# questions

title: let-0-4-and-t-1-tan-t-a-n-t-2-tan-c-o-t-and-t-4-cot-t-a-n-then
slug: let-0-4-and-t-1-tan-t-a-n-t-2-tan-c-o-t-and-t-4-cot-t-a-n-then-71133
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\theta \in\left(0, \frac{\pi}{4}\right)$ and $t_1=(\tan \theta)^{\tan \theta}, t_2=(\tan \theta)^{\cot \theta}$ and $t_4=(\cot \theta)^{\tan \theta}$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$t_1&gt;t_2&gt;t_3&gt;t_4$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$t_4&gt;t_3&gt;t_1&gt;t_2$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$t_3&gt;t_1&gt;t_2&gt;t_4$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$t_2&gt;t_3&gt;t_1&gt;t_4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$t_4&gt;t_3&gt;t_1&gt;t_2$<br/></span> </div>

answer:
<div class="solution">As when $\theta \in\left(0, \frac{\pi}{4}\right)$<br/>$\tan \theta &lt; \cot \theta$<br/>Since, $\tan \theta &lt; 1$ and $\cot \theta&gt;1$<br/>$\therefore(\tan \theta)^{\cot \theta} &lt; 1$ and $(\cot \theta)^{\tan \theta}&gt;1$<br/>$\therefore t_4&gt;t_1$ which only holds in option (b).</div>


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-1-be-a-cube-root-of-unity-and-s-be-the-set-of-all-nonsingular-matrices-of-the-form-1-2-a-1-b-c-1-where-each-of-a-b-and-c-is-either-or-2-then-the-n

slug:
let-1-be-a-cube-root-of-unity-and-s-be-the-set-of-all-nonsingular-matrices-of-the-form-1-2-a-1-b-c-1-where-each-of-a-b-and-c-is-either-or-2-then-the-n-71815

question:
<div class="question">Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\left[\begin{array}{ccc}1 &amp; a &amp; b \\ \omega &amp; 1 &amp; c \\ \omega^2 &amp; \omega &amp; 1\end{array}\right]$, where each of $a, b$ and $c$ is either $\omega$ or $\omega^2$. Then, the number of distinct matrices in the set $S$ is</div>

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

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

answer:
<div class="solution">$|A| \neq 0$, as non-singular.<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad\left|\begin{array}{ccc}<br/>1 &amp; a &amp; b \\<br/>\omega &amp; 1 &amp; c \\<br/>\omega^2 &amp; \omega &amp; 1<br/>\end{array}\right| \neq 0 \\<br/>&amp; \Rightarrow \quad 1(1-c \omega)-a\left(\omega-c \omega^2\right) \\<br/>&amp; +b\left(\omega^2-\omega^2\right) \neq 0 \\<br/>&amp; \Rightarrow \quad 1-c \omega-a \omega+a c \omega^2 \neq 0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad(1-c \omega)(1-a \omega) \neq 0 \\<br/>&amp; \Rightarrow \quad a \neq \frac{1}{\omega}, c \neq \frac{1}{\omega} \Rightarrow a=\omega, c=\omega<br/>\end{aligned}<br/>$$<br/>and $b \in\left\{\omega, \omega^2\right\} \Rightarrow 2$ solutions</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-1-a-2-a-3-a-11-be-real-numbers-satisfying-a-1-15-27-2-a-2-0-and-a-k-2-a-k-1-a-k-2-for-k-3-4-11-if-11-a-1-2-a-2-2-a-11-2-90-then-the-value-of-11-

slug:
let-a-1-a-2-a-3-a-11-be-real-numbers-satisfying-a-1-15-27-2-a-2-0-and-a-k-2-a-k-1-a-k-2-for-k-3-4-11-if-11-a-1-2-a-2-2-a-11-2-90-then-the-value-of-11-66431

question:
<div class="question">Let $a_1, a_2, a_3, \ldots, a_{11}$ be real numbers satisfying $a_1=15,27-2 a_2&gt;0$ and $a_k=2 a_{k-1}-a_{k-2}$ for $k=3,4, \ldots, 11$.<br/>If $\frac{a_1^2+a_2^2+\ldots+a_{11}^2}{11}=90$, then the value of $\frac{a_1+a_2+\ldots+a_{11}}{11}$ is equal to</div>

answer:
<div class="solution">$$<br/>\text { } \begin{aligned}<br/>&amp; a_k=2 a_{k-1}-a_{k-2} \\<br/>&amp; \Rightarrow \quad a_1, a_2, \ldots, a_{11} \text { are in AP } \\<br/>&amp; \therefore \frac{a_1^2+a_2^2+\ldots+a_{11}^2}{11} \\<br/>&amp;=\frac{11 a^2+35 \times 11 d^2+10 a d}{11}=90 \\<br/>&amp; \Rightarrow \quad 225+35 d^2+150 d=90 \\<br/>&amp; 35 d^2+150 d+135=0 \\<br/>&amp; \Rightarrow \quad d=-3,-\frac{9}{7}<br/>\end{aligned}<br/>$$<br/>Given, $a_2 &lt; \frac{27}{2} \therefore d=-3$ and $d \neq-\frac{9}{7}$<br/>$$<br/>\Rightarrow \quad \frac{a_1+a_2+\ldots+a_{11}}{11}<br/>$$<br/>$$<br/>=\frac{11}{2}[30-10 \times 3]=0<br/>$$</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-1-a-2-a-3-a-100-be-an-arithmetic-progression-with-a-1-3-and-s-p-i-1-p-a-i-1-p-100-for-any-integer-n-with-1-n-20-let-m-5-n-if-s-n-s-m-does-not-de

slug:
let-a-1-a-2-a-3-a-100-be-an-arithmetic-progression-with-a-1-3-and-s-p-i-1-p-a-i-1-p-100-for-any-integer-n-with-1-n-20-let-m-5-n-if-s-n-s-m-does-not-de-24201

question:
<div class="question">Let $a_1, a_2, a_3, \ldots ., a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\sum_{i=1}^p a_i$, $1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m=5 n$. If $\frac{S_m}{S_n}$ does not depend on $n$, then $a_2$ is</div>

answer:
<div class="solution">Given, $a_1=3, m=5 n$ and $a_1, a_2, \ldots$ are in AP. $\therefore \frac{S_m}{S_n}=\frac{S_{5 n}}{S_n}$ is independent of $n$. Now, $\quad \frac{\frac{5 n}{2}[2 \times 3+(5 n-1) d]}{\frac{n}{2}[2 \times 3+(n-1) d]}$ $\Rightarrow \quad \frac{f\{(6-d)+5 n\}}{(6-d)+n}$ independent of $n$, if $\begin{aligned} \quad 6-d= &amp; 0 \Rightarrow d=6 \\ \therefore \quad a_2= &amp; a_1+d=3+6=9 \\ &amp; \text { Or }\end{aligned}$<br/>If $d=0$ $\frac{S_m}{S_n}$ is independent of $n$.<br/>$$<br/>\therefore \quad a_2=3<br/>$$</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">25</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-1-a-2-a-3-be-in-harmonic-progression-with-a-1-5-and-a-20-25-the-least-positive-integer-n-for-which-a-n-0-is

slug:
let-a-1-a-2-a-3-be-in-harmonic-progression-with-a-1-5-and-a-20-25-the-least-positive-integer-n-for-which-a-n-0-is-70554

question:
<div class="question">Let $a_{1}, a_{2}, a_{3}, \ldots . .$ be in harmonic progression with $a_{1}=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_{n} &lt; 0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">22</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">23</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">24</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">25</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\because a_{1}, a_{2}, a_{3}, \ldots \ldots$ are in H.P.
<br/> <br/>$\begin{array}{l}
<br/> <br/>\therefore \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}} \ldots \ldots \text { are in A.P. } \\
<br/> <br/>\therefore \quad \frac{1}{a_{1}}=\frac{1}{5} \text { and } \frac{1}{a_{20}}=\frac{1}{25} \\
<br/> <br/>\frac{1}{a_{1}}+19 d=\frac{1}{a_{20}} \Rightarrow \frac{1}{5}+19 d=\frac{1}{25} \Rightarrow d=\frac{-4}{475} \\
<br/> <br/>\text { Now } \frac{1}{a_{n}}=\frac{1}{5}+(n-1)\left(\frac{-4}{475}\right) \\
<br/> <br/>\text { Clearly } a_{n} &lt; 0, \text { if } \frac{1}{a_{n}} &lt; 0 \Rightarrow \frac{1}{5}-\frac{4 n}{475}+\frac{4}{475} &lt; 0 \\
<br/> <br/>\Rightarrow \quad-4 n &lt; -99 \text { or } n&gt;\frac{99}{4}=24 \frac{3}{4} \quad \therefore n \geq 25 \\
<br/> <br/>\therefore \quad \text { Least value of } n \text { is } 25 .
<br/> <br/>\end{array}$</div>


tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-a-and-a-be-the-roots-of-the-equation-3-1-a-1-x-2-1-a-1-x-6-1-a-1-0-where-a-1-then-lim-a-0-a-and-lim-x-0-a-are

slug:
let-a-and-a-be-the-roots-of-the-equation-3-1-a-1-x-2-1-a-1-x-6-1-a-1-0-where-a-1-then-lim-a-0-a-and-lim-x-0-a-are-32087

question:
<div class="question">Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation $(\sqrt[3]{1+a}-1) x^{2}+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0$ where $a$ $&gt;-1$. Then $\lim _{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _{x \rightarrow 0^{+}} \beta(a)$ are</div>

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

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

answer:
<div class="solution">$(\sqrt[3]{1+a}-1) x^{2}+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0$
<br/> <br/>Let $a+1=y$, then equation reduces to $\left(y^{1 / 3}-1\right) x^{2}+\left(y^{1 / 2}-1\right) x+\left(y^{1 / 6}-1\right)=0$
<br/> <br/>On dividing both sides by $y-1$, we get $\left(\frac{y^{1 / 3}-1}{y-1}\right) x^{2}+\left(\frac{y^{1 / 2}-1}{y-1}\right) x+\left(\frac{y^{1 / 6}-1}{y-1}\right)=0$
<br/> <br/>On taking limit as $y \rightarrow 1$ i.e. $a \rightarrow 0$ on both sides, we get $\frac{1}{3} x^{2}+\frac{1}{2} x+\frac{1}{6}=0 \Rightarrow 2 x^{2}+3 x+1=0$
<br/> <br/>$\Rightarrow x=-1,-\frac{1}{2}$ (roots of the equation)
<br/> <br/>$\therefore \quad \lim _{a \rightarrow 0^{+}} \alpha(a)=-1, \lim _{a \rightarrow 0^{+}} \beta(a)=-\frac{1}{2}$</div>


tags: ['Mathematics', 'Pair of Lines', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-nonzero-and-real-numbers-then-the-equation-a-x-2-b-y-2-c-x-2-5-x-y-6-y-2-0-represents

slug:
let-a-and-b-be-nonzero-and-real-numbers-then-the-equation-a-x-2-b-y-2-c-x-2-5-x-y-6-y-2-0-represents-53393

question:
<div class="question">Let $a$ and $b$ be non-zero and real numbers. Then, the equation $\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0$ represents</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Two straight lines and a circle, when $a=b$ and $c$ is of sign opposite to that of $a$<br/></span> </div>

answer:
<div class="solution">Let $a$ and $b$ be non- zero real numbers.<br/>Therefore, the given equation $\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0$ implies either<br/>$$<br/>\begin{array}{rc} <br/>&amp; x^2-5 x y+6 y^2=0 \\<br/>\Rightarrow &amp; (x-2 y)(x-3 y)=0 \\<br/>\Rightarrow &amp; x=2 y \text { and } x=3 y<br/>\end{array}<br/>$$<br/>represent two straight lines passing through origin.<br/>Or $\quad a x^2+b y^2+c=0$<br/>When $c=0$ and $a$ and $b$ are of same signs, then<br/>$$<br/>\begin{aligned}<br/>&amp; a x^2+b y^2+c=0 \\<br/>\Rightarrow &amp; x=0 \text { and } y=0<br/>\end{aligned}<br/>$$<br/>which is a point specified as the origin.<br/>When $a=b$ and $c$ is of sign opposite to that of $a, a x^2+b y^2+c=0$ represents a circle.<br/>Hence, the given equation $\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0$<br/>may represent two straight lines and a circle.</div>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-two-distinct-points-on-the-parabola-y-2-4-x-if-the-axis-of-the-parabola-touches-a-circle-of-radius-r-having-a-b-as-its-diameter-then-th

slug:
let-a-and-b-be-two-distinct-points-on-the-parabola-y-2-4-x-if-the-axis-of-the-parabola-touches-a-circle-of-radius-r-having-a-b-as-its-diameter-then-th-50269

question:
<div class="question">Let $A$ and $B$ be two distinct points on the parabola $y^2=4 x$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining $A$ and $B$ can be</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\frac{2}{r}$<br/>, <br/>$-\frac{2}{r}$</span> </div>

answer:
<div class="solution">Here, coordinate $M=\left(\frac{t_1^2+t_2^2}{2}, t_1+t_2\right) i e$, mid point of chord $A B$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5OP5-giT0HF2WPAeV49UhRy4TxSM3EtXoEcLRC-CfFc.original.fullsize.png"/><br/><br/>$M P=t_1+t_2=r$ Also, $\quad m_{A B}=\frac{2 t_2-2 t_1}{t_2^2-t_1^2}=\frac{2^{\ldots(i)}}{t_2+t_1}$ (when $A B$ is chord)<br/>$$<br/>\Rightarrow \quad m_{A B}=\frac{2}{r}<br/>$$<br/>[from Eq. (i)]<br/>Also, $\quad m_{A^{\prime} B^{\prime}}=-\frac{2}{r}$ (when $A^{\prime} B^{\prime}$ is chord)<br/>Hence, (c, d) is the correct option.</div>


title: let-a-b-c-be-unit-vectors-such-that-a-b-c-0-which-one-of-the-following-is-correct
slug: let-a-b-c-be-unit-vectors-such-that-a-b-c-0-which-one-of-the-following-is-correct-97227
tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors such that $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$. Which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a} \neq \mathbf{0}$<br/></span> </div>

answer:
<div class="solution">Since $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$ $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ represent an equilateral triangle.<br/>$$<br/>\therefore \quad \mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a} \neq \mathbf{0} .<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-a-b-c-p-q-be-real-numbers-suppose-are-the-roots-of-the-equation-x-2-2-p-x-q-0-and-1-are-the-roots-of-the-equation-a-x-2-2-b-x-c-0-where-2-1-0-1-st

slug:
let-a-b-c-p-q-be-real-numbers-suppose-are-the-roots-of-the-equation-x-2-2-p-x-q-0-and-1-are-the-roots-of-the-equation-a-x-2-2-b-x-c-0-where-2-1-0-1-st-10949

question:
<div class="question">Let $a, b, c, p, q$ be real numbers.<br/>Suppose, $\alpha, \beta$ are the roots of the equation $x^2+2 p x+q=0$ and $\alpha, \frac{1}{\beta}$ are the roots of the equation $a x^2+2 b x+c=0$, where $\beta^2 \notin\{-1,0,1\}$.<br/>Statement $1\left(p^2-q\right)\left(b^2-a c\right) \geq 0$.<br/>Statement $2 b \neq p a$ or $c \neq q a$.</div>

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

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

answer:
<div class="solution">Given, $\alpha$ and $\beta$ are the roots of $x^2+2 p x+q=0$.<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; \therefore &amp; \alpha+\beta &amp; =-2 p \\<br/>\text { and } &amp; \alpha \beta &amp; =q<br/>\end{array}<br/>$$<br/>$\alpha$ and $\frac{1}{\beta}$ are the roots of $a x^2+2 b x+c=0$<br/>and<br/>$$<br/>\alpha+\frac{1}{\beta}=\frac{-2 b}{a}<br/>$$<br/><br/>and<br/>$$<br/>\frac{\alpha}{\beta}=\frac{c}{a}<br/>$$<br/>$$<br/>\text { Now, } \begin{aligned}<br/>\left(p^2-q\right)\left(b^2-a c\right) &amp; =\left[\left(\frac{\alpha+\beta}{-2}\right)^2-\alpha \beta\right]\left[\left(\frac{\alpha+\frac{1}{\beta}}{2}\right)^2-\frac{\alpha}{\beta}\right] a^2 \\<br/>&amp; =\frac{(\alpha-\beta)^2}{16}\left(\alpha-\frac{1}{\beta}\right)^2 \cdot a^2 \geq 0<br/>\end{aligned}<br/>$$<br/>Statement 1 is true.<br/>$$<br/>\begin{array}{ll}<br/>\text { Again now, } &amp; p a=-\left(\frac{\alpha+\beta}{2}\right) a=-\frac{a}{2}(\alpha+\beta) \\<br/>\text { and } &amp; b=-\frac{a}{2}\left(\alpha+\frac{1}{\beta}\right) \\<br/>&amp; p a \neq b \Rightarrow \alpha+\frac{1}{\beta} \neq \alpha+\beta \\<br/>\Rightarrow &amp; \beta \neq 1 \\<br/>\because &amp; \beta^2 \neq\{-1,0,1\}, \text { correct. } \\<br/>\text { Similarly, if } &amp; c \neq q x \\<br/>\Rightarrow &amp; a \frac{\alpha}{\beta} \neq a \alpha \beta \Rightarrow \alpha\left(\beta-\frac{1}{\beta}\right) \neq 0<br/>\end{array}<br/>$$<br/><br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; \alpha \neq 0 \text { and } \beta-\frac{1}{\beta} \neq 0 \\<br/>\Rightarrow &amp; \beta \neq\{-1,0,1\}<br/>\end{array}<br/>$$<br/>Statement 2 is true.<br/>Both Statement 1 and Statement 2 are true. But Statement 2 does not explain Statement 1.</div>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-bc-and-a-b-c-be-two-noncongruent-triangles-with-sides-a-b-4-a-c-a-c-2-2-and-angle-b-3-0-the-absolute-value-of-the-difference-between-the-areas-o

slug:
let-a-bc-and-a-b-c-be-two-noncongruent-triangles-with-sides-a-b-4-a-c-a-c-2-2-and-angle-b-3-0-the-absolute-value-of-the-difference-between-the-areas-o-15439

question:
<div class="question">Let $A B C$ and $A B C^{\prime}$ be two non-congruent triangles with sides $A B=4$, $A C=A C^{\prime}=2 \sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is</div>

answer:
<div class="solution">In $\triangle A B C$, by sine rule,<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{a}{\sin A}=\frac{2 \sqrt{2}}{\sin 30^{\circ}}=\frac{4}{\sin C} \\<br/>\Rightarrow \quad &amp; C=45^{\circ}, C^{\prime}=135^{\circ} \\<br/>\Rightarrow &amp; A=180^{\circ}-\left(45^{\circ}+30^{\circ}\right)=105^{\circ}<br/>\end{aligned}<br/>$$<br/><br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BUgaugr3-2iN2vBBsw8lq7vfyVWLNvDQoBTiS3mnVkM.original.fullsize.png"/><br/><br/><br/><br/>When $\quad C^{\prime}=135^{\circ}$, then<br/>$$<br/>\begin{aligned}<br/>&amp; \quad A=180^{\circ}-\left(135^{\circ}+30^{\circ}\right)=15^{\circ} \\<br/>&amp; \text { Area of } \triangle A B C=\frac{1}{2} A B \times A C \sin A \\<br/>&amp; =\frac{1}{2} \times 4 \times 2 \sqrt{2} \sin \left(105^{\circ}\right) \\<br/>&amp; =4 \sqrt{2} \times \frac{\sqrt{3}+1}{2 \sqrt{2}}=2(\sqrt{3}+1) \\<br/>&amp; \text { Area of } \triangle A B C^{\prime}=\frac{1}{2} A B \times A C \sin A \\<br/>&amp; =\frac{1}{2} \times 4 \times 2 \sqrt{2} \sin \left(15^{\circ}\right)=2(\sqrt{3}-1)<br/>\end{aligned}<br/>$$<br/>Difference of areas of triangles<br/>$$<br/>=|2(\sqrt{3}+1)-2(\sqrt{3}-1)|=4<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/7lJ98Kv3OvF_T4cGscG-lVOXIIQusN3nwnSa4WPdPhg.original.fullsize.png"/><br/><br/>$$<br/>A D=2, D C=2<br/>$$<br/>Difference of areas of $\triangle A B C$ and $\triangle A B C^{\prime}$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; =\text { Area of } \triangle A C C^{\prime} \\<br/>&amp; =\frac{1}{2} A D \times C C^{\prime}=\frac{1}{2} \times 2 \times 4=4<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-a-bc-be-a-triangle-such-that-a-cb-6-and-let-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-a-b-and-c-respectively-the-values-of-x-for-which

slug:
let-a-bc-be-a-triangle-such-that-a-cb-6-and-let-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-a-b-and-c-respectively-the-values-of-x-for-which-66914

question:
<div class="question">Let $A B C$ be a triangle such that $\angle A C B=\frac{\pi}{6}$ and let $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively. The value(s) of $x$ for which $a=x^2+x+1, b=x^2-1$ and $c=2 x+1$ is (are)</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$1+\sqrt{3}$<br/></span> </div>

answer:
<div class="solution">Using, $\cos C=\frac{a^2+b^2-c^2}{2 a b}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/89PQ-836ICof7ENzJeD4KV6Ji82KVNt2qE7sAheUO2c.original.fullsize.png"/><br/><br/>$$<br/>\begin{gathered}<br/>\Rightarrow \frac{\sqrt{3}}{2}=\frac{\left(x^2+x+1\right)^2+\left(x^2-1\right)^2-(2 x+1)^2}{2\left(x^2+x+1\right)\left(x^2-1\right)} \\<br/>\Rightarrow(x+2)(x+1)(x-1) x+\left(x^2-1\right)^2 \\<br/>=\sqrt{3}\left(x^2+x+1\right)\left(x^2-1\right) \\<br/>\Rightarrow x^2+2 x+\left(x^2-1\right)=\sqrt{3}\left(x^2+x+1\right) \\<br/>\Rightarrow(2-\sqrt{3}) x^2+(2-\sqrt{3}) x-(\sqrt{3}+1)=0<br/>\end{gathered}<br/>$$<br/><br/>$\Rightarrow x=-(2+\sqrt{3})$ and $x=1+\sqrt{3}$<br/>But, $x=-(2+\sqrt{3}) \Rightarrow c$ is negative. $\therefore \quad x=1+\sqrt{3}$ is the only solution.<br/>Hence, (b) is the correct option.</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
let-a-bc-d-be-a-quadrilateral-with-area-18-with-side-a-b-parallel-to-the-side-c-d-and-a-b-2-c-d-let-a-d-be-perpendicular-to-a-b-and-c-d-if-a-circle-is-1

slug:
let-a-bc-d-be-a-quadrilateral-with-area-18-with-side-a-b-parallel-to-the-side-c-d-and-a-b-2-c-d-let-a-d-be-perpendicular-to-a-b-and-c-d-if-a-circle-is-1-51680

question:
<div class="question">Let $A B C D$ be a quadrilateral with area 18 , with side $A B$ parallel to the side $C D$ and $A B=2 C D$. Let $A D$ be perpendicular to $A B$ and $C D$. If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is</div>

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

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

answer:
<div class="solution">$$<br/>\text { } 18=\frac{1}{2}(3 \alpha)(2 r) \Rightarrow \alpha r=6<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/t_pMgp4UiyFCXUuzb7KvmMfhX6yIJXigJy_YvZXXXZU.original.fullsize.png"/><br/><br/>Line, $y=-\frac{2 r}{\alpha}(x-2 \alpha)$ is tangent to circle<br/>$$<br/>\begin{aligned}<br/>(x-r)^2+(y-r)^2 &amp; =r^2 \\<br/>2 \alpha &amp; =3 r \text { and } \alpha=6 \\<br/>r &amp; =2<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>\frac{1}{2}(x+2 x) \times 2 r &amp; =18 \\<br/>x r &amp; =6 \\<br/>\tan \theta &amp; =\frac{x-r}{r} \\<br/>\tan \left(90^{\circ}-\theta\right) &amp; =\frac{2 x-r}{r} \\<br/>\frac{x-r}{r} &amp; =\frac{r}{2 x-r} \times(2 x-3 r)=0<br/>\end{aligned}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RYKRVuCPEMy3tbKorSAXw9Jpm6M4NNkl3z1Zw8MBA1Y.original.fullsize.png"/><br/><br/>$$<br/>x=\frac{3 r}{2}<br/>$$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>r=2<br/>$$</div>


tags: ['Mathematics', 'Circle', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-a-bc-d-be-a-quadrilateral-with-area-18-with-side-a-b-parallel-to-the-side-c-d-and-a-b-2-c-d-let-a-d-be-perpendicular-to-a-b-and-c-d-if-a-circle-is

slug:
let-a-bc-d-be-a-quadrilateral-with-area-18-with-side-a-b-parallel-to-the-side-c-d-and-a-b-2-c-d-let-a-d-be-perpendicular-to-a-b-and-c-d-if-a-circle-is-49338

question:
<div class="question">Let $A B C D$ be a quadrilateral with area 18 , with side $A B$ parallel to the side $C D$ and $A B=2 C D$. Let $A D$ be perpendicular to $A B$ and $C D$. If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is</div>

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

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

answer:
<div class="solution">$$<br/>\text { } 18=\frac{1}{2}(3 \alpha)(2 r) \Rightarrow \alpha r=6<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/t_pMgp4UiyFCXUuzb7KvmMfhX6yIJXigJy_YvZXXXZU.original.fullsize.png"/><br/><br/>Line, $y=-\frac{2 r}{\alpha}(x-2 \alpha)$ is tangent to circle<br/>$$<br/>\begin{aligned}<br/>(x-r)^2+(y-r)^2 &amp; =r^2 \\<br/>2 \alpha &amp; =3 r \text { and } \alpha=6 \\<br/>r &amp; =2<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>\frac{1}{2}(x+2 x) \times 2 r &amp; =18 \\<br/>x r &amp; =6 \\<br/>\tan \theta &amp; =\frac{x-r}{r} \\<br/>\tan \left(90^{\circ}-\theta\right) &amp; =\frac{2 x-r}{r} \\<br/>\frac{x-r}{r} &amp; =\frac{r}{2 x-r} \times(2 x-3 r)=0<br/>\end{aligned}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RYKRVuCPEMy3tbKorSAXw9Jpm6M4NNkl3z1Zw8MBA1Y.original.fullsize.png"/><br/><br/>$$<br/>x=\frac{3 r}{2}<br/>$$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>r=2<br/>$$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
let-a-be-vector-parallel-to-line-of-intersection-of-planes-p-1-and-p-2-through-origin-p-1-is-parallel-to-the-vectors-2-j-3-k-and-4-j-3-k-and-p-2-is-pa

slug:
let-a-be-vector-parallel-to-line-of-intersection-of-planes-p-1-and-p-2-through-origin-p-1-is-parallel-to-the-vectors-2-j-3-k-and-4-j-3-k-and-p-2-is-pa-96897

question:
<div class="question">Let $\mathbf{A}$ be vector parallel to line of intersection of planes $P_1$ and $P_2$ through origin. $P_1$ is parallel to the vectors $2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $P_2$ is parallel to $\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$, then the angle between vector $\mathbf{A}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ is</div>

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

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

answer:
<div class="solution">Let vector $\mathbf{A O}$ be parallel to line of intersection of planes $P_1$ and $P_2$ through, i.e.<br/>$$<br/>\begin{aligned}<br/>&amp; {[(2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \times(4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})] \times[(\hat{\mathbf{j}}-\hat{\mathbf{k}}) \times(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}})]=54(\hat{\mathbf{j}}-\hat{\mathbf{k}}) .} \\<br/>&amp; \therefore \text { Angle between } 54(\hat{\mathbf{j}}-\hat{\mathbf{k}}) \text { and }(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \\<br/>&amp; \Rightarrow \cos \theta=\pm\left(\frac{54+108}{3.54 \cdot \sqrt{2}}\right)=\pm \frac{1}{\sqrt{2}} \\<br/>&amp; \therefore \quad \theta=\frac{\pi}{4}, \frac{3 \pi}{4}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
let-a-i-2-j-k-b-i-j-k-c-i-j-k-a-vector-coplanar-to-a-and-b-has-a-projection-along-c-of-magnitude-3-1-then-the-vector-is

slug:
let-a-i-2-j-k-b-i-j-k-c-i-j-k-a-vector-coplanar-to-a-and-b-has-a-projection-along-c-of-magnitude-3-1-then-the-vector-is-34659

question:
<div class="question">Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$. A vector coplanar to $\mathbf{a}$ and $\mathbf{b}$ has a projection along c of magnitude $\frac{1}{\sqrt{3}}$, then the vector is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}$<br/></span> </div>

answer:
<div class="solution">Let vector $\mathbf{r}$ be coplanar to $\mathbf{a}$ and $\mathbf{b}$.<br/>$$<br/>\begin{array}{ll}<br/>\therefore &amp; \mathbf{r}=\mathbf{a}+t \mathbf{b} \\<br/>\Rightarrow &amp; \mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+t(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\hat{\mathbf{i}}(1+t)+\hat{\mathbf{j}}(2-t)+\hat{\mathbf{k}}(1+t)<br/>\end{array}<br/>$$<br/>The projection of $\mathbf{r}$ on $\mathbf{c}=\frac{1}{\sqrt{3}} \Rightarrow \frac{\mathbf{r} \cdot \mathbf{c}}{|\mathbf{c}|}=\frac{1}{\sqrt{3}}$ $\Rightarrow \frac{|1 \cdot(1+t)+1 \cdot(2-t)-1 \cdot(1+t)|}{\sqrt{3}}=\frac{1}{\sqrt{3}}$ $\Rightarrow|2-t|=\pm 1 \Rightarrow t=1$ or 3<br/>When, $t=1$ we have $\mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$<br/>When, $t=3$ we have $\mathbf{r}=4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-a-i-j-k-b-i-j-k-and-c-i-j-k-be-three-vectors-a-vector-v-in-the-plane-of-a-and-b-whose-projection-of-c-is-3-1-is-given-by

slug:
let-a-i-j-k-b-i-j-k-and-c-i-j-k-be-three-vectors-a-vector-v-in-the-plane-of-a-and-b-whose-projection-of-c-is-3-1-is-given-by-15759

question:
<div class="question">Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ be three vectors. A vector $\mathbf{v}$ in the plane of $\mathbf{a}$ and $\mathbf{b}$ whose projection of $\mathbf{c}$ is $\frac{1}{\sqrt{3}}$, is given by</div>

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

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

answer:
<div class="solution">Let $\mathbf{v}=\mathbf{a}+\lambda \mathbf{b}$<br/>$$<br/>\mathbf{v}=(1+\lambda) \hat{\mathbf{i}}+(1-\lambda) \hat{\mathbf{j}}+(1+\lambda) \hat{\mathbf{k}}<br/>$$<br/>Projection of $\mathbf{v}$ on $\mathbf{c}=\frac{1}{\sqrt{3}}$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow &amp; \frac{\mathbf{v} \cdot \mathbf{c}}{|\mathbf{c}|}=\frac{1}{\sqrt{3}} \\<br/>\Rightarrow &amp; \frac{(1+\lambda)-(1-\lambda)-(1+\lambda)}{\sqrt{3}}=\frac{1}{\sqrt{3}} \\<br/>\Rightarrow &amp; 1+\lambda-1+\lambda-1-\lambda=1 \\<br/>\Rightarrow &amp; \lambda-1=1 \Rightarrow \lambda=2 \\<br/>\therefore &amp; \mathbf{v}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-i-k-b-i-j-and-c-i-2-j-3-k-be-three-given-vectors-if-r-is-a-vector-such-that-r-b-c-b-and-r-a-0-then-the-value-of-r-b-is

slug:
let-a-i-k-b-i-j-and-c-i-2-j-3-k-be-three-given-vectors-if-r-is-a-vector-such-that-r-b-c-b-and-r-a-0-then-the-value-of-r-b-is-88519

question:
<div class="question">Let $\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ be three given vectors. If $\mathbf{r}$ is a vector such that $\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}$ and $\mathbf{r} \cdot \mathbf{a}=0$, then the value of $\mathbf{r} \cdot \mathbf{b}$ is</div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b} \\<br/>&amp; \Rightarrow \quad(\mathbf{r}-\mathbf{c}) \times \mathbf{b}=0 \Rightarrow \mathbf{r}-\mathbf{c}+\lambda \mathbf{b} \\<br/>&amp; \text { or } \quad \mathbf{r}=\mathbf{c}+\lambda \mathbf{b} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>Given, $\mathbf{r} \cdot \mathbf{a}=0$, taking dot product with a for Eq. (i).<br/>Now, $\mathbf{r} \cdot \mathbf{a}=\mathbf{a} \cdot \mathbf{c}+\lambda \mathbf{a} \cdot \mathbf{b}$<br/>$\therefore \quad \lambda=\frac{-\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \quad[\because \vec{r} \cdot \vec{a}=0] \ldots$ (i)<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>\mathbf{r}=\mathbf{c}-\frac{\mathbf{a} \cdot \mathbf{c}}{\mathbf{a} \cdot \mathbf{b}} \mathbf{b}<br/>$$<br/>Taking dot with $\mathbf{b}$, we get<br/>$$<br/>\mathbf{r} \cdot \mathbf{b}=\mathbf{c} \cdot \mathbf{b}-\frac{\mathbf{a} \cdot \mathbf{c}}{\mathbf{a} \cdot \mathbf{b}}(\mathbf{b} \cdot \mathbf{b})<br/>$$<br/><br/>where, $\left[\begin{array}{l}\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}} \\ \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}} \\ \mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\end{array}\right]$<br/>$$<br/>=(-1+2)-\frac{(-1-3)}{(1)}(1+1)=1+8=9<br/>$$</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2006']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-n-4-3-4-3-2-4-3-3-1-n-1-4-3-n-and-b-n-1-a-n-find-a-least-odd-natural-number-n-0-so-that-b-n-a-n-n-n-0

slug:
let-a-n-4-3-4-3-2-4-3-3-1-n-1-4-3-n-and-b-n-1-a-n-find-a-least-odd-natural-number-n-0-so-that-b-n-a-n-n-n-0-69545

question:
<div class="question">Let $A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $B_n=1-A_n$. Find a least odd natural number $n_0$, so that $B_n&gt;A_n, \forall n \geq n_0$.</div>

answer:
<div class="solution">$B_n=1-A_n&gt;A_n \Rightarrow A_n &lt; \frac{1}{2}$ Now, $A_n=\frac{\frac{3}{4}\left(1-\left(-\frac{3}{4}\right)^n\right)}{1+\frac{3}{4}} &lt; \frac{1}{2} \Rightarrow\left(-\frac{3}{4}\right)&gt;-\frac{1}{6}$<br/>Obviously, it is true for all even values of $n$. But, for<br/>$$<br/>\begin{aligned}<br/>n &amp; =1, &amp; \frac{3}{4} &amp; &lt; \frac{1}{6} \\<br/>n &amp; =3, &amp; \frac{27}{64} &amp; &lt; \frac{1}{6} \\<br/>n &amp; =5, &amp; \frac{243}{1024} &amp; &gt;\frac{1}{6}<br/>\end{aligned}<br/>$$<br/>which is true for $n=7$.<br/>Obviously, $n_0=7$</div>


tags: ['Mathematics', 'Permutation Combination', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-a-n-denote-the-number-of-all-n-digit-positive-integers-formed-by-the-digits-01-or-both-such-that-no-consecutive-digits-in-them-are-0-let-b-n-the-n-1

slug:
let-a-n-denote-the-number-of-all-n-digit-positive-integers-formed-by-the-digits-01-or-both-such-that-no-consecutive-digits-in-them-are-0-let-b-n-the-n-1-55664

question:
<div class="question">Let $a_{n}$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let $b_{n}=$ the number of such $n$-digit integers ending with digit 1 and $c_{n}=$ the number of such $n$-digit integers ending with digit 0 .
<br/> <br/><strong> Question:</strong> Which of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$a_{17}=a_{16}+a_{15}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$c_{17} \neq c_{16}+c_{15}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$b_{17} \neq b_{16}+c_{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a_{17}=c_{17}+b_{16}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a_{17}=a_{16}+a_{15}$</span> </div>

answer:
<div class="solution">By recurring formula, $a_{17}=a_{16}+a_{15}$ is correct
<br/> <br/>Also $c_{17} \neq c_{16}+c_{15}$
<br/> <br/>$\Rightarrow a_{15} \neq a_{14}+a_{13}\left(\because c_{n}=a_{n-2}\right)$
<br/> <br/>$\therefore$ Incorrect
<br/> <br/>Similarly, other parts are also incorrect.</div>


tags: ['Mathematics', 'Permutation Combination', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-n-denote-the-number-of-all-n-digit-positive-integers-formed-by-the-digits-01-or-both-such-that-no-consecutive-digits-in-them-are-0-let-b-n-the-n

slug:
let-a-n-denote-the-number-of-all-n-digit-positive-integers-formed-by-the-digits-01-or-both-such-that-no-consecutive-digits-in-them-are-0-let-b-n-the-n-10346

question:
<div class="question">Let $a_{n}$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let $b_{n}=$ the number of such $n$-digit integers ending with digit 1 and $c_{n}=$ the number of such $n$-digit integers ending with digit 0 .
<br/> <br/><strong> Question:</strong> The value of $b_{6}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">11</span> </li> </ul>

answer:
<div class="solution">$\because a_{n}=$ number of all $n$ digit +ve integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 .
<br/> <br/>and $b_{n}=$ number of such $n$ digit integers ending with 1 $c_{n}=$ number of such $n$ digit integers ending with 0
<br/> <br/>Clearly, $a_{n}=b_{n}+c_{n}\left(\because a_{n}\right.$ can end with 0 or 1)
<br/> <br/>Also $b_{n}=a_{n-1}$ and $c_{n}=a_{n-2}[\because$ if last digit is 0 , second last has to be 1]
<br/> <br/>$\therefore$ We get $a_{n}=a_{n-1}+a_{n-2}, n \geq 3$
<br/> <br/>Also $a_{1}=1, a_{2}=2$,
<br/> <br/>Now by this recurring formula, we get
<br/> <br/>$\begin{aligned} a_{3} &amp;=a_{2}+a_{1}=3 \\ a_{4} &amp;=a_{3}+a_{2}=3+2=5 \\ a_{5} &amp;=a_{4}+a_{3}=5+3=8 \\ \text { Also } b_{6} &amp;=a_{5}=8 \end{aligned}$</div>


tags: ['Mathematics', 'Differential Equations', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-a-solution-y-y-x-of-the-differential-equation-x-x-2-1-d-y-y-y-2-1-d-x-0-satisfy-y-2-3-2-statement-1-y-x-sec-sec-1-x-6-statement-2-y-x-is-given-by-

slug:
let-a-solution-y-y-x-of-the-differential-equation-x-x-2-1-d-y-y-y-2-1-d-x-0-satisfy-y-2-3-2-statement-1-y-x-sec-sec-1-x-6-statement-2-y-x-is-given-by-74693

question:
<div class="question">Let a solution $y=y(x)$ of the differential equation $x \sqrt{x^2-1} d y-y \sqrt{y^2-1} d x=0$ satisfy $y(2)=\frac{2}{\sqrt{3}}$<br/>Statement $1 y(x)=\sec \left(\sec ^{-1} x-\frac{\pi}{6}\right)$<br/>Statement $2 y(x)$ is given by $\frac{1}{y}=\frac{2 \sqrt{3}}{x}-\sqrt{1-\frac{1}{x^2}}$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement 1 is true, Statement 2 is false.<br/></span> </div>

answer:
<div class="solution">$\because \frac{d y}{d x}=\frac{y \sqrt{y^2-1}}{x \sqrt{x^2-1}} \Rightarrow \int \frac{d y}{y \sqrt{y^2-1}}=\int \frac{d x}{x \sqrt{x^2-1}}$ $\Rightarrow \quad \sec ^{-1} y=\sec ^{-1} x+C$ At $\quad x=2, y=\frac{2}{\sqrt{3}}$ $$ \frac{\pi}{6}=\frac{\pi}{3}+C \Rightarrow C=-\frac{\pi}{6} $$<br/>Now, $y=\sec \left(\sec ^{-1} x-\frac{\pi}{6}\right)$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; =\cos \left[\cos ^{-1} \frac{1}{x}-\cos ^{-1} \frac{\sqrt{3}}{2}\right] \\<br/>&amp; =\cos \left[\cos ^{-1}\left(\frac{\sqrt{3}}{2 x}+\sqrt{1-\frac{1}{x^2}} \sqrt{1-\frac{3}{4}}\right)\right] \\<br/>\Rightarrow \quad \frac{1}{y} &amp; =\frac{\sqrt{3}}{2 x}+\frac{1}{2} \sqrt{1-\frac{1}{x^2}} \\</div>


title: let-and-be-the-roots-of-x-2-6-x-2-0-with-if-a-n-n-n-for-n-1-then-the-value-of-2-a-9-a-10-2-a-8-is
tags: ['Mathematics', 'Quadratic Equation', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P1.db
slug:
let-and-be-the-roots-of-x-2-6-x-2-0-with-if-a-n-n-n-for-n-1-then-the-value-of-2-a-9-a-10-2-a-8-is-46905

question:
<div class="question">Let $\alpha$ and $\beta$ be the roots of $x^2-6 x-2=0$ with $\alpha&gt;\beta$. If $a_n=\alpha^n-\beta^n$ for $n \geq 1$, then the value of $\frac{a_{10}-2 a_8}{2 a_9}$ is</div>

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

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

answer:
<div class="solution">$\frac{a_{10}-2 a_8}{2 a_9}=\frac{\left(\alpha^{10}-\beta^{10}\right)-2\left(\alpha^8-\beta^8\right)}{2\left(\alpha^9-\beta^9\right)}$ $=\frac{\alpha^8\left(\alpha^2-2\right)-\beta^8\left(\beta^2-2\right)}{2\left(\alpha^9-\beta^9\right)}$ $\left[\because \alpha\right.$ is root of $x^2-6 x-2=0$ $\left.\alpha^2-2=6 \alpha\right]$<br/>Also, $\beta$ is root of $x^2-6 x-2=0$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad \beta^2-2=6 \beta \\<br/>&amp; =\frac{\alpha^8(6 \alpha)-\beta^8(6 \beta)}{2\left(\alpha^9-\beta^9\right)}=\frac{6\left(\alpha^9-\beta^9\right)}{2\left(\alpha^9-\beta^9\right)}=3<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-be-a-complex-cube-root-of-unity-with-1-a-fair-die-is-thrown-three-times-if-r-1-r-2-and-r-3-are-the-numbers-obtained-on-the-die-then-the-probabilit

slug:
let-be-a-complex-cube-root-of-unity-with-1-a-fair-die-is-thrown-three-times-if-r-1-r-2-and-r-3-are-the-numbers-obtained-on-the-die-then-the-probabilit-32393

question:
<div class="question">Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. A fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{1}{18}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{1}{9}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{2}{9}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{1}{36}$</span> </li> </ul>

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

answer:
<div class="solution">Sample space A dice is thrown thrice, $n(s)=6 \times 6 \times 6$.<br/>Favorable events $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ ie, $\left(r_1, r_2, r_3\right)$ are ordered 3-triples which can take values, $\left.\begin{array}{llll}(1,2,3), &amp; (1,5,3), &amp; (4,2,3), &amp; (4,5,3) \\ (1,2,6), &amp; (1,5,6), &amp; (4,2,6), &amp; (4,5,6)\end{array}\right\}$ ie, 8 ordered pairs and each can be arranged in 3 ! ways $=6$<br/>$$<br/>\begin{array}{ll}<br/>\therefore &amp; n(E)=8 \times 6 \\<br/>\Rightarrow &amp; P(E)=\frac{8 \times 6}{6 \times 6 \times 6}=\frac{2}{9}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
let-be-the-complex-number-cos-3-2-i-sin-3-2-then-the-number-of-distinct-complex-number-z-satisfying-z-1-2-z-2-1-2-1-z-0-is-equal-to

slug:
let-be-the-complex-number-cos-3-2-i-sin-3-2-then-the-number-of-distinct-complex-number-z-satisfying-z-1-2-z-2-1-2-1-z-0-is-equal-to-72006

question:
<div class="question">Let $\omega$ be the complex number $\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}$. Then the number of distinct complex number $z$ satisfying $\left|\begin{array}{ccc}z+1 &amp; \omega &amp; \omega^2 \\ \omega &amp; z+\omega^2 &amp; 1 \\ \omega^2 &amp; 1 &amp; z+\omega\end{array}\right|=0$ is equal to</div>

answer:
<div class="solution">Let $A=\left[\begin{array}{ccc}1 &amp; \omega &amp; \omega^2 \\ \omega &amp; \omega^2 &amp; 1 \\ \omega^2 &amp; 1 &amp; \omega\end{array}\right]$<br/>Now, $\quad A^2=\left[\begin{array}{lll}0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0\end{array}\right]$<br/>and $\operatorname{Tr}(A)=0,|A|=0$<br/>$\therefore \quad A^3=0$<br/><br/>$\Rightarrow\left|\begin{array}{ccc}z+1 &amp; \omega &amp; \omega^2 \\ \omega &amp; z+\omega^2 &amp; 1 \\ \omega^2 &amp; 1 &amp; z+\omega\end{array}\right|=|A+z I|=0$<br/>$\Rightarrow \quad z^3=0$<br/>$\Rightarrow z=0$, the number of $z$ satisfying the given equation is 1 .</div>


tags: ['Mathematics', 'Quadratic Equation', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-be-the-roots-of-the-equation-x-2-p-x-r-0-and-2-2-be-the-roots-of-the-equation-x-2-q-x-r-0-then-the-value-of-r-is

slug:
let-be-the-roots-of-the-equation-x-2-p-x-r-0-and-2-2-be-the-roots-of-the-equation-x-2-q-x-r-0-then-the-value-of-r-is-80075

question:
<div class="question">Let $\alpha, \beta$ be the roots of the equation $x^2-p x+r=0$ and $\frac{\alpha}{2}, 2 \beta$ be the roots of the equation $x^2-q x+r=0$. Then, the value of $r$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{2}{9}(2 p-q)(2 q-p)$</span> </div>

answer:
<div class="solution">The equation $x^2-p x+r=0$ has roots $(\alpha, \beta)$ and the equation $x^2-q x+r=0$ has roots<br/>$$<br/>\begin{array}{ll}<br/>\left(\frac{\alpha}{2}, 2 \beta\right) . &amp; \\<br/>\Rightarrow &amp; r=\alpha \beta \text { and } \alpha+\beta=p \text { and } \frac{\alpha}{2}+2 \beta=q \\<br/>\Rightarrow &amp; \beta=\frac{2 q-p}{3} \text { and } \alpha=\frac{2(2 p-q)}{3} \\<br/>\Rightarrow &amp; \alpha \beta=r=\frac{2}{9}(2 q-p)(2 p-q)<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-e-and-f-be-two-independent-events-the-probability-that-exactly-one-of-them-occurs-is-1125-and-the-probability-of-none-of-them-occurring-is-225-if-

slug:
let-e-and-f-be-two-independent-events-the-probability-that-exactly-one-of-them-occurs-is-1125-and-the-probability-of-none-of-them-occurring-is-225-if-64101

question:
<div class="question">Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $11 / 25$ and the probability of none of them occurring is $2 / 25$. If $P(T)$ denotes the probability of occurrence of the event $T$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$P(E)=\frac{4}{5}, P(F)=\frac{3}{5}$<br/>, <br/>$P(E)=\frac{3}{5}, P(F)=\frac{4}{5}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/npte3oGDly9GUN3k8gBrVXSvn3jwoEwWiCLTtMUQ5Ng.original.fullsize.png"/><br/><br/><br/>$$<br/>P(E \cup F)-P(E \cap F)=\frac{11}{25}<br/>$$<br/>(i.e. only $E$ or only $F$ )<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/mBHGp2I9tOtkfP60SSq3iGSoDXC_ovlYuNpy4rtKhKE.original.fullsize.png"/><br/><br/>Neither of them occurs $=\frac{2}{25}$ $\Rightarrow \quad P(\bar{E} \cap \bar{F})=\frac{2}{25}$<br/>From Eq. (i), we get<br/>$$<br/>P(E)+P(F)-2 P(E \cap F)=\frac{11}{25}<br/>$$<br/>From Eq. (ii), we get<br/>$$<br/>\begin{gathered}<br/>(1-P(E))(1-P(F))=\frac{2}{25} \\<br/>\Rightarrow 1-P(E)-P(F)+P(E) \cdot P(F)=\frac{2}{25}<br/>\end{gathered}<br/>$$<br/>(iv)<br/>From Eqs. (iii) and (iv), we get<br/>$$<br/>\begin{aligned}<br/>&amp; P(E)+P(F)=\frac{7}{5} \text { and } P(E) \cdot P(F)=\frac{12}{25} \\<br/>&amp; \therefore \quad P(E) \cdot\left\{\frac{7}{5}-P(E)\right\}=\frac{12}{25}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad(P(E))^2-\frac{7}{5} P(E)+\frac{12}{25}=0 \\<br/>&amp; \Rightarrow \quad\left(P(E)-\frac{3}{5}\right)\left(P(E)-\frac{4}{5}\right)=0 \\<br/>&amp; \therefore \quad P(E)=\frac{3}{4} \text { or } \frac{4}{5} \Rightarrow P(F)=\frac{4}{5} \text { or } \frac{3}{5}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-e-c-denotes-the-complement-of-an-event-e-let-e-f-g-be-pairwise-independent-events-with-p-g-0-and-p-e-f-g-0-then-p-e-c-f-c-g-equals

slug:
let-e-c-denotes-the-complement-of-an-event-e-let-e-f-g-be-pairwise-independent-events-with-p-g-0-and-p-e-f-g-0-then-p-e-c-f-c-g-equals-50316

question:
<div class="question">Let $E^c$ denotes the complement of an event $E$. Let $E, F, G$ be pairwise independent events with $P(G)&gt;0$ and $P(E \cap F \cap G)=0$. Then, $P\left(E^c \cap F^c \mid G\right)$ equals</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$P\left(E^c\right)-P(F)$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { } \begin{aligned}<br/>P\left(\frac{E^c \cap F^c}{G}\right) &amp; =\frac{P\left(E^c \cap F^c \cap G\right)}{P(G)} \\<br/>&amp; =\frac{P(G)-P(E \cap G)-P(G \cap F)}{P(G)} \\<br/>&amp; =\frac{P(G)[1-P(E)-P(F)]}{P(G)} \\<br/>&amp; =1-P(E)-P(F) \\<br/>&amp; =P\left(E^c\right)-P(F) .<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
let-e-i-3-and-a-b-c-x-y-z-be-nonzero-complex-numbers-such-that-a-b-c-x-a-b-c-2-y-a-b-2-c-z-then-the-value-of-a-2-b-2-c-2-x-2-y-2-z-2-is

slug:
let-e-i-3-and-a-b-c-x-y-z-be-nonzero-complex-numbers-such-that-a-b-c-x-a-b-c-2-y-a-b-2-c-z-then-the-value-of-a-2-b-2-c-2-x-2-y-2-z-2-is-35077

question:
<div class="question">Let $\omega=e^{i \pi / 3}$ and $a, b, c, x, y, z$ be non-zero complex numbers such that $a+b+c=x, a+b \omega+c \omega^2=y, a+b \omega^2+c \omega=z$.<br/>Then, the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is</div>

answer:
<div class="solution">Here, $\omega=e^{i 2 \pi / 3}$, then only integer solution exists.<br/>Then, $\frac{\left|x^2\right|+\left|y^2\right|+\left|z^2\right|}{\left|a^2\right|+\left|b^2\right|+\left|c^2\right|}=3$</div>


title: let-f-0-1-r-be-defined-by-f-x-1-b-x-b-x-where-b-is-a-constant-such-that-0-b-1-then
slug: let-f-0-1-r-be-defined-by-f-x-1-b-x-b-x-where-b-is-a-constant-such-that-0-b-1-then-66708
tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f:(0,1) \rightarrow R$ be defined by $f(x)=\frac{b-x}{1-b x}$, where $b$ is a constant such that $0 &lt; b &lt; 1$. Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$f$ if not invertible on $(0,1)$<br/></span> </div>

answer:
<div class="solution">Here, $f(x)=\frac{b-x}{1-b x}$ where, $0 &lt; b &lt; 1,0 &lt; x &lt; 1$ For function to be invertible it should be one-one onto.<br/>$\therefore$ Check range :<br/>Let $\quad f(x)=y \Rightarrow y=\frac{b-x}{1-b x}$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow y-b x y=b-x \Rightarrow x(1-b y)=b-y \\<br/>&amp; \Rightarrow \quad x=\frac{b-y}{1-b y}<br/>\end{aligned}<br/>$$<br/><br/>where, $0 &lt; x &lt; 1$<br/>$$<br/>\begin{gathered}<br/>\therefore \quad 0 &lt; \frac{b-y}{1-b y} &lt; 1 \\<br/>\frac{b-y}{1-b y}&gt;0 \text { and } \frac{b-y}{1-b y} &lt; 1 \\<br/>\quad+\quad-\quad+ \\<br/>\Rightarrow \quad b \\<br/>y &lt; b \text { or } y&gt;\frac{1}{b} \\<br/>\frac{(b-1)(y+1)}{1-b y} &lt; -1 &lt; y &lt; \frac{1}{b}<br/>\end{gathered}<br/>$$<br/>From Eqs. (i) and (ii), we get $y \in\left(-1, \frac{1}{b}\right) \subset$ Codomain<br/>Thus, $f(x)$ is not invertible.</div>


title: let-f-1-1-i-r-be-such-that-f-cos-4-2-s-e-c-2-2-for-0-4-4-2-then-the-value-s-of-f-3-1-is-are
slug: let-f-1-1-i-r-be-such-that-f-cos-4-2-s-e-c-2-2-for-0-4-4-2-then-the-value-s-of-f-3-1-is-are-66058
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f:(-1,1) \rightarrow I R$ be such that $f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}$ for $\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value (s) of $f\left(\frac{1}{3}\right)$ is (are)</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$1-\sqrt{\frac{3}{2}}$, $1+\sqrt{\frac{3}{2}}$</span> </div>

answer:
<div class="solution">Given : $f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}=\frac{2 \cos ^{2} \theta}{2 \cos ^{2} \theta-1}$
<br/> <br/>$=\frac{1+\cos 2 \theta}{\cos 2 \theta}=1+\frac{1}{\cos 2 \theta}$
<br/> <br/>Let $\cos 4 \theta=\frac{1}{3} \Rightarrow 2 \cos ^{2} 2 \theta-1=\frac{1}{3} \Rightarrow \cos 2 \theta=\pm \sqrt{\frac{2}{3}}$
<br/> <br/>$\therefore f(\cos 4 \theta)=1+\frac{1}{\cos 2 \theta}=1 \pm \sqrt{\frac{3}{2}}$ or $f\left(\frac{1}{3}\right)=1 \pm \sqrt{\frac{3}{2}}$</div>


tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-f-1-2-0-be-a-continuous-function-such-that-f-x-f-1-x-for-all-x-1-2-let-r-1-1-2-x-f-x-d-x-and-r-2-be-the-area-of-the-region-bounded-by-y-f-x-x-1-x-

slug:
let-f-1-2-0-be-a-continuous-function-such-that-f-x-f-1-x-for-all-x-1-2-let-r-1-1-2-x-f-x-d-x-and-r-2-be-the-area-of-the-region-bounded-by-y-f-x-x-1-x-68472

question:
<div class="question">Let $f:[-1,2] \rightarrow[0, \infty)$ be a continuous function such that $f(x)=f(1-x)$ for all $x \in[-1,2]$. Let $R_1=\int_{-1}^2 x f(x) d x$ and $R_2$ be the area of the region bounded by $y=f(x), x=-1, x=2$ and the $X$-axis. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$R_1=2 R_2$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$R_1=3 R_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$2 R_1=R_2$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$3 R_1=R_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$2 R_1=R_2$<br/></span> </div>

answer:
<div class="solution">Here, $R_1=\int_{-1}^2 x f(x) d x$<br/>Using, $\quad \int_a^b f(x) d x=\int_a^b f(a+b-x) d x$<br/>$$<br/>\begin{aligned}<br/>&amp; R_1=\int_{-1}^2(1-x) f(1-x) d x, \\<br/>\therefore \quad &amp; \left.R_1=\int_{-1}^2(1-x) f(x) d x \quad \ldots(1-x)\right]<br/>\end{aligned}<br/>$$<br/>Given, $R_2$ is area bounded by<br/>$$<br/>\begin{aligned}<br/>&amp; f(x), x-1 \text { and } x=2 \\<br/>\therefore &amp; R_2=\int_{-1}^2 f(x) d x<br/>\end{aligned}<br/>$$<br/>Adding Eqs. (i) and (ii), we get<br/>$$<br/>2 R_1=\int_{-1}^2 f(x) d x<br/>$$<br/><br/>$\therefore$ From Eqs. (iii) and (iv), we get<br/>$$<br/>2 R_1=R_2<br/>$$</div>


tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2.667</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-1-2-be-a-differentiable-function-such-that-f-1-2-if-6-1-x-f-t-d-t-3-x-f-x-x-3-for-all-x-1-then-the-value-of-f-2-is

slug:
let-f-1-2-be-a-differentiable-function-such-that-f-1-2-if-6-1-x-f-t-d-t-3-x-f-x-x-3-for-all-x-1-then-the-value-of-f-2-is-18706

question:
<div class="question">Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function such that $f(1)=2$. If $6 \int_1^x f(t) d t=3 x f(x)-x^3$ for all $x \geq 1$, then the value of $f(2)$ is</div>

answer:
<div class="solution">Given, $f(1)=\frac{1}{3}$ and $6 \int_1^x f(t) d t$ $=3 x f(x)-x^3$, for all $x \geq 1$<br/>Using (Newton-Leibnitz formula),<br/>On Differentiating both sides,<br/>$$<br/>\begin{aligned}<br/>&amp; 6 f(x) \cdot 1-0-3 f(x)+3 x f^{\prime}(x)-3 x^2 \\<br/>\Rightarrow &amp; 3 x f^{\prime}(x)-3 f(x)=3 x^2 \\<br/>\Rightarrow &amp; f^{\prime}(x)-\frac{1}{x} f(x)=x<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\Rightarrow \frac{x f^{\prime}(x)-f(x)}{x^2}=1 \Rightarrow \frac{d}{d x}\left\{\frac{f(x)}{x}\right\}=1<br/>$$<br/>On integrating both sides,<br/>$$<br/>\begin{array}{ll} <br/>&amp; \frac{f(x)}{x}=x+C \quad\left[\because f(1)=\frac{1}{3}\right] \\<br/>\Rightarrow \quad &amp; \frac{1}{3}=1+C \Rightarrow \quad C=-\frac{2}{3} \\<br/>&amp; \text { Now, } \quad f(x)=x^2-\frac{2}{3} x \\<br/>\Rightarrow \quad &amp; f(2)=4-\frac{4}{3}=\frac{8}{3}<br/>\end{array}<br/>$$<br/>Note Here, $f(1)=2$ does not satisfy given function.<br/>$$<br/>\therefore \quad f(1)=\frac{1}{3}<br/>$$<br/>For that, $f(x)=x^2-\frac{2}{3} x$ and $\quad f(2)=4-\frac{4}{3}=\frac{8}{3}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-and-g-be-real-valued-functions-defined-on-interval-1-1-such-that-g-x-is-continuous-g-0-0-g-0-0-g-0-0-and-f-x-g-x-sin-x-statement-1-lim-x-0-g-x-c

slug:
let-f-and-g-be-real-valued-functions-defined-on-interval-1-1-such-that-g-x-is-continuous-g-0-0-g-0-0-g-0-0-and-f-x-g-x-sin-x-statement-1-lim-x-0-g-x-c-95023

question:
<div class="question">Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g^{\prime \prime}(x)$ is continuous, $g(0) \neq 0, g^{\prime}(0)=0, g^{\prime \prime}(0) \neq 0$ and $f(x)=g(x) \sin x$.<br/>Statement $1 \lim _{x \rightarrow 0}[g(x) \cos x-g(0) \operatorname{cosec} x]=f^{\prime \prime}(0)$.<br/>Statement $2 f^{\prime}(0)=g(0)$.</div>

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

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

answer:
<div class="solution">We have,<br/>$$<br/>\begin{aligned}<br/>&amp; \lim _{x \rightarrow 0} \frac{g(x) \cos x-g(0)}{\sin x} \\<br/>&amp; =\lim _{x \rightarrow 0} \frac{g^{\prime}(x) \cos x-g(x) \sin x}{\cos x}=0<br/>\end{aligned}<br/>$$<br/>Since, $f(x)=g(x) \sin x$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad f^{\prime}(x)=g^{\prime}(x) \sin x+g(x) \cos x \\<br/>&amp; \Rightarrow \quad f^{\prime \prime}(x)=g^{\prime \prime}(x) \sin x+2 g^{\prime}(x) \cos x-g(x) \sin x \\<br/>&amp; \Rightarrow \quad f^{\prime \prime}(0)=0<br/>\end{aligned}<br/>$$<br/>Thus, $\lim _{x \rightarrow 0}[g(x) \cos x-g(0) \operatorname{cosec} x]=0=f^{\prime \prime}(0)$<br/>$\Rightarrow$ Statement 1 is true.<br/>Statement 2. $f^{\prime}(x)=g^{\prime}(x) \sin x+g(x) \cos x$<br/>$$<br/>\Rightarrow \quad f^{\prime}(0)=g(0)<br/>$$<br/>Statement 2 is not a correct explanation of Statement 1.</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-be-a-function-defined-on-r-the-set-of-all-real-numbers-such-that-f-x-2010-x-2009-x-2010-2-x-2011-3-x-2012-4-for-all-x-r-if-g-is-a-function-defin

slug:
let-f-be-a-function-defined-on-r-the-set-of-all-real-numbers-such-that-f-x-2010-x-2009-x-2010-2-x-2011-3-x-2012-4-for-all-x-r-if-g-is-a-function-defin-19026

question:
<div class="question">Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)$ $(x-2010)^2(x-2011)^3(x-2012)^4$, for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln (g(x))$, for all $x \in R$, then the number of points in $R$ at which $g$ has a local maximum is</div>

answer:
<div class="solution">Let $g(x)=e^{f(x)}, \forall x \in R$<br/>$$<br/>\Rightarrow g^{\prime}(x)=e^{f(x)} \cdot f^{\prime}(x)<br/>$$<br/>$\Rightarrow f^{\prime}(x)$ changes its sign from positive to negative in the neighbourhood of $x=2009$<br/>$\Rightarrow f(x)$ has local maxima at $x=2009$<br/>So, the number of local maximum is one.</div>


tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-be-a-nonnegative-function-defined-on-the-interval-0-1-if-0-x-1-f-t-2-d-t-0-x-f-t-d-t-0-x-1-and-f-0-0-then

slug:
let-f-be-a-nonnegative-function-defined-on-the-interval-0-1-if-0-x-1-f-t-2-d-t-0-x-f-t-d-t-0-x-1-and-f-0-0-then-72949

question:
<div class="question">Let $f$ be a non-negative function defined on the interval $[0,1]$. If $\int_0^x \sqrt{1-\left\{f^{\prime}(t)\right\}^2} d t=\int_0^x f(t) d t, \quad 0 \leq x \leq 1$<br/>$$<br/>\text { and } f(0)=0 \text {, then }<br/>$$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$f\left(\frac{1}{2}\right) &lt; \frac{1}{2}$ and $f\left(\frac{1}{3}\right)&gt;\frac{1}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$f\left(\frac{1}{2}\right)&gt;\frac{1}{2}$ and $f\left(\frac{1}{3}\right)&gt;\frac{1}{3}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$f\left(\frac{1}{2}\right) &lt; \frac{1}{2}$ and $f\left(\frac{1}{3}\right) &lt; \frac{1}{3}$</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"><br/>$f\left(\frac{1}{2}\right)&gt;\frac{1}{2}$ and $f\left(\frac{1}{3}\right) &lt; \frac{1}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$f\left(\frac{1}{2}\right) &lt; \frac{1}{2}$ and $f\left(\frac{1}{3}\right) &lt; \frac{1}{3}$</span> </div>

answer:
<div class="solution">Given $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} d t=\int_0^x f(t) d t$,<br/>$$<br/>0 \leq x \leq 1<br/>$$<br/>Applying Leibnitz theorem, we get<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; \sqrt{1-\left(f^{\prime}(x)\right)^2} &amp; =f(x) \\<br/>\Rightarrow &amp; &amp; 1-\left(f^{\prime}(x)\right)^2 &amp; =f^2(x) \\<br/>\Rightarrow &amp; &amp; \left(f^{\prime}(x)\right)^2 &amp; =1-f^2(x) \\<br/>\Rightarrow &amp; &amp; f^{\prime}(x) &amp; =\pm \sqrt{1-f^2(x)} \\<br/>\Rightarrow &amp; &amp; \quad \frac{d y}{d x} &amp; =\pm \sqrt{1-y^2} \\<br/>&amp; \text { where } y=f(x) \Rightarrow &amp; \frac{d y}{\sqrt{1-y^2}}=\pm d x<br/>\end{array}<br/>$$<br/>On integrating both sides, we get<br/>$$<br/>\begin{aligned}<br/>&amp; \sin ^{-1}(y)=\pm x+C \\<br/>&amp; \because \quad f(0)=0 \Rightarrow C=0 \Rightarrow y=\pm \sin x \\<br/>&amp; y=\sin x=f(x) \text { given } f(x) \geq 0 \text { for } \\<br/>&amp; x \in[0,1]<br/>\end{aligned}<br/>$$<br/>It is known that $\sin x &lt; x, \forall x \in R^{+}$<br/>$\therefore \quad \sin \left(\frac{1}{2}\right) &lt; \frac{1}{2} \Rightarrow f\left(\frac{1}{2}\right) &lt; \frac{1}{2}$<br/>and $\sin \left(\frac{1}{3}\right) &lt; \frac{1}{3} \Rightarrow f\left(\frac{1}{3}\right) &lt; \frac{1}{3}$</div>


tags: ['Mathematics', 'Differential Equations', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-be-a-realvalued-differentiable-function-on-r-the-set-of-all-real-numbers-such-that-f-1-1-if-the-y-intercept-of-the-tangent-at-any-point-p-x-y-on

slug:
let-f-be-a-realvalued-differentiable-function-on-r-the-set-of-all-real-numbers-such-that-f-1-1-if-the-y-intercept-of-the-tangent-at-any-point-p-x-y-on-54781

question:
<div class="question">Let $f$ be a real-valued differentiable function on $R$ (the set of all real numbers) such that $f(1)=1$. If the $y$-intercept of the tangent at any point $P(x, y)$ on the curve $y=f(x)$ is equal to the cube of the abscissa of $P$, then the value of $f(-3)$ is equal to</div>

answer:
<div class="solution">The equation of the tangent at $(x, y)$ to the given curve $y=f(x)$ is<br/>$$<br/>\begin{gathered}<br/>Y-y=\frac{d y}{d x}(X-x) \\<br/>Y \text {-intercept }=y-x \frac{d y}{d x}<br/>\end{gathered}<br/>$$<br/>According to the question<br/>$$<br/>x^3=y-x \frac{d y}{d x}<br/>$$<br/><br/>$$<br/>\Rightarrow \quad \frac{d y}{d x}-\frac{y}{x}=-x^2<br/>$$<br/>which is linear in $x$.<br/>$$<br/>\text { IF }=e^{\int \frac{-1}{x} d x}=\frac{1}{x}<br/>$$<br/>$\therefore$ Required solution is<br/>$$<br/>\begin{aligned}<br/>&amp; y \cdot \frac{1}{x}=\int-x^2 \cdot \frac{1}{x} d x \\<br/>&amp; \Rightarrow \quad \frac{y}{x}=\frac{-x^2}{2}+c \\<br/>&amp; \Rightarrow \quad y=\frac{-x^3}{2}+c x \\<br/>&amp; \text { At } x=1, y=1 \text {, } \\<br/>&amp; 1=\frac{-1}{2}+c \\<br/>&amp; \Rightarrow \quad c=\frac{3}{2} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>Now, $\begin{aligned} f(-3) &amp; =\frac{27}{2}+\frac{3}{2}(-3) \\ &amp; =\frac{27-9}{2}=9\end{aligned}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-be-a-realvalued-function-defined-on-the-interval-0-by-f-x-ln-x-0-x-1-sin-t-d-t-then-which-of-the-following-statements-is-are-true

slug:
let-f-be-a-realvalued-function-defined-on-the-interval-0-by-f-x-ln-x-0-x-1-sin-t-d-t-then-which-of-the-following-statements-is-are-true-89713

question:
<div class="question">Let $f$ be a real-valued function defined on the interval $(0, \infty)$, by $f(x)=\ln x+\int_0^x \sqrt{1+\sin t} d t$. Then which of the following statement(s) is (are) true ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$, but not differentiable on $(0, \infty)$<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="correct"> <span class="option-label">C</span> <span class="option-data"><br/>there exists $\alpha&gt;1$ such that $\left|f^{\prime}(x)\right| &lt; |f(x)|$ for all $x \in(\alpha, \infty)$</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">D</span> <span class="option-data"><br/>there exists $\beta&gt;0$ such that $|f(x)|+\left|f^{\prime}(x)\right| \leq \beta \quad$ from all $x \in(0, \infty)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$, but not differentiable on $(0, \infty)$<br/>, <br/>there exists $\alpha&gt;1$ such that $\left|f^{\prime}(x)\right| &lt; |f(x)|$ for all $x \in(\alpha, \infty)$</span> </div>

answer:
<div class="solution">Here, $f^{\prime}(x)=\frac{1}{x}+\sqrt{1+\sin x}, x&gt;0$ but $f(x)$ is not differentiable in $(0, \infty)$ as $\sin x$ may be $-1$ and then $f^{\prime \prime}(x)=-\frac{1}{x^2}+\frac{\cos x}{2 \sqrt{1+\sin x}}$ will not exists.<br/>$\Rightarrow f^{\prime}(x)$ is continuous for all $x \in(0, \infty)$ but $f^{\prime}(x)$ is not differentiable on $(0, \infty)$.<br/>$\therefore$ Option (b) is true.<br/>Also, $\quad f^{\prime}(x) \leq 3$, if $x&gt;1$ and $\quad f(x)&gt;3$, if $x&gt;e^3$<br/>$\therefore$ Let $\alpha=e^3$<br/>$\Rightarrow$ Option (c) is true.<br/>(d) is not possible as $f(x) \rightarrow \infty$ when $x \rightarrow \infty$.<br/>Hence, (b, c) is the correct option.</div>


tags: ['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-f-be-a-realvalued-function-defined-on-the-interval-1-1-such-that-e-x-f-x-2-0-x-t-4-1-d-t-for-all-x-1-1-and-let-f-1-be-the-inverse-function-of-f-th

slug:
let-f-be-a-realvalued-function-defined-on-the-interval-1-1-such-that-e-x-f-x-2-0-x-t-4-1-d-t-for-all-x-1-1-and-let-f-1-be-the-inverse-function-of-f-th-66400

question:
<div class="question">Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad$ for all $x \in(-1,1)$ and let $f^{-1}$ be the inverse function of $f$. Then $\left(f^{-1}\right)^{\prime}(2)$ is equal to</div>

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

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

answer:
<div class="solution">We have, $e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t x \in(-1,1)$<br/>On differentiating w.r.t. $x$, we get<br/>$$<br/>\begin{array}{ll} <br/>&amp; e^{-x}\left(f^{\prime}(x)-f(x)\right)=\sqrt{x^4+1} \\<br/>\Rightarrow \quad &amp; f^{\prime}(x)=f(x)+\sqrt{x^4+1} e^x \\<br/>\because &amp; f^{-1} \text { is the inverse of } f \\<br/>\therefore &amp; f^{-1}(f(x))=x \\<br/>\Rightarrow &amp; f^{-1^{\prime}}(f(x)) f^{\prime}(x)=1 \\<br/>\Rightarrow &amp; f^{-1^{\prime}}(f(x))=\frac{1}{f^{\prime}(x)} \\<br/>\Rightarrow \quad &amp; f^{-1^{\prime}}(f(x))=\frac{1}{f(x)+\sqrt{x^4+1} e^x} \\<br/>\text { At } \quad &amp; x=0, f(x)=2 \\<br/>&amp; f^{-1^{\prime}}(2)=\frac{1}{2+1}=\frac{1}{3}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-g-and-h-be-realvalued-functions-defined-on-the-interval-0-1-by-f-x-e-x-2-e-x-2-g-x-x-e-x-2-e-x-2-and-h-x-x-2-e-x-2-e-x-2-if-a-b-and-c-denote-res

slug:
let-f-g-and-h-be-realvalued-functions-defined-on-the-interval-0-1-by-f-x-e-x-2-e-x-2-g-x-x-e-x-2-e-x-2-and-h-x-x-2-e-x-2-e-x-2-if-a-b-and-c-denote-res-21179

question:
<div class="question">Let $f, g$ and $h$ be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2}, \quad g(x)=x e^{x^2}+e^{-x^2}$ and $h(x)=x^2 e^{x^2}+e^{-x^2}$. If $a, b$ and $c$ denote respectively, the absolute maximum of $f, g$ and $h$ on $[0,1]$, then</div>

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

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

answer:
<div class="solution">Given function,<br/>$$<br/>\begin{aligned}<br/>&amp; f(x)=e^{x^2}+e^{-x^2}, \\<br/>&amp; g(x)=x e^{x^2}+e^{-x^2} \text { and } \\<br/>&amp; h(x)=x^2 e^{x^2}+e^{-x^2} \text { are } \quad \text { strictly }<br/>\end{aligned}<br/>$$<br/>increasing on $[0,1]$. Hence, at $x=1$, the given function attains absolute maximum all equal to $e+\frac{1}{e}$.<br/>$$<br/>\Rightarrow \quad a=b=c<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-i-r-i-r-be-defined-as-f-x-x-x-2-1-the-total-number-of-points-at-which-f-attains-either-a-local-maximum-or-a-local-minimum-is

slug:
let-f-i-r-i-r-be-defined-as-f-x-x-x-2-1-the-total-number-of-points-at-which-f-attains-either-a-local-maximum-or-a-local-minimum-is-74307

question:
<div class="question">Let $f: I R \rightarrow I R$ be defined as $f(x)=|x|+\left|x^{2}-1\right|$. The total number of points at which $f$ attains either a local maximum or a local minimum is</div>

answer:
<div class="solution">$f(x)=|x|+\left|x^{2}-1\right|=\left\{\begin{array}{c}-x+x^{2}-1, x &lt; -1 \\ -x-x^{2}+1,-1 \leq x \leq 0 \\ x-x^{2}+1,0 &lt; x &lt; 1 \\ x^{2}+x-1, \quad x \geq 1\end{array}\right.$
<br/> <br/>$\therefore \quad f^{\prime}(x)=\left[\begin{array}{ccc}
<br/> <br/>2 x-1 &amp; , &amp; x &lt; -1 \\
<br/> <br/>-2 x-1 &amp; , &amp; -1 \leq x \leq 0 \\
<br/> <br/>-2 x+1 &amp; , &amp; 0 &lt; x &lt; 1 \\
<br/> <br/>2 x+1 &amp; , &amp; x&gt;1
<br/> <br/>\end{array}\right.$
<br/> <br/>Critical points are $\frac{1}{2}, \frac{-1}{2},-1,0$ and 1 .
<br/> <br/>We observe at five points $f^{\prime}(x)$ changes its sign $\therefore$ There are 5 points at which either local maximum or local minimum.</div>


title: let-f-r-r-be-a-continuous-function-which-satisfies-f-x-0-x-f-t-d-t-then-the-value-of-f-ln-5-is
tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
slug:
let-f-r-r-be-a-continuous-function-which-satisfies-f-x-0-x-f-t-d-t-then-the-value-of-f-ln-5-is-54869

question:
<div class="question">Let $f: R \rightarrow R$ be a continuous function, which satisfies $f(x)=\int_0^x f(t) d t$. Then, the value of $f(\ln 5)$ is</div>

answer:
<div class="solution">From given integral equation, $f(0)=0$. Also, differentiating the given integral equation w.r.t. $x$, we get<br/>$$<br/>\begin{gathered}<br/>f^{\prime}(x)=f(x) \\<br/>\text { If } \quad f(x) \neq 0 \Rightarrow \frac{f^{\prime}(x)}{f(x)}=1 \\<br/>\Rightarrow \quad \log f(x)=x+C \Rightarrow f(x)=e^C e^x \\<br/>\because f(0)=0 \Rightarrow e^C=0, \text { a contradiction } \\<br/>\therefore f(x)=0, \forall x \in R \Rightarrow f(\ln 5)=0<br/>\end{gathered}<br/>$$</div>


title: let-f-r-r-be-a-function-such-that-f-x-y-f-x-f-y-x-y-r-if-f-x-is-differentiable-at-x-0-then
slug: let-f-r-r-be-a-function-such-that-f-x-y-f-x-f-y-x-y-r-if-f-x-is-differentiable-at-x-0-then-45424
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y), \forall x, y \in R$. If $f(x)$ is differentiable at $x=0$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$f(x)$ is continuous, $\forall x \in R$<br/>, <br/>$f^{\prime}(x)$ is constant, $\forall x \in R$<br/></span> </div>

answer:
<div class="solution">$f(x+y)=f(x)+f(y)$, as $f(x)$ is differentiable at $x=0$.<br/>$$<br/>\begin{aligned}<br/>\Rightarrow &amp; f^{\prime}(0)=k \\<br/>\text { Now, } f^{\prime}(x) &amp; =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\<br/>&amp; =\lim _{h \rightarrow 0} \frac{f(x)+f(h)-f(x)}{h} \\<br/>&amp; =\lim _{h \rightarrow 0} \frac{f(h)}{h} \quad\left[\frac{0}{0} \text { from }\right]<br/>\end{aligned}<br/>$$<br/>Given, $f(x+y)=f(x)+f(y), \forall x, y$<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore &amp; f(0) &amp; =f(0)+f(0), \\<br/>&amp; \text { when } &amp; x &amp; =y=0 \Rightarrow f(0)=0<br/>\end{aligned}<br/>$$<br/>Using L'Hospital's rule,<br/>$$<br/>\lim _{h \rightarrow 0} \frac{f^{\prime}(h)}{1}=f^{\prime}(0)=k<br/>$$<br/>$\Rightarrow f^{\prime}(x)=k$, on integrating both sides, $f(x)=k x+C$, as $f(0)=0 \Rightarrow C=0$<br/>So, $f(x)=k x$<br/>$\therefore f(x)$ is continuous for all $x \in R$ and $f^{\prime}(x)=k$, i.e. constant for all $x \in R$.<br/>Hence, both (b) and (c) are correct.<br/>Solutions (Q. Nos. 12-13)<br/>$$<br/>\left.\left(\begin{array}{l}<br/>3 \mathrm{~W} \\<br/>2 \mathrm{R}<br/>\end{array}\right)(\underbrace{1 \mathrm{~W}}_{u_1})\right\} \text { Initial }<br/>$$<br/>Head appears.<br/>$\left(\begin{array}{l}2 \mathrm{~W} \\ 2 \mathrm{R}\end{array}\right) \underset{u_1}{\mathrm{wW}}(\underbrace{2 \mathrm{~W}}_{u_2})$<br/>or $\left.\left(\begin{array}{l}3 \mathrm{~W} \\ 1 \mathrm{R}\end{array}\right) \underset{u_1}{\stackrel{1 \mathrm{R}}{u_1}}\left(\begin{array}{c}1 \mathrm{~W} \\ 1 \mathrm{R}\end{array}\right)\right)_{u_2}^{u_2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/GBdGyDD_0UwU3UqT52VW0IalBzIUtwdpq7GUfZVxGHE.original.fullsize.png"/><br/></div>


title: let-f-sin-tan-1-c-o-s-2-s-i-n-where-4-4-then-the-value-of-d-t-a-n-d-f-is
slug: let-f-sin-tan-1-c-o-s-2-s-i-n-where-4-4-then-the-value-of-d-t-a-n-d-f-is-70148
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(\theta)=\sin \left[\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right]$, where $-\frac{\pi}{4} &lt; \theta &lt; \frac{\pi}{4}$. Then, the value of $\frac{d}{d(\tan \theta)}(f(\theta))$ is</div>

answer:
<div class="solution">$f(\theta)=\sin \left(\tan ^{-1} \frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right),-\frac{\pi}{4} &lt; \theta &lt; \frac{\pi}{4}$<br/>Let $\tan ^{-1} \frac{\sin \theta}{\sqrt{\cos 2 \theta}}=\phi$<br/>$$<br/>\Rightarrow \quad \tan \phi=\frac{\sin \theta}{\sqrt{\cos 2 \theta}}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/bbRBtOL8gHVb5FgMH9HblcPKawLYnY879QtTMljq-9M.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>\therefore \quad \sin \phi &amp; =\frac{\sin \theta}{\sqrt{\sin ^2 \theta+\cos 2 \theta}} \\<br/>&amp; =\frac{\sin \theta}{\sqrt{1-\sin ^2 \theta}}=\frac{\sin \theta}{\cos \theta}=\tan \theta<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{array}{ll}<br/>\therefore &amp; f(\theta)=\sin \phi=\tan \theta \\<br/>\Rightarrow &amp; \frac{d f(\theta)}{d(\tan \theta)}=1<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-1-x-2-sin-2-x-x-2-for-all-x-i-r-and-let-g-x-1-x-t-1-2-t-1-ln-t-f-t-d-t-for-all-x-1-question-consider-the-statements-p-there-exists-some-x-r-su

slug:
let-f-x-1-x-2-sin-2-x-x-2-for-all-x-i-r-and-let-g-x-1-x-t-1-2-t-1-ln-t-f-t-d-t-for-all-x-1-question-consider-the-statements-p-there-exists-some-x-r-su-55128

question:
<div class="question">Let $f(x)=(1-x)^{2} \sin ^{2} x+x^{2}$ for all $x \in I R$ and let $g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t$ for all $x \in(1, \infty)$.
<br/> <br/><strong> Question:</strong> Consider the statements:
<br/> <br/>$P$ : There exists some $x \in \mathrm{R}$ such that $f(x)+2 x$
<br/> <br/>$=2\left(1+x^{2}\right)$
<br/> <br/>$Q$ : There exists some $x \in \mathrm{R}$ such that $2 f(x)+1$ $=2 x(1+x)$
<br/> <br/>Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$P$ is false and $Q$ is true</span> </div>

answer:
<div class="solution">For the statement $P, f(x)+2 x=2\left(1+x^{2}\right)$
<br/> <br/>$\Rightarrow(1-x)^{2} \sin ^{2} x+x^{2}+2 x=2\left(1+x^{2}\right)$
<br/> <br/>$\Rightarrow(1-x)^{2} \sin ^{2} x=x^{2}-2 x+1+1$
<br/> <br/>$\Rightarrow(1-x)^{2} \sin ^{2} x=(1-x)^{2}+1$
<br/> <br/>$\Rightarrow(1-x)^{2} \cos ^{2} x=-1$, which is not possible for any real value of $x$.
<br/> <br/>Hence $P$ is not true.
<br/> <br/>Let $H(x)=2 f(x)+1-2 x(1+x)$
<br/> <br/>$H(0)=2 f(0)+1-0=1$
<br/> <br/>and $H(1)=2 f(1)+1-4=-3$
<br/> <br/>Hence, $H(x)$ has a solution in $(0,1)$
<br/> <br/>Therefore, $Q$ is true.</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-1-x-2-sin-2-x-x-2-for-all-x-i-r-and-let-g-x-1-x-t-1-2-t-1-ln-t-f-t-d-t-for-all-x-1-question-which-of-the-following-is-true

slug:
let-f-x-1-x-2-sin-2-x-x-2-for-all-x-i-r-and-let-g-x-1-x-t-1-2-t-1-ln-t-f-t-d-t-for-all-x-1-question-which-of-the-following-is-true-48953

question:
<div class="question">Let $f(x)=(1-x)^{2} \sin ^{2} x+x^{2}$ for all $x \in I R$ and let $g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t$ for all $x \in(1, \infty)$.
<br/> <br/><strong> Question:</strong> Which of the following is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$g$ is increasing on $(1, \infty)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$g$ is decreasing on $(1, \infty)$</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">$g$ is increasing on $(1,2)$ and decreasing on $(2, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$g$ is decreasing on $(1,2)$ and increasing on $(2, \infty)$</span> </li> </ul>

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

answer:
<div class="solution">$g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t$, $\therefore \quad g^{\prime}(x)=\left[\frac{2(x-1)}{x+1}-\ln x\right] f(x)$
<br/> <br/>Here $f(x)&gt;0, \forall x \in(1, \infty)$
<br/> <br/>Let $h(x)=\frac{2(x-1)}{x+1}-\ln x$
<br/> <br/>$\therefore \quad h^{\prime}(x)=\frac{4}{(x+1)^{2}}-\frac{1}{x}=\frac{-(x-1)^{2}}{(x+1)^{2} x} &lt; 0, x \in(1, \infty)$
<br/> <br/>$\Rightarrow h(x)$ is decreasing function.
<br/> <br/>Hence, for $x&gt;1, h(x) &lt; h(1) \Rightarrow h(x) &lt; 0 \forall x&gt;1$
<br/> <br/>$\Rightarrow g^{\prime}(x) &lt; 0 \forall x \in(1, \infty)$
<br/> <br/>Therefore, $g(x)$ is decreasing on $(1, \infty)$.</div>


title: let-f-x-1-x-n-1-n-x-for-n-2-and-g-x-f-occurs-n-times-f-f-o-o-f-x-then-x-n-2-g-x-d-x-equals
slug: let-f-x-1-x-n-1-n-x-for-n-2-and-g-x-f-occurs-n-times-f-f-o-o-f-x-then-x-n-2-g-x-d-x-equals-83727
tags: ['Mathematics', 'Indefinite Integration', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\frac{x}{\left(1+x^n\right)^{1 / n}}$ for $n \geq 2$ and $g(x)=\underbrace{(f \circ f o \ldots o f)}_{f \text { occurs } n \text { times }}(x)$. Then, $\int x^{n-2} g(x) d x$ equals</div>

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

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

answer:
<div class="solution">Here, $\begin{array}{rlrl} &amp; \text { and } &amp; f f(x) &amp; =\frac{f(x)}{\left[1+f(x)^n\right]^{1 / n}}=\frac{x}{\left(1+2 x^n\right)^{1 / n}} \\ &amp; \therefore \quad f f f(x) &amp; =\frac{x}{\left(1+3 x^n\right)^{1 / n}} \\ &amp; \text { Let } &amp; g(x) &amp; =\underbrace{(f \circ f o \ldots o f)}_{n \text { times }}(x)=\frac{x}{\left(1+n x^n\right)^{1 / n}} \\ I &amp; =\int x^{n-2} g(x) d x=\int \frac{x^{n-1} d x}{\left(1+n x^n\right)^{1 / n}} \\ &amp; &amp; =\frac{1}{n^2} \int \frac{n^2 x^{n-1} d x}{\left(1+n x^n\right)^{1 / n}}=\frac{1}{n^2} \int \frac{\frac{d}{d x}\left(1+n x^n\right)}{\left(1+n x^n\right)^{1 / n}} d x\end{array}$<br/>$\therefore \quad I=\frac{1}{n(n-1)}\left(1+n x^n\right)^{1-\frac{1}{n}}+k$.</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-2-cos-x-for-all-real-x-statement-i-for-each-real-t-there-exists-a-point-c-in-t-t-such-that-f-c-0-statement-ii-f-t-f-t-2-for-each-real-t

slug:
let-f-x-2-cos-x-for-all-real-x-statement-i-for-each-real-t-there-exists-a-point-c-in-t-t-such-that-f-c-0-statement-ii-f-t-f-t-2-for-each-real-t-67226

question:
<div class="question">Let $f(x)=2+\cos x$ for all real $x$.<br/>Statement I For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f^{\prime}(c)=0$. <br/>Statement II $f(t)=f(t+2 \pi)$ for each real $t$.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is True, Statement II is true; Statement II is not a correct explanation for Statement I<br/></span> </div>

answer:
<div class="solution">$f(x)=2+\cos x \forall x \in R$<br/>Statement I There exists a point $c \in[t, t+\pi]$, where $f^{\prime}(c)=0$ Hence, Statement 1 is true.<br/>Statement II $f(t)=f(t+2 \pi)$ is true.<br/>But Statement II is not a correct explanation for Statement I.</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-x-be-a-nonconstant-twice-differentiable-function-defined-on-such-that-f-x-f-1-x-and-f-4-1-0-then

slug:
let-f-x-be-a-nonconstant-twice-differentiable-function-defined-on-such-that-f-x-f-1-x-and-f-4-1-0-then-78446

question:
<div class="question">Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty, \infty)$, such that $f(x)=f(1-x)$ and $f^{\prime}\left(\frac{1}{4}\right)=0$. Then,</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$f^{\prime}(x)$ vanishes atleast twice on $[0,1]$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$f^{\prime}\left(\frac{1}{2}\right)=0$<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="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\int_{\frac{1}{2}}^{\frac{1}{2}} f\left(x+\frac{1}{2}\right) \sin x d x=0$<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="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\int_0^{\frac{1}{2}} f(t) e^{\sin \pi t} d t=\int_{\frac{1}{2}}^1 f(1-t) e^{\sin \pi t} d t$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$f^{\prime}(x)$ vanishes atleast twice on $[0,1]$<br/>, <br/>$f^{\prime}\left(\frac{1}{2}\right)=0$<br/>, <br/>$\int_{\frac{1}{2}}^{\frac{1}{2}} f\left(x+\frac{1}{2}\right) \sin x d x=0$<br/>, <br/>$\int_0^{\frac{1}{2}} f(t) e^{\sin \pi t} d t=\int_{\frac{1}{2}}^1 f(1-t) e^{\sin \pi t} d t$</span> </div>

answer:
<div class="solution">Given that, $f(x)=f(1-x)$<br/>On differentiating w.r.t. $x$, we get<br/>$$<br/>f^{\prime}(x)=-f^{\prime}(1-x)<br/>$$<br/>Let us put $\quad x=\frac{1}{2}$<br/>$$<br/>\Rightarrow \quad 2 f^{\prime}\left(\frac{1}{2}\right)=0 \Rightarrow f^{\prime}\left(\frac{1}{2}\right)=0<br/>$$<br/>Since, $f^{\prime}\left(\frac{1}{2}\right)=0$ and $f^{\prime}\left(\frac{1}{4}\right)=0$<br/>$\Rightarrow f^{\prime \prime}(x)=0$ at two points in $[0,1]$.<br/>Now, $\int_{-1 / 2}^{1 / 2} f\left(x+\frac{1}{2}\right) \sin x d x=0$<br/>As, $f\left(x+\frac{1}{2}\right) \sin x$ is an odd function which is clear from the following explanation.<br/>Let $g(x)=f\left(x+\frac{1}{2}\right) \sin x$,<br/>$$<br/>g(-x)=f\left(\frac{1}{2}-x\right) \sin (-x)=-\sin x f\left(1-\left(\frac{1}{2}-x\right)\right)=-\sin x f\left(\frac{1}{2}+x\right)=-g(x)<br/>$$<br/>Moreover, $\int_{1 / 2}^1 f(1-t) e^{\sin (\pi t)} d t=\int_0^{1 / 2} f(u) \cdot e^{\sin \pi u} d u$ where, $1-t=u$.</div>


tags: ['Mathematics', 'Indefinite Integration', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-x-be-an-indefinite-integral-of-sin-2-x-statement-i-the-function-f-x-satisfies-f-x-f-x-for-all-real-x-statement-ii-sin-2-x-sin-2-x-for-all-real-x

slug:
let-f-x-be-an-indefinite-integral-of-sin-2-x-statement-i-the-function-f-x-satisfies-f-x-f-x-for-all-real-x-statement-ii-sin-2-x-sin-2-x-for-all-real-x-64347

question:
<div class="question">Let $F(x)$ be an indefinite integral of $\sin ^2 x$.<br/>Statement I The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$.<br/>Statement II $\sin ^2(x+\pi)=\sin ^2 x$ for all real $x$.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is false, Statement II is true</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; F(x)=\int \sin ^2 x d x=\int \frac{1-\cos 2 x}{2} d x \\<br/>&amp; \Rightarrow F(x)=\frac{1}{4}(2 x-\sin 2 x)+c<br/>\end{aligned}<br/>$$<br/>Since, $F(x+\pi) \neq F(x)$.<br/>Hence, Statement I is false.<br/>But Statement II is true as $\sin ^2 x$ is periodic with period $\pi$.</div>


tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-f-x-be-differentiable-on-the-interval-0-such-that-f-1-1-and-lim-t-x-t-x-t-2-f-x-x-2-f-t-1-for-each-x-0-then-f-x-is

slug:
let-f-x-be-differentiable-on-the-interval-0-such-that-f-1-1-and-lim-t-x-t-x-t-2-f-x-x-2-f-t-1-for-each-x-0-then-f-x-is-83213

question:
<div class="question">Let $f(x)$ be differentiable on the interval $(0, \infty)$ such that $f(1)=1$, and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x&gt;0$. Then, $f(x)$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{1}{3 x}+\frac{2 x^2}{3}$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { } \lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow &amp; x^2 f^{\prime}(x)-2 x f(x)+1 &amp; =0 \\<br/>&amp; \Rightarrow &amp; \frac{x^2 f^{\prime}(x)-2 x f(x)}{\left(x^2\right)^2}+\frac{1}{x^4} &amp; =0 \\<br/>&amp; \Rightarrow &amp; \frac{d}{d x}\left(\frac{f(x)}{x^2}\right) &amp; =-\frac{1}{x^4} \\<br/>\Rightarrow &amp; &amp; f(x) &amp; =c x^2+\frac{1}{3 x} \text { also } f(1)=1 \Rightarrow c=\frac{2}{3} . \\<br/>&amp; \text { Hence, } &amp; f(x) &amp; =\frac{2}{3} x^2+\frac{1}{3 x}<br/>\end{aligned}<br/>$$</div>


title: let-f-x-x-2-5-x-6-x-2-6-x-5-1
slug: let-f-x-x-2-5-x-6-x-2-6-x-5-1-60880
tags: ['Mathematics', 'Application of Derivatives', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\frac{x^2-6 x+5}{x^2-5 x+6}$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5LpElX78gcLAqJVL43i4iELoQ4DU2RBjX9Jh8inUrTA.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p; B-q; C-q; D-p<br/></span> </div>

answer:
<div class="solution">$$<br/>f(x)=\frac{(x-1)(x-5)}{(x-2)(x-3)}<br/>$$<br/>The graph of $f(x)$ is shown<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wvp0nANowQRUUqEKwDE04wK9eFTqRImRXTt4B2hjHOM.original.fullsize.png"/><br/><br/>(A) If $-1 &lt; x &lt; 1 \Rightarrow 0 &lt; f(x) &lt; 1$<br/>(B) If $1 &lt; x &lt; 2 \Rightarrow f(x) &lt; 0$<br/>(C) If $3 &lt; x &lt; 5 \Rightarrow f(x) &lt; 0$<br/>(D) If $x&gt;5 \Rightarrow 0 &lt; f(x) &lt; 1$</div>


tags: ['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-2-and-g-x-sin-x-for-all-x-r-then-the-set-of-all-x-satisfying-fogogof-x-gogof-x-where-f-g-x-f-g-x-is

slug:
let-f-x-x-2-and-g-x-sin-x-for-all-x-r-then-the-set-of-all-x-satisfying-fogogof-x-gogof-x-where-f-g-x-f-g-x-is-40956

question:
<div class="question">Let $f(x)=x^2$ and $g(x)=\sin x$ for all $x \in R$. Then, the set of all $x$ satisfying $($ fogogof $)(x)=(\operatorname{gogof})(x)$, where $(f \circ g)(x)=f(g(x))$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\pm \sqrt{n \pi}, n \in\{1,2, \ldots\}$<br/></span> </div>

answer:
<div class="solution">$f(x)=x^2, g(x)=\sin x$<br/>$(g \circ f)(x)=\sin x^2$<br/>go(gof) $(x)=\sin \left(\sin x^2\right)$<br/>$(f \circ g \circ g \circ f)(x)=\left(\sin \left(\sin x^2\right)\right)^2$<br/>Again, $(g \circ f)(x)=\sin x^2$<br/>(gogof) $(x)=\sin \left(\sin x^2\right)$<br/>Given, $($ fogogof $)(x)=($ gogof $)(x)$<br/>$\Rightarrow \quad\left(\sin \left(\sin x^2\right)\right)^2=\sin \left(\sin x^2\right)$<br/>$\Rightarrow \sin \left(\sin x^2\right)\left\{\sin \left(\sin x^2\right)-1\right\}=0$<br/>$\Rightarrow \sin \left(\sin x^2\right)=0$ or $\sin \left(\sin x^2\right)=1$<br/>$\Rightarrow \sin x^2=0$ or $\sin x^2=\frac{\pi}{2}$<br/>$\therefore \quad x^2=n \pi$<br/>(i.e. not possible as $-1 \leq \sin \theta \leq 1$ )</div>


tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-g-x-l-o-g-c-o-s-m-x-1-x-1-n-0-x-2-m-and-n-are-integers-m-0-n-0-and-let-p-be-the-left-hand-derivative-of-x-1-at-x-1-if-lim-x-1-g-x-p-then

slug:
let-g-x-l-o-g-c-o-s-m-x-1-x-1-n-0-x-2-m-and-n-are-integers-m-0-n-0-and-let-p-be-the-left-hand-derivative-of-x-1-at-x-1-if-lim-x-1-g-x-p-then-93537

question:
<div class="question">Let $g(x)=\frac{(x-1)^n}{\log \cos ^m(x-1)} ; 0 &lt; x &lt; 2, m$ and $n$ are integers, $m \neq 0, n&gt;0$ and let $p$ be the left hand derivative of $|x-1|$ at $x=1$. If $\lim _{x \rightarrow 1^{+}} g(x)=p$, then</div>

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

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

answer:
<div class="solution">Given, $g(x)=\frac{(x-1)^n}{\log \cos ^m(x-1)} ; 0 &lt; x &lt; 2, m \neq 0, n$ are integers and $|x-1|=\left\{\begin{array}{l}x-1 ; x \geq 1 \\ 1-x ; x &lt; 1\end{array}\right.$<br/>The left hand derivative of $|x-1|$ at $x=1$ is $p=-1$.<br/>Also,<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad \lim _{h \rightarrow 0} \frac{(1+h-1)^n}{\log \cos ^m(1+h-1)}=-1 \Rightarrow \lim _{h \rightarrow 0} \frac{h^n}{\log \cos ^m h}=-1 \\<br/>&amp; \Rightarrow \quad \lim _{h \rightarrow 0} \frac{h^n}{m \log \cos h}=-1<br/>\end{aligned}<br/>$$<br/>[Using L' Hospital rule]<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; \quad \lim _{h \rightarrow 0} \frac{n \cdot h^{n-1}}{m \frac{1}{\cos h}(-\sin h)}=-1 \\<br/>\Rightarrow \quad &amp; \quad \lim _{h \rightarrow 0}\left(-\frac{n}{m}\right) \cdot \frac{h^{n-2}}{\left(\frac{\tan h}{h}\right)}=-1 \\<br/>\Rightarrow \quad &amp; \quad\left(\frac{n}{m}\right) \lim _{h \rightarrow 0} \frac{h^{n-2}}{\left(\frac{\tan h}{h}\right)}=1 \\<br/>\Rightarrow &amp; n=2 \text { and } \frac{n}{m}=1 \\<br/>\therefore &amp; \quad m=n=2<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-g-x-lo-g-f-x-where-f-x-is-a-twice-differentiable-positive-function-on-0-such-that-f-x-1-x-f-x-then-for-n-1-2-3-g-n-2-1-g-2-1-is-equal-to

slug:
let-g-x-lo-g-f-x-where-f-x-is-a-twice-differentiable-positive-function-on-0-such-that-f-x-1-x-f-x-then-for-n-1-2-3-g-n-2-1-g-2-1-is-equal-to-69403

question:
<div class="question">Let $g(x)=\log f(x)$, where $f(x)$ is a twice differentiable positive function on $(0, \infty)$ such that $f(x+1)=x f(x)$. Then, for $N=1,2,3, \ldots . g^{\prime \prime}\left(N+\frac{1}{2}\right)-g^{\prime \prime}\left(\frac{1}{2}\right)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$-4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$-4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N+1)^2}\right\}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N+1)^2}\right\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$-4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\}$<br/></span> </div>

answer:
<div class="solution">Since, $\quad f(x)=e^{g(x)}$<br/>$$<br/>\begin{aligned}<br/>\Rightarrow \quad e^{g(x+1)} &amp; =f(x+1) \\<br/>&amp; =x f(x) \\<br/>&amp; =x e^{g(x)}<br/>\end{aligned}<br/>$$<br/>and<br/>$$<br/>g(x+1)=\log x+g(x)<br/>$$<br/>$$<br/>\Rightarrow \quad g(x+1)-g(x)=\log x<br/>$$<br/>Replacing $x$ by $x-\frac{1}{2}$, we get<br/>$$<br/>\begin{aligned}<br/>g\left(x+\frac{1}{2}\right)-g\left(x-\frac{1}{2}\right) &amp; =\log \left(x-\frac{1}{2}\right)=\log (2 x-1)-\log 2 \\<br/>\therefore g^{\prime \prime}\left(x+\frac{1}{2}\right)-g^{\prime \prime}\left(x-\frac{1}{2}\right) &amp; =-\frac{4}{(2 x-1)^2}<br/>\end{aligned}<br/>$$<br/>Substituting, $x=1,2,3, \ldots, N$ in Eq. (ii) and adding, we get<br/>$$<br/>g^{\prime \prime}\left(N+\frac{1}{2}\right)-g^{\prime \prime}\left(\frac{1}{2}\right)=-4\left\{1+\frac{1}{9}+\frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\} .<br/>$$</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-h-1-h-2-h-n-be-mutually-exclusive-events-with-p-h-i-0-i-1-2-n-let-e-be-any-other-event-with-0-p-e-1-statement-i-p-h-i-e-p-e-h-i-p-h-i-for-i-1-2-n-

slug:
let-h-1-h-2-h-n-be-mutually-exclusive-events-with-p-h-i-0-i-1-2-n-let-e-be-any-other-event-with-0-p-e-1-statement-i-p-h-i-e-p-e-h-i-p-h-i-for-i-1-2-n-81492

question:
<div class="question">Let $H_1, H_2, \ldots, H_n$ be mutually exclusive events with $P\left(H_i\right)&gt;0, i=1,2, \ldots, n$. Let $E$ be any other event with $0 &lt; P(E) &lt; 1$.<br/>Statement I $P\left(H_i / E\right)&gt;P\left(E / H_i\right) P\left(H_i\right)$ for $i=1,2, \ldots, n$.<br/>Statement II $\sum_{i=1}^n P\left(H_i\right)=1$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is false, Statement II is true</span> </div>

answer:
<div class="solution">Statement : I If $P\left(H_i \cap E\right)=0$ for some $i$, then<br/>$$<br/>P\left(\frac{H_i}{E}\right)=P\left(\frac{E}{H_i}\right)=0<br/>$$<br/>If $P\left(H_i \cap E\right) \neq 0$ for $\forall i=1,2, \ldots . n$, then<br/>$$<br/>\begin{aligned}<br/>P\left(\frac{H_i}{E}\right) &amp; =\frac{P\left(H_i \cap E\right)}{P\left(H_i\right)} \times \frac{P\left(H_i\right)}{P(E)} \\<br/>&amp; \left.=\frac{P\left(\frac{E}{H_i}\right) \times P\left(H_i\right)}{P(E)}&gt;P\left(\frac{E}{H_i}\right) \cdot P\left(H_i\right) \quad \text { [as } 0 &lt; P(E) &lt; 1\right]<br/>\end{aligned}<br/>$$<br/>Hence, Statement I may not always be true.<br/>Statement II<br/>Clearly, $H_1 \cup H_2 \cup \ldots . \cup H_n=S$ (sample space)<br/>$$<br/>\Rightarrow \quad P\left(H_1\right)+P\left(H_2\right)+\ldots+P\left(H_n\right)=1 .<br/>$$<br/>Hence, Statement II is true.</div>


tags: ['Mathematics', 'Indefinite Integration', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-i-e-4-x-e-2-x-1-e-x-d-x-j-e-4-x-e-2-x-1-e-x-d-x-then-for-an-arbitrary-constant-c-the-value-of-j-i-equals

slug:
let-i-e-4-x-e-2-x-1-e-x-d-x-j-e-4-x-e-2-x-1-e-x-d-x-then-for-an-arbitrary-constant-c-the-value-of-j-i-equals-56578

question:
<div class="question">Let $I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x$.<br/>Then, for an arbitrary constant $C$, the value of $J-I$ equals</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{1}{2} \log \left|\frac{e^{2 x}-e^x+1}{e^{2 x}+e^x+1}\right|+C$<br/></span> </div>

answer:
<div class="solution">Since, $J=\int \frac{e^{3 x}}{1+e^{2 x}+e^{4 x}} d x$<br/>$$<br/>\begin{aligned}<br/>\therefore J-I &amp; =\int \frac{\left(e^{3 x}-e^x\right)}{1+e^{2 x}+e^{4 x}} d x=\int \frac{\left(u^2-1\right)}{1+u^2+u^4} d u \\<br/>&amp; =\int \frac{\left(1-\frac{1}{u^2}\right)}{1+\frac{1}{u^2}+u^2} d u=\int \frac{\left(1-\frac{1}{u^2}\right)}{\left(u+\frac{1}{u}\right)^2-1} d u \\<br/>&amp; =\int \frac{d t}{t^2-1} \\<br/>&amp; =\frac{1}{2} \log \left|\frac{t-1}{t+1}\right|+C \\<br/>&amp; =\frac{1}{2} \log \left|\frac{u^2-u+1}{u^2+u+1}\right|+C=\frac{1}{2} \log \left|\frac{e^{2 x}-e^x+1}{e^{2 x}+e^x+1}\right|+C<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
let-k-be-a-positive-real-number-and-let-a-b-2-k-1-2-k-2-k-2-k-1-2-k-2-k-2-k-1-and-0-1-2-k-k-2-k-1-0-2-k-k-2-k-0-if-det-adj-a-det-adj-b-1-0-6-then-k-is

slug:
let-k-be-a-positive-real-number-and-let-a-b-2-k-1-2-k-2-k-2-k-1-2-k-2-k-2-k-1-and-0-1-2-k-k-2-k-1-0-2-k-k-2-k-0-if-det-adj-a-det-adj-b-1-0-6-then-k-is-39129

question:
<div class="question">Let $k$ be a positive real number and let $\begin{aligned} A &amp; =\left[\begin{array}{ccc}2 k-1 &amp; 2 \sqrt{k} &amp; 2 \sqrt{k} \\ 2 \sqrt{k} &amp; 1 &amp; -2 k \\ -2 \sqrt{k} &amp; 2 k &amp; -1\end{array}\right] \text { and } \\ B &amp; =\left[\begin{array}{ccc}0 &amp; 2 k-1 &amp; \sqrt{k} \\ 1-2 k &amp; 0 &amp; 2 \sqrt{k} \\ -\sqrt{k} &amp; -2 \sqrt{k} &amp; 0\end{array}\right]\end{aligned}$<br/>If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$ is equal to<br/>[Note : adj $M$ denotes the adjoint of a square matrix $M$ and $[k]$ denotes the largest integer less than or equal to $k$ ].</div>

answer:
<div class="solution">$|A|=(2 k+1)^3,|B|=0$<br/>But $\operatorname{det}(\operatorname{adj} A)=\operatorname{det}(\operatorname{adj} B)=10^6$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow(2 k+1)^6=10^6 \\<br/>&amp; \Rightarrow k=\frac{9}{2} \Rightarrow[k]=4<br/>\end{aligned}<br/>$$</div>


title: let-l-be-a-normal-to-the-parabola-y-2-4-x-if-l-passes-through-the-point-9-6-then-l-is-given-by
tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
slug:
let-l-be-a-normal-to-the-parabola-y-2-4-x-if-l-passes-through-the-point-9-6-then-l-is-given-by-41823

question:
<div class="question">Let $L$ be a normal to the parabola $y^2=4 x$. If $L$ passes through the point $(9,6)$, then $L$ is given by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$y-x+3=0$<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="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$y+3 x-33=0$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$y+x-15=0$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$y-2 x+12=0$</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 answers are:
<span class="option-value"><br/>$y-x+3=0$<br/>, <br/>$y+3 x-33=0$<br/>, <br/>$y-2 x+12=0$</span> </div>

answer:
<div class="solution">Normal to $y^2=4 x$, is $y-m x-2 m-m^3$ which passes through $(9,6)$.<br/>Now, $\quad 6=9 m-2 m-m^3$ $\Rightarrow m^3-7 m+6=0 \Rightarrow m=1,2,-3$<br/>$\therefore$ Equation of normals are<br/>$y-x+3=0$<br/>$y+3 x-33=0$ and $y-2 x+12=0$</div>


title: let-l-lim-x-0-x-4-a-a-2-x-2-4-x-2-a-0-if-l-is-finite-then
slug: let-l-lim-x-0-x-4-a-a-2-x-2-4-x-2-a-0-if-l-is-finite-then-27156
tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $L=\lim _{x \rightarrow 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}$, $a&gt;0$. If $L$ is finite, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$a=2$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$a=1$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$L=\frac{1}{64}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$L=\frac{1}{32}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$a=2$<br/>, <br/>$L=\frac{1}{64}$<br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OppXIo5ahAMcXNcH9uV52GAIpIQkQcAKb8c0qNyv8nQ.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
let-m-and-n-be-two-3-3-nonsingular-skewsymmetric-matrices-such-that-mn-nm-if-p-t-denotes-the-transpose-of-p-then-m-2-n-2-m-t-n-1-m-n-1-t-is-equal-to

slug:
let-m-and-n-be-two-3-3-nonsingular-skewsymmetric-matrices-such-that-mn-nm-if-p-t-denotes-the-transpose-of-p-then-m-2-n-2-m-t-n-1-m-n-1-t-is-equal-to-93774

question:
<div class="question">Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N=N M$. If $P^T$ denotes the transpose of $P$, then $M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$-M^2$<br/></span> </div>

answer:
<div class="solution">Given, $M^T=-M, N^T=-N$<br/>and $\quad M N=N M$<br/>$$<br/>\begin{aligned}<br/>\therefore &amp; M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T \\<br/>&amp; =M^2 N^2 N^{-1}\left(M^T\right)^{-1}\left(N^{-1}\right)^T \cdot M^T \\<br/>&amp; =M^2 N\left(N N^{-1}\right)(-M)^{-1}\left(N^T\right)^{-1}(-M) \\<br/>&amp; =M^2 N\left(-M^{-1}\right)(-N)^{-1}(-M) \\<br/>&amp; =-M^2 N M^{-1} N^{-1} M \\<br/>&amp; =-M \cdot(M N) M^{-1} N^{-1} M \\<br/>&amp; =-M(N M) M^{-1} N^{-1} M \\<br/>&amp; =-M N\left(N M^{-1}\right) N^{-1} M \\<br/>&amp; =-M\left(N N^{-1}\right) M=-M^2<br/>\end{aligned}<br/>$$<br/>Note Here, non-singular word should not be used, since there is no non-singular $3 \times 3$ skew-symmetric matrix.</div>


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
title:
let-m-be-a-3-3-matrix-satisfying-m-0-1-0-1-2-3-m-1-1-0-1-1-1-and-m-1-1-1-0-0-12-t-h-e-n-t-h-es-u-m-o-f-t-h-e-d-ia-g-o-na-l-e-n-t-r-i-eso-f-m-is

slug:
let-m-be-a-3-3-matrix-satisfying-m-0-1-0-1-2-3-m-1-1-0-1-1-1-and-m-1-1-1-0-0-12-t-h-e-n-t-h-es-u-m-o-f-t-h-e-d-ia-g-o-na-l-e-n-t-r-i-eso-f-m-is-39211

question:
<div class="question">$$<br/>\text { Let } M \text { be a } 3 \times 3 \text { matrix satisfying }<br/>$$<br/><br/>$$<br/>M\left[\begin{array}{l}<br/>0 \\<br/>1 \\<br/>0<br/>\end{array}\right]=\left[\begin{array}{c}<br/>-1 \\<br/>2 \\<br/>3<br/>\end{array}\right], M\left[\begin{array}{c}<br/>1 \\<br/>-1 \\<br/>0<br/>\end{array}\right]=\left[\begin{array}{c}<br/>1 \\<br/>1 \\<br/>-1<br/>\end{array}\right] \text { and } M\left[\begin{array}{l}<br/>1 \\<br/>1 \\<br/>1<br/>\end{array}\right]=\left[\begin{array}{c}<br/>0 \\<br/>0 \\<br/>12<br/>\end{array}\right]<br/>$$<br/>Then, the sum of the diagonal entries of $M$ is</div>

answer:
<div class="solution">Let $M=\left[\begin{array}{lll}a_1 &amp; a_2 &amp; a_3 \\ b_1 &amp; b_2 &amp; b_3 \\ c_1 &amp; c_2 &amp; c_3\end{array}\right]$<br/>$\therefore M\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right], M\left[\begin{array}{r}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{r}1 \\ 1 \\ -1\end{array}\right]$,<br/>$M\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{r}0 \\ 0 \\ 12\end{array}\right]$,<br/>$$<br/>\begin{aligned}<br/>\Rightarrow &amp; {\left[\begin{array}{l}<br/>a_2 \\<br/>b_2 \\<br/>c_2<br/>\end{array}\right]=\left[\begin{array}{r}<br/>-1 \\<br/>2 \\<br/>3<br/>\end{array}\right],\left[\begin{array}{l}<br/>a_1-a_2 \\<br/>b_1-b_2 \\<br/>c_1-c_2<br/>\end{array}\right]=\left[\begin{array}{r}<br/>1 \\<br/>1 \\<br/>-1<br/>\end{array}\right], } \\<br/>&amp; {\left[\begin{array}{l}<br/>a_1+a_2+a_3 \\<br/>b_1+b_2+b_3 \\<br/>c_1+c_2+c_3<br/>\end{array}\right]=\left[\begin{array}{r}<br/>0 \\<br/>0 \\<br/>12<br/>\end{array}\right] } \\<br/>\Rightarrow &amp; a_2=-1, b_2=2, c_2=3, a_1-a_2=1, \\<br/>\Rightarrow &amp; b_1-b_2=1, c_1-c_2=-1 \\<br/>\Rightarrow \quad &amp; a_1+a_2+a_3=0, b_1+b_2+b_3=0, \\<br/>\therefore &amp; c_1+c_2+c_3=12 \\<br/>\therefore &amp; a_1=0, b_2=2 \text { and } c_3=7<br/>\end{aligned}<br/>$$<br/>Hence, sum of diagonal elements<br/>$$<br/>=0+2+7=9<br/>$$</div>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
let-o-0-0-p-3-4-and-q-6-0-be-the-vertices-of-the-opq-the-point-r-inside-the-opq-is-such-that-the-opr-pqr-oqr-are-of-equal-area-the-coordinates-of-r-ar

slug:
let-o-0-0-p-3-4-and-q-6-0-be-the-vertices-of-the-opq-the-point-r-inside-the-opq-is-such-that-the-opr-pqr-oqr-are-of-equal-area-the-coordinates-of-r-ar-17740

question:
<div class="question">Let $O(0,0), P(3,4)$ and $Q(6,0)$ be the vertices of the $\triangle O P Q$. The point $R$ inside the $\triangle O P Q$ is such that the $\triangle O P R, \triangle P Q R, \triangle O Q R$ are of equal area. The coordinates of $R$ are</div>

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

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

answer:
<div class="solution">$$<br/>\text { 7. Since, triangle is on isosceles, hence centroid is the desired point. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/mPt6BiU7se3e8XRpvnhJ0CQd1IFrMT3sobkeFYbTv-Y.original.fullsize.png"/><br/><br/>$\therefore$ Coordinates of $R\left(3, \frac{4}{3}\right)$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-p-3-2-6-be-a-point-in-space-and-q-be-a-point-on-the-line-r-i-j-2-k-3-i-j-5-k-then-the-value-of-for-which-the-vector-pq-is-parallel-to-the-plane-x-

slug:
let-p-3-2-6-be-a-point-in-space-and-q-be-a-point-on-the-line-r-i-j-2-k-3-i-j-5-k-then-the-value-of-for-which-the-vector-pq-is-parallel-to-the-plane-x-38163

question:
<div class="question">Let $P(3,2,6)$ be a point in space and $Q$ be a point on the line $\mathbf{r}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}})$.<br/>Then, the value of $\mu$ for which the vector $\mathbf{P Q}$ is parallel to the plane $x-4 y+3 z=1$ is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xhSyjGZAfvOYhd3cVOPzs1UpROnoGZGmyZU8hofeEj4.original.fullsize.png"/><br/><br/>$\mathbf{O Q}=(1-3 \mu) \mathbf{i}+(\mu-1) \mathbf{j}+(5 \mu+2) \mathbf{k}$ and $\mathbf{O P}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$, where $O$ is origin.<br/>$$<br/>\begin{aligned}<br/>&amp; \text { Now, } \mathbf{P Q}=(1-3 \mu-3) \hat{\mathbf{i}}+(\mu-1-2) \hat{\mathbf{j}} \\<br/>&amp; +(5 \mu+2-6) \hat{\mathbf{k}} \\<br/>&amp; =(2-3 \mu) \hat{\mathbf{i}}+(\mu-3) \hat{\mathbf{j}}+(5 \mu-4) \hat{\mathbf{k}} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>$\because \mathbf{P Q}$ is parallel to the plane<br/>$$<br/>\begin{array}{rr} <br/>&amp; x-4 y+3 z=1 . \\<br/>\therefore &amp; -2-3 \mu-4 \mu+12+15 \mu-12=0 \\<br/>\Rightarrow &amp; 8 \mu=2 \Rightarrow \mu=\frac{1}{4}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
let-p-6-3-be-a-point-on-the-hyperbola-a-2-x-2-b-2-y-2-1-if-the-normal-at-the-point-p-intersects-the-x-axis-at-9-0-then-the-eccentricity-of-the-hyperbo

slug:
let-p-6-3-be-a-point-on-the-hyperbola-a-2-x-2-b-2-y-2-1-if-the-normal-at-the-point-p-intersects-the-x-axis-at-9-0-then-the-eccentricity-of-the-hyperbo-46862

question:
<div class="question">Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at the point $P$ intersects the $X$-axis at $(9,0)$, then the eccentricity of the hyperbola is</div>

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

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

answer:
<div class="solution">Equation of normal to hyperbola at $\left(x_1, y_1\right)$ is<br/>$$<br/>\begin{gathered}<br/>\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2 \\<br/>\therefore \text { At }(6,3), \frac{a^2 x}{6}+\frac{b^2 y}{3}=a^2+b^2<br/>\end{gathered}<br/>$$<br/>It passes throught $(9,0)$.<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \text { Now, } \quad \frac{a^2 \cdot 9}{6}=a^2+b^2 \\<br/>&amp; \Rightarrow \quad \frac{3 a^2}{2}-a^2=b^2 \Rightarrow \frac{a^2}{b^2}=2 \\<br/>&amp; \therefore \quad e^2=1+\frac{b^2}{a^2}=1+\frac{1}{2} \Rightarrow e=\sqrt{\frac{3}{2}}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-p-a-ij-be-a-3-3-matrix-and-let-q-b-ij-where-b-ij-2-i-j-a-ij-for-1-i-j-3-if-the-determinant-of-p-is-2-then-the-determinant-of-the-matrix-q-is

slug:
let-p-a-ij-be-a-3-3-matrix-and-let-q-b-ij-where-b-ij-2-i-j-a-ij-for-1-i-j-3-if-the-determinant-of-p-is-2-then-the-determinant-of-the-matrix-q-is-86042

question:
<div class="question">Let $P=\left[a_{i j}\right]$ be a $3 \times 3$ matrix and let $Q=\left[b_{i j}\right]$, where $b_{i j}=2^{i+j} a_{i j}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{10}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2^{11}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{12}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2^{13}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$2^{13}$</span> </div>

answer:
<div class="solution">$|Q|=\left|\begin{array}{lll}2^{2} a_{11} &amp; 2^{3} a_{12} &amp; 2^{4} a_{13} \\ 2^{3} a_{21} &amp; 2^{4} a_{22} &amp; 2^{5} a_{23} \\ 2^{4} a_{31} &amp; 2^{5} a_{32} &amp; 2^{6} a_{33}\end{array}\right|$
<br/> <br/>$=2^{2} \cdot 2^{3} \cdot 2^{4}\left|\begin{array}{ccc}a_{11} &amp; a_{12} &amp; a_{13} \\ 2 a_{21} &amp; 2 a_{22} &amp; 2 a_{23} \\ 2^{2} a_{31} &amp; 2^{2} a_{32} &amp; 2^{2} a_{33}\end{array}\right|$
<br/> <br/>$=2^{9} \cdot 2 \cdot 2^{2}\left|\begin{array}{lll}a_{11} &amp; a_{12} &amp; a_{13} \\ a_{21} &amp; a_{22} &amp; a_{23} \\ a_{31} &amp; a_{32} &amp; a_{33}\end{array}\right|$
<br/> <br/>$=2^{12} \times|P|=2^{12} \times 2=2^{13}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-p-and-q-be-real-numbers-such-that-p-0-p-3-q-and-p-3-q-if-and-are-nonzero-complex-numbers-satisfying-p-and-3-3-q-then-a-quadratic-equation-having-a

slug:
let-p-and-q-be-real-numbers-such-that-p-0-p-3-q-and-p-3-q-if-and-are-nonzero-complex-numbers-satisfying-p-and-3-3-q-then-a-quadratic-equation-having-a-65635

question:
<div class="question">Let $p$ and $q$ be real numbers such that $p \neq 0, p^3 \neq q$ and $p^3 \neq-q$. If $\alpha$ and $\beta$ are non-zero complex numbers satisfying $\alpha+\beta=-p$ and $\alpha^3+\beta^3=q$, then $a$ quadratic equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\left(p^3+q\right) x^2-\left(p^3-2 q\right) x$<br/>$$<br/>+\left(p^3+q\right)=0<br/>$$<br/></span> </div>

answer:
<div class="solution">Sum of roots $=\frac{\alpha^2+\beta^2}{\alpha \beta}$ and product $=1$<br/>Given, $\alpha+\beta=-p$ and $\alpha^3+\beta^3=q$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow(\alpha+\beta)\left(\alpha^2-\alpha \beta+\beta^2\right)=q \\<br/>&amp; \therefore \quad \alpha^2+\beta^2-\alpha \beta=\frac{-q}{p}<br/>\end{aligned}<br/>$$<br/>and $\quad(\alpha+\beta)^2=p^2$<br/>$$<br/>\Rightarrow \quad \alpha^2+\beta^2+2 \alpha \beta=p^2<br/>$$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>\alpha^2+\beta^2=\frac{p^3-2 q}{3 p}<br/>$$<br/>and $\alpha \beta=\frac{p^3+q}{3 p}$<br/>$\therefore$ Required equation is<br/>$$<br/>\begin{gathered}<br/>x^2-\frac{\left(p^3-2 q\right) x}{\left(p^3+q\right)}+1=0 \\<br/>\Rightarrow\left(p^3+q\right) x^2-\left(p^3-2 q\right) x+\left(p^3+q\right)=0<br/>\end{gathered}<br/>$$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-p-q-r-and-s-be-the-points-on-the-plane-with-position-vectors-2-i-j-4-i-3-i-3-j-an-d-3-i-2-j-respectively-the-quadrilateral-pqrs-must-be-a

slug:
let-p-q-r-and-s-be-the-points-on-the-plane-with-position-vectors-2-i-j-4-i-3-i-3-j-an-d-3-i-2-j-respectively-the-quadrilateral-pqrs-must-be-a-58541

question:
<div class="question">Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 4 \hat{\mathbf{i}}, 3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ an $\mathrm{d}-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ respectively. The quadrilateral $P Q R S$ must be a</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>parallelogram, which is neither a rhombus nor a rectangle<br/></span> </div>

answer:
<div class="solution">$m_{P Q}=\frac{1}{6}, m_{S R}=\frac{1}{6}, \quad m_{R Q}=-3$, $m_{S P}=-3$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZpLK0MicUbuHy0zRsNvkTlJbp96CIu7p_UQH6TC2DEQ.original.fullsize.png"/><br/><br/>$\Rightarrow$ Parallelogram<br/>But neither $P R=S Q$ nor $P R \perp S Q$.<br/>$\therefore$ Parallelogram, which is neither a rhombus nor a rectangle.</div>


title: let-p-sin-cos-2-cos-and-q-sin-cos-2-sin-be-two-sets-then-1
slug: let-p-sin-cos-2-cos-and-q-sin-cos-2-sin-be-two-sets-then-1-90689
tags: ['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}$ and $Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}$ be two sets. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$P \subset Q$ and $Q-P \neq \Phi$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$Q \not \subset P$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$P \not \subset Q$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$P=Q$</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> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\} \\<br/>&amp; \Rightarrow \quad \cos \theta(\sqrt{2}+1)=\sin \theta \\<br/>&amp; \Rightarrow \quad \tan \theta=\sqrt{2}+1 \\<br/>&amp; Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\} \\<br/>&amp; \Rightarrow \quad \sin \theta(\sqrt{2}-1)=\cos \theta \\<br/>&amp; \Rightarrow \quad \tan \theta=\frac{1}{\sqrt{2}-1} \times \frac{\sqrt{2}+1}{\sqrt{2}+1} \\<br/>&amp; =(\sqrt{2}+1) \\<br/>&amp; \therefore \quad P=Q \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Ellipse', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-p-x-1-y-1-and-q-x-2-y-2-y-1-0-y-2-0-be-the-end-points-of-the-latusrectum-of-the-ellipse-x-2-4-y-2-4-the-equations-of-parabolas-with-latusrectum-pq

slug:
let-p-x-1-y-1-and-q-x-2-y-2-y-1-0-y-2-0-be-the-end-points-of-the-latusrectum-of-the-ellipse-x-2-4-y-2-4-the-equations-of-parabolas-with-latusrectum-pq-45989

question:
<div class="question">Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), y_1 &lt; 0, y_2 &lt; 0$, be the end points of the latusrectum of the ellipse $x^2+4 y^2=4$. The equations of parabolas with latusrectum $P Q$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$x^2-2 \sqrt{3} y=3+\sqrt{3}$<br/>, <br/>$x^2+2 \sqrt{3} y=3-\sqrt{3}$<br/></span> </div>

answer:
<div class="solution">The equation $x^2+4 y^2=4$ represents an ellipse with 2 and 1 as semi-major and semi-minor axes and eccentricity $\frac{\sqrt{3}}{2}$. Thus, the ends of latusrect are $\left(\sqrt{3}, \frac{1}{2}\right)$ and $\left(\sqrt{3},-\frac{1}{2}\right),\left(-\sqrt{3},-\frac{1}{2}\right)$ and $\left(\sqrt{3},-\frac{1}{2}\right)$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xEg5w8HkzhfUb5VNonqRlTHouyfpQlt8WmCSapQNpQ8.original.fullsize.png"/><br/><br/>According to the question, we consider only $P\left(-\sqrt{3},-\frac{1}{2}\right)$ and $Q\left(\sqrt{3},-\frac{1}{2}\right)$.<br/>Now, $\quad P Q=2 \sqrt{3}$<br/>Thus, the coordinates of the vertex of the parabolas are $A\left(0, \frac{-1+\sqrt{3}}{2}\right)$ and $A^{\prime}\left(0, \frac{-1-\sqrt{3}}{2}\right)$ and corresponding equations are and<br/>$$<br/>(x-0)^2=-4 \cdot \frac{\sqrt{3}}{2}\left(y+\frac{1-\sqrt{3}}{2}\right)<br/>$$<br/>i.e., $(x-0)^2=4 \cdot \frac{\sqrt{3}}{2}\left(y-\frac{-1-\sqrt{3}}{2}\right)$ and<br/>$$<br/>\begin{aligned}<br/>&amp; x^2+2 \sqrt{3} y=3-\sqrt{3} \\<br/>&amp; x^2-2 \sqrt{3} y=3+\sqrt{3}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P2.db
title:
let-p-x-be-a-polynomial-of-degree-4-having-extremum-at-x-1-2-and-lim-x-0-1-x-2-p-x-2-then-the-value-of-p-2-is

slug:
let-p-x-be-a-polynomial-of-degree-4-having-extremum-at-x-1-2-and-lim-x-0-1-x-2-p-x-2-then-the-value-of-p-2-is-16434

question:
<div class="question">Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and $\lim _{x \rightarrow 0}\left[1+\frac{p(x)}{x^2}\right]=2$. Then, the value of $p(2)$ is</div>

answer:
<div class="solution">Let $p(x)=a x^4+b x^3+c x^2+d x+e$<br/>$$<br/>\Rightarrow \quad p^{\prime}(x)=4 a x^3+3 b x^2+2 c x+d<br/>$$<br/>$$<br/>\therefore \quad p^{\prime}(1)=4 a+3 b+2 c+d=0<br/>$$<br/>and $p^{\prime}(2)=32 a+12 b+4 c+d=0$<br/>Since, $\lim _{x \rightarrow 0}\left(1+\frac{p(x)}{x^2}\right)=2 \quad$ [given]<br/>$$<br/>\lim _{x \rightarrow 0} \frac{a x^4+b x^3+(c+1) x^2+d x+e}{x^2}=2<br/>$$<br/><br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow &amp; &amp; x+1=2, d &amp; =0, e=0 \\<br/>\Rightarrow &amp; c &amp; =1<br/>\end{array}<br/>$$<br/>From Eqs. (i) and (ii)<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; 4 a+3 b=-2 &amp; \text { and } 32 a+12 b=-4 \\<br/>\Rightarrow &amp; &amp; a=\frac{1}{4} \text { and } b=-1 \\<br/>\therefore &amp; &amp; p(x)=\frac{x^4}{4}-x^3+x^2 \\<br/>\Rightarrow &amp; &amp; p(2)=\frac{16}{4}-8+4<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P2.db
title:
let-p-x-be-a-real-polynomial-of-least-degree-which-has-a-local-maximum-at-x-1-and-a-local-minimum-at-x-3-if-p-1-6-and-p-3-2-then-p-0-is

slug:
let-p-x-be-a-real-polynomial-of-least-degree-which-has-a-local-maximum-at-x-1-and-a-local-minimum-at-x-3-if-p-1-6-and-p-3-2-then-p-0-is-75729

question:
<div class="question">Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is</div>

answer:
<div class="solution">Since, $p(x)$ has a local maximum at $x=1$ and a local minimum at $x=3$ and $p(x)$ is a real polynomial of least degree.
<br/> <br/>Hence, let $p^{\prime}(x)=k(x-1)(x-3)=k\left(x^{2}-4 x+3\right)$
<br/> <br/>$\Rightarrow p(x)=k\left(\frac{x^{3}}{3}-2 x^{2}+3 x\right)+c$
<br/> <br/>Now, $p(1)=6$ and $p(3)=2$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow \frac{4}{3} k+C=6 \text { and } 0+C=2 \Rightarrow k=3 \\
<br/> <br/>\therefore \quad p^{\prime}(x)=3(x-1)(x-3) \Rightarrow p^{\prime}(0)=9
<br/> <br/>\end{array}$</div>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
let-pqr-be-a-triangle-of-area-with-a-2-b-2-7-and-c-2-5-where-a-b-and-c-are-the-lengths-of-the-sides-of-the-triangle-opposite-to-the-angles-at-p-q-and-

slug:
let-pqr-be-a-triangle-of-area-with-a-2-b-2-7-and-c-2-5-where-a-b-and-c-are-the-lengths-of-the-sides-of-the-triangle-opposite-to-the-angles-at-p-q-and-53112

question:
<div class="question">Let $P Q R$ be a triangle of area $\Delta$ with $a=2, b=\frac{7}{2}$ and $c=\frac{5}{2} ;$ where $a, b$, and $c$ are the lengths of the sides of the triangle opposite to the angles at $P, Q$ and $R$ respectively. Then $\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}$ equals.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{4 \Delta}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{45}{4 \Delta}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{3}{4 \Delta}\right)^{2}$</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">$\left(\frac{45}{4 \Lambda}\right)^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{3}{4 \Delta}\right)^{2}$</span> </div>

answer:
<div class="solution">$\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}=\frac{2 \sin P-2 \sin P \cos P}{2 \sin P+2 \sin P \cos P}=\frac{1-\cos P}{1-\sin P}$
<br/> <br/>$=\frac{2 \sin ^{2} \frac{P}{2}}{2 \cos ^{2} \frac{P}{2}}=\tan ^{2} \frac{P}{2}=\frac{(s-b)(s-c)}{s(s-a)}$ where $s=\frac{a+b+c}{2}$
<br/> <br/>$=\frac{(s-b)^{2}(s-c)^{2}}{s(s-a)(s-b)(s-c)}=\frac{(a+c-b)^{2}(a+b-c)^{2}}{16 . \Delta^{2}}$
<br/> <br/>$=\frac{\left(2+\frac{5}{2}-\frac{7}{2}\right)^{2}\left(2+\frac{7}{2}-\frac{5}{2}\right)^{2}}{16 \Delta^{2}}=\frac{1 \times 9}{16 \Delta^{2}}=\left(\frac{3}{4 \Delta}\right)^{2}$</div>


title: let-s-1-2-3-4-the-total-number-of-unordered-pairs-of-disjoint-subsets-of-s-is-equal-to
slug: let-s-1-2-3-4-the-total-number-of-unordered-pairs-of-disjoint-subsets-of-s-is-equal-to-35136
tags: ['Mathematics', 'Sets and Relations', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>25<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>34<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>42<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>41</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>41</span> </div>

answer:
<div class="solution">Let $A \cap B=\phi, A, B \subset S$<br/>$$<br/>\begin{aligned}<br/>&amp; 3^4=\frac{81+1}{2}=41 \\<br/>\Rightarrow \quad &amp; \frac{3^4+1}{2}=41<br/>\end{aligned}<br/>$$</div>


title: let-s-be-the-area-of-the-region-enclosed-by-y-e-x-2-y-0-x-0-and-x-1-then
slug: let-s-be-the-area-of-the-region-enclosed-by-y-e-x-2-y-0-x-0-and-x-1-then-66852
tags: ['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $\mathrm{S}$ be the area of the region enclosed by $y=e^{-x^{2}}$, $y=0, x=0$ and $x=1$; then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$S \geq \frac{1}{e}$, $S \geq 1-\frac{1}{e}$, $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$</span> </div>

answer:
<div class="solution">The given curve $y=e^{-x^{2}}$ Draw a rough sketch of curve at $x=0, y=1$ and at $x=1, y=1 / e$ $\because y=e^{-x^{2}}$ $\Rightarrow \frac{d y}{d x}=-2 x e^{-x^{2}} &lt; 0 \quad \forall x \in(0,1)$
<br/> <br/>$\therefore y=e^{-x^{2}}$ is decreasing on $(0,1)$
<br/> <br/>Hence its graph is as shown in figure given below<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/EsW5pZYCySy0dpsUUr6Q7PG_G-p7oYK6ONlT1YJfaKg.original.fullsize.png"/><br/> <br/> <br/>Now, $\mathrm{S}=$ area exclosed by curve $=\mathrm{XYCO}$
<br/> <br/>and area of rectangle $\mathrm{OCYL}=\frac{1}{e}$
<br/> <br/>Clearly $S&gt;\frac{1}{e} \quad \therefore$ a is true.
<br/> <br/>For $x \in[0,1] \Rightarrow x^{2} &lt; x$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow-x^{2}&gt;-x \quad \Rightarrow e^{-x^{2}} \geq e^{-x} \forall x \in[0,1] \\
<br/> <br/>\Rightarrow \int_{0}^{1} e^{-x^{2}} d x&gt;\int_{0}^{1} e^{-x} d x=1-\frac{1}{e} \\
<br/> <br/>\Rightarrow S&gt;1-\frac{1}{e} \quad \therefore \text { (b) is true. }
<br/> <br/>\end{array}$
<br/> <br/>Now $S &lt; $ area of rectangle $\mathrm{XADO}+$ area of rectangle $\mathrm{ZDCN}$ $\Rightarrow S &lt; \frac{1}{\sqrt{2}} \times 1+\left(1-\frac{1}{\sqrt{2}}\right) \frac{1}{\sqrt{e}}$ $\therefore S &lt; \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right) \quad \because(\mathrm{d})$ is true
<br/> <br/>Also as $\frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right) &lt; 1-\frac{1}{e} \quad \therefore$ (c) is incorrect.</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P2.db
title:
let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian-1

slug:
let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian-1-68798

question:
<div class="question">Let $S$ be the focus of the parabola $y^{2}=8 \mathrm{x}$ and let $P Q$ be the common chord of the circle $x^{2}+y^{2}-2 x-4 y=0$ and the given parabola. The area of the triangle $P Q S$ is</div>

answer:
<div class="solution">Given parabola $y^{2}=8 x$
<br/> <br/>and circle $x^{2}+y^{2}-2 x-4 y=0$ pass through the origin
<br/> <br/>$\therefore$ One end of common chord PQ is origin. Say $\mathrm{P}(0,0)$
<br/> <br/>Let $\mathrm{Q}$ be the point $\left(2 t^{2}, 4 t\right)$, then it will satisfy the equation of circle.
<br/> <br/>$\therefore \quad 4 t^{4}+16 t^{2}-4 t^{2}-16 t=0$
<br/> <br/>$\Rightarrow t^{4}+3 t^{2}-4 t=0 \Rightarrow t\left(t^{3}+3 t-4\right)=0$
<br/> <br/>$\Rightarrow t(t-1)\left(t^{2}+t-4\right)=0 \Rightarrow t=0$ or 1
<br/> <br/>For $t=0$, we get point $P$, therefore $t=1$ gives point $Q$ as $(2,4)$.
<br/> <br/>We also observe here that $\mathrm{P}(0,0)$ and $\mathrm{Q}(2,4)$ are end points of diameter of the given circle and focus of the parabola is the point $S(2,0)$.
<br/> <br/>$\therefore \quad$ area $(\Delta \mathrm{PQS})=\frac{1}{2} \times P S \times Q S=\frac{1}{2} \times 2 \times 4=4$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eU4fQJz3adHaoGP7hh6deAA3XZ2LwdvTG4BfAwf634Y.original.fullsize.png"/></div>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P2.db
title:
let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian

slug:
let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian-89208

question:
<div class="question">Let $S$ be the focus of the parabola $y^{2}=8 \mathrm{x}$ and let $P Q$ be the common chord of the circle $x^{2}+y^{2}-2 x-4 y=0$ and the given parabola. The area of the triangle $P Q S$ is</div>

answer:
<div class="solution">Given parabola $y^{2}=8 x$
<br/> <br/>and circle $x^{2}+y^{2}-2 x-4 y=0$ pass through the origin
<br/> <br/>$\therefore$ One end of common chord PQ is origin. Say $\mathrm{P}(0,0)$
<br/> <br/>Let $\mathrm{Q}$ be the point $\left(2 t^{2}, 4 t\right)$, then it will satisfy the equation of circle.
<br/> <br/>$\therefore \quad 4 t^{4}+16 t^{2}-4 t^{2}-16 t=0$
<br/> <br/>$\Rightarrow t^{4}+3 t^{2}-4 t=0 \Rightarrow t\left(t^{3}+3 t-4\right)=0$
<br/> <br/>$\Rightarrow t(t-1)\left(t^{2}+t-4\right)=0 \Rightarrow t=0$ or 1
<br/> <br/>For $t=0$, we get point $P$, therefore $t=1$ gives point $Q$ as $(2,4)$.
<br/> <br/>We also observe here that $\mathrm{P}(0,0)$ and $\mathrm{Q}(2,4)$ are end points of diameter of the given circle and focus of the parabola is the point $S(2,0)$.
<br/> <br/>$\therefore \quad$ area $(\Delta \mathrm{PQS})=\frac{1}{2} \times P S \times Q S=\frac{1}{2} \times 2 \times 4=4$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eU4fQJz3adHaoGP7hh6deAA3XZ2LwdvTG4BfAwf634Y.original.fullsize.png"/></div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P2.db
title:
let-s-k-k-1-2-100-denote-the-sum-of-the-infinite-geometric-series-whose-first-term-is-k-k-1-and-the-common-ratio-is-k-1-then-the-value-of-100-10-0-2-k

slug:
let-s-k-k-1-2-100-denote-the-sum-of-the-infinite-geometric-series-whose-first-term-is-k-k-1-and-the-common-ratio-is-k-1-then-the-value-of-100-10-0-2-k-38259

question:
<div class="question">Let $S_k, k=1,2, \ldots, 100$, denote the sum of the infinite geometric series whose first term is $\frac{k-1}{k !}$ and the common ratio is $\frac{1}{k}$. Then the value of $\frac{100^2}{100 !}+\sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right|$ is</div>

answer:
<div class="solution">We have, $S_k=\frac{\frac{k-1}{k !}}{1-\frac{1}{k}}=\frac{1}{(k-1) !}$<br/>Now,<br/>$$<br/>\begin{aligned}<br/>&amp; \left(k^2-3 k+1\right) S_k=\{(k-2)(k-1)-1\} \\<br/>&amp; =\frac{1}{(k-3) !}-\frac{1}{(k-1) !} \\<br/>&amp; \Rightarrow \sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right| \\<br/>&amp; =1+1+2-\left(\frac{1}{99 !}+\frac{1}{98 !}\right)=4-\frac{100^2}{100 !}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\Rightarrow \quad \frac{100^2}{100 !}+\sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right|=4<br/>$$</div>


title: let-s-n-k-0-n-n-2-kn-k-2-n-and-t-n-k-0-n-1-n-2-kn-k-2-n-for-n-1-2-3-then
slug: let-s-n-k-0-n-n-2-kn-k-2-n-and-t-n-k-0-n-1-n-2-kn-k-2-n-for-n-1-2-3-then-34514
tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}$ and $T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}$, for $n=1,2,3, \ldots$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$S_n &lt; \frac{\pi}{3 \sqrt{3}}$, <br/>$T_n&gt;\frac{\pi}{3 \sqrt{3}}$</span> </div>

answer:
<div class="solution">Given,<br/>$$<br/>\begin{aligned}<br/>S_n &amp; =\sum_{k=0}^n \frac{n}{n^2+k n+k^2} \\<br/>&amp; =\sum_{k=0}^n \frac{1}{n} \cdot\left(\frac{1}{1+\frac{k}{n}+\frac{k^2}{n^2}}\right) &lt; \lim _{n \rightarrow \infty} \sum_{k=0}^n \frac{1}{n}\left(\frac{1}{1+\frac{k}{n}+\left(\frac{k}{n}\right)^2}\right) \\<br/>&amp; =\int_0^1 \frac{1}{1+x+x^2} d x=\left[\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2}{\sqrt{3}}\left(x+\frac{1}{2}\right)\right)\right]_0^1 \\<br/>&amp; =\frac{2}{\sqrt{3}} \cdot\left(\frac{\pi}{3}-\frac{\pi}{6}\right)=\frac{\pi}{3 \sqrt{3}}<br/>\end{aligned}<br/>$$<br/>i.e., $\quad S_n &lt; \frac{\pi}{3 \sqrt{3}}$<br/>Similarly, $T_n&gt;\frac{\pi}{3 \sqrt{3}}$.</div>


tags: ['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-the-eccentricity-of-the-hyperbola-a-2-x-2-b-2-y-2-1-be-reciprocal-to-that-of-the-ellipse-x-2-4-y-2-4-if-the-hyperbola-passes-through-a-focus-of-th

slug:
let-the-eccentricity-of-the-hyperbola-a-2-x-2-b-2-y-2-1-be-reciprocal-to-that-of-the-ellipse-x-2-4-y-2-4-if-the-hyperbola-passes-through-a-focus-of-th-82215

question:
<div class="question">Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4 y^2=4$. If the hyperbola passes through a focus of the ellipse, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>a focus of the hyperbola is $(2,0)$<br/>, <br/>the equation of the hyperbola is $x^2-3 y^2=3$</span> </div>

answer:
<div class="solution">Here, equation of ellipse<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{x^2}{4}+\frac{y^2}{1}=1 \\<br/>&amp; \Rightarrow \quad e^2=1-\frac{b^2}{a^2}=1-\frac{1}{4}=\frac{3}{4} \\<br/>&amp; \therefore \quad e=\frac{\sqrt{3}}{2} \text { and focus }(\pm a e, 0) \\<br/>&amp; =(\pm \sqrt{3}, 0) \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>For hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, $e_1^2=1+\frac{b^2}{a^2}$ where, $e_1^2=\frac{1}{e^2}=\frac{4}{3}$<br/>$$<br/>\Rightarrow \quad 1+\frac{b^2}{a^2}=\frac{4}{3} \Rightarrow \frac{b^2}{a^2}=\frac{1}{3}<br/>$$<br/>and hyperbola passes through $(\pm \sqrt{3}, 0)$. Now, $\quad \frac{3}{a^2}=1 \Rightarrow a^2=3$<br/>From Eqs. (i) and (ii), we get $b^2=1$<br/>$\therefore$ Equation of hyperbola is<br/>$$<br/>\frac{x^2}{3}-\frac{y^2}{1}=1<br/>$$<br/>Focus is $(\pm a e, 0)$.<br/>Now, $\quad\left(\pm \sqrt{3} \cdot \frac{2}{\sqrt{3}}, 0\right) \Rightarrow(\pm 2,0)$<br/>Hence, both options (b) and (d) are correct.</div>


title: let-the-function-g-2-2-be-given-by-g-u-2-tan-1-e-u-2-then-g-is
slug: let-the-function-g-2-2-be-given-by-g-u-2-tan-1-e-u-2-then-g-is-79924
tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let the function $g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$. Then, $g$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>odd and is strictly increasing in $(-\infty, \infty)$<br/></span> </div>

answer:
<div class="solution">$\therefore \quad g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$<br/>for $\quad u \in(-\infty, \infty)$<br/>and<br/>$$<br/>\begin{aligned}<br/>g(-u) &amp; =2 \tan ^{-1}\left(e^{-u}\right)-\frac{\pi}{2} \\<br/>&amp; =2\left(\cot ^{-1}\left(e^u\right)\right)-\frac{\pi}{2} \\<br/>&amp; =2\left(\frac{\pi}{2}-\tan ^{-1}\left(e^u\right)\right)-\frac{\pi}{2} \\<br/>&amp; =\frac{\pi}{2}-2 \tan ^{-1}\left(e^u\right)=-g(u)<br/>\end{aligned}<br/>$$<br/>$\therefore \quad g(-u)=-g(u)$<br/>$\Rightarrow g(u)$ is an odd function.</div>


tags: ['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-the-straight-line-x-b-divide-the-area-enclosed-by-y-1-x-2-y-0-and-x-0-into-two-parts-r-1-0-x-b-and-r-2-b-x-1-such-that-r-1-r-2-4-1-then-b-equals-t

slug:
let-the-straight-line-x-b-divide-the-area-enclosed-by-y-1-x-2-y-0-and-x-0-into-two-parts-r-1-0-x-b-and-r-2-b-x-1-such-that-r-1-r-2-4-1-then-b-equals-t-70961

question:
<div class="question">Let the straight line $x=b$ divide the area enclosed by $y=(1-x)^2, y=0$ and $x=0$ into two parts $R_1(0 \leq x \leq b)$ and $R_2(b \leq x \leq 1)$ such that $R_1-R_2=\frac{1}{4}$. Then, $b$ equals to</div>

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

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

answer:
<div class="solution">Here, area between 0 to $b$ is $R_1$ and $b$ to<br/>$$<br/>\begin{aligned}<br/>&amp; 1 \text { is } R_2 \\<br/>&amp; \therefore \int_0^b(1-x)^2 d x-\int_b^1(1-x)^2 d x=\frac{1}{4} \\<br/>&amp; \Rightarrow \quad\left(\frac{(1-x)^3}{-3}\right)_0^b-\left(\frac{(1-x)^3}{-3}\right)_b^1=\frac{1}{4} \\<br/>&amp; \Rightarrow-\frac{1}{3}\left\{(1-b)^3-1\right\}+\frac{1}{3}\left\{0-(1-b)^3\right\} \\<br/>&amp; =\frac{1}{4}<br/>\end{aligned}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad-\frac{2}{3}(1-b)^3=-\frac{1}{3}+\frac{1}{4}=-\frac{1}{12} \\<br/>&amp; \Rightarrow \quad(1-b)^3=\frac{1}{8} \\<br/>&amp; \Rightarrow \quad(1-b)=\frac{1}{2} \Rightarrow b=\frac{1}{2} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-the-vectors-pq-qr-rs-st-tu-and-up-represent-the-sides-of-a-regular-hexagon-statement-i-pq-rs-st-0-statement-ii-pq-rs-0-and-pq-sq-0

slug:
let-the-vectors-pq-qr-rs-st-tu-and-up-represent-the-sides-of-a-regular-hexagon-statement-i-pq-rs-st-0-statement-ii-pq-rs-0-and-pq-sq-0-36339

question:
<div class="question">Let the vectors $\mathbf{P Q}, \mathbf{Q R}, \mathbf{R S}, \mathbf{S T}, \mathbf{T U}$ and UP represent the sides of a regular hexagon.<br/>Statement I PQ $\times(\mathbf{R S}+\mathbf{S T}) \neq \mathbf{0}$<br/>Statement II $\mathbf{P Q} \times \mathbf{R S}=\mathbf{0}$ and $\mathbf{P Q} \times \mathbf{S Q} \neq \mathbf{0}$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is true, Statement II is false<br/></span> </div>

answer:
<div class="solution">Since, $\mathbf{P Q}$ is not parallel to $\mathbf{T R}$<br/>$\because$ TR is resultant of $\mathbf{S R}$ and $\mathbf{S T}$ vectors.<br/>$\Rightarrow \quad \mathbf{P Q} \times(\mathbf{R S}+\mathbf{S T}) \neq \mathbf{0}$<br/>But for Statement II, we have $\mathbf{P Q} \times \mathbf{R S}=\mathbf{0}$<br/>which is not possible as $\mathbf{P Q}$ not parallel to $\mathbf{R S}$.<br/>Hence, Statement I is true and Statement II is false.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jQ8Sa0ufMUOmwUe0xgUS2AYCWxDpsiUAFOd6R-w_QmA.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
let-two-noncollinear-unit-vectors-a-and-b-form-an-acute-angle-a-point-p-moves-so-that-at-any-time-t-the-position-vector-op-where-o-is-the-origin-is-gi

slug:
let-two-noncollinear-unit-vectors-a-and-b-form-an-acute-angle-a-point-p-moves-so-that-at-any-time-t-the-position-vector-op-where-o-is-the-origin-is-gi-10387

question:
<div class="question">Let two non-collinear unit vectors $\mathbf{a}$ and $\hat{\mathbf{b}}$ form an acute angle.<br/>A point $P$ moves so that at any time $t$ the position vector $\mathbf{O P}$ (where, $O$ is the origin) is given by $\mathbf{a} \cos t+\hat{\mathbf{b}} \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\mathbf{O P}$ and $\hat{\mathbf{u}}$ be the unit vector along $\mathbf{O P}$. Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$<br/></span> </div>

answer:
<div class="solution">$\mathbf{O P}=\hat{\mathbf{a}} \cos t+\hat{\mathbf{b}} \sin t$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad|\mathbf{O P}|=\sqrt{\left(\hat{\mathbf{a}} \cdot \hat{\mathbf{a}} \cos ^2 t+\hat{\mathbf{b}} \cdot \hat{\mathbf{b}} \sin ^2 t+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \sin t \cos t\right)} \\<br/>&amp; \Rightarrow \quad|\mathbf{O P}|=\sqrt{1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \cdot \sin t \cos t} \\<br/>&amp; \Rightarrow \quad|\mathbf{O P}|=\sqrt{1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \cdot \sin 2 t} \\<br/>&amp; \Rightarrow \quad|\mathbf{O P}|_{\max }=\sqrt{1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}} \text { at } \sin 2 t=1 \Rightarrow t=\frac{\pi}{4} \\<br/>&amp; \Rightarrow \mathbf{O P}\left(\text { at } t=\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}(\hat{\mathbf{a}}+\hat{\mathbf{b}}) \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>$\therefore$ Unit vector along $\mathbf{O P}$ at $\left(t=\frac{\pi}{4}\right)=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$</div>


tags: ['Mathematics', 'Basic of Mathematics', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-x-0-y-0-be-the-solution-of-the-following-equations-2-x-l-o-g-2-3-y-l-o-g-3-3-l-o-g-x-2-l-o-g-y-then-x-0-is-equal-to

slug:
let-x-0-y-0-be-the-solution-of-the-following-equations-2-x-l-o-g-2-3-y-l-o-g-3-3-l-o-g-x-2-l-o-g-y-then-x-0-is-equal-to-82320

question:
<div class="question">Let $\left(x_0, y_0\right)$ be the solution of the following equations $(2 x)^{\log 2}=(3 y)^{\log 3}$, $3^{\log x}=2^{\log y}$, then $x_0$ is equal to</div>

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

answer:
<div class="solution">Taking log on both sides,<br/>$$<br/>\log 2 \cdot \log (2 x)=\log 3(\log 3 y)<br/>$$<br/><br/><br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; \log 2\{\log 2+\log x\} \\<br/>&amp; =\log 3\{\log 3+\log y\} \\<br/>\text { and } &amp; \log x \cdot \log 3=\log y \log 2 \\<br/>&amp; \log y=\frac{\log x \cdot \log 3}{\log 2}<br/>\end{array}<br/>$$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>\begin{aligned}<br/>&amp; \log 2\{\log 2+\log x\} \\<br/>&amp;=\log 3 \cdot\left\{\log 3+\frac{\log x \cdot \log 3}{\log 2}\right\} \\<br/>&amp; \Rightarrow \quad(\log 2)^2+\log 2 \cdot \log x \\<br/>&amp;=(\log 3)^2+\frac{(\log 3)^2}{(\log 2)} \cdot \log x \\<br/>&amp; \Rightarrow \quad \log x\left\{\frac{(\log 3)^2}{\log 2}-\log 2\right\} \\<br/>&amp;=(\log 2)^2-(\log 3)^2<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad \log x=-\log 2=\log 2^{-1} \\<br/>&amp; \therefore \quad x=\frac{1}{2}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
let-x-and-y-be-two-events-such-that-p-x-y-2-1-p-y-x-3-1-and-p-x-y-6-1-which-of-the-following-is-are-correct

slug:
let-x-and-y-be-two-events-such-that-p-x-y-2-1-p-y-x-3-1-and-p-x-y-6-1-which-of-the-following-is-are-correct-99603

question:
<div class="question">Let $X$ and $Y$ be two events such that $P(X \mid Y)=\frac{1}{2}$, $P(Y / X)=\frac{1}{3}$ and $P(X \cap Y)=\frac{1}{6}$. Which of the following is (are) correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$P(X \cup Y)=\frac{2}{3}$, $X$ and $Y$ are independent</span> </div>

answer:
<div class="solution">$\because P(X / Y)=\frac{P(X \cap Y)}{P(Y)} \Rightarrow \frac{1}{2}=\frac{1 / 6}{P(Y)} \Rightarrow P(Y)=\frac{1}{3}$
<br/> <br/>Similarly, $\mathrm{P}(Y / X)=\frac{P(X \cap Y)}{P(X)}$
<br/> <br/>$\Rightarrow \frac{1}{3}=\frac{1 / 6}{P(X)} \Rightarrow P(X)=\frac{1}{2}$
<br/> <br/>(a) $P(X \cup Y)=P(X)+P(Y)-P(X \cap Y)=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}$
<br/> <br/>$\therefore$ (a) is true.
<br/> <br/>(b) $\because P(X \cap Y)=P(X) P(Y)$
<br/> <br/>$\Rightarrow X$ and $Y$ are independent events.
<br/> <br/>$\therefore$ (b) is true.
<br/> <br/>But (c) is not true.
<br/> <br/>(d) $\mathrm{P}\left(X^{C} \cap Y\right)=P\left(X^{C}\right) \times P(Y)=\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$
<br/> <br/>$\therefore$ (d) is not true.</div>


title: let-x-x-2-cos-x-x-0-0-x-0-x-r-then-is
slug: let-x-x-2-cos-x-x-0-0-x-0-x-r-then-is-54281
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.$ then $\int$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">differentiable both at $x=0$ and at $x=2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">differentiable at $x=0$ but not differentiable at $x=2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">not differentiable at $x=0$ but differentiable at $x=2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">differentiable neither at $x=0$ nor at $x=2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">differentiable at $x=0$ but not differentiable at $x=2$</span> </div>

answer:
<div class="solution">$f^{\prime}\left(0^{+}\right)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$
<br/> <br/>$\begin{aligned}
<br/> <br/>=\lim _{h \rightarrow 0} \frac{h^{2}\left|\cos \frac{\pi}{h}\right|}{h} &amp;=\lim _{h \rightarrow 0} h\left|\cos \frac{\pi}{h}\right| \\
<br/> <br/>&amp;=0 \times \text { some finite value }=0
<br/> <br/>\end{aligned}$
<br/> <br/>$\begin{array}{c}
<br/> <br/>=0 \times \text { some finite value }=0 \\
<br/> <br/>\text { and } f^{\prime}\left(0^{-}\right)=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0} \frac{h^{2}\left|\cos \frac{\pi}{-h}\right|}{-h} \\
<br/> <br/>=\lim _{h \rightarrow 0}-h\left|\cos \frac{\pi}{h}\right|=0 \times \text { some finite value }=0
<br/> <br/>\end{array}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\because f^{\prime}\left(0^{+}\right)=f^{\prime}\left(0^{-}\right) \therefore f \text { is differentiable at } x=0 \\
<br/> <br/>\text { Now } f^{\prime}\left(2^{+}\right)=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h} \\
<br/> <br/>\quad=\lim _{h \rightarrow 0} \frac{(2+h)^{2}\left|\cos \frac{\pi}{2+h}\right|-4\left|\cos \frac{\pi}{2}\right|}{h}
<br/> <br/>\end{array}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}\left(\cos \frac{\pi}{2+h}\right)}{h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}}{h} \sin \left(\frac{\pi}{2}-\frac{\pi}{2+h}\right)$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}}{h} \sin \left(\frac{\pi h}{2(2+h)}\right)$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}}{h} \times \frac{\sin \left(\frac{\pi h}{2(2+h)}\right)}{\left(\frac{\pi h}{2(2+h)}\right)} \times \frac{\pi h}{2(2+h)}=\pi$
<br/> <br/>and $f^{\prime}\left(2^{-}\right)=\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2-h)^{2}\left|\cos \left(\frac{\pi}{2-h}\right)\right|-0}{-h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{-(2-h)^{2} \cos \left(\frac{\pi}{2-h}\right)}{-h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2-h)^{2} \sin \left(\frac{\pi}{2}-\frac{\pi}{2-h}\right)}{h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2-h)^{2}}{h} \times \frac{\sin \left(\frac{-\pi h}{2(2-h)}\right)}{\left(\frac{-\pi h}{2(2-h)}\right)} \times\left(\frac{-\pi h}{2(2-h)}\right)=-\pi$
<br/> <br/>$\because f^{\prime}\left(2^{+}\right) \neq f^{\prime}\left(2^{-}\right), \therefore f$ is not differentiable at $x=2 .$</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is-1

slug:
let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is-1-36623

question:
<div class="question">Let $(x, y)$ be any point on the parabola $y^2=4 x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x, y)$ in the ratio $1: 3$. Then, the locus of $P$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$y^2=x$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { By section formula, }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NZeJW46lYjK_Tb4Ljhd5REr04ncqMpNoSecFzLx2gXo.original.fullsize.png"/><br/><br/>$h=\frac{x+0}{4}, k=\frac{y+0}{4}$<br/>$\therefore \quad x=4 h$ and $y=4 k$<br/>Substituting in $y^2=4 x$,<br/>$(4 k)^2=4(4 h) \Rightarrow k^2=h$<br/>or $y^2=x$ is required locus.</div>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is

slug:
let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is-57991

question:
<div class="question">Let $(x, y)$ be any point on the parabola $y^2=4 x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x, y)$ in the ratio $1: 3$. Then, the locus of $P$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$y^2=x$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { By section formula, }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NZeJW46lYjK_Tb4Ljhd5REr04ncqMpNoSecFzLx2gXo.original.fullsize.png"/><br/><br/>$h=\frac{x+0}{4}, k=\frac{y+0}{4}$<br/>$\therefore \quad x=4 h$ and $y=4 k$<br/>Substituting in $y^2=4 x$,<br/>$(4 k)^2=4(4 h) \Rightarrow k^2=h$<br/>or $y^2=x$ is required locus.</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii-1

slug:
let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii-1-92023

question:
<div class="question">Let $(x, y)$ be such that $\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}$. Match the statements in Column I with the values in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/-gTv06R5XWK5EKREPVN42FLFgZnm7GEW-trEZSNPICY.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p; B-q; C-p; D-s<br/></span> </div>

answer:
<div class="solution">(A) If $a=1$ and $b=0$, then $\sin ^{-1} x+\cos ^{-1} y=0$<br/>$$<br/>\Rightarrow \quad \sin ^{-1} x=-\cos ^{-1} y \Rightarrow x^2+y^2=1<br/>$$<br/>(B) If $a=1$ and $b=1$, then<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1} x y &amp; =\frac{\pi}{2} \\<br/>\Rightarrow &amp; &amp; \cos ^{-1} x-\cos ^{-1} y &amp; =\cos ^{-1} x y \\<br/>\Rightarrow &amp; x y+\sqrt{1-x^2} \sqrt{1-y^2} &amp; =x y \quad \text { [taking sine on both the sides] }<br/>\end{array}<br/>$$<br/>(C) If $a=1$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>&amp; &amp; \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1}(2 x y) &amp; =\frac{\pi}{2} \\<br/>\Rightarrow &amp; &amp; \cos ^{-1} x-\cos ^{-1} y &amp; =\cos ^{-1}(2 x y) \\<br/>\Rightarrow &amp; &amp; x y+\sqrt{1-x^2} \sqrt{1-y^2} &amp; =2 x y \\<br/>\Rightarrow &amp; &amp; x^2+y^2 &amp; =1<br/>\end{aligned}<br/>$$<br/>[on squaring]<br/>(D) If $a=2$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>&amp; &amp; \sin ^{-1}(2 x)+\cos ^{-1}(y)+\cos ^{-1}(2 x y) &amp; =\frac{\pi}{2} \\<br/>\Rightarrow &amp; &amp; 2 x y+\sqrt{1+4 x^2} \sqrt{1-y^2} &amp; =2 x y \\<br/>\Rightarrow &amp; &amp; \left(4 x^2-1\right)\left(y^2-1\right) &amp; =0 .<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii

slug:
let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii-13694

question:
<div class="question">Let $(x, y)$ be such that $\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}$. Match the statements in Column I with the values in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/-gTv06R5XWK5EKREPVN42FLFgZnm7GEW-trEZSNPICY.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p; B-q; C-p; D-s<br/></span> </div>

answer:
<div class="solution">(A) If $a=1$ and $b=0$, then $\sin ^{-1} x+\cos ^{-1} y=0$<br/>$$<br/>\Rightarrow \quad \sin ^{-1} x=-\cos ^{-1} y \Rightarrow x^2+y^2=1<br/>$$<br/>(B) If $a=1$ and $b=1$, then<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1} x y &amp; =\frac{\pi}{2} \\<br/>\Rightarrow &amp; &amp; \cos ^{-1} x-\cos ^{-1} y &amp; =\cos ^{-1} x y \\<br/>\Rightarrow &amp; x y+\sqrt{1-x^2} \sqrt{1-y^2} &amp; =x y \quad \text { [taking sine on both the sides] }<br/>\end{array}<br/>$$<br/>(C) If $a=1$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>&amp; &amp; \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1}(2 x y) &amp; =\frac{\pi}{2} \\<br/>\Rightarrow &amp; &amp; \cos ^{-1} x-\cos ^{-1} y &amp; =\cos ^{-1}(2 x y) \\<br/>\Rightarrow &amp; &amp; x y+\sqrt{1-x^2} \sqrt{1-y^2} &amp; =2 x y \\<br/>\Rightarrow &amp; &amp; x^2+y^2 &amp; =1<br/>\end{aligned}<br/>$$<br/>[on squaring]<br/>(D) If $a=2$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>&amp; &amp; \sin ^{-1}(2 x)+\cos ^{-1}(y)+\cos ^{-1}(2 x y) &amp; =\frac{\pi}{2} \\<br/>\Rightarrow &amp; &amp; 2 x y+\sqrt{1+4 x^2} \sqrt{1-y^2} &amp; =2 x y \\<br/>\Rightarrow &amp; &amp; \left(4 x^2-1\right)\left(y^2-1\right) &amp; =0 .<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P2.db
title:
let-x-y-z-be-points-with-integer-coordinates-satisfying-the-system-of-homogeneous-equations-3-x-y-z-0-3-x-z-0-3-x-2-y-z-0-then-the-number-of-such-poin

slug:
let-x-y-z-be-points-with-integer-coordinates-satisfying-the-system-of-homogeneous-equations-3-x-y-z-0-3-x-z-0-3-x-2-y-z-0-then-the-number-of-such-poin-25448

question:
<div class="question">Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations $3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0$. Then, the number of such points for which $x^2+y^2+z^2 \leq 100$ is</div>

answer:
<div class="solution">Given, $\quad 3 x-y-z=0$<br/>and<br/>$$<br/>\begin{array}{r}<br/>-3 x+2 y+z=0 \\<br/>-3 x+z=0<br/>\end{array}<br/>$$<br/>On adding Eqs. (i) and (ii), we get $y=0$<br/>So, $\quad 3 x=z$<br/>Now, $\quad x^2+y^2+z^2 \leq 100$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow &amp; x^2+(3 x)^2+0 \leq 100 \\<br/>\Rightarrow &amp; 10 x^2 \leq 100 \\<br/>\Rightarrow &amp; x^2 \leq 10 \\<br/>\therefore &amp; x=-3,-2,-1,0,1,2,3<br/>\end{array}<br/>$$<br/>So, number of such 7 points are possible.</div>


tags: ['Mathematics', 'Differential Equations', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P2.db
title:
let-y-x-y-x-g-x-g-x-g-x-y-0-0-x-r-where-f-x-denotes-d-x-df-x-and-g-x-is-a-given-nonconstant-differentiable-function-on-r-with-g-0-g-2-0-then-the-value

slug:
let-y-x-y-x-g-x-g-x-g-x-y-0-0-x-r-where-f-x-denotes-d-x-df-x-and-g-x-is-a-given-nonconstant-differentiable-function-on-r-with-g-0-g-2-0-then-the-value-37046

question:
<div class="question">Let $y^{\prime}(x)+y(x) g^{\prime}(x)=g(x) g^{\prime}(x) y(0)=0, x \in R$, where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $R$ with $g(0)=g(2)=0$. Then, the value of $y$ (2) is...</div>

answer:
<div class="solution">$\frac{d y}{d x}+y \cdot g^{\prime}(x)=g(x) g^{\prime}(x)$<br/>$$<br/>\mathrm{IF}=e^{\int g^{\prime}(x) d x}=e^{g(x)}<br/>$$<br/>$\therefore$ Solution is<br/><br/>$$<br/>y\left(e^{g(x)}\right)=\int g(x) \cdot g^{\prime}(x) \cdot e^{g(x)} d x+C<br/>$$<br/>Put $g(x)=t, g^{\prime}(x) d x=d t$<br/>$$<br/>\begin{aligned}<br/>&amp; y\left(e^{g(x)}\right)=\int t \cdot e^t d t+C \\<br/>= &amp; t \cdot e^t-\int 1 \cdot e^t d t+C=t \cdot e^t-e^t+C \\<br/>&amp; y e^{g(x)}=(g(x)-1) e^{g(x)}+C \quad \ldots(\mathrm{i})<br/>\end{aligned}<br/>$$<br/>Given, $y(0)=0, g(0)=g(2)=0$<br/><br/>$\therefore$ Eq. (i) becomes,<br/>$$<br/>\begin{aligned}<br/>&amp; \quad y(0) \cdot e^{g(0)}=(g(0)-1) \cdot e^{g(0)}+C \\<br/>&amp; \Rightarrow \quad 0=(-1) \cdot 1+C \Rightarrow C=1 \\<br/>&amp; \therefore \quad y(x) \cdot e^{g(x)}=(g(x)-1) e^{g(x)}+1 \\<br/>&amp; \Rightarrow \quad y(2) \cdot e^{g(2)}=(g(2)-1) e^{g(2)}+1 \\<br/>&amp; \text { where, } \quad g(2)=0 \\<br/>&amp; \Rightarrow \quad y(2) \cdot 1=(-1) \cdot 1+1 \Rightarrow y(2)=0<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-z-1-and-z-2-be-two-distinct-complex-numbers-and-let-z-1-t-z-1-t-z-2-for-some-real-number-t-with-0-t-1-if-ar-g-w-denotes-the-principal-argument-of-

slug:
let-z-1-and-z-2-be-two-distinct-complex-numbers-and-let-z-1-t-z-1-t-z-2-for-some-real-number-t-with-0-t-1-if-ar-g-w-denotes-the-principal-argument-of-39811

question:
<div class="question">Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t) z_1+t z_2$ for some real number $t$ with $0 &lt; t &lt; 1$. If $\arg (w)$ denotes the principal argument of a non-zero complex number $w$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\arg \left(z-z_1\right)=\arg \left(z-z_2\right)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\left|\begin{array}{cc}z-z_1 &amp; \bar{z}-\bar{z}_1 \\ z_2-z_1 &amp; \bar{z}_2-\bar{z}_1\end{array}\right|=0$<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="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\arg \left(z-z_1\right)=\arg \left(z_2-z_1\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$<br/>, <br/>$\left|\begin{array}{cc}z-z_1 &amp; \bar{z}-\bar{z}_1 \\ z_2-z_1 &amp; \bar{z}_2-\bar{z}_1\end{array}\right|=0$<br/>, <br/>$\arg \left(z-z_1\right)=\arg \left(z_2-z_1\right)$</span> </div>

answer:
<div class="solution">Given, $z=\frac{(1-t) z_1+t z_2}{(1-t)+t}$<br/>Clearly, $z$ divides $z_1$ and $z_2$ in the ratio of $t:(1-t), 0 &lt; t &lt; 1$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad A P+B P=A B \\<br/>&amp; \text { ie, }\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right| \\<br/>&amp; \Rightarrow \text { Option (a) is true. } \\<br/>&amp; \text { and } \begin{aligned}<br/>\arg \left(z-z_1\right) &amp; =\arg \left(z_2-z\right) \\<br/>&amp; =\arg \left(z_2-z_1\right)<br/>\end{aligned}<br/>\end{aligned}<br/>$$<br/>$\Rightarrow$ (b) is false and (d) is true.<br/>Also, $\arg \left(z-z_1\right)=\arg \left(z_2-z_1\right)$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad \arg \left(\frac{z-z_1}{z_2-z_1}\right)=0 \\<br/>&amp; \therefore \frac{z-z_1}{z_2-z_1} \text { is purely real. } \\<br/>&amp; \Rightarrow \quad \frac{z-z_1}{z_2-z_1}=\frac{\bar{z}-\bar{z}_1}{\bar{z}_2-\bar{z}_1} \\<br/>&amp; \text { or }\left|\begin{array}{cc}<br/>z-z_1 &amp; \bar{z}-\bar{z}_1 \\<br/>z_2-z_1 &amp; \bar{z}_2-\bar{z}_1<br/>\end{array}\right|=0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>$\therefore$ Option (c) is correct.<br/>Hence, (a, c, d) is the correct option.</div>


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-z-be-a-complex-number-such-that-the-imaginary-part-of-z-is-nonzero-and-a-z-2-z-1-is-real-then-a-cannot-take-the-value

slug:
let-z-be-a-complex-number-such-that-the-imaginary-part-of-z-is-nonzero-and-a-z-2-z-1-is-real-then-a-cannot-take-the-value-48085

question:
<div class="question">Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a=z^{2}+z+1$ is real. Then a cannot take the value</div>

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

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

answer:
<div class="solution">$\because \quad \operatorname{Im}(z) \neq 0$ $\Rightarrow \quad z$ is non-real
<br/> <br/>and equation $z^{2}+z+(1-a)=0$
<br/> <br/>will have non-real roots, if $D &lt; 0$
<br/> <br/>$\Rightarrow \quad 1-4(1-a) &lt; 0$ $\Rightarrow \quad 4 a &lt; 3 \Rightarrow a &lt; \frac{3}{4}$
<br/> <br/>$\therefore \quad a$ can not take the value $\frac{3}{4}$.</div>


title: let-z-cos-i-sin-then-the-value-of-m-1-15-im-z-2-m-1-at-2-is
slug: let-z-cos-i-sin-then-the-value-of-m-1-15-im-z-2-m-1-at-2-is-49013
tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Let $z=\cos \theta+i \sin \theta$. Then, the value of $\sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right)$ at $\theta=2^{\circ}$ is</div>

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

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

answer:
<div class="solution">Given that $z=\cos \theta+i \sin \theta=e^{i \theta}$<br/>$$<br/>\begin{aligned}<br/>\therefore \sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right) &amp; =\sum_{m=1}^{15} \operatorname{Im}\left(e^{i \theta}\right)^{2 m-1}=\sum_{m=1}^{15} \operatorname{Im} e^{i(2 m-1) \theta} \\<br/>&amp; =\sin \theta+\sin 3 \theta+\sin 5 \theta+\ldots+\sin 29 \theta \\<br/>&amp; =\frac{\sin \left(\frac{\theta+29 \theta}{2}\right) \sin \left(\frac{15 \times 2 \theta}{2}\right)}{\sin \left(\frac{2 \theta}{2}\right)}=\frac{\sin (15 \theta) \sin (15 \theta)}{\sin \theta}=\frac{1}{4 \sin 2^{\circ}}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
let-z-x-i-y-be-a-complex-number-where-x-and-y-are-integers-then-the-area-of-the-rectangle-whose-vertices-are-the-roots-of-the-equation-z-z-3-z-z-3-350

slug:
let-z-x-i-y-be-a-complex-number-where-x-and-y-are-integers-then-the-area-of-the-rectangle-whose-vertices-are-the-roots-of-the-equation-z-z-3-z-z-3-350-55502

question:
<div class="question">Let $z=x+i y$ be a complex number, where $x$ and $y$ are integers. Then, the area of the rectangle whose vertices are the roots of the equation $z \bar{z}^3+\bar{z} z^3=350$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>48<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>32<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>40<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>80</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>48<br/></span> </div>

answer:
<div class="solution">Since, $z \bar{z}\left(z^2+\bar{z}^2\right)=250$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad 2\left(x^2+y^2\right)\left(x^2-y^2\right)=350 \\<br/>&amp; \Rightarrow \quad\left(x^2+y^2\right)\left(x^2-y^2\right)=175<br/>\end{aligned}<br/>$$<br/>Since, $x, y \in I$, the only possible case which gives integral solution, is and $\quad \begin{aligned} x^2+y^2 &amp; =25 \\ x^2-y^2 &amp; =7\end{aligned}$<br/>From Eqs. (i) and (ii), we get $x^2=16 ; y^2=9 \Rightarrow x=\pm 4 ; y=\pm 3$<br/>$\therefore$ Area of rectangle $=8 \times 6=48$</div>


title: lim-x-4-x-2-16-2-2-s-e-c-2-x-f-t-d-t-equals
slug: lim-x-4-x-2-16-2-2-s-e-c-2-x-f-t-d-t-equals-27793
tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{8}{\pi} f(2)$<br/></span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow \frac{x}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ $\left[\therefore \frac{0}{0}\right.$ form $]$<br/>Let<br/>$$<br/>\begin{array}{ll}<br/>\text { Let } &amp; L=\lim _{x \rightarrow \frac{x}{4}} \frac{f\left(\sec ^2 x\right) 2 \sec x \sec x \tan x}{2 x} \\<br/>\therefore &amp; L=\frac{2 f(2)}{\pi / 4}=\frac{8 f(2)}{\pi}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
lines-l-1-y-x-0-and-l-2-2-x-y-0-intersect-the-line-l-3-y-2-0-at-p-and-q-respectively-the-bisector-of-the-acute-angle-between-l-1-and-l-2-intersects-l-

slug:
lines-l-1-y-x-0-and-l-2-2-x-y-0-intersect-the-line-l-3-y-2-0-at-p-and-q-respectively-the-bisector-of-the-acute-angle-between-l-1-and-l-2-intersects-l-69591

question:
<div class="question">Lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.<br/>Statement I The ratio $P R: R Q$ equals $2 \sqrt{2}: \sqrt{5}$.<br/>Statement II In any triangle, bisector of an angle divides the triangle into two similar triangles.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I<br/></span> </div>

answer:
<div class="solution">In $\triangle O P Q$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/39h5Z-uPmRqp6qXFqjROF92ujcwMsaH5BCQ92r-Xcbk.original.fullsize.png"/><br/><br/>Clearly,<br/>$$<br/>\frac{P R}{R Q}=\frac{O P}{O Q}=\frac{2 \sqrt{2}}{\sqrt{5}}<br/>$$</div>


tags: ['Physics', 'Work Power Energy', 'JEE Advanced', 'JEE Advanced 2022 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
list-i-describes-four-systems-each-with-two-particles-a-and-b-in-relative-motion-as-shown-in-figure-list-ii-gives-possible-magnitudes-of-their-relativ

slug:
list-i-describes-four-systems-each-with-two-particles-a-and-b-in-relative-motion-as-shown-in-figure-list-ii-gives-possible-magnitudes-of-their-relativ-99030

question:
<div class="question"><p>List I describes four systems, each with two particles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>) at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math>.</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"><strong>List-I</strong></td> <td style="text-align: center;"> </td> <td style="text-align: center;"><strong>List-II</strong></td> </tr> <tr> <td>(I)</td> <td> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are moving on a horizontal circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> with uniform angular speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo> </mo><mi>rad</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. The initial angular positions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math> respectively.</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ff049b50-ab5e-454c-8c66-cb6a46d2edb2-image.png" style="width: 140px; height: 128px;"/></p> </td> <td>(P)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></math></td> </tr> <tr> <td>(II)</td> <td> <p>Projectiles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are fired (in the same vertical plane) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math> respectively, with the same speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math> from the horizontal plane. The initial separation between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is large enough so that they do not collide, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math>.</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/8c0f53b4-2b00-4b3b-b08e-11aa472a0f03-image.png" style="width: 200px; height: 146px;"/></p> </td> <td>(Q)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mfenced><mrow><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced><msqrt><mn>2</mn></msqrt></mfrac></math></td> </tr> <tr> <td>(III)</td> <td> <p>Two harmonic oscillators <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> moving in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction according to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>A</mi></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfrac><mi>t</mi><msub><mi>t</mi><mn>0</mn></msub></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>B</mi></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfenced><mrow><mfrac><mi>t</mi><msub><mi>t</mi><mn>0</mn></msub></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></mfenced></math> respectively, starting from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>. Take <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>,</mo><msub><mi>t</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math>.</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/0e65e96f-c87f-4250-9818-e1145eb3b0a7-image.png" style="width: 280px; height: 145px;"/></p> </td> <td>(R)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>10</mn></msqrt></math></td> </tr> <tr> <td>(IV)</td> <td> <p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is rotating in a horizontal circular path of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math> plane, with constant angular speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo> </mo><mi>rad</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is moving up at a constant speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in the vertical direction as shown in the figure. (Ignore gravity.)</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/d19263f1-da07-447d-8586-560b6286677b-image.png" style="width: 190px; height: 123px;"/></p> </td> <td>(S)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt></math></td> </tr> <tr> <td> </td> <td> </td> <td>(T)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>25</mn><msup><mi>π</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math></td> </tr> </tbody></table><p>Which one of the following options is correct?</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">R</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">S</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">Q</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">R</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">R</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">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">R</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">S</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 mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">R</mi></math></span> </div>

answer:
<div class="solution"><blockquote><p>(I)</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/90b87b12-93ae-4b20-a396-6511d7ecc0da-image.png" style="width: 180px; height: 166px;"/></p><p>Angular velocity of both the particles is same. Therefore, angle between them will remain <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>v</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub></mfenced><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>ω</mi><mi>R</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi></math></p><p>(II)</p><p>The velocity of A at any time t can be written as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac><mi>cos</mi><mn>45</mn><mo>°</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced close="]" open="["><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo>×</mo><mi>sin</mi><mn>45</mn><mo>°</mo><mo>-</mo><mi>g</mi><mo>×</mo><mi>t</mi></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>g</mi><mi>t</mi></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p>Velocity of B at any time t</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><mo>-</mo><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>g</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p>Relative velocity at time t will be,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>v</mi><mo>→</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub><mo>=</mo><mn>5</mn><mi>π</mi><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mi>g</mi><mo>×</mo><mn>0</mn><mo>.</mo><mn>1</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mover><mi>v</mi><mo>→</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub></mfenced><mo>=</mo><msqrt><mn>25</mn><msup><mi>π</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>II</mi><mo>→</mo><mi>T</mi></math></p><p>(III) </p><p>Relative position of A wrt B can be written as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msub><mi>x</mi><mi>A</mi></msub><mo>-</mo><msub><mi>x</mi><mi>B</mi></msub></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mi>t</mi><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfenced><mrow><mi>t</mi><mo>+</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></mfenced></math></p><p>Using trigonometric relation</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p><p>Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>rel</mi></msub><mo>=</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><msub><mi>x</mi><mn>0</mn></msub><mi>cos</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p><p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mfenced><mrow><mfrac><mi>π</mi><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>×</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>=</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>III</mi><mo>→</mo><mi>P</mi></math></p><p>(IV) </p><p>The velocity of A is in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math> plane and it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>v</mi><mi>A</mi></msub></mfenced><mo>=</mo><mi>ω</mi><mi>r</mi><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and the velocity of B is along <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> axis and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>v</mi><mi>B</mi></msub></mfenced><mo>=</mo><mn>3</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>rel</mi></msub><mo>=</mo><msqrt><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mn>10</mn></msqrt><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>IV</mi><mo>→</mo><mi>R</mi></math></p></blockquote></div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-1

slug:
look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-1-67535

question:
<div class="question">Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xNFfaYiituH7Vz9vRPx1wgTgYD4I1Vbby5q7GRnc9Fs.original.fullsize.png"/><br/><br/>The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6 \mathrm{~m}$. The coordinates of the centres of the different parts are, outer circle 10 , $0)$, left inner circle $(0,0)$, right inner circle $(a, a)$, vertical line $(0,0)$ and horizontal line $(0,-a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{a}{10}$<br/></span> </div>

answer:
<div class="solution">$y_{C M}$<br/>$$<br/>\begin{gathered}<br/>=\frac{m_1 y_1+m_2 y_2+m_3 y_3+m_4 y_4+m_5 y_5}{m_1+m_2+m_3+m_4+m_5} \\<br/>=\frac{(6 m)(0)+(m)(a)+m(a)+m(0)}{6 m+m+m+m+m}=\frac{a}{10}<br/>\end{gathered}<br/>$$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-2

slug:
look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-2-65763

question:
<div class="question">Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xNFfaYiituH7Vz9vRPx1wgTgYD4I1Vbby5q7GRnc9Fs.original.fullsize.png"/><br/><br/>The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6 \mathrm{~m}$. The coordinates of the centres of the different parts are, outer circle 10 , $0)$, left inner circle $(0,0)$, right inner circle $(a, a)$, vertical line $(0,0)$ and horizontal line $(0,-a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{a}{10}$<br/></span> </div>

answer:
<div class="solution">$y_{C M}$<br/>$$<br/>\begin{gathered}<br/>=\frac{m_1 y_1+m_2 y_2+m_3 y_3+m_4 y_4+m_5 y_5}{m_1+m_2+m_3+m_4+m_5} \\<br/>=\frac{(6 m)(0)+(m)(a)+m(a)+m(0)}{6 m+m+m+m+m}=\frac{a}{10}<br/>\end{gathered}<br/>$$</div>


title: mark-the-incorrect-option-1
slug: mark-the-incorrect-option-1-25833
question: <div class="question">Mark the incorrect option.</div>
tags: ['Biology', 'Animal Kingdom', 'NEET']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The larva of <em>Scoliodon</em> exists.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Lateral line is absent in <em>Bangarus.</em></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Air bladder is present in <em>Betta.</em></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All chordates are deuterostomes.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The larva of <em>Scoliodon</em> exists.</span> </div>

answer:
<div class="solution"><p style="text-align: justify;"><em>Scoliodon</em> is a predator of small bony fishes and invertebrates. This species exhibits the most advanced mode of viviparity of any fish, in which the developed embryos form a highly complex placental connection to the mother at a very small size.</p></div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
match-each-of-the-compounds-given-in-column-i-with-the-reaction-s-that-they-can-undergo-given-in-column-ii

slug:
match-each-of-the-compounds-given-in-column-i-with-the-reaction-s-that-they-can-undergo-given-in-column-ii-19907

question:
<div class="question">Match each of the compounds given in Column I with the reaction (s) that they can undergo, given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/emrlHMSRSxJBfoD-QedlMy-2bSSQ9SuGiI61DsRkYMY.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,s, (B) q,s, (C) r,t, (D) q,t<br/></span> </div>


title: match-each-of-the-compounds-is-column-i-with-its-characteristic-reactions-is-column-ii
slug: match-each-of-the-compounds-is-column-i-with-its-characteristic-reactions-is-column-ii-19219
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
question:
<div class="question">$$<br/>\text { Match each of the compounds is Column I with its characteristic reaction(s) is Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fjpzYZkIB7A3picoT0unS5p28yTVroAGAOEWZI8JLYs.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,s,t, (B) p,s,t, (C) p, (D) r<br/></span> </div>


title: match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii-1
slug: match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii-1-27838
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Main']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Match each of the diatomic molecules in Column I with its property/properties in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FPjqA-UDTF1p1emxoGgSvmMJsFjrHiranmdfbinxMBY.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,r,t, (B) s,t, (C) p,q,r, (D) p,r,s<br/></span> </div>


title: match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii
slug: match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii-90686
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
question:
<div class="question">Match each of the diatomic molecules in Column I with its property/properties in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FPjqA-UDTF1p1emxoGgSvmMJsFjrHiranmdfbinxMBY.original.fullsize.png"/><br/></div>

tags:
['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,r,t, (B) s,t, (C) p,q,r, (D) p,r,s<br/></span> </div>


title: match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii-1
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Main']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
slug:
match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii-1-45944

question:
<div class="question">Match each of the reactions given in Column I with the corresponding products(s) given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/V3GuxJAZ7_VpCvRcy3XmvJlvyEmLatpDGSsk2j_7RAw.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,t, (B) p,s,t, (C) r,s, (D) p<br/></span> </div>


title: match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
slug:
match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii-49904

question:
<div class="question">Match each of the reactions given in Column I with the corresponding products(s) given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/V3GuxJAZ7_VpCvRcy3XmvJlvyEmLatpDGSsk2j_7RAw.original.fullsize.png"/><br/></div>

tags:
['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,t, (B) p,s,t, (C) r,s, (D) p<br/></span> </div>


tags: ['Chemistry', 'Polymers', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bond-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-4

slug:
match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bond-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-4-22019

question:
<div class="question">Match the chemical substance in Column I with type of polymers/type of bond in Column II. Indicate your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/aBcz_-koYhRdo9Zs0JRtXZYX4tgTBARB13Q2LAWna7c.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-q, r; C-p, r; D-s<br/></span> </div>

answer:
<div class="solution">(A) $p, s$<br/>(B) $q, r$<br/>(C) $p, r$<br/>(D) $s$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/0u0UXSHYX2s-C2PZe6qS4EkMFVTL9LWZNY5ja52juUQ.original.fullsize.png"/><br/><br/>Sucrose (diasaccharide) &lt; smiles&gt;OCC1OC(CO)C(O)C(O)C1O &lt; /smiles&gt;<br/>Sucrose/glucosidic as well as fructosidic linkage</div>


tags: ['Chemistry', 'General Organic Chemistry', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of--1

slug:
match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-1-75635

question:
<div class="question">Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/w4x9XLhiCV8vUdID4JoLSlbgktj2IlXjSR4y-LbxBJs.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, q, s; B-q; C-r, s, s; D-r<br/></span> </div>

answer:
<div class="solution">A-p, $q, s, \quad \mathrm{~B}-q, \quad \mathrm{C}-r, s, \quad \mathrm{D}-r$</div>


tags: ['Chemistry', 'General Organic Chemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-

slug:
match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-76065

question:
<div class="question">Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/w4x9XLhiCV8vUdID4JoLSlbgktj2IlXjSR4y-LbxBJs.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, q, s; B-q; C-r, s, s; D-r<br/></span> </div>

answer:
<div class="solution">A-p, $q, s, \quad \mathrm{~B}-q, \quad \mathrm{C}-r, s, \quad \mathrm{D}-r$</div>


title: match-the-column-normals-at-p-q-r-are-drawn-to-y-2-4-x-which-intersect-at-3-0-then
slug: match-the-column-normals-at-p-q-r-are-drawn-to-y-2-4-x-which-intersect-at-3-0-then-58100
tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the column.<br/>Normals at $P, Q, R$ are drawn to $y^2=4 x$ which intersect at $(3,0)$. Then,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/V8pddult167dBNpS7Z8V5Cw_zSQ4Lu0oQfWh1qEbXzw.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (a), (ii) (b), (iii) (d), (iv) (c)<br/></span> </div>

answer:
<div class="solution">Since, equation of normal is $y+x t=2 a t+a t^3$ passes through $(3,0)$.<br/>$$<br/>\Rightarrow \quad 3 t=2 t+t^3 \Rightarrow t=0,1,-1 \text {. }<br/>$$<br/>$\therefore P(1,2), Q(0,0), R(1,-2)$. Thus,<br/>(i) Area of $\triangle P Q R=\frac{1}{2} \times 1 \times 4=2$<br/>(ii) Centroid of $\triangle P Q R=\left(\frac{2}{3}, 0\right)$<br/>Equation of circle passing through $P, Q, R$ is<br/>$$<br/>\begin{aligned}<br/>&amp; \quad(x-1)(x-1)+(y-2)(y+2)+\lambda(x-1)=0 \\<br/>&amp; \Rightarrow \quad 1-4-\lambda=0 \Rightarrow \lambda=-3 \\<br/>&amp; \therefore \text { Required equation of circle is } x^2+y^2-5 x-1=0 \\<br/>&amp; \therefore \text { Centre }\left(\frac{5}{2}, 0\right) \text { and radius } \frac{5}{2} .<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix--1

slug:
match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-1-89933

question:
<div class="question">Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u3ZyoDHANLps94g_Bgxxf_Fo4PLbZbkKoAmBhvW_0C8.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-r; C-p, q; D-p, s</span> </div>

answer:
<div class="solution">A-p,s B-r, C-p,q, D-p,s,<br/>$$<br/>\begin{aligned}<br/>2 \mathrm{CrO}_4^{2-} \longrightarrow \mathrm{Cr}_2 \mathrm{O}_7^{2-} \\<br/>\mathrm{MnO}_4^{-}+\mathrm{NO}_2^{-}+\mathrm{H}^{+} \longrightarrow \mathrm{Mn}^{2+}+\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{O} \\<br/>\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{SO}_4+\mathrm{Fe}^{2+} \longrightarrow \mathrm{Fe}^{3+}+\mathrm{NO}+\mathrm{SO}_4^{2-}+\mathrm{H}_2 \mathrm{O}<br/>\end{aligned}<br/>$$</div>


