and their standard deviation is 12 , the mean of $w_i$ 's is 55 and standard deviation of $w_i^{\prime}$ 's is 15 , then the value of $l$ and $k$ should be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$l=1.25, k=-5 &amp; \text$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$l=-1.25, k=5\end{array}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$l=2.5, k=-5$ <br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$l=2.5, k=5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$l=1.25, k=-5 &amp; \text$<br/></span> </div>

answer:
<div class="solution">Since, $w_i=l x_l+k, \bar{x}_i=48, \sigma x_i=12, w_i=55$ and $\sigma w_i=15$<br/>So, $\bar{w}_i=l \bar{x}_i+k$<br/>$\Rightarrow 55=48 l+k$<br/>Now, $\sigma \omega_i=l \sigma x_i \Rightarrow l=\frac{15}{2}=1.25$<br/>After putting the value of 1 in equation (i), we get $k=55-1.25 \times 48=-5$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-and-y-be-two-nonempty-sets-and-let-r-1-and-r-2-be-two-relationsfrom-x-into-y-then-which-one-of-the-following-is-correct

slug:
let-x-and-y-be-two-nonempty-sets-and-let-r-1-and-r-2-be-two-relationsfrom-x-into-y-then-which-one-of-the-following-is-correct-78153

question:
<div class="question">Let $X$ and $Y$ be two non-empty sets and let $R_{1}$ and $R_{2}$ be two relationsfrom $\mathrm{X}$ into $\mathrm{Y}$. Then, which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\mathrm{R}_{1} \cap \mathrm{R}_{2}\right)^{-1}=\mathrm{R}_{1}^{-1} \cup \mathrm{R}_{2}^{-1}$</span> </div>

answer:
<div class="solution">The correct relations as per De Morgan's theorem is $\left(\mathrm{R}_{1} \cap \mathrm{R}_{2}\right)^{-1}=\mathrm{R}_{1}^{-1} \cup \mathrm{R}_{2}^{-1}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-x-and-y-be-two-related-variables-the-two-regression-lines-are-given-by-x-y-1-0-and-2-x-y-4-0-the-two-regression-lines-pass-through-the-point

slug:
let-x-and-y-be-two-related-variables-the-two-regression-lines-are-given-by-x-y-1-0-and-2-x-y-4-0-the-two-regression-lines-pass-through-the-point-59106

question:
<div class="question">Let $\mathrm{X}$ and $\mathrm{Y}$ be two related variables. The two regression lines are given by $\mathrm{x}-\mathrm{y}+1=0$ and $2 \mathrm{x}-\mathrm{y}+4=0$. The two regression lines pass through the point : $\quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(-3,-2)$</span> </div>

answer:
<div class="solution">Given regression lines are $x-y+1=0$ and $2 x-y+4$ $=0$ only point $(-3,-2)$ given in option (d) satisfies the both equations. Hence, The two regression lines pass through the point $(-3,-2)$</div>


tags: ['Mathematics', 'Probability']
options: None
correct_option: None
source_db: MarksBatch2_P2.db
title:
let-x-be-a-discrete-random-variable-whose-probability-distribution-is-defined-as-follows-p-x-x-k-x-1-2-kx-0-for-x-1-2-3-4-for-x-5-6-7-otherwise-w-h-er

slug:
let-x-be-a-discrete-random-variable-whose-probability-distribution-is-defined-as-follows-p-x-x-k-x-1-2-kx-0-for-x-1-2-3-4-for-x-5-6-7-otherwise-w-h-er-25494

question:
<div class="question">Let $\mathrm{X}$ be a discrete random variable whose probability distribution is defined as follows.<br/>$$<br/>\mathrm{P}(\mathrm{X}=\mathrm{x})= \begin{cases}\mathrm{k}(\mathrm{x}+1), &amp; \text { for } \mathrm{x}=1,2,3,4 \\ 2 \mathrm{kx}, &amp; \text { for } \mathrm{x}=5,6,7 \\ 0, &amp; \text { otherwise }\end{cases}<br/>$$<br/>where, $\mathrm{k}$ is a constant, calculate<br/>(i) the value of $k$. (ii) $\mathrm{E}(\mathrm{X})$<br/>(iii) standard deviation of X.</div>

answer:
<div class="solution">We have following table<br/>\begin{array}{|l|c|c|c|c|c|c|c|}<br/>\hline \mathbf{X} &amp; \mathbf{1} &amp; \mathbf{2} &amp; \mathbf{3} &amp; \mathbf{4} &amp; \mathbf{5} &amp; \mathbf{6} &amp; \mathbf{7} \\<br/>\hline \mathbf{P}(\mathbf{X}) &amp; 2 \mathrm{k} &amp; 3 \mathrm{k} &amp; 4 \mathrm{k} &amp; 5 \mathrm{k} &amp; 10 \mathrm{k} &amp; 12 \mathrm{k} &amp; 14 \mathrm{k} \\<br/>\mathbf{X P}(\mathbf{X}) &amp; 2 \mathrm{k} &amp; 6 \mathrm{k} &amp; 12 \mathrm{k} &amp; 20 \mathrm{k} &amp; 50 \mathrm{k} &amp; 72 \mathrm{k} &amp; 98 \mathrm{k} \\<br/>\mathbf{X} \mathbf{P}(\mathbf{X}) &amp; 2 \mathrm{k} &amp; 12 \mathrm{k} &amp; 36 \mathrm{k} &amp; 80 \mathrm{k} &amp; 250 \mathrm{k} &amp; 432 \mathrm{k} &amp; 686 \mathrm{k} \\<br/>\hline<br/>\end{array}<br/>(i) We know that<br/>$\Rightarrow \mathrm{k}(2+3+4+5+10+12+14)=1$<br/>$\Rightarrow \mathrm{k}=\frac{1}{50} \quad\left[\because \Sigma \mathrm{P}_{\mathrm{i}}=1\right]$<br/>(ii) $\because \mathrm{E}(\mathrm{X})=\Sigma \mathrm{XP}(\mathrm{X})$<br/>$\begin{aligned} \therefore \mathrm{E}(\mathrm{X})=2 \mathrm{k}+6 \mathrm{k}+12 \mathrm{k}+20 \mathrm{k}+&amp; 50 \mathrm{k} \\ &amp;+72 \mathrm{k}+98 \mathrm{k}+0=260 \mathrm{k} \end{aligned}$<br/>$=260 \times \frac{1}{50}=\frac{26}{5}=5.2$<br/>(iii) We know that,<br/>$$<br/>\begin{aligned}<br/>&amp; \operatorname{Var}(\mathrm{X})=\Sigma \mathrm{X}^2 \mathrm{P}(\mathrm{X})-\left[\sum\{\mathrm{XP}(\mathrm{X})\}\right]^2 \\<br/>=&amp; {[2 \mathrm{k}+12 \mathrm{k}+36 \mathrm{k}+80 \mathrm{k}+250 \mathrm{k}+432 \mathrm{k}} \\<br/>&amp;+686 \mathrm{k}+0]-[5.2]^2[\mathrm{Using} \text { Eq. (i) }] \\<br/>=&amp; {\left[1498 \times \frac{1}{50}\right]-27.04\left[\because \mathrm{k}=\frac{1}{50}\right] } \\<br/>&amp;=29.96-24.04=2.92<br/>\end{aligned}<br/>$$<br/>And standard deviation of $X=\sqrt{\operatorname{Var}(X)}$ $=\sqrt{2.92}=1.7088=1.7$ (approx)</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2019 (Phase 1)']
answer: <div class="solution">Statements $(2)$ and $(3)$ are correct.</div>
source_db: MarksBatch2_P2.db
title:
let-x-be-a-nonempty-set-and-let-a-b-c-be-subsets-of-x-consider-the-following-statments-1-a-c-a-b-c-b-a-b-c-b-2-a-b-c-b-for-all-sets-b-a-c-3-a-b-c-b-fo

slug:
let-x-be-a-nonempty-set-and-let-a-b-c-be-subsets-of-x-consider-the-following-statments-1-a-c-a-b-c-b-a-b-c-b-2-a-b-c-b-for-all-sets-b-a-c-3-a-b-c-b-fo-44261

question:
<div class="question">Let $\mathrm{X}$ be a non-empty set and let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be subsets of $\mathrm{X}$. Consider the following statments:<br/>(1) $\mathrm{A} \subset \mathrm{C} \Rightarrow(\mathrm{A} \cap \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})$<br/>(2) $(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cap \mathrm{B})$ for all sets $\mathrm{B} \Rightarrow \mathrm{A} \subset \mathrm{C}$<br/>(3) $(\mathrm{A} \cup \mathrm{B}) \subset(\mathrm{C} \cup \mathrm{B})$ for all sets $\mathrm{B} \Rightarrow \mathrm{A} \subset \mathrm{C}$<br/>Which of the above statements are correct?</div>


title: let-x-be-any-nonempty-set-containing-n-elements-then-what-is-the-number-of-relations-on-x
slug: let-x-be-any-nonempty-set-containing-n-elements-then-what-is-the-number-of-relations-on-x-62870
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $\mathrm{X}$ be any non-empty set containing $\mathrm{n}$ elements. Then what is the number of relations on $\mathrm{X}$ ?</div>

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

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

answer:
<div class="solution">Number of elements in $\mathrm{X}$, is $\mathrm{n}$, then the number of relations on $\mathrm{X}$ means, number of elements of cartesian product $\mathrm{X} \times \mathrm{X}$<br/>Since, $n(X)=n$ So, $\mathrm{n}(\mathrm{X} \times \mathrm{X})=\mathrm{n}, \mathrm{n}$<br/>then the total number of relations is $2^{<br/>\mathrm{n} \mathrm{n}}=2^{\mathrm{n}^{2}}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-be-the-mean-of-n-observations-x-1-x-2-x-n-if-a-b-is-added-to-each-observation-then-what-is-the-mean-of-new-set-of-observations

slug:
let-x-be-the-mean-of-n-observations-x-1-x-2-x-n-if-a-b-is-added-to-each-observation-then-what-is-the-mean-of-new-set-of-observations-60812

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

answer:
<div class="solution">Let $\bar{x}$ is the mean of $n$ observation $x_{1}, x_{2}, \ldots ., x_{n}$<br/>$\Rightarrow \bar{x}=\frac{x_{1}+x_{2}+x_{3}+\ldots .+x_{n}}{n}$<br/>Now, $(a-b)$ is added to each term. New mean $=\frac{\mathrm{x}_{1}+(\mathrm{a}-\mathrm{b})+\mathrm{x}_{2}+(\mathrm{a}-\mathrm{b})+\ldots . .+\mathrm{x}_{\mathrm{n}}+(\mathrm{a}-\mathrm{b})}{\mathrm{n}}$<br/>$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}+\frac{\mathrm{n}(\mathrm{a}-\mathrm{b})}{\mathrm{n}}$<br/>$=\overline{\mathrm{x}}+(\mathrm{a}-\mathrm{b})$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-be-the-mean-of-x-1-x-2-x-3-x-n-if-x-i-a-c-y-i-for-some-constants-a-and-c-then-what-will-be-the-mear-y-2-y-3-y-n

slug:
let-x-be-the-mean-of-x-1-x-2-x-3-x-n-if-x-i-a-c-y-i-for-some-constants-a-and-c-then-what-will-be-the-mear-y-2-y-3-y-n-22650

question:
<div class="question">Let $\bar{x}$ be the mean of $x_{1}, x_{2}, x_{3}, \ldots ., x_{n} .$ If $x_{i}=a+c y_{i}$ for some constants a and $\mathrm{c}$, then what will be the mear $\mathrm{y}_{2}, \mathrm{y}_{3}, \ldots, \mathrm{y}_{\mathrm{n}}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad a+c \bar{x}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{a}-\frac{1}{\mathrm{c}} \overline{\mathrm{x}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \frac{1}{x} \bar{x}-a$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\underline{\mathrm{X}-}$</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> </ul>

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

answer:
<div class="solution">Given, Mean of $x_{i}=\bar{x}$ Also, Given $x_{i}=a+c y_{i}$ Mean of $\mathrm{a}+\mathrm{cy}_{\mathrm{i}}=\overline{\mathrm{x}}$<br/>$\Rightarrow$ Mean of $\mathrm{cy}_{\mathrm{i}}=\overline{\mathrm{x}}-\mathrm{a}$<br/>$\Rightarrow$ Mean of $y_{i}=\frac{\bar{x}-a}{c}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2481</span> </div>
source_db: MarksBatch2_P2.db
title:
let-x-be-the-number-of-integers-lying-between-2999-and-8001-which-have-at-least-two-digits-equal-then-x-is-equal-to

slug:
let-x-be-the-number-of-integers-lying-between-2999-and-8001-which-have-at-least-two-digits-equal-then-x-is-equal-to-82778

question:
<div class="question">Let $x$ be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then $\mathrm{x}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2480</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2481</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">2482</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2483</span> </li> </ul>

answer:
<div class="solution">Number of numbers between 2999 and 8001 $=8001-2999-1=5001$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/-zQfIAOft-FRu67myfvQJmO1WnJY1smJnFY8fqWZYKQ.original.fullsize.png"/><br/><br/>Number ofnumbers with all digitdistinct and having<br/>as starting digit<img src="https://cdn-question-pool.getmarks.app/pyq/nda/XNRWijBmbOhWY46WRlUr1T4io1evyIP7g8e62teWY9Y.original.fullsize.png"/><br/><br/>Number ofnumbers with all digitdistinct and having 4 as starting digi $4=50$<br/>Similarly number of numbers with starting digit 5,6 and 7 respectively are 504,504 and 504 . Total numbers $=5 \times 504=2520$<br/>Hence, required number $=5001-2520=2481$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-be-the-set-of-all-citizensof-india-elements-x-y-in-x-are-said-to-be-related-if-the-difference-of-their-age-is-5-years-which-one-of-the-following

slug:
let-x-be-the-set-of-all-citizensof-india-elements-x-y-in-x-are-said-to-be-related-if-the-difference-of-their-age-is-5-years-which-one-of-the-following-30891

question:
<div class="question">Let $\mathrm{X}$ be the set of all citizensof India. Elements $\mathrm{x}, \mathrm{y}$ in $\mathrm{X}$ are said to be related if the difference of their age is 5 years Which one of the following is correct?</div>

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

answer:
<div class="solution">$\mathrm{X}=$ Set of all citizens of India $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{X},|\mathrm{x}-\mathrm{y}|=5\}$<br/>$|\mathrm{x}-\mathrm{x}|=0 \neq 5 \quad(\mathrm{R}$ is not reflexive)<br/>$x R y \Rightarrow|x-y|=5$<br/>$\Rightarrow|y-x|=5 \quad$ (R is symmetric) $\mathrm{xRy} \Rightarrow|\mathrm{x}-\mathrm{y}|=5$<br/>$y R z \Rightarrow|y-z|=5$<br/>But $|x-z| \neq 5$<br/>(R is not transitive)</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-be-the-set-of-all-graduates-in-india-elements-x-and-y-in-x-are-said-to-berelated-if-they-are-graduates-of-the-sameuniversity-which-one-of-the-fo

slug:
let-x-be-the-set-of-all-graduates-in-india-elements-x-and-y-in-x-are-said-to-berelated-if-they-are-graduates-of-the-sameuniversity-which-one-of-the-fo-58563

question:
<div class="question">Let $X$ be the set of all graduates in India. Elements $x$ and $y$ in $X$ are said to berelated if they are graduates of the sameuniversity. Which one of the following statements is correct?</div>

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

answer:
<div class="solution">$x R y \Leftrightarrow x$ and $y$ are graduates of the same university.<br/>Reflexive $x R x \Leftrightarrow x$ and $x$ are graduates of the same<br/>university. Relation is reflexive.<br/>Symmetric $x R y \Leftrightarrow x$ and $y$ are graduates of the<br/>same university<br/>$\Rightarrow y R x \Leftrightarrow y$ and $x$ are graduates of the same<br/>university. $\therefore$ Relation issymmetric. Transitive $x R y, y R z \Leftrightarrow x R z$<br/>It means $x$ and $y, y$ and $z$ are graduates of the same university, then $x$ and $z$ are also graduates of the same university.<br/>- Relation is transitive Hence, relation is reflexive, symmetric and transitive</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-be-the-set-of-all-persons-living-in-a-city-persons-x-in-x-are-said-to-be-related-as-x-y-if-y-is-at-least-5-years-older-than-x-which-one-of-the-f

slug:
let-x-be-the-set-of-all-persons-living-in-a-city-persons-x-in-x-are-said-to-be-related-as-x-y-if-y-is-at-least-5-years-older-than-x-which-one-of-the-f-22770

question:
<div class="question">Let $X$ be the set of all persons living in a city. Persons $x$, in $X$ are said to be related as $x   &lt; y$ if $y$ is at least 5 years older than $x$. Which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The relation is transitive but neither reflexive nor symmetric</span> </div>

answer:
<div class="solution">Given that $x   &lt; y$ if $y \geq x+5$ For Reflexive:<br/>$x &lt; x$<br/>Hence, relation is not reflexive For Symmetry:<br/>if $x &lt; y$<br/>then $y &lt; x$<br/>Hence, relation is not symmetry.<br/>ForTransitive:<br/>if $x &lt; y$ and $y &lt; z$<br/>then $\mathrm{x} &lt; 2$<br/>Hence, relation is transitive</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
let-x-be-theset-of-all-persons-living-in-delhi-thepersons-a-and-b-in-x-are-said-to-be-related-if-the-difference-in-their-ages-is-at-most-5-years-the-r

slug:
let-x-be-theset-of-all-persons-living-in-delhi-thepersons-a-and-b-in-x-are-said-to-be-related-if-the-difference-in-their-ages-is-at-most-5-years-the-r-91097

question:
<div class="question">Let $\mathrm{X}$ be theset of all persons living in Delhi. Thepersons $a$ and $b$ in $X$ are said to be related if the difference in their ages<br/>is at most 5 years. The relation is</div>

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

answer:
<div class="solution">$a-b \leq 5$ Let $b-c \leq 5$<br/>so $a-c \leq 10$<br/>as $(a-b) \in R$ and $(b-c) \in R$<br/>but $(a-c) \notin R$<br/>So it is not a transitive relation. By considering all the options, we come to the conclusion that only option (d) is correct.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">One</span> </div>
source_db: MarksBatch2_P2.db
title:
let-x-denote-the-greatest-integer-function-what-is-the-number-of-solutions-of-the-equation-x-2-4-x-x-0-in-the-interval-02

slug:
let-x-denote-the-greatest-integer-function-what-is-the-number-of-solutions-of-the-equation-x-2-4-x-x-0-in-the-interval-02-95194

question:
<div class="question">Let $[\mathrm{x}]$ denote the greatest integer function. What is the number of solutions of the equation $x^{2}-4 x+[x]=0$ in the interval [0,2]?</div>

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

answer:
<div class="solution">$\mathrm{x}^{2}-4 \mathrm{x}+[\mathrm{x}]=0$<br/>Given internal, $[0,2]$<br/>Case $1:$ Let $0 \leq \mathrm{x}   &lt; 1$<br/>$[\mathrm{x}]=0$<br/>$\therefore \mathrm{x}^{2}-4 \mathrm{x}+0=0 \Rightarrow \mathrm{x}(\mathrm{x}-4)=0 \Rightarrow \mathrm{x}=0, \mathrm{x}=4$<br/>$\mathrm{x}=4$ can't be taken in $0 \leq \mathrm{x} &lt; 1$<br/>$x=0$<br/>Case $2:$ Let $1 \leq x &lt; 2$ $[\mathrm{x}]=1$<br/>$\therefore \mathrm{x}^{2}-4 \mathrm{x}+1=0$<br/>roots are $\frac{4 \pm \sqrt{16-4}}{2}=\frac{4 \pm \sqrt{12}}{2}=2 \pm \sqrt{3}$<br/>In internal $1 \leq \mathrm{x} &lt; 2,2 \pm \sqrt{3}$ are not the roots. Case $3:$ Let $x=2$ $[\mathrm{x}]=2$<br/>$\therefore x^{2}-4 x+2=0$<br/>roots are $\frac{4 \pm \sqrt{16-8}}{2}=\frac{4 \pm \sqrt{8}}{2}=2 \pm \sqrt{2}$<br/>Since, $x=2$, roots can't be $2 \pm \sqrt{2}$<br/>$\therefore$ There is only one solution, $x=0$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-denote-the-number-of-scores-which-exceed-4-in-18-tosses-of-a-symmetrical-die-consider-the-following-statements-2014-i-1-the-arithmetic-mean-of-x

slug:
let-x-denote-the-number-of-scores-which-exceed-4-in-18-tosses-of-a-symmetrical-die-consider-the-following-statements-2014-i-1-the-arithmetic-mean-of-x-37012

question:
<div class="question">Let $X$ denote the number of scores which exceed 4 in 18 , tosses of a symmetrical die. Consider the following statements : $\quad[2014-I]$<br/> 1 . The arithmetic mean of $X$ is 6<br/> 2 . The standard deviation of $X$ is 2 . Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement 1 $n(X)=2$<br/>$\mathrm{p}=\frac{\mathrm{n}(\mathrm{X})}{\mathrm{n}(\mathrm{S})}=\frac{2}{6}=\frac{1}{3}$<br/>$q=1-p=1-\frac{1}{3}=\frac{2}{3}$<br/>arithmetric mean of $\mathrm{X}=\mathrm{n} \mathrm{p}=18 \times \frac{1}{3}=6$<br/>Statement $2:$ Standard deviation of<br/>$\mathrm{x}-\sqrt{\text { variance of } \mathrm{X}}-\sqrt{18 \times \frac{1}{3} \times \frac{2}{3}}-\sqrt{4}-2$<br/>Hence, statements 1 and 2 both are correct.<br/>Sol. $(128-130):$ \begin{array}{|c|cc|c|}<br/>\hline Numbers (\mathrm{x}) &amp; Frequency (\mathrm{f}) &amp; cf. &amp; \Sigma \mathrm{fx} \\<br/>\hline 0 &amp; 14 &amp; 14 &amp; 0 \\<br/>1 &amp; 21 &amp; 35 &amp; 21 \\<br/>2 &amp; 25 &amp; 60 &amp; 50 \\<br/>3 &amp; 43 &amp; 103 &amp; 129 \\<br/>4 &amp; 51 &amp; 154 &amp; 204 \\<br/>5 &amp; 40 &amp; 194 &amp; 200 \\<br/>6 &amp; 39 &amp; 233 &amp; 234 \\<br/>7 &amp; 12 &amp; 245 &amp; 84 \\<br/>\hline &amp; \mathrm{N}=245 &amp; &amp; \Sigma \mathrm{fx} =922 \\<br/>\hline<br/>\end{array}</div>


tags: ['Mathematics', 'Probability']
options: None
correct_option: None
source_db: MarksBatch2_P2.db
title:
let-x-denote-the-sum-of-the-numbers-obtained-when-two-fair-dice-are-rolled-find-the-variance-and-standard-deviation-of-x

slug:
let-x-denote-the-sum-of-the-numbers-obtained-when-two-fair-dice-are-rolled-find-the-variance-and-standard-deviation-of-x-62206

question:
<div class="question">Let $\mathbf{X}$ denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of $\mathbf{X}$.</div>

answer:
<div class="solution">$S=\left[\begin{array}{llllll}11 &amp; 21 &amp; 31 &amp; 41 &amp; 51 &amp; 61 \\ 12 &amp; 22 &amp; 32 &amp; 42 &amp; 52 &amp; 62 \\ 13 &amp; 23 &amp; 33 &amp; 43 &amp; 53 &amp; 63 \\ 14 &amp; 24 &amp; 34 &amp; 44 &amp; 54 &amp; 64 \\ 15 &amp; 25 &amp; 35 &amp; 45 &amp; 55 &amp; 65 \\ 16 &amp; 26 &amp; 36 &amp; 46 &amp; 56 &amp; 66\end{array}\right]$<br/>$$<br/>n(S)=36<br/>$$<br/>Let A denotes the sum of the numbers $=2$<br/>$B$ denotes the sum of the numbers $=3$<br/>$\mathrm{C}$ denotes the sum of the numbers $=4$<br/>D denotes the sum of the numbers $=5$<br/>E denotes the sum of the numbers $=6$<br/>$F$ denotes the sum of the numbers $=7$<br/>$\mathrm{G}$ denotes the sum of the numbers $=8$<br/>$\mathrm{H}$ denotes the sum of the numbers $=9$<br/>I denotes the sum of the numbers $=10$<br/>$J$ denotes the sum of the numbers $=11$<br/>$\mathrm{K}$ denotes the sum of the numbers $=12$<br/>$A=\{11\}, P(A)=\frac{1}{36}$<br/>$$<br/>\mathrm{B}=\{12,21\}, \mathrm{P}(\mathrm{B})=\frac{2}{36}<br/>$$<br/>$$<br/>\mathrm{C}=\{13,22,31\}, \mathrm{P}(\mathrm{C})=\frac{3}{36}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp;\mathrm{D}=\{14,23,32,41\}, P(D)=\frac{4}{36} \\<br/>&amp;E=\{15,24,33,42,51\}, P(E)=\frac{5}{36} \\<br/>&amp;F=\{16,25,34,43,52,61\}, P(F)=\frac{6}{36}<br/>\end{aligned}<br/>$$<br/>$$<br/>\mathrm{G}=\{26,35,44,53,62\}, \mathrm{P}(\mathrm{G})=\frac{5}{36}<br/>$$<br/>$$<br/>\mathrm{H}=\{36,45,54,63\}, \mathrm{P}(\mathrm{H})=\frac{4}{36}<br/>$$<br/>$$<br/>\mathrm{I}=\{46,55,64\}, \mathrm{P}(\mathrm{I})=\frac{3}{36}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp;J=\{56,65\}, P(J)=\frac{2}{36} \\<br/>&amp;K=\{6,6\}, P(K)=\frac{1}{36}<br/>\end{aligned}<br/>$$<br/>Probability distribution:<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/IaXJE1jmBUkicT9-0zVfR21bRnpxz9DwXCYMjelbApQ.original.fullsize.png"/><br/><br/>Mean $=\mu=\sum p_i x_i$<br/>$=\frac{1}{36}(2+6+12+20+30+42+40+36+30+22+12)=\frac{252}{36}=7$<br/>Now, $\sum p_i x_i^2$<br/>$=\frac{1}{36}[4+18+48+100+180+294+320+$<br/>$=\frac{1}{36} \times 1974=\frac{329}{6}$<br/>Variance $=\sum p_i x_i^2-\left(\sum p_i x_i\right)^2$<br/>$=54.83-49=5.83$<br/>Standard deviation $=\sqrt{8.83}=2.4$ (nearly).</div>


tags: ['Mathematics', 'Probability', 'VITEEE', 'VITEEE 2014']
source_db: MarksBatch2_P2.db
title:
let-x-denote-the-sum-of-the-numbers-obtained-when-two-fair-dice-are-rolled-the-variance-and-standard-deviation-of-x-are

slug:
let-x-denote-the-sum-of-the-numbers-obtained-when-two-fair-dice-are-rolled-the-variance-and-standard-deviation-of-x-are-37411

question:
<div class="question">Let $X$ denote the sum of the numbers obtained when two fair dice are rolled. The variance and standard deviation of $X$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{31}{6}$ and $\sqrt{\frac{31}{6}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{35}{6}$ and $\sqrt{\frac{35}{6}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{17}{6}$ and $\sqrt{\frac{17}{6}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{31}{6}$ and $\sqrt{\frac{35}{6}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{35}{6}$ and $\sqrt{\frac{35}{6}}$</span> </div>

answer:
<div class="solution">Let $\mathbf{x}$ denote the sum of the numbers obtained when two fair dice are rolled. So, X may have values $2,3,4,5,6,7,8,9$, 10,11 and 12 .<br/>$\begin{array}{l}<br/>\mathrm{P}(\mathrm{X}=2)=\mathrm{P}(1,1)=\frac{1}{36} \\<br/>\mathrm{P}(\mathrm{X}=3)=\mathrm{P}\{(1,2),(2,1)\}=\frac{2}{36} \\<br/>\mathrm{P}(\mathrm{X}=4)=\frac{3}{36}, \mathrm{P}(\mathrm{X}=5)=\frac{4}{36}: \\<br/>\mathrm{P}(\mathrm{X}=6)=\frac{5}{36} ; \mathrm{P}(\mathrm{X}=7)=\frac{6}{36} ; \\<br/>\mathrm{P}(\mathrm{X}=9)=\frac{4}{36} ; \mathrm{P}(\mathrm{X}=10)=\frac{3}{36} ; \mathrm{P}(\mathrm{X}=11) \\<br/>=\frac{2}{36} ; \\<br/>\mathrm{P}(\mathrm{X}=12)=\frac{1}{36}<br/>\end{array}$<br/>$\therefore \quad$ Probability distribution table is given below<br/>\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c}<br/>\hline \mathbf{X} &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 &amp; 11 &amp; 12 \\<br/>\hline \mathbf{P}(\mathbf{X}) &amp; \frac{1}{36} &amp; \frac{2}{36} &amp; \frac{3}{36} &amp; \frac{4}{36} &amp; \frac{5}{36} &amp; \frac{6}{36} &amp; \frac{5}{36} &amp; \frac{4}{36} &amp; \frac{3}{36} &amp; \frac{2}{36} &amp; \frac{1}{36} \\<br/>\hline<br/>\end{array}<br/>Mean $\bar{X}=\sum X P(X)$<br/>$=\frac{\left[\begin{array}{l}2 \times 1+3 \times 2+4 \times 3+5 \times 4+6 \times 5+7 \times 6+7 \\ 8 \times 5+9 \times 4+10 \times 3+11 \times 2+12 \times 1\end{array}\right]}{36}$<br/>$=\frac{252}{36}=7$<br/>Variance $=\sum X^{2} P(X)-\bar{X}^{2}$<br/>$\begin{aligned}<br/>&amp;\left[\begin{array}{l}<br/>2^{2} \times 1+3^{2} \times 2+4^{2} \times 3+5^{2} \times 4 \\<br/>+6^{2} \times 5+7^{2} \times 6+8^{2} \times 5+9^{2} \times \\<br/>4+10^{2} \times 3+11^{2} \times 2+12^{2} \times 1<br/>\end{array}\right] \\<br/>=&amp; \frac{36}{1974}-7^{2} \\<br/>=&amp; \frac{36}{1974-1764} \\<br/>=&amp; \frac{36}{36}=\frac{35}{6} \\<br/>\therefore &amp; \text { Variance }=\frac{35}{6}<br/>\end{aligned}$<br/>And, $S D=\sqrt{\frac{35}{6}}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
answer: <div class="solution">All statements are correct.</div>
source_db: MarksBatch2_P2.db
title:
let-x-y-be-two-real-numbers-and-z-r-z-0-consider-the-following-1-x-z-y-z-and-x-z-yz-2-x-z-y-z-and-x-z-y-z-3-xz-yzand-z-x-z-y-4-x-z-y-z-and-z-x-z-y-whi

slug:
let-x-y-be-two-real-numbers-and-z-r-z-0-consider-the-following-1-x-z-y-z-and-x-z-yz-2-x-z-y-z-and-x-z-y-z-3-xz-yzand-z-x-z-y-4-x-z-y-z-and-z-x-z-y-whi-59693

question:
<div class="question">Let $x&gt;y$ be two real numbers and $z \in R, z \neq 0$. Consider the following:<br/>1.$\quad x+z&gt;y+z$ and $x z&gt;y z$<br/>$2 \quad x+z&gt;y-z$ and $x-z&gt;y-z$<br/>3.$\mathrm{xz}&gt;$ yzand $\frac{\mathrm{x}}{\mathrm{z}}&gt;\frac{\mathrm{y}}{\mathrm{z}}$<br/>4.$x-z&gt;y-z$ and $\frac{x}{z}&gt;\frac{y}{z}$<br/>Which of the above is/are correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1,2,3$ and 4</span> </div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-x-y-z-be-positive-real-numbers-such-that-x-y-z-are-in-gp-and-tan-1-x-tan-1-y-and-tan-1-z-are-in-ap-then-which-one-of-the-following-is-correct

slug:
let-x-y-z-be-positive-real-numbers-such-that-x-y-z-are-in-gp-and-tan-1-x-tan-1-y-and-tan-1-z-are-in-ap-then-which-one-of-the-following-is-correct-55409

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

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

answer:
<div class="solution">$2 \tan ^{-1} \mathrm{y}=\tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{z}$<br/>$\left(\therefore \tan ^{-1} \mathrm{x}, \tan ^{-1} \mathrm{y}, \tan ^{-1} \mathrm{z}\right.$ are in A.P)<br/>and $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in $\mathrm{G} \mathrm{P} \quad \therefore \quad \mathrm{y}^{2}=\mathrm{xz}$ ...(i)<br/>$\Rightarrow \frac{2 y}{1-y^{2}}=\frac{x+z}{1-x z}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/YPZmaqUgvyP6hgJGuUgqApn0qBiiVAfqYXSNP6S-wnQ.original.fullsize.png"/><br/><br/>$\Rightarrow 2 \mathrm{y}=\mathrm{x}+\mathrm{z}$<br/>$\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in A.P.<br/>Given $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are also in G.P<br/>So, $x=y=z$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">30</span> </div>
source_db: MarksBatch2_P2.db
title:
let-x1-x2-ldots-ldots-ldots-ldots-x100-are-100-observations-such-that-sum-xi0-sum1-leq-i-j-leq-100leftxi-xjright80000-mean-deviation-from-their-mean-i

slug:
let-x1-x2-ldots-ldots-ldots-ldots-x100-are-100-observations-such-that-sum-xi0-sum1-leq-i-j-leq-100leftxi-xjright80000-mean-deviation-from-their-mean-i-64150

question:
<div class="question">Let \( x_{1}, x_{2}, \ldots \ldots \ldots \ldots x_{100} \) are \( 100 \) observations such that \( \sum x_{i}=0, \sum_{1 \leq i &lt; j \leq 100}\left|x_{i} x_{j}\right|=80000 \&amp; \) mean deviation from their mean is \( 5 \), then their standard deviation is</div>

answer:
<div class="solution"><math><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow><mrow><mn>100</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> &amp;<br/><math><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><mfenced close="|" open="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow></mfenced></mrow></mrow></mrow><mrow><mn>100</mn></mrow></mfrac><mo>=</mo><mn>5</mn><mo>⇒</mo><mrow><mo stretchy="false">∑</mo><mrow><mfenced close="|" open="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mrow><mo>=</mo><mn>500</mn></math><br/><math><mo>⇒</mo><mrow><mo stretchy="false">∑</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mn>2</mn><mi> </mi></mrow></mrow><mrow><munder><mo stretchy="false">∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mn>100</mn></mrow></munder><mrow><mfenced close="|" open="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn>500</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></math><br/><math><mo>⇒</mo><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi> </mi></mrow></mrow></mrow><mrow><mn>100</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mfenced separators="|"><mrow><mn>500</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>2</mn><mrow><mo stretchy="false">∑</mo><mrow><mfenced close="|" open="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mrow></mrow><mrow><mn>100</mn></mrow></mfrac><mo>=</mo><mn>2500</mn><mo>-</mo><mn>1600</mn></math><br/><math><mi>S</mi><mo>.</mo><mi>D</mi><mo>.</mo><mi> </mi><mo>=</mo><msqrt><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mn>100</mn></mrow></mfrac></msqrt><mo>=</mo><msqrt><mn>900</mn></msqrt><mo>=</mo><mn>30</mn></math></div>


title: let-y-x-a-x-n-and-y-denote-small-change-in-y-what-is-limit-of-x-y-as-x-0
slug: let-y-x-a-x-n-and-y-denote-small-change-in-y-what-is-limit-of-x-y-as-x-0-12760
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $y(x)=a x^{n}$ and $\delta y$ denote small change in $y$. What is limit <br/>of $\frac{\delta y}{\delta x}$ as $\delta x \rightarrow 0 ?$</div>

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

answer:
<div class="solution">We know $\lim _{\delta x \rightarrow 0} \frac{\delta y}{\delta x}=\left(\frac{d y}{d x}\right)_{\text {at } x=0}$<br/>$\quad=\frac{d}{d x}\left(a x^{n}\right)_{\text {at } x=0}=\left(\operatorname{an} x^{n-1}\right)_{\mathrm{at} x=0}=0$</div>


title: let-z-1-and-z-2-be-any-two-complex-number-statement-1-z-1-z-2-z-1-z-2-statement-2-z-1-z-2-z-1-z-2
tags: ['Mathematics', 'Complex Number', 'JEE Main', 'JEE Main 2012 (07 May Online)']
source_db: MarksBatch2_P2.db
slug:
let-z-1-and-z-2-be-any-two-complex-number-statement-1-z-1-z-2-z-1-z-2-statement-2-z-1-z-2-z-1-z-2-68018

question:
<div class="question">Let $Z_1$ and $Z_2$ be any two complex number.<br/>Statement 1: $\left|Z_1-Z_2\right| \geq\left|Z_1\right|-\left|Z_2\right|$<br/>Statement 2: $\left|Z_1+Z_2\right| \leq\left|Z_1\right|+\left|Z_2\right|$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is false.<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Statement 1 is false, Statement 2 is true.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.<br/></span> </div>

answer:
<div class="solution">Statement $-1$ and 2 both are true.<br/>It is fundamental property.<br/>But Statement - 2 is not correct explanation for Statement $-1$.</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P2.db
title:
let-z-1-and-z-2-be-two-nonzero-complex-numbers-such-that-z-1-z-2-z-1-1-z-2-1-2-what-is-the-value-of-z-1-z-2

slug:
let-z-1-and-z-2-be-two-nonzero-complex-numbers-such-that-z-1-z-2-z-1-1-z-2-1-2-what-is-the-value-of-z-1-z-2-42948

question:
<div class="question">Let $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ be two non-zero complex numbers such that<br/>$\left|z_{1}\right|=\left|z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2$<br/>What is the value of $\left|z_{1}+z_{2}\right|$ ?</div>

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

answer:
<div class="solution">As given $\left|z_{1}\right|=\left|z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2$<br/>$\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2 \Rightarrow \frac{z_{2}+z_{1}}{z_{1} z_{2}}=2$<br/>$\Rightarrow \frac{\left|z_{1}+z_{2}\right|}{\left|z_{1}\right|\left|z_{2}\right|}=2$<br/>$\Rightarrow\left|z_{1}+z_{2}\right|=2\left|z_{1} \| z_{2}\right|$<br/>$\Rightarrow\left|z_{1}+z_{2}\right|=2.2 .2$<br/>$\Rightarrow\left|z_{1}+z_{2}\right|=8$</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
let-z-1-z-2-and-z-3-be-nonzero-complex-numbers-satisfying-z-2-i-z-where-i-1-consider-the-following-statements-1-z-1-z-2-z-3-is-purely-imaginary-2-z-1-

slug:
let-z-1-z-2-and-z-3-be-nonzero-complex-numbers-satisfying-z-2-i-z-where-i-1-consider-the-following-statements-1-z-1-z-2-z-3-is-purely-imaginary-2-z-1-90944

question:
<div class="question">Let $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ be non-zero complex numbers satisfying $\mathrm{z}^{2}=\mathrm{i} \overline{\mathrm{z}}$, where $\mathrm{i}=\sqrt{-1}$.<br/>Consider the following statements:<br/>1- $\quad \mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}_{3}$ is purely imaginary.<br/>2- $\quad z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}$ is purelyreal. Which of the above statements is/are correct?</div>

answer:
<div class="solution">$z_{1} z_{2} z_{3}=(-i)\left(\frac{\sqrt{3}+i}{2}\right)\left(\frac{-\sqrt{3}+i}{2}\right)$<br/>$=\frac{-i}{4}\left(i^{2}-(\sqrt{3})^{2}\right)$<br/>$=\frac{-i}{4}(-3-1)=i$<br/>Hence $z_{1} z_{2} z_{3}$ is purely imaginary. $z_{1} z_{2}=-i\left(\frac{\sqrt{3}+i}{2}\right)=\frac{-\sqrt{3} i+1}{2}$<br/>$\begin{aligned} z_{2} z_{3} &amp;=\frac{(\sqrt{3}+i)(-\sqrt{3}+i)}{4} \\ &amp;=\left(\frac{-3-\sqrt{3} i+\sqrt{3} i+i^{2}}{4}\right)=-1 \end{aligned}$<br/>$z_{3} z_{1}=\frac{(-\sqrt{3}+i)(-i)}{2}=\frac{+\sqrt{3} i+1}{2}$<br/>$z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=\left(\frac{-\sqrt{3} i+1}{2}\right)+(-1)+\left(\frac{\sqrt{3} i+1}{2}\right)$<br/>$=\left(\frac{-\sqrt{3} i+1+\sqrt{3} i+1}{2}\right)-1$<br/>$=0 \in R$<br/>Hence $z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=0$ is purelyreal. Hence both statements are correct.</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P2.db
title:
let-z-1-z-2-and-z-3-be-nonzero-complex-numbers-satisfying-z-2-i-z-where-i-1-what-is-z-1-z-2-z-3-equal-to

slug:
let-z-1-z-2-and-z-3-be-nonzero-complex-numbers-satisfying-z-2-i-z-where-i-1-what-is-z-1-z-2-z-3-equal-to-93143

question:
<div class="question">Let $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ be non-zero complex numbers satisfying $\mathrm{z}^{2}=\mathrm{i} \overline{\mathrm{z}}$, where $\mathrm{i}=\sqrt{-1}$.<br/>What is $\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}$ 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">$-\mathrm{i}$</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 $z^{2}=i \bar{z}$<br/>Let us suppose that $z=x+i y$<br/>$\Rightarrow(x+i y)^{2}=i(x-i y)$<br/>$\Rightarrow x^{2}-y^{2}+2 x y i=i x+y$<br/>Comparing real and imaginary part of both sides<br/>$x^{2}-y^{2}=y$ and $2 x y=x$.<br/>Taking $2 x y=x$ $(2 y-1) x=0$<br/>$\Rightarrow x=0$ and $y=\frac{1}{2}$<br/>if $x=0$ $\therefore y+y^{2}=0 \quad\left(\because x^{2}-y^{2}=y\right)$<br/>$y(y+1)=0$<br/>$y=0$ and $-1$ $x=0 \quad y=0$ and $-1$<br/>If $y=\frac{1}{2}$<br/>$x^{2}-\frac{1}{4}=\frac{1}{2}$<br/>$\Rightarrow x^{2}=\frac{3}{4}$<br/>$\Rightarrow x=\pm \frac{\sqrt{3}}{2}$<br/>Since given numbers are non zero complex numbers.<br/>So, $z_{1}=0+(-1) i=-i$<br/>$z_{2}=\frac{\sqrt{3}}{2}+\frac{1}{2} i \quad z_{3}=\frac{-\sqrt{3}}{2}+\frac{1}{2} i$<br/>$z_{1}+z_{2}+z_{3}=(-i)+\left(\frac{\sqrt{3}}{2}+\frac{1}{2} i\right)+\left(-\frac{\sqrt{3}}{2}+\frac{1}{2} i\right)=0$<br/>Hence $z_{1}+z_{2}+z_{3}=0$</div>


title: let-z-be-a-complex-number-satisfying-z-8-z-4-1-and-z-2-z-2-3-what-is-z-6-z-6-equal-to
slug: let-z-be-a-complex-number-satisfying-z-8-z-4-1-and-z-2-z-2-3-what-is-z-6-z-6-equal-to-88921
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">$\left|\frac{z-6}{z+6}\right|=\left|\frac{6-6}{6+6}\right|=0$</div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $z$ be a complex number satisfying $\left|\frac{z-4}{z-8}\right|=1$ and $\left|\frac{z}{z-2}\right|=\frac{3}{2}$<br/>What is $\left|\frac{z-6}{z+6}\right|$ equal to?</div>

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


title: let-z-be-a-complex-number-satisfying-z-8-z-4-1-and-z-2-z-2-3-what-is-z-equal-to
slug: let-z-be-a-complex-number-satisfying-z-8-z-4-1-and-z-2-z-2-3-what-is-z-equal-to-90060
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $z$ be a complex number satisfying$\left|\frac{z-4}{z-8}\right|=1$ and $\left|\frac{z}{z-2}\right|=\frac{3}{2}$<br/>What is $|z|$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">18</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">36</span> </li> </ul>

answer:
<div class="solution">$\left|\frac{z-4}{z-8}\right|=1$ and $\left|\frac{z}{z-2}\right|=\frac{3}{2}$<br/>$\Rightarrow|z-4|=|z-8|$<br/>Let $z=x+i y$ $|x+i y-4|=|x+i y-8|$<br/>Squaring both sides, we get<br/>$\left[(x-4)^{2}+y^{2}\right]=\left[(x-8)^{2}+y^{2}\right]$<br/>$(x-4)^{2}=(x-8)^{2}$<br/>$\Rightarrow x^{2}+16-8 x=x^{2}+64-16 x$<br/>$\Rightarrow 8 x=48 \Rightarrow x=6$<br/>when $\left|\frac{z}{z-2}\right|=\frac{3}{2}$<br/>$\Rightarrow 2|z|=3|z-2|$<br/>Squaring both sides, we get $4\left(x^{2}+y^{2}\right)=9\left[(x-2)^{2}+y^{2}\right]$<br/>$\Rightarrow 4 x^{2}+4 y^{2}=9 x^{2}+36-36 x+9 y^{2}$<br/>$\Rightarrow 5 x^{2}+5 y^{2}-36 x+36=0$<br/>as we know $x=6$<br/>$5(6)^{2}+5 y^{2}-36 \times 6+36=0$<br/>$\Rightarrow 5 y^{2}=0 \Rightarrow y=0$<br/>Hence $x=6$ and $y=0$. $\Rightarrow z=6$<br/>$|z|=6$</div>


title: let-z-be-a-complex-number-such-that-z-4-and-arg-z-6-5-where-i-1-what-is-z-equal-to
slug: let-z-be-a-complex-number-such-that-z-4-and-arg-z-6-5-where-i-1-what-is-z-equal-to-67348
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $z$ be a complex number such that $|z|=4$ and arg $z=\frac{5 \pi}{6}$.<br/>Where $i=\sqrt{-1}$. What is $z$ equal to ?</div>

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

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

answer:
<div class="solution">$\begin{aligned} \text { Let } z &amp;=\mathrm{K}(\cos \theta+i \sin \theta) \\ &amp;|\mathrm{z}|=\sqrt{\mathrm{K}^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)}=4 \\ &amp; \therefore \mathrm{K}=4 \\ &amp; \text { Again } \operatorname{Arg}(\mathrm{z})=\frac{5 \pi}{6} \\ \text { So, } \theta=\frac{5 \pi}{6} \\ &amp; \text { Now, } \mathrm{Z}=4\left(\cos \left(\frac{5 \pi}{6}\right)+\mathrm{i} \sin \left(\frac{5 \pi}{6}\right)\right) \\ &amp;=4\left(\frac{-\sqrt{3}}{2}+\mathrm{i} \frac{(1)}{2}\right)=-2 \sqrt{3}+2 \mathrm{i} \end{aligned}$</div>


title: let-z-be-a-nonzero-complex-number-then-what-is-z-1-multiplicative-inverse-of-z-equal-to
slug: let-z-be-a-nonzero-complex-number-then-what-is-z-1-multiplicative-inverse-of-z-equal-to-70174
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $z$ be a non-zero complex number. Then, what is $z^{-1}$ (multiplicative inverse of z) equal to?</div>

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

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

answer:
<div class="solution">Let $z$ be a non-zero complex number, such that $\mathrm{z}=\mathrm{x}+$ iy where $\mathrm{x}, \mathrm{y} \in \mathrm{R}$<br/>Then $\mathrm{z}^{-1}=\frac{1}{\mathrm{x}+\mathrm{iy}}$<br/>So, $z^{-1}=\frac{x-i y}{(x+i y)(x-i y)}=\frac{x-i y}{x^{2}+y^{2}}=\frac{\bar{z}}{|z|^{2}}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P2.db
title:
let-z-be-the-set-of-integers-and-a-r-b-where-a-b-z-if-anc-only-if-a-b-is-divisible-by-5-consider-the-following-statements-1the-relation-r-partitions-z

slug:
let-z-be-the-set-of-integers-and-a-r-b-where-a-b-z-if-anc-only-if-a-b-is-divisible-by-5-consider-the-following-statements-1the-relation-r-partitions-z-14976

question:
<div class="question">Let $Z$ be the set of integers and $a R b$, where $a, b \in Z$ if anc only if $(a-b)$ is divisible by $5 .$ Consider the following statements:<br/>1.The relation $R$ partitions $Z$ into five equivalentclasses<br/>2.Any twoequivalent classes are either equal or disjoint Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1  only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and 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">Neither  1 nor 2</span> </li> </ul>


title: let-z-i-3-1-i-be-a-complex-number-what-is-its-argument
slug: let-z-i-3-1-i-be-a-complex-number-what-is-its-argument-46627
question: <div class="question">Let $z=i^{3}(1+i)$ be a complex number. What is its argument?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{\pi}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5 \pi}{4}$</span> </li> </ul>

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

answer:
<div class="solution">The complex number is $z=i^{3}(1+i)=-i(1+i)$ $\begin{aligned} &amp;=-i-\mathrm{i}^{2} \\ \Rightarrow \mathrm{z} &amp;=1-\mathrm{i} \end{aligned}$<br/>This gives, $\arg (z)=\tan ^{-1}\left(\frac{-1}{1}\right)=\tan ^{-1}(-1)=-\frac{\pi}{4}$</div>


title: let-z-x-i-y-where-x-y-are-real-variables-i-1-if-2-z-1-z-2-then-the-point-z-describes
slug: let-z-x-i-y-where-x-y-are-real-variables-i-1-if-2-z-1-z-2-then-the-point-z-describes-19951
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $z=x+i y$ Where $x, y$ are real variables $i=\sqrt{-1}$. If<br/>$|2 z-1|=|z-2|$, then the point $z$ describes:</div>

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

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

answer:
<div class="solution">$|2 z-1|=|z-2|$<br/>$|2(\mathrm{x}+\mathrm{iy})-1|=|\mathrm{x}+\mathrm{iy}-2|$<br/>$|(2 x-1)+2 y i|=|(x-2)+i y|$<br/>$\sqrt{(2 \mathrm{x}-1)^{2}+\mathrm{y}^{2}}=\sqrt{(\mathrm{x}-2)^{2}+\mathrm{y}^{2}}$<br/>Squaring both sides $4 x^{2}+1-4 x+4 y^{2}=x^{2}+4-4 x+y^{2}$<br/>$\Rightarrow 3 x^{2}+3 y^{2}=3$<br/>$\Rightarrow x^{2}+y^{2}=1$<br/>It is the equation of a circle. $\therefore$ The point $z$ describes a circle.</div>


title: level-of-estrogen-and-progesterone-are-minimum-at-the-time-of
slug: level-of-estrogen-and-progesterone-are-minimum-at-the-time-of-28211
question: <div class="question">Level of estrogen and progesterone are minimum at the time of</div>
tags: ['Biology', 'Human Reproduction', 'NEET']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">follicular phase</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">ovulation</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">secretory phase</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">onset of menstrual phase</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">onset of menstrual phase</span> </div>

answer:
<div class="solution">The menstrual cycle consists of changes in the ovary and the uterus simultaneously. It begins with the menstrual or bleeding phase. This represents the shedding of the endometrium of the uterus and unfertilized secondary oocyte. This occurs because the corpus luteum of the previous cycle is degenerating. Due to this the hormones secreted by corpus luteum also start decreasing. These hormones include estrogen and progesterone. These hormones are responsible for maintaining the uterine lining and since these are decreasing the uterine lining too starts shedding. Thus levels of these hormones are minimum at the beginning of the menstrual cycle. During the follicular phase, FSH is increasing and estrogen and progesterone have started the proliferation of the endometrium. During ovulation LH is at its peak and estrogen is also increasing. The secretory phase in uterus coincides with luteal phase in the ovary and thus because of the formation of corpus luteum, estrogen and progesterone are also high.</div>


title: leydig-cells-produce-a-group-of-hormones-called
slug: leydig-cells-produce-a-group-of-hormones-called-89910
question: <div class="question">Leydig cells produce a group of hormones called</div>
tags: ['Biology', 'Chemical Coordination and Integration']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>androgens<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>estrogens<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>aldosterone<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>gonadotropins</span> </li> </ul>

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

answer:
<div class="solution">The Leydig cells or interstitial cells, which are present in the interstitial spaces of testis produce a group of hormones called androgens mainly testosterone.<br/>Estrogen is secreted by the growing follicles in ovaries and stimulates growth and development of female secondary sex organs.<br/>Aldosterone is secreted by the adrenal gland helps in maintaining electrolytes in the body fluid, volume, osmotic pressure and blood pressure.<br/>Gonadotropins are secreted by anterior pituitary gland and stimulate the gonadal activity. These include LH and FSH.</div>


title: li-occupies-higher-position-in-the-electrochemical-series-of-metals-as-compared-to-cu-since
slug: li-occupies-higher-position-in-the-electrochemical-series-of-metals-as-compared-to-cu-since-84115
tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">Li occupies higher position in the electrochemical series of metals as compared to Cu since</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the standard reduction potential of $\mathrm{Li}^{+} / \mathrm{Li}$ is lower than that of $\mathrm{Cu}^{2+} / \mathrm{Cu}$</span> </div>

answer:
<div class="solution">In the electrochemical series, metals are arranged in increasing order of their standard reduction potential. The standard reduction potentials of $L^{+} / L$ and $C u^{2+} / C u$ are -3.05 V and $+.034 \mathrm{V}$ respectively. Therefore, Li occupies higher position in series as compared to Cu.</div>


title: lim-x-0-e-1-x-is-equal-to
slug: lim-x-0-e-1-x-is-equal-to-11721
question: <div class="question">$\lim _{x \rightarrow 0} e^{-1 / x}$ is equal to</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\infty$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">e</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">does not exist</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">does not exist</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{LHL} &amp;=\lim _{h \rightarrow 0^{-}} e^{\frac{-1}{(0-h)}}=\lim _{h \rightarrow 0} \mathrm{e}^{\frac{1}{h}} \\=\mathrm{e}^{\infty}=\infty \end{aligned}$</div>


title: lim-x-0-x-2-1-c-o-s-3-4-x-is-equal-to
slug: lim-x-0-x-2-1-c-o-s-3-4-x-is-equal-to-51557
question: <div class="question">$\lim _{x \rightarrow 0} \frac{1-\cos ^{3} 4 x}{x^{2}}$ is equal to</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">36</span> </li> </ul>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{1-\cos ^{3} 4 x}{x^{2}}$<br/>The given limit is in $\frac{0}{0}$ form. So, apply L'hospital rule.<br/>$\lim _{x \rightarrow 0} \frac{-3 \cos ^{2} 4 x(-\sin 4 x) 4}{2 x}$<br/>$=\lim _{x \rightarrow 0} \frac{12 \cos ^{2} 4 x \sin 4 x}{(2 x) 2} \times 2$<br/>$=24 \cdot \lim _{x \rightarrow 0} \cos ^{2} 4 x \cdot \frac{\sin 4 x}{4 x}$<br/>$=24(1)=24$</div>


title: lim-x-0-x-s-i-n-x-t-a-n-x-equal-to
slug: lim-x-0-x-s-i-n-x-t-a-n-x-equal-to-57233
question: <div class="question">$\lim _{x \rightarrow 0} \frac{\sin x-\tan x}{x}$ equal to?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1/2</span> </li> </ul>

answer:
<div class="solution">Consider $\lim _{x \rightarrow 0} \frac{\sin x-\tan x}{x}$<br/>$=\lim _{x \rightarrow 0} \frac{\cos x-\sec ^{2} x}{1}\left(\right.$ By $L^{\prime}$ Hospital Rule $)$<br/>$=\frac{\cos 0-\sec ^{2} 0}{1}=\frac{1-1}{1}=0$</div>


title: lim-x-2-2-x-2-1-c-o-s-2-x
slug: lim-x-2-2-x-2-1-c-o-s-2-x-27305
question: <div class="question">$\lim _{x \rightarrow \pi / 2} \frac{1+\cos 2 x}{(\pi-2 x)^2}=$</div>
tags: ['Mathematics', 'Limits', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1/2$</span> </div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1/2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">First, let's substitute \(y=\pi-2 x\). As \(x \rightarrow \frac{\pi}{2}, y \rightarrow 0\).
<br/>So, the limit becomes:
<br/>\(\lim _{y \rightarrow 0} \frac{1+\cos (\pi-y)}{y^2}\)
<br/>Using the trigonometric identity \(\cos (\pi-y)=-\cos (y)\), we can rewrite the limit as:
<br/>\(\lim _{y \rightarrow 0} \frac{1-\cos (y)}{y^2}\)
<br/>Now, we know the standard limit:
<br/>\(\lim _{y \rightarrow 0} \frac{1-\cos (y)}{y^2}=\frac{1}{2}\)</div>


title: lipids-are-insoluble-in-water-because-lipids-molecules-are
slug: lipids-are-insoluble-in-water-because-lipids-molecules-are-75967
question: <div class="question">Lipids are insoluble in water because lipids molecules are:</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET', 'NEET 2002']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">hydrophilic</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">hydrophobic</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">neutral</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zwitter ions</span> </li> </ul>

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

answer:
<div class="solution">Lipid have long nonpolar chains of carbon atoms which adversely affects its solubility in water. Thus, they are insoluble in water because they are hydrophobic.<br/>Related Theory<br/>Allofthelipidmoleculesincellmembranesare amphipathic that is, they have a hydrophilic or polar end and a hydrophobic or nonpolar end. The most abundant membrane lipids are the phospholipids. These have a polar head group and two hydrophobic hydrocarbon tails.</div>


tags: ['Biology', 'Evolution']
options: None
correct_option: None
source_db: MarksBatch2_P2.db
title:
list-10-modernday-animals-and-using-the-internet-resources-link-it-to-a-corresponding-ancient-fossil-name-both

slug:
list-10-modernday-animals-and-using-the-internet-resources-link-it-to-a-corresponding-ancient-fossil-name-both-66464

question:
<div class="question">List 10 modern-day animals and using the internet resources link it to a corresponding ancient fossil. Name both.</div>

answer:
<div class="solution">(i) Cockroach, Limulus (king crab), Neopilina, Latimaria (Fish) are fossil that has remain unchanged over years.<br/>(ii) Trilobites - fossil arthropods<br/>(iii) Lung fishes - connecting link between fishes and amphibians<br/>(iv) Peripatus - connecting link between annelids and arthropods<br/>(v) Woody mammoth - ice fossils<br/>(vi) Gastropods - mould and cast fossil<br/>(vii) Giant elk - amber fossil of asphalt<br/>(viii) Dinosaur footprint-imprints</div>


title: list-of-endangered-species-was-released-by
slug: list-of-endangered-species-was-released-by-13647
question: <div class="question">List of endangered species was released by</div>
tags: ['Biology', 'Biodiversity and its Conservation', 'NEET', 'NEET 2024']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">IUCN</span> </div>
answer: <div class="solution">List of endangered species was released by - IUCN.</div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">WWF</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">FOAM</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">IUCN</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">GEAC</span> </li> </ul>


title: list-the-harmful-effects-caused-by-alcoholdrug-abuse
slug: list-the-harmful-effects-caused-by-alcoholdrug-abuse-14663
question: <div class="question">List the harmful effects caused by alcohol/drug abuse.</div>
tags: ['Biology', 'Human Health and Diseases']
options: None
correct_option: None
source_db: MarksBatch2_P2.db
answer:
<div class="solution">Harmful effects caused by alcohol abuse are-<br/>(i) Alcohol generates more energy mostly in the form of heat, but at the same time, it dilates the blood vessels. Consequently the heat generated is rapidly lost. Due to constant dilation, the arterial walls soon become brittle $\&amp;$ rigid. Such a change in the property of blood vessels $\&amp;$ deposition of alcoholic fat affect the working of heart.<br/>(ii) Alcoholism leads to gastric ulcers \&amp; gastritis.<br/>(iii) In chronic alcoholism, the axon of the nerve inflame thus causing neuritis.<br/>(iv) Permanent damage to liver cells occur due to deposition of fats. The liver dries up \&amp; harden (cirrhosis).<br/>Harmful effects caused by drug abuse are-<br/>(i) Excessive doses of drugs may lead to coma \&amp; death due to respiratory failure, heart failure \&amp; cerebral haemorrhage.<br/>(ii) Lack of interest in personal hygiene, withdrawal, isolation, depression, fatigue aggressive \&amp; rebellious behaviour etc.<br/>(iii) Acquire serious infections like AIDS \&amp; hepatitis B, who take drugs intravenously.<br/>(iv) The adverse effects of drugs are manifested in the form of reckless behaviour, vandalism \&amp; violence.</div>


title: lodoform-reaction-is-answered-by-all-except
slug: lodoform-reaction-is-answered-by-all-except-74277
question: <div class="question">lodoform reaction is answered by all, except</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'KCET', 'KCET 2014']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( \mathrm{CH}_{3} \mathrm{CHO} \)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">(1) \( \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2} \mathrm{OH} \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/ques_21.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{OH} \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(1) \( \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2} \mathrm{OH} \)</span> </div>

answer:
<div class="solution">For alcohols secondary and tertiary alcohols gives iodoform reaction and ethanol is the only primary alcohol which gives<br/>iodoform reaction, therefore, propanol will not give iodoform reaction.<br/>(1) \( \underset{\text { Acetaldehyde }}{\mathrm{C}_{3} \mathrm{CH} \mathrm{O}}+3 \mathrm{NaOI} \rightarrow \underset{\text { Yellow ppt }}{\mathrm{CH}}+ \) HCOON \( a+2 \mathrm{NaOH} \)<br/>(2) \( \begin{array}{c}\text { Acetaldehyde } \\ \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{OH}\end{array}+\mathrm{NaOI}_{2}-\mathrm{NaOH}_{\mathrm{S}_{2}}&gt; \) do not give iodoform reaction.<br/>(3) \( \mathrm{CH}_{3}-\stackrel{\mathrm{OH}}{\mathrm{CH}}-\mathrm{CH}_{2} \mathrm{COOH} \) NaOI \( \underset{\text { Yellowppt. }}{\mathrm{CHI}_{3}}+\mathrm{NaOCCH}_{2} \mathrm{COOH} \)<br/>\( \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}-\mathrm{NaOH}_{\mathrm{I}_{2}} \rightarrow \mathrm{CH}_{3} \mathrm{CHO}+\mathrm{NaI}+\mathrm{H}_{2} \mathrm{O} \)<br/>\( \mathrm{CH}_{3} \mathrm{CHO}+3 \mathrm{NaOI} \rightarrow \mathrm{CHI}_{3}+\mathrm{HCOON} a+2 \mathrm{NaOH} \)</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P2.db
title:
looking-from-the-top-of-a-20-m-high-building-the-angle-of-elevation-of-the-top-of-a-tower-is-6-0-and-the-angle-of-depression-of-its-bottom-is-3-0-what

slug:
looking-from-the-top-of-a-20-m-high-building-the-angle-of-elevation-of-the-top-of-a-tower-is-6-0-and-the-angle-of-depression-of-its-bottom-is-3-0-what-68441

question:
<div class="question">Looking from the top of a $20 \mathrm{~m}$ high building, the angle of elevation of the top of a tower is $60^{\circ}$ and the angle of depression of its bottom is $30^{\circ}$. What is the height of the tower? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$50 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$60 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$70 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$80 \mathrm{~m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$80 \mathrm{~m}$</span> </div>

answer:
<div class="solution">$\operatorname{In} \Delta A B D, \tan 30^{\circ}=\frac{A D}{A B}$<br/>$\Rightarrow \frac{1}{\sqrt{3}}=\frac{A D}{A B}$<br/>$\Rightarrow \frac{1}{\sqrt{3}}=\frac{20}{A B}$<br/>$\Rightarrow A B=20 \sqrt{3} \mathrm{~m} .$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/AMqepUZPjenPvElKniL-d_6BNTcNBnNeQA4m2m8si-o.original.fullsize.png"/><br/><br/>$\operatorname{In} \Delta D C F, \tan 60^{\circ}=\frac{h}{D C} \Rightarrow \sqrt{3}=\frac{b}{D C}$<br/>$\Rightarrow h=(\sqrt{3})(20 \sqrt{3})$<br/>$(\because A B=D C=20 \sqrt{3})$<br/>$\Rightarrow h=60 \mathrm{~m}$<br/>$\therefore$ Height of tower, $B F=60+20=80 \mathrm{~m}$</div>


title: lucas-reagent-is-1
slug: lucas-reagent-is-1-13669
question: <div class="question">Lucas reagent is</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JIPMER', 'JIPMER 2008']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">anhy. \(\mathrm{ZnCl}_2\) and \(\mathrm{NH}_3\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">anhy. \(\mathrm{ZnCl}_2\) and \(\mathrm{CaCl}_2\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">anhy. \(\mathrm{ZnCl}_2\) and conc. HCl</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">anhy. \(\mathrm{ZnCl}_2\) and HCl gas</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">anhy. \(\mathrm{ZnCl}_2\) and conc. HCl</span> </div>

answer:
<div class="solution">Anhy. \(\mathrm{ZnCl}_2\) and conc. HCl is called Lucas reagent. It is used to distinguish the primary, secondary and tertiary alcohol.</div>


title: lucas-test-is-associated-with
slug: lucas-test-is-associated-with-88314
question: <div class="question">Lucas test is associated with</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'KCET', 'KCET 2009']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">aldehydes</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">phenols</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">carboxylic acids</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">alcohols</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">alcohols</span> </div>

answer:
<div class="solution">Lucas test is used to distinguish primary, secondary and tertiary alcohols.</div>


title: lucas-test-is-used-for-the-determination-of-1
slug: lucas-test-is-used-for-the-determination-of-1-63823
question: <div class="question">Lucas test is used for the determination of</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'AIIMS', 'AIIMS 2002']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">alcohols</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">alkyl halides</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">phenols</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">aldehydes.</span> </li> </ul>

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

answer:
<div class="solution">Lucas test-Alcohol reacts with concentrated hydrochloric acid in presence of anhydrous $\mathrm{ZnCl}_2$ to form alkyl halides. The three type of alcohols undergo this reaction at different rates. Order of rate of reaction is : tertiary $&gt;$ secondary $&gt;$ primary.<br/>$$<br/>R-\mathrm{OH}+\mathrm{HCl} \underset{\mathrm{ZnCl}_2}{\longrightarrow} R-\mathrm{Cl}+\mathrm{H}_2 \mathrm{O}<br/>$$<br/>Alkyl chloride</div>


title: lucas-test-is-used-for-the-determination-of
slug: lucas-test-is-used-for-the-determination-of-44600
question: <div class="question">Lucas test is used for the determination of</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'AP EAMCET', 'AP EAMCET 2021 (23 Aug Shift 1)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">aldehydes</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">phenols</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">carboxylic acid</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">alcohols</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">alcohols</span> </div>

answer:
<div class="solution">Lucas test is used to distinguish alcohols.<br/>Lucas reagent is a solution of anhydrous zinc chloride in concentrated $\mathrm{HCl}$.<br/>Lucas reagent converts alcohols to alkyl chlorides. Tertiary alcohols give immediate reaction with Lucas reagent, secondary alcohols give reaction within five minutes, while primary alcohol do not give reaction with it.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
m-is-a-matrix-with-real-entries-given-by-m-4-6-2-k-3-t-0-0-k-which-of-the-following-conditions-guarantee-the-invertibility-of-m-1-k-2-2-k-0-3-t-0-4-t-

slug:
m-is-a-matrix-with-real-entries-given-by-m-4-6-2-k-3-t-0-0-k-which-of-the-following-conditions-guarantee-the-invertibility-of-m-1-k-2-2-k-0-3-t-0-4-t-15630

question:
<div class="question">$\mathrm{M}$ is a matrix with real entries given by<br/>$\mathrm{M}=\left[\begin{array}{lll}<br/>4 &amp; \mathrm{k} &amp; 0 \\<br/>6 &amp; 3 &amp; 0 \\<br/>2 &amp; \mathrm{t} &amp; \mathrm{k}<br/>\end{array}\right]$<br/>Which of the following conditions guarantee the invertibility of $\mathrm{M}$ ?<br/>$1. k \neq 2$<br/>$2. \mathrm{k} \neq 0$<br/>$3. \mathrm{t} \neq 0$<br/>$4. \mathrm{t} \neq 1$<br/>Select the correct answer using the code given below:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 and 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 and 3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3 and 4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 2</span> </div>

answer:
<div class="solution">As given $\mathrm{M}=\left[\begin{array}{lll}4 &amp; \mathrm{k} &amp; 0 \\ 6 &amp; 3 &amp; 0 \\ 2 &amp; \mathrm{t} &amp; \mathrm{k}\end{array}\right]$<br/>$\mathrm{M}$ will be invertible, if<br/>$\left[\begin{array}{lll}4 &amp; \mathrm{k} &amp; 0 \\ 6 &amp; 3 &amp; 0 \\ 2 &amp; \mathrm{t} &amp; \mathrm{k}\end{array}\right] \neq 0$<br/>$\Rightarrow \mathrm{k} \neq 0$ or $\mathrm{k}(12-6 \mathrm{k}) \neq 0$<br/>$\Rightarrow \mathrm{k} \neq 0, \mathrm{k} \neq 2$<br/>Thus, statement (1) and (2) are correct.</div>


tags: ['Chemistry', 's Block Elements', 'NEET', 'NEET 2018']
source_db: MarksBatch2_P2.db
title:
magnesium-reacts-with-an-element-x-to-form-an-ionic-compound-if-the-ground-state-electronic-configuration-of-x-is-1-s-2-2-s-2-2-p-3-the-simplest-formu-1

slug:
magnesium-reacts-with-an-element-x-to-form-an-ionic-compound-if-the-ground-state-electronic-configuration-of-x-is-1-s-2-2-s-2-2-p-3-the-simplest-formu-1-44777

question:
<div class="question">Magnesium reacts with an element <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mi mathvariant="normal">X</mi></mfenced></math> to form an ionic compound. If the ground state electronic configuration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mi mathvariant="normal">X</mi></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><msup><mi mathvariant="normal">s</mi><mn>2</mn></msup><mi mathvariant="normal"> </mi><mo>,</mo><mo> </mo><mn>2</mn><msup><mi mathvariant="normal">s</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mi mathvariant="normal"> </mi><mn>2</mn><msup><mi mathvariant="normal">p</mi><mn>3</mn></msup></math>, the simplest formula for this compound 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"><msub><mi>Mg</mi><mn>2</mn></msub><mi mathvariant="normal">X</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>MgX</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Mg</mi><mn>2</mn></msub><msub><mi mathvariant="normal">X</mi><mn>3</mn></msub></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"><msub><mi>Mg</mi><mn>3</mn></msub><msub><mi mathvariant="normal">X</mi><mn>2</mn></msub></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"><msub><mi>Mg</mi><mn>3</mn></msub><msub><mi mathvariant="normal">X</mi><mn>2</mn></msub></math></span> </div>

answer:
<div class="solution"><p>Magnesium ion <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>Mg</mi><mrow><mo>+</mo><mn>2</mn></mrow></msup></math><br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/neet/nlezjuwtqcfmajvxwjeabtrcjxotceishswutigqvzfjnqlmyl.jpeg"/></p><p>The electron configuration of an element describes how electrons are distributed in its atomic orbitals. Electron configurations of atoms follow a standard notation in which all electron-containing atomic subshells (with the number of electrons they hold written in superscript) are placed in a sequence.</p></div>


