# questions

tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-points-with-the-coordinates-a-ma-b-m-1-b-c-m-2-c-are-collinear-then-which-one-of-the-following-is-correct

slug:
if-the-points-with-the-coordinates-a-ma-b-m-1-b-c-m-2-c-are-collinear-then-which-one-of-the-following-is-correct-53217

question:
<div class="question">If the points with the coordinates $(\mathrm{a}, \mathrm{ma}),\{\mathrm{b},(\mathrm{m}+1) \mathrm{b}\}$, $\{\mathrm{c},(\mathrm{m}+2) \mathrm{c}\}$ are collinear, 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">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in arithmetic progression for all $\mathrm{m}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in geometric progression for all $\mathrm{m}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in harmonic progression for all $\mathrm{m}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in arithmetic progression only for $\mathrm{m}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in harmonic progression for all $\mathrm{m}$</span> </div>

answer:
<div class="solution">As given :<br/>Points $\mathrm{A}(\mathrm{a}, \mathrm{ma}), \mathrm{B}[\mathrm{b},(\mathrm{m}+1) \mathrm{b}]$ and $\mathrm{C}[\mathrm{c},(\mathrm{m}+2) \mathrm{c}]$ are<br/>collinear. $\Rightarrow \quad \mathrm{a}\{(\mathrm{m}+1) \mathrm{b}-(\mathrm{m}+2) \mathrm{c}\}+\mathrm{b}\{(\mathrm{m}+2) \mathrm{c}-\mathrm{ma}\}$<br/>$+\mathrm{c}\{\mathrm{ma}-(\mathrm{m}+1) \mathrm{b}\}=0$<br/>$\Rightarrow \quad \mathrm{mab}+\mathrm{ab}-\mathrm{mac}-2 \mathrm{ac}+\mathrm{mbc}+2 \mathrm{bc}-\mathrm{mab}+\mathrm{mac}$<br/>$\quad-\mathrm{mbc}-\mathrm{bc}=0$<br/>$\Rightarrow \mathrm{ab}-2 \mathrm{ac}+2 \mathrm{bc}-\mathrm{bc}=0$<br/>$\Rightarrow \mathrm{ab}+\mathrm{bc}=2 \mathrm{ac}$<br/>Dividing both the sides by abc, we get<br/>$\frac{1}{c}+\frac{1}{a}=\frac{2}{b}$<br/>$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P<br/>$\Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c}$, are in Harmonic progression for all $\mathrm{m}$.</div>


title: if-the-position-vector-a-of-the-point-5-n-is-such-that-a-13-then-the-valuevalues-of-n-an-be
slug: if-the-position-vector-a-of-the-point-5-n-is-such-that-a-13-then-the-valuevalues-of-n-an-be-92879
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the position vector $\vec{a}$ of the point $(5, \mathrm{n})$ is such that $|\overrightarrow{\mathrm{a}}|=13$, then the value/values of $n$ an be</div>

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

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

answer:
<div class="solution">$\sqrt{(5-0)^{2}+(n-0)^{2}}=13$<br/>$25+n^{2}=169$<br/>$n^{2}=144$<br/>$n=\pm 12$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-position-vector-of-a-point-p-with-respect-to-origin-o-is-i-3-j-2-k-and-that-of-a-point-q-is-3-i-j-2-k-then-what-is-the-position-vector-of-a-poi

slug:
if-the-position-vector-of-a-point-p-with-respect-to-origin-o-is-i-3-j-2-k-and-that-of-a-point-q-is-3-i-j-2-k-then-what-is-the-position-vector-of-a-poi-70267

question:
<div class="question">If the position vector of a point $P$ with respect to origin $O$ is<br/>$\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and that of a point $Q$ is $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, then<br/>what is the position vector of a point on the bisector of the angle $P O Q$ ?</div>

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

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

answer:
<div class="solution">Let $\overrightarrow{\mathrm{OP}}=\hat{i}+3 \hat{j}-2 \hat{k}$ and $\overrightarrow{\mathrm{OQ}}=3 \hat{i}+\hat{j}-2 \hat{k}$<br/>Let $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ be required position vector of the bisector of the angle $P O Q$ since, it is the bisector of<br/>$\angle P O Q$ therefore. It will make equal angles with $\overline{O P}$<br/>and $\overrightarrow{\mathrm{OQ}}$.<br/>Let Angle between $\hat{i}+3 \hat{j}-2 \hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$ is<br/>$\theta=\cos ^{-1}\left(\frac{1+3+2}{\sqrt{1+9+4} \sqrt{1+1+1}}\right)$<br/>$=\cos ^{-1}\left(\frac{6}{\sqrt{14} \sqrt{3}}\right)$<br/>and angle between $3 \hat{i}+\hat{j}-2 \hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$, is<br/>$\phi=\cos ^{-1}\left(\frac{1+3+2}{\sqrt{9+1+4} \sqrt{1+1+1}}\right)=\cos ^{-1}\left(\frac{6}{\sqrt{14} \sqrt{3}}\right)$<br/>Hence, $\theta=\phi$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-position-vectors-of-points-a-and-b-are-3-i-2-j-k-and-2-i-4-j-3-k-respectively-then-what-is-the-length-of-a-b

slug:
if-the-position-vectors-of-points-a-and-b-are-3-i-2-j-k-and-2-i-4-j-3-k-respectively-then-what-is-the-length-of-a-b-87727

question:
<div class="question">If the position vectors of points $\mathrm{A}$ and $\mathrm{B}$ are $3 \hat{i}-2 \hat{j}+\hat{k}$<br/>and $2 \hat{i}+4 \hat{j}-3 \hat{k}$ respectively, then what is the length of<br/>$\overrightarrow{A B} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{14}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{29}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{43}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\sqrt{53}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{53}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \text { Given, } \overrightarrow{O A}=3 \hat{i}-2 \hat{j}+\hat{k} \\ &amp; \overrightarrow{O B}=2 \hat{i}+4 \hat{j}-3 \hat{k} \\ &amp; \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A} \\ &amp;=(2 \hat{i}+4 \hat{j}-3 \hat{k})-(3 \hat{i}-2 \hat{j}+\hat{k}) \\ &amp;=\hat{i}(2-3)+\hat{j}(4+2)+\hat{k}(-3-1) \\ &amp;=-\hat{i}+6 \hat{j}-4 \hat{k} \\ &amp; \text { Length of } \mathrm{AB}=\sqrt{(-1)^{2}+(6)^{2}+(-4)^{2}} \\ &amp;=\sqrt{1+36+16}=\sqrt{53} \end{aligned}$</div>


title: if-the-positive-integers-a-b-c-d-are-in-a-p-then-the-numbers-ab-c-ab-d-a-c-d-b-c-d-are-in
slug: if-the-positive-integers-a-b-c-d-are-in-a-p-then-the-numbers-ab-c-ab-d-a-c-d-b-c-d-are-in-11273
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$H P$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the positive integers $a, b, c, d$ are in $A P$, then the numbers $a b c, a b d, a c d, b c d$ are in</div>

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

answer:
<div class="solution">Given, $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in A.P.<br/>$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in H.P.<br/>$\Rightarrow \frac{1}{d}, \frac{1}{c}, \frac{1}{b}, \frac{1}{a}$ are also in H.P.<br/>Now, multiply each term by abcd.<br/>$\frac{\text { abcd }}{\mathrm{d}}, \frac{\text { abcd }}{\mathrm{c}}, \frac{\text { abcd }}{\mathrm{b}}, \frac{\text { abcd }}{\mathrm{a}}$<br/>abc, abd, acd, bed, are in H.P.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-probability-of-simultaneous-occurrence-of-two-events-a-and-b-is-p-and-the-probability-that-exactly-one-of-a-b-occurs-is-q-then-which-of-the-fol

slug:
if-the-probability-of-simultaneous-occurrence-of-two-events-a-and-b-is-p-and-the-probability-that-exactly-one-of-a-b-occurs-is-q-then-which-of-the-fol-10395

question:
<div class="question">If the probability of simultaneous occurrence of two events A and B is p and the probability that exactly one of $\mathrm{A}$. B occurs is q, then which of the following is/are correct?<br/>$1.$ $\quad \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2 \mathrm{p}-\mathrm{q}$<br/>$2.$ $\quad \mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=1-\mathrm{p}-\mathrm{q}$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">$\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=q$<br/>$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=p$<br/>$\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=2 p+q$<br/>$1-\mathrm{P}(\overline{\mathrm{A}})+1-\mathrm{P}(\overline{\mathrm{B}})=p+q$<br/>$\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2 p-q$<br/>$\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})$<br/>$=1-(q+p)=1-p-q$</div>


title: if-the-product-of-the-roots-of-the-equation-x-2-5-x-k-15-is-3-then-what-is-the-value-of-k
slug: if-the-product-of-the-roots-of-the-equation-x-2-5-x-k-15-is-3-then-what-is-the-value-of-k-43926
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the product of the roots of the equation $x^{2}-5 x+k=15$ is<br/>$-3$, then what is the value of $k$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">15</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">18</span> </li> </ul>

answer:
<div class="solution">The given quadratic equation is<br/>$x^{2}-5 x+k=15 \Rightarrow x^{2}-5 x+k-15=0$<br/>Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-5 x+k-15=0$<br/>Now, product of roots $=\alpha \beta=k-15$ But $\alpha \beta=-3 \Rightarrow-3=k-15$<br/>$\Rightarrow k=15-3=12$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-projections-of-a-straight-line-segment-on-the-coordinate-axes-are-2-3-6-then-the-length-of-the-segment-is

slug:
if-the-projections-of-a-straight-line-segment-on-the-coordinate-axes-are-2-3-6-then-the-length-of-the-segment-is-45098

question:
<div class="question">If the projections of a straight line segment on the coordinate axes are $2,3,6$, then the length of the segment is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5 units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">7 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">11 units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">49 units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">7 units</span> </div>

answer:
<div class="solution">Position vector of line segment $=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$<br/>length $=\sqrt{2^{2}+3^{2}+6^{2}}=\sqrt{49}=7$ units</div>


title: if-the-radius-of-the-sphere-x-2-y-2-z-2-6-x-8-y-10-z-0-is-unity-what-is-the-value-of
slug: if-the-radius-of-the-sphere-x-2-y-2-z-2-6-x-8-y-10-z-0-is-unity-what-is-the-value-of-45661
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">49</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the radius of the sphere $x^{2}+y^{2}+z^{2}-6 x-8 y+10 z+\lambda=0$ is unity, what is the value of $\lambda ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">49</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-49$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-7$</span> </li> </ul>

answer:
<div class="solution">Given sphere : $x^{2}+y^{2}+z^{2}-6 x-8 y+10 z+\lambda=0$ Its radius $=1$<br/>$\Rightarrow \sqrt{(-3)^{2}+\left(-4^{2}\right)+(5)^{2}-\lambda}=1$<br/>$\Rightarrow 9+16+25-\lambda=1$<br/>$\therefore \lambda=49$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-rate-of-change-in-volume-of-spherical-soap-bubble-is-uniform-then-the-rate-of-change-of-surface-area-varies-as

slug:
if-the-rate-of-change-in-volume-of-spherical-soap-bubble-is-uniform-then-the-rate-of-change-of-surface-area-varies-as-25038

question:
<div class="question">If the rate of change in volume of spherical soap bubble is uniform, then the rate of change of surface area varies as</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">inversely proportional to radius</span> </div>

answer:
<div class="solution">Let volume $=\mathrm{V}=\frac{4}{3} \pi r^{3}$ and surface area $=\mathrm{S}=4 \pi r^{2}$<br/>Now, $(1) \Rightarrow \frac{d v}{d t}=\frac{4}{3} \times 3 \pi r^{2} \times \frac{d r}{d t}$<br/>$\quad=4 \pi r^{2} \frac{d r}{d t}$<br/>(2) $\Rightarrow \frac{d s}{d t}=4 \pi \times 2 \times r \frac{d r}{d t}=\frac{8 \pi r^{2}}{r} \frac{d r}{d t}$<br/>$=\frac{2}{r}\left[4 \pi r^{2} \frac{d r}{d t}\right]=\frac{2}{r} \frac{d v}{d t} \quad($ from 3$)$</div>


title: if-the-ratio-of-am-to-gm-of-two-positive-numbers-a-and-b-is-5-3-then-a-b-is-equal-to
slug: if-the-ratio-of-am-to-gm-of-two-positive-numbers-a-and-b-is-5-3-then-a-b-is-equal-to-91321
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$9: 1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the ratio of AM to GM of two positive numbers $a$ and $b$ is $5: 3$, then $\mathrm{a}: \mathrm{b}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3: 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2: 9$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$9: 1$</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">$5: 3$</span> </li> </ul>

answer:
<div class="solution">A.M of two numbers a, $\mathrm{b}=\frac{\mathrm{a}+\mathrm{b}}{2}$<br/>G. $\mathrm{M}$ of two numbers $\mathrm{a}, \mathrm{b}=\sqrt{\mathrm{ab}}$.<br/>Given, $\frac{\mathrm{A} \cdot \mathrm{M}}{\mathrm{G} \cdot \mathrm{M}}=\frac{5}{3}$<br/>$\Rightarrow \frac{\left(\frac{a+b}{2}\right)}{\sqrt{a b}}=\frac{5}{3}$<br/>$\Rightarrow \frac{a+b}{\sqrt{a b}}=\frac{10}{3}$<br/>$\Rightarrow \frac{\mathrm{a}^{2}+\mathrm{b}^{2}+2 \mathrm{ab}}{\mathrm{ab}}=\frac{100}{9} \quad$ (Squaring on both sides)<br/>$\Rightarrow \quad 9 a^{2}+9 b^{2}+18 a b=100 a b$<br/>$\Rightarrow \quad 9 a^{2}-82 a b+9 b^{2}=0$<br/>$\Rightarrow \quad 9 a^{2}-81 a b-a b+9 b^{2}=0$<br/>$\Rightarrow \quad 9 a(a-9 b)-b(a-9 b)=0$<br/>$\Rightarrow \quad(9 a-b)(a-9 b)=0$<br/>$\Rightarrow \quad 9 a-b=0 ; a-9 b=0$<br/>$\Rightarrow \mathrm{b}=9 \mathrm{a} ; \mathrm{a}=9 \mathrm{~b}$<br/>$\Rightarrow \frac{\mathrm{a}}{\mathrm{b}}=\frac{1}{9} ; \frac{\mathrm{a}}{\mathrm{b}}=\frac{9}{1}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-regression-coefficient-of-x-on-vand-x-are-2-1-and-2-1-respectively-then-what-is-the-correlation-coefficient-between-x-and-y

slug:
if-the-regression-coefficient-of-x-on-vand-x-are-2-1-and-2-1-respectively-then-what-is-the-correlation-coefficient-between-x-and-y-15053

question:
<div class="question">If the regression coefficient of $x$ on $\mathrm{vand}$ $x$ are $-\frac{1}{2}$<br/>and $-\frac{1}{2}$ respectively, then what is the correlation<br/>coefficient between $\mathrm{x}$ and $\mathrm{y} ?$</div>

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

answer:
<div class="solution">Correlation coefficient $=\sqrt{\frac{-1}{2} \times \frac{-1}{8}}$<br/>$\sqrt{\frac{1}{16}}=-\frac{1}{4}$<br/>Since, both the regression coefficients are negative, correlation coefficient is negative.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-regression-coefficient-of-yon-x-is-6-and-the-correlation-coefficient-between-x-and-y-2-1-then-the-regression-coefficient-of-x-on-y-would-be

slug:
if-the-regression-coefficient-of-yon-x-is-6-and-the-correlation-coefficient-between-x-and-y-2-1-then-the-regression-coefficient-of-x-on-y-would-be-17640

question:
<div class="question">If the regression coefficient of Yon $\mathrm{X}$ is $-6$, and the correlation coefficient between $\mathrm{X}$ and $\mathrm{Y}-\frac{1}{2}$, then the regression coefficient of $X$ on Y would be $\quad$</div>

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

answer:
<div class="solution">$b_{y x}=-6, r=-\frac{1}{2}$<br/>$\left(-\frac{1}{2}\right)^{2}=-6 \times b_{x y}$<br/>$\Rightarrow b_{x y}=-\frac{1}{24}$</div>


title: if-the-roots-of-a-x-2-b-x-c-0-are-sin-and-cos-for-some-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-roots-of-a-x-2-b-x-c-0-are-sin-and-cos-for-some-then-which-one-of-the-following-is-correct-40688

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

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

answer:
<div class="solution">Let $\sin \alpha$ and $\cos \alpha$ be the roots of $a x^{2}+b x+c=0$ Now, $\sin \alpha+\cos \alpha=\frac{-b}{a}$ and $\sin \alpha \cos \alpha=\frac{c}{a}$<br/>Consider $\sin \alpha+\cos \alpha=\frac{-b}{a}$<br/>Squaring both side, $(\sin \alpha+\cos \alpha)^{2}=\frac{b^{2}}{a^{2}}$<br/>$\Rightarrow \sin ^{2} \alpha+\cos ^{2} \alpha+2 \sin \alpha \cos \alpha=\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}$<br/>$\Rightarrow 1+\frac{2 c}{a}=\frac{b^{2}}{a^{2}}$<br/>$\Rightarrow \frac{a+2 c}{a}=\frac{b^{2}}{a^{2}} \Rightarrow a+2 c=\frac{\mathrm{b}^{2}}{\mathrm{a}}$<br/>$\Rightarrow a^{2}+2 a c=b^{2} <br/>\Rightarrow b^{2}-a^{2}=2 a c$</div>


title: if-the-roots-of-the-equation-3-x-2-5-x-q-0-are-equal-then-what-is-the-value-of-q
slug: if-the-roots-of-the-equation-3-x-2-5-x-q-0-are-equal-then-what-is-the-value-of-q-65203
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the roots of the equation $3 x^{2}-5 x+q=0$ are equal, then what is the value of $q$ ?</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">$5 \sqrt{1}$.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$12 / 25$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$25 / 12$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$25 / 12$</span> </div>

answer:
<div class="solution">Givenquadratic equation is $3 x^{2}-5 x+q=0$<br/>Since, roots of this equation are equal therefo $b^{2}-4 a c=0$<br/>$\Rightarrow(-5)^{2}-4(3)(q)=0$<br/>$\Rightarrow 25-12 q=0$<br/>$\Rightarrow q=\frac{25}{12}$</div>


title: if-the-roots-of-the-equation-4-2-2-0-are-of-the-from-k-1-k-and-k-2-k-1-then-what-is-the-value-of
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-roots-of-the-equation-4-2-2-0-are-of-the-from-k-1-k-and-k-2-k-1-then-what-is-the-value-of-81687

question:
<div class="question">If the roots of the equation $4 \beta^{2}+\lambda \beta-2=0$ are of the from<br/>$\frac{\mathrm{k}}{\mathrm{k}+1}$ and $\frac{\mathrm{k}+1}{\mathrm{k}+2}$, then what is the value of $\lambda$ ?</div>

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

answer:
<div class="solution">Let $\frac{\mathrm{k}}{\mathrm{k}+1}$ and $\frac{\mathrm{k}+1}{\mathrm{k}+2}$ are the roots of the equation $4 \beta^{2}+\lambda, \beta-2=0$, then<br/>Sum of the roots $=\frac{k}{k+1}+\frac{k+1}{k+2}=-\frac{\lambda}{4}$...(i)<br/>and product of the roots, $\frac{k}{k+1} \times \frac{k+1}{k+2}=-\frac{2}{4}$<br/>$\Rightarrow \frac{\mathrm{k}}{\mathrm{k}+2}=-\frac{1}{2} \Rightarrow 2 \mathrm{k}=-\mathrm{k}-2 \Rightarrow \mathrm{k}=-\frac{2}{3}$<br/>Putting the value of $\mathrm{k}$ in(i), we get $\frac{-\frac{2}{3}}{-\frac{2}{3}+1}+\frac{-\frac{2}{3}+1}{-\frac{2}{3}+2}=-\frac{\lambda}{4}$<br/>$\Rightarrow \frac{-\frac{2}{3}}{\frac{1}{3}}+\frac{\frac{1}{3}}{\frac{4}{3}}=-\frac{\lambda}{4} \Rightarrow-2+\frac{1}{4}=-\frac{\lambda}{4}$<br/>$\Rightarrow \lambda=7$</div>


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

slug:
if-the-roots-of-the-equation-a-2-b-2-x-2-2-b-a-c-x-b-2-c-2-0-are-equal-then-which-one-of-the-following-is-correct-81043

question:
<div class="question">If the roots of the equation $\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0$ are equal, then<br/>which one of the following is correct?</div>

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

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

answer:
<div class="solution">Since, the roots of the equation $\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0$ are equal.<br/>therefore, discriminant $=0$<br/>$\Rightarrow[2 b(a+c)]^{2}-4\left(a^{2}+b^{2}\right)\left(b^{2}+c^{2}\right)=0$<br/>$4 b^{2}(a+c)^{2}=4\left(a^{2}+b^{2}\right)\left(b^{2}+c^{2}\right)$<br/>$\Rightarrow b^{2}\left(a^{2}+c^{2}+2 a c\right)=a^{2} b^{2}+b^{2} c^{2}+a^{2} c^{2}+b^{4}$<br/>$\Rightarrow a^{2} b^{2}+b^{2} c^{2}+2 a c b^{2}=a^{2} b^{2}+b^{2} c^{2}+a^{2} c^{2}+b^{4}$<br/>$\Rightarrow b^{4}+a^{2} c^{2}-2 a c b^{2}=0$<br/>$\Rightarrow\left(b^{2}-a c\right)^{2}=0 \Rightarrow b^{2}=a c$</div>


title: if-the-roots-of-the-equation-x-2-4-x-lo-g-3-n-0-are-real-then-what-is-the-minimum-value-of-n
slug: if-the-roots-of-the-equation-x-2-4-x-lo-g-3-n-0-are-real-then-what-is-the-minimum-value-of-n-84892
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the roots of the equation $x^{2}-4 x-\log _{3} N=0$ are real, then what is the minimum value of $N ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 256$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 27$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 81$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 6$</span> </div>

answer:
<div class="solution">Given equation is $x^{2}-4 x-\log _{3} N=0$<br/>Since, roots are real $\therefore b^{2}-4 a c=0 \Rightarrow(4)^{2}-4\left(-\log _{3} N\right) \geq 0$<br/>$\Rightarrow 16 \geq-4 \log _{3} N$<br/>$\Rightarrow 4 \geq-\log _{3} N$<br/>$\Rightarrow 4 \geq \log _{3} N^{-1}$<br/>$\Rightarrow N^{-1} \geq 3^{4} \geq 81$<br/>$\Rightarrow N \geq \frac{1}{81}$<br/>Hence, minimum val ue of $\mathrm{N}$ is $\frac{1}{81}$.</div>


title: if-the-roots-of-the-equation-x-2-n-x-m-0-differ-by-1-then
slug: if-the-roots-of-the-equation-x-2-n-x-m-0-differ-by-1-then-37356
question: <div class="question">If the roots of the equation $x^{2}-n x+m=0$ differ by 1, then</div>
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{n}^{2}-4 \mathrm{~m}-1=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{n}^{2}+4 \mathrm{~m}-1=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{m}^{2}+4 \mathrm{n}+1=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{m}^{2}-4 \mathrm{n}-1=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{n}^{2}-4 \mathrm{~m}-1=0$</span> </div>

answer:
<div class="solution">Let the root be $\alpha$ and $\beta$ $x^{2}-n x+m=0$<br/>$\Rightarrow \alpha+\beta=n ; \alpha . \beta=\mathrm{m}$<br/>$\alpha-\beta=$<br/>$\Rightarrow(\alpha+\beta)^{2}=(\alpha-\beta)^{2}+4 \alpha \beta$<br/>$\Rightarrow \mathrm{n}^{2}=1+4 \mathrm{~m}$<br/>- $n^{2}-4 m-1=0$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-roots-of-the-equation-x-2-p-x-q-0-are-in-the-same-ratio-as-those-of-the-equation-x-2-b-x-m-0-then-which-one-of-the-following-is-correct

slug:
if-the-roots-of-the-equation-x-2-p-x-q-0-are-in-the-same-ratio-as-those-of-the-equation-x-2-b-x-m-0-then-which-one-of-the-following-is-correct-87401

question:
<div class="question">If the roots of the equation $x^{2}+p x+q=0$ are in the same ratio as those of the equation $x^{2}+b x+m=0$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{p}^{2} \mathrm{~m}=R^{2} \mathrm{q}$</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">B</span> <span class="option-data">$\mathrm{m}^{2} \mathrm{p}=\hat{\mathrm{p}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$m^{2} p=q^{2} y$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$m^{2} p^{2}=R^{2} q$</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{a}, \mathrm{b}$ be the roots of $x^{2}+p / x^{2}+q=0$<br/>$\mathrm{So}, \mathrm{a}+\mathrm{b}=-\mathrm{p} / \mathrm{x}^{2}, \mathrm{ab}=\mathrm{q}$...(i)<br/>Let $c, d$ be the roots of $x^{2}+b x+m=0$<br/>So, $\mathrm{c}+\mathrm{d}=-1, \mathrm{~cd}=\mathrm{m}$...(ii)<br/>Given that roots of both the equations are in the<br/>Same ratio<br/>So, $\frac{a}{b}=\frac{c}{d} \quad$...(iii)<br/>$\Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{d}}{\mathrm{c}}\ldots(\mathrm{iv})$<br/>$(\mathrm{iii})+(\mathrm{iv}) \Rightarrow \frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{d}}+\frac{\mathrm{d}}{\mathrm{c}}$<br/>$\Rightarrow \frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{ab}}=\frac{\mathrm{c}^{2}+\mathrm{d}^{2}}{\mathrm{ed}} \Rightarrow \frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{ab}}+2=\frac{\mathrm{c}^{2}+\mathrm{d}^{2}}{\mathrm{~cd}}+2$<br/>$\Rightarrow \frac{\mathrm{a}^{2}+\mathrm{b}^{2}+2 \mathrm{ab}}{\mathrm{ab}}=\frac{\mathrm{c}^{2}+\mathrm{d}^{2}+2 \mathrm{~cd}}{\mathrm{~cd}}$<br/>$\Rightarrow \frac{(\mathrm{a}+\mathrm{b})^{2}}{\mathrm{ab}}=\frac{(\mathrm{c}+\mathrm{d})^{2}}{\mathrm{ed}}$<br/>$\Rightarrow \frac{(-\mathrm{P})^{2}}{\mathrm{q}}=\frac{(-1)^{2}}{\mathrm{~m}} \quad($ from (i) and(ii))<br/>$\Rightarrow \mathrm{P}^{2} \mathrm{~m}=1^{2} \mathrm{q}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-roots-of-the-equation-x-2-p-x-q-0-are-tan-1-9-and-tan-2-6-then-which-one-of-the-following-is-correct

slug:
if-the-roots-of-the-equation-x-2-p-x-q-0-are-tan-1-9-and-tan-2-6-then-which-one-of-the-following-is-correct-88605

question:
<div class="question">If the roots of the equation $x^{2}+p x+q=0$ are $\tan 19^{\circ}$ and $\tan 26^{\circ}$, 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">$q-p=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}-\mathrm{q}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{p}+\mathrm{q}=2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{p}+\mathrm{q}=3$</span> </li> </ul>

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

answer:
<div class="solution">Given, $\tan 19^{\circ}$ and $\tan 26^{\circ}$ are roots of $x^{2}+p x+q=0$<br/>$\therefore \tan 19^{\circ}+\tan 26^{\circ}=\frac{-p}{1}=-p$<br/>$\left(\tan 19^{\circ}\right)\left(\tan 26^{\circ}\right)=\frac{q}{1}=q$<br/>$\tan \left(19^{\circ}+26^{\circ}\right)=\frac{\tan 19^{\circ}+\tan 26^{\circ}}{1-\tan 19^{\circ} \tan 26^{\circ}}$<br/>$\tan 45^{\circ}=\frac{-p}{1-q} \Rightarrow 1=\frac{-p}{1-q}$<br/>$\Rightarrow 1-q=-p$<br/>$\Rightarrow q-p=1$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-roots-of-the-equations-x-2-a-1-x-a-b-0-and-a-x-2-2-x-b-0-are-identical-then-what-are-the-values-of-a-and-b

slug:
if-the-roots-of-the-equations-x-2-a-1-x-a-b-0-and-a-x-2-2-x-b-0-are-identical-then-what-are-the-values-of-a-and-b-92204

question:
<div class="question">If the roots of the equations $x^{2}-(a-1) x+(a+b)=0$ and $a x^{2}-2 x+b=0$ are identical, then what are the values of a and $\mathrm{b}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a=2, b=4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a=2, b=-4$</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">$\mathrm{a}=1, \mathrm{~b}=\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{a}=-1, \mathrm{~b}=-\frac{1}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Let $\alpha$ and $\beta$ be the roots of both the equations $x^{2}-(a-1) x+(a+b)=0$<br/>$\Rightarrow \alpha+\beta=(a-1)$ and $\alpha \beta=(a+b)$<br/>and $a x^{2}-2 x+b=0$<br/>$\Rightarrow \alpha+\beta=\frac{2}{a}$ and $\alpha \beta=\frac{b}{a}$<br/>Equating the sums of roots<br/>$a^{2}-a-2=0 \Rightarrow a=-1,2$<br/>Equating the products of roots and<br/>and if $a=2, b=$ $1=$<br/>From the given option, $=-4$ matches.</div>


title: if-the-roots-of-x-2-2-m-x-m-2-1-0-lie-between-2-and-4-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-the-roots-of-x-2-2-m-x-m-2-1-0-lie-between-2-and-4-then-which-one-of-the-following-is-correct-48144

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

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

answer:
<div class="solution">Since, the roots of $x^{2}-2 m x+m^{2}-1=0$ lie between<br/>$-2$ and 4 i.e., $\mathrm{b} 2-4 \mathrm{ac} \geq 0$ and $-2   &lt; \frac{-\mathrm{b}}{2 \mathrm{a}} &lt; 4$<br/>$(2 m)^{2}-4\left(m^{2}-1\right) \geq 0$...(1)<br/>and $-2 &lt; \frac{2 \mathrm{~m}}{2} &lt; 4$<br/>$\Rightarrow-2 &lt; m &lt; 4$<br/>From (i) $4 m^{2}-4 m^{2}+4 \geq 0$<br/>$\Rightarrow m \in R$<br/>and $\mathrm{f}(-2)&gt;0$, also $\mathrm{f}(4)&gt;0$<br/>$4+4 m+m^{2}-1&gt;0,16-8 m+m^{2}-1&gt;0$<br/>$\Rightarrow \mathrm{m}^{2}+4 \mathrm{~m}+3&gt;0, \mathrm{~m}^{2}-8 \mathrm{~m}+15&gt;0$<br/>$\Rightarrow(m+1)(m+3)&gt;0,(m-3)(m-5)&gt;0$<br/>$\Rightarrow-3 &lt; \mathrm{m} &lt; -1$ and $3 &lt; \mathrm{m} &lt; 5$<br/>Thus, the inteval in which it lies is $-1 \leq \mathrm{m} \leq 5$</div>


title: if-the-second-term-of-a-gp-is-2-and-the-sum-of-its-infinite-term-is-8-then-the-gp-is
slug: if-the-second-term-of-a-gp-is-2-and-the-sum-of-its-infinite-term-is-8-then-the-gp-is-49642
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the second term of a GP is 2 and the sum of its infinite term is 8, then the GP is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8,2, \frac{1}{2}, \frac{1}{8}, \ldots . .$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10,2, \frac{2}{5}, \frac{2}{25}, \ldots .$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4,2,1, \frac{1}{2}, \frac{1}{2^{2}}, \ldots . .$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$6,3, \frac{3}{2}, \frac{3}{4}, \ldots . .$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4,2,1, \frac{1}{2}, \frac{1}{2^{2}}, \ldots . .$</span> </div>

answer:
<div class="solution">Let the first term and common ratio of the G.P. is a and<br/>r respectively.<br/>$\begin{aligned} \text { Then, } a r=2, \frac{a}{1-r}=&amp; 8 \\ \Rightarrow \quad \frac{a r}{r-r^{2}}=8 \quad \Rightarrow \quad 2=8 r-8 r^{2} \\ \Rightarrow \quad 4 r^{2}-4 r+1=0 \quad \Rightarrow \quad r=\frac{1}{2} \\ \therefore \text { GP. is } 4,2,1, \frac{1}{2}, \ldots . \end{aligned}$</div>


title: if-the-sequence-s-n-is-a-geometric-progression-and-s-2-s-11-s-p-s-8-then-what-is-the-value-of-p
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-sequence-s-n-is-a-geometric-progression-and-s-2-s-11-s-p-s-8-then-what-is-the-value-of-p-57710

question:
<div class="question">If the sequence $\left\{\mathrm{S}_{\mathrm{n}}\right\}$ is a geometric progression and $\mathrm{S}_{2} \mathrm{~S}_{11}=\mathrm{S}_{\mathrm{p}} \mathrm{S}_{8}$, 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">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">cannot be determined</span> </li> </ul>

answer:
<div class="solution">Given $\mathrm{S}_{2} \mathrm{~S}_{11}=\mathrm{S}_{\mathrm{p}} \mathrm{S}_{8}$<br/>$\Rightarrow($ ar $)\left(\mathrm{ar}^{10}\right)=\mathrm{ar}^{\mathrm{P}-1}\left(\mathrm{ar}^{7}\right)$<br/>where 'a' is the first term and ' $\mathrm{r}$ ' is the common ratio of GP. $\Rightarrow r^{11}=r^{7+p-1} \Rightarrow r^{11}=r^{6+p}$<br/>$\Rightarrow 11=6+\mathrm{p} \Rightarrow \mathrm{p}=5$</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-sides-of-a-triangle-are-6-cm-10-cm-and-14-cm-then-what-is-the-largest-angle-included-by-the-sides

slug:
if-the-sides-of-a-triangle-are-6-cm-10-cm-and-14-cm-then-what-is-the-largest-angle-included-by-the-sides-44129

question:
<div class="question">If the sides of a triangle are $6 \mathrm{~cm}, 10 \mathrm{~cm}$ and $14 \mathrm{~cm}$, then what is the largest angle included by the sides?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$90^{\circ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$120^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$135^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$150^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$120^{\circ}$</span> </div>

answer:
<div class="solution">We know that largest side has greatest angle opposite it. $\therefore \mathrm{a}=14 \mathrm{~cm}, \mathrm{~b}=10 \mathrm{~cm}$ and $\mathrm{c}=6 \mathrm{~cm}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/HchwuJBvEvyoZM6aS5VW7rJ8eoSGkFqOsi-WqWOb4iE.original.fullsize.png"/><br/><br/>$\therefore \cos A=\frac{c^{2}+b^{2}-a^{2}}{2 b c}$<br/>$=\frac{36+100-196}{2 \times 6 \times 10}=-\frac{1}{2}=\cos 120^{\circ}$<br/>$\Rightarrow \angle \mathrm{A}=120^{\circ}$</div>


title: if-the-sides-of-a-triangle-are-in-the-ratio-2-6-1-3-then-what-is-the-smallest-angle-of-the-triangle
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-sides-of-a-triangle-are-in-the-ratio-2-6-1-3-then-what-is-the-smallest-angle-of-the-triangle-22360

question:
<div class="question">If the sides of a triangle are in the ratio $2: \sqrt{6}: 1+\sqrt{3}$, then what is the smallest angle of the triangle?</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/SSqKRXT-jX-viH66FFXzj48B4IPuZNiJmVFtqC9hg-I.original.fullsize.png"/><br/><br/>Let $\mathrm{ABC}$ be a triangle with sides $\mathrm{a}=1+\sqrt{3}, \mathrm{~b}=2$ and<br/>$c=\sqrt{6}$<br/>So, $\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}=\frac{(1+\sqrt{3})^{2}+(\sqrt{6})^{2}-4}{2(1+\sqrt{3})(\sqrt{6})}$<br/>$=\frac{2 \sqrt{3}+6}{2 \sqrt{6}+\sqrt{18}}=\frac{3+\sqrt{3}}{\sqrt{6}+3 \sqrt{2}}$<br/>$=\frac{\sqrt{3} \sqrt{3}+\sqrt{3}}{(\sqrt{3}+\sqrt{3} \sqrt{3}) \sqrt{2}}=\frac{1}{\sqrt{2}}$<br/>$\Rightarrow B=45^{\circ}$ is the smallest angle. <br/>$(\because$ smallest side is $b=2$ )</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-slopes-of-the-line-of-regression-of-y-and-x-and-of-x-and-y-are-3-0-and-6-0-respectively-then-r-x-y-is

slug:
if-the-slopes-of-the-line-of-regression-of-y-and-x-and-of-x-and-y-are-3-0-and-6-0-respectively-then-r-x-y-is-93604

question:
<div class="question">If the slopes of the line of regression of $Y$ and $X$ and of $X$ and $\mathrm{Y}$ are $30^{\circ}$ and $60^{\circ}$ respectively, then $\mathrm{r}(\mathrm{X}, \mathrm{Y})$ is:</div>

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

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

answer:
<div class="solution">Slope of line of regression of $\mathrm{Y}$ and $\mathrm{X}$, is $30^{\circ}$. So $\mathrm{b}_{\mathrm{yx}}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$ and for $\mathrm{X}$ and $\mathrm{Y}$ it is $60^{\circ}$<br/>Hence, $\frac{1}{\mathrm{~b}_{\mathrm{xy}}}=\tan 60^{\circ}=\sqrt{3}$<br/>$b_{y x}=\frac{1}{\sqrt{3}}$ and $b_{x y}=\frac{1}{\sqrt{3}}$<br/>$\mathrm{r}(\mathrm{x}, \mathrm{y})=\mathrm{r}^{2}=\mathrm{b}_{\mathrm{yx}} \cdot \mathrm{b}_{\mathrm{xy}}=\frac{1}{3}$<br/>so, $\mathrm{r}=\pm \frac{1}{\sqrt{3}}$<br/>Since, $b_{y x}$ and $b_{x y}$ are both positive, $r=+\frac{1}{\sqrt{3}}$</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-2$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-solution-of-the-differential-equation-d-x-d-y-2-y-f-a-x-3-represents-a-circle-then-what-is-the-value-of-a

slug:
if-the-solution-of-the-differential-equation-d-x-d-y-2-y-f-a-x-3-represents-a-circle-then-what-is-the-value-of-a-59803

question:
<div class="question">If the solution of the differential equation $\quad$<br/>$\frac{d y}{d x}=\frac{a x+3}{2 y+f}$<br/>represents a circle, then what is the value of $a$ ?</div>

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

answer:
<div class="solution">Given differential equation is<br/>$$<br/>\frac{d y}{d x}=\frac{a x+3}{2 y+f}<br/>$$<br/>By separating the variable we get $(2 y+f) d y=(a x+3) d x$<br/>Integrate on both side,<br/>$$<br/>\begin{array}{l}<br/>\int(2 y+f) d y=\int(a x+3) d x \\<br/>\Rightarrow \quad y^{2}+f y=\frac{a x^{2}}{2}+3 x<br/>\end{array}<br/>$$<br/>This equation represents a circle, if<br/>$$<br/>-1=\frac{a}{2} \Rightarrow a=-2<br/>$$</div>


tags: ['Physics', 'Units and Dimensions', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-the-speed-of-light-c-acceleration-due-to-gravity-g-and-pressure-p-are-taken-as-the-fundamental-quantities-then-the-dimension-of-gravitational-const

slug:
if-the-speed-of-light-c-acceleration-due-to-gravity-g-and-pressure-p-are-taken-as-the-fundamental-quantities-then-the-dimension-of-gravitational-const-60604

question:
<div class="question">If the speed of light <math> <mrow> <mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math>, acceleration due to gravity <math> <mrow> <mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow> </math> and pressure <math> <mrow> <mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow> </math> are taken as the fundamental quantities, then the dimension of gravitational constant is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <math> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>g</mi> <mn>0</mn> </msup> <msup> <mi>p</mi> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> </mrow> </math> </span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math> <mrow> <msup> <mi>c</mi> <mn>0</mn> </msup> <msup> <mi>g</mi> <mn>2</mn> </msup> <msup> <mi>p</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> </mrow> </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> <mrow> <mi>c</mi><msup> <mi>g</mi> <mn>3</mn> </msup> <msup> <mi>p</mi> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> </mrow> </math> </span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <math> <mrow> <msup> <mi>c</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <msup> <mi>g</mi> <mn>0</mn> </msup> <msup> <mi>p</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> </mrow> </math> </span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math> <mrow> <msup> <mi>c</mi> <mn>0</mn> </msup> <msup> <mi>g</mi> <mn>2</mn> </msup> <msup> <mi>p</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> </mrow> </math></span> </div>

answer:
<div class="solution"><p>Let <math> <mrow> <mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>∝</mo><msup> <mi>c</mi> <mi>x</mi> </msup> <msup> <mi>g</mi> <mi>y</mi> </msup> <msup> <mi>p</mi> <mi>z</mi> </msup> </mrow> </math></p><p>by substituting the following dimensions :</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi mathvariant="normal">G</mi><mo>]</mo><mo>=</mo><mo>[</mo><msup><mi mathvariant="normal">M</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">L</mi><mn>3</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>]</mo><mo>,</mo><mspace linebreak="newline"></mspace><mo>[</mo><mi mathvariant="normal">c</mi><mo>]</mo><mo>=</mo><mo>[</mo><msup><mi>LT</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>,</mo><mspace linebreak="newline"></mspace><mo>[</mo><mi mathvariant="normal">g</mi><mo>]</mo><mo>=</mo><mo>[</mo><msup><mi>LT</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi mathvariant="normal">p</mi><mo>]</mo><mo>=</mo><mo>[</mo><msup><mi>ML</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>]</mo></math></p><p>and by comparing the powers of both sides we can get <math> <mrow> <mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mtext> </mtext><mi>z</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow> </math><br/>∴ <math> <mrow> <mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>∝</mo><msup> <mi>c</mi> <mn>0</mn> </msup> <msup> <mi>g</mi> <mn>2</mn> </msup> <msup> <mi>p</mi> <mrow> <mo>−</mo><mn>1</mn></mrow> </msup> </mrow> </math></p></div>


title: if-the-standard-deviation-of-0-1-2-3-wes-9-is-k-then-the-standard-deviation-of-10-11-12-13-19-is
tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$K$</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-standard-deviation-of-0-1-2-3-wes-9-is-k-then-the-standard-deviation-of-10-11-12-13-19-is-38542

question:
<div class="question">If the standard deviation of $0,1,2,3$, wes 9 is $K$, then the standard deviation of $10,11,12,13 \ldots . .19$ is</div>

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

answer:
<div class="solution">Since each data increases by 10 ,<br/>There will be no change in standard deviation because $\left(\mathrm{x}_{\mathrm{i}}-\mathrm{x}\right.$) remains same.<br/>$\therefore$ The standard deviation of $10,11,12,13 \ldots\ldots$ 19 will be will be $K$.</div>


title: if-the-standard-deviation-of-0-1-2-9-is-k-then-the-standard-deviation-of-10-11-12-19-is
slug: if-the-standard-deviation-of-0-1-2-9-is-k-then-the-standard-deviation-of-10-11-12-19-is-60645
tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>If the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mo> </mo><mn>9</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, then the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>,</mo><mo> </mo><mn>11</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo>,</mo><mo> </mo><mn>19</mn></math> is</p> </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"><mi>k</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mn>10</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math> <mrow> <mtext>k</mtext> <mo>+</mo> <msqrt> <mn>1</mn> <mn>0</mn> </msqrt> </mrow> </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>10</mn><mi>k</mi></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>k</mi></math></span> </div>

answer:
<div class="solution">Here, we see that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> is added in each observation of the first data.<br/><br/>Since we know that SD does not depend on change of origin.<br/><br/>Hence, SD of second data is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</div>


title: if-the-standard-deviation-of-data-is-12-and-mean-is-72-then-coefficient-of-variation-is-1
slug: if-the-standard-deviation-of-data-is-12-and-mean-is-72-then-coefficient-of-variation-is-1-17469
tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the standard deviation of data is 12 and mean is 72 , then coefficient of variation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$15.67 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$14.67 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$13.67 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$16.67 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$16.67 \%$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \text { Coefficient of variation }=\frac{\text { Standard Deviation }}{\text { Mean }} \times 100 \\ &amp; \frac{12}{72} \times 100 \%=16.67 \%\end{aligned}$</div>


title: if-the-standard-deviation-of-first-n-natural-numbers-is-2-then-the-value-of-n-is-1
slug: if-the-standard-deviation-of-first-n-natural-numbers-is-2-then-the-value-of-n-is-1-19219
tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the standard deviation of first $n$ natural numbers is 2 , then the value of $n$ is</div>

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

answer:
<div class="solution">$\begin{aligned} &amp; \text { S.D. }=\sqrt{\frac{\sum x_i^2}{n}-(\bar{x})^2} \\ &amp; \Rightarrow 2=\sqrt{\frac{\sum n^2}{n}-\left(\frac{\sum n}{n}\right)^2} \\ &amp; \Rightarrow 4=\frac{n(n+1)(2 n+1)}{6 n}-\left(\frac{n(n+1)}{2 n}\right)^2 \\ &amp; \Rightarrow 4=\frac{n+1}{2}\left\{\frac{2 n+1}{3}-\frac{n+1}{2}\right\} \\ &amp; \Rightarrow 4=\frac{n+1}{2}\left(\frac{n-1}{6}\right) \\ &amp; \Rightarrow n^2=49 \Rightarrow n=7\end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-the-standard-deviation-of-n-observations-x-1-x-2-x-n-is-5-and-for-another-set-of-n-observations-y-1-y-2-y-n-is-4-then-the-standard-deviation-of-n-o

slug:
if-the-standard-deviation-of-n-observations-x-1-x-2-x-n-is-5-and-for-another-set-of-n-observations-y-1-y-2-y-n-is-4-then-the-standard-deviation-of-n-o-17296

question:
<div class="question">If the standard deviation of <math><mi>n</mi></math> observations <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is <math><mn>5</mn></math> and for another set of <math><mi>n</mi></math> observations <math><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo></math> then the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> observations <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is</div>

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

answer:
<div class="solution"><p><math><msup><mrow><mfenced separators="|"><mrow><mi>S</mi><mo>.</mo><mi>D</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover><mo>+</mo><mover accent="false"><mrow><mi>y</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></math><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>y</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>y</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow></mrow></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>-</mo><mn>2</mn><mi>α</mi><mo> </mo><mfenced close="}" open="{" separators="|"><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mo>,</mo><mo> </mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mfenced separators="|"><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>-</mo><mover><mi>y</mi><mo>¯</mo></mover></mrow></mfenced><mo>=</mo><mi>α</mi></mrow></mfenced></math><br/>As <math><mi>α</mi></math> is not known, therefore we cannot find <math><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math> or <math><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></msub><mo>.</mo></math></p><p>Hence, data is insufficient. </p></div>


title: if-the-standard-deviation-of-the-numbers-1-0-1-k-is-5-where-k-0-then-k-is-equal-to-1
slug: if-the-standard-deviation-of-the-numbers-1-0-1-k-is-5-where-k-0-then-k-is-equal-to-1-80665
tags: ['Mathematics', 'Statistics', 'KCET', 'KCET 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the standard deviation of the numbers -1 , $0,1, k$ is $\sqrt{5}$ where $k&gt;0$, then $k$ is equal to</div>

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

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

answer:
<div class="solution">Given, numbers are $-1,0,1, k$.<br/>Standard deviation, $\sigma=\sqrt{5}$<br/>$$<br/>\sigma^2=\frac{\Sigma x_i^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad 5=\frac{1+0+1+k^2}{4}-\left(\frac{-1+0+1+k}{4}\right)^2 \\<br/>&amp; \Rightarrow \quad 5=\frac{2+k^2}{4}-\frac{k^2}{16} \Rightarrow 80=8+4 k^2-k^2 \\<br/>&amp; \Rightarrow \quad 72=3 k^2 \Rightarrow k^2=24 \Rightarrow k=2 \sqrt{6} .<br/>\end{aligned}<br/>$$</div>


title: if-the-standard-deviation-of-the-numbers-1-0-1-k-is-5-where-k-0-then-k-is-equal-to
slug: if-the-standard-deviation-of-the-numbers-1-0-1-k-is-5-where-k-0-then-k-is-equal-to-83588
tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2019 (09 Apr Shift 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the standard deviation of the numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1,</mn><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mi>k</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>5</mn></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo></math> then <math><mi>k</mi></math> is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>2</mn><msqrt><mn>6</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>.</mo><mo>=</mo><msqrt><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mi>n</mi></mfrac><mo>-</mo><msup><mfenced separators="|"><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mn>5</mn></msqrt></math></p><p><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></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>5</mn></math></p><p><math><mo>⇒</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>0</mn><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mo>-</mo><mn>1</mn><mo>+</mo><mn>0</mn><mo>+</mo><mn>1</mn><mo>+</mo><mi>k</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>5</mn></math></p><p><math><mo>⇒</mo><mfrac><mrow><mn>2</mn><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>5</mn><mi> </mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>8</mn><mo>+</mo><mn>4</mn><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><msup><mi>k</mi><mn>2</mn></msup></mrow><mn>16</mn></mfrac><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>8</mn><mo>+</mo><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><mn>80</mn></math></p><p><math><mo>⇒</mo><mn>3</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>72</mn><mi> </mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>⇒</mo><mi>k</mi></mrow><mn>2</mn></msup><mo>=</mo><mn>24</mn><mi> </mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>k</mi><mo>=</mo><mo>±</mo><mn>2</mn><msqrt><mn>6</mn></msqrt></math></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>⇒</mo><mi>k</mi><mo>=</mo><mn>2</mn><msqrt><mn>6</mn></msqrt><mo>.</mo></math></p></div>


title: if-the-standard-deviation-of-the-numbers-2-3-a-and-11-is-3-5-then-which-of-the-following-is-true
tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2016 (03 Apr)']
source_db: MarksBatch2_P1.db
slug:
if-the-standard-deviation-of-the-numbers-2-3-a-and-11-is-3-5-then-which-of-the-following-is-true-62368

question:
<div class="question">If the standard deviation of the numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn></math>, then which of the following is true ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>34</mn><mi>a</mi><mo>+</mo><mn>91</mn><mo>=</mo><mn>0.</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>23</mn><mi>a</mi><mo>+</mo><mn>44</mn><mo>=</mo><mn>0.</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>26</mn><mi>a</mi><mo>+</mo><mn>55</mn><mo>=</mo><mn>0.</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>+</mo><mn>84</mn><mo>=</mo><mn>0.</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>+</mo><mn>84</mn><mo>=</mo><mn>0.</mn></math></span> </div>

answer:
<div class="solution"><table border="1"> <tbody> <tr> <td><math><mi>x</mi></math></td> <td><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></td> </tr> <tr> <td><math><mn>2</mn></math></td> <td><math><mn>4</mn></math></td> </tr> <tr> <td><math><mn>3</mn></math></td> <td><math><mn>9</mn></math></td> </tr> <tr> <td><math><mi>a</mi></math></td> <td><math><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></math></td> </tr> <tr> <td><math><mn>11</mn></math></td> <td><math><mn>121</mn></math></td> </tr> <tr> <td><math><mtext>Sum</mtext><mo>â</mo><mn>16</mn><mo>+</mo><mi>a</mi></math></td> <td><math><mtext>Sum</mtext><mo>â</mo><mn>134</mn><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></math></td> </tr> </tbody></table><br/><br/><br/><math><mi>Ï</mi><mo>=</mo><msqrt><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mn>4</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><br/><math><msqrt><mfrac><mrow><mn>134</mn><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>16</mn><mo>+</mo><mi>a</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt><mo>=</mo><mfrac><mrow><mn>35</mn></mrow><mrow><mn>10</mn></mrow></mfrac></math><br/><br/><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msqrt><mn>134</mn><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mfrac><mrow><msup><mrow><mfenced separators="|"><mrow><mn>16</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn></mrow></mfrac></msqrt><mo>=</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math><br/><br/><math><msqrt><mn>536</mn><mo>+</mo><mn>4</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>256</mn><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi></msqrt><mo>=14</mo></math><br/><br/><math><mo>â</mo><mn>280</mn><mo>+</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>=</mo><mn>196</mn></math><br/><br/><math><mo>â</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>+</mo><mn>84</mn><mo>=</mo><mn>0.</mn></math></div>


title: if-the-standard-deviation-of-the-numbers-2-4-a-and-10-is-35-then-which-of-the-following-is-true
tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P1.db
slug:
if-the-standard-deviation-of-the-numbers-2-4-a-and-10-is-35-then-which-of-the-following-is-true-24511

question:
<div class="question">If the standard deviation of the numbers <math><mn>2</mn><mo>,</mo><mi mathvariant="normal"></mi><mn>4</mn><mo>,</mo><mi mathvariant="normal"></mi><mi>a</mi></math> and <math><mn>10</mn></math> is <math><mn>3.5</mn><mo>,</mo></math> then which of the following is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>23</mn><mi>a</mi><mo>+</mo><mn>24</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>26</mn><mi>a</mi><mo>+</mo><mn>46</mn><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>+</mo><mn>28</mn><mo>=</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>34</mn><mi>a</mi><mo>+</mo><mn>45</mn><mo>=</mo><mn>0</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>+</mo><mn>28</mn><mo>=</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution">S.D. <math><mo>=</mo><msqrt><mfrac><mrow><mo>â</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mo>â</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>â´</mo><mfrac><mrow><mn>49</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>+</mo><mn>16</mn><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>100</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>16</mn><mo>+</mo><mi>a</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>â</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>32</mn><mi>a</mi><mo>+</mo><mn>28</mn><mo>=</mo><mn>0</mn></math><br/></div>


title: if-the-standard-deviation-of-the-numbers-23-2-x-and-11-is-35-find-the-possible-values-of-x
slug: if-the-standard-deviation-of-the-numbers-23-2-x-and-11-is-35-find-the-possible-values-of-x-27154
tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2020 (17 Sep Shift 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the standard deviation of the numbers 2,3 , \(2 x\) and 11 is 3.5. Find the possible values of \(x\).</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(3 \cdot \frac{7}{3}\)</span> </div>

answer:
<div class="solution">\(\sigma=3.5\)<br/>\(\begin{gathered}<br/>\text {Mean }(\bar{x})=\frac{2+3+2 x+11}{4}=\left(4+\frac{x}{2}\right) \\<br/>\sigma^2=\frac{1}{n} \sum\left(x_i-\bar{x}\right)^2 \\<br/>\Rightarrow \quad(3 \cdot 5)^2=\frac{1}{4}\left[\left(-2-\frac{x}{2}\right)^2+\left(-1-\frac{x}{2}\right)^2+\left(\frac{3 x}{2}-4\right)^2+\left(7-\frac{x}{2}\right)^2\right] \\<br/>\Rightarrow 3 x^2-16 x+21=0 \Rightarrow x=\frac{7}{3}, 3<br/>\end{gathered}\)</div>


title: if-the-standard-deviation-of-the-random-variable-x-is-3-p-q-and-mean-is-3p-then-e-x-2
slug: if-the-standard-deviation-of-the-random-variable-x-is-3-p-q-and-mean-is-3p-then-e-x-2-94966
tags: ['Mathematics', 'Probability', 'MHT CET', 'MHT CET 2019 (Shift 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the standard deviation of the random variable X is <math><msqrt><mn>3</mn><mi>p</mi><mi>q</mi></msqrt></math> and mean is 3p then <math><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mo>â¦</mo><mo>.</mo></math></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>3</mn><mi>p</mi><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>p</mi></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Key Idea: Use p +q =1 and <math><mi>V</mi><mi>a</mi><mi>r</mi><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>=</mo><mi>E</mi><msup><mrow><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> <br/>We have standard deviation of X = <math><msqrt><mn>3</mn><mi>p</mi><mi>q</mi></msqrt></math> <br/><math><mo>⇒</mo><mi>V</mi><mi>a</mi><mi>r</mi><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>=</mo><mn>3</mn><mi>p</mi><mi>q</mi></math> <br/>and mean, <math><mi>E</mi><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>=</mo><mn>3</mn><mi>p</mi></math> <br/>Now, <math><mn>3</mn><mi>p</mi><mi>q</mi><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn>3</mn><mi>p</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> <br/><math><mfenced separators="|"><mrow><mo>âµ</mo><mi>V</mi><mi>a</mi><mi>r</mi><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>-</mo><mfenced separators="|"><mrow><mi>E</mi><msup><mrow><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfenced></math> <br/><math><mo>⇒</mo><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn>3</mn><mi>p</mi><mi>q</mi><mo>+</mo><mn>9</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math> <br/><math><mo>=</mo><mn>3</mn><mi>p</mi><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced><mo>+</mo><mn>9</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi></mi></mrow></msup><mfenced separators="|"><mrow><mo>âµ</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math> <br/><math><mo>=</mo><mn>3</mn><mi>p</mi><mo>-</mo><mn>3</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>9</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math> <br/><math><mo>⇒</mo><mn>3</mn><mi>p</mi><mo>+</mo><mn>6</mn><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⇒</mo><mn>3</mn><mi>p</mi><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>p</mi></mrow></mfenced></math></div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2019 (09 Jan Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-the-standard-electrode-potential-for-a-cell-is-2-v-at-300-k-the-equilibrium-constant-k-for-the-reaction-z-n-s-c-u-2-a-q-z-n-2-a-q-c-u-s-at-300-k-is

slug:
if-the-standard-electrode-potential-for-a-cell-is-2-v-at-300-k-the-equilibrium-constant-k-for-the-reaction-z-n-s-c-u-2-a-q-z-n-2-a-q-c-u-s-at-300-k-is-84402

question:
<div class="question">If the standard electrode potential for a cell is <math><mn>2</mn><mi> </mi><mi>V</mi></math> at <math><mn>300</mn><mi> </mi><mi>K</mi><mo>,</mo></math> the equilibrium constant <math><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></math> for the reaction.<br/><br/><math><mi>Z</mi><mi>n</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>+</mo><mi>C</mi><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mfenced separators="|"><mrow><mi>a</mi><mi>q</mi></mrow></mfenced><mo>⇌</mo><mi>Z</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mfenced separators="|"><mrow><mi>a</mi><mi>q</mi></mrow></mfenced><mo>+</mo><mi>C</mi><mi>u</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mi> </mi></math><br/><br/>at <math><mn>300</mn><mi> </mi><mi>K</mi></math> is approximately:<br/><br/><math><mfenced separators="|"><mrow><mi>R</mi><mo>=</mo><mn>8</mn><mi> </mi><mi>J</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>m</mi><mi>o</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mi> </mi><mi>F</mi><mo>=</mo><mn>96000</mn><mi> </mi><mi>C</mi><mi> </mi><mi>m</mi><mi>o</mi><msup><mrow><mi>l</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mn>160</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mn>80</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><msup><mrow><mi>e</mi></mrow><mrow><mn>160</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><msup><mrow><mi>e</mi></mrow><mrow><mn>320</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msup><mrow><mi>e</mi></mrow><mrow><mn>160</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><blockquote><p><math><mo>∆</mo><mi>G</mi><mo>=</mo><mo>∆</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>o</mi></mrow></msup><mo>+</mo><mi>R</mi><mi>T</mi><mi>l</mi><mi>n</mi><mi>Q</mi></math><br/>At equilibrium <math><mo>∆</mo><mi>G</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi> </mi><mi>Q</mi><mo>=</mo><mi>k</mi></math><br/><math><mn>0</mn><mo>=</mo><mo>∆</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>o</mi></mrow></msup><mo>+</mo><mi>R</mi><mi>T</mi><mi> </mi><mi>l</mi><mi>n</mi><mi>K</mi></math><br/><math><mo>∆</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>o</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>R</mi><mi>T</mi><mi> </mi><mi>l</mi><mi>n</mi><mi>K</mi><mi> </mi><mfenced separators="|"><mrow><mo>∴</mo><mo>∆</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>o</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>n</mi><mi>F</mi><msup><mrow><mi>E</mi></mrow><mrow><mi>o</mi></mrow></msup></mrow></mfenced></math><br/>So <math><mo>-</mo><mi>n</mi><mi>F</mi><msup><mrow><mi>E</mi></mrow><mrow><mi>o</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>R</mi><mi>T</mi><mi> </mi><mi>l</mi><mi>n</mi><mi>k</mi></math><br/><math><mi>′</mi><mi>n</mi><mi>′</mi></math> factor for the above cell <math><mo>=</mo><mn>2</mn></math><br/>So <math><mo>-</mo><mn>2</mn><mo>×</mo><mn>96000</mn><mo>×</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>8</mn><mo>×</mo><mn>300</mn><mi> </mi><mi>l</mi><mi>n</mi><mi>k</mi></math><br/><math><mi>l</mi><mi>n</mi><mi>k</mi><mo>=</mo><mn>160</mn></math><br/><math><mi>k</mi><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>160</mn></mrow></msup></math></p></blockquote></div>


tags: ['Chemistry', 'Thermodynamics (C)', 'AP EAMCET', 'AP EAMCET 2023 (15 May Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-standard-enthalpy-change-r-h-for-a-certain-reaction-at-298-k-and-constant-pressure-is-1860-kj-mol-1-the-standard-entropy-change-sys-s-of-the-sa

slug:
if-the-standard-enthalpy-change-r-h-for-a-certain-reaction-at-298-k-and-constant-pressure-is-1860-kj-mol-1-the-standard-entropy-change-sys-s-of-the-sa-33000

question:
<div class="question">If the standard enthalpy change $\left(\Delta_{\mathrm{r}} \mathrm{H}^\theta\right)$ for a certain reaction at $298 \mathrm{~K}$ and constant pressure is $-1860 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$, the standard entropy change $\left(\Delta_{\text {sys }} \mathrm{s}^\theta\right)$ of the same reaction is $-550 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, 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">$\left(\Delta_{\text {sys }} S^\theta\right)+\Delta_{\text {surr }} S^\theta=-7692 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, the reaction is spontaneous</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\Delta_{\text {sys }} \mathrm{S}^\theta\right)+\Delta_{\text {surr }} \mathrm{S}^\theta=-5692 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, the reaction is non-spontaneous</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\Delta_{\text {sys }} \mathrm{S}^\theta\right)+\Delta_{\text {surr }} \mathrm{S}^\theta=+5692 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, the reaction is spontaneous</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\Delta_{\text {sys }} S^\theta\right)+\Delta_{\text {surr }} S^\theta=+7692 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, the reaction is non-spontaneous</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\Delta_{\text {sys }} \mathrm{S}^\theta\right)+\Delta_{\text {surr }} \mathrm{S}^\theta=+5692 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, the reaction is spontaneous</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text {  } \Delta \mathrm{G}^{\circ}=\Delta \mathrm{H}^{\circ}-\mathrm{T} \Delta \mathrm{s}^{\circ} \\<br/>&amp; =\left(-1860 \times 10^3\right)-298(-550) \\<br/>&amp; =-1696.1 \mathrm{~kJ} \mathrm{Mol}^{-1}<br/>\end{aligned}<br/>$$<br/>Thus, the reaction is Spontaneous as $\Delta \mathrm{G}^{\circ}$ is negative. Now, Since the system loses the heat, the surrounding must gain it and therefore,<br/>$$<br/>\begin{aligned}<br/>&amp; \Delta \mathrm{H}^{\circ}{ }_{\text {Surrounding }}=+1860 \times 10^3 \mathrm{~J} \mathrm{~mol}^{-1} . \\<br/>&amp; \Rightarrow \Delta \mathrm{S}_{\text {surrounding }}^{\circ}=\frac{\Delta \mathrm{H}_{\text {surrounding }}^{\circ}}{\mathrm{T}}<br/>\end{aligned}<br/>$$<br/>$\begin{aligned} &amp; =\frac{+1860 \times 10^3}{298} \\ &amp; =6241.6 \mathrm{~J} \\ &amp; \Rightarrow \Delta S_{\text {total }}^{\circ}=\Delta S_{\text {system }}^{\circ}+\Delta S_{\text {surrounding }}^{\circ} \\ &amp; =(-550)+(6241.6) \\ &amp; =+5692 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} .\end{aligned}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'AP EAMCET', 'AP EAMCET 2023 (15 May Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-36.4</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-standard-enthalpy-change-r-h-for-reaction-h-2-g-br-2-l-2-hbr-g-is-728-kj-the-standard-enthalpy-of-formation-f-h-of-hbr-g-in-kj-mol-1-is

slug:
if-the-standard-enthalpy-change-r-h-for-reaction-h-2-g-br-2-l-2-hbr-g-is-728-kj-the-standard-enthalpy-of-formation-f-h-of-hbr-g-in-kj-mol-1-is-75695

question:
<div class="question">If the standard enthalpy change $\left(\Delta_{\mathrm{r}} \mathrm{H}^\theta\right)$ for reaction $\mathrm{H}_2(\mathrm{~g})+\mathrm{Br}_2(\mathrm{l}) \rightarrow 2 \mathrm{HBr}(\mathrm{g})$ is $-72.8 \mathrm{~kJ}$, the standard enthalpy of formation $\left(\Delta_{\mathrm{f}} \mathrm{H}^\theta\right)$ of $\mathrm{HBr}(\mathrm{g})$ (in $\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-36.4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">+36.4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-18.2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-18.2</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \text {  } \Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left(\mathrm{H}_2 \text { gas }\right)=0.0 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left(\mathrm{Br}_2 \text { liquid) }\right. \\ &amp; =0.0 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ &amp; \Delta \mathrm{H}_{\mathrm{T}}^{\circ}=-72.8 \mathrm{~kJ} \text { for } 2 \text { moles of } \mathrm{HBr} \text {. } \\ &amp; \Rightarrow \Delta \mathrm{H}_{\mathrm{r}}^{\circ}=\left[2 \Delta \mathrm{H}_{\mathrm{f}}^{\circ} \text { ( } \mathrm{H} \mathrm{Br} \text { gas }\right)-\left(\Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left(\mathrm{H}_2 \text { gas }\right)\right) \\ &amp; \left.+\left(\Delta \mathrm{H}_{\mathrm{f}}{ }^0\left(\mathrm{Br}_2 \text { liq. }\right)\right)\right] \\ &amp; -72.8=2 \Delta \mathrm{H}_{\mathrm{f}}^{\circ}(\mathrm{HBr} \text { gas })-(0.0+0.0) \\ &amp; =2 \Delta \mathrm{H}_{\mathrm{f}}^{\circ}(\mathrm{HBr} \text { gas }) \\ &amp; \Rightarrow \Delta \mathrm{H}_{\mathrm{f}}^{\circ}(\mathrm{HBr} \text { gas })=\frac{-72.8}{2}=,-36.4 \mathrm{~kJ} \mathrm{~mol}^{-1} \text {. } \\ &amp; \end{aligned}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'AP EAMCET', 'AP EAMCET 2023 (16 May Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-standard-enthalpy-of-sublimation-gos-h-of-solid-co-2-naphthalene-li-and-na-are-252-730-162-108-kj-mol-respectively-the-order-of-sublimation-of-

slug:
if-the-standard-enthalpy-of-sublimation-gos-h-of-solid-co-2-naphthalene-li-and-na-are-252-730-162-108-kj-mol-respectively-the-order-of-sublimation-of-48225

question:
<div class="question">If the standard enthalpy of sublimation $\left(\Delta_{\mathrm{gos}} \mathrm{H}^{\circ}\right)$ of solid $\mathrm{CO}_2$, naphthalene, $\mathrm{Li}$ and $\mathrm{Na}$ are $25.2,73.0,162,108$ $\mathrm{kJ}$ mol respectively, the order of sublimation of these substances with temperature is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Solid $\mathrm{CO}_2&gt;$ Naphthalene $&gt;\mathrm{Na}&gt;\mathrm{Li}$</span> </div>

answer:
<div class="solution">The lower the sublimation enthalpy, the lower is the heat required to sublime it and easier for that substance to get sublimed.<br/>Thus, the order of ease of sublimation will be:solid $\mathrm{CO}_2&gt;\mathrm{Na}&gt;\mathrm{Na}&gt;\mathrm{Li}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Main', 'JEE Main 2021 (22 Jul Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">21</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-standard-molar-enthalpy-change-for-combustion-of-graphite-powder-is-2-48-10-2-kj-mol-1-the-amount-of-heat-generated-on-combustion-of-1-g-of-gra

slug:
if-the-standard-molar-enthalpy-change-for-combustion-of-graphite-powder-is-2-48-10-2-kj-mol-1-the-amount-of-heat-generated-on-combustion-of-1-g-of-gra-12667

question:
<div class="question">If the standard molar enthalpy change for combustion of graphite powder is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>.</mo><mn>48</mn><mo>×</mo><msup><mn>10</mn><mn>2</mn></msup><mo> </mo><mi>kJ</mi><msup><mrow><mo> </mo><mi>mol</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, the amount of heat generated on combustion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">g</mi></math> of graphite powder in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>kJ</mi></math> (Nearest integer):</div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>mol</mi></math> graphite <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn><mo> </mo><mi>gmC</mi></math><br/>Ans. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>248</mn><mn>12</mn></mfrac><mo>=</mo><mn>20</mn><mo>.</mo><mn>67</mn><mo> </mo><mi>kJ</mi><mo>/</mo><mi>gm</mi></math> heat evolved</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-straight-linc-x-x-0-m-y-y-0-n-z-z-0-is-parallcl-to-the-plane-a-x-b-y-cz-d-0-then-which-one-of-the-following-is-correct

slug:
if-the-straight-linc-x-x-0-m-y-y-0-n-z-z-0-is-parallcl-to-the-plane-a-x-b-y-cz-d-0-then-which-one-of-the-following-is-correct-79812

question:
<div class="question">If the straight linc $\frac{x-x_{0}}{\ell}=\frac{y-y_{0}}{m}=\frac{z-z_{0}}{n}$ is parallcl to the plane $a x+b y+c z+d=0$ then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ell+\mathrm{m}+\mathrm{n}=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a+b+c=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{a}}{\ell}+\frac{\mathrm{b}}{\mathrm{m}}+\frac{\mathrm{c}}{\mathrm{n}}=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{a} \ell+\mathrm{bm}+\mathrm{cn}=0$</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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{a} \ell+\mathrm{bm}+\mathrm{cn}=0$</span> </div>

answer:
<div class="solution">If the line is parallel to the plane then $\mathrm{a} \ell+\mathrm{bm}+\mathrm{cn}=0$</div>


title: if-the-straight-lines-x-2-y-0-and-k-x-y-1-intersect-at-the-point-1-2-1-then-what-is-the-value-of-k
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-straight-lines-x-2-y-0-and-k-x-y-1-intersect-at-the-point-1-2-1-then-what-is-the-value-of-k-42587

question:
<div class="question">If the straight lines $x-2 y=0$ and $k x+y=1$ intersect at the<br/>point $\left(1, \frac{1}{2}\right)$, then what is the value of $k ?$</div>

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

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

answer:
<div class="solution">Since. the straight lines $x-2 y=0$ and $k x+y=1$ intersect at the point $\left(1, \frac{1}{2}\right)$.<br/>$\therefore \quad$ The point $\left(1, \frac{1}{2}\right)$ satisfies the equation $k x+y=1$<br/>$\therefore \quad$ Put $x=1$, and $y=\frac{1}{2}$ in $\mathrm{eq}^{\mathrm{n}} k x+y=1$<br/>we get $k .1+\frac{1}{2}=1 \Rightarrow k=\frac{1}{2}$</div>


title: if-the-sum-of-m-terms-of-an-ap-is-n-and-the-sum-of-n-terms-is-m-then-the-sum-of-m-n-terms-is
slug: if-the-sum-of-m-terms-of-an-ap-is-n-and-the-sum-of-n-terms-is-m-then-the-sum-of-m-n-terms-is-14046
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the sum of $\mathrm{m}$ terms of an AP is $\mathrm{n}$ and the sum of $\mathrm{n}$ terms is $\mathrm{m}$, then the sum of $(\mathrm{m}+\mathrm{n})$ terms is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-(m+n)$</span> </div>

answer:
<div class="solution">Given, $\mathrm{S}_{\mathrm{m}}=\mathrm{n}$ and $\mathrm{S}_{\mathrm{n}}=\mathrm{m}$<br/>This is direct formula $\mathrm{S}_{\mathrm{m}+\mathrm{n}}=-(\mathrm{m}+\mathrm{n})$</div>


title: if-the-sum-of-n-terms-of-an-arithmetic-progression-is-n-2-2-n-then-what-is-the-n-th-term
slug: if-the-sum-of-n-terms-of-an-arithmetic-progression-is-n-2-2-n-then-what-is-the-n-th-term-77402
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the sum of ' $n$ ' terms of an arithmetic progression is $n^{2}-2 n$, then what is the $n^{\text {th }}$ term?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 n-n^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 n-3$</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">$2 n+3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 n-5$</span> </li> </ul>

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

answer:
<div class="solution">Given, <br/>$\begin{aligned} &amp; \mathrm{S}_{\mathrm{n}}=\mathrm{n}^{2}-2 \mathrm{n} \\ \therefore &amp; \quad \mathrm{  a}_{\mathrm{n}}=\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1} \\=&amp; \mathrm{n}^{2}-2 \mathrm{n}-\left[(\mathrm{n}-1)^{2}-2(\mathrm{n}-1)\right] \\=&amp; \mathrm{n}^{2}-2 \mathrm{n}-\left[\mathrm{n}^{2}+1-2 \mathrm{n}-2 \mathrm{n}+2\right]=2 \mathrm{n}-3 \end{aligned}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">486</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-the-first-two-terms-and-the-sum-of-the-first-four-terms-of-a-geometric-progression-with-positive-common-ratio-are-8-and-80-respectively-

slug:
if-the-sum-of-the-first-two-terms-and-the-sum-of-the-first-four-terms-of-a-geometric-progression-with-positive-common-ratio-are-8-and-80-respectively-16892

question:
<div class="question">If the sum of the first two terms and the sum of the first four terms of a geometric progression with positive common ratio are 8 and 80 respectively, then what is the 6 th term?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">88</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">243</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">486</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1458</span> </li> </ul>

answer:
<div class="solution">Let the Geometric progression be a, ar, ar $^{2}$, ar $^{3}, \ldots$ with common ratio $\mathrm{r}$ and first term 'a '. According to the question, we have<br/>$$<br/>\mathrm{a}+\mathrm{ar}=8 \Rightarrow \mathrm{a}(1+\mathrm{r})=8 ...(i)<br/>$$ <br/>and $\mathrm{a}+\mathrm{ar}+\mathrm{ar}^{2}+\mathrm{ar}^{3}=80$<br/>$\Rightarrow \mathrm{a}(1+\mathrm{r})+\operatorname{ar}^{2}(1+\mathrm{r})=80$<br/>$\Rightarrow \mathrm{a}(1+\mathrm{r})\left(1+\mathrm{r}^{2}\right)=80$<br/>$\Rightarrow 8\left(1+r^{2}\right)=80 \quad$ (from (i))<br/>$\Rightarrow 1+r^{2}=\frac{80}{8}=10$<br/>$\Rightarrow \mathrm{r}^{2}=10-1=9$<br/>$\Rightarrow \mathrm{r}=3 \quad(\because \mathrm{r}&gt;0)$<br/>From eq. (i), $a(1+3)=8$ $\Rightarrow \mathrm{a}=2$<br/>Now, $6^{\text {th }}$ term $=$ ar $^{5}=2(3)^{5}=2 \times 243=486$</div>


title: if-the-sum-of-the-matrices-x-x-y-y-y-z-and-z-0-0-is-the-matrix-10-5-5-then-what-is-the-value-of-y
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-sum-of-the-matrices-x-x-y-y-y-z-and-z-0-0-is-the-matrix-10-5-5-then-what-is-the-value-of-y-39658

question:
<div class="question">If the sum of the matrices $\left[\begin{array}{l}\mathrm{x} \\ \mathrm{x} \\ \mathrm{y}\end{array}\right],\left[\begin{array}{l}\mathrm{y} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]$ and $\left[\begin{array}{l}\mathrm{z} \\ 0 \\ 0\end{array}\right]$ is the matrix $\left[\begin{array}{c}10 \\ 5 \\ 5\end{array}\right]$, then what is the value of $\mathrm{y} ?$</div>

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

answer:
<div class="solution">We have,<br/>$\left[\begin{array}{l}\mathrm{x} \\ \mathrm{x} \\ \mathrm{y}\end{array}\right]+\left[\begin{array}{l}\mathrm{y} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]+\left[\begin{array}{l}\mathrm{z} \\ 0 \\ 0\end{array}\right]=\left[\begin{array}{c}10 \\ 5 \\ 5\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{c}x+y+z \\ x+y \\ y+z\end{array}\right]=\left[\begin{array}{c}10 \\ 5 \\ 5\end{array}\right]$<br/>$\Rightarrow x+y+z=10$ ...(i)<br/>$x+y=5$ ...(ii)<br/>$y+z=5$ ...(iii)<br/>from (ii), $x=5-y$ from (iii), $z=5-y$ $\therefore$ from(i),<br/>$5-y+y+5-y=10$<br/>$\Rightarrow 10-y=10 \Rightarrow y=0$</div>


title: if-the-sum-of-the-roots-of-a-quadratic-equation-is-3-and-the-product-is-2-then-the-equation-is
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-the-sum-of-the-roots-of-a-quadratic-equation-is-3-and-the-product-is-2-then-the-equation-is-94574

question:
<div class="question">If the sum of the roots of a quadratic equation is 3 and the product is 2, then the equation is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}-3 x+2=0$</span> </div>

answer:
<div class="solution">Quadratic equation can be given as $x^{2}-($ sum of roots $) x+($ product of roots $)=0$ Hence, Required quadratic equation is $x^{2}-3 x+2=0$</div>


title: if-the-sum-of-the-rootsof-the-equation-a-x-2-b-x-c-0-is-equal-to-the-sum-of-their-squares-then
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-sum-of-the-rootsof-the-equation-a-x-2-b-x-c-0-is-equal-to-the-sum-of-their-squares-then-32429

question:
<div class="question">If the sum of the rootsof the equation $a x^{2}+b x+c=0$ is equal to the sum of their squares, then</div>

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

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

answer:
<div class="solution">$a \mathrm{x}^{2}+\mathrm{bx}+\mathrm{c}=0$<br/>Let the root be $\alpha$ and $\beta$.<br/>$\alpha+\beta=-\frac{\mathrm{b}}{\mathrm{a}}, \alpha, \beta=\frac{\mathrm{c}}{\mathrm{a}}$<br/>$\Rightarrow \alpha+\beta=\alpha^{2}+\beta^{2} \quad \ldots$ (given)<br/>$\Rightarrow \alpha+\beta=(\alpha+\beta)^{2}-2 \alpha \beta$<br/>$-\frac{\mathrm{b}}{\mathrm{a}}=\left(-\frac{\mathrm{b}}{\mathrm{a}}\right)^{2}-\frac{2 \mathrm{c}}{\mathrm{a}}$<br/>$\Rightarrow \mathrm{b}^{2}+\mathrm{ab}=2 \mathrm{ac}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-the-squares-of-the-distance-of-the-point-x-y-z-from-the-points-a-0-0-and-a-0-0-is-2-c-2-then-which-one-of-the-following-is-correct

slug:
if-the-sum-of-the-squares-of-the-distance-of-the-point-x-y-z-from-the-points-a-0-0-and-a-0-0-is-2-c-2-then-which-one-of-the-following-is-correct-48036

question:
<div class="question">If the sum of the squares of the distance of the point $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ from the points $(\mathrm{a}, 0,0)$ and $(-\mathrm{a}, 0,0)$ is $2 \mathrm{c}^{2}$, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">Let the point be $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ and two points, $(\mathrm{a}, 0,0)$ and $(-a, 0,0)$ be $A$ and $B$<br/>As given in the problem, $\mathrm{PA}^{2}+\mathrm{PB}^{2}=2 \mathrm{c}^{2}$<br/>$\mathrm{so},(\mathrm{x}+\mathrm{a})^{2}+(\mathrm{y}-0)^{2}+(\mathrm{z}-0)^{2}$<br/>$+(\mathrm{x}-\mathrm{a})^{2}+(\mathrm{y}-0)^{2}+(\mathrm{z}-0)^{2}=2 \mathrm{c}^{2}$<br/>or, $(\mathrm{x}+\mathrm{a})^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+(\mathrm{x}-\mathrm{a})^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=2 \mathrm{c}^{2}$<br/>$\mathrm{x}^{2}+2 \mathrm{ax}+\mathrm{a}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+\mathrm{x}^{2}-2 \mathrm{ax}+\mathrm{a}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=2 \mathrm{c}^{2}$<br/>$=2\left(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+\mathrm{a}^{2}\right)=2 \mathrm{c}^{2}$<br/>$=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+\mathrm{a}^{2}=\mathrm{c}^{2}$<br/>$=\mathrm{x}^{2}+\mathrm{a}^{2}=\mathrm{c}^{2}-\mathrm{y}^{2}-\mathrm{z}^{2}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-sum-of-the-squares-of-the-distances-of-the-point-x-y-from-the-points-a-0-and-a-0-is-2-b-2-then-which-one-of-the-following-is-correct

slug:
if-the-sum-of-the-squares-of-the-distances-of-the-point-x-y-from-the-points-a-0-and-a-0-is-2-b-2-then-which-one-of-the-following-is-correct-75591

question:
<div class="question">If the sum of the squares of the distances of the point $(x, y)$ from the points $(a, 0)$ and $(-a, 0)$ is $2 b^{2}$, then which one of the following is correct?</div>

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

answer:
<div class="solution">Let $\mathrm{P}(x, y)$ be a point and $\mathrm{A}=(a, 0), \mathrm{B}=(-a, 0)$. Now, $\mathrm{PA}^{2}=(x-a)^{2}+y^{2}$<br/>$\mathrm{PB}^{2}=(x+a)^{2}+y^{2}$<br/>Since the sum of the distances of the point $\mathrm{P}(x, y)$ from the points $\mathrm{A}(a, 0)$ and $\mathrm{B}(-a, 0)$ is $2 b^{2}$. $\therefore \quad \mathrm{PA}^{2}+\mathrm{PB}^{2}=2 b^{2}$<br/>$(x-a)^{2}+(y)^{2}+(x+a)^{2}+(y)^{2}=2 b^{2}$<br/>$\Rightarrow x^{2}+a^{2}-2 a x+y^{2}+x^{2}+a^{2}+2 a x+y^{2}=2 b^{2}$<br/>$\Rightarrow x^{2}+a^{2}+y^{2}=b^{2}$<br/>$\Rightarrow x^{2}+a^{2}=b^{2}-y^{2}$</div>


title: if-the-sum-of-the-squares-of-the-roots-of-x-2-p-2-x-p-1-0-p-r-is-5-then-what-is-the-value-of-p
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-sum-of-the-squares-of-the-roots-of-x-2-p-2-x-p-1-0-p-r-is-5-then-what-is-the-value-of-p-57092

question:
<div class="question">If the sum of the squares of the roots of $\mathrm{x}^{2}-(\mathrm{p}-2) \mathrm{x}-(\mathrm{p}+1)=0(\mathrm{p} \in \mathrm{R})$ is 5<br/>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">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">$\frac{3}{2}$</span> </li> </ul>

answer:
<div class="solution">Let $\alpha$ and $\beta$ be the roots of $x^{2}-(p-2) x-(p+1)=0$<br/>Then $\alpha+\beta=(p-2)$ and $\alpha \beta=-(p+1)$<br/>$\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2 \alpha \beta$<br/>$\Rightarrow(\alpha+\beta)^{2}-2 \alpha \beta=5$<br/>$\Rightarrow(p-2)^{2}+2(p+1)=5$<br/>$\Rightarrow \mathrm{p}^{2}-4 \mathrm{p}+4+2 \mathrm{p}+2=5$<br/>$\Rightarrow \mathrm{p}^{2}-2 \mathrm{p}+1=0$<br/>$\Rightarrow(p-1)^{2}=0$<br/>$\Rightarrow \mathrm{p}=1$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-system-of-equations-2-x-3-y-7-and-2-a-x-a-b-y-28-has-infinitely-many-solutions-then-which-one-of-the-following-is-correct

slug:
if-the-system-of-equations-2-x-3-y-7-and-2-a-x-a-b-y-28-has-infinitely-many-solutions-then-which-one-of-the-following-is-correct-20583

question:
<div class="question">If the system of equations $2 x+3 y=7$ and $2 a x+(a+b) y=28$ has infinitely many solutions, 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">$a=2 b$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$b=2 a$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a=-2 b$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$b=-2 a$</span> </li> </ul>

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

answer:
<div class="solution">The given system of equations has infinitely many solution, then<br/>$\frac{2}{2 a}=\frac{3}{a+b}=\frac{7}{28}$<br/>$\Rightarrow \quad a=4$ and $12=a+b$<br/>$\Rightarrow \mathrm{a}=4$ and $b=8 \Rightarrow b=2 a$<br/>$ALTERNATE SOLUTION$ : Given equations are $2 x+3 y=7$<br/>$2 a x+(a+b) y=28$<br/>Matrix form by these equations is $\left[\begin{array}{cc}2 &amp; 3 \\ 2 a &amp; (a+b)\end{array}\right]$<br/>As we know if value of determinant is zero then system of equations have infinitely many solutions.<br/>So, $\left|\begin{array}{cc}2 &amp; 3 \\ 2 a &amp; a+b\end{array}\right|=0$<br/>$\Rightarrow 2 a+2 b-3 \times 2 a=0$<br/>$\Rightarrow 2 a+2 b-6 a=0$<br/>$\Rightarrow \quad 2 b-4 a=0 \Rightarrow b=2 a$</div>


title: if-the-tangent-at-p-and-q-on-the-parabola-meet-in-t-then-s-p-s-t-and-s-q-are-in
slug: if-the-tangent-at-p-and-q-on-the-parabola-meet-in-t-then-s-p-s-t-and-s-q-are-in-36148
tags: ['Mathematics', 'Parabola', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">GP</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the tangent at \( P \) and \( Q \) on the parabola meet in \( T \), then \( S P, S T \) and \( S Q \) are in</div>

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

answer:
<div class="solution"><p>Since, tangent at <math><mi>P</mi></math> and <math><mi>Q</mi></math> on the parabola meet in <math><mi>T</mi></math></p><p>Taking parametric co-ordinates of the point of tangency, let the coordinates of <math><mi>P</mi></math> and <math><mi>Q</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><mi> </mi><mn>2</mn><mi>a</mi><msub><mi>t</mi><mn>1</mn></msub></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup><mo>,</mo><mi> </mi><mn>2</mn><mi>a</mi><msub><mi>t</mi><mn>2</mn></msub></mrow></mfenced></math> respectively, then coordinates of <math><mi>T</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mrow><mi>a</mi><msub><mi>t</mi><mn>1</mn></msub><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><mi>a</mi><mfenced><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></mfenced></mrow></mfenced></math></p><p>Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> being the focus of the standard parabola <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> i.e. co-ordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> being <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>,</mo><mn>0</mn></mrow></mfenced></math>.</p><p>By distance formula, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi>S</mi><mi>P</mi><mo>=</mo><msqrt><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>a</mi><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>-</mo><mn>2</mn><mi>a</mi><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>1</mn><mn>4</mn></msubsup><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup></msqrt><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>1</mn><mn>4</mn></msubsup><mo>+</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup></msqrt><mo>=</mo><mi>a</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfenced></math></p><p>Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>Q</mi><mo>=</mo><msqrt><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>a</mi><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>-</mo><mn>2</mn><mi>a</mi><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>2</mn><mn>4</mn></msubsup><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></msqrt><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>2</mn><mn>4</mn></msubsup><mo>+</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></msqrt><mo>=</mo><mi>a</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfenced></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>T</mi><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub><msub><mi>t</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mfenced><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup></msqrt></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>+</mo><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfenced></msqrt></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>a</mi><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mfenced><mi>a</mi><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfenced><mo>=</mo><mi>S</mi><mi>P</mi><mo>×</mo><mi>S</mi><mi>Q</mi></math><br/>Thus, <math><mi>S</mi><mi>P</mi><mo>,</mo><mi> </mi><mi>S</mi><mi>T</mi><mo>,</mo><mi> </mi><mi>S</mi><mi>Q</mi></math> are in GP</p></div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-three-consecutive-vertices-of-a-parallelogram-are-2-1-1-0-and-4-3-then-what-are-the-coordinates-of-the-fourth-vertex

slug:
if-the-three-consecutive-vertices-of-a-parallelogram-are-2-1-1-0-and-4-3-then-what-are-the-coordinates-of-the-fourth-vertex-37303

question:
<div class="question">If the three consecutive vertices of a parallelogram are $(-2,-1),(1,0)$ and $(4,3)$, then what are the coordinates of the fourth vertex?</div>

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

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

answer:
<div class="solution">Given vertices of parallelogram are $(-2,-1),(1,0),(4,3)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Ov3NwTZ__Gvumf1PdFPmg9K5NsuW0RSwy-zmUBWHghs.original.fullsize.png"/><br/><br/>Let the fourth vertex be $(\mathrm{x}, \mathrm{y})$ We know, in a parallelogram diagonals bisect each other.<br/>i.e, Midpoint of $\mathrm{AC}=$ Midpoint of $\mathrm{BD}$ $\Rightarrow\left(\frac{-2+4}{2}, \frac{-1+3}{2}\right)=\left(\frac{\mathrm{X}+1}{2}, \frac{\mathrm{Y}+0}{2}\right)$<br/>$\Rightarrow(1,1)=\left(\frac{\mathrm{X}+1}{2}, \frac{\mathrm{Y}+0}{2}\right)$<br/>$\Rightarrow \mathrm{X}+1=2 ; \mathrm{Y}+0=2$<br/>$\Rightarrow \mathrm{X}=1, \mathrm{Y}=2$</div>


title: if-the-three-observations-are-3-6-and-6-then-what-is-their-harmonic-mean
slug: if-the-three-observations-are-3-6-and-6-then-what-is-their-harmonic-mean-96736
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the three observations are $3,-6$ and $-6$, then what is their harmonic mean?</div>

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

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

answer:
<div class="solution">Harmonic mean of three number $\mathrm{x} 1, \mathrm{x} 2, \mathrm{x} 3$ is<br/>$\frac{3}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}}$<br/>$\therefore$ H. $M=\frac{3}{\frac{1}{3}+\left(\frac{-1}{6}\right)+\left(\frac{-1}{6}\right)}=\frac{3}{\frac{1}{3}-\frac{1}{3}}=\frac{3}{0}=\infty$</div>


title: if-the-three-vertices-of-the-parallelogram-a-bc-d-are-a-1-a-b-3-a-c-2-b-then-d-is-equal-to
slug: if-the-three-vertices-of-the-parallelogram-a-bc-d-are-a-1-a-b-3-a-c-2-b-then-d-is-equal-to-22603
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the three vertices of the parallelogram $A B C D$ are $A(1, a), B$ $(3, a), C(2, b)$, then $D$ is equal to</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/NjizEiVxpl0naaJ2Mr2x1IVFzRWGHQAWiyh0lBFRPHI.original.fullsize.png"/><br/><br/>$\mathrm{ABCD}$ is a parallelogram.<br/>$\mathrm{AB} \| \mathrm{DC}$, then slope of line $\mathrm{AB}=$ slope of line $\mathrm{DC} \frac{\mathrm{a}-\mathrm{a}}{3-1}$<br/>$=\frac{b-y}{2-x} \Rightarrow y=b$<br/>Also, $\mathrm{AD} \| \mathrm{BC}, \frac{\mathrm{a}-\mathrm{y}}{1-\mathrm{x}}=\frac{\mathrm{a}-\mathrm{b}}{3-2}$<br/>$\frac{a-b}{1-x}=\frac{a-b}{3-2} \Rightarrow \frac{a-b}{1-x} \times \frac{1}{a-b}=1$<br/>$1-x=1$<br/>$\mathrm{x}=0$<br/>Then points $\mathrm{D}$ are $(0, \mathrm{~b})$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1600</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-total-number-of-observations-is-20-x-i-1000-and-x-i-2-84000-then-what-is-the-variance-of-the-distribution

slug:
if-the-total-number-of-observations-is-20-x-i-1000-and-x-i-2-84000-then-what-is-the-variance-of-the-distribution-12919

question:
<div class="question">If the total number of observations is $20, \Sigma x_{i}=1000$ and<br/>$\Sigma x_{i}^{2}=84000$, then what is the variance of the distribution?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1500</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1700</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1600</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">180</span> </li> </ul>

answer:
<div class="solution">Total no. of observation $(n)=20$<br/>$\Sigma x_{i}=1000$<br/>$\bar{x}=\frac{\Sigma x_{i}}{n}=\frac{1000}{20}=50$<br/>Variance $=s d$<br/>$s d=\sqrt{\frac{1}{n} \Sigma x_{i}^{2}-(\bar{x})^{2}}$<br/>$(s d)^{2}=\frac{1}{n} \Sigma x_{i}^{2}-(\bar{x})^{2}=\frac{84000}{20}-(50)^{2}$<br/>$=4200-2500=1700$.<br/>Variance $=1700$</div>


tags: ['Physics', 'Units and Dimensions', 'AIIMS', 'AIIMS 2014']
source_db: MarksBatch2_P1.db
title:
if-the-units-of-length-mass-and-force-are-chosen-as-fundamental-units-the-dimensions-of-time-would-be

slug:
if-the-units-of-length-mass-and-force-are-chosen-as-fundamental-units-the-dimensions-of-time-would-be-70336

question:
<div class="question">If the units of length, mass and force are chosen as fundamental units, the dimensions of time would be</div>

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

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

answer:
<div class="solution">$\begin{aligned} \text { As } F=\left[\mathrm{MLT}^{-2}\right] \text { or } &amp; {\left[\mathrm{T}^2\right]=\frac{[\mathrm{ML}]}{[\mathrm{F}]} } \\ \text { or }[\mathrm{T}] &amp; =\left[\mathrm{M}^{1 / 2} \mathrm{~L}^{1 / 2} \mathrm{~F}^{-1 / 2}\right]\end{aligned}$</div>


title: if-the-value-of-the-determinant-a-1-1-1-b-1-1-1-c-is-positive-where-a-b-c-then-the-value-of-abc
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-value-of-the-determinant-a-1-1-1-b-1-1-1-c-is-positive-where-a-b-c-then-the-value-of-abc-17169

question:
<div class="question">If the value of the determinant $\left|\begin{array}{lll}a &amp; 1 &amp; 1 \\ 1 &amp; b &amp; 1 \\ 1 &amp; 1 &amp; c\end{array}\right|$ is positive, where $\mathrm{a} \neq \mathrm{b} \neq \mathrm{c}$, then the value of abc</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">cannot be less than 1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">is greater than-8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is less than $-8$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">must be greater than 8</span> </li> </ul>

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

answer:
<div class="solution">$\left|\begin{array}{lll}\mathrm{a} &amp; 1 &amp; 1 \\ 1 &amp; \mathrm{~b} &amp; 1 \\ 1 &amp; 1 &amp; \mathrm{c}\end{array}\right|&gt;0$<br/>$\Rightarrow a(b c-1)-1(c-1)+1(1-b)&gt;0$<br/>$\Rightarrow a b c-a-c+1+1-b&gt;0$<br/>$\Rightarrow a b c+2-(a+b+c)&gt;0$<br/>$\Rightarrow a b c&gt;(a+b+c)-2$<br/>Let; $\mathrm{a}=-1 ; \mathrm{b}=0 \&amp; \mathrm{c}=1$<br/>Then; $0&gt;-2$ [which is correct $]$ Hence, $\mathrm{abc}=0$<br/>$\therefore$ After considering all the option; (b) is correct option.s</div>


title: if-the-values-of-a-set-are-measured-in-cm-what-will-be-the-unit-of-variance
slug: if-the-values-of-a-set-are-measured-in-cm-what-will-be-the-unit-of-variance-87638
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the values of a set are measured in $\mathrm{cm}$, what will be the $\begin{array}{ll}\text { unit of variance? } &amp; {}\end{array}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{cm}^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{cm}^{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">No unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">No unit</span> </div>

answer:
<div class="solution">If the values of a set are measured in $\mathrm{cm}$, then there will not be unit of variance</div>


title: if-the-variance-of-6-7-8-9-10-11-is-2-then-the-variance-of-12-14-16-18-20-22-is
slug: if-the-variance-of-6-7-8-9-10-11-is-2-then-the-variance-of-12-14-16-18-20-22-is-41647
tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2018 (24 Apr Shift 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the variance of $6,7,8,9,10,11$ is $\sigma^2$, then the variance of $12,14,16,18,20,22$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \sigma^2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4 \sigma^2$</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">$100+2 \sigma^2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$100+4 \sigma^2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \sigma^2$</span> </div>

answer:
<div class="solution">Its a fundamental concept if each data is doubled then variance is 4 times.<br/>So, new variance $=(2)^2 \sigma^2=4 \sigma^2$</div>


title: if-the-variance-of-a-data-is-121-then-the-standard-deviation-of-the-data-is
slug: if-the-variance-of-a-data-is-121-then-the-standard-deviation-of-the-data-is-14003
tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
answer: <div class="solution">11</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the variance of a data is 121 , then the standard deviation of the data is</div>


title: if-the-variance-of-the-data-2-4-5-6-17-is-v-then-what-is-the-variance-of-the-data-4-8-10-12-34
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 v$</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-variance-of-the-data-2-4-5-6-17-is-v-then-what-is-the-variance-of-the-data-4-8-10-12-34-48433

question:
<div class="question">If the variance of the data $2,4,5,6,17$ is $\mathrm{v}$, then what is the variance of the data $4,8,10,12,34 ?$</div>

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

answer:
<div class="solution">Given, variance of the data $2,4,5,6,17$ is $v$ Variance of the data $4,8,10,12,34$ is $2 v$. because when each observation is multiplied by 2, ther variance is also multiplied by 2</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">183.2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-variance-of-the-distribution-beginarraycccccccc-hline-boldsymbolxboldsymboli-4-8-11-17-20-24-32-hline-boldsymbolfboldsymboli-3-5-9-5-4-3-1-enda-1

slug:
if-the-variance-of-the-distribution-beginarraycccccccc-hline-boldsymbolxboldsymboli-4-8-11-17-20-24-32-hline-boldsymbolfboldsymboli-3-5-9-5-4-3-1-enda-1-75919

question:
<div class="question">If the variance of the distribution<br/>\begin{array}{cccccccc}<br/>\hline \boldsymbol{x}_{\boldsymbol{i}} &amp; 4 &amp; 8 &amp; 11 &amp; 17 &amp; 20 &amp; 24 &amp; 32 \\<br/>\hline \boldsymbol{f}_{\boldsymbol{i}} &amp; 3 &amp; 5 &amp; 9 &amp; 5 &amp; 4 &amp; 3 &amp; 1<br/>\end{array}<br/>is 45.8 , then the variance of the distribution.<br/>\begin{array}{llllllll}<br/>\hline \boldsymbol{x}_{\boldsymbol{i}} &amp; 10 &amp; 18 &amp; 24 &amp; 36 &amp; 42 &amp; 50 &amp; 66 \\<br/>\hline \boldsymbol{f}_{\boldsymbol{i}} &amp; 3 &amp; 5 &amp; 9 &amp; 5 &amp; 4 &amp; 3 &amp; 1 \\<br/>\hline<br/>\end{array}</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">93.6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{93.9}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">183.2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{183.2}$</span> </li> </ul>

answer:
<div class="solution">Mean of given observation $\left(x_i\right)$ is<br/>$$<br/>\begin{array}{r}<br/>\frac{(4 \times 3)+(8 \times 5)+(11 \times 9)+(17 \times 5)+(20 \times 4)}{+(24 \times 3)+(32 \times 1)} \\<br/>=\frac{420}{30}=14<br/>\end{array}<br/>$$<br/>So,<br/>$$<br/>\begin{aligned}<br/>\mathrm{ce} &amp; =k \frac{10^2+6^2+3^2+3^2+6^2+10^2+18^2}{30} \\<br/>&amp; =45.8 \text { (given) }<br/>\end{aligned}<br/>$$<br/>Now, mean of given observation $\left(y_i\right)$ is<br/>$$<br/>\begin{array}{r}<br/>\begin{array}{r}<br/>(10 \times 3)+(18 \times 5)+(24 \times 9)+(36 \times 5)+(42 \times 4) \\<br/>+(50 \times 3)+(66 \times 1)<br/>\end{array} \\<br/>\begin{aligned}<br/>30 &amp; \frac{900}{30}=30<br/>\end{aligned}<br/>\end{array}<br/>$$<br/>So, variance<br/>$$<br/>\begin{aligned}<br/>&amp; =k \frac{20^2+12^2+6^2+6^2+12^2+20^2+36^2}{30} \\<br/>&amp; =4 \times k\left(\frac{10^2+6^2+3^2+3^2+6^2+10^2+18^2}{30}\right) \\<br/>&amp; =4 \times 45.8=183.2<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2018 (23 Apr Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">183.2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-variance-of-the-distribution-beginarraycccccccc-hline-boldsymbolxboldsymboli-4-8-11-17-20-24-32-hline-boldsymbolfboldsymboli-3-5-9-5-4-3-1-enda

slug:
if-the-variance-of-the-distribution-beginarraycccccccc-hline-boldsymbolxboldsymboli-4-8-11-17-20-24-32-hline-boldsymbolfboldsymboli-3-5-9-5-4-3-1-enda-78679

question:
<div class="question">If the variance of the distribution<br/>\begin{array}{cccccccc}<br/>\hline \boldsymbol{x}_{\boldsymbol{i}} &amp; 4 &amp; 8 &amp; 11 &amp; 17 &amp; 20 &amp; 24 &amp; 32 \\<br/>\hline \boldsymbol{f}_{\boldsymbol{i}} &amp; 3 &amp; 5 &amp; 9 &amp; 5 &amp; 4 &amp; 3 &amp; 1<br/>\end{array}<br/>is 45.8 , then the variance of the distribution.<br/>\begin{array}{llllllll}<br/>\hline \boldsymbol{x}_{\boldsymbol{i}} &amp; 10 &amp; 18 &amp; 24 &amp; 36 &amp; 42 &amp; 50 &amp; 66 \\<br/>\hline \boldsymbol{f}_{\boldsymbol{i}} &amp; 3 &amp; 5 &amp; 9 &amp; 5 &amp; 4 &amp; 3 &amp; 1 \\<br/>\hline<br/>\end{array}</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">93.6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{93.9}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">183.2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{183.2}$</span> </li> </ul>

answer:
<div class="solution">Mean of given observation $\left(x_i\right)$ is<br/>$$<br/>\begin{array}{r}<br/>\frac{(4 \times 3)+(8 \times 5)+(11 \times 9)+(17 \times 5)+(20 \times 4)}{+(24 \times 3)+(32 \times 1)} \\<br/>=\frac{420}{30}=14<br/>\end{array}<br/>$$<br/>So,<br/>$$<br/>\begin{aligned}<br/>\mathrm{ce} &amp; =k \frac{10^2+6^2+3^2+3^2+6^2+10^2+18^2}{30} \\<br/>&amp; =45.8 \text { (given) }<br/>\end{aligned}<br/>$$<br/>Now, mean of given observation $\left(y_i\right)$ is<br/>$$<br/>\begin{array}{r}<br/>\begin{array}{r}<br/>(10 \times 3)+(18 \times 5)+(24 \times 9)+(36 \times 5)+(42 \times 4) \\<br/>+(50 \times 3)+(66 \times 1)<br/>\end{array} \\<br/>\begin{aligned}<br/>30 &amp; \frac{900}{30}=30<br/>\end{aligned}<br/>\end{array}<br/>$$<br/>So, variance<br/>$$<br/>\begin{aligned}<br/>&amp; =k \frac{20^2+12^2+6^2+6^2+12^2+20^2+36^2}{30} \\<br/>&amp; =4 \times k\left(\frac{10^2+6^2+3^2+3^2+6^2+10^2+18^2}{30}\right) \\<br/>&amp; =4 \times 45.8=183.2<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-vectors-i-2-x-j-3-y-k-and-i-3-x-j-2-y-k-are-orthogonal-to-each-other-then-what-is-the-locus-of-the-point-x-y-1

slug:
if-the-vectors-i-2-x-j-3-y-k-and-i-3-x-j-2-y-k-are-orthogonal-to-each-other-then-what-is-the-locus-of-the-point-x-y-1-91976

question:
<div class="question">If the vectors $\hat{\mathrm{i}}-2 \mathrm{x} \hat{\mathrm{j}}-3 \mathrm{y} \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+3 \mathrm{x} \hat{\mathrm{j}}+2 \mathrm{y} \hat{\mathrm{k}}$ are<br/>orthogonal to each other, then what is the locus of the point $(\mathrm{x}, \mathrm{y}) ?$</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 parabola</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A hyperbola</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">Let the given vectors $\hat{\mathrm{i}}-2 \mathrm{x} \hat{\mathrm{j}}-3 \mathrm{y} \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+3 \mathrm{x} \hat{\mathrm{j}}+2 \mathrm{y} \widehat{\mathrm{k}}$.be $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$<br/>respectively, and $\theta$ be the angle between<br/>them, so, $\cos \theta=\frac{\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}}{|\overrightarrow{\mathrm{A}} \| \overrightarrow{\mathrm{B}}|}$<br/>These are orthogonal to each other, $\theta=\pi / 2$ so, $\overrightarrow{\mathrm{A} \cdot \overrightarrow{\mathrm{B}}}=0$<br/>$\Rightarrow 1-(2 x)(3 x)-(3 y)(2 y)=0$<br/>$\Rightarrow 1-6 x^{2}-6 y^{2}=0$<br/>$\Rightarrow x^{2}+y^{2}=\frac{1}{6}$<br/>This equation represents an equation of a circle which is the locus of the point $(\mathrm{x}, \mathrm{y})$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-vectors-i-2-x-j-3-y-k-and-i-3-x-j-2-y-k-are-orthogonal-to-each-other-then-what-is-the-locus-of-the-point-x-y

slug:
if-the-vectors-i-2-x-j-3-y-k-and-i-3-x-j-2-y-k-are-orthogonal-to-each-other-then-what-is-the-locus-of-the-point-x-y-35433

question:
<div class="question">If the vectors $-\hat{i}-2 x \hat{j}-3 y \hat{k}$ and $\hat{i}-3 x \hat{j}-2 y \hat{k}$ are<br/>orthogonal to each other, then what is the locus of the point $(x, y) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a straight line</span> </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 parabola</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

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

answer:
<div class="solution">Given vectors $-\hat{i}-2 x \hat{j}-3 y \hat{k}$ and $\hat{i}-3 x \hat{j}-2 y \hat{k}$<br/>are orthogonal to each other.<br/>$\therefore(-\hat{i}-2 x \hat{j}-3 y \hat{k}) \cdot(\hat{i}-3 x \hat{j}-2 y \hat{k})=0$<br/>$\Rightarrow(-1)(1)+(-2 x)(-3 x)+(-3 y)(-2 y)=0$<br/>$\Rightarrow-1+6 x^{2}+6 y^{2}=0$<br/>$\Rightarrow 6 x^{2}+6 y^{2}=1$<br/>$\Rightarrow x^{2}+y^{2}=\left(\frac{1}{\sqrt{6}}\right)^{2}$<br/>Hence, locus of $(x, y)$ is a circle.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-vectors-i-j-k-i-b-k-and-i-j-c-k-a-b-c-1-are-coplanar-then-the-value-of-1-a-1-1-b-1-1-c-1-is-equal-to

slug:
if-the-vectors-i-j-k-i-b-k-and-i-j-c-k-a-b-c-1-are-coplanar-then-the-value-of-1-a-1-1-b-1-1-c-1-is-equal-to-32157

question:
<div class="question">If the vectors $\hat{\mathbf{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{b}}+\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}=\hat{\mathrm{j}}=\mathrm{c} \hat{\mathrm{k}}(\mathrm{a}, \mathrm{b}, \mathrm{c} \neq 1)$<br/>are coplanar, then the value of $\frac{1}{1-\mathrm{a}}+\frac{1}{1-\mathrm{b}}+\frac{1}{1-\mathrm{c}}$ is equal to</div>

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

answer:
<div class="solution">$a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c \hat{k}$ are coplanar.<br/>i.e., $\left|\begin{array}{lll}\mathrm{a} &amp; 1 &amp; 1 \\ 1 &amp; \mathrm{~b} &amp; 1 \\ 1 &amp; 1 &amp; \mathrm{c}\end{array}\right|=0$<br/>$c_{2} \rightarrow c_{2}-c_{3} ; c_{3} \rightarrow c_{3}-c_{1}$<br/>$\left|\begin{array}{ccc}a &amp; 1-a &amp; 1-a \\ 1 &amp; b-1 &amp; 0 \\ 1 &amp; 0 &amp; c-1\end{array}\right|=0$<br/>$\Rightarrow \mathrm{a}(\mathrm{b}-1)(\mathrm{c}-1)-(1-\mathrm{a})(\mathrm{c}-1)-(1-\mathrm{a})(\mathrm{b}-1)=0$<br/>Divide both sides by $(1-a)(1-b)(1-c)$<br/>$\begin{aligned} \Rightarrow &amp; \frac{\mathrm{a}(\mathrm{b}-1)(\mathrm{c}-1)}{(1-\mathrm{a})(1-\mathrm{b})(1-\mathrm{c})}-\frac{(1-\mathrm{a})(\mathrm{c}-1)}{(1-\mathrm{a})(1-\mathrm{b})(1-\mathrm{c})} \\ &amp; \quad-\frac{(1-\mathrm{a})(\mathrm{b}-1)}{(1-\mathrm{a})(1-\mathrm{b})(1-\mathrm{c})}=\frac{0}{(1-\mathrm{a})(1-\mathrm{b})(1-\mathrm{c})} \\ \Rightarrow &amp; \frac{\mathrm{a}}{1-\mathrm{a}}+\frac{1}{1-\mathrm{b}}+\frac{1}{1-\mathrm{c}}=0 \\ \Rightarrow &amp; \frac{1}{1-\mathrm{b}}+\frac{1}{1-\mathrm{c}}=-\frac{\mathrm{a}}{1-\mathrm{a}} \end{aligned}$<br/>Add $\frac{1}{1-\mathrm{a}}$ on both sides.<br/>$\Rightarrow \frac{1}{1-\mathrm{a}}+\frac{1}{1-\mathrm{b}}+\frac{1}{1-\mathrm{c}}=\frac{1}{1-\mathrm{a}}-\frac{\mathrm{a}}{1-\mathrm{a}}=\frac{1-\mathrm{a}}{1-\mathrm{a}}=1 .$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-vectors-i-j-k-i-k-and-i-j-k-lie-on-a-plane-where-and-are-distinct-nonnegative-numbers-then-is

slug:
if-the-vectors-i-j-k-i-k-and-i-j-k-lie-on-a-plane-where-and-are-distinct-nonnegative-numbers-then-is-84695

question:
<div class="question">If the vectors $\alpha \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{k}}$ and $\gamma \hat{\mathrm{i}}+\hat{\mathrm{j}}+\beta \hat{\mathrm{k}}$ lie on a<br/>plane, where $\alpha, \beta$ and $\gamma$ are distinct non-negative numbers, then $\gamma$ is</div>

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

answer:
<div class="solution">If three vectors are co-planar.<br/>$\Rightarrow\left|\begin{array}{lll}\alpha &amp; \alpha &amp; \gamma \\ 1 &amp; 0 &amp; 1 \\ \gamma &amp; \gamma &amp; \beta\end{array}\right|=0$<br/>$\Rightarrow \quad \alpha[0-\gamma]-\alpha[\beta-\gamma]+\gamma[\gamma-0]=0$<br/>$\Rightarrow-\alpha \gamma-\alpha \beta+\alpha \gamma+\gamma^{2}=0$<br/>$\Rightarrow \quad \gamma^{2}=\alpha \beta$<br/>$\Rightarrow \quad$ So $\alpha, \gamma, \beta$ are in G.P.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-vectors-i-x-j-y-k-and-i-x-j-y-k-are-orthogonal-to-each-other-then-what-is-the-locus-of-the-point-x-y

slug:
if-the-vectors-i-x-j-y-k-and-i-x-j-y-k-are-orthogonal-to-each-other-then-what-is-the-locus-of-the-point-x-y-33536

question:
<div class="question">If the vectors $\hat{\mathrm{i}}-\mathrm{x} \hat{\mathrm{j}}-\mathrm{y} \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\mathrm{x} \hat{\mathrm{j}}+\mathrm{y} \hat{\mathrm{k}}$ are orthogonal to<br/>each other, then what is the locus of the point $(\mathrm{x}, \mathrm{y})$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a parabola</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">an ellipse</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">a straight line</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">Since both vectors are orthogonal $\therefore$ their dot product is zero.<br/>$\therefore 1(1)+(-x)(x)+(-y)(y)=0$<br/>$\Rightarrow 1-x^{2}-y^{2}=0$<br/>$\Rightarrow x^{2}+y^{2}=1$<br/>Which is a circle.</div>


title: if-the-vectors-k-and-a-are-parallel-to-each-other-then-what-is-kk-a-equal-to
slug: if-the-vectors-k-and-a-are-parallel-to-each-other-then-what-is-kk-a-equal-to-67039
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the vectors $\overrightarrow{\mathrm{k}}$ and $\overrightarrow{\mathrm{A}}$ are parallel to each other, then what<br/>is $\overrightarrow{k k} \times \vec{A}$ equal to?</div>

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

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

answer:
<div class="solution">Given, $\overrightarrow{\mathrm{k}}$ and $\overrightarrow{\mathrm{A}}$ are parallel vectors.<br/>we know, cross product of parallel vectors is $\overrightarrow{\mathrm{O}}$.</div>


tags: ['Physics', 'Units and Dimensions', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-the-velocity-of-light-c-gravitational-constant-g-and-plancks-constant-h-are-chosen-as-fundamental-units-then-the-dimensions-of-mass-in-this-new-sys

slug:
if-the-velocity-of-light-c-gravitational-constant-g-and-plancks-constant-h-are-chosen-as-fundamental-units-then-the-dimensions-of-mass-in-this-new-sys-87626

question:
<div class="question">If the velocity of light <math> <mrow> <mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow> </math> , gravitational constant <math> <mrow> <mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow> </math> and Planck's constant <math> <mrow> <mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow> </math> are chosen as fundamental units, then the dimensions of mass in this new system is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <math> <mrow> <msup> <mi>c</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>G</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>h</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> </mrow> </math> </span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <math> <mrow> <msup> <mi>c</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>G</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>h</mi> <mrow> <mo>â</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> </mrow> </math> </span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math> <mrow> <msup> <mi>c</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>G</mi> <mrow> <mo>â</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>h</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> </mrow> </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> <mrow> <msup> <mi>c</mi> <mrow> <mo>â</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>G</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>h</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> </mrow> </math> </span> </li> </ul>

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

answer:
<div class="solution"><p>Let <math> <mrow> <mi>m</mi><mo>â</mo><msup> <mi>C</mi> <mi>x</mi> </msup> <msup> <mi>G</mi> <mi>y</mi> </msup> <msup> <mi>h</mi> <mi>z</mi> </msup> </mrow> </math></p><p>By substituting the following dimensions : <math> <mrow> <mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">]</mo><mo>=</mo><mi>L</mi><msup> <mi>T</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <mo>;</mo><mtext> </mtext><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msup> <mi>M</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <msup> <mi>L</mi> <mn>3</mn> </msup> <msup> <mi>T</mi> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> <mo stretchy="false">]</mo></mrow> </math> and <math> <mrow> <mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>M</mi><msup> <mi>L</mi> <mn>2</mn> </msup> <msup> <mi>T</mi> <mrow> <mo>â</mo><mn>1</mn></mrow> </msup> <mo stretchy="false">]</mo></mrow> </math></p><p>Now, comparing both sides, we get</p><p><math> <mrow> <mi>x</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>;</mo><mtext> </mtext><mi>y</mi><mo>=</mo><mo>â</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mtext> </mtext><mi>z</mi><mo>=</mo><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow> </math></p><p>So, <math> <mrow> <mi>m</mi><mo>â</mo><msup> <mi>c</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>G</mi> <mrow> <mo>â</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> <msup> <mi>h</mi> <mrow> <mn>1</mn><mo>/</mo><mn>2</mn></mrow> </msup> </mrow> </math></p></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-the-velocity-of-light-c-the-gravitational-constant-g-and-plancks-constant-h-are-chosen-as-the-fundamental-units-the-dimension-of-density-in-the-new-1

slug:
if-the-velocity-of-light-c-the-gravitational-constant-g-and-plancks-constant-h-are-chosen-as-the-fundamental-units-the-dimension-of-density-in-the-new-1-72820

question:
<div class="question">If the velocity of light $C$, the gravitational constant $G$ and Planck's constant $h$ are chosen as the fundamental units, the dimension of density in the new system is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$C^3 G^{-2} h^1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$C^5 G^{-2} h^{-1}$</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">$C^{-3 / 2} G^{-1 / 2} h^{1 / 2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$C^{9 / 2} G^{-1 / 2} h^{-1 / 2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$C^5 G^{-2} h^{-1}$</span> </div>

answer:
<div class="solution">Let $\rho=\mathrm{kC}^{\mathrm{a}} \mathrm{G}^{\mathrm{b}} \mathrm{h}^{\mathrm{c}}$<br/>$\begin{aligned}<br/>&amp; \Rightarrow \quad[\rho]=[\mathrm{C}]^{\mathrm{a}}[\mathrm{G}]^{\mathrm{b}}[\mathrm{h}]^{\mathrm{c}} \\<br/>&amp; \Rightarrow \quad \mathrm{ML}^{-3}=\mathrm{L}^{\mathrm{a}} \mathrm{T}^{-\mathrm{a}} \mathrm{M}^{-\mathrm{b}} \mathrm{L}^{3 \mathrm{~b}} \mathrm{~T}^{-2 \mathrm{~b}} \mathrm{M}^{\mathrm{c}} \mathrm{L}^{2 \mathrm{c}} \mathrm{T}^{-\mathrm{c}} \\<br/>&amp; \Rightarrow \quad 1=-\mathrm{b}+\mathrm{c},-3=\mathrm{a}+3 \mathrm{~b}+2 \mathrm{c}, 0=-\mathrm{a}-2 \mathrm{~b}-\mathrm{c} \\<br/>&amp; \Rightarrow \quad \mathrm{a}=5, \mathrm{~b}=-2 \text { and } \mathrm{c}=-1<br/>\end{aligned}$</div>


tags: ['Physics', 'Units and Dimensions', 'TS EAMCET', 'TS EAMCET 2022 (20 Jul Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-velocity-of-light-c-the-gravitational-constant-g-and-plancks-constant-h-are-chosen-as-the-fundamental-units-the-dimension-of-density-in-the-new

slug:
if-the-velocity-of-light-c-the-gravitational-constant-g-and-plancks-constant-h-are-chosen-as-the-fundamental-units-the-dimension-of-density-in-the-new-58952

question:
<div class="question">If the velocity of light $C$, the gravitational constant $G$ and Planck's constant $h$ are chosen as the fundamental units, the dimension of density in the new system is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$C^3 G^{-2} h^1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$C^5 G^{-2} h^{-1}$</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">$C^{-3 / 2} G^{-1 / 2} h^{1 / 2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$C^{9 / 2} G^{-1 / 2} h^{-1 / 2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$C^5 G^{-2} h^{-1}$</span> </div>

answer:
<div class="solution">Let $\rho=\mathrm{kC}^{\mathrm{a}} \mathrm{G}^{\mathrm{b}} \mathrm{h}^{\mathrm{c}}$<br/>$\begin{aligned}<br/>&amp; \Rightarrow \quad[\rho]=[\mathrm{C}]^{\mathrm{a}}[\mathrm{G}]^{\mathrm{b}}[\mathrm{h}]^{\mathrm{c}} \\<br/>&amp; \Rightarrow \quad \mathrm{ML}^{-3}=\mathrm{L}^{\mathrm{a}} \mathrm{T}^{-\mathrm{a}} \mathrm{M}^{-\mathrm{b}} \mathrm{L}^{3 \mathrm{~b}} \mathrm{~T}^{-2 \mathrm{~b}} \mathrm{M}^{\mathrm{c}} \mathrm{L}^{2 \mathrm{c}} \mathrm{T}^{-\mathrm{c}} \\<br/>&amp; \Rightarrow \quad 1=-\mathrm{b}+\mathrm{c},-3=\mathrm{a}+3 \mathrm{~b}+2 \mathrm{c}, 0=-\mathrm{a}-2 \mathrm{~b}-\mathrm{c} \\<br/>&amp; \Rightarrow \quad \mathrm{a}=5, \mathrm{~b}=-2 \text { and } \mathrm{c}=-1<br/>\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">350 Hz</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-velocity-of-sound-in-air-is-350-m-s-then-the-fundamental-frequency-of-an-open-organ-pipe-of-length-50-cm-will-be

slug:
if-the-velocity-of-sound-in-air-is-350-m-s-then-the-fundamental-frequency-of-an-open-organ-pipe-of-length-50-cm-will-be-13521

question:
<div class="question">If the velocity of sound in air is $350 \mathrm{~m} / \mathrm{s}$. Then the fundamental frequency of an open organ pipe of length $50 \mathrm{~cm}$, will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">350 Hz</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">175 Hz</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">900 Hz</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">750 Hz</span> </li> </ul>

answer:
<div class="solution">Given $; l=50 \mathrm{~cm}, v=350 \mathrm{~m} / \mathrm{s}$<br/>Fundamental frequency of open pipe<br/>$\nu=\frac{v}{2 l}=\frac{350}{2 \times 0.5}=350 \mathrm{~Hz}$</div>


tags: ['Physics', 'Alternating Current', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-the-voltage-in-the-primary-coil-of-a-stepup-transformer-with-a-transformation-ratio-3-2-is-30-v-then-the-voltage-in-the-secondary-coil-is

slug:
if-the-voltage-in-the-primary-coil-of-a-stepup-transformer-with-a-transformation-ratio-3-2-is-30-v-then-the-voltage-in-the-secondary-coil-is-68910

question:
<div class="question">If the voltage in the primary coil of a step-up transformer with a transformation ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>:</mo><mn>2</mn></math> is<strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><mi mathvariant="normal">V</mi></math></strong>,<strong> </strong>then the voltage in the secondary coil is</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>45</mn><mi mathvariant="normal"> V</mi></math><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"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mi mathvariant="normal"> V</mi></math><br/> </span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mi mathvariant="normal"> V</mi></math><br/> </span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mi mathvariant="normal"> V</mi></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>45</mn><mi mathvariant="normal"> V</mi></math><br/> </span> </div>

answer:
<div class="solution"><p>Transformation ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">k</mi><mo>=</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">s</mi></msub><mo>/</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">p</mi></msub><mo>,</mo><mo>  </mo></math></p><p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">s</mi></msub><mo>/</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">p</mi></msub><mo>=</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">s</mi></msub><mo>/</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">p</mi></msub><mo>â´</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">s</mi></msub><mo>=</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">s</mi></msub><mo>/</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">p</mi></msub><mo>Ã</mo><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">p</mi></msub></math></p></div>


tags: ['Mathematics', 'Parabola', 'JEE Main', 'JEE Main 2023 (01 Feb Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-x-intercept-of-a-focal-chord-of-the-parabola-y-2-8-x-4-y-4-is-3-then-the-length-of-this-chord-is-equal-to

slug:
if-the-x-intercept-of-a-focal-chord-of-the-parabola-y-2-8-x-4-y-4-is-3-then-the-length-of-this-chord-is-equal-to-15465

question:
<div class="question">If the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept of a focal chord of the parabola <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>4</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>, then the length of this chord is equal to _____ .</div>

answer:
<div class="solution"><p>Given,</p><p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept of a focal chord of the parabola <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>4</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>,</p><p>Now simplifying equation of the given parabola we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>8</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p><p>Now on comparing with standard parabola <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,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>-</mo><mn>2</mn></math></p><p>Hence, the vertex is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math> and focus is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math></p><p>So, equation of line passing through focus is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mi>m</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p>Now, putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> {as given <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo></math>intercept of focal chord is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>} in the above line we get, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> </p><p>So, equation of chord will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>x</mi></math>,</p><p>Now putting the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>x</mi></math> in parabola <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>8</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>8</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>-</mo><mn>7</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="|" open="|"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>-</mo><msub><mi>x</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><msqrt><msup><mn>10</mn><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>×</mo><mn>7</mn></msqrt><mo>=</mo><msqrt><mn>128</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="|" open="|"><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>-</mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mfenced close="|" open="|"><mrow><mn>3</mn><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><msub><mi>x</mi><mn>2</mn></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced close="|" open="|"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfenced><mo>=</mo><msqrt><mn>128</mn></msqrt></math></p><p>Length of focal chord <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mfenced close="|" open="|"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced close="|" open="|"><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>-</mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mn>256</mn></msqrt><mo>=</mo><mn>16</mn></math>.</p></div>


tags: ['Mathematics', 'Probability', 'JEE Main']
answer: <div class="solution">it is a fundamental concept</div>
source_db: MarksBatch2_P1.db
title:
if-there-are-n-independent-trials-p-and-q-the-probability-of-success-and-failure-respectively-then-probability-of-exactly-r-successes-or-let-p-be-the-

slug:
if-there-are-n-independent-trials-p-and-q-the-probability-of-success-and-failure-respectively-then-probability-of-exactly-r-successes-or-let-p-be-the-64709

question:
<div class="question">If there are $n$ independent trials, $p$ and $q$ the probability of success and failure respectively, then probability of exactly $r$ successes<br/>or <br/>Let $p$ be the probability of happening an event and $q$ its failure, then the total chance of $r$ successes in $n$ trials is</div>

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

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


title: if-three-events-a-b-c-are-mutually-exclusive-then-which-one-of-the-following-is-correct
slug: if-three-events-a-b-c-are-mutually-exclusive-then-which-one-of-the-following-is-correct-20792
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If three events $A, B, C$ are mutually exclusive, then which one of the following is correct ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad P(A \cap B \cap C)=0$</span> </div>

answer:
<div class="solution">A, B and C are mutually exclusive $A \cap B \cap C=0$<br/>$\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2021 (20 Jul Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-time-t-velocity-v-and-angular-momentum-l-are-taken-as-the-fundamental-units-then-the-dimension-of-mass-m-in-terms-of-t-v-and-l-is

slug:
if-time-t-velocity-v-and-angular-momentum-l-are-taken-as-the-fundamental-units-then-the-dimension-of-mass-m-in-terms-of-t-v-and-l-is-88516

question:
<div class="question">If time <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>t</mi></mfenced><mo>,</mo></math> velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>v</mi></mfenced><mo>,</mo></math> and angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>l</mi></mfenced></math> are taken as the fundamental units. Then the dimension of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>m</mi><mo>)</mo></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>,</mo><mo> </mo><mi>v</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> is:</div>

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

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

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>∝</mo><msup><mi>t</mi><mi>a</mi></msup><msup><mi>v</mi><mi>b</mi></msup><msup><mi>ℓ</mi><mi>c</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>∝</mo><mo>[</mo><mi mathvariant="normal">T</mi><msup><mo>]</mo><mi mathvariant="normal">a</mi></msup><msup><mfenced close="]" open="["><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mi mathvariant="normal">b</mi></msup><msup><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><msup><mrow><mo> </mo><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mi mathvariant="normal">c</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">M</mi><mn>1</mn></msup><msup><mrow><mo> </mo><mi mathvariant="normal">L</mi></mrow><mn>0</mn></msup><msup><mrow><mo> </mo><mi mathvariant="normal">T</mi></mrow><mn>0</mn></msup><mo>=</mo><msup><mi mathvariant="normal">M</mi><mi mathvariant="normal">c</mi></msup><msup><mi mathvariant="normal">L</mi><mrow><mi mathvariant="normal">b</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">c</mi></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mi mathvariant="normal">a</mi><mo>-</mo><mi mathvariant="normal">b</mi><mo>-</mo><mi mathvariant="normal">c</mi></mrow></msup></math></p><p>Comparing powers, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>∝</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>v</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mi>ℓ</mi><mn>1</mn></msup></math></p></div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-two-circles-a-b-of-equal-radii-pass-through-the-centres-of-each-other-then-what-is-the-ratio-of-the-length-of-the-smaller-arc-to-the-circumference-

slug:
if-two-circles-a-b-of-equal-radii-pass-through-the-centres-of-each-other-then-what-is-the-ratio-of-the-length-of-the-smaller-arc-to-the-circumference-93130

question:
<div class="question">If two circles A, B of equal radii pass through the centres of each other, then what is the ratio of the length of the smaller arc to the circumference of the circle A cut off by the circle $\mathrm{B} ?$</div>

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

answer:
<div class="solution">When two circles A and B of equal radii pass through the centres of each other, the angle made by arc of $\mathrm{B}$ at<br/>the centre of $\mathrm{B}$ is $90^{\circ}$.<br/>So, length of small $\operatorname{arc}$ of $\mathrm{B}=\frac{2 \pi \mathrm{r} 90^{\circ}}{360^{\circ}}=\frac{\pi \mathrm{r}}{2}$<br/>Hence, circumference of A cut off by the circle $\mathrm{B}$<br/>$=2 \pi r-\frac{\pi r}{2}=\frac{3 \pi r}{2}$<br/>$\therefore \quad$ Required ratio $=\frac{\pi \mathrm{r} / 2}{3 \pi \mathrm{r} / 2}=\frac{1}{3}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-two-dice-are-thrown-and-at-least-one-of-the-dice-shows-5-then-the-probability-that-the-sum-is-10-or-more-is

slug:
if-two-dice-are-thrown-and-at-least-one-of-the-dice-shows-5-then-the-probability-that-the-sum-is-10-or-more-is-43317

question:
<div class="question">If two dice are thrown and at least one of the dice shows 5 . then the probability that the sum is 10 or more is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{6}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{4}{11}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3}{11}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{11}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{11}$</span> </div>

answer:
<div class="solution">A = Event of showing 5 and atleast one dice $=(1,5),(2,5),(3,5),(4,5)(5,5),(6,5),(5,1)$<br/>$(5,2),(5,3),(5,4),(5,6)$<br/>$n(\mathrm{~A})=11$<br/>and $n(s)=6 \times 6=36$<br/>$\mathrm{B}=$ Event of showing sum 10 or more when atleast one dice shows 5 $=(5,5),(5,6),(6,5)$<br/>$n(\mathrm{~B})=3$<br/>$\Rightarrow \quad n(A \cap B)=3$<br/>Now, $\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}$<br/>$=\frac{\frac{n(\mathrm{~A} \cap \mathrm{B})}{n(\mathrm{~S})}}{\frac{n(\mathrm{~A})}{n(\mathrm{~S})}}=\frac{\frac{3}{36}}{\frac{11}{36}}=\frac{3}{11}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-two-dice-are-thrown-then-what-is-the-probability-that-the-sum-on-the-two-faces-is-greater-than-or-equal-to-4

slug:
if-two-dice-are-thrown-then-what-is-the-probability-that-the-sum-on-the-two-faces-is-greater-than-or-equal-to-4-45705

question:
<div class="question">If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to $4$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{13}{18}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5}{6}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{11}{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{35}{36}$</span> </li> </ul>

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

answer:
<div class="solution">Since two dice are thrown so number of outcomes are $36 .$<br/>No. of ways when sum on two faces less than $4=3$. $[(1,1),(1,2),(2,1)]$<br/>Hence prob of getting sum on two faces less than 4<br/>$$<br/>=\frac{3}{36}=\frac{1}{12}<br/>$$<br/>Thus required prob. that sum on the two faces is greater<br/>than or equal to $4=1-\frac{1}{12}=\frac{11}{12}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-two-fair-dice-are-rolled-then-what-is-the-conditional-probability-that-the-first-dice-lands-on-6-given-that-the-sum-of-numbers-on-the-dice-is-8

slug:
if-two-fair-dice-are-rolled-then-what-is-the-conditional-probability-that-the-first-dice-lands-on-6-given-that-the-sum-of-numbers-on-the-dice-is-8-69760

question:
<div class="question">If two fair dice are rolled then what is the conditional probability that the first dice lands on 6 given that the sum of numbers on the dice is 8 ?</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">Total number of cases that sum is 8 are $(2,6),(3,5),(4,4),(5,3),(6,2)$<br/>favourable case $=(6,2)$<br/>$\therefore$ Probability $=\frac{1}{5}$</div>


title: if-two-fair-dice-are-thrown-then-what-is-the-probability-that-the-sum-is-neither-8-nor-9
slug: if-two-fair-dice-are-thrown-then-what-is-the-probability-that-the-sum-is-neither-8-nor-9-70693
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If two fair dice are thrown, then what is the probability that the sum is neither 8 nor $9$ ?</div>

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

answer:
<div class="solution">Total no of outcomes when two dice are thrown $=6 \times 6=36$<br/>outcomes when sum is $8=(2,6)(6,2)(3,5)(5,3)(4,4)$ $=5$<br/>Outcomes when sum is 9 $=(3,6)(6,3)(4,5)(5,4)$<br/>$=4$<br/>Possible number of outcomes $=36-(5+4)=36-9=27$<br/>$\therefore \quad$ Required probability $=\frac{27}{36}=\frac{3}{4}$</div>


title: if-two-lines-of-regression-are-perpendicular-then-the-correlation-coefficient-r-is
slug: if-two-lines-of-regression-are-perpendicular-then-the-correlation-coefficient-r-is-56466
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">It is a property</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If two lines of regression are perpendicular, then the correlation coefficient $\mathrm{r}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2} \quad$</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">None of the above</span> </li> </ul>


title: if-two-organisms-cant-interbreed-to-produce-fertile-offsprings-you-cant-place-them-under-a
slug: if-two-organisms-cant-interbreed-to-produce-fertile-offsprings-you-cant-place-them-under-a-99715
tags: ['Biology', 'The Living World', 'NEET']
source_db: MarksBatch2_P1.db
question:
<div class="question">If two organisms can't interbreed to produce fertile offsprings, you can't place them under a</div>

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

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

answer:
<div class="solution">The classification of all living organisms has been done by dividing them into different levels based on their features and characteristics. There is a hierarchy in this classification and each step represents a rank or category known as a taxonomic category. There are around seven main taxonomic ranks or categories - species, genus, family, order, class, phylum/division, and kingdom. Groups of individuals having similar fundamental qualities are called species. Species contain a group of interacting and interbreeding organisms. When the group of related species has more common characters among them they are referred to as genus. Families that exhibit few similar characters are placed under the same Order. All the fishes, amphibians, reptiles, birds along with mammals constitute the taxonomic level phylum.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-two-regression-lines-between-height-x-and-weight-y-are-4-y-15-x-410-0-and-30-x-2-y-825-0-then-what-will-be-the-correlation-coefficient-between-heig

slug:
if-two-regression-lines-between-height-x-and-weight-y-are-4-y-15-x-410-0-and-30-x-2-y-825-0-then-what-will-be-the-correlation-coefficient-between-heig-57698

question:
<div class="question">If two regression lines between height $(\mathrm{x})$ and weight $(\mathrm{y})$ are $4 y-15 x+410=0$ and $30 x-2 y-825=0$, then what<br/>will be the correlation coefficient between height ang weight?</div>

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

answer:
<div class="solution">$4 y-15 x+410=0$<br/>$\Rightarrow y-\frac{15}{4} x+\frac{410}{4}=0 \Rightarrow y=\frac{15}{4} x-\frac{410}{4}$<br/>$\mathrm{b}_{\mathrm{yx}}=\frac{15}{4}$<br/>$30 x-2 y-825=0 \Rightarrow x=\frac{2}{30} y+\frac{825}{30}$<br/>$b_{x y}=\frac{2}{30}$<br/>Correlation coefficient $=\sqrt{\left(\mathrm{b}_{\mathrm{yx}}\right)\left(\mathrm{b}_{\mathrm{xy}}\right)}$<br/>$=\sqrt{\frac{15}{4} \times \frac{2}{30}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$</div>


title: if-two-rows-of-a-determinant-are-identical-then-what-is-the-value-of-the-determinant
slug: if-two-rows-of-a-determinant-are-identical-then-what-is-the-value-of-the-determinant-91847
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">It is a property.</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If two rows of a determinant are identical, then what is the value of the determinant?</div>

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


title: if-two-unit-vectors-p-and-q-make-an-angle-3-with-each-other-what-is-the-magnitude-of-p-2-1-q
slug: if-two-unit-vectors-p-and-q-make-an-angle-3-with-each-other-what-is-the-magnitude-of-p-2-1-q-76759
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If two unit vectors $\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ make an angle $\frac{\pi}{3}$ with each<br/>other, what is the magnitude of $\vec{p}-\frac{1}{2} \vec{q}$ ?</div>

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

answer:
<div class="solution">If $\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ are unit vectors which make an angle $\frac{\pi}{3}$ with each other.<br/>Then, $\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{q}}=|\overrightarrow{\mathrm{p}}||\overrightarrow{\mathrm{q}}| \cos \frac{\pi}{3}=\frac{1}{2}$<br/>Now, $\left|\vec{p}-\frac{1}{2} \vec{q}\right|^{2}=|\vec{p}|^{2}+\frac{1}{4}|\vec{q}|^{2}-\frac{2}{2} \vec{p} \cdot \vec{q}$<br/>$=1+\frac{1}{4}-\frac{1}{2} \quad[$ since $|\overrightarrow{\mathrm{p}}|=|\overrightarrow{\mathrm{q}}|=1]$<br/>$=\frac{5}{4}-\frac{1}{2}=\frac{5-2}{4}=\frac{3}{4}$<br/>So, $\left|\vec{p}-\frac{1}{2} \vec{q}\right|=\frac{\sqrt{3}}{2}$</div>


title: if-two-variables-x-and-y-are-independent-then-what-is-the-correlation-coefficient-between-them
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">The correlation coefficient of two independent events is zero.</div>
source_db: MarksBatch2_P1.db
slug:
if-two-variables-x-and-y-are-independent-then-what-is-the-correlation-coefficient-between-them-16667

question:
<div class="question">If two variables $\mathrm{X}$ and $\mathrm{Y}$ are independent, then what is the correlation coefficient between them?</div>

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


title: if-u-e-ax-sin-bx-and-v-e-ax-cos-bx-then-what-is-u-dx-du-v-dx-dv-equal-to
slug: if-u-e-ax-sin-bx-and-v-e-ax-cos-bx-then-what-is-u-dx-du-v-dx-dv-equal-to-85417
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{u}=\mathrm{e}^{\mathrm{ax}} \sin \mathrm{bx}$ and $\mathrm{v}=\mathrm{e}^{\mathrm{ax}} \cos \mathrm{bx}$, then what is $\mathrm{u}$<br/>$\frac{\mathrm{du}}{\mathrm{dx}}+\mathrm{v} \frac{\mathrm{dv}}{\mathrm{dx}}$ equal to?</div>

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

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

answer:
<div class="solution">$\begin{aligned} u &amp; \frac{d u}{d x}+v \frac{d v}{d x}=e^{a x} \sin b x\left[a e^{a x} \sin b x+b e^{a x} \cos b x\right] \\ &amp;+e^{a x} \cos b x\left[a e^{a x} \cos b x-b e^{a x} \sin b x\right] \\=&amp; e^{2 a x}\left[a \sin ^{2} b x+b \sin b x \cos b x+\right.\\ \left.\quad a \cos ^{2} b x-b \sin b x \cos b x\right] \\=&amp; a e^{2 a x} \end{aligned}$</div>


title: if-u-sin-1-x-y-x-3-t-y-4-t-3-then-what-is-the-derivative-of-u-with-respect-to-t
slug: if-u-sin-1-x-y-x-3-t-y-4-t-3-then-what-is-the-derivative-of-u-with-respect-to-t-37438
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{u}=\sin ^{-1}(\mathrm{x}-\mathrm{y}), \mathrm{x}=3 \mathrm{t}, \mathrm{y}=4 \mathrm{t}^{3}$, then what is the derivative<br/>of $\mathrm{u}$ with respect to $\mathrm{t}$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3\left(1-t^{2}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3\left(1-t^{2}\right)^{-\frac{1}{2}}$</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">$5\left(1-\mathrm{t}^{2}\right)^{\frac{1}{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5\left(1-t^{2}\right)$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{u}=\sin ^{-1}(\mathrm{x}-\mathrm{y})$ and $\mathrm{x}=3 \mathrm{t}, \mathrm{y}=4 \mathrm{t}^{3}$<br/>So, $\mathrm{u}=\sin ^{-1}\left(3 \mathrm{t}-4 \mathrm{t}^{3}\right)$<br/>Let $\mathrm{t}=\sin \theta \Rightarrow \theta=\sin ^{-1} \mathrm{t}$<br/>So, $\mathrm{u}=\sin ^{-1}\left(3 \sin \theta-4 \sin ^{3} \theta\right)$<br/>$\quad=\sin ^{-1}(\sin 3 \theta)=3 \theta$<br/>Hence, $\mathrm{u}=3 \sin ^{-1} \mathrm{t}$<br/>$\frac{\mathrm{du}}{\mathrm{dt}}=3 \frac{1}{\sqrt{1-\mathrm{t}^{2}}}=3\left(1-\mathrm{t}^{2}\right)^{-1 / 2}$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-u-vand-w-all-positive-are-the-p-th-q-th-and-r-th-terms-of-a-gp-the-determinant-of-the-matrix-ln-up-1-ln-vq-1-ln-wr-1-is

slug:
if-u-vand-w-all-positive-are-the-p-th-q-th-and-r-th-terms-of-a-gp-the-determinant-of-the-matrix-ln-up-1-ln-vq-1-ln-wr-1-is-74328

question:
<div class="question">If u, vand w (all positive) are the $\mathrm{p}^{\mathrm{th}}, \mathrm{q}^{\text {th }}$ and $\mathrm{r}^{\text {th }}$ terms of a GP, the determinant of the matrix <br/>$\left(\begin{array}{l}\ln \mathrm{u} \mathrm{p} 1 \\ \ln \mathrm{v} \mathrm{q} 1 \\ \ln \mathrm{w} \mathrm{r} 1\end{array}\right)$ is</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">$(\mathrm{p}-\mathrm{q})(\mathrm{q}-\mathrm{r})(\mathrm{r}-\mathrm{p})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\ln \mathrm{u} \times 1 \mathrm{n} \mathrm{v} \times \ln \mathrm{w}$</span> </li> </ul>

answer:
<div class="solution">Let $u, v$ and $w$ are $\mathrm{p}^{\text {th }}, q^{\text {th }}$ and $\mathrm{r}^{\text {th }}$ term of the G.P. with first term ' $a$ ' and common ratio ' $d$ '.<br/>then, $u=a .(d)^{p-1} \Rightarrow \operatorname{In} u=\operatorname{In}(a)+(p-1) \operatorname{In}(d)$<br/>$v=a \cdot(d)^{q-1} \Rightarrow \operatorname{In} v=\operatorname{In}(a)+(q-1) \operatorname{In}(d)$<br/>$w=a .(d)^{r-1} \Rightarrow \operatorname{In} w=\operatorname{In}(a)+(r-1) \operatorname{In}(d)$<br/>Now, $\operatorname{In} u-\operatorname{In} v=(p-q) \operatorname{In}(d)$<br/>In $u-\operatorname{In} w=(p-r) \operatorname{In}(d)$<br/>$\left|\begin{array}{ccc}\operatorname{In} u &amp; p &amp; 1 \\ \operatorname{In} v &amp; q &amp; 1 \\ \operatorname{In} w &amp; r &amp; 1\end{array}\right|$<br/>$\left|\begin{array}{ccc}\operatorname{In} u &amp; p &amp; 1 \\ \operatorname{In} u-\operatorname{In} v &amp; (p-q) &amp; 0 \\ \operatorname{In} u-\operatorname{In} w &amp; (p-r) &amp; 0\end{array}\right|$<br/>\{Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ and $\left.\mathrm{R}_{3} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{3}\right\}$<br/>$\left|\begin{array}{ccc}\operatorname{In} u &amp; p &amp; 1 \\ (p-q) \operatorname{In}(d) &amp; (p-q) &amp; 0 \\ (p-r) \operatorname{In}(d) &amp; (p-r) &amp; 0\end{array}\right|$<br/>$=(p-q)(p-r)[\operatorname{In}(d)-\operatorname{In}(d)]=0$</div>


tags: ['Physics', 'Units and Dimensions', 'TS EAMCET', 'TS EAMCET 2018 (05 May Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-v-0-is-the-volume-of-a-standard-unit-cell-of-germanium-crystal-containing-n-0-atoms-then-the-expression-for-the-mass-m-of-a-volume-v-in-terms-of-v-

slug:
if-v-0-is-the-volume-of-a-standard-unit-cell-of-germanium-crystal-containing-n-0-atoms-then-the-expression-for-the-mass-m-of-a-volume-v-in-terms-of-v-90768

question:
<div class="question">If $V_0$ is the volume of a standard unit cell of germanium crystal containing $N_0$ atoms, then the expression for the mass $m$ of a volume $V$ in terms of $V_0, N_0, M_{\text {mol }}$ and $N_A$ is [here, $M$ is the molar mass of germanium and $N_A$ is the Avogadro's constant]</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$M \frac{V}{V_0} \frac{N_A}{N_0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{N_{\mathrm{A}}}{N_0} \frac{V_0}{V} M$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$M \frac{V}{V_0} \frac{N_0}{N_A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$M \frac{V_0}{V} \frac{N_0}{N_A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$M \frac{V}{V_0} \frac{N_0}{N_A}$</span> </div>

answer:
<div class="solution">Number of unit cells in volume $V$<br/>$=\frac{\text { Total volume }}{\text { Volume of a unit cell }}=\frac{V}{V_0}$<br/>Number of atoms in volume $V$<br/>$=$ Number of unit cells $\times$ Number of atoms in 1 unit cell $=\frac{V}{V_0} \times N_0$<br/>Number of moles in volume $V$<br/>$=\frac{\text { Number of a toms }}{\text { Avagadro number }}=\frac{V}{V_0} \times \frac{N_0}{N_A}$<br/>Mass $n$ of given sample volume<br/>$=$ Number of moles $\times$ Molar mass<br/>$\Rightarrow m=\frac{V}{V_0} \times \frac{N_0}{N_A} \times M$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-velocity-force-and-time-are-taken-as-the-fundamental-quantities-then-using-dimensional-analysis-choose-the-correct-dimensional-formula-for-mass-amo-1

slug:
if-velocity-force-and-time-are-taken-as-the-fundamental-quantities-then-using-dimensional-analysis-choose-the-correct-dimensional-formula-for-mass-amo-1-97212

question:
<div class="question">If velocity, force and time are taken as the fundamental quantities, then using dimensional analysis choose the correct dimensional formula for mass among the following. <math> <mo stretchy="false">[</mo> </math> <math> <mi>K</mi> </math> is a dimensionless constant<math> <mo stretchy="false">]</mo> </math></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"><mi>Q</mi><mo>=</mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mn>3</mn></msup><mo> </mo><mi>F</mi><mo> </mo><msup><mi>T</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mn>3</mn><mo> </mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mn>2</mn></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></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>Q</mi><mo>=</mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></math></span> </div>

answer:
<div class="solution"><p>Let the quantity be Q, then,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>T</mi><mo>)</mo></math><br/><br/>Assuming that the function is the product of power functions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>,</mo><mi>F</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">=</mo><mo mathvariant="italic"> </mo><mi>K</mi><msup><mi>v</mi><mi mathvariant="italic">x</mi></msup><mo mathvariant="italic"> </mo><msup><mi>F</mi><mi mathvariant="italic">y</mi></msup><mo mathvariant="italic"> </mo><msup><mi>T</mi><mi mathvariant="italic">z</mi></msup><mo mathvariant="italic"> </mo><mo mathvariant="italic"> </mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math>     <br/><br/>where K is a dimensionless constant of proportionality. The above equation dimensionally becomes<br/><br/>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>[</mo><mi>Q</mi><mo>]</mo><mo> </mo><mo>=</mo><mo> </mo><mo>[</mo><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mo>]</mo><mi>x</mi></msup><mo>[</mo><mi>M</mi><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mo>]</mo><mi>y</mi></msup><mo>[</mo><mi>T</mi><msup><mo>]</mo><mi>z</mi></msup></math><br/><br/>i.e.,   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>Q</mi><mo>]</mo><mo> </mo><mo>=</mo><mo> </mo><mo>[</mo><msup><mi>M</mi><mi>y</mi></msup><msup><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></msup><msup><mi>T</mi><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>)</mo></mrow></msup><mo>]</mo></math>, âNow<br/><br/>Q = mass i.e.,   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>Q</mi><mo>]</mo><mo>=</mo><mo>[</mo><mi>M</mi><mo>]</mo></math><br/><br/>So Equation (ii) becomes<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>M</mi><mo>]</mo><mo> </mo><mo>=</mo><mo> </mo><mo>[</mo><msup><mi>M</mi><mi>y</mi></msup><msup><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></msup><msup><mi>T</mi><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>)</mo></mrow></msup><mo>]</mo></math><br/>its dimensional correctness requires <br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo> </mo></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mo> </mo><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi>y</mi><mo> </mo><mo>+</mo><mo> </mo><mi>z</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math><br/><br/>which on solving yields<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn><mo> </mo></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn></math><br/><br/>Substituting it in Equation (i), we get<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo mathvariant="italic">=</mo><mi>K</mi><mo mathvariant="italic"> </mo><msup><mi>v</mi><mrow><mo mathvariant="italic">-</mo><mn mathvariant="italic">1</mn></mrow></msup><mo mathvariant="italic"> </mo><mi>F</mi><mo mathvariant="italic"> </mo><mi>T</mi></math></p></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-velocity-force-and-time-are-taken-as-the-fundamental-quantities-then-using-dimensional-analysis-choose-the-correct-dimensional-formula-for-mass-amo

slug:
if-velocity-force-and-time-are-taken-as-the-fundamental-quantities-then-using-dimensional-analysis-choose-the-correct-dimensional-formula-for-mass-amo-96099

question:
<div class="question">If velocity, force and time are taken as the fundamental quantities, then using dimensional analysis choose the correct dimensional formula for mass among the following. <math> <mo stretchy="false">[</mo> </math> <math> <mi>K</mi> </math> is a dimensionless constant<math> <mo stretchy="false">]</mo> </math></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"><mi>Q</mi><mo>=</mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mn>3</mn></msup><mo> </mo><mi>F</mi><mo> </mo><msup><mi>T</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mn>3</mn><mo> </mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mn>2</mn></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></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>Q</mi><mo>=</mo><mi>K</mi><mo> </mo><msup><mi>v</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo> </mo><mi>F</mi><mo> </mo><mi>T</mi></math></span> </div>

answer:
<div class="solution"><p>Let the quantity be Q, then,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>T</mi><mo>)</mo></math><br/><br/>Assuming that the function is the product of power functions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>,</mo><mi>F</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo mathvariant="italic"> </mo><mo mathvariant="italic">=</mo><mo mathvariant="italic"> </mo><mi>K</mi><msup><mi>v</mi><mi mathvariant="italic">x</mi></msup><mo mathvariant="italic"> </mo><msup><mi>F</mi><mi mathvariant="italic">y</mi></msup><mo mathvariant="italic"> </mo><msup><mi>T</mi><mi mathvariant="italic">z</mi></msup><mo mathvariant="italic"> </mo><mo mathvariant="italic"> </mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo mathvariant="italic">.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math>     <br/><br/>where K is a dimensionless constant of proportionality. The above equation dimensionally becomes<br/><br/>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>[</mo><mi>Q</mi><mo>]</mo><mo> </mo><mo>=</mo><mo> </mo><mo>[</mo><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mo>]</mo><mi>x</mi></msup><mo>[</mo><mi>M</mi><mi>L</mi><msup><mi>T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mo>]</mo><mi>y</mi></msup><mo>[</mo><mi>T</mi><msup><mo>]</mo><mi>z</mi></msup></math><br/><br/>i.e.,   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>Q</mi><mo>]</mo><mo> </mo><mo>=</mo><mo> </mo><mo>[</mo><msup><mi>M</mi><mi>y</mi></msup><msup><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></msup><msup><mi>T</mi><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>)</mo></mrow></msup><mo>]</mo></math>, âNow<br/><br/>Q = mass i.e.,   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>Q</mi><mo>]</mo><mo>=</mo><mo>[</mo><mi>M</mi><mo>]</mo></math><br/><br/>So Equation (ii) becomes<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>M</mi><mo>]</mo><mo> </mo><mo>=</mo><mo> </mo><mo>[</mo><msup><mi>M</mi><mi>y</mi></msup><msup><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></msup><msup><mi>T</mi><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>)</mo></mrow></msup><mo>]</mo></math><br/>its dimensional correctness requires <br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo> </mo></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mo> </mo><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi>y</mi><mo> </mo><mo>+</mo><mo> </mo><mi>z</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math><br/><br/>which on solving yields<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn><mo> </mo></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn></math><br/><br/>Substituting it in Equation (i), we get<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo mathvariant="italic">=</mo><mi>K</mi><mo mathvariant="italic"> </mo><msup><mi>v</mi><mrow><mo mathvariant="italic">-</mo><mn mathvariant="italic">1</mn></mrow></msup><mo mathvariant="italic"> </mo><mi>F</mi><mo mathvariant="italic"> </mo><mi>T</mi></math></p></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-velocity-force-and-time-are-taken-to-be-fundamental-quantities-then-the-dimensional-formula-for-mass-is

slug:
if-velocity-force-and-time-are-taken-to-be-fundamental-quantities-then-the-dimensional-formula-for-mass-is-98916

question:
<div class="question">If velocity, force and time are taken to be fundamental quantities, then the dimensional formula for mass is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Q = Kv<sup>-1</sup> FT</span> </div>

answer:
<div class="solution">Let the quantity be Q, then,<br/><br/>Q = f (v, F, T)<br/><br/>Assuming that the function is the product of power functions of v, F and T,<br/><br/>Q = Kv<sup>x </sup>F<sup>y </sup>T<sup>z</sup>                                ...(i)<br/><br/>where K is a dimensionless constant of proportionality. The above equation dimensionally becomes<br/><br/>        [Q] = [LT<sup>-1</sup>]<sup>x</sup>[MLT<sup>-2</sup>]<sup>y</sup>[T]<sup>z</sup><br/><br/>i.e.,   [Q] = [M<sup>y</sup>][L<sup>x</sup><sup>+</sup><sup>y</sup>T<sup>-x-2y</sup><sup>+z</sup>]            ...(ii)<br/><br/>âNow<br/><br/>Q = mass i.e.,   [Q] = [M]<br/><br/>So Equation (ii) becomes<br/><br/>[M] = [M<sup>y</sup>L<sup>x</sup><sup>+</sup><sup>y</sup>T<sup>-x-2y</sup><sup>+z</sup>]<br/><br/>its dimensional correctness requires<br/><br/>y = 1,     x + y = 0   and   - x - 2y + z = 0<br/><br/>which on solving yields<br/><br/>x = -1,    y = 1   and   z = 1<br/><br/>Substituting it in Equation (i), we get<br/><br/>Q = Kv<sup>-1</sup> FT</div>


tags: ['Physics', 'Units and Dimensions']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
title:
if-velocity-of-light-c-plancks-constant-h-and-gravitational-constant-g-are-taken-as-fundamental-quantities-then-express-mass-length-and-time-in-terms-

slug:
if-velocity-of-light-c-plancks-constant-h-and-gravitational-constant-g-are-taken-as-fundamental-quantities-then-express-mass-length-and-time-in-terms-45808

question:
<div class="question">If velocity of light $c$, Planck's constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express mass, length and time in terms of dimensions of these quantities.</div>

answer:
<div class="solution">As we know that, principle of homogeneity of dimensions that is LHS and RHS of an equation will have same dimensions.<br/>We also know that, dimensions of $(h)$<br/>$$<br/>=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]<br/>$$<br/>$\therefore$ Dimensions of $(c)=\left[\mathrm{LT}^{-1}\right]$<br/>Dimensions of gravitation constant $(G)$<br/>$$<br/>\begin{aligned}<br/>&amp;=\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^2\right]\left[\mathrm{M}^{-2}\right] \\<br/>&amp;=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]<br/>\end{aligned}<br/>$$<br/>(i) Let we consider mass, $M \propto c^x h^y G^z$<br/>$$<br/>\Rightarrow M=k c^x h^y G^z<br/>$$<br/>where, $k$ is a dimensionless constant of proportianlity. Substituting dimensions of each term in Eq. (i), we get<br/>$$<br/>\begin{aligned}<br/>{\left[\mathrm{ML}^0 \mathrm{~T}^0\right] } &amp;=\left[\mathrm{LT}^{-1}\right]^x \times\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^y\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^z \\<br/>&amp;=\left[\mathrm{M}^{y-z} \mathrm{~L}^{x+2 y+3 z} \mathrm{~T}^{-x-y-2 z}\right]<br/>\end{aligned}<br/>$$<br/>By principle of homogeneity.<br/>Comparing powers of same terms on both sides, we have<br/>$$<br/>\begin{aligned}<br/>&amp;y-z=1 \\<br/>&amp;x+2 y+3 z=0 \\<br/>&amp;-x-y-2 z=0<br/>\end{aligned}<br/>$$<br/>Adding Eqs. (ii), (iii) and (iv), we have<br/>$$<br/>2 y=1 \Rightarrow y=\frac{1}{2}<br/>$$<br/>Substituting value of $y$ in Eq. (ii), we have<br/>$$<br/>z=-\frac{1}{2}<br/>$$<br/>From Eq. (iv)<br/>$$<br/>x=-y-2 z<br/>$$<br/>Substituting values of $y$ and $z$, we have<br/>$$<br/>x=-\frac{1}{2}-2\left(-\frac{1}{2}\right)=\frac{1}{2}<br/>$$<br/>Putting values of $x, y$ and $z$ in Eq. (i), we have<br/>$$<br/>\begin{aligned}<br/>&amp;m=k c^{1 / 2} h^{1 / 2} G^{-1 / 2} \\<br/>&amp;\Rightarrow m=k \sqrt{\frac{c h}{G}}<br/>\end{aligned}<br/>$$<br/>(ii) Let us consider $(L)$,<br/>$$<br/>\begin{gathered}<br/>L \propto c^x h^y G^z \\<br/>\Rightarrow L=k c^x h^y G^z \ldots \text { (i) }<br/>\end{gathered}<br/>$$<br/>where, $k$ is a dimensionless constant.<br/>Substituting dimensions of each term in Eq. (v), we get<br/>$$<br/>\begin{aligned}<br/>{\left[\mathrm{M}^0 \mathrm{LT}^0\right] } &amp;=\left[\mathrm{LT}^{-1}\right]^x \times\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right\}^y \times\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^z \\<br/>&amp;=\left[\mathrm{M}^{y-z} \mathrm{~L}^{x+2 y+3 z} \mathrm{~T}^{x-y-2 z}\right]<br/>\end{aligned}<br/>$$<br/>By the principle of homogeneity comparing powers of same terms M, L and T, we get<br/>$$<br/>\begin{array}{ll}<br/>y-z=0 &amp; \text {...(ii) } \\<br/>x+2 y+3 z=1 &amp; \text {...(iii) } \\<br/>-x-y-2 z=0 &amp; \text {...(iv) }<br/>\end{array}<br/>$$<br/>Adding Eqs. (ii), (iii) and (iv), we have<br/>$$<br/>2 y=1<br/>$$<br/>So, $y=\frac{1}{2}$<br/>Substituting value of $y$ in Eq. (iii), we have<br/>$$<br/>z=\frac{1}{2}<br/>$$<br/>From Eq. (iv),<br/>$$<br/>x=-y-2 z<br/>$$<br/>Substituting values of $y$ and $z$, we have<br/>$$<br/>x=-\frac{1}{2}-2\left(\frac{1}{2}\right)=-\frac{3}{2}<br/>$$<br/>Putting value of $x, y$ and $z$ in Eq. (i), we have<br/>$$<br/>\begin{aligned}<br/>&amp;L=k c^{-3 / 2} h^{1 / 2} G^{1 / 2} \\<br/>&amp;L=k \sqrt{\frac{h G}{c^3}}<br/>\end{aligned}<br/>$$<br/>(iii) Let us consider, $T \propto c^x h^y G^z$<br/>So, $T=k c^x h^y G^z$...(i)<br/>where, $k$ is a dimensionless constant.<br/>Substituting dimensions of each term in Eq. (i), we get<br/>$$<br/>\begin{aligned}<br/>{\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^1\right] } &amp;=\left[\mathrm{LT}^{-1}\right]^x \times\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^y \\<br/>&amp; \times\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right] z \\<br/>&amp;=\left[\mathrm{M}^{y-z} \mathrm{~L}^{x+2 y+3 z} \mathrm{~T}^{-x-y-2 z}\right]<br/>\end{aligned}<br/>$$<br/>By the principle of homogeneity comparing powers of same terms, as $M, L$ and $T$ so we get<br/>$$<br/>\begin{aligned}<br/>&amp;y-z=0 \\<br/>&amp;x+2 y+3 z=0 \\<br/>&amp;-x-y-2 z=1<br/>\end{aligned}<br/>$$<br/>Adding Eqs. (ii), (iii) and (iv) we have<br/>$$<br/>2 y=1<br/>$$<br/>So, $y=\frac{1}{2}$<br/>Substituting value of $y$ in Eq. (ii), we have<br/>$$<br/>z=y=\frac{1}{2}<br/>$$<br/>From Eq. (iv)<br/>$$<br/>x=-y-2 z-1<br/>$$<br/>Substituting values of $y$ and $z$, we have<br/>$$<br/>x=-\frac{1}{2}-2\left(\frac{1}{2}\right)-1=-\frac{5}{2}<br/>$$<br/>Putting values of $x, y$ and $z$ in Eq. (i), we get<br/>$$<br/>\begin{aligned}<br/>&amp;T=k c^{-5 / 2} h^{1 / 2} G^{1 / 2} \\<br/>&amp;T=k \sqrt{\frac{h G}{c^5}}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (20 Oct Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-we-add-3-kg-load-to-the-hanger-of-sonometer-the-fundamental-frequency-becomes-two-times-its-initial-value-the-initial-load-must-be

slug:
if-we-add-3-kg-load-to-the-hanger-of-sonometer-the-fundamental-frequency-becomes-two-times-its-initial-value-the-initial-load-must-be-30389

question:
<div class="question">If we add $3 \mathrm{~kg}$ load to the hanger of sonometer, the fundamental frequency<br/>becomes two times its initial value. The initial load must be</div>

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

answer:
<div class="solution">$\mathrm{n} \propto \sqrt{\mathrm{T}} \quad \frac{\mathrm{n}_{1}}{\mathrm{n}_{2}}=\sqrt{\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}}$<br/>$\frac{1}{2}=\left(\frac{T}{T+3}\right)^{1 / 2} \Rightarrow \frac{1}{4}=\frac{T}{T+3}$<br/>$1+\frac{3}{T}=4$<br/>$\frac{3}{T}=3 \Rightarrow T=1 \mathrm{~kg}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-we-define-a-relation-r-on-the-set-n-n-as-a-b-r-c-d-a-d-b-c-for-a-ll-a-b-c-d-n-n-then-the-relation-is

slug:
if-we-define-a-relation-r-on-the-set-n-n-as-a-b-r-c-d-a-d-b-c-for-a-ll-a-b-c-d-n-n-then-the-relation-is-86760

question:
<div class="question">If we define a relation $\mathrm{R}$ on the set $\mathrm{N} \times \mathrm{N}$ as $(a, b) R(c, d) \Leftrightarrow a+d=b+c$ for $a l l(a, b),(c, d) \in N \times N$, then<br/>the relation is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">symmetric only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">symmetric and transitive only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">reflexive only</span> </li> </ul>

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

answer:
<div class="solution">$(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d}) \Leftrightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}$<br/>(i) $\mathrm{a}+\mathrm{a}=\mathrm{a}+\mathrm{a}$.<br/>$(\mathrm{a}, \mathrm{a}) \mathrm{R}(\mathrm{a}, \mathrm{a}) \Rightarrow \mathrm{R}$ is reflexive<br/>(ii) $(a, b) R(c, d) \Rightarrow a+d=b+c$<br/>$(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow \mathrm{c}+\mathrm{b}=\mathrm{d}+\mathrm{a}$<br/>$\mathrm{R}$ is symmetric.<br/>(iii) Let $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ and $(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{e}, \mathrm{f})$<br/>$\Rightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}$ and $\mathrm{c}+\mathrm{f}=\mathrm{d}+\mathrm{e}$<br/>$\Rightarrow a+d+c+f=b+c+d+e$<br/>$\Rightarrow \mathrm{a}+\mathrm{f}=\mathrm{b}+\mathrm{e}$<br/>$\Rightarrow(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{e}, \mathrm{f})$<br/>R is transitive.<br/>from (i), (ii), (iii) R is anequivalence relation.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-we-join-the-mid-points-of-the-upper-horizontal-sides-of-each-rectangle-of-a-histogram-by-straight-lines-what-is-the-figure-so-obtained-known-as

slug:
if-we-join-the-mid-points-of-the-upper-horizontal-sides-of-each-rectangle-of-a-histogram-by-straight-lines-what-is-the-figure-so-obtained-known-as-84579

question:
<div class="question">If we join the mid points of the upper horizontal sides of each rectangle of a histogram by straight lines, what is the figure so obtained known as?</div>

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

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

answer:
<div class="solution">Joining the mid points of the upper horizontal sides of each rectangle of a histogram by straight lines, the figure so obtained is known as frequency polygon.</div>


title: if-we-study-the-vibration-of-a-pipe-open-at-both-ends-then-the-following-statements-is-not-true
tags: ['Physics', 'Waves and Sound', 'NEET', 'NEET 2013 (All India)']
source_db: MarksBatch2_P1.db
slug:
if-we-study-the-vibration-of-a-pipe-open-at-both-ends-then-the-following-statements-is-not-true-15059

question:
<div class="question">If we study the vibration of a pipe open at both ends, then the following statements is not true</div>

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

answer:
<div class="solution">Statement (d) is not true, because at the open ends pressure change will be zero.</div>


title: if-x-0-5-then-what-is-the-probability-that-x-2-3-x-2-0
slug: if-x-0-5-then-what-is-the-probability-that-x-2-3-x-2-0-25061
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x} \in[0,5]$, then what is the probability that $\mathrm{x}^{2}-3 \mathrm{x}+2 \geq 0$ ?</div>

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

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

answer:
<div class="solution">Let $x^{2}-3 x+2=0$<br/>$\Rightarrow \mathrm{x}=1.2$<br/>$\therefore \mathrm{x}^{2}-3 \mathrm{x}+2 \geq 0$ for $\mathrm{x} \in[0,1] \cup[2,3] \cup[3,4] \cup[4,5] .$<br/>It is given that :<br/>$\mathrm{x} \in[0,1] \cup[1,2] \cup[2,3] \cup[3,4] \cup[4,5]$<br/>$\therefore$ Required probability $=\frac{4}{5}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/9XOPq6K38bBaJjhpz_Z7WZgRr_mfw8Z9eb15BM-8r6I.original.fullsize.png"/></div>


title: if-x-0-y-y-x-0-0-y-x-0-then-which-one-of-the-following-is-correct
slug: if-x-0-y-y-x-0-0-y-x-0-then-which-one-of-the-following-is-correct-77960
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{lll}\mathrm{x} &amp; \mathrm{y} &amp; 0 \\ 0 &amp; \mathrm{x} &amp; \mathrm{y} \\ \mathrm{y} &amp; 0 &amp; \mathrm{x}\end{array}\right|=0$, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x}{y}$ is one of the cube roots of $-1$</span> </div>

answer:
<div class="solution">$\left|\begin{array}{lll}x &amp; y &amp; 0 \\ 0 &amp; x &amp; y \\ y &amp; 0 &amp; x\end{array}\right|=0 \Rightarrow x\left(x^{2}-0\right)-y\left(0-y^{2}\right)+0=0$<br/>$\Rightarrow x^{3}+y^{3}=0$<br/>$\Rightarrow x^{3}=-y^{3}$<br/>$\Rightarrow \frac{x^{3}}{y^{3}}=-1 \Rightarrow\left(\frac{x}{y}\right)^{3}=-1 \Rightarrow \frac{x}{y}=\sqrt{-1}$</div>


title: if-x-1-0-2-3-and-i-is-a-2-2-identity-matrix-then-x-2-2-x-3-i-equals-to-which-one-of-the-following
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 X$</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-x-1-0-2-3-and-i-is-a-2-2-identity-matrix-then-x-2-2-x-3-i-equals-to-which-one-of-the-following-45835

question:
<div class="question">If $\mathrm{X}=\left[\begin{array}{cc}1 &amp; -2 \\ 0 &amp; 3\end{array}\right]$, and $\mathrm{I}$ is a $2 \times 2$ identity matrix, then $\mathrm{X}^{2}-2 \mathrm{X}$ $+3$ I equals to which one of the following?</div>

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

answer:
<div class="solution">Given matrix is :<br/>$\mathrm{X}=\left[\begin{array}{cc}1 &amp; -2 \\ 0 &amp; 3\end{array}\right]$<br/>$\therefore \mathrm{X}^{2}=\left[\begin{array}{cc}1 &amp; -2 \\ 0 &amp; 3\end{array}\right]\left[\begin{array}{cc}1 &amp; -2 \\ 0 &amp; 3\end{array}\right]$<br/>$=\left[\begin{array}{cc}1 &amp; -2-6 \\ 0 &amp; 9\end{array}\right]=\left[\begin{array}{cc}1 &amp; -8 \\ 0 &amp; 9\end{array}\right]$<br/>So, the given expression is:<br/>$\mathrm{X}^{2}-2 \mathrm{X}+3 \mathrm{I}=\left[\begin{array}{cc}1 &amp; -8 \\ 0 &amp; 9\end{array}\right]-2\left[\begin{array}{cc}1 &amp; -2 \\ 0 &amp; 3\end{array}\right]+3\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{cc}1 &amp; -8 \\ 0 &amp; 9\end{array}\right]+\left[\begin{array}{cc}-2 &amp; +4 \\ 0 &amp; -6\end{array}\right]+\left[\begin{array}{ll}3 &amp; 0 \\ 0 &amp; 3\end{array}\right]$<br/>$=\left[\begin{array}{cc}1-2+3 &amp; -8+4 \\ 0 &amp; 9-6+3\end{array}\right]=\left[\begin{array}{cc}2 &amp; -4 \\ 0 &amp; 6\end{array}\right]=2\left[\begin{array}{cc}1 &amp; -2 \\ 0 &amp; 3\end{array}\right]=2 \mathrm{X}$</div>


title: if-x-1-2-y-2-y-2-2-y-3-where-y-2-what-is-1-y-equal-to
slug: if-x-1-2-y-2-y-2-2-y-3-where-y-2-what-is-1-y-equal-to-81431
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=1+\frac{y}{2}+\left(\frac{y}{2}\right)^{2}+\left(\frac{y}{2}\right)^{3}+\ldots .$ where $|y|   &lt; 2$, what is $1 y$<br/>equal to ?</div>

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

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

answer:
<div class="solution">$\mathrm{x}=1+\frac{\mathrm{y}}{2}+\left(\frac{\mathrm{y}}{2}\right)^{2}+\left(\frac{\mathrm{y}}{2}\right)^{3}+\ldots . .$<br/>Here, $\frac{\mathrm{y}}{2}   &lt; 1$ and this a G. with first term $=1$ and common<br/>ratio $=\frac{\mathrm{y}}{2}$ so,<br/>$\Rightarrow \quad x=\frac{1}{1-\frac{y}{2}} \Rightarrow x=\frac{2}{2-y}$<br/>$\Rightarrow 2 x-x y=2 \Rightarrow y=\frac{2 x-2}{x}$</div>


title: if-x-1-4-x-1-and-8-x-1-are-in-geometric-progression-then-what-is-the-nontrivial-value-of-x
slug: if-x-1-4-x-1-and-8-x-1-are-in-geometric-progression-then-what-is-the-nontrivial-value-of-x-69066
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x+1,4 x+1$, and $8 x+1$ are in geometric progression, then what is the non-trivial value of $\mathrm{x}$ ?</div>

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

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

answer:
<div class="solution">$\operatorname{If}(\mathrm{x}+1),(4 \mathrm{x}+1)$ and $(8 \mathrm{x}+1)$ are in GP.<br/>then $(4 x+1)^{2}=(x+1)(8 x+1)$<br/>$\Rightarrow 16 x^{2}+8 x+1=8 x^{2}+x+8 x+1$<br/>$\Rightarrow 8 x^{2}-x=0 \Rightarrow x(8 x-1)=0$<br/>$\Rightarrow \mathrm{x}=0, \frac{1}{8},\left[\frac{1}{8}\right.$ is non-trivial value]</div>


title: if-x-1-and-lo-g-2-x-lo-g-3-x-lo-g-x-16-are-in-gp-then-what-is-x-equal-to
slug: if-x-1-and-lo-g-2-x-lo-g-3-x-lo-g-x-16-are-in-gp-then-what-is-x-equal-to-68821
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x&gt;1$ and $\log _{2} x, \log _{3} x, \log _{x} 16$ are in GP, then what is $x$<br/>equal to?</div>

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

answer:
<div class="solution">$\log _{2} x, \log _{3} x, \log _{x} 16$ are in G.P.<br/>$\therefore \quad \frac{\log _{3} x}{\log _{2} x}=\frac{\log _{x} 16}{\log _{3} x}$<br/>$\Rightarrow\left(\log _{3} x\right)^{2}=\log _{2} x \cdot \log x 16$<br/>$\Rightarrow 2 \times \log _{3} x=\log 2 x \cdot \log x 2^{4}$<br/>$\Rightarrow 2 \times \log _{3} x=4 \times \log _{2} x \cdot \log _{x} 2$<br/>$\Rightarrow \log _{3} x=2\left(\log _{2} x \times \log _{x} 2\right)$<br/>$\Rightarrow \log _{3} x=2\left[\frac{\log _{2} x}{\log _{2} x}\right] \quad\left(\because \log _{b} a=\frac{1}{\log _{a} b}\right)$<br/>$\Rightarrow \log _{3} x=2 \Rightarrow x=3^{2}=9$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-x-1-and-x-2-are-positive-quantities-then-the-condition-for-the-difference-between-the-arithmetic-mean-and-the-geometric-mean-to-be-greater-than-1-i

slug:
if-x-1-and-x-2-are-positive-quantities-then-the-condition-for-the-difference-between-the-arithmetic-mean-and-the-geometric-mean-to-be-greater-than-1-i-11039

question:
<div class="question">If $\mathrm{x}_{1}$ and $\mathrm{x}_{2}$ are positive quantities, then the condition for the difference between the arithmetic mean and the geometric mean to be greater than 1 is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left|\sqrt{\mathrm{x}_{1}}+\sqrt{\mathrm{x}_{2}}\right|&gt;\sqrt{2}$</span> </div>

answer:
<div class="solution">Arithmetic mean of $\mathrm{x}_{1}, \mathrm{x}_{2}=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}}{2}$<br/>Geometric mean of $\mathrm{x}_{1}, \mathrm{x}_{2}=\sqrt{\mathrm{x}_{1} \mathrm{x}_{2}}$<br/>Given, $\frac{\mathrm{x}_{1}+\mathrm{x}_{2}}{2}-\sqrt{\mathrm{x}_{1} \mathrm{x}_{2}}&gt;1$<br/>$\Rightarrow \frac{\mathrm{x}_{1}+\mathrm{x}_{2}}{2}&gt;\sqrt{\mathrm{x}_{1} \mathrm{x}_{2}}+1$<br/>$\Rightarrow \mathrm{x}_{1}+\mathrm{x}_{2}&gt;2 \sqrt{\mathrm{x}_{1} \mathrm{x}_{2}}+2$<br/>$\Rightarrow \mathrm{x}_{1}+\mathrm{x}_{2}-2 \sqrt{\mathrm{x}_{1} \mathrm{x}_{2}}&gt;2$<br/>$\Rightarrow\left(\sqrt{\mathrm{x}_{1}}\right)^{2}+\left(\sqrt{\mathrm{x}_{2}}\right)^{2}-2 \sqrt{\mathrm{x}_{1}} \sqrt{\mathrm{x}_{2}}&gt;2$<br/>$\Rightarrow\left(\sqrt{\mathrm{x}_{1}}-\sqrt{\mathrm{x}_{2}}\right)^{2}&gt;2$<br/>$\Rightarrow\left|\sqrt{\mathrm{x}_{1}}-\sqrt{\mathrm{x}_{2}}\right|&gt;\sqrt{2}$<br/>Correct option (c).</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-x-1-x-2-and-x-3-are-enthalpies-of-h-h-o-o-and-o-h-bonds-respectively-and-x-4-is-the-enthalpy-of-vaporization-of-water-estimate-the-standard-enthalp

slug:
if-x-1-x-2-and-x-3-are-enthalpies-of-h-h-o-o-and-o-h-bonds-respectively-and-x-4-is-the-enthalpy-of-vaporization-of-water-estimate-the-standard-enthalp-36496

question:
<div class="question">If <math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and <math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub></math> are enthalpies of <math><mi mathvariant="normal">H</mi><mo>-</mo><mi mathvariant="normal">H</mi><mo>,</mo><mi mathvariant="normal">O</mi><mo>=</mo><mi mathvariant="normal">O</mi></math> and <math><mi mathvariant="normal">O</mi><mo>-</mo><mi mathvariant="normal">H</mi></math> bonds respectively, and <math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub></math> is the enthalpy of vaporization of water, estimate the standard enthalpy of combustion of hydrogen.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></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><li class=""> <span class="option-label">C</span> <span class="option-data"><math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>2</mn><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> </div>

answer:
<div class="solution"><math><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo><mo>â</mo><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">l</mi><mo>)</mo></math> â¦(i)<br/><math><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo><mo>â</mo><mn>2</mn><mi mathvariant="normal">H</mi><mo>(</mo><mi mathvariant="normal">g</mi><mo>);</mo></math> <math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">H</mi><mo>=</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub></math><br/><math><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo><mo>â</mo><mn>2</mn><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">g</mi><mo>);</mo></math> <math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">H</mi><mo>=</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><math><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">l</mi><mo>)</mo><mo>â</mo><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">g</mi><mo>);</mo></math> <math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">H</mi><mo>=</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub></math><br/><math><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo><mo>â</mo><mn>2</mn><mi mathvariant="normal">H</mi><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">g</mi><mo>);</mo></math> <math><mi mathvariant="normal">Î</mi><mi mathvariant="normal">H</mi><mo>=</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub></math><br/>Using all four, we (find) (i)<br/><math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>3</mn></mrow></msub></math></div>


