# questions

title: let-f-x-be-a-differentiable-function-and-f-4-5-then-lim-x-2-x-2-f-4-f-x-2-equals
slug: let-f-x-be-a-differentiable-function-and-f-4-5-then-lim-x-2-x-2-f-4-f-x-2-equals-60104
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-20</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)$ be a differentiable function and $f^{\prime}(4)=5 .$ Then, $\lim _{x \rightarrow 2} \frac{f(4) - f\left(x^{2}\right)}{x-2}$ equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">20</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">-20</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, $f^{\prime}(4)=5$
<br/>$$
<br/>\begin{array}{ll}
<br/>\text { Now, } \lim _{x \rightarrow 2} \frac{f(4)-f\left(x^{2}\right)}{x-2} &amp; {\left[\frac{0}{0} \text { form }\right]} \\
<br/>=\lim _{x \rightarrow 2} \frac{0-f^{\prime}\left(x^{2}\right) \cdot 2 x}{1} \\
<br/>=\frac{-f^{\prime}(4) \cdot 2 \times 2}{1} \\
<br/>=-(5) \times 4=-20
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">28</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-f-x-be-a-differentiable-function-in-2-7-if-f-2-3-and-f-x-5-for-all-x-in-2-7-then-the-maximum-posaible-value-of-f-x-at-x-7-is

slug:
let-f-x-be-a-differentiable-function-in-2-7-if-f-2-3-and-f-x-5-for-all-x-in-2-7-then-the-maximum-posaible-value-of-f-x-at-x-7-is-72641

question:
<div class="question">Let $f(x)$ be a differentiable function in $[2,7]$. If $f(2)=3$ and $f^{\prime}(x) \leq 5$ for all $x$ in $(2,7),$ then the maximum posaible value of $f(x)$ at $x=7$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">15</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">28</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">14</span> </li> </ul>

answer:
<div class="solution">Given, $\quad f^{\prime}(x) \leq 5$
<br/>$\frac{f(7)-f(2)}{x-2} \leq 5$
<br/>(by Lagrange mean value theorem] $\begin{array}{ll}\Rightarrow &amp; \frac{f(7)-3}{7-2} \leq 5 \\ \Rightarrow &amp; f(7) \leq 5(7-2)+3 \\ \therefore &amp; f(7) \leq 28\end{array}$</div>


title: let-f-x-cos-x-x-0-then-assuming-k-as-an-integer
slug: let-f-x-cos-x-x-0-then-assuming-k-as-an-integer-21569
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\cos \left(\frac{\pi}{x}\right), x \neq 0,$ then assuming $k$ as an integer,</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f(x)$ increases in the interval $\left(\frac{1}{2 k+1}, \frac{1}{2 k}\right)$, $f(x)$ decreases in the interval $\left(\frac{1}{2 k+2}, \frac{1}{2 k+1}\right)$</span> </div>

answer:
<div class="solution">$f(x)=\cos \left(\frac{\pi}{x}\right)$
<br/>$\Rightarrow \quad f'(x)=-\sin \left(\frac{\pi}{x}\right)\left(\frac{-\pi}{x^{2}}\right)=\frac{\pi}{x^{2}} \sin \frac{\pi}{x}$
<br/>For increasing function, $f'(x)&gt;0$ $\Rightarrow \sin \left(\frac{\pi}{x}\right)&gt;0 \Rightarrow 2 k \pi    &lt; \frac{\pi}{x} &lt; (2 k+1) \pi$
<br/>$\Rightarrow \quad \frac{1}{2 k}&gt;x&gt;\frac{1}{2 k+1}$
<br/>For decreasing function, $f(x) &lt; 0$ $\Rightarrow \quad \sin \left(\frac{\pi}{x}\right) &lt; 0$
<br/>$\Rightarrow \frac{\pi}{x} \in[(2 k+1) \pi,(2 k+2) \pi] \Rightarrow x \in\left(\frac{1}{2 k+2}, \frac{1}{2 k+1}\right)$</div>


title: let-f-x-denotes-the-fractional-part-of-a-real-number-x-then-the-value-of-0-3-f-x-2-d-x-is
slug: let-f-x-denotes-the-fractional-part-of-a-real-number-x-then-the-value-of-0-3-f-x-2-d-x-is-55890
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)$ denotes the fractional part of a real number $x$. Then, the value of $\int_{0}^{\sqrt{3}} f\left(x^{2}\right) d x$ is</div>

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

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

answer:
<div class="solution">Let $I=\int_{0}^{\sqrt{3}} f\left(x^{2}\right) d x=\int_{0}^{\sqrt{3}}\left\{x^{2}\right\} d x$
<br/>$\left.=\int_{0}^{\sqrt{3}}\left(x^{2}-\mid x^{2}\right)\right) d x$
<br/>$=\int_{0}^{\sqrt{3}} x^{2} d x-\int_{0}^{\sqrt{3}}\left|x^{2}\right| d x$
<br/>$=\left[\frac{x^{3}}{3}\right]_{0}^{\sqrt{3}}-1 \int_{0}^{1}\left|x^{2}\right| d x+\int_{1}^{\sqrt{2}}\left|x^{2}\right| d x+\int_{\sqrt{2}}\left|x^{2}\right| d x \mid$
<br/>$=\sqrt{3}-\left[\int_{0}^{1} 0 d x+\int_{1}^{\sqrt{2}} 1 d x+\int_{\sqrt{2}}^{\sqrt{3}} 2 d x\right]$
<br/>$=\sqrt{3}-\left\{0+|x|_{i}^{\sqrt{2}}+|2 x| \sqrt{2}\right\}$
<br/>$=\sqrt{3}-\sqrt{2}+1-2 \sqrt{3}+2 \sqrt{2}$
<br/>$=\sqrt{2}-\sqrt{3}+1$</div>


title: let-f-x-e-x-g-x-e-x-and-h-x-g-f-x-where-x-is-a-real-variable-then-d-x-d-h-at-x-0-is
slug: let-f-x-e-x-g-x-e-x-and-h-x-g-f-x-where-x-is-a-real-variable-then-d-x-d-h-at-x-0-is-40928
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $F(x)=e^{x}, G(x)=e^{-x}$ and $H(x)=G(F(x)),$ where $x$ is a real variable. Then, $\frac{d H}{d x}$ at $x=0$ is</div>

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

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

answer:
<div class="solution">We have, $F(x)=e^{x}, G(x)=e^{-x}$ $\therefore \quad
<br/>H(x)=G(F(x))$ $\begin{aligned} &amp;=G\left(e^{x}\right) \\ &amp;=e^{-e^{x}} \\ \frac{d H}{d x} &amp;=e^{-e^{x}} \cdot\left(-e^{x}\right) \\ &amp;=-e^{x} e^{-e^{x}} \end{aligned}$
<br/>$\left.\therefore \quad \frac{d H}{d x}\right|_{x=0}=-e^{0} \cdot e^{-e^{0}}$
<br/>$=-1 . e^{-1}$
<br/>$=-e^{-1}$
<br/>$=\frac{-1}{e}$</div>


title: let-f-x-max-x-x-x-x-where-x-denotes-the-greateat-integer-x-then-the-value-of-3-3-f-x-d-x-is
slug: let-f-x-max-x-x-x-x-where-x-denotes-the-greateat-integer-x-then-the-value-of-3-3-f-x-d-x-is-46361
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\max \{x+|x|, x-[x]\}, \quad$ where $\quad \mid x]$ denotes the greateat integer $\leq x$. Then, the value of $\int_{-3}^{3} f(x) d x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$51 / 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$21 / 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$21 / 2$</span> </div>

answer:
<div class="solution">Given, $f(x)=\max \{x+|x|, x-[x]\}$
<br/>$$
<br/>\begin{aligned}
<br/>=\left\{\begin{array}{ll}
<br/>2 x, &amp; x \geq 0 \\
<br/>x-[x], &amp; x \leq 0
<br/>\end{array}\right.\\
<br/>\begin{aligned}
<br/>\int_{-3}^{3} f(x) d x &amp;=\int_{-3}^{0} x-[x] d x+\int_{0}^{3} 2 x d x \\
<br/>&amp;=3 \int_{-1}^{0}(1+x) d x+2 \int_{0}^{3} x d x
<br/>\end{aligned}
<br/>\end{aligned}
<br/>$$
<br/>[$\because$ $x-[x]$ is a periodic function at $x=1]$
<br/>$$
<br/>\begin{array}{l}
<br/>=3\left[x+\frac{x^{2}}{2}\right]_{-1}^{0}+2\left[\frac{x^{2}}{2}\right]_{0}^{3} \\
<br/>=3\left[0-0-\left(-1+\frac{1}{2}\right)\right]+3^{2}-0 \\
<br/>=3\left[\frac{1}{2}\right]+9=\frac{3}{2}+9 \\
<br/>=\frac{21}{2}
<br/>\end{array}
<br/>$$</div>


title: let-f-x-s-i-n-x-c-o-s-x-e-t-2-d-t-then-f-4-equals
slug: let-f-x-s-i-n-x-c-o-s-x-e-t-2-d-t-then-f-4-equals-61880
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\int_{\sin x}^{\cos x} e^{-t^2} d t$. Then $f^{\prime}\left(\frac{\pi}{4}\right)$ equals</div>

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

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

answer:
<div class="solution">$f^{\prime}(x)=-e^{-\cos ^2 x} \sin x-e^{-\sin ^2 x} \cos x$<br/>$\therefore f^{\prime}\left(\frac{\pi}{4}\right)=-e^{-\frac{1}{2}} \frac{1}{\sqrt{2}}-e^{-\frac{1}{2}} \frac{1}{\sqrt{2}}=-\frac{2}{\sqrt{2 e}}=-\sqrt{\frac{2}{e}}$</div>


tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-f-x-s-i-n-x-q-x-p-0-if-0-x-2-if-x-0-p-q-r-then-lagranges-mean-value-theorem-is-applicable-to-f-x-in-closed-interval-0-x

slug:
let-f-x-s-i-n-x-q-x-p-0-if-0-x-2-if-x-0-p-q-r-then-lagranges-mean-value-theorem-is-applicable-to-f-x-in-closed-interval-0-x-36226

question:
<div class="question">Let $f(x)=\left\{\begin{array}{ccc}\frac{x^{p}}{(\sin x)^{q}} &amp; , &amp; \text { if } 0    &lt; x \leq \frac{\pi}{2} \\ 0 &amp; , &amp; \text { if } x=0\end{array}\right.$
<br/>$(p, q \in R) .$ Then, Lagrange's mean value theorem is applicable to $f(x)$ in closed interval $[0, x]$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">only when $p&gt;q$</span> </div>

answer:
<div class="solution">Since, Lagrange's mean value theorem is applicable on
<br/>$\begin{array}{ll}\therefore &amp; \lim _{x \rightarrow 0} \frac{x^{P}}{(\sin x)^{q}}=f(0) \\ \Rightarrow \quad &amp; \lim _{x \rightarrow 6} \frac{x^{p}}{(\sin x)^{q}}=0\end{array}$
<br/>Above equation holds only when $p&gt;q$.</div>


title: let-f-x-sin-x-cos-a-x-be-periodic-function-then
slug: let-f-x-sin-x-cos-a-x-be-periodic-function-then-44678
question: <div class="question">Let $f(x)=\sin x+\cos a x$ be periodic function. Then</div>
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2020']
answer: <div class="solution">Hint: <br/>LCM of rational multiple of same irrational is defined</div>
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">'a' is any real number</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">'a' is any irrational number</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">'a' is rational number</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a=0$</span> </li> </ul>

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


title: let-f-x-x-1-x-3-then-the-value-of-x-3-0-lt-f-x-is
slug: let-f-x-x-1-x-3-then-the-value-of-x-3-0-lt-f-x-is-46203
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">Hints : Because on left hand side of 3 function is not defined.</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\frac{\sqrt{x+3}}{x+1}$ then the value of $\underset{x \rightarrow-3-0}{\operatorname{Lt}} f(x)$ is</div>

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

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


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
let-f-x-x-2-3-x-2-and-g-x-x-be-two-given-functions-if-s-be-the-domain-of-f-g-and-t-be-the-domain-of-g-of-then

slug:
let-f-x-x-2-3-x-2-and-g-x-x-be-two-given-functions-if-s-be-the-domain-of-f-g-and-t-be-the-domain-of-g-of-then-31668

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}$ and $\mathrm{g}(\mathrm{x})=\sqrt{\mathrm{x}}$ be two given functions. If $\mathrm{S}$ be the domain of $\mathrm{f} \circ \mathrm{g}$ and $\mathrm{T}$ be the domain of $\mathrm{g}$ of then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{S} \cap \mathrm{T}$ is an interval</span> </div>

answer:
<div class="solution">Hint: 
<br/>$f(x)=\sqrt{(x-1)(x-2)}, g(x)=\sqrt{x}$
<br/>$f(g(x))=\sqrt{(\sqrt{x}-1)(\sqrt{x}-2)}$
<br/>$S=\{x: x \in[0,1] \cup[4, \infty)\}$
<br/>$g(f(x))=\sqrt{\sqrt{(x-1)(x-2)}}$
<br/>$T=\{x: x \in(-\infty, 1] \cup[2, \infty)\}$</div>


title: let-f-x-x-2-17-x-5-24-then
slug: let-f-x-x-2-17-x-5-24-then-53028
question: <div class="question">Let $f(x)=(x-2)^{17}(x+5)^{24}$. Then</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f$ does not have a critical point at $x=2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f$ has a minimum at $x=2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f$ has neither a maximum nor a minimum 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">D</span> <span class="option-data">f has a maximum at $x=2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ has neither a maximum nor a minimum at $x=2$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; f(x)=(x-2)^{17}(x+5)^{24} \\ \Rightarrow &amp; f^{\prime}(x)=17(x-2)^{16}(x+5)^{24}+24(x-2)^{17}(x+5)^{23}=(x-2)^{16}(x+5)^{23}(17 x+85+24 x-48)=(x-2)^{16}(x+5)^{23}(41 x+37) \end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/FTARjmXMyCr7h61cW37DaJd7sHAFhwU-Vp0t6btf-Oc.original.fullsize.png"/></div>


title: let-f-x-x-2-x-sin-x-cos-x-then
slug: let-f-x-x-2-x-sin-x-cos-x-then-39704
question: <div class="question">Let $f(x)=x^2+x \sin x-\cos x$. Then</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(x)=0$ has at least one real root</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(x)=0$ has no real root</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f(x)=0$ has at least one positive root</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">$f(x)=0$ has at least one negative root</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$f(x)=0$ has at least one real root, $f(x)=0$ has at least one positive root, $f(x)=0$ has at least one negative root</span> </div>

answer:
<div class="solution">$f^{\prime}(x)=2 x+x \cos x+\sin x+\sin x=2(x+\sin x)+x \cos x$<br/>$\Rightarrow f^{\prime}(x) &gt; 0 \quad \forall x &gt; 0, \quad f^{\prime}(x)  &lt; 0 \quad \forall x &lt; 0 \Rightarrow f(0)=-1$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6lWKwjS99W4DsQpgke3d5yfY-t6on1RTAXyZ4WA73qg.original.fullsize.png"/></div>


title: let-f-x-x-4-4-x-3-4-x-2-c-c-r-then
slug: let-f-x-x-4-4-x-3-4-x-2-c-c-r-then-61607
question: <div class="question">Let $f(x)=x^{4}-4 x^{3}+4 x^{2}+c, c \in R .$ Then,</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)$ has infinitely many zeros in (1,2) for all $c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f(x)$ has exactly one zero in (1,2) if $-1    &lt; c &lt; 0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(x)$ has double zeros in (1,2) if $-1    &lt; c &lt; 0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">whatever be the value of $c, f(x)$ has no zero in (1,2)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)$ has exactly one zero in (1,2) if $-1    &lt; c &lt; 0$</span> </div>

answer:
<div class="solution">Given,
<br/>$$
<br/>\begin{array}{ll}
<br/>f(x) &amp; =x^{4}-4 x^{3}+4 x^{2}+c \\
<br/>f(1) &amp; =1+c \\
<br/>\text { and } \quad f(2) &amp; =2^{4}-4(2)^{3}+4(2)^{2}+c
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>\begin{aligned}
<br/>=&amp; 16-32+16+c \\
<br/>&amp;=c
<br/>\end{aligned} \\
<br/>\text { Now, } f(1) \cdot f(2)=c(1+c) \\
<br/>\text { Let } f(1) \cdot f(2)    &lt; 0 \\
<br/>\Rightarrow \quad c(1+c) &lt; 0 \Rightarrow c \in(-1,0)
<br/>\end{array}
<br/>$$
<br/>For $c \in(-1,0), f(x)$ has exactly one zero is (1,2)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-f-x-x-12-then-the-number-of-real-values-of-x-for-which-the-three-unequal-terms-f-x-f-2-x-f-4-x-are-in-hp-is

slug:
let-f-x-x-12-then-the-number-of-real-values-of-x-for-which-the-three-unequal-terms-f-x-f-2-x-f-4-x-are-in-hp-is-85785

question:
<div class="question">Let $f(x)=x+1 / 2$. Then, the number of real values of $x$ for which the three unequal terms $f(x), f(2 x), f(4 x)$ are in $\mathrm{HP}$ is</div>

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

answer:
<div class="solution">Given, $f(x)=x+\frac{1}{2}=\frac{2 x+1}{2}$
<br/>$\therefore$
<br/>$f(2 x)=2 x+\frac{1}{2}$
<br/>$\Rightarrow$
<br/>$f(2 x)=\frac{4 x+1}{2}$
<br/>and $f(4 x)=4 x+\frac{1}{2} \Rightarrow f(4 x)=\frac{8 x+1}{2}$
<br/>since, $f(x), f(2 x)$ and $f(4 x)$ are in $\mathrm{HP}$. $\frac{1}{f(x)} \cdot \frac{1}{f(2 x)}$ and $\frac{1}{f(4 x)}$ are in AP.
<br/>$\Rightarrow \quad \frac{1}{f(2 x)}=\frac{\frac{1}{f(x)}+\frac{1}{f(4 x)}}{2}$
<br/>$\Rightarrow \quad \frac{2}{4 x+1}=\frac{\frac{2}{2 x+1}+\frac{2}{8 x+1}}{2}$
<br/>$\Rightarrow \quad-\frac{2}{4 x+1}=\frac{10 x+2}{(2 x+1)(8 x+1)}$
<br/>$\Rightarrow(2 x+1)(8 x+1)=(5 x+1)(4 x+1)$
<br/>$\Rightarrow 16 x^{2}+10 x+1=20 x^{2}+9 x+1$
<br/>$\Rightarrow \quad 4 x^{2}-x=0 \Rightarrow x(4 x-1)=0$
<br/>$\Rightarrow \quad x=\frac{1}{4} \quad[\because x \neq 0$
<br/>Hence, one real value of $x$ for which the thrce unequal terms are in HP.</div>


title: let-f-x-x-13-x-11-x-9-x-7-x-5-x-3-x-12-then
slug: let-f-x-x-13-x-11-x-9-x-7-x-5-x-3-x-12-then-50679
question: <div class="question">Let $f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+12$. Then</div>
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)$ has 13 non-zero real roots</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ has exactly one real root</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ has exactly one pair of imaginary roots</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(x)$ has no real root</span> </li> </ul>

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

answer:
<div class="solution">Hint: 
<br/>$f^{\prime}(x)&gt;0 \quad \forall x \in R$
<br/>$f(x)=0$ has exactly one real root</div>


title: let-f-x-x-13-x-11-x-9-x-7-x-5-x-3-x-19-then-f-x-0-has
slug: let-f-x-x-13-x-11-x-9-x-7-x-5-x-3-x-19-then-f-x-0-has-52224
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}$
<br/>$+x^{3}+x+19 .$ Then, $f(x)=0$ has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">13 real roots</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">only one positive and only two negative real roots</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">not more than one real root</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">has two positive and one negative real root</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">not more than one real root</span> </div>

answer:
<div class="solution">We have, $f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+19$
<br/>$$
<br/>\begin{array}{r}
<br/>\Rightarrow f^{\prime}(x)=13 x^{12}+11 x^{10}+9 x^{4} \\
<br/>+7 x^{6}+5 x^{4}+3 x^{2}+1
<br/>\end{array}
<br/>$$
<br/>$f^{\prime}(x)$ has no real root.
<br/>$\therefore f(x)=0$ has not more than one real root.</div>


title: let-f-x-x-be-such-that-f-f-x-x-for-all-x-x-and-x-r-then
slug: let-f-x-x-be-such-that-f-f-x-x-for-all-x-x-and-x-r-then-94529
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: X \rightarrow X$ be such that $f[f(x)]=x,$ for all
<br/>$x \in X$ and $X \subseteq R,$ then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f$ is one-to-one, $f$ is onto</span> </div>

answer:
<div class="solution">Given, $f(f(x)]=x$ 
<br/>Now, $\quad f^{-1}(x)=f(x)$
<br/>Le. $f(x)$ is bijective. 
<br/>Hence, $f(x)$ has to be one-one and onto.</div>


title: let-f-x-y-and-a-b-are-nonvoid-subsets-of-y-then-where-the-symbols-have-their-usual-interpretation
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
slug:
let-f-x-y-and-a-b-are-nonvoid-subsets-of-y-then-where-the-symbols-have-their-usual-interpretation-76506

question:
<div class="question">Let $f: X \rightarrow Y$ and $A, B$ are non-void subsets of $Y$, then (where the symbols have their usual interpretation)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f^{-1}(A-B)=f^{-1}(A)-f^{-1}(B)$</span> </div>

answer:
<div class="solution">Let A={1,2,3}
<br/>and  B={2,3,4}
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a58.png"/> <br/>here, $f^{-1}(3)=1$
<br/>$f^{-1}(4)=2$ and $f^{-1}(2)=3$
<br/>It is clear that option (C) $f^{-1}(A-B)=f^{-1}(A)$
<br/>$-f^{-1}(B)$ is correct.</div>


title: let-f13-rightarrow-mathbbr-be-continuous-and-be-derivable-in-13-and-fprimexfx24-forall-x-in13-then
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
slug:
let-f13-rightarrow-mathbbr-be-continuous-and-be-derivable-in-13-and-fprimexfx24-forall-x-in13-then-92394

question:
<div class="question">Let \(f:[1,3] \rightarrow \mathbb{R}\) be continuous and be derivable in \((1,3)\) and \(f^{\prime}(x)=[f(x)]^2+4 \forall x \in(1,3)\). Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(f(3)-f(1)=5\) holds</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(f(3)-f(1)=5\) does not hold</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(f(3)-f(1)=3\) holds</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(f(3)-f(1)=4\) holds</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(f(3)-f(1)=5\) does not hold</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hint : } f^{\prime}(c)=(f(c))^2+4=\frac{f(3)-f(1)}{2} \\<br/>&amp; \Rightarrow f(3)-f(1)=2(f(c))^2+8<br/>\end{aligned}\)</div>


title: let-for-all-x-0-f-x-lim-n-n-x-1-n-1-then
slug: let-for-all-x-0-f-x-lim-n-n-x-1-n-1-then-55891
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let for all $x&gt;0, f(x)=\lim _{n \rightarrow \infty} n\left(x^{1 / n}-1\right),$ then</div>

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

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

answer:
<div class="solution">We have, $f(x)=\lim _{n \rightarrow \infty} n\left(x^{1 / n}-1\right)$
<br/>$=\lim _{n \rightarrow \infty} \frac{x^{1 / n}-1}{1 / n}$
<br/>Let $\quad \frac{1}{n}=y$
<br/>$\begin{aligned} \therefore \quad f(x)=&amp; \lim _{y \rightarrow 0} \frac{x^{y}-1}{y} \\ &amp;=\log x \end{aligned}$
<br/>$\therefore \quad f(x y)=\log (x, y)$
<br/>$=\log x+\log y$
<br/>$=f(x)+f(y)$</div>


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

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

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

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


title: let-fxleftx2right-sin-pi-x-x-0-then
slug: let-fxleftx2right-sin-pi-x-x-0-then-12702
question: <div class="question">Let \(f(x)=\left[x^2\right] \sin \pi x, x &gt; 0\). Then</div>
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(f\) is discontinuous everywhere.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(f\) is continuous everywhere.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(f\) is continuous at only those points which are perfect squares.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">None of these</span> </div>

answer:
<div class="solution">Hint: \(\underset{\downarrow \atop \large \text{discontinuous}}{\left[x^2\right]} \sin \pi x\)<br/>at all points where \(x^2\) is integer<br/>If \(x^2\) is integer and \(x\) is also integer then \(f(x)\) will be continuous, but if \(x^2\) is integer and \(x\) is not integer then \(f(x)\) will be discontinuous.</div>


title: let-fxx3-e3-x-x-0-then-the-maximum-value-of-fx-is
slug: let-fxx3-e3-x-x-0-then-the-maximum-value-of-fx-is-83033
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let \(f(x)=x^3 e^{-3 x}, x &gt; 0\). Then the maximum value of \(f(x)\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\mathrm{e}^{-3}\)</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">\(3 \mathrm{e}^{-3}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(27 \mathrm{e}^{-9}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\infty\)</span> </li> </ul>

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hints: } f(x)=x^3 \cdot e^{-3 x} \\<br/>&amp; =f^{\prime}(x)=3 x^2 e^{-3 x}+x^3 e^{-3 x}(-3) \\<br/>&amp; =x^2 3 e^{-3 x}[1-x]=0, x=1 \\<br/>&amp; \text { Maximum at } x=1 \\<br/>&amp; f(1)=e^{-3}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2023']
correct_option: <div class="correct-answer"> The correct answers are: <span class="option-value">0, 2</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-fxxm-m-being-a-nonnegative-integer-the-value-of-m-so-that-the-equality-fprimeabfprimeafprimeb-is-valid-for-all-a-b-0-is

slug:
let-fxxm-m-being-a-nonnegative-integer-the-value-of-m-so-that-the-equality-fprimeabfprimeafprimeb-is-valid-for-all-a-b-0-is-92454

question:
<div class="question">Let \(f(x)=x^m, m\) being a non-negative integer. The value of \(m\) so that the equality \(f^{\prime}(a+b)=f^{\prime}(a)+f^{\prime}(b)\) is valid for all \(a\), \(b &gt; 0\) is</div>

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

answer:
<div class="solution">Hint : Let \(f(x)=x^m, m \geq 0, m \in I\)<br/>\(\begin{aligned}<br/>&amp; f^{\prime}(a+b)=f^{\prime}(a)+f^{\prime}(b), a, b &gt; 0 \\<br/>&amp; \Rightarrow m(a+b)^{m-1}=m a^{m-1}+m \cdot b^{m-1} \\<br/>&amp; (a+b)^{m-1}=a^{m-1}+b^{m-1} \quad f^{\prime}(x)=m n^{m-1} \\<br/>&amp; \text { for } m=0 \Rightarrow 0=0 \\<br/>&amp; (a+b)^{-1}=a^{-1}+b^{-1}<br/>\end{aligned}\)<br/>Not satisfied<br/>for \(m=1\)<br/>\(\begin{aligned}<br/>&amp; (a+b)^0=a^{1-1}+b^{1-1} \\<br/>&amp; 1=2<br/>\end{aligned}\)<br/>Not satisfied<br/>for \(m=2\)<br/>\(\begin{aligned}<br/>&amp; (a+b)^{2-1}=a^{2-1}+b^{2-1} \\<br/>&amp; a+b=a+b \text { (satisfied) } \\<br/>&amp; \text { for } m=3 \\<br/>&amp; (a+b)^{3-1}=a^{3-1}+b^{3-1}<br/>\end{aligned}\)<br/>Not satisfied</div>


title: let-g-x-x-2-x-t-f-t-d-t-where-x-0-and-f-be-continuous-function-and-f-2-x-f-x-then
slug: let-g-x-x-2-x-t-f-t-d-t-where-x-0-and-f-be-continuous-function-and-f-2-x-f-x-then-28854
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $g(x)=\int_{x}^{2 x} \frac{f(t)}{t} d t$ where $x&gt;0$ and $f$ be continuous function and $f(2 x)=f(x)$, then</div>

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

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

answer:
<div class="solution">$g(x)=\int_{x}^{2 x} \frac{f(t)}{t} d t$
<br/>$g^{\prime}(x)=\frac{f(2 x)}{2 x} \cdot 2^{\prime}-\frac{f(x)}{x} \cdot 1=\frac{f(2 x)-f(x)}{x}=\frac{f(x)-f(x)}{x}[\because f(2 x)=f(x)]$
<br/>$g^{\prime}(x)=0$
<br/>$g(x)=$ constant.</div>


title: let-i-0-1-2-x-2-x-3-c-o-s-3-x-d-x-then
slug: let-i-0-1-2-x-2-x-3-c-o-s-3-x-d-x-then-87504
question: <div class="question">Let $I=\int_{0}^{1} \frac{x^{3} \cos 3 x}{2+x^{2}} d x$. Then</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{2}    &lt; I &lt; \frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{3}    &lt; I &lt; \frac{1}{3}$</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">C</span> <span class="option-data">$-1    &lt; I &lt; 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{3}{2}    &lt; I &lt; \frac{3}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$-\frac{1}{3}    &lt; I &lt; \frac{1}{3}$</span> </div>

answer:
<div class="solution">$I=\int_{0}^{1} \frac{x^{3} \cos 3 x}{2+x^{2}} d x$
<br/>Here, $\quad-1    &lt; \cos 3 x &lt; 1$
<br/>$\Rightarrow \quad-x^{3} &lt; x^{3} \cos 3 x &lt; x^{3}$
<br/>$\Rightarrow \quad \frac{-x^{3}}{x^{2}} &lt; \frac{-x^{3}}{x} &lt; \frac{-x^{3}}{2+x^{2}} &lt; \frac{x^{3} \cos 3 x}{2+x^{2}}$
<br/>$ &lt; \frac{x^{3}}{2+x^{2}} &lt; \frac{x^{3}}{x} &lt; \frac{x^{3}}{x^{2}}$
<br/>$\Rightarrow \quad \int_{0}^{1}-x^{2} d x &lt; I &lt; \int_{0}^{1} x^{2} d x$
<br/>$\Rightarrow$
<br/>$\left(\frac{-x^{3}}{3}\right)_{0}^{1} &lt; I &lt; \left(\frac{x^{3}}{3}\right)_{0}^{1}$
<br/>$\frac{-1}{3} &lt; I &lt; \frac{1}{3}$</div>


title: let-i-0-100-1-cos-2-x-d-x-then
slug: let-i-0-100-1-cos-2-x-d-x-then-53609
question: <div class="question">Let $I=\int_{0}^{{100 \pi }} \sqrt{(1-\cos 2 x)} d x$, then</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$I=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$I=200 \sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$I=\pi \sqrt{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$I=100$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$I=200 \sqrt{2}$</span> </div>

answer:
<div class="solution">$I=\int_{0}^{1004} \sqrt{1-\cos 2 x} d x$
<br/>$=\int_{0}^{100 \pi} \sqrt{2 \sin ^{2} x} d x$
<br/>$=\sqrt{2} \int_{0}^{100 \pi} \sin x \mid d x$
<br/>$=\sqrt{2} \times 100 \int_{0}^{n} \sin x|d x|$
<br/>$[\sin x \mid$ has period of $\pi]$
<br/>$=100 \sqrt{2} \int_{0}^{\pi} sin x d x$
<br/>$=100 \sqrt{2}[-\cos x]_{0}^{\pi}$
<br/>$=100 \sqrt{2}\{(-\cos \pi)-(-\cos 0)]$
<br/>$=100 \sqrt{2}[-(-1)-(-1)]$
<br/>$=100 \sqrt{2} \times 2$
<br/>$=200 \sqrt{2}$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$n-1$</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-i-1-0-n-x-d-x-and-i-2-0-n-x-d-x-where-x-and-x-are-integral-and-fractional-parts-of-x-and-n-n-1-then-i-1-i-2-is-equal-to

slug:
let-i-1-0-n-x-d-x-and-i-2-0-n-x-d-x-where-x-and-x-are-integral-and-fractional-parts-of-x-and-n-n-1-then-i-1-i-2-is-equal-to-44621

question:
<div class="question">Let $I_{1}=\int_{0}^{n}[x] d x$ and $I_{2}=\int_{0}^{n}\{x\} d x,$ where $[x]$
<br/>and $\{x\}$ are integral and fractional parts of $x$ and $n \in N-\{1\} .$ Then, $I_{1} / I_{2}$ is equal to</div>

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

answer:
<div class="solution">We have, $I_{1}=\int_{0}^{n}[x] d x$
<br/>$=\int_{0}^{1}[x] d x+\int_{1}^{2}[x] d x$
<br/>$\quad+\int_{2}^{3}[x] d x+\ldots+\int_{n-1}^{n}[x] d x$
<br/>$=\int_{0}^{1} 0 d x+\int_{2}^{3} 1 d x+\int_{2}^{3} 2 d x$
<br/>$\quad+\ldots+\int_{n-1}^{n}(n-1) d x$
<br/>$=0+[x]_{1}^{2}+2[x]_{2}^{3}+\ldots+(n-1)[x]_{n-1}^{n}$
<br/>$=(2-1)+2(3-2)+\ldots+$
<br/>$=\frac{n(n-1)}{2}$
<br/>$\begin{aligned} \text { Now, } I_{2} &amp;=\int_{0}^{n}\{x\} d x=\int_{0}^{n} x-[x] d x \\ &amp;=\int_{0}^{n} x d x-\int_{0}^{n}[x] d x \\ &amp;=\left[\frac{x^{2}}{2}\right]_{0}^{n}-I_{1} \\ &amp;=\frac{n^{2}}{2}-\frac{n(n-1)}{2} \\ &amp;=\frac{n^{2}-n^{2}+n}{2} \\ &amp;=\frac{n}{2} \\ \therefore \quad \frac{I_{1}}{I_{2}}=\frac{\frac{n(n-1)}{2}}{\frac{n}{2}} \\=&amp;(n-1) \end{aligned}$</div>


title: let-i-4-3-x-s-i-n-x-d-x-then
slug: let-i-4-3-x-s-i-n-x-d-x-then-93624
question: <div class="question">Let $I=\int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} d x$. Then</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \leq 1 \leq 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \leq 1 \leq 2 \sqrt{30}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}}{8} \leq 1 \leq \frac{\sqrt{2}}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 \leq 1 \leq \frac{2 \sqrt{3}}{\sqrt{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3}}{8} \leq 1 \leq \frac{\sqrt{2}}{6}$</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>I=\int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} d x
<br/>$$
<br/>since, $\frac{\sin x}{x}$ is a decreasing function.
<br/>$$
<br/>\begin{array}{c}
<br/>\frac{\pi}{12} \times \frac{\sin \pi / 3}{\pi / 3} \leq 1 \leq \frac{\pi}{12} \times \frac{\sin \pi / 4}{\pi / 4} \\
<br/>\frac{\sqrt{3}}{8} \leq I \leq \frac{\sqrt{2}}{6}
<br/>\end{array}
<br/>$$</div>


title: let-i-4-3-x-s-i-n-x-dx-then
slug: let-i-4-3-x-s-i-n-x-dx-then-43927
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{I}=\int_{\pi / 4}^{\pi / 3} \frac{\sin \mathrm{x}}{\mathrm{x}} \mathrm{dx}$. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{3}}{8} \leq \mathrm{I} \leq \frac{\sqrt{2}}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{3}}{2 \pi} \leq \mathrm{I} \leq \frac{2 \sqrt{3}}{\pi}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}}{9} \leq \mathrm{I} \leq \frac{\sqrt{2}}{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi \leq \mathrm{I} \leq \frac{4 \pi}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{3}}{8} \leq \mathrm{I} \leq \frac{\sqrt{2}}{6}$</span> </div>

answer:
<div class="solution">$f(x)=\frac{\sin x}{x}$ (decreasing function)
<br/>$\Rightarrow f\left(\frac{\pi}{3}\right)     &lt; f(x) &lt; f\left(\frac{\pi}{4}\right)$
<br/>$\Rightarrow \frac{\sqrt{3}}{2} \times \frac{3}{\pi} \int_{\pi / 4}^{\pi / 3} d x &lt; I &lt; \frac{1}{\sqrt{2}} \times \frac{4}{\pi} \times \int_{\pi / 4}^{\pi / 3} d x$
<br/>$\Rightarrow \frac{\sqrt{3}}{8} &lt; I &lt; \frac{\sqrt{2}}{6}$</div>


title: let-i-10-19-1-x-6-s-i-n-x-d-x-then
slug: let-i-10-19-1-x-6-s-i-n-x-d-x-then-48323
question: <div class="question">Let $I=\int_{10}^{19} \frac{\sin x}{1+x^{6}} d x$. Then,</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$|I|    &lt; 10^{-9}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$|I|    &lt; 10^{-7}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$|I|    &lt; 10^{-5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|I|&gt;10^{-7}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|I|    &lt; 10^{-7}$</span> </div>

answer:
<div class="solution">For $x&gt;10,$ we have
<br/>$|\sin x|    &lt; 1$ and $1+x^{8}&gt;10^{8}$
<br/>$\Rightarrow \quad \frac{1}{1+x^{8}} \leq 10^{-8}$
<br/>$\therefore \quad\left|\int_{10}^{19} \frac{\sin x}{1+x^{8}} d x\right| \leq \int_{10}^{19} \frac{|\sin x|}{1+x^{8}} d x$
<br/>$\quad \leq \int_{10}^{19} 10^{-8}=9 \times 10^{-8} &lt; 10^{-7}$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
let-i-denote-the-3-3-identity-matrix-and-p-be-a-matrix-obtained-by-rearranging-the-columns-of-i-then

slug:
let-i-denote-the-3-3-identity-matrix-and-p-be-a-matrix-obtained-by-rearranging-the-columns-of-i-then-73964

question:
<div class="question">Let $I$ denote the $3 \times 3$ identity matrix and $P$ be a matrix obtained by rearranging the columns of
<br/>$I$. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">there are six distinct choices for $P$ and $\operatorname{det}(P)=1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">there are six distinct choices for $P$ and $\operatorname{det}(P)=\pm 1$</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">there are more than one choices for $P$ and some of them are not invertible</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">there are more than one choices for $P$ and $P^{-1}=1$ in each choice</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there are six distinct choices for $P$ and $\operatorname{det}(P)=\pm 1$</span> </div>

answer:
<div class="solution">Given, $I=\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>Then, $\quad \operatorname{det}(I)=1$
<br/>If we take $I$ as
<br/>$$
<br/>A_{1}=\left[\begin{array}{lll}
<br/>0 &amp; 1 &amp; 0 \\
<br/>1 &amp; 0 &amp; 0 \\
<br/>0 &amp; 0 &amp; 1
<br/>\end{array}\right]
<br/>$$
<br/>Then, $\quad \operatorname{det}\left(I_{1}\right)=-1$
<br/>Similarly, there are four other possibilities,
<br/>$$
<br/>\left[\begin{array}{lll}
<br/>1 &amp; 0 &amp; 0 \\
<br/>0 &amp; 0 &amp; 1 \\
<br/>0 &amp; 1 &amp; 0
<br/>\end{array}\right] \cdot\left[\begin{array}{lll}
<br/>0 &amp; 0 &amp; 1 \\
<br/>1 &amp; 0 &amp; 0 \\
<br/>0 &amp; 1 &amp; 0
<br/>\end{array}\right]
<br/>$$
<br/>$$
<br/>\left[\begin{array}{lll}
<br/>0 &amp; 1 &amp; 0 \\
<br/>0 &amp; 0 &amp; 1 \\
<br/>1 &amp; 0 &amp; 0
<br/>\end{array}\right] \cdot\left[\begin{array}{lll}
<br/>0 &amp; 0 &amp; 1 \\
<br/>0 &amp; 1 &amp; 0 \\
<br/>1 &amp; 0 &amp; 0
<br/>\end{array}\right]
<br/>$$
<br/>who will give a determinant either -1 or 1 . Hence, there are six distinct choices for $P$ and
<br/>$\operatorname{det}(P)=\pm 1$</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
let-i-j-k-i-j-k-and-i-j-k-be-three-vectors-a-vector-in-the-plane-of-and-whose-projection-on-is-3-1-is-given-by

slug:
let-i-j-k-i-j-k-and-i-j-k-be-three-vectors-a-vector-in-the-plane-of-and-whose-projection-on-is-3-1-is-given-by-64662

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

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

answer:
<div class="solution">We have,
<br/>$\overrightarrow{8}=\vec{a}+\lambda \vec{\beta}$
<br/>$=(\hat{i}+\hat{j}+\hat{k})+\lambda(\hat{i}-\hat{j}-\hat{k})$
<br/>$\Rightarrow \quad \vec{\delta}=(1+\lambda) \hat{i}+(1-\lambda) \hat{j}+(1-\lambda) \hat{k}$
<br/>Given, $\quad \frac{\vec{\delta} \cdot \vec{\gamma}}{|\vec{\gamma}|}=\frac{1}{\sqrt{3}}$
<br/>$\Rightarrow \frac{(1+\lambda)(-1)+(1-\lambda)(1)+(1-\lambda)(-1)}{\sqrt{3}}=\frac{1}{\sqrt{3}}$
<br/>$\Rightarrow \quad \lambda=-2$
<br/>$\therefore$
<br/>$\vec{\delta}=-\hat{i}+3 \hat{j}+3 \hat{k}$</div>


title: let-i-n-0-1-x-n-tan-1-x-d-x-if-a-n-i-n-2-b-n-i-n-c-n-for-all-n-1-then
slug: let-i-n-0-1-x-n-tan-1-x-d-x-if-a-n-i-n-2-b-n-i-n-c-n-for-all-n-1-then-90247
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $I_{n}=\int_{0}^{1} x^{n} \tan ^{-1} x d x .$ If $a_{n} I_{n+2}+b_{n} I_{n}=c_{n}$ for all $n \geq 1,$ then</div>

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

answer:
<div class="solution">We have,
<br/>$I_{n}=\int_{0}^{1} x^{n} \tan ^{-1} x d x$
<br/>$=\left[\tan ^{-1} x \cdot \frac{x^{n+1}}{n+1}\right]_{0}^{1}-\int_{0}^{1} \frac{x^{n+1}}{n+1} \cdot \frac{1}{1+x^{2}} d x$
<br/>$=\frac{\pi}{4(n+1)}-\frac{1}{n+1} \int_{0}^{1} \frac{x^{2} \cdot x^{n-1}}{1+x^{2}} d x$
<br/>$=\frac{\pi}{4(n+1)}-\frac{1}{n+1} \int_{0}^{1} \frac{\left(1+x^{2}\right) x^{n-1}-x^{n-1}}{1+x^{2}} d x$
<br/>$=\frac{\pi}{4(n+1)}-\frac{1}{n+1} \int_{0}^{1} x^{n-1} d x+\frac{1}{n+1} \int_{0}^{1} \frac{x^{n-1}}{1+x^{2}} d x$
<br/>$=\frac{\pi}{4(n+1)}-\frac{1}{n+1}\left[\frac{x^{n}}{n}\right]_{0}^{1}+\frac{1}{n+1}$
<br/>$\left[x^{n-1} \cdot \tan ^{-1} x\right]_{0}^{1}-\int_{0}^{1} \tan ^{-1} x \cdot(n-1) x^{n-2} d x$
<br/>$\begin{aligned}=\frac{\pi}{4(n+1)}-\frac{1}{n(n+1)}+\frac{\pi}{4(n+1)}-\frac{n-1}{n+1} &amp; \int_{0}^{1} x^{n-2} \tan ^{-1} x d x \end{aligned}$
<br/>$\Rightarrow \quad I_{n}=\frac{\pi}{2(n+1)}-\frac{1}{n(n+1)}-\frac{n-1}{n+2} I_{n-2}$
<br/>$\Rightarrow \quad I_{n}+\frac{n-1}{n+2} I_{n-2}=\frac{\pi}{2(n+1)}-\frac{1}{n(n+1)}$
<br/>Put $n=n+2,$ we get $I_{n+2}+\frac{n+1}{n+3} I_{n}=\frac{\pi}{2(n+3)}-\frac{1}{(n+2)(n+3)}$
<br/>$\Rightarrow(n+3) I_{n+2}+(n+1) I_{n}=\frac{\pi}{2}-\frac{1}{n+2}$
<br/>$\therefore a_{n}=n+3, b_{n}=n+1, c_{n}=\frac{\pi}{2}-\frac{1}{n+2}$
<br/>$\therefore a_{n}$ and $b_{n}$ are in $\mathrm{AP}$.</div>


title: let-i-n-n-n-j-n-135-2-n-1-for-all-n-1-n-n-then
slug: let-i-n-n-n-j-n-135-2-n-1-for-all-n-1-n-n-then-99846
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{I}(\mathrm{n})=\mathrm{n}^{\mathrm{n}}, \mathrm{J}(\mathrm{n})=1.3 .5 \ldots . .(2 \mathrm{n}-1)$ for all $(\mathrm{n}&gt;1), \mathrm{n} \in \mathrm{N}$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{I}(\mathrm{n})&gt;\mathrm{J}(\mathrm{n})$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\mathrm{AM} \geq \mathrm{GM}$
<br/>$\frac{1+3+5+7+\ldots+(2 n-1)}{\mathrm{n}}&gt;(\mathrm{J}(\mathrm{n}))^{\frac{1}{n}}, \quad \frac{\mathrm{n}^{2}}{\mathrm{n}}&gt;(\mathrm{J}(\mathrm{n}))^{\frac{1}{n}}, \quad \mathrm{n}^{\mathrm{n}}&gt;\mathrm{J}(\mathrm{n}), \mathrm{I}(\mathrm{n})&gt;\mathrm{J}(\mathrm{n})$</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
title:
let-l-be-the-length-and-d-be-the-diameter-of-crosssection-of-a-wire-wires-of-the-same-material-with-different-l-and-d-are-subjected-to-the-same-tensio

slug:
let-l-be-the-length-and-d-be-the-diameter-of-crosssection-of-a-wire-wires-of-the-same-material-with-different-l-and-d-are-subjected-to-the-same-tensio-70832

question:
<div class="question">Let $L$ be the length and d be the diameter of cross-section of a wire. Wires of the same material with different $L$ and $d$ are subjected to the same tension along the length of the wire. In which of the following cases, the extension of wire will be the maximum?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$L=200 \mathrm{cm}, \mathrm{d}=0.5 \mathrm{mm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$L=300 \mathrm{cm}, \mathrm{d}=1.0 \mathrm{mm}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$L=50 \mathrm{cm}, \quad d=0.05 \mathrm{mm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$L=100 \mathrm{cm}, \mathrm{d}=0.2 \mathrm{mm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$L=50 \mathrm{cm}, \quad d=0.05 \mathrm{mm}$</span> </div>

answer:
<div class="solution">We know
<br/>$$
<br/>\Delta l=\frac{F \times l}{A \times Y}=\frac{4 L}{\pi d^{2}}
<br/>$$
<br/>Here $\frac{4}{\pi}$ is a constant.
<br/>Hence, $\Delta l \propto \frac{L}{d^{2}}$</div>


title: let-lim-c-0-c-x-t-2-b-t-c-o-s-4-t-a-s-i-n-4-t-d-t-x-a-s-i-n-4-x-1-0-x-4-then-a-and-b-are-given-by
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
slug:
let-lim-c-0-c-x-t-2-b-t-c-o-s-4-t-a-s-i-n-4-t-d-t-x-a-s-i-n-4-x-1-0-x-4-then-a-and-b-are-given-by-38505

question:
<div class="question">Let $\lim _{c \rightarrow 0} \int_c^x \frac{b t \cos 4 t-a \sin 4 t}{t^2} d t=\frac{a \sin 4 x}{x}-1,(0 &lt; x &lt; \pi / 4)$. Then a and $b$ are given by</div>

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

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

answer:
<div class="solution">Let $g(x)=\lim _{c \rightarrow 0} \int_E^x \frac{b t \cos 4 t-a \sin 4 t}{t^2} d t=\frac{a \sin 4 x}{x}-1$<br/>$\lim _{x \rightarrow 0} g(x)=0=4 a-1 \quad \Rightarrow a=1 / 4<br/>$$<br/>$$<br/>g^{\prime}(x)=\frac{b x \cos 4 x-a \sin 4 x}{x^2}$<br/>Comparing, $b=4 a=1$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
let-mathrma-mathrmb-mathrmc-be-three-real-numbers-such-that-mathrma2-mathrmb4-mathrmc0-then-the-equation-mathrmax2mathrmbxmathrmc0

slug:
let-mathrma-mathrmb-mathrmc-be-three-real-numbers-such-that-mathrma2-mathrmb4-mathrmc0-then-the-equation-mathrmax2mathrmbxmathrmc0-66276

question:
<div class="question">Let \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) be three real numbers such that \(\mathrm{a}+2 \mathrm{~b}+4 \mathrm{c}=0\). Then the equation \(\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}=0\)</div>

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

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

answer:
<div class="solution">Hints: \(\frac{1}{4} \mathrm{a}+\frac{1}{2} \mathrm{~b}+\mathrm{c}=0\)<br/>\(\begin{aligned}<br/>&amp; \left(\frac{1}{2}\right)^2 a+\left(\frac{1}{2}\right) b+c=0 \\<br/>&amp; \therefore x=\frac{1}{2}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">+1</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-mathrmfmathrmxtan-1-mathrmx-then-mathrmfprimemathrmxmathrmfprime-primemathrmx-is-0-when-mathrmx-is-equal-to

slug:
let-mathrmfmathrmxtan-1-mathrmx-then-mathrmfprimemathrmxmathrmfprime-primemathrmx-is-0-when-mathrmx-is-equal-to-44354

question:
<div class="question">Let \(\mathrm{f}(\mathrm{x})=\tan ^{-1} \mathrm{x}\). Then \(\mathrm{f}^{\prime}(\mathrm{x})+\mathrm{f}^{\prime \prime}(\mathrm{x})\) is \(=0\), when \(\mathrm{x}\) is equal to</div>

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

answer:
<div class="solution">Hints: \(f(x)=\tan ^{-1} x\)<br/>\(\begin{aligned}<br/>&amp; f^{\prime}(x)=\frac{1}{1+x^2} \\<br/>&amp; f^{\prime \prime}(x)=\frac{-1}{\left(1+x^2\right)} \cdot 2 x, \frac{1}{1+x^2}=\frac{2 x}{\left(1+x^2\right)^2} \\<br/>&amp; 1+x^2=2 x,(x-1)^2=0 \\<br/>&amp; x=1<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
let-n-2-be-an-integer-a-cos-2-n-sin-2-n-0-sin-2-n-cos-2-n-0-0-0-1-and-i-is-the-identity-matrix-of-order-3-then

slug:
let-n-2-be-an-integer-a-cos-2-n-sin-2-n-0-sin-2-n-cos-2-n-0-0-0-1-and-i-is-the-identity-matrix-of-order-3-then-67444

question:
<div class="question">Let $n \geq 2$ be an integer. $A=\left[\begin{array}{ccc}\cos (2 \pi / n) &amp; \sin (2 \pi / n) &amp; 0 \\ -\sin (2 \pi / n) &amp; \cos (2 \pi / n) &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$ and $I$ is the identity matrix of order 3 . Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A^{n}=I$ and $A^{n-1} \neq I$</span> </div>

answer:
<div class="solution">Given,
<br/>$A=\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) &amp; \sin \left(\frac{2 \pi}{n}\right) &amp; 0 \\ -\sin \left(\frac{2 \pi}{n}\right) &amp; \cos \left(\frac{2 \pi}{n}\right) &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>Now, $A \times A=\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) &amp; \sin \left(\frac{2 \pi}{n}\right) &amp; 0 \\ -\sin \left(\frac{2 \pi}{n}\right) &amp; \cos \left(\frac{2 \pi}{n}\right) &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right] \times$$\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) &amp; \sin \left(\frac{2 \pi}{n}\right) &amp; 0 \\ -\sin \left(\frac{2 \pi}{n}\right) &amp; \cos \left(\frac{2 \pi}{n}\right) &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$\left[\begin{array}{ccc}\cos ^{2}\left(\frac{2 \pi}{n}\right)-\sin ^{2}\left(\frac{2 \pi}{n}\right) &amp; \cos \left(\frac{2 \pi}{n}\right) \sin \left(\frac{2 \pi}{n}\right)+\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) &amp; 0\\ -\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right)-\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) &amp; -\sin ^{2}\left(\frac{2 \pi}{n}\right)+\cos ^{2}\left(\frac{2 \pi}{n}\right) &amp; 0\\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$=\left[\begin{array}{cccc}\cos \left(2 \times \frac{2 \pi}{n}\right) &amp; 2 \sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) &amp; 0 \\ -2 \sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) &amp; \cos \left(2 \times \frac{2 \pi}{n}\right) &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$=\left[\begin{array}{ccc}
<br/>\cos \left(2 \times \frac{2 \pi}{n}\right) &amp; \sin \left(2 \times \frac{2 \pi}{n}\right) &amp; 0 \\
<br/>-\sin \left(2 \times \frac{2 \pi}{n}\right) &amp; \cos \left(2 \times \frac{2 \pi}{n}\right) &amp; 0 \\
<br/>0 &amp; 0 &amp; 1
<br/>\end{array}\right]$
<br/>Similarly,
<br/>$A^{n}=\left[\begin{array}{cccc}\cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) &amp; \sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) &amp; 0 \\ -\sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) &amp; \cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$=\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]=I$
<br/>and
<br/>$\begin{aligned}
<br/>A^{n-1}=&amp;\left[\begin{array}{cccc}
<br/>\cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) &amp; \sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) &amp; 0 \\
<br/>-\sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) &amp; \cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) &amp; 0 \\
<br/>0 &amp; 0 &amp; 1
<br/>\end{array}\right] \\
<br/>\neq I
<br/>\end{aligned}$
<br/>$\therefore A^{n-1} \neq I$ for any positive integer $m$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">21</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-n-be-a-positive-even-integer-if-the-ratio-of-the-largest-coefficient-and-the-2nd-largest-coefficient-in-the-expansion-of-1-x-n-is-11-10-then-the-n

slug:
let-n-be-a-positive-even-integer-if-the-ratio-of-the-largest-coefficient-and-the-2nd-largest-coefficient-in-the-expansion-of-1-x-n-is-11-10-then-the-n-95538

question:
<div class="question">Let $n$ be a positive even integer. If the ratio of the largest coefficient and the 2nd largest coefficient in the expansion of $(1+x)^{n}$ is $11: 10 .$ Then, the number of terms in the expansion of $(1+x)^{n}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">21</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">10</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">11</span> </li> </ul>

answer:
<div class="solution">Let the number of terms, $n=2 \mathrm{m}$ 
<br/>Now, by condition
<br/>$\frac{\text { Largest coefficient in }(1+x)^{\circ}}{\text { Second largest coefficient in }(1+x)^{n}}=\frac{11}{10}$
<br/>(given)
<br/>$\Rightarrow$
<br/>$$
<br/>\begin{aligned}
<br/>\frac{2 m}{2 m} C_{m} &amp;=\frac{11}{10} \\
<br/>\frac{(m-1) !(m+1) !}{m ! m !} &amp;=\frac{11}{10}
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow \quad \frac{1}{m} \frac{m(m-1) !(m+1) m !}{m ! m !}=\frac{11}{10}$
<br/>$\Rightarrow \quad \frac{m+1}{m} \frac{m+m !}{m ! m !}=\frac{11}{10}$
<br/>$\Rightarrow \quad 10 m+10=11 m \Rightarrow m=10$
<br/>$\therefore \quad n=20$
<br/>Hence, total number of term
<br/>$$
<br/>=n+1=20+1=21
<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
let-o-be-the-vertex-q-be-any-point-on-the-parabola-x28-y-if-the-point-p-divides-the-line-segment-o-q-internally-in-the-ratio-1-3-then-the-locus-of-p-i

slug:
let-o-be-the-vertex-q-be-any-point-on-the-parabola-x28-y-if-the-point-p-divides-the-line-segment-o-q-internally-in-the-ratio-1-3-then-the-locus-of-p-i-57012

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

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dvP82l3qYX6D_hwKSEueC8yymWNh14bP38QXoJ_wMxs.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; O P: P Q=1: 3 \\ &amp; \Rightarrow \mathrm{h}=\mathrm{t}, \mathrm{k}=\frac{\mathrm{t}^2}{2} \\ &amp; \Rightarrow \mathrm{k}=\frac{\mathrm{h}^2}{2}\end{aligned}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-p-2-3-q-2-1-be-the-vertices-of-the-pqr-if-the-centroid-of-pqr-lies-on-the-line-2-x-3-y-1-then-the-locus-of-r-is

slug:
let-p-2-3-q-2-1-be-the-vertices-of-the-pqr-if-the-centroid-of-pqr-lies-on-the-line-2-x-3-y-1-then-the-locus-of-r-is-52611

question:
<div class="question">Let $P(2,-3), Q(-2,1)$ be the vertices of the $\Delta P Q R .$ If the centroid of $\Delta P Q R$ lies on the line $2 x+3 y=1,$ then the locus of $R$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 x+3 y=9$</span> </div>

answer:
<div class="solution">Let the vertices of $R$ be $(h, k)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a49.png"/> <br/> <br/>$\therefore$ Centroid of triangle is given by
<br/>$$
<br/>\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)
<br/>$$
<br/>Le, $\left(\frac{2+h-2}{3}, \frac{-3+1+k}{3}\right)$
<br/>Le, $\left(\frac{h}{3}, \frac{-2+k}{3}\right)$
<br/>since, the point $\left(\frac{h}{3}, \frac{-2+k}{3}\right)$ lies on the line $2 x+3 y=1,$ therefore the point $\left(\frac{h}{3}, \frac{-2+k}{3}\right)$ must be satisfy the equation of the given line.
<br/>Le., $2\left(\frac{h}{3}\right)+3\left(\frac{-2+k}{3}\right)=1$
<br/>$\Rightarrow \quad \frac{2 h-6+3 k}{3}=1$
<br/>$\Rightarrow \quad 2 h-6+3 k=3$
<br/>$\Rightarrow h+3 k=9$
<br/>Replace $h=x$ and $k=y,$ we get the required locus of $R$. ie, $\quad 2 x+3 y=9$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
let-p-3-sec-2-tan-q-3-sec-2-tan-be-two-points-on-9-x-2-4-y-2-1-such-that-2-0-2-then-the-ordinate-of-the-point-of-intersection-of-the-normals-at-p-and-

slug:
let-p-3-sec-2-tan-q-3-sec-2-tan-be-two-points-on-9-x-2-4-y-2-1-such-that-2-0-2-then-the-ordinate-of-the-point-of-intersection-of-the-normals-at-p-and-57627

question:
<div class="question">Let $\mathrm{P}(3 \sec \theta, 2 \tan \theta) \mathrm{Q}(3 \sec \phi, 2 \tan \phi)$ be two points on $\frac{\mathrm{x}^2}{9}-\frac{\mathrm{y}^2}{4}=1$ such that $\theta+\phi=\frac{\pi}{2} .0 &lt; \theta, \phi &lt; \frac{\pi}{2}$. Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{13}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{13}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{5}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{5}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{13}{2}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2, P(3 \sec \theta, 2 \tan \theta) \\ \therefore &amp; \frac{9 x}{3 \sec \theta}+\frac{4 y}{2 \tan \theta}=9+4=13 \\ \Rightarrow &amp; 3 x \cos \theta+2 y \cot \theta=13 \end{aligned}$<br/>Similarly, $3 x \cos \phi+2 y \cot \phi=13$<br/>$3 x \cos \left(\frac{\pi}{2}-\theta\right)+2 y \cot \left(\frac{\pi}{2}-\theta\right)=13$<br/>$\Rightarrow 3 x \sin \theta+2 y \tan \theta=13 \ldots \ldots(2)$<br/>$\therefore \frac{13-2 \mathrm{y} \cot \theta}{3 \cos \theta}=\frac{13-2 \mathrm{y} \tan \theta}{3 \sin \theta}$<br/>$\Rightarrow 13 \sin \theta-2 y \cos \theta=13 \cos \theta-2 y \sin \theta$<br/>$\Rightarrow 13 \tan \theta-2 y=13-2 y \tan \theta$<br/>$\Rightarrow 13(\tan \theta-1)=-2 y(\tan \theta-1)$<br/>$\therefore \mathrm{y}=-\frac{13}{2}$<br/>$\therefore$ Ordinate $=-\frac{13}{2}$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-p-4-3-be-a-point-on-the-hyperbola-a-2-x-2-b-2-y-2-1-if-the-normal-at-p-intersects-the-x-axis-at-160-then-the-eccentricity-of-the-hyperbola-is

slug:
let-p-4-3-be-a-point-on-the-hyperbola-a-2-x-2-b-2-y-2-1-if-the-normal-at-p-intersects-the-x-axis-at-160-then-the-eccentricity-of-the-hyperbola-is-53324

question:
<div class="question">Let $P(4,3)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=\mathrm{1}$. If the normal at $P$ intersects the $X$ -axis at (16,0) , then the eccentricity of the hyperbola is</div>

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

answer:
<div class="solution">Given hyperbola equation
<br/>$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
<br/>$P(4,3) $ lie on $H$
<br/>$\Rightarrow$ $\frac{16}{a^{2}}-\frac{9}{b^{2}}=1$
<br/>Normal equation at $P(4,3)$ for $H$
<br/>$$
<br/>a^{2} y_{1}\left(x-x_{1}\right)+b^{2} x_{1}\left(y-y_{1}\right)=0
<br/>$$
<br/>$\Rightarrow \quad 3 a^{2}(x-4)+4 b^{2}(y-3)=0$
<br/>Normal cuts the $X$ -axis at (16,0) $\Rightarrow \quad 3 a^{2}(16-4)+4 b^{2}(0-3)=0$
<br/>$\Rightarrow \quad 3 a^{2} 12+4 b^{2}(-3)=0$
<br/>$\Rightarrow$ $36 a^{2}-12 b^{2}=0$
<br/>$\Rightarrow$ $3 a^{2}-b^{2}=0$ 
<br/>$\Rightarrow$ $3 a^{2}=b^{2}$
<br/>$\therefore$ Eccentricity of the hyperbola is $e=\frac{\sqrt{a^{2}+b^{2}}}{a}$
<br/>$=\frac{\sqrt{a^{2}+3 a^{2}}}{a}=\frac{\sqrt{4 a^{2}}}{a}=\frac{2 a}{a}=2$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
let-p-a-t-2-2-a-t-q-rar-2-2-a-r-be-three-points-on-a-parabola-y-2-4-a-x-if-pq-is-the-focal-chord-and-p-k-qr-are-parallel-where-the-coordinates-of-k-is

slug:
let-p-a-t-2-2-a-t-q-rar-2-2-a-r-be-three-points-on-a-parabola-y-2-4-a-x-if-pq-is-the-focal-chord-and-p-k-qr-are-parallel-where-the-coordinates-of-k-is-17094

question:
<div class="question">Let $P\left(a t^{2}, 2 a t\right), Q,$ R(ar $^{2}, 2ar)$ be three points on a parabola $y^{2}=4 a x$. If $P Q$ is the focal chord and $P K, Q R$ are parallel where the co-ordinates of $K$ is $(2 a, 0)$ then the value of $r$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{t}{1-t^{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1-t^{2}}{t}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{t^{2}+1}{t}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{t^{2}-1}{t}$</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 answer is:
<span class="option-value">$\frac{t^{2}-1}{t}$</span> </div>

answer:
<div class="solution">Here, coordinate of $Q$ will be $\left(\frac{a}{t^{2}}, \frac{-2 a}{t}\right)$
<br/>Slope of $Q R=\frac{2}{r-\frac{1}{t}}$
<br/>Slope of $P K=\frac{2 a t}{a t^{2}-2 a}=\frac{2 t}{t^{2}-2}$
<br/>since, slope of $Q R=$ slope of $P K$
<br/>$\therefore \quad \frac{2}{r-\frac{1}{t}}=\frac{2 t}{t^{2}-2}$
<br/>$\Rightarrow \quad r=\frac{t^{2}-1}{t}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a47.png"/></div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-p-and-q-be-the-points-on-the-parabola-y-2-4-x-so-that-the-line-segment-pq-subtends-right-angle-at-the-vertex-if-pq-intersects-the-axis-of-the-para

slug:
let-p-and-q-be-the-points-on-the-parabola-y-2-4-x-so-that-the-line-segment-pq-subtends-right-angle-at-the-vertex-if-pq-intersects-the-axis-of-the-para-92502

question:
<div class="question">Let $P$ and $Q$ be the points on the parabola $y^{2}=4 x$ so that the line segment $P Q$ subtends right angle at the vertex. If $P Q$ intersects the axis of the parabola at $R,$ then the distance of the vertex from $R$ is</div>

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

answer:
<div class="solution">According to question, it is given that slope of $P X$ and $Q X$ are perpendicular to each other, i.e., slope of $P X \times$ slope of $X Q=-1$ $\Rightarrow \quad \frac{2 t}{t^{2}} \times \frac{2 m}{m^{2}}=-1$
<br/>$\Rightarrow \quad t_{m}=-4$
<br/>$P Q: \frac{x-t^{2}}{t^{2}-m^{2}}=\frac{y-2 t}{2 t-2 m}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a67.png"/> <br/> <br/>Let $P Q$ meets axis of parabola i.e, $x$ -axis at $R(\alpha, 0),$ then
<br/>$\frac{\alpha-t^{2}}{(t+m)(t-m)}=\frac{0-2 t}{2(t-m)}$
<br/>$\Rightarrow \quad \alpha-t^{2}=-t^{2}-t m=-t^{2}+4$
<br/>$\Rightarrow \quad \alpha=4 \Rightarrow A R=4$</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$P$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-p-and-t-be-the-subsets-of-x-y-plane-defined-by-p-x-y-x-0-y-0-and-x-2-y-2-1-t-x-y-x-0-y-0-and-x-8-y-8-1-then-p-t-is

slug:
let-p-and-t-be-the-subsets-of-x-y-plane-defined-by-p-x-y-x-0-y-0-and-x-2-y-2-1-t-x-y-x-0-y-0-and-x-8-y-8-1-then-p-t-is-56613

question:
<div class="question">Let $P$ and $T$ be the subsets of $x$, $y$ -plane defined by $P=\left\{(x, y): x&gt;0, y&gt;0\right.$ and $\left.x^{2}+y^{2}=1\right\}$, $T=\left\{(x, y): x&gt;0, y&gt;0\right.$ and $\left.x^{8}+y^{8}    &lt; 1\right\}$. Then, $P \cap T$ is</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a26.png"/> <br/>From the graph, it is clear that there is common region which satisfies $x^{2}+y^{2}=1$ and $x^{8}+y^{8} &lt; 1$
<br/>$P \cap T=P$</div>


tags: ['Mathematics', 'Ellipse', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
let-p-be-a-point-on-the-ellipse-9-x-2-4-y-2-1-and-the-line-through-p-parallel-to-the-y-axis-mects-the-circle-x-2-y-2-9-at-q-where-p-q-are-on-the-same--1

slug:
let-p-be-a-point-on-the-ellipse-9-x-2-4-y-2-1-and-the-line-through-p-parallel-to-the-y-axis-mects-the-circle-x-2-y-2-9-at-q-where-p-q-are-on-the-same-1-22136

question:
<div class="question">Let $P$ be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the line through $P$ parallel to the $Y$ -axis mects the circle $x^{2}+y^{2}=9$ at $Q$ where $P, Q$ are on the same side of the $X$ -axis. If $R$ is a point on $P Q$ such that $\frac{P R}{R Q}=\frac{1}{2},$ then the locus of $R$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{9}+\frac{9 y^{2}}{49}=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{x^{2}}{49}+\frac{y^{2}}{9}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x^{2}}{9}+\frac{y^{2}}{49}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9 x^{2}}{49}+\frac{y^{2}}{9}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x^{2}}{9}+\frac{9 y^{2}}{49}=1$</span> </div>

answer:
<div class="solution">Since, point $P$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
<br/>$\therefore$
<br/>$P(3 \cos \theta, 2 \sin \theta)$
<br/>Now, equation of line parallel of $Y$ -axis is
<br/>$$
<br/>x=3 \cos \theta
<br/>$$
<br/>and above line meets circle at $Q$
<br/>$\therefore$
<br/>$Q(3 \cos \theta, 3 \sin \theta)$
<br/>Given, $\frac{P R}{R Q}=\frac{1}{2}$
<br/>$\therefore h=\frac{3 \cos \theta+6 \cos \theta}{3}, k=\frac{3 \sin \theta+4 \sin \theta}{3}$
<br/>$\Rightarrow \quad h=3 \cos \theta, \quad k=\frac{7}{3} \sin \theta$
<br/>$\Rightarrow \quad \cos \theta=h / 3, \sin \theta=\frac{3 k}{7}$
<br/>Now, $\cos ^{2} \theta+\sin ^{2} \theta=h^{2} / 9+\frac{9 k^{2}}{49}=1$
<br/>Hence, locus of a point is $\frac{x^{2}}{9}+\frac{9 y^{2}}{49}=1$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a48.png"/></div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
let-p-be-a-point-on-the-ellipse-9-x-2-4-y-2-1-and-the-line-through-p-parallel-to-the-y-axis-mects-the-circle-x-2-y-2-9-at-q-where-p-q-are-on-the-same-

slug:
let-p-be-a-point-on-the-ellipse-9-x-2-4-y-2-1-and-the-line-through-p-parallel-to-the-y-axis-mects-the-circle-x-2-y-2-9-at-q-where-p-q-are-on-the-same-92153

question:
<div class="question">Let $P$ be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the line through $P$ parallel to the $Y$ -axis mects the circle $x^{2}+y^{2}=9$ at $Q$ where $P, Q$ are on the same side of the $X$ -axis. If $R$ is a point on $P Q$ such that $\frac{P R}{R Q}=\frac{1}{2},$ then the locus of $R$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{9}+\frac{9 y^{2}}{49}=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{x^{2}}{49}+\frac{y^{2}}{9}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x^{2}}{9}+\frac{y^{2}}{49}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9 x^{2}}{49}+\frac{y^{2}}{9}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x^{2}}{9}+\frac{9 y^{2}}{49}=1$</span> </div>

answer:
<div class="solution">Since, point $P$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
<br/>$\therefore$
<br/>$P(3 \cos \theta, 2 \sin \theta)$
<br/>Now, equation of line parallel of $Y$ -axis is
<br/>$$
<br/>x=3 \cos \theta
<br/>$$
<br/>and above line meets circle at $Q$
<br/>$\therefore$
<br/>$Q(3 \cos \theta, 3 \sin \theta)$
<br/>Given, $\frac{P R}{R Q}=\frac{1}{2}$
<br/>$\therefore h=\frac{3 \cos \theta+6 \cos \theta}{3}, k=\frac{3 \sin \theta+4 \sin \theta}{3}$
<br/>$\Rightarrow \quad h=3 \cos \theta, \quad k=\frac{7}{3} \sin \theta$
<br/>$\Rightarrow \quad \cos \theta=h / 3, \sin \theta=\frac{3 k}{7}$
<br/>Now, $\cos ^{2} \theta+\sin ^{2} \theta=h^{2} / 9+\frac{9 k^{2}}{49}=1$
<br/>Hence, locus of a point is $\frac{x^{2}}{9}+\frac{9 y^{2}}{49}=1$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a48.png"/></div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
let-p-be-a-variable-point-on-a-circle-c-and-q-be-a-fixed-point-outside-c-if-r-is-the-midpoint-of-the-line-segment-pq-then-locus-of-r-is

slug:
let-p-be-a-variable-point-on-a-circle-c-and-q-be-a-fixed-point-outside-c-if-r-is-the-midpoint-of-the-line-segment-pq-then-locus-of-r-is-67689

question:
<div class="question">Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the midpoint of the line segment $P Q$, then locus of $R$ is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/w32zBs17aWbX1Fh7yxKEdyjo_urT3caim1U537y4JxQ.original.fullsize.png"/><br/> <br/>Let circle be $\left(x-x_{1}\right)=r \cos \theta$
<br/>$y-y_{1}=r \sin \theta$
<br/>$h=\frac{a+x_{1}+r \cos \theta}{2}$
<br/>$k=\frac{b+y_{1}+r \sin \theta}{2}$
<br/>$\Rightarrow\left(h-\frac{\left(a+x_{1}\right)}{2}\right)^{2}+\left(k-\frac{\left(b+y_{1}\right)}{2}\right)^{2}=\frac{r^{2}}{4}$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$a b$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-p-be-the-foot-of-the-perpendicular-from-focus-s-of-hyperbola-a-2-x-2-b-2-y-2-1-on-the-line-b-x-a-y-0-and-let-c-be-the-centre-of-the-hyperbola-then

slug:
let-p-be-the-foot-of-the-perpendicular-from-focus-s-of-hyperbola-a-2-x-2-b-2-y-2-1-on-the-line-b-x-a-y-0-and-let-c-be-the-centre-of-the-hyperbola-then-76725

question:
<div class="question">Let $P$ be the foot of the perpendicular from focus $S$ of hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ on the line $b x-a y=0$ and let $C$ be the centre of the hyperbola. Then, the area of the rectangle whose sides are equal to that of $S P$ and $C P$ is</div>

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

answer:
<div class="solution">Given, equation of hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a21.png"/> <br/> <br/>From figure, 
<br/>$\begin{aligned} S P &amp;=\mid \frac{a b e}{\sqrt{b^{2}+a^{2}}} \\ &amp;=\left|\frac{a b e}{a e}\right|=b \end{aligned}$
<br/>and $C S=a e$ 
<br/>Again, $\Delta S P C$ is right angled triangle at $P$.
<br/>$$
<br/>\begin{aligned}
<br/>\therefore \quad C P &amp;=\sqrt{C S^{2}-S P^{2}} \\
<br/>&amp;=\sqrt{a^{2} e^{2}-b^{2}} \\
<br/>&amp;=\sqrt{a^{2}\left(1+\frac{b^{2}}{a^{2}}\right)-b^{2}} \\
<br/>&amp;=\sqrt{a^{2}+b^{2}-b^{2}}=a
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Area of rectangle $=C P \times S P$ $=a b$</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
let-p-be-the-midpoint-of-a-chord-joining-the-vertex-of-the-parabola-y-2-8-x-to-another-point-on-it-then-the-locus-of-p-is-1

slug:
let-p-be-the-midpoint-of-a-chord-joining-the-vertex-of-the-parabola-y-2-8-x-to-another-point-on-it-then-the-locus-of-p-is-1-46776

question:
<div class="question">Let $P$ be the mid-point of a chord joining the vertex of the parabola $y^{2}=8 x$ to another point on it. Then, the locus of $P$ is</div>

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

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

answer:
<div class="solution">Let the chord be MN. Let other end of chord joining from vertex to it lying on $y^{2}=8 x$ be $N\left(2 t^{2}, 4 t\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a54.png"/> <br/> <br/>$\therefore$ Mid-point of $M N=\left(t^{2}, 2 t\right)$ $\therefore \quad x=t^{2}, y=2 t \Rightarrow x=\left(\frac{y}{2}\right)^{2}$
<br/>$\Rightarrow \quad y^{2}=4 x$
<br/>is the required locus of $P$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-p-be-the-midpoint-of-a-chord-joining-the-vertex-of-the-parabola-y-2-8-x-to-another-point-on-it-then-the-locus-of-p-is

slug:
let-p-be-the-midpoint-of-a-chord-joining-the-vertex-of-the-parabola-y-2-8-x-to-another-point-on-it-then-the-locus-of-p-is-32297

question:
<div class="question">Let $P$ be the mid-point of a chord joining the vertex of the parabola $y^{2}=8 x$ to another point on it. Then, the locus of $P$ is</div>

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

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

answer:
<div class="solution">Let the chord be MN. Let other end of chord joining from vertex to it lying on $y^{2}=8 x$ be $N\left(2 t^{2}, 4 t\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a54.png"/> <br/> <br/>$\therefore$ Mid-point of $M N=\left(t^{2}, 2 t\right)$ $\therefore \quad x=t^{2}, y=2 t \Rightarrow x=\left(\frac{y}{2}\right)^{2}$
<br/>$\Rightarrow \quad y^{2}=4 x$
<br/>is the required locus of $P$</div>


title: let-p-be-the-point-2-0-and-q-be-a-variable-point-on-y-6-2-2-x-4-then-the-locus-of-midpoint-of-pq-is
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
slug:
let-p-be-the-point-2-0-and-q-be-a-variable-point-on-y-6-2-2-x-4-then-the-locus-of-midpoint-of-pq-is-56347

question:
<div class="question">Let $P$ be the point $(2,0)$ and $Q$ be a variable point on $(y-6)^2=2(x-4)$. Then the locus of mid-point of $P Q$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y^2+x+6 y+12=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y^2-x+6 y+12=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y^2+x-6 y+12=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$y^2-x-6 y+12=0$</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 answer is:
<span class="option-value">$y^2-x-6 y+12=0$</span> </div>

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>\mathrm{P}(2,0) \\<br/>\mathrm{Q}\left(4+\frac{1}{2} \mathrm{t}^2, 6+\mathrm{t}\right)<br/>\end{gathered}<br/>$$<br/>Let midpoint of $\mathrm{PQ}$ be R(h, k)<br/>$$<br/>\begin{aligned}<br/>&amp;\mathrm{h}=3+\frac{1}{4} \mathrm{t}^2, \mathrm{k}=3+\frac{\mathrm{t}}{2} \\<br/>&amp;\Rightarrow(\mathrm{k}-3)^2=(\mathrm{h}-3) \\<br/>&amp;\Rightarrow \mathrm{y}^2-6 \mathrm{y}-\mathrm{x}+12=0<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
let-p-be-the-set-of-all-nonsingular-matrices-of-order-3-over-r-and-q-be-the-set-of-all-orthogonal-matrices-of-order-3-over-r-then

slug:
let-p-be-the-set-of-all-nonsingular-matrices-of-order-3-over-r-and-q-be-the-set-of-all-orthogonal-matrices-of-order-3-over-r-then-46531

question:
<div class="question">Let $P$ be the set of all non-singular matrices of order 3 over $R$ and $Q$ be the set of all orthogonal matrices of order 3 over $R .$ Then,</div>

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

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

answer:
<div class="solution">All the orthogonal matrix are non-singular matrix. $\therefore Q$ is proper subset of $P$</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-p-q-and-r-be-the-altitudes-of-a-triangle-with-area-s-and-perimeter-2-t-then-the-value-of-p-1-q-1-r-1-is

slug:
let-p-q-and-r-be-the-altitudes-of-a-triangle-with-area-s-and-perimeter-2-t-then-the-value-of-p-1-q-1-r-1-is-40637

question:
<div class="question">Let $p, q$ and $r$ be the altitudes of a triangle with area $S$ and perimeter $2 t .$ Then, the value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$ is</div>

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

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

answer:
<div class="solution">According to question, $S=\frac{1}{2} a \cdot p$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a64.png"/> <br/> <br/>$$
<br/>\begin{array}{c}
<br/>=\frac{1}{2} b \cdot q=\frac{1}{2} c \cdot r \\
<br/>\therefore \quad \frac{1}{p}=\frac{a}{2 S}, \frac{1}{q}=\frac{b}{2 S}, \frac{1}{r}=\frac{c}{2 S} \\
<br/>\therefore \quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{2 S}(a+b+c)=\frac{2 t}{2 S}=\frac{t}{S}
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-p-q-and-r-be-the-side4s-opposite-to-the-angles-p-q-and-r-respectively-in-a-pqr-then-2-p-r-sin-2-p-q-r-equals

slug:
let-p-q-and-r-be-the-side4s-opposite-to-the-angles-p-q-and-r-respectively-in-a-pqr-then-2-p-r-sin-2-p-q-r-equals-32815

question:
<div class="question">Let $p, q$ and $r$ be the side4s opposite to the angles $P, Q$ and $R,$ respectively in a $\Delta P Q R$. Then, $2 p r \sin \left(\frac{p-Q+R}{2}\right)$ equals</div>

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

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

answer:
<div class="solution">In $\Delta P Q R, P+Q+R=180^{\circ}$
<br/>$\begin{aligned} \therefore 2 p r \sin \left(\frac{P-Q+R}{2}\right) &amp;=2 p r \sin \frac{180^{\circ}-Q-Q}{2} \\ &amp;=2 p r \sin \left(90^{\circ}-Q\right) \\ &amp;=2 p r \cos Q \\=&amp;\left(\quad \ln \Delta P Q R \cos Q=\frac{p^{2}+r^{2}-q^{2}}{2 p r}\right) \\=&amp; 2 p r\left(\frac{p^{2}+r^{2}-q^{2}}{2 p r}\right)=p^{2}+r^{2}-q^{2} \end{aligned}$</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-p-q-and-r-be-the-sides-opposite-to-the-angles-p-q-and-r-respectively-in-a-pqr-if-r-2-sin-p-sin-q-pq-then-the-triangle-is

slug:
let-p-q-and-r-be-the-sides-opposite-to-the-angles-p-q-and-r-respectively-in-a-pqr-if-r-2-sin-p-sin-q-pq-then-the-triangle-is-14747

question:
<div class="question">Let $p, q$ and $r$ be the sides opposite to the angles $P, Q$ and $R,$ respectively in a $\Delta P Q R .$ If $r^{2} \sin P \sin Q=p q,$ then the triangle is</div>

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

answer:
<div class="solution">We know that in $\Delta A B C$ $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$
<br/>$\therefore \quad r^{2} \sin P \sin Q=p q$
<br/>$\Rightarrow r^{2} \cdot \frac{p}{2 R_{1}} \cdot \frac{q}{2 R_{1}}=p q,$ where $R_{1}$ is circumras
<br/>of $\Delta P Q R$. $\begin{array}{ll}\Rightarrow &amp; r^{2}=4 R_{1}^{2} \\ \Rightarrow &amp; r=2 R_{1} \\ \Rightarrow &amp; 2 R_{1} \sin R=2 R_{1}\end{array}$
<br/>$\Rightarrow \quad R_{1}=90^{\circ}$
<br/>$\therefore \Delta P Q R$ is right angled.</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
let-p-q-be-real-numbers-if-is-the-root-of-x-2-3-p-2-x-5-q-2-0-is-a-root-of-x-2-9-p-2-x-15-q-2-0-and-0-then-the-equation-x-2-6-p-2-x-10-q-2-0-has-a-roo

slug:
let-p-q-be-real-numbers-if-is-the-root-of-x-2-3-p-2-x-5-q-2-0-is-a-root-of-x-2-9-p-2-x-15-q-2-0-and-0-then-the-equation-x-2-6-p-2-x-10-q-2-0-has-a-roo-97909

question:
<div class="question">Let $p, q$ be real numbers. If $\alpha$ is the root of $x^{2}+3 p^{2} x+5 q^{2}=0, \quad \beta \quad$ is $\quad$ a $\quad$ root $\quad$ of $x^{2}+9 p^{2} x+15 q^{2}=0$ and $0    &lt; \alpha &lt; \beta,$ then the equation $x^{2}+6 p^{2} x+10 q^{2}=0$ has a root $\gamma$ that always satisfies</div>

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

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

answer:
<div class="solution">Since, $\alpha$ is a root of
<br/>$\therefore$
<br/>$$
<br/>\begin{array}{l}
<br/>x^{2}+3 p^{2} x+5 q^{2}=0 \\
<br/>\alpha^{2}+3 p^{2} \alpha+5 q^{2}=0
<br/>\end{array}
<br/>$$
<br/>and $\beta$ is a root of
<br/>$\therefore$
<br/>$$
<br/>\begin{array}{l}
<br/>x^{2}+9 p^{2} x+15 q^{2}=0 \\
<br/>\beta^{2}+9 p^{2} \beta+15 q^{2}=0
<br/>\end{array}
<br/>$$
<br/>Let $f(x)=x^{2}+6 p^{2} x+10 q^{2}$
<br/>$\begin{aligned} \text { Then, } f(\alpha) &amp;=\alpha^{2}+6 p^{2} \alpha+10 q^{2} \\ &amp;=\left(\alpha^{2}+3 p^{2} \alpha+5 q^{2}\right)+3 p^{2} \alpha+5 q^{2} \end{aligned}$
<br/>$=0+3 p^{2} \alpha+5 q^{2}$
<br/>$\Rightarrow \quad f(\alpha)&gt;0$
<br/>and $\quad f(\beta)=\beta^{2}+6 p^{2} \beta+10 q^{2}$
<br/>$=\left(\beta^{2}+9 p^{2} \beta+15 q^{2}\right)-\left(3 p^{2} \beta+5 q^{2}\right)$
<br/>$=0-\left(3 p^{2} \beta+5 q^{2}\right) \quad$ I from Eq. (ii)]
<br/>$\Rightarrow \quad f(\beta)    &lt; 0$
<br/>Thus, $f(x)$ is a polynomial such that $f(\alpha)&gt;0$ and $f(\beta) &lt; 0$
<br/>Therefore, there exists $\gamma$ satisfying $\alpha &lt; \gamma &lt; \beta$ such that $f(\gamma)=0$</div>


tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
let-p-x-be-a-polynomial-which-when-divided-by-x-3-and-x-5-leaves-remainders-10-and-6-respectively-if-the-polynomial-is-divided-by-x-3-x-5-thes-the-rem

slug:
let-p-x-be-a-polynomial-which-when-divided-by-x-3-and-x-5-leaves-remainders-10-and-6-respectively-if-the-polynomial-is-divided-by-x-3-x-5-thes-the-rem-86419

question:
<div class="question">Let $P(x)$ be a polynomial, which when $\begin{array}{llll}\text { divided by } &amp; (x-3) &amp; \text { and } &amp; (x-5) &amp; \text { leaves }\end{array}$ remainders 10 and 6, respectively. If the polynomial is divided by $(x-3)(x-5),$ thes the remainder is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-2 x+16$</span> </div>

answer:
<div class="solution">$\because P(x)=(x-3)(x-5)$
<br/>$=O(x)+(a x+b)$
<br/>Given, $\quad P(3)=10$ and $P(5)=6$
<br/>$\Rightarrow \quad 3 a+b=10$
<br/>and $\quad 5 a+b=6$
<br/>On solving Bqs.
<br/>(i) and (ii), we get $a=-2$ and $b=16$
<br/>$\therefore$ Remainder $=-2 x+16$</div>


title: let-p-x-be-a-polynomial-with-real-coefficients-p-0-1-and-p-x-0-for-all-x-r-then
slug: let-p-x-be-a-polynomial-with-real-coefficients-p-0-1-and-p-x-0-for-all-x-r-then-14488
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">$P(x)=0$ has exactly one negative real root.</div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $p(x)$ be a polynomial with real co-efficients, $p(0)=1$ and $p^{\prime}(x) &gt; 0$ for all $x \in \mathbb{R}$. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{p}(\mathrm{x})$ may have negative real root</span> </div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
let-p-x-be-a-quadratic-polynomial-with-constant-term-1-suppose-p-x-when-divided-by-x-1-leaves-remainder-2-and-when-divided-by-x-1-leaves-remainder-4-t

slug:
let-p-x-be-a-quadratic-polynomial-with-constant-term-1-suppose-p-x-when-divided-by-x-1-leaves-remainder-2-and-when-divided-by-x-1-leaves-remainder-4-t-48304

question:
<div class="question">Let $p(x)$ be a quadratic polynomial with constant term 1. Suppose $p(x),$ when divided by $x-1$ leaves remainder 2 and when divided by $x+1$ leaves remainder $4 .$ Then, the sum of the roots of $p(x)=0$ is</div>

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

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

answer:
<div class="solution">Let $\quad p(x)=a x^{2}+b x+c$
<br/>Given, constant term 'c' $=1$
<br/>$\therefore \quad p(x)=a x^{2}+b x+1$
<br/>Now, by given condition, $p(1)=2$ (remainder)
<br/>$\begin{array}{lr}\Rightarrow &amp; a+b+1=2 \\ \Rightarrow &amp; a+b=1\end{array}$
<br/>$\begin{array}{ll}\text { and } &amp; p(-1)=4 \quad \text { (remainder) }\end{array}$
<br/>$\begin{array}{lr}\Rightarrow &amp; a-b+1=4 \\ \Rightarrow &amp; a-b=3\end{array}$
<br/>On adding Eqs. (iii) and (iv), we get
<br/>$2 a=4 \Rightarrow a=2$ form eys
<br/>(iii) $b=-1$
<br/>On putting the values of a and b in Eq. (ii), we get $\begin{aligned} p(x)=2 x^{2}-x+1=0 \\ \text { Sum of the roots } &amp;=-\frac{\text { (Coefficient of } x)}{\text { (Coefficient of } x^{2} \text { ) }}=\frac{-(-1)}{2} \\ &amp;=\frac{1}{2} \end{aligned}$</div>


title: let-pn32-n12n2-where-n-in-mathbbn-then
slug: let-pn32-n12n2-where-n-in-mathbbn-then-71994
question: <div class="question">Let \(P(n)=3^{2 n+1}+2^{n+2}\) where \(n \in \mathbb{N}\). Then</div>
tags: ['Mathematics', 'Mathematical Induction', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{P}(\mathrm{n})\) is not divisible by any prime integer.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">there exists prime integer which divides \(P(n)\).</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(P(n)\) is divisible by 5 for all \(n \in \mathbb{N}\).</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{P}(\mathrm{n})\) is divisible by 3 for all \(\mathrm{n} \in \mathbb{N}\).</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there exists prime integer which divides \(P(n)\).</span> </div>

answer:
<div class="solution">Hint : \(P(1)=3^3+2^3\) which is divisible by \((3+2)=5\) which is prime<br/>\(\therefore\) There exists prime integer which divides \(\mathrm{P}(\mathrm{n})\)</div>


title: let-q-cos-4-sin-4-sin-4-cos-4-and-x-2-1-2-1-then-q-3-x-is-equal-to
slug: let-q-cos-4-sin-4-sin-4-cos-4-and-x-2-1-2-1-then-q-3-x-is-equal-to-15741
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $Q=\left(\begin{array}{cc}\cos \frac{\pi}{4} &amp; -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} &amp; \cos \frac{\pi}{4}\end{array}\right)$ and$x=\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)$ then $Q^{3} x$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\begin{array}{c}-1 \\ 0\end{array}\right)$</span> </div>

answer:
<div class="solution">Given,
<br/>$Q=\left[\begin{array}{cc}\cos \frac{\pi}{4} &amp; -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} &amp; \cos \frac{\pi}{4}\end{array}\right]$ and $x=\left[\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right]$
<br/>$\begin{aligned} \text { Let } Q(\theta) &amp;=\left[\begin{array}{cc}\cos \theta &amp; -\sin \theta \\ \sin \theta &amp; \cos \theta\end{array}\right] \\ Q^{3}(\theta) &amp;=Q(3 \theta) \end{aligned}$
<br/>$\therefore Q^{3}\left(\frac{\pi}{4}\right)=\left[\begin{array}{ll}\cos \frac{3 \pi}{4} &amp; -\sin \frac{3 \pi}{4} \\ \sin \frac{3 \pi}{4} &amp; \cos \frac{3 \pi}{4}\end{array}\right]=\left[\begin{array}{cc}-1 / \sqrt{2} &amp; -1 / \sqrt{2} \\ 1 / \sqrt{2} &amp; -1 / \sqrt{2}\end{array}\right]$
<br/>Now, $Q^{3}\left(\frac{\pi}{4}\right) x=\left[\begin{array}{cc}\frac{-1}{\sqrt{2}} &amp; -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} &amp; -\frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right]=\left[\begin{array}{c}-\frac{1}{2}-\frac{1}{2} \\ \frac{1}{2}-\frac{1}{2}\end{array}\right]$
<br/>$\therefore \quad Q^{3}\left(\frac{\pi}{4}\right) x=\left[\begin{array}{c}-1 \\ 0\end{array}\right]$</div>


title: let-r-and-s-be-two-equivalence-relations-on-a-nonvoid-set-a-then
slug: let-r-and-s-be-two-equivalence-relations-on-a-nonvoid-set-a-then-15302
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">(B) $\mathrm{R} \cap \mathrm{S}$ is equivalence relation</div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $R$ and $S$ be two equivalence relations on a non-void set $A$. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{R} \cap \mathrm{S}$ is equivalence relation</span> </div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
let-r-be-a-relation-defined-on-the-set-z-of-all-integers-and-x-r-y-when-x-2-y-is-divisible-by-3-then

slug:
let-r-be-a-relation-defined-on-the-set-z-of-all-integers-and-x-r-y-when-x-2-y-is-divisible-by-3-then-33523

question:
<div class="question">Let $R$ be a relation defined on the set $Z$ of all integers and $x R y,$ when $x+2 y$ is divisible by 3 , then</div>

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

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

answer:
<div class="solution">Reflexivity For reflexive $(x, x) \in R .$<br/>$x+2 x=3 x$<br/>which is divisible by 3.<br/>$\Rightarrow$ xRx $\in R$ Hence, xRy is reflexive.<br/>Symmetric Let $x+2 y=3 \lambda$<br/>$\Rightarrow \quad x=3 \lambda-2 y$<br/>$\begin{aligned}<br/>\text {Now, } &amp; \begin{aligned}<br/>y+2 x &amp;=y+2(3 \lambda-2 y) \\<br/>&amp;=y+6 \lambda-4 y \\<br/>&amp;=6 \lambda-3 y<br/>\end{aligned}<br/>\end{aligned}$<br/>$\Rightarrow \quad y+2 x=3(2 \lambda-y)$<br/>which is divisible by 3.<br/>$\therefore$ $(x, y) \in R \Rightarrow(y, x) \in R$<br/>Hence, xRy is symmetric.<br/>$\begin{array}{ll}\text {Transifive } &amp; x+2 y=3 \lambda \\ \text { and } &amp; y+2 z=3 \mu\end{array}$<br/>$x+3 y+2 z=3 x+3 \mu$<br/>$\Rightarrow \quad x+2 z=3(\lambda+\mu-y)$<br/>which is divisible by $3 .$<br/>$\because (x, y) \in$ R and $(y, z) \in R$<br/>$\Rightarrow \quad(x, z) \in R$<br/>Hence, xRy is transitive<br/>$\because$ R is reflexive, symmetric and transitive.<br/>$\therefore$ R is an equivalence relation.</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2021']
answer: <div class="solution">$\mathrm{T}$ is an equivalence but $\mathrm{S}$ is not</div>
source_db: db\MarksBatch2_P2.db
title:
let-r-be-the-real-line-let-the-relations-s-and-t-on-r-be-defined-by-s-x-y-y-x-1-0-x-2-t-x-y-x-y-is-an-integer-then

slug:
let-r-be-the-real-line-let-the-relations-s-and-t-on-r-be-defined-by-s-x-y-y-x-1-0-x-2-t-x-y-x-y-is-an-integer-then-39254

question:
<div class="question">Let $R$ be the real line. Let the relations $S$ and $T$ on $R$ be defined by $S=\{(x, y): y=x+1,0     &lt; x &lt; 2\}, T=\{(x, y):(x-y)$ is an integer $\}$. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{T}$ is an equivalence on $\mathrm{R}$ but $\mathrm{S}$ is not</span> </div>


title: let-r-be-the-set-of-all-real-numbers-and-f-1-1-r-be-defined-f-x-x-sin-x-1-0-x-0-then-x-0
slug: let-r-be-the-set-of-all-real-numbers-and-f-1-1-r-be-defined-f-x-x-sin-x-1-0-x-0-then-x-0-19213
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $R$ be the set of all real numbers and	$f:[-1,1] \rightarrow R$ be defined $f(x)=\left\{\begin{array}{ll}x \sin \frac{1}{x}, &amp; x \neq 0 \text { . Then } \\ 0, &amp; x=0\end{array}\right.$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ satisfies the conditions of Lagrange's mean value theorem on [0,1]</span> </div>

answer:
<div class="solution">Given, $\quad f(x)=\left\{\begin{array}{cl}x \sin \frac{1}{x}, &amp; x \neq 0 \\ 0, &amp; x=0\end{array}\right.$
<br/>Continuity at $x=0$. $\mathrm{LHL}=\lim _{x \rightarrow 0^{-}} x \sin \frac{1}{x}=0$
<br/>$\mathrm{RHL}=\lim _{x \rightarrow 0^{*}} x \sin \frac{1}{x}=0$
<br/>and
<br/>$\mathrm{LHL}=\mathrm{RHL}=f(0)$
<br/>Hence, $f(x)$ is continuous for all values of $x$. Differentiability at $x=0$
<br/>$$
<br/>\begin{aligned}
<br/>\mathrm{LHD} &amp;=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=\lim _{h \rightarrow 0} \frac{h \sin \frac{1}{h}}{h} \\
<br/>&amp;=-\lim _{h \rightarrow 0} \sin \frac{1}{h} \\
<br/>&amp;=-\sin \left(\frac{1}{0}\right)=\text { some definite value }
<br/>\end{aligned}
<br/>$$
<br/>Thus, $f(x)$ is not differentiable at $x=0 .$ Now, $f(0)=0$ and $f(1)=1 \sin 1$
<br/>$\therefore f(0) \neq f(1)$. Rolle's theorem is not satisfy.
<br/>Now.
<br/>$$
<br/>\begin{aligned}
<br/>f^{\prime}(c) &amp;=\frac{f(1)-f(0)}{b-0} \\
<br/>&amp;=\frac{1 \sin (1)-0}{1-0}=\sin 1=\sin 1
<br/>\end{aligned}
<br/>$$
<br/>which lies between 0 to 1 .
<br/>Hence, there exist a real number $c \in(0,1)$ Hence, 
<br/>Lagrange's mean value theorem is satisfy on $(0,1)$.</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-r-be-the-set-of-real-numbers-and-the-functions-f-r-r-and-g-r-r-be-defined-by-f-x-x-2-2-x-3-and-g-x-x-1-then-the-value-of-x-for-which-f-g-x-g-f-x-i

slug:
let-r-be-the-set-of-real-numbers-and-the-functions-f-r-r-and-g-r-r-be-defined-by-f-x-x-2-2-x-3-and-g-x-x-1-then-the-value-of-x-for-which-f-g-x-g-f-x-i-46800

question:
<div class="question">Let $R$ be the set of real numbers and the functions $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x^{2}+2 x-3$ and $g(x)=x+1 .$ Then, the value of $x$ for which $f(g(x))=g(f(x))$ is</div>

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

answer:
<div class="solution">According to question,
<br/>$$
<br/>f(g(x))=g(f(x))
<br/>$$
<br/>$\Rightarrow \quad f(x+1)=g\left(x^{2}+2 x-3\right)$
<br/>$\Rightarrow \quad(x+1)^{2}+2(x+1)-3=x^{2}+2 x-3+1$
<br/>$\Rightarrow \quad x^{2}+1+2 x+2 x+2-3=x^{2}+2 x-2$
<br/>$\Rightarrow \quad x^{2}+4 x=x^{2}+2 x-2$
<br/>$\Rightarrow \quad x^{2}+4 x-x^{2}-2 x+2=0$
<br/>$\Rightarrow \quad 2 x+2=0$
<br/>$\Rightarrow \quad 2 x=-2$
<br/>$$
<br/>\Rightarrow \quad x=-1
<br/>$$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-44</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-r-be-the-set-of-real-numbers-and-the-mapping-f-r-r-and-g-r-r-be-defined-by-f-x-5-x-2-and-g-x-3-x-4-then-the-value-of-f-g-1-is

slug:
let-r-be-the-set-of-real-numbers-and-the-mapping-f-r-r-and-g-r-r-be-defined-by-f-x-5-x-2-and-g-x-3-x-4-then-the-value-of-f-g-1-is-68896

question:
<div class="question">Let $R$ be the set of real numbers and the mapping $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=5-x^2$ and $g(x)=3 x-4$, then the value of $(f \circ g)(-1)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-44</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">-54</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-32</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-64</span> </li> </ul>

answer:
<div class="solution">Hints: $\mathrm{f}(\mathrm{g}(-1))=\mathrm{f}(-3-4)=\mathrm{f}(-7)=5-49=-44$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">vr</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-r-v-e-be-the-radius-of-orbit-speed-of-electron-and-total-energy-of-electron-respectively-in-a-h-atom-which-of-the-following-quantities-according-t

slug:
let-r-v-e-be-the-radius-of-orbit-speed-of-electron-and-total-energy-of-electron-respectively-in-a-h-atom-which-of-the-following-quantities-according-t-52871

question:
<div class="question">Let $\mathrm{r}, \mathrm{v}, \mathrm{E}$ be the radius of orbit, speed of electron and total energy of electron respectively in a $\mathrm{H}$-atom. Which of the following quantities according to Bohr theory, is proportional to the quantum number $\mathrm{n}$ ?</div>

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

answer:
<div class="solution">$\mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \quad \therefore \mathrm{n}=\frac{\mathrm{m} 2 \pi}{\mathrm{h}} \cdot \mathrm{vr} \quad \therefore \mathrm{n} \propto \mathrm{vr}$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
let-rho-be-a-relation-defined-on-set-of-natural-numbers-mathbbn-as-rhox-y-in-mathbbn-times-mathbbn-2-xy41-then-domain-a-and-range-b-are

slug:
let-rho-be-a-relation-defined-on-set-of-natural-numbers-mathbbn-as-rhox-y-in-mathbbn-times-mathbbn-2-xy41-then-domain-a-and-range-b-are-45040

question:
<div class="question">Let \(\rho\) be a relation defined on set of natural numbers \(\mathbb{N}\), as \(\rho=\{(x, y) \in \mathbb{N} \times \mathbb{N}: 2 x+y=41\}\). Then domain \(A\) and range B are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\mathrm{A} \subset\{\mathrm{x} \in \mathbb{N}: 1 \leq \mathrm{x} \leq 20\}\) and \(B \subset\{\mathrm{y} \in \mathbb{N}: 1 \leq \mathrm{y} \leq 39\}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(A=\{x \in \mathbb{N}: 1 \leq x \leq 15\}\) and \(B=\{y \in \mathbb{N}: 2 \leq \mathrm{y} \leq 30\}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\mathrm{A} \equiv \mathbb{N}, \mathrm{B} \equiv \mathbb{Q}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(A \equiv \mathbb{Q}, B \equiv \mathbb{Q}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{A} \subset\{\mathrm{x} \in \mathbb{N}: 1 \leq \mathrm{x} \leq 20\}\) and \(B \subset\{\mathrm{y} \in \mathbb{N}: 1 \leq \mathrm{y} \leq 39\}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned} \text {Hint: } &amp;<br/>2 x  +y=41 \\<br/>&amp; y=41-2 x \\<br/>\therefore &amp; x=\{1,2,3,4, \ldots .20\} \\<br/>&amp; y=\{1,3,5, \ldots .39\}<br/>\end{aligned}\)</div>


title: let-s-1-2-n-c-0-2-2-2-n-c-1-3-2-3-n-c-2-n-1-2-n-1-n-c-n-then-s-equals
slug: let-s-1-2-n-c-0-2-2-2-n-c-1-3-2-3-n-c-2-n-1-2-n-1-n-c-n-then-s-equals-89194
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $S=\frac{2}{1}{ }^{n} C_{0}+\frac{2^{2}}{2}{ }^{n} C_{1}+\frac{2^{3}}{3}{ }^{n} C_{2}$ $+\ldots+\frac{2^{n+1}}{n+1}{ }^{n} C_{n} .$ Then, $S$ equals</div>

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

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

answer:
<div class="solution">We know that $(1+x)^{n}={ }^{n} C_{0}+x{ }^{-} C_{1}+x^{2}{ }^{n} C_{2}+\ldots+x^{n}{ }^{n} C_{n}$
<br/>On integrating both sides from 0 to 2 , we get
<br/>$\left[\frac{(1+x)^{n+1}}{n+1}\right]_{0}^{2}$
<br/>$=\left[x^{n} C_{0}+\frac{x^{2}}{2}^{n} C_{1}+\frac{x^{3}}{3}^{n} C_{2}+\ldots+\frac{x^{n+1}}{n+1}^{n} C_{n}\right]_{0}^{2}$
<br/>$\Rightarrow \frac{(3)^{n+1}}{n+1}-\frac{1}{n+1}=2{ }^{n} C_{0}+\frac{2^{2}}{2}{ }^{n} C_{1}+\frac{2^{3}}{3}{ }^{n} C_{2}$
<br/>$\quad+\ldots+\frac{2^{n+1}}{n+1}{ }^{n} C_{n}-0$
<br/>$\Rightarrow \frac{2}{1}{ }^{n} C_{0}+\frac{2^{2}}{2}{ }^{n} C_{1}+\frac{2^{3}}{3}{ }^{n} C_{2}+\ldots+\frac{2^{n+1}}{n+1}{ }^{n} C_{n}$
<br/>$\quad=\frac{3^{n+1}-1}{n+1}$</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-s-a-b-c-n-n-n-a-b-c-21-a-b-c-and-t-a-b-c-n-n-n-a-b-c-are-in-a-p-where-n-is-the-set-of-all-natural-numbers-then-the-number-of-elements-in-the-set-s

slug:
let-s-a-b-c-n-n-n-a-b-c-21-a-b-c-and-t-a-b-c-n-n-n-a-b-c-are-in-a-p-where-n-is-the-set-of-all-natural-numbers-then-the-number-of-elements-in-the-set-s-43389

question:
<div class="question">Let $S=\{(a, b, c) \in N \times N \times N: a+b+c=21$, $a \leq b \leq c\}$ and $T=\{(a, b, c) \in N \times N \times N: a, b, c$ are in $A P \}$ where $N$ is the set of all natural numbers. Then, the number of elements in the set $\mathrm{S} \cap T$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">13</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">14</span> </li> </ul>

answer:
<div class="solution">We have, $a+b+c=21$ and $\frac{a+c}{2}=b$ $\Rightarrow \quad a+c+\frac{a+c}{2}=21$
<br/>$\begin{array}{ll}\Rightarrow &amp; a+c=14 \Rightarrow \frac{a+c}{2}=7 \\ \Rightarrow &amp; b=7\end{array}$
<br/>So, a can take values from 1 to 6 . C can have values from 8 to 13
<br/>or a=b=c=7
<br/>So, there are 7 such triplets.</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
let-s-be-the-sample-space-of-the-random-experiment-of-throwing-simultaneously-two-unbiased-dice-and-eka-b-in-s-a-bk-if-pkpleftekright-then-the-correct

slug:
let-s-be-the-sample-space-of-the-random-experiment-of-throwing-simultaneously-two-unbiased-dice-and-eka-b-in-s-a-bk-if-pkpleftekright-then-the-correct-25241

question:
<div class="question">Let \(S\) be the sample space of the random experiment of throwing simultaneously two unbiased dice and \(E_k=\{a, b) \in\) \(S: a b=k\}\). If \(p_k=P\left(E_k\right)\), then the correct among the following is :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(p_1  &lt; p_{10} &lt; p_4\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{p}_2  &lt; \mathrm{p}_8 &lt; \mathrm{p}_{14}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\mathrm{p}_4  &lt; \mathrm{p}_8 &lt; \mathrm{p}_{17}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{p}_2  &lt; \mathrm{p}_{16} &lt; \mathrm{p}_5\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(p_1  &lt; p_{10} &lt; p_4\)</span> </div>

answer:
<div class="solution">Hint: \(E_1=\{(1,1)\}, \quad E_4=\{(2,2),(1,4),(4,1)\} \quad E_{10}=\{(2,5),(5,2)\}\)<br/>\(p_1=\frac{1}{36}\)<br/>\(p_4=\frac{3}{36}=\frac{1}{12}\),<br/>\(p_{10}=\frac{2}{36}=\frac{1}{18}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-s-be-the-set-of-points-whose-abscissae-and-ordinates-are-natural-numbers-let-p-s-such-that-the-sum-of-the-distance-of-p-from-80-and-012-is-minimum

slug:
let-s-be-the-set-of-points-whose-abscissae-and-ordinates-are-natural-numbers-let-p-s-such-that-the-sum-of-the-distance-of-p-from-80-and-012-is-minimum-42717

question:
<div class="question">Let $S$ be the set of points, whose abscissae and ordinates are natural numbers. Let $P \in S$, such that the sum of the distance of $P$ from (8,0) and (0,12) is minimum among all elements in $S .$ Then, the number of such points $P$ in $S$ is</div>

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

answer:
<div class="solution">Sum of distances of point $P$ from point (8,0) and (0,12) will be minimum, if points are collinear.
<br/>Equation of line at point (8,0) and (0,12)
<br/>$$
<br/>\begin{aligned}
<br/>\frac{x}{8}+\frac{y}{12} &amp;=1 \\
<br/>\Rightarrow \quad \frac{y}{12} &amp;=1-\frac{x}{8} \\
<br/>\Rightarrow \quad y &amp;=12-\frac{x}{8} \times 12 \\
<br/>\Rightarrow \quad y &amp;=12-\frac{3}{2} x
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Points $(x, y)=(2,9),(4,6)$ and (6,3) . Hence, the number of such points $P$ in S is 3.</div>


title: let-s-denote-the-sum-of-the-infinite-series-1-2-8-3-21-4-40-5-65-then
slug: let-s-denote-the-sum-of-the-infinite-series-1-2-8-3-21-4-40-5-65-then-71347
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $S$ denote the sum of the infinite series $1+\frac{8}{2 !}+\frac{21}{3 !}+\frac{40}{4 !}+\frac{65}{5 !}+\ldots .$ Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8    &lt; S &lt; 12$</span> </div>

answer:
<div class="solution">Let $S=1+\frac{8}{2 !}+\frac{21}{3 !}+\frac{40}{4 !}+\frac{65}{5 !}+\ldots$
<br/>Again, let $S_{1}=1+8+21+40+65+\ldots+T_{n}$
<br/>and $\quad S_{1}=+1+8+21+40+\ldots+T_{n}$
<br/>$$
<br/>\begin{aligned} 0=1+7+13+19+25+\ldots-T_{n} \\ T_{n} &amp;=1+7+13+19+25+\ldots+n \text { terms } \\ &amp;=\frac{n}{2}[2(1)+(n-1) 6] \\ &amp;=n[1+3(n-1)]=n(3 n-2) \end{aligned}
<br/>$$
<br/>$S=\sum \frac{3 n-2}{(n-1) !}$
<br/>$=\sum \frac{3 n-3+1}{(n-1) !}$
<br/>$=\sum \frac{3}{(n-2) !}+\sum \frac{1}{(n-1) !}$
<br/>$\quad=3 e+e \quad\left[\because e=1+\frac{1}{1 !}+\frac{1}{2 !}+\ldots\right]$
<br/>$=4 e$
<br/>We know $\quad 2    &lt; e &lt; 3$
<br/>$\therefore \quad 8 &lt; 4 e &lt; 12$
<br/>$\Rightarrow \quad 8 &lt; S &lt; 12$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-s-k-be-the-sum-of-an-infinite-gp-series-whose-first-term-is-k-and-common-ratio-is-k-1-k-k-0-then-the-value-of-k-1-s-k-1-k-is-equal-to

slug:
let-s-k-be-the-sum-of-an-infinite-gp-series-whose-first-term-is-k-and-common-ratio-is-k-1-k-k-0-then-the-value-of-k-1-s-k-1-k-is-equal-to-73325

question:
<div class="question">Let $S_{k}$ be the sum of an infinite GP series whose first term is $k$ and common ratio is $\frac{k}{k+1}(k&gt;0)$ Then, the value of $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{S_{k}}$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1-\log _{e} 4$</span> </div>

answer:
<div class="solution">\(S_{1}=1+\frac{1}{2}+\frac{1}{2^{2}}+\infty=\frac{1}{1-\frac{1}{2}}=2\)
<br/>\(S_{2}=2+2 \cdot \frac{2}{3}+2\left(\frac{2}{3}\right)^{2}+\ldots \infty=\frac{2}{1-\frac{2}{3}}=6\)
<br/>\(S_{3}=3+3\left(\frac{3}{4}\right)+3\left(\frac{3}{4}\right)^{2}+\infty=\frac{3}{1-\frac{3}{4}}=12\)
<br/>\(S_{4}=4+4\left(\frac{4}{5}\right)+4\left(\frac{4}{5}\right)^{2}+\infty=\frac{4}{1-\frac{4}{5}}=20\)
<br/>\(\therefore \sum_{k=1}^{\infty} \frac{(-1)^{k}}{S_{k}}=-\frac{1}{S_{1}}+\frac{1}{S_{2}}-\frac{1}{S_{3}}+\frac{1}{S_{4}}-\infty\)
<br/>\(=-\frac{1}{2}+\frac{1}{6}-\frac{1}{12}+\frac{1}{20}-\infty\)
<br/>\(=-\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}-\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}-\infty\)
<br/>\(=-\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)-\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)-\ldots \infty\)
<br/>\(=-1+2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\infty\right)\)
<br/>\(=-1-2\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots \infty\right)+2 \infty\)
<br/>\(=-1-2 \log _{e} 2+2=1-\log _{e} 4\)</div>


title: let-s-n-cot-1-2-cot-1-8-cot-1-18-cot-1-32-to-n-th-term-then-lim-n-s-n-is
slug: let-s-n-cot-1-2-cot-1-8-cot-1-18-cot-1-32-to-n-th-term-then-lim-n-s-n-is-12988
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $S_{n}=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18+\cot ^{-1} 32+\ldots \ldots .$ to $n^{\text {th }}$ term. Then $\lim _{n \rightarrow \infty} S_{n}$ is</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; t_{n}=\cot ^{-1} 2 n^{2} \\=&amp; \tan ^{-1} \frac{1}{2 n^{2}}=\tan ^{-1} \frac{(2 n+1)-(2 n-1)}{1+(2 n+1)(2 n-1)} \\=&amp; \tan ^{-1}(2 n+1)-\tan ^{-1}(2 n-1) \\ \therefore &amp; S_{n}=\tan ^{-1}(2 n+1)-\tan ^{-1} 1 \\ \therefore &amp; \operatorname{Lim}_{n \rightarrow \infty} S_{n}=\pi / 2-\pi / 4=\pi / 4 \end{aligned}$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
let-s-t-u-be-three-nonvoid-sets-and-f-s-t-g-t-u-and-composed-mapping-gof-s-u-be-defined-let-gof-be-injective-mapping-then

slug:
let-s-t-u-be-three-nonvoid-sets-and-f-s-t-g-t-u-and-composed-mapping-gof-s-u-be-defined-let-gof-be-injective-mapping-then-59351

question:
<div class="question">Let $\mathrm{S}, \mathrm{T}, \mathrm{U}$ be three non-void sets and $\mathrm{f}: \mathrm{S} \rightarrow \mathrm{T}, \mathrm{g}: \mathrm{T} \rightarrow \mathrm{U}$ and composed mapping $\mathrm{gof}: \mathrm{S} \rightarrow U$ be defined. Let $\mathrm{gof}$ be injective mapping. Then</div>

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

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

answer:
<div class="solution">Let, $x_1, x_2 \in S$ and $f\left(x_1\right)=f\left(x_2\right)$<br/>$\Rightarrow g \circ f\left(x_1\right)=g\left(f\left(x_1\right)\right)=g\left(f\left(x_2\right)\right)=g$ of $\left(x_2\right)$<br/>$\Rightarrow \mathrm{x}_1=\mathrm{x}_2$ (as gof is injective)<br/>$\Rightarrow \mathrm{f}(\mathrm{x})$ is obviously injective.</div>


title: let-s-t-u-be-three-nonvoid-sets-and-f-s-t-g-t-u-be-so-that-g-o-f-s-u-is-surjective-then
slug: let-s-t-u-be-three-nonvoid-sets-and-f-s-t-g-t-u-be-so-that-g-o-f-s-u-is-surjective-then-38378
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $S, T, U$ be three non-void sets and $f: S \rightarrow T, g: T \rightarrow U$ be so that $g o f: s \rightarrow U$ is surjective. Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$g$ is surjective, $f$ may not be so</span> </div>

answer:
<div class="solution">We have,
<br/>gof : $S \rightarrow U$ is an onto function.
<br/>Let $Z$ be any arbitrary element such that $Z \in U$. Now, $g o f: S \rightarrow U$ is onto $\Rightarrow g_{0} f(x)=Z,$ for $x \in S$
<br/>$\Rightarrow g(f(x))=Z$
<br/>$\Rightarrow g(y)=Z,$ where $y=f(x) \in T$ for all $Z \in U$
<br/>$\therefore$ For all $Z \in U,$ there exists $y=f(x) \in T$ such that $g(y)=z$
<br/>$\therefore g: T \rightarrow U$ is an onto function.</div>


title: let-sin-cos-cos-cos-sin-sin-sin-sin-cos-sin-sin-cos-cos-sin-0-then
slug: let-sin-cos-cos-cos-sin-sin-sin-sin-cos-sin-sin-cos-cos-sin-0-then-52837
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $\Delta=\left|\begin{array}{ccc}\sin \theta \cos \phi &amp; \sin \theta \sin \phi &amp; \cos \theta \\ \cos \theta \cos \phi &amp; \cos \theta \sin \phi &amp; -\sin \theta \\ -\sin \theta \sin \phi &amp; \sin \theta \cos \phi &amp; 0\end{array}\right|$. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\Delta$ is independent of $\varphi$, $\left(\frac{d \Delta}{d \theta}\right)_{\theta-\frac{\pi}{2}}=0$</span> </div>

answer:
<div class="solution">$\Delta=\left|\begin{array}{ccc}\sin \theta \cos \phi &amp; \sin \theta \sin \phi &amp; \cos \theta \\ \cos \theta \cos \phi &amp; \cos \theta \sin \phi &amp; -\sin \theta \\ -\sin \theta \sin \phi &amp; \sin \theta \cos \phi &amp; 0\end{array}\right|$<br/>$\begin{aligned}<br/>\mathrm{C}_1 \longrightarrow \mathrm{C}_1-\left(\frac{\cos \phi}{\sin \phi}\right) \mathrm{C}_2 \Rightarrow \Delta=\left|\begin{array}{ccc}<br/>0 &amp; \sin \theta \sin \phi &amp; \cos \theta \\<br/>0 &amp; \cos \theta \sin \phi &amp; -\sin \theta \\<br/>-\frac{\sin \theta}{\sin \phi} &amp; \sin \theta \cos \phi &amp; \theta<br/>\end{array}\right| \\<br/>&amp;=\frac{\sin \theta}{\sin \phi} \cdot \sin \phi\left(\cos ^2 \theta+\sin ^2 \theta\right)= \sin \theta \rightarrow \text { Independent of } \phi \\<br/>\frac{\mathrm{d} \Delta}{\mathrm{d} \theta}=\left.\cos \theta \Rightarrow \frac{\mathrm{d} \Delta}{\mathrm{d} \theta}\right|_{\theta-\frac{\pi}{2}}=0<br/>\end{aligned}$</div>


title: let-t-n-denotes-the-nth-term-of-the-infinite-series-1-1-2-10-3-21-4-34-5-49-then-lim-n-t-n-is
slug: let-t-n-denotes-the-nth-term-of-the-infinite-series-1-1-2-10-3-21-4-34-5-49-then-lim-n-t-n-is-26382
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0$</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $t_{n}$ denotes the nth term of the infinite series $\frac{1}{1 !}+\frac{10}{2 !}+\frac{21}{3 !}+\frac{34}{4 !}+\frac{49}{5 !}+\ldots .$ Then $\lim _{n \rightarrow \infty} t_{n}$ is</div>

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

answer:
<div class="solution">Let $\quad S=1+10+21+34+49+\ldots+t_{n}^{\prime}$
<br/>and $S=1+10+21+34+\ldots+t_{n}^{\prime}$
<br/>$0=1+9+11+13+15+\ldots-t_{n}^{\prime}$
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow t_{n}^{\prime} &amp;=1+[9+11+13+15+\ldots(n-1) \text { term }] \\
<br/>&amp;=1+\left[\frac{n-1}{2}\{2 \times 9+(n-2) 2\}\right] \\
<br/>&amp;=1+(n-1)[9+n-2] \\
<br/>&amp;=1+(n-1)(n+7) \\
<br/>\therefore \quad t_{n} &amp;=\frac{t^{\prime}}{n !}=\frac{1+(n-1)(n+7)}{n !} \\
<br/>&amp;=\frac{1+(n-1)(n+7)}{n !} \\
<br/>&amp;=\frac{1+n^{2}+6 n-7}{n !} \\
<br/>&amp;=\frac{n}{(n-1) !}+\frac{6}{(n-1) !}-\frac{n^{2}+6 n-6}{n !} \\
<br/>&amp;=\frac{n-1+1}{(n-1) !}+\frac{6}{(n-1) !}-\frac{1}{n !} \\
<br/>&amp;=\frac{1}{(n-2) !}+\frac{7}{(n-1) !}-\frac{1}{n !} \\
<br/>&amp;=\frac{1}{n \rightarrow \infty} t_{n}=\lim _{n \rightarrow \infty}\left[\frac{1}{(n-2) !}+\frac{7}{(n-1) !}-\frac{1}{n !}\right]=0
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
let-t-u-be-the-set-of-all-orthogonal-matrices-of-order-3-over-r-the-set-of-all-nonsingular-matrices-of-order-3-over-r-respectively-let-a-1-0-1-then

slug:
let-t-u-be-the-set-of-all-orthogonal-matrices-of-order-3-over-r-the-set-of-all-nonsingular-matrices-of-order-3-over-r-respectively-let-a-1-0-1-then-16550

question:
<div class="question">Let $T \&amp; U$ be the set of all orthogonal matrices of order 3 over $R \&amp;$ the set of all non-singular matrices of order 3 over $\mathrm{R}$ respectively
<br/>Let $A=\{-1,0,1\}$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there does not exist bijective mapping between $\mathrm{A}$ and $\mathrm{T}, \mathrm{U}$.</span> </div>

answer:
<div class="solution">As $\mathrm{n}(\mathrm{A}) \neq \mathrm{n}(\mathrm{T})$ and $\mathrm{n}(\mathrm{A}) \neq \mathrm{n}(\mathrm{u})$</div>


title: let-tan-alphafracaa1-and-tan-betafrac12-a1-then-alphabeta-is
slug: let-tan-alphafracaa1-and-tan-betafrac12-a1-then-alphabeta-is-46712
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let \(\tan \alpha=\frac{a}{a+1}\) and \(\tan \beta=\frac{1}{2 a+1}\) then \(\alpha+\beta\) is</div>

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

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hints: } \tan \alpha=\frac{a}{a+1}, \tan \beta=\frac{1}{2 a+1} \\<br/>&amp; \tan (\alpha+\beta)=\frac{\frac{a}{a+1}+\frac{1}{2 a+1}}{1-\frac{a}{(a+1)(2 a+1)}}=\frac{\frac{a(2 a+1)+a+1}{(a+1)(2 a+1)}}{\frac{(a+1)(2 a+1)-a}{(a+1)(2 a+1)}}=\frac{2 a^2+2 a+1}{2 a^2+2 a+1}=1 \\<br/>&amp; \alpha+\beta=\frac{\pi}{4}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">64</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-the-coefficients-of-powers-of-x-in-the-2-n-d-3rd-and-4-th-terms-in-the-expansion-of-1-x-n-where-n-is-a-positive-integer-be-in-arithmetic-progressi

slug:
let-the-coefficients-of-powers-of-x-in-the-2-n-d-3rd-and-4-th-terms-in-the-expansion-of-1-x-n-where-n-is-a-positive-integer-be-in-arithmetic-progressi-77403

question:
<div class="question">Let the coefficients of powers of $x$ in the $2 n d$, 3rd and 4 th terms in the expansion of $(1+x)^{n}$ where $n$ is a positive integer, be in arithmetic progression. Then, the sum of the coefficients of odd powers of $x$ in the expansion is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">32</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">64</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">128</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">256</span> </li> </ul>

answer:
<div class="solution">According to question, ${ }^{n} C_{1},{ }^{n} C_{2}$ and ${ }^{n} C_{3}$ are in $\mathrm{AP}$
<br/>$\Rightarrow \quad \frac{2 n(n-1)}{2 !}=n+\frac{n(n-1)(n-2)}{3 !}$
<br/>$\Rightarrow \quad n^{2}-9 n+14=0$
<br/>$\Rightarrow \quad(n-7)(n-2)=0$
<br/>$\Rightarrow \quad n=7$ since $\underline{n \neq 2}$
<br/>Â¿The sum of the coefficients of odd powers of $x$ in the expansion of $(1+x)^{n}$ is $\frac{2^{n}}{2}=\frac{2^{\gamma}}{2}=2^{6}=64$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8: 1</span> </div>
source_db: 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-2-9-y-2-9-then-the-ratio-a-2-b-2-equals

slug:
let-the-eccentricity-of-the-hyperbola-a-2-x-2-b-2-y-2-1-be-reciprocal-to-that-of-the-ellipse-x-2-9-y-2-9-then-the-ratio-a-2-b-2-equals-56743

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 $x^{2}+9 y^{2}=9,$ then the ratio $a^{2}: b^{2}$ equals</div>

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

answer:
<div class="solution">Given equation of ellipses is $x^{2}+9 y^{2}=9$
<br/>$\Rightarrow \quad \frac{x^{2}}{9}+\frac{y^{2}}{1}=1$
<br/>Here, $a=3, b=1$
<br/>$c=\sqrt{(3)^{2}-(1)^{2}}=\sqrt{8}$
<br/>$\therefore$ Eccentricity of ellipse, $e=\frac{c}{a}$ $e=\frac{\sqrt{8}}{3}$
<br/>$\therefore$ Eccentricity of hyperbola $=\frac{3}{\sqrt{8}}$
<br/>$\Rightarrow$
<br/>$1+\frac{b^{2}}{a^{2}}=\frac{9}{8}$
<br/>$\Rightarrow$
<br/>$\frac{b^{2}}{a^{2}}=\frac{1}{8}$
<br/>$a^{2}: b^{2}=8: 1$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-the-equation-of-an-ellipse-be-144-x-2-25-y-2-1-then-the-radius-of-the-circle-with-centre-0-2-and-passing-through-the-foci-of-the-ellipse-is

slug:
let-the-equation-of-an-ellipse-be-144-x-2-25-y-2-1-then-the-radius-of-the-circle-with-centre-0-2-and-passing-through-the-foci-of-the-ellipse-is-98313

question:
<div class="question">Let the equation of an ellipse be $\frac{x^{2}}{144}+\frac{y^{2}}{25}=1$ Then, the radius of the circle with centre $(0, \sqrt{2})$ and passing through the foci of the ellipse is</div>

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

answer:
<div class="solution">Given equation of ellipse is
<br/>Here,
<br/>$$
<br/>\begin{array}{r}
<br/>\frac{x^{2}}{144}+\frac{y^{2}}{25}=1 \\
<br/>a^{2}=144 \text { and } b^{2}=25
<br/>\end{array}
<br/>$$
<br/>Now.
<br/>$$
<br/>e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{25}{144}}
<br/>$$
<br/>$$
<br/>=\sqrt{\frac{119}{144}}=\frac{\sqrt{119}}{12}
<br/>$$
<br/>Foci of an ellipse $=(\pm a e, 0)$
<br/>$$
<br/>\begin{array}{l}
<br/>=\left(\pm 12 \times \frac{\sqrt{119}}{12}, 0\right) \\
<br/>=(\pm \sqrt{119}, 0)
<br/>\end{array}
<br/>$$
<br/>Since, the circle with centre $(0, \sqrt{2})$ and passing through foci $(\pm \sqrt{119}, 0)$ of the ellipse.
<br/>$\therefore$ Radius of circle $=\sqrt{(\sqrt{119}-0)^{2}+(0-\sqrt{2})^{2}}$
<br/>$$
<br/>=\sqrt{119+2}=\sqrt{121}=11
<br/>$$</div>


title: let-the-foci-of-the-ellipse-9-x-2-y-2-1-subtend-a-right-angle-at-a-point-p-then-the-locus-of-p-is
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
slug:
let-the-foci-of-the-ellipse-9-x-2-y-2-1-subtend-a-right-angle-at-a-point-p-then-the-locus-of-p-is-90876

question:
<div class="question">Let the foci of the ellipse $\frac{x^{2}}{9}+y^{2}=1$ subtend a right angle at a point $P$. Then, the locus of $P$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}=8$</span> </div>

answer:
<div class="solution">$\frac{x^{2}}{9}+\frac{y^{2}}{1}=1 \Rightarrow e=\sqrt{1-\frac{1}{9}}=\frac{2 \sqrt{2}}{3}$
<br/>0) ie, $(\pm 2 \sqrt{2}, 0)$ Two foci are $(\pm a e,$ Let $P(h, k)$ by any point on the ellipse
<br/>$\therefore$
<br/>$\frac{k-0}{h-2 \sqrt{2}} \times \frac{k-0}{h+2 \sqrt{2}}=-1$
<br/>[From given condition]
<br/>$\Rightarrow \quad h^{2}-8=-k^{2}$
<br/>$\Rightarrow \quad x^{2}+y^{2}=8$</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
let-the-number-of-elements-of-the-sets-a-and-be-p-and-q-respectively-then-the-number-relations-from-the-set-a-to-the-set-b-is

slug:
let-the-number-of-elements-of-the-sets-a-and-be-p-and-q-respectively-then-the-number-relations-from-the-set-a-to-the-set-b-is-64302

question:
<div class="question">Let the number of elements of the sets $A$ and be $p$ and $q$, respectively. Then, the number relations from the set $A$ to the set $B$ is</div>

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

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

answer:
<div class="solution">Given number of elements of $A$ and $B$ are $p$ and q, respectively.
<br/>$\therefore$ The number of relations from the set $A$ to the $\operatorname{set} B$ is $2^{p q}$</div>


title: let-the-relation-r-1-be-defined-on-r-as-a-r-1-b-if-1-ab-0-then
slug: let-the-relation-r-1-be-defined-on-r-as-a-r-1-b-if-1-ab-0-then-67836
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let the relation $R_{1}$ be defined on $R$ as $a R_{1}b$ if $1+a b &gt; 0 .$ Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$R_{1}$ is reflexive and symmetric but not transitive.</span> </div>

answer:
<div class="solution">We observe the following properties:
<br/>Reflexivity : Let a be an arbitrary element of $R$, Then, $a \in R$ $\Rightarrow 1+a \cdot a=1+a^{2}&gt;0 \quad\left[\because a^{2}&gt;0\right.$ for all $\left.a \in R\right]$
<br/>$\Rightarrow \quad(a, a) \in R_{1}$
<br/>$\left[\right.$ By def. of $\left.R_{1}\right]$ 
<br/>Thus, $(a, a) \in R_{1}$ for all $a \in R$. So, $R_{1}$ is reflexive on $R$. 
<br/>Symmetry : Let $(a, b) \in R$ Then, $(a, b) \in R_{1}$
<br/>$\Rightarrow \quad 1+a b&gt;0$
<br/>$[\because a b=b a$ for all $]$
<br/>$\Rightarrow \quad 1+b a&gt;0 \Rightarrow(b, a) \in R_{1}$
<br/>Thus, $(a, b) \in R_{1} \neq(b, a) \in R,$ for all $a, b \in R$ 
<br/>So, $R_{1}$ is symmetric on $R$. 
<br/>Transitivity : we observe that $\left(1, \frac{1}{2}\right) \in R_{1}$ and $\left(\frac{1}{2},-1\right) \in R_{1}$ but (1,-1)$\notin R_{1}$ because
<br/>$1+1 \times(-1)=0 \gg 0$
<br/>So, $R_{1}$ is not transitive on $R$.</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
let-the-tangent-and-normal-at-any-point-p-a-t-2-2at-a-0-on-the-parabola-y-2-4-a-x-meet-the-axis-of-the-parabola-at-t-and-g-respectively-then-the-radiu

slug:
let-the-tangent-and-normal-at-any-point-p-a-t-2-2at-a-0-on-the-parabola-y-2-4-a-x-meet-the-axis-of-the-parabola-at-t-and-g-respectively-then-the-radiu-51269

question:
<div class="question">Let the tangent and normal at any point $P\left(at^2\right.$, 2at), $(a &gt; 0)$, on the parabola $y^2=4 a x$ meet the axis of the parabola at $T$ and $\mathrm{G}$ respectively. Then the radius of the circle through $\mathrm{P}, \mathrm{T}$ and $\mathrm{G}$ is</div>

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

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

answer:
<div class="solution">$P\left(a t^2, 2 a t\right), T\left(-a t^2, 0\right), G\left(2 a+a t^2, 0\right)$<br/>Slope of $P T \times$ Slope of $P G=-1$<br/>$\therefore$ TG is a diameter of the circle through points $\mathrm{P}, \mathrm{T}$ and $\mathrm{G}$.<br/>$\therefore \text { radius }=\frac{1}{2} \mathrm{TG}=\mathrm{a}\left(1+\mathrm{t}^2\right)$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
let-v-n-and-e-n-be-the-respective-speed-and-energy-of-an-electron-in-the-n-th-orbit-of-radius-r-n-in-a-hydrogen-atom-as-predicted-by-bohrs-model-then

slug:
let-v-n-and-e-n-be-the-respective-speed-and-energy-of-an-electron-in-the-n-th-orbit-of-radius-r-n-in-a-hydrogen-atom-as-predicted-by-bohrs-model-then-59027

question:
<div class="question">Let $v_{n}$ and $E_{n}$ be the respective speed and energy of an electron in the $n$ th orbit of radius $r_{n}$, in a hydrogen atom, as predicted by Bohr's model. Then.</div>

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

answer:
<div class="solution">We know that, $v \propto \frac{1}{n}$ (speed)<br/>$E_{n} \propto \frac{1}{n^{2}} \quad(\text { energy })$<br/>$r_{n} \propto n^{2}$<br/>$\therefore E_{n T_{n}} \propto n^{0}$<br/>$\therefore E_{n} r_{n} \propto E_{1} r_{1}$<br/>$\frac{E_{1} r_{1}}{E_{1} r_{1}}$<br/>$r_{n} v_{s}=n^{2} \times \frac{1}{n} \propto n = constant$<br/>$\therefore \frac{r_{n} v_{n}}{r_{1} v_{1}}=n$<br/>$r_{\mathrm{e}} \sim n^{2}$<br/>$\therefore \frac{r_{\mathrm{n}}}{r_{1}}=n^{2}$<br/>$\log \left(\frac{r_{\mathrm{n}}}{r_{1}}\right)=2 \log n \quad(\because$ slope $=2)$<br/>$\frac{r_{n}}{E_{n}}=n^{4}$<br/>$\therefore \frac{r_{n}}{E_{n}} \times \frac{E_{1}}{r_{1}}=n^{4}$<br/>$\log \left(\frac{r_{1} E_{1}}{E_{\alpha} r_{1}}\right)=4 \log (n)(\because$ slope $=4)$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answers are: <span class="option-value">64</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-x-1-x-2-be-the-roots-of-x-2-3-x-a-0-and-x-3-x-4-be-the-roots-of-x-2-12-x-b-0-if-x-1-x-2-x-3-x-4-and-x-1-x-2-x-3-x-4-are-in-gp-then-ab-equals

slug:
let-x-1-x-2-be-the-roots-of-x-2-3-x-a-0-and-x-3-x-4-be-the-roots-of-x-2-12-x-b-0-if-x-1-x-2-x-3-x-4-and-x-1-x-2-x-3-x-4-are-in-gp-then-ab-equals-11089

question:
<div class="question">Let $x_{1}, x_{2}$ be the roots of $x^{2}-3 x+a=0$ and $x_{3}, x_{4}$ be the roots of $x^{2}-12 x+b=0$ If $x_{1}    &lt; x_{2} &lt; x_{3} &lt; x_{4}$ and $x_{1}, x_{2}, x_{3}, x_{4}$ are in GP. then ab equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{24}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">64</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">16</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">We have,
<br/>$x_{1}, x_{2}$ be the roots of equation $x^{2}-3 x+a=0$
<br/>$\therefore \quad x_{1}+x_{2}=3$ and $x_{1} x_{2}=a$
<br/>Also, $x_{3}, x_{4}$ be the roots of equation $x^{2}-12 x+b=0$
<br/>$\therefore x_{3}+x_{4}=12$ and $x_{3} x_{4}=b$
<br/>Again, $x_{1}, x_{2}, x_{3}, x_{4}$ are in GP
<br/>$\therefore \quad x_{1}=A, x_{2}=A R, x_{3}=A R^{2}, x_{4}=A R^{3}$
<br/>Now, $x_{1}+x_{2}=3$
<br/>$\Rightarrow \quad A(1+R)=3$
<br/>and $\quad x_{3}+x_{4}=12$
<br/>$\Rightarrow \quad A R^{2}(1+R)=12$
<br/>From Eqs. (i) and (ii), we have
<br/>$R^{2}=4 \Rightarrow R=\pm 2$
<br/>When, $R=2, A=1$ and when, $R=-2,A=-3$
<br/>$\therefore$ Numbers are either 1,2,4,8 or -3,6,-12,24 But $x_{1}    &lt; x_{2} &lt; x_{3} &lt; x_{4}$
<br/>So, $x_{1}=1, x_{2}=2, x_{3}=4, x_{4}=8$
<br/>$\begin{aligned} \therefore a b &amp;=x_{1} x_{2} x_{3} x_{4} \\ &amp;=1 \times 2 \times 4 \times 8=64 \end{aligned}$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-x-1-x-2-x-13-be-15-distinct-numbers-chosen-from-1-2-3-15-then-the-value-of-x-1-1-x-2-1-x-3-1-x-15-1-is

slug:
let-x-1-x-2-x-13-be-15-distinct-numbers-chosen-from-1-2-3-15-then-the-value-of-x-1-1-x-2-1-x-3-1-x-15-1-is-36766

question:
<div class="question">Let $x_{1}, x_{2}, \ldots, x_{13}$ be 15 distinct numbers chosen from $1,2,3, \ldots, 15 .$ Then, the value of $\left(x_{1}-1\right)\left(x_{2}-1\right)\left(x_{3}-1\right) \ldots\left(x_{15}-1\right)$ is</div>

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

answer:
<div class="solution">Since, $x_{1}, x_{2} \ldots, x_{15}$ are among $1,2,3, \ldots, 15$. So, $x_{1}$ can take any value from $1,2,3, \ldots, 15$. Among these values, one of the number must be 1, hence product will be 0</div>


title: let-x-be-a-nonvoid-set-if-rho1-and-rho2-be-the-transitive-relations-on-x-then
slug: let-x-be-a-nonvoid-set-if-rho1-and-rho2-be-the-transitive-relations-on-x-then-11972
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint: Fact</div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let \(X\) be a nonvoid set. If \(\rho_1\) and \(\rho_2\) be the transitive relations on \(X\), then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\rho_1, \rho_2\) is transitive relation</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\rho_1 \cdot \rho_2\) is not transitive relation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\rho_1 \cdot \rho_2\) is equivalence relation</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\rho_1 \cdot \rho_2\) is not any relation on \(X\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\rho_1 \cdot \rho_2\) is not transitive relation</span> </div>


tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
let-x-denote-the-greatest-integer-less-than-or-equal-to-x-for-any-real-number-x-then-lim-n-n-n-2-is-equal-to

slug:
let-x-denote-the-greatest-integer-less-than-or-equal-to-x-for-any-real-number-x-then-lim-n-n-n-2-is-equal-to-56687

question:
<div class="question">Let $[x]$ denote the greatest integer less than or equal to $x$ for any real number $x$. Then, $\lim _{n \rightarrow \infty} \frac{[n \sqrt{2}]}{n}$ is equal to</div>

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

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

answer:
<div class="solution">We have. $n \sqrt{2}-1    &lt; [n \sqrt{2]} \leq n \sqrt{2} \quad[\because x-1 \leq[x] \leq x]$
<br/>$\Rightarrow \quad \sqrt{2}-\frac{1}{n} &lt; [n \sqrt{2}] \leq 1$
<br/>$\therefore$ By Sandwich theorem. $\lim _{n \rightarrow \infty}\left(\sqrt{2}-\frac{1}{n}\right)=\sqrt{2}$</div>


tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-x-denote-the-greatest-integer-less-than-or-equal-to-x-then-the-value-of-the-integral-1-1-x-2-x-d-x-is-equal-to

slug:
let-x-denote-the-greatest-integer-less-than-or-equal-to-x-then-the-value-of-the-integral-1-1-x-2-x-d-x-is-equal-to-23070

question:
<div class="question">Let $[x]$ denote the greatest integer less than or equal to $x$, then the value of the integral $\int_{-1}^{1}(|x|-2[x]) d x$ is equal to</div>

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

answer:
<div class="solution">Let $I=\int_{-1}^{1}(|x|-2[x]) d x$
<br/>$=\int_{-1}^{0}(|x|-2[x]) d x+\int_{0}^{1}(|x|-2[x]) d x$
<br/>$=\int_{-1}^{0}(-x-2(-1)) d x+\int_{0}^{1}(x-2(0)) d x$
<br/>$\quad=\int_{-1}^{0}(-x+2) d x+\int_{0}^{1} x d x$
<br/>$\quad=\left(-\frac{x^{2}}{2}+2 x\right)_{-1}^{0}+\left[\frac{x^{2}}{2}\right]_{0}$
<br/>$\quad=-\left(-\frac{1}{2}+2(-1)\right)+\frac{1}{2}$
<br/>$\quad=\frac{1}{2}+2+\frac{1}{2}=1+2=3$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
let-x-denotes-the-greatest-integer-less-than-or-equal-to-x-then-the-value-of-for-which-the-function-f-x-x-2-s-i-n-x-2-x-0-x-0-is-continuous-at-x-0-is

slug:
let-x-denotes-the-greatest-integer-less-than-or-equal-to-x-then-the-value-of-for-which-the-function-f-x-x-2-s-i-n-x-2-x-0-x-0-is-continuous-at-x-0-is-53397

question:
<div class="question">Let $[x]$ denotes the greatest integer less than or equal to $x$. Then, the value of $\alpha$ for which the function $f(x)=\left\{\begin{array}{c}\frac{\sin \left[-x^{2}\right]}{\left[-x^{2}\right]}, x \neq 0 \\ \alpha, \quad x=0\end{array}\right.$ is continuous at $x=0,$ is</div>

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

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

answer:
<div class="solution">We have, $f(x)=\left\{\begin{array}{ll}\frac{\sin \left(-x^{2}\right)}{[-x^{2}]} &amp; , x \neq 0 \\ \alpha, &amp; x=0\end{array}\right.$
<br/>Now, $\lim _{x \rightarrow 0} \frac{\sin \left(-x^{2}\right)}{\left.\mid-x^{2}\right]}=\frac{\sin (-1)}{(-1)}=\sin (1)$
<br/>Since, $f(x)$ is continuous at $x=0$.
<br/>$\Rightarrow \lim _{x \rightarrow 0} f(x)=f(0) \Rightarrow \sin (1)=\alpha$</div>


title: let-x-f-x-f-1-x-and-f-x-0-in-0-1-then
slug: let-x-f-x-f-1-x-and-f-x-0-in-0-1-then-43498
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $\phi(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x)    &lt; 0$ in $[0,1]$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\phi$ is monotonic increasing in $\left[0, \frac{1}{2}\right]$ and monotonic decrasing in $\left[\frac{1}{2}, 1\right]$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$\phi$ is monotonic increasing in $\left[\frac{1}{2}, 1\right]$ and monotonic decrasing in $\left[0, \frac{1}{2}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\phi$ is neither increasing nor decreasing in any sub interval of $[0,1]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\phi$ is increasing $[0,1]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\phi$ is monotonic increasing in $\left[0, \frac{1}{2}\right]$ and monotonic decrasing in $\left[\frac{1}{2}, 1\right]$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\phi^{\prime}(x)=f^{\prime}(x)-f^{\prime}(1-x)$
<br/>$f^{\prime}(x)-f^{\prime}(1-x) \geq 0$ (for monotonic increasing)
<br/>$f^{\prime}(x) \geq f^{\prime}(1-x), x \leq 1-x\left(\because f^{\prime}(x)\right.$ is decreasing)
<br/>$\mathrm{x} \leq \frac{1}{2} \Rightarrow \phi(\mathrm{x})$ is monotonic increasing in $\left[0, \frac{1}{2}\right]$ and monotonic decreasing in $\left[\frac{1}{2}, 1\right]$</div>


title: let-x-n-1-3-1-2-1-6-1-2-1-10-1-2-1-2-n-n-1-1-2-n-2-then-the-value-of-lim-n-x-n-is
slug: let-x-n-1-3-1-2-1-6-1-2-1-10-1-2-1-2-n-n-1-1-2-n-2-then-the-value-of-lim-n-x-n-is-84001
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $x_{n}=\left(1-\frac{1}{3}\right)^{2}\left(1-\frac{1}{6}\right)^{2}\left(1-\frac{1}{10}\right)^{2} \ldots$ $\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^{2}, n \geq 2$ Then, the value of $\lim _{n \rightarrow \infty} x_{n}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 / 9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 81$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 9$</span> </div>

answer:
<div class="solution">We have, $x_{n}=\left[\left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right)\right.$ $$
<br/>\left.\left(1-\frac{1}{10}\right) \ldots\left(1-\frac{2}{n(n+1)}\right)\right]^{2}
<br/>$$
<br/>$\begin{aligned} \Rightarrow x_{n} &amp;=\left[\prod_{n=2}^{n}\left(\frac{n^{2}+n-2}{n(n+1)}\right)\right]^{2} \\ &amp;=\left[\prod_{n=2}^{n}\left(\frac{(n+2)(n-1)}{n(n+1)}\right)\right]^{2} \\ &amp;=\left[\prod_{n=2}^{n}\left(\frac{n+2}{n+1}\right) \cdot \prod_{n=2}^{n}\left(\frac{n-1}{n}\right)\right]^{2} \\ \Rightarrow &amp;\left[\prod_{n=2}^{n}\left(\frac{n+2}{n+1}\right)\right]^{2}\left[\prod_{n=2}^{n}\left(\frac{n-1}{n}\right)\right]^{2} \\ \Rightarrow \quad x_{n}=&amp;\left(\frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \ldots \frac{n+2}{n+1}\right)^{2}\left(\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \ldots \frac{n-1}{n}\right)^{2} \\ \Rightarrow \quad &amp; x_{n}=\left(\frac{n+2}{3}\right)^{2}\left(\frac{1}{n}\right)^{2} \Rightarrow x_{n}=\frac{1}{9}\left(\frac{n+2}{n}\right)^{2} \\=&amp; \quad \lim _{n \rightarrow \infty} x_{n}=\frac{1}{9}(1+0)^{2}=\frac{1}{9} \end{aligned}$</div>


title: let-x-n-z-x-i-y-z-2-n-1-for-all-integers-n-1-then-n-1-x-n-is
slug: let-x-n-z-x-i-y-z-2-n-1-for-all-integers-n-1-then-n-1-x-n-is-30522
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $X_{n}=\left\{z=x+i y:|z|^{2} \leq \frac{1}{n}\right\}$ for all integers $n \geq 1 .$ Then, $\underset{n=1}{\cap} X_{n}$ is</div>

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

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

answer:
<div class="solution">Given, $X_{n}=\left\{z=x+i y:|z|^{2} \leq \frac{1}{n}\right\}$
<br/>$=\left\{x^{2}+y^{2} \leq \frac{1}{n}\right\}$
<br/>$\therefore$
<br/>$X_{1}=\left\{x^{2}+y^{2} \leq 1\right\}$
<br/>$\begin{array}{c}
<br/>X_{2}=\left\{x^{2}+y^{2} \leq \frac{1}{2}\right\} \\
<br/>X_{3}=\left\{x^{2}+y^{2} \leq \frac{1}{3}\right\} \\
<br/>\ldots \\
<br/>\qquad \begin{aligned}
<br/>X_{\infty} &amp;=\left\{x^{2}+y^2 \leq 0\right\} \\
<br/>\bigcap_{n=1}^{\infty} x_n &amp;=X_{1} \cap X_{2} \cap X_{3} \cap \ldots \cap X_{\infty} \\
<br/>&amp;=\left\{x^{2}+y^{2}=0\right\}
<br/>\end{aligned}
<br/>\end{array}$
<br/>Hence, $\underset{n=1}{\cap} X_{n}$ is a singleton set.</div>


title: let-y-1-x-l-n-x-1-then
slug: let-y-1-x-l-n-x-1-then-53708
question: <div class="question">Let $\mathrm{y}=\frac{1}{1+\mathrm{x}+\ln \mathrm{x}}$, Then</div>
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2020']
answer: <div class="solution">Hint: <br/>$x \frac{d y}{d x}=y(y \ln x-1)$</div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x \frac{d y}{d x}+y=x$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x \frac{d y}{d x}=y(y \ln x-1)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2} \frac{d y}{d x}=y^{2}+1-x^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x\left(\frac{d y}{d x}\right)^{2}=y-x$</span> </li> </ul>

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


title: let-y-3-x-1-3-x-1-sin-x-lo-g-e-1-x-x-1-then-at-x-0-d-x-d-y-equals
slug: let-y-3-x-1-3-x-1-sin-x-lo-g-e-1-x-x-1-then-at-x-0-d-x-d-y-equals-66862
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $y=\left(\frac{3^{x}-1}{3^{x}+1}\right) \sin x+\log _{e}(1+x) \quad x&gt;-1$ Then, at $x=0, \frac{d y}{d x}$ equals</div>

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

answer:
<div class="solution">Given, $y=\left(\frac{3^{x}-1}{3^{x}+1}\right) \sin x+\log _{e}(1+x), x&gt;-1$
<br/>Differentiating w.r.t. $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\left(\frac{3^{x}-1}{3^{x}+1}\right) \sin x\right]+\frac{d}{d x} \log _{e}(1+x)$
<br/>$=\left(\frac{3^{x}-1}{3^{x}+1}\right) \frac{d}{d x} \sin x+\sin x \frac{d}{d x}\left(\frac{3^{x}-1}{3^{x}+1}\right)$
<br/>$=\frac{1}{1+x} \frac{d}{d x}(1+x)$
<br/>$=\left(\frac{3^{x}-1}{3^{x}+1}\right) \cos x+\sin x$
<br/>$\frac{\left(3^{x}+1\right) \frac{d}{d x}\left(3^{x}-1\right)-\left(3^{x}-1\right) \frac{d}{d x}\left(3^{x}+1\right)}{\left(3^{x}+1\right)^{2}}$
<br/>$+\frac{1}{1+x}$
<br/>$=\left(\frac{3^{x}-1}{3^{x}+1}\right) \cos x+\sin x\left(3^{x}+1\right)\left(3^{x} \log _{e} 3\right)$
<br/>$\quad \frac{-\left(3^{x}-1\right)\left(3^{x} \log _{t} 3\right)}{\left(3^{x}+1\right)^{2}}+\frac{1}{1+x}$
<br/>$\therefore \quad\left(\frac{d y}{d x}\right)_{\text {at } x=0}=0+0+\frac{1}{1+0}=1$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
let-y-be-the-solution-of-the-differential-equation-x-d-x-d-y-1-y-l-o-g-x-y-2-satisfying-y-1-1-then-y-satisfies

slug:
let-y-be-the-solution-of-the-differential-equation-x-d-x-d-y-1-y-l-o-g-x-y-2-satisfying-y-1-1-then-y-satisfies-72370

question:
<div class="question">Let $y$ be the solution of the differential equation
<br/>$x \frac{d y}{d x}=\frac{y^{2}}{1-y \log x}$ satisfying $y(1)=1 .$ Then, $y$
<br/>satisfies</div>

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

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

answer:
<div class="solution">Let $y=x^{y} \Rightarrow \log y=y \log x$
<br/>Differentiating both sides w.r.t. $x$, we get $\frac{1}{y} \frac{d y}{d x}=y \times \frac{1}{x}+\log x \frac{d y}{d x}$
<br/>$\Rightarrow \quad \frac{1}{y} \frac{d y}{d x}-\log x \frac{d y}{d x}=\frac{y}{x}$
<br/>$\Rightarrow \quad \frac{d y}{d x}\left(\frac{1}{y}-\log x\right)=\frac{y}{x}$
<br/>$\Rightarrow \quad \frac{d y}{d x}\left(\frac{1-y \log x}{y}\right)=\frac{y}{x}$
<br/>$\Rightarrow \quad \frac{x d y}{d x}=\frac{y^{2}}{(1-y \log x)}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P2.db
title:
let-y-e-x-2-and-y-e-x-2-sin-x-be-two-given-curves-then-angle-between-the-tangents-to-the-curves-at-any-point-of-their-intersection

slug:
let-y-e-x-2-and-y-e-x-2-sin-x-be-two-given-curves-then-angle-between-the-tangents-to-the-curves-at-any-point-of-their-intersection-97950

question:
<div class="question">Let $y=e^{x^{2}}$ and $y=e^{x^{2}} \sin x$ be two given curves. Then, angle between the tangents to the curves at any point of their intersection</div>

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

answer:
<div class="solution">For intersecting points, $e^{x^{2}}=e^{x^{2}} \sin x$ $\Rightarrow \quad e^{x^{2}}(\sin x-1)=0$
<br/>$\Rightarrow \quad e^{x^{2}}=0 \quad$ or $\sin x=1$
<br/>But $\quad e^{x^{2}} \neq 0 \Rightarrow \sin x=1$
<br/>$\Rightarrow$
<br/>$$
<br/>x=\frac{\pi}{2}
<br/>$$
<br/>Now.
<br/>$$
<br/>y=e^{x^{2}}
<br/>$$
<br/>$\therefore \quad \frac{d y}{d x}=e^{x^{2}} \cdot 2 x=2 x e^{x^{2}}$
<br/>Also, $y=e^{x^{2}} \sin x$
<br/>$\frac{d y}{d x}=e^{x^{2}} \cdot 2 x \sin x+e^{x^{2}} \cos x=2 x e^{x^{2}}$
<br/>$[\because \sin x=1, \cos x=0]$
<br/>Since, both the curves has equal slope. Hence, angle between the tangents at intersecting point is $0 .$</div>


title: let-y-f-x-2-x-2-3-x-2-the-differential-of-y-when-x-changes-from-2-to-199-is
slug: let-y-f-x-2-x-2-3-x-2-the-differential-of-y-when-x-changes-from-2-to-199-is-67304
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $y=f(x)=2 x^{2}-3 x+2$. The differential of $y$ when $x$ changes from 2 to $1.99$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.01$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.18$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-0.05$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.07$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-0.05$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\Delta x=-0.01 \Rightarrow \Delta y=f^{\prime}(x) \Delta x \Rightarrow \Delta y=f^{\prime}(2)(-0.01) \Rightarrow \Delta y=-5 \times 0.01 \Rightarrow \Delta y=-0.05$</div>


title: let-y-x-1-2-x-2-x-2-then-dx-2-d-2-y-is
slug: let-y-x-1-2-x-2-x-2-then-dx-2-d-2-y-is-57531
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $\mathrm{y}=\frac{\mathrm{x}^{2}}{(\mathrm{x}+1)^{2}(\mathrm{x}+2)}$. Then $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$2\left[\frac{3}{(x+1)^{4}}-\frac{3}{(x+1)^{3}}+\frac{4}{(x+2)^{3}}\right]$</span> </div>

answer:
<div class="solution">Hint: 
<br/>By partial fraction technique
<br/>$\begin{array}{l}
<br/>y=\frac{x^{2}}{(x+1)^{2}(x+2)}=\frac{4}{(x+2)}-\frac{3}{(x+1)}+\frac{1}{(x+1)^{2}} \\
<br/>\Rightarrow y^{\prime \prime}=\frac{6}{(x+1)^{4}}-\frac{6}{(x+1)^{3}}+\frac{8}{(x+2)^{3}}
<br/>\end{array}$</div>


title: let-y-x-be-a-solution-of-1-x-2-d-x-d-y-2-x-y-4-x-2-0-and-y-0-1-then-y1-s-equal-to
slug: let-y-x-be-a-solution-of-1-x-2-d-x-d-y-2-x-y-4-x-2-0-and-y-0-1-then-y1-s-equal-to-62504
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Let $y(x)$ be a solution of $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y-4 x^{2}=0$ and $y(0)=-1 .$ Then y(1) s equal to</div>

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

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

answer:
<div class="solution">We have. $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y-4 x^{2}=0$
<br/>$\Rightarrow \quad \frac{d y}{d x}+\left(\frac{2 x}{1+x^{2}}\right) y=\frac{4 x^{2}}{1+x^{2}}$
<br/>Here. $\mathrm{IF}=e^{\frac{1}{1+x^{2}}}=e^{\log \left(1+x^{2}\right)}=1+x^{2}$
<br/>$\therefore \quad y\left(1+x^{2}\right)=\int\left(1+x^{2}\right) \times \frac{4 x^{2}}{\left(1+x^{2}\right)} d x+C$
<br/>$\Rightarrow y\left(1+x^{2}\right)=\int 4 x^{2} d x+C$
<br/>$\Rightarrow \quad y\left(1+x^{2}\right)=\frac{4 x^{3}}{3}+C$
<br/>$\Rightarrow \quad y\left(1+x^{2}\right)=\frac{4 x^{3}}{3}-1$
<br/>$$
<br/>[y(0)=-1]
<br/>$$
<br/>$\Rightarrow \quad y=\frac{4 x^{3}}{3\left(1+x^{2}\right)}-\frac{1}{1+x^{2}}$
<br/>$\therefore \quad y(1)=\frac{4}{6}-\frac{1}{2}=\frac{1}{6}$</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
let-z-1-and-z-2-be-complex-numbers-such-that-z-1-z-2-and-z-1-z-2-if-re-z-1-0-and-im-z-2-0-then-z-1-z-2-z-1-z-2-is

slug:
let-z-1-and-z-2-be-complex-numbers-such-that-z-1-z-2-and-z-1-z-2-if-re-z-1-0-and-im-z-2-0-then-z-1-z-2-z-1-z-2-is-39450

question:
<div class="question">Let $z_{1}$ and $z_{2}$ be complex numbers such that $z_{1} \neq z_{2}$ and $\left|z_{1}\right|=\left|z_{2}\right| .$ If $\operatorname{Re}\left(z_{1}\right)&gt;0$ and
<br/>$\operatorname{Im}\left(z_{2}\right)    &lt; 0,$ then $\frac{z_{1}+z_{2}}{z_{1}-z_{2}}$ is</div>

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

answer:
<div class="solution">Let $z_{1}=x_{1}+i y_{1}$ and $z_{2}=x_{2}+i y_{2}$
<br/>$\operatorname{Re}\left(z_{1}\right)&gt;0 \Rightarrow x_{1}&gt;0$
<br/>$\begin{array}{lr}\text { and } &amp; \operatorname{Im}\left(z_{2}\right)    &lt; 0 \\ \Rightarrow &amp; y_{2} &lt; 0 \\ \text { Given, } &amp; \left|z_{1}\right|=\left|z_{2}\right|\end{array}$
<br/>$\Rightarrow \quad\left|z_{1}\right|^{2}=\left|z_{2}^{2}\right|$
<br/>$\Rightarrow \quad z_{1} \bar{z}_{1}=z_{2} \bar{z}_{2}$
<br/>Now, $\left(\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\right)+\left(\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\right)$
<br/>$=\left(\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\right)+\left(\frac{\bar{z}_{1}+\bar{z}_{2}}{\bar{z}_{1}-\bar{z}_{2}}\right)$
<br/>$=\frac{z_{1} \bar{z}_{1}+z_{2} \bar{z}_{1}-z_{1} \bar{z}_{2}-z_{2} \bar{z}_{2}+z_{1} \bar{z}_{1}+z_{1} \bar{z}_{2}-z_{2} \bar{z}_{1}+z_{2} \bar{z}_{2}}{\left(z_{1}-z_{2}\right)\left(\bar{z}_{1}-\bar{z}_{2}\right)}$
<br/>$=\frac{2\left(\left|z_{1}\right|^{2}-\left|z_{2}\right|^{2}\right)}{\left(z_{1}-z_{2}\right)\left(\bar{z}_{1}-\bar{z}_{2}\right)}=0 \quad\left(\because\left|z_{1}\right|^{2}=\left|z_{2}\right|^{2}\right)$
<br/>$=\frac{z_{1}+z_{2}}{z_{1}-z_{2}}$ is purely imaginary.</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
let-z-1-and-z-2-be-two-imaginary-roots-of-z-2-p-z-q-0-where-p-and-q-are-real-the-points-z-1-z-2-and-origin-form-an-equilateral-triangle-if

slug:
let-z-1-and-z-2-be-two-imaginary-roots-of-z-2-p-z-q-0-where-p-and-q-are-real-the-points-z-1-z-2-and-origin-form-an-equilateral-triangle-if-65430

question:
<div class="question">Let $z_{1}$ and $z_{2}$ be two imaginary roots of $z^{2}+p z+q=0$, where $p$ and $q$ are real. The points $z_{1}, z_{2}$ and origin form an equilateral triangle if</div>

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

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

answer:
<div class="solution">Hint: 
<br/>$\begin{aligned} &amp; \mathrm{O}^{2}+\mathrm{z}_{1}^{2}+\mathrm{z}_{2}^{2}=\mathrm{z}_{1} \mathrm{z}_{2} \\ &amp; \Rightarrow \mathrm{z}_{1}^{2}+\mathrm{z}_{2}^{2}=\mathrm{z}_{1} \mathrm{z}_{2} \\ &amp; \Rightarrow\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)^{2}=3 \mathrm{z}_{1} \mathrm{z}_{2} \\ &amp; \Rightarrow \mathrm{p}^{2}=3 \mathrm{q} \end{aligned}$</div>


title: let-z-1-and-z-2-be-two-nonzero-complex-numbers-then
slug: let-z-1-and-z-2-be-two-nonzero-complex-numbers-then-52374
question: <div class="question">Let $z_1$ and $z_2$ be two non-zero complex numbers. Then</div>
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Principal value of $\arg \left(z_1 z_2\right)$ may not be equal to Principal value of arg $z_1+$ Principal value of arg $z_2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Principal value of $\arg \left(z_1 z_2\right)=$ Principal value of $\arg z_1+$ Principal value of $\arg z_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Principal value of $\arg \left(z_1 / z_2\right)=$ Principal value of $\arg z_1-$ Principal value of $\operatorname{argz} z_2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Principal value of $\arg \left(z_1 / z_2\right)$ may not be $\arg z_1-\arg z_2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Principal value of $\arg \left(z_1 z_2\right)$ may not be equal to Principal value of arg $z_1+$ Principal value of arg $z_2$, Principal value of $\arg \left(z_1 / z_2\right)$ may not be $\arg z_1-\arg z_2$</span> </div>

answer:
<div class="solution">$\arg \left(z_1 z_2\right)=\operatorname{argz_{1}}+\operatorname{argz_{2}}+2 k \pi$<br/>$\arg \left(z_1 / z_2\right)=\operatorname{argz_{1}}-\operatorname{argz_{2}}+2 \mathrm{k} \pi$</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
let-z-1-be-a-fixed-point-on-the-circle-of-radius-1-centred-at-the-origin-in-the-argand-plane-and-z-1-1-consider-an-equilateral-triangle-inscribed-in-t

slug:
let-z-1-be-a-fixed-point-on-the-circle-of-radius-1-centred-at-the-origin-in-the-argand-plane-and-z-1-1-consider-an-equilateral-triangle-inscribed-in-t-28211

question:
<div class="question">Let $z_{1}$ be a fixed point on the circle of radius 1 centred at the origin in the argand plane and $z_{1} \neq \pm 1 .$ Consider an equilateral triangle inscribed in the circle with $z_{1}, z_{2}, z_{3}$ as the
<br/>direction. Then, $z_{1} z_{2} z_{3}$ is equal to</div>

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

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

answer:
<div class="solution">Given, $z_{1} \neq\pm 1$ 
<br/>Since, $z_{2}$ and $z_{3}$ can be obtained by rotating
<br/>vector representing through $\frac{2 \pi}{3}$ and $\frac{4 \pi}{3}$
<br/>respectively
<br/>$\therefore \quad z_{2}=z_{1} \omega$
<br/>and
<br/>$$
<br/>z_{3}=z_{1} \omega^{2}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>\therefore \quad z, z_{2} z_{3} &amp;=z_{1} \times z_{1} \omega \times z_{1} \omega^{2} \\
<br/>&amp;=z_{1}^{3} \omega^{3} \\
<br/>&amp;=z_{1}^{3}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
let-z-1-z-2-be-two-fixed-complex-numbers-in-the-argand-plane-and-z-be-an-arbitrary-point-satisfying-z-z-1-z-z-2-2-z-1-z-2-then-the-locus-of-z-will-be

slug:
let-z-1-z-2-be-two-fixed-complex-numbers-in-the-argand-plane-and-z-be-an-arbitrary-point-satisfying-z-z-1-z-z-2-2-z-1-z-2-then-the-locus-of-z-will-be-81987

question:
<div class="question">Let $z_{1}, z_{2}$ be two fixed complex numbers in the argand plane and $z$ be an arbitrary point satisfying $\left|z-z_{1}\right|+\left|z-z_{2}\right|=2\left|z_{1}-z_{2}\right|$. Then, the locus of $z$ will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">an ellipse</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a straight line joining $z_{1}$ and $z_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a parabola</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a bisector of the line segment joining $z_1$ and $z_2$</span> </li> </ul>

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

answer:
<div class="solution">We know that $\left|z-z_{1}\right|+\left|z-z_{2}\right|=k$ will represent an ellipse, if $\left|z_{1}-z_{2}\right|    &lt; k$
<br/>Hence. the equation $\left|z-z_{1}\right|+\left|z-z_{2}\right|$ $=2\left|z_{1}-z_{2}\right|$ represent an ellipse.</div>


title: let-z-be-a-complex-number-such-that-the-principal-value-of-argument-arg-z-0-then-arg-z-ar-g-z-is
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: db\MarksBatch2_P2.db
slug:
let-z-be-a-complex-number-such-that-the-principal-value-of-argument-arg-z-0-then-arg-z-ar-g-z-is-82774

question:
<div class="question">Let $z$ be a complex number such that the principal value of argument, arg $z&gt;0$ Then, arg $z-\arg (-z)$ is</div>

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

answer:
<div class="solution">$\begin{aligned} \therefore \arg (z)-\arg (-z) &amp;=\theta-(-\pi+\theta) \\ &amp;=\pi \end{aligned}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a15.png"/></div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
let-z-x-i-y-where-x-and-y-are-real-the-points-x-y-in-the-x-y-plane-for-which-z-i-z-i-is-purely-imaginary-lie-on

slug:
let-z-x-i-y-where-x-and-y-are-real-the-points-x-y-in-the-x-y-plane-for-which-z-i-z-i-is-purely-imaginary-lie-on-85096

question:
<div class="question">Let $z=x+i y,$ where $x$ and $y$ are real. The points $(x, y)$ in the $X-Y$ plane for which $\frac{z+i}{z-i}$
<br/>is purely imaginary, lie on</div>

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

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

answer:
<div class="solution">$\frac{z+i}{z-i}=\frac{(x+j)+i}{(x+i y)-i}$
<br/>$=\frac{x+(1+y) i}{x+(y-1) i}$
<br/>$=\frac{x+(y+1) i}{x+(y-1) i} \times \frac{x-(y-1) i}{x-(y-1) i}$
<br/>$=\frac{x^{2}-x(y-1) i+x(y+1) i-(y+1)(y-1) i^{2}}{x^{2}-(y-1)^{2} i^{2}}$
<br/>$=\frac{x^{2}+i[-x y+x+x y+x]+\left(y^{2}-1\right)}{x^{2}+(y-1)^{2}}$
<br/>$=\frac{x^{2}+y^{2}-1}{x^{2}+(y-1)^{2}}+\frac{2 x}{x^{2}+(y-1)^{2}} i$
<br/>Now, $\frac{z+i}{z-i}$ is purely imaginary.
<br/>$\therefore$
<br/>$\operatorname{Re}\left(\frac{z+i}{z-i}\right)=0$
<br/>$\begin{array}{lr}\Rightarrow &amp; \frac{x^{2}+y^{2}-1}{x^{2}+(y-1)^{2}}=0 \\ \Rightarrow &amp; x^{2}+y^{2}=1\end{array}$
<br/>$\therefore(x, y)$ lies on a circle.</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
letf-r-r-be-twice-continuously-differentiable-or-f-exists-and-is-continuous-such-that-f-0-f-1-f-0-0-then

slug:
letf-r-r-be-twice-continuously-differentiable-or-f-exists-and-is-continuous-such-that-f-0-f-1-f-0-0-then-20632

question:
<div class="question">Letf $: \mathbb{R} \rightarrow \mathbb{R}$, be twice continuously differentiable (or $f$ " exists and is continuous) such that $f(0)=f(1)=f^{\prime}(0)=0$. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}^{\prime \prime}(\mathrm{c})=0$ for some $\mathrm{c} \in \mathbb{R}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$f(0)=f(1)=0$
<br/>By Rolle's theorem $\mathrm{f}^{\prime}(\mathrm{c})=0$ for some $\mathrm{c} \in(0,1)$
<br/>Now, $\mathrm{f}^{\prime}(0)=\mathrm{f}^{\prime}(\mathrm{c})=0$. Again by Rolle's theorem $\mathrm{f}^{\prime \prime}\left(\mathrm{c}_{1}\right)=0 \quad \mathrm{c}_{1} \in(0, \mathrm{c})$</div>


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

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

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

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


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
light-travels-through-a-glass-plate-of-thickness-t-and-having-refractive-index-if-c-be-the-velocity-of-light-in-vacuum-time-taken-by-the-light-to-trav

slug:
light-travels-through-a-glass-plate-of-thickness-t-and-having-refractive-index-if-c-be-the-velocity-of-light-in-vacuum-time-taken-by-the-light-to-trav-80563

question:
<div class="question">Light travels through a glass plate of thickness $t$ and having refractive index $\mu$. If $c$ be the velocity of light in vacuum, time taken by the light to travel through this thickness of glass is</div>

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

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

answer:
<div class="solution">Thickness of glass plate $=t$
<br/>Refractive index $=\mu$ 
<br/>Velocity of light = c
<br/>$$
<br/>\begin{array}{r}
<br/>\text { Speed of light }=\frac{\text { distance }}{\text { time }}=\frac{t}{T} \\
<br/>\qquad T=\frac{t}{\text { speed of light }}
<br/>\end{array}
<br/>$$
<br/>Speed of light in glass plate
<br/>$$
<br/>\begin{array}{l}
<br/>v=\frac{c}{\mu} \\
<br/>T=\frac{t}{c / \mu}=\frac{\mu t}{c}
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\left(\because \mu=\frac{c}{v}\right)
<br/>$$</div>


title: lim-n-n-3-1-n-3-n-n-6-n-n-9-n-n-3-1-n-1
slug: lim-n-n-3-1-n-3-n-n-6-n-n-9-n-n-3-1-n-1-80318
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">is 2</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\lim _{n \rightarrow \infty} \frac{3}{n}\left\{1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}\right.$$+\ldots+\sqrt{\frac{1}{n+3}(n-1)}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">does not exist</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is 1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is 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">is 3</span> </li> </ul>

answer:
<div class="solution">$\lim _{n \rightarrow \infty} \frac{3}{n}\left[1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}+\ldots\right.+\sqrt{\frac{n}{n+3(n-1)}}$
<br/>$=\lim _{n \rightarrow \infty} \frac{3}{n}\left[\sqrt{\frac{n}{n+3 \times 0}}+\sqrt{\frac{n}{n+3 \times 1}}+\sqrt{\frac{n}{n+3 \times 2}}\right.$
<br/>$+\sqrt{\frac{n}{n+3 \times 3}}+\ldots+\sqrt{\frac{n}{n+3(n-1)}}$
<br/>$=\lim _{n \rightarrow \infty} \frac{3}{n}\left[\sum_{r=0}^{n-1} \sqrt{\frac{n}{n+3 r}}\right]$
<br/>$=\lim _{n \rightarrow \infty} \frac{3}{n}\left[\sum_{r=0}^{n-1}\left[\frac{1}{1+3\left(\frac{r}{n}\right)}\right]\right.$
<br/>$=3 \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{n-1} \left(\sqrt{1+3\left(\frac{r}{n}\right)}\right)^{-\frac{1}{2}}$
<br/>$=3 \int_{0}^{1} \sqrt{\frac{1}{1+3 x}} d x$
<br/>$=3 \int_{0}^{1}(1+3 x)^{-1 / 2} d x$
<br/>$=3\left[\frac{2 \sqrt{1+3 x}}{3}\right]_{0}^{1}$
<br/>$=[4-2]$
<br/>$=2$</div>


title: lim-n-n-3-n-n-4-3-n-n-8-3-n-n-4-n-1-3-n-is
slug: lim-n-n-3-n-n-4-3-n-n-8-3-n-n-4-n-1-3-n-is-33063
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\lim _{n \rightarrow \infty} \frac{\sqrt{n}}{\sqrt{\left(n^{3}\right)}}+\frac{\sqrt{n}}{\sqrt{(n+4)^{3}}}+\frac{\sqrt{n}}{\sqrt{(n+8)^{3}}}+\cdots \cdots \cdots+\frac{\sqrt{n}}{\sqrt{[n+4(n-1)]^{3}}}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{5-\sqrt{5}}{10}$</span> </div>

answer:
<div class="solution">$\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{\sqrt{n}}{\sqrt{(n+4 r)^{3}}}$
<br/>$=\sum_{r=0}^{n-1} \frac{1}{n}\left(\frac{n \sqrt{n}}{\sqrt{(n+4 r)^{3}}}\right)$
<br/>$=\sum_{r=0}^{n-1} \frac{1}{n}\left(\frac{1}{\left(1+\frac{4 r}{n}\right)^{3 / 2}}\right)$
<br/>$=\int_{0}^{1} \frac{d x}{(1+4 x)^{3 / 2}}$
<br/>$=\frac{1}{4} \int_{1}^{5} \frac{d z}{z^{3 / 2}}=\left(1 / 4\left(\frac{-2}{\sqrt{z}}\right)\right)_{1}^{5}=\frac{5-\sqrt{5}}{10}$</div>


