# questions

tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-a-string-suspended-from-the-ceilling-is-given-a-downward-force-f-1-its-length-becomes-l-1-its-length-is-l-2-if-the-downward-force-is-f-2-what-is-it

slug:
if-a-string-suspended-from-the-ceilling-is-given-a-downward-force-f-1-its-length-becomes-l-1-its-length-is-l-2-if-the-downward-force-is-f-2-what-is-it-51156

question:
<div class="question">If a string, suspended from the ceilling is given a downward force $F_1$, its length becomes $L_1$, Its length is $L_2$, if the downward force is $\mathrm{F}_2$. What is its actual length?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{L_1+L_2}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{L_1 L_2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{F}_2 \mathrm{~L}_1+\mathrm{F}_1 \mathrm{~L}_2}{\mathrm{~F}_2+\mathrm{F}_1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{F}_2 \mathrm{~L}_1-\mathrm{F}_1 \mathrm{~L}_2}{\mathrm{~F}_2-\mathrm{F}_1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{F}_2 \mathrm{~L}_1-\mathrm{F}_1 \mathrm{~L}_2}{\mathrm{~F}_2-\mathrm{F}_1}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp;\left(L_1-L\right)=\frac{F_1 L}{A y} ; L_2-L=\frac{F_2 L}{A y} \\ \Rightarrow &amp; \frac{L_1-L}{L_2-L}=\frac{F_1}{F_2} \Rightarrow F_2 L_1-F_2 L=F_1 L_2-F_1 L \Rightarrow L=\frac{F_2 L_1-F_1 L_2}{F_2-F_1} \end{aligned}$</div>


title: if-a-x-are-real-numbers-and-a-1-x-1-then-1-1-a-x-1-a-a-2-x-2-is-equal-to
slug: if-a-x-are-real-numbers-and-a-1-x-1-then-1-1-a-x-1-a-a-2-x-2-is-equal-to-25330
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $a, x$ are real numbers and $|a|    &lt; 1,|x| &lt; 1$ then $1 (1+a) x+\left(1+a+a^{2}\right) x^{2}+\ldots \infty \quad$ is equal to</div>

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

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

answer:
<div class="solution">We have. $1+(1+a) x+\left(1+a+a^{2}\right) x^{2}+\ldots \infty$
<br/>$=\sum_{n=1}^{\infty}\left(1+a+a^{2}+\ldots+a^{n-1}\right) x^{n-1}$
<br/>$=\sum_{n=1}^{\infty}\left(\frac{1-a^{n}}{1-a}\right) x^{n-1}$
<br/>$=\sum_{n=1}^{\infty} \frac{x^{n-1}}{1-a}-\sum_{n=1}^{\infty} \frac{a^{n} x^{n-1}}{1-a}$
<br/>$=\frac{1}{1-a} \sum_{n=1}^{\infty} x^{n-1}-\frac{a}{1-a} \sum_{n=1}^{\infty}(a x)^{n-1}$
<br/>$=\frac{1}{1-a}\left[\left(1+x+x^{2}+\ldots \infty\right)\right]$ $-\frac{a}{1-a}\left[\left\{1+a x+(a x)^{2}+\ldots \infty\right\}\right]$
<br/>$=\frac{1}{(1-a)} \cdot \frac{1}{(1-x)}-\frac{a}{(1-a)(1-a x)}$
<br/>$=\frac{1}{(1-a x)(1-x)}$</div>


title: if-aleftbeginarraycc3-x1-2-x3-x2endarrayright-is-a-symmetric-matrix-then-the-value-of-x-is
slug: if-aleftbeginarraycc3-x1-2-x3-x2endarrayright-is-a-symmetric-matrix-then-the-value-of-x-is-91454
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(A=\left(\begin{array}{cc}3 &amp; x-1 \\ 2 x+3 &amp; x+2\end{array}\right)\) is a symmetric matrix, then the value of \(x\) is</div>

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

answer:
<div class="solution">Hints: \(A=A^T\)<br/>\(\begin{aligned}<br/>&amp; \left(\begin{array}{cc}<br/>3 &amp; x-1 \\<br/>2 x+3 &amp; x+2<br/>\end{array}\right)=\left(\begin{array}{cc}<br/>3 &amp; 2 x+3 \\<br/>x-1 &amp; x+2<br/>\end{array}\right) \\<br/>&amp; \Rightarrow x-1=2 x+3 \text { or } x=-4<br/>\end{aligned}\)</div>


tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: db\MarksBatch2_P1.db
title:
if-all-the-nucleophilic-substitution-reactions-at-saturated-carbon-atoms-in-the-above-sequence-of-reactions-follow-mathrmsmathrmn-2-mechanism-then-und-1

slug:
if-all-the-nucleophilic-substitution-reactions-at-saturated-carbon-atoms-in-the-above-sequence-of-reactions-follow-mathrmsmathrmn-2-mechanism-then-und-1-74929

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5JT-CZ-_ZAgZ25eWrtN7_8EeESMZ9vXxwKGIXCjBUXo.original.fullsize.png"/><br/><br/>If all the nucleophilic substitution reactions at saturated carbon atoms in the above sequence of reactions follow \(\mathrm{S}_{\mathrm{N}} 2\) mechanism, then \(\underline{E}\) and \(\underline{E}\) will be respectively,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Mhg3UuiUkatMDhdUBaNq87zmtW_XYoggw7WxwgcHLYs.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/WoOkasQgClr5xhLGCNQpkIBwkp2fo1qfOZ3bcYxHv6M.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/a5Lj-o0f5mhIYXEwaV4nhTCFO2bgdsAPnLc3ir5zF5A.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/202tP1pngiwVgTMqJqWsoscFH37g0mPQgw8wuqTJorM.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/202tP1pngiwVgTMqJqWsoscFH37g0mPQgw8wuqTJorM.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/O5JpyQwChQ8oOi28eMtuIvQbposgaN7VDNYVc2Tl5o4.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/r8H7mzqD-BObXhv_S4rSImwPsaWx9NM6g5aZQhk59MY.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kUjV-gZ_L5Evra5zjIUmlEXOlmZzShk3LyCTWSfliYM.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-all-the-nucleophilic-substitution-reactions-at-saturated-carbon-atoms-in-the-above-sequence-of-reactions-follow-mathrmsmathrmn-2-mechanism-then-und

slug:
if-all-the-nucleophilic-substitution-reactions-at-saturated-carbon-atoms-in-the-above-sequence-of-reactions-follow-mathrmsmathrmn-2-mechanism-then-und-42550

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5JT-CZ-_ZAgZ25eWrtN7_8EeESMZ9vXxwKGIXCjBUXo.original.fullsize.png"/><br/><br/>If all the nucleophilic substitution reactions at saturated carbon atoms in the above sequence of reactions follow \(\mathrm{S}_{\mathrm{N}} 2\) mechanism, then \(\underline{E}\) and \(\underline{E}\) will be respectively,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Mhg3UuiUkatMDhdUBaNq87zmtW_XYoggw7WxwgcHLYs.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/WoOkasQgClr5xhLGCNQpkIBwkp2fo1qfOZ3bcYxHv6M.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/a5Lj-o0f5mhIYXEwaV4nhTCFO2bgdsAPnLc3ir5zF5A.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/202tP1pngiwVgTMqJqWsoscFH37g0mPQgw8wuqTJorM.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/202tP1pngiwVgTMqJqWsoscFH37g0mPQgw8wuqTJorM.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/O5JpyQwChQ8oOi28eMtuIvQbposgaN7VDNYVc2Tl5o4.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/r8H7mzqD-BObXhv_S4rSImwPsaWx9NM6g5aZQhk59MY.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kUjV-gZ_L5Evra5zjIUmlEXOlmZzShk3LyCTWSfliYM.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-alpha-and-beta-are-the-roots-of-the-equation-x2x10-then-the-equation-whose-roots-are-alpha19-and-beta7-is

slug:
if-alpha-and-beta-are-the-roots-of-the-equation-x2x10-then-the-equation-whose-roots-are-alpha19-and-beta7-is-72265

question:
<div class="question">If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+x+1=0\), then the equation whose roots are \(\alpha^{19}\) and \(\beta^7\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x^2+x+1=0\)</span> </div>

answer:
<div class="solution">Hints: \(\alpha\) and \(\beta\) are the roots of \(x^2+x+1=0\)<br/>\(\begin{array}{ll}<br/>\alpha=\omega &amp; \beta=\omega^2 \\<br/>\alpha^{19}=\omega &amp; \beta^7=\omega^2 \\<br/>x^2-\left(\alpha^{19}+\beta^7\right) x+\alpha^{19} \beta^7=0<br/>\end{array}\)<br/>Thou,<br/>\(\begin{aligned}<br/>&amp; x^2-\left(\omega+\omega^2\right) x+\omega \cdot \omega^2=0 \\<br/>&amp; x^2+x+1=0<br/>\end{aligned}\)</div>


title: if-and-are-roots-of-a-x-2-b-x-c-0-then-the-equation-whose-roots-are-2-and-2-is
slug: if-and-are-roots-of-a-x-2-b-x-c-0-then-the-equation-whose-roots-are-2-and-2-is-50791
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ and $\beta$ are roots of $a x^{2}+b x+c=0,$ then
<br/>the equation whose roots are $\alpha^{2}$ and $\beta^{2},$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a^{2} x^{2}-\left(b^{2}-2 a c\right) x+c^{2}=0$</span> </div>

answer:
<div class="solution">Given. $\alpha$ and $\beta$ are roots of $a x^{2}+b x+c=0$
<br/>$\therefore \quad \alpha+\beta=\frac{-b}{a}$
<br/>and $\quad a \beta=\frac{c}{a}$
<br/>Now, if the roots are $\alpha^{2}$ and $\beta^{2}$, then
<br/>$$
<br/>\begin{aligned}
<br/>a^{2}+\beta^{2} &amp;=(\alpha+\beta)^{2}-2 \alpha \beta \\
<br/>&amp;=\left(\frac{-b}{a}\right)^{2}-\frac{2c}{a}=\frac{b^{2}}{a^{2}}-\frac{2 c}{a} \\
<br/>\text { And } \quad \alpha^{2} \beta^{2} &amp;=(\alpha \beta)^{2}=\left(\frac{c}{a}\right)^{2}=\frac{c^{2}}{a^{2}}
<br/>\end{aligned}
<br/>$$
<br/>The equation whose roots are $\alpha^{2}$ and $\beta^{2}$. is
<br/>$$
<br/>x^{2}-\left(\frac{b^{2}}{a^{2}}-\frac{2 c}{a}\right) x+\frac{c^{2}}{a^{2}}=0
<br/>$$
<br/>$\therefore \quad a^{2} x^{2}-\left(b^{2}-2 a c\right) x+c^{2}=0$</div>


title: if-and-are-roots-of-x-2-x-1-0-then-the-value-of-2013-2013-is
slug: if-and-are-roots-of-x-2-x-1-0-then-the-value-of-2013-2013-is-33943
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ and $\beta$ are roots of $x^{2}-x+1=0,$ then the value of $\alpha^{2013}+\beta^{2013}$ is</div>

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

answer:
<div class="solution">Given equation is $x^{2}-x+1=0$
<br/>$$
<br/>\begin{aligned}
<br/>x &amp;=\frac{1 \pm \sqrt{1-4}}{2} \quad\left(\because x=\frac{b \pm \sqrt{b^{2}-4 a c}}{2 a}\right) \\
<br/>&amp;=\frac{1 \pm i \sqrt{3}}{2}=\frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2} \\
<br/>\Rightarrow-x &amp;=\frac{-1+i \sqrt{3}}{2}, \frac{-1-i \sqrt{3}}{2} \\
<br/>\Rightarrow+x &amp;=\omega,-\omega^{2}
<br/>\end{aligned}
<br/>$$
<br/>Since, $(\alpha, \beta)$ are the roots of given equation. Then, $\alpha=-\omega$ and $\beta=-\omega^{2}$
<br/>$$
<br/>\begin{aligned}
<br/>\therefore \alpha^{2013}+\beta^{2013} &amp;=-(\omega)^{2013}+\left(-\omega^{2}\right)^{2013} \\
<br/>&amp;=-\omega^{2013}-\omega^{4026} \\
<br/>&amp;=-\left(\omega^{3}\right)^{671}-\left(\omega^{3}\right)^{1342} \\
<br/>&amp;=-(1)^{671}-(1)^{1342} \\
<br/>&amp;=-1-1=-2
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
title:
if-and-are-the-roots-of-the-equation-x-2-p-x-q-0-where-p-and-q-are-real-then-the-roots-of-the-equation-p-2-4-q-p-2-x-2-4-p-x-16-q-0-are

slug:
if-and-are-the-roots-of-the-equation-x-2-p-x-q-0-where-p-and-q-are-real-then-the-roots-of-the-equation-p-2-4-q-p-2-x-2-4-p-x-16-q-0-are-76978

question:
<div class="question">If $(\alpha+\sqrt{\beta})$ and $(\alpha-\sqrt{\beta})$ are the roots of the equation $x^{2}+p x+q=0,$ where $\alpha, \beta, p$ and $q$ are real, then the roots of the equation $\left(p^{2}-4 q\right)\left(p^{2} x^{2}+4 p x\right)-16 q=0$ are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\frac{1}{\alpha}+\frac{1}{\sqrt{\beta}}\right)$ and $\left(\frac{1}{\alpha}-\frac{1}{\sqrt{\beta}}\right)$</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">$\left(\frac{1}{\sqrt{\alpha}}+\frac{1}{\beta}\right)$ and $\left(\frac{1}{\sqrt{\alpha}}-\frac{1}{\beta}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{\sqrt{\alpha}}+\frac{1}{\sqrt{\beta}}\right)$ and $\left(\frac{1}{\sqrt{\alpha}}-\frac{1}{\sqrt{\beta}}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(\sqrt{\alpha}+\sqrt{\beta})$ and $(\sqrt{\alpha}-\sqrt{\beta})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{1}{\alpha}+\frac{1}{\sqrt{\beta}}\right)$ and $\left(\frac{1}{\alpha}-\frac{1}{\sqrt{\beta}}\right)$</span> </div>

answer:
<div class="solution">Since, $(\alpha+\sqrt{\beta})$ and $(\alpha-\sqrt{\beta})$ are the roots of the equation $x^{2}+p x+q=0$
<br/>$\therefore \quad$ Sum of roots $=-p$
<br/>$\Rightarrow \quad(\alpha+\sqrt{\beta})+(\alpha-\sqrt{\beta})=-P$
<br/>$\Rightarrow$
<br/>$\mathbf{2}\alpha=-p \Rightarrow \alpha=-\frac{\underline{p}}{2}$
<br/>and produet of roots $=\mathrm{q}$ 
<br/>$(\alpha+\sqrt{\beta})(\alpha-\sqrt{\beta})=q$
<br/>$\Rightarrow \alpha^{2}-\beta=q$
<br/>$\Rightarrow \quad \beta=\alpha^{2}-q=\left(-\frac{p}{2}\right)^{2}-q=\frac{p^{2}}{4}-q$
<br/>$\Rightarrow \quad p^{2}-4 q=4 \beta$
<br/>$\therefore$ The given equation $\left(p^{2}-4 q\right)\left(p^{2} x^{2}+4 p x\right)-16 q=0$
<br/>$4 \beta\left(p^{2} x^{2}+4 p x\right)-16\left(\alpha^{2}-\beta\right)=0$
<br/>$\left[\alpha^{2}-\beta=q\right]$
<br/>$\beta\left(4 a^{2} x^{2}-8 x x\right)-4\left(a^{2}-\beta\right)=0$
<br/>$\alpha^{2} \beta x^{2}-2 x f\left(x+\beta=\alpha^{2}\right.$
<br/>$\Rightarrow \quad(\alpha x \sqrt{\beta}-\sqrt{\beta})^{2}=\alpha^{2}$
<br/>$\Rightarrow \quad \operatorname{cox} \sqrt{\beta}-\sqrt{\beta}=\pm \alpha$
<br/>$\therefore$
<br/>$x=\frac{1}{\alpha} \pm \frac{1}{\sqrt{\beta}}$
<br/>$\Rightarrow x=\left(\frac{1}{\alpha}+\frac{1}{\sqrt{\beta}}\right)$ and $\left(\frac{1}{\alpha}-\frac{1}{\sqrt{\beta}}\right)$</div>


title: if-and-are-the-roots-of-the-quadratic-equation-a-x-2-b-x-c-0-and-3-b-2-16-a-c-then
slug: if-and-are-the-roots-of-the-quadratic-equation-a-x-2-b-x-c-0-and-3-b-2-16-a-c-then-14199
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x^{2}+b x+c=0$ and $3 b^{2}=16 a c$ then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha=3 \beta$ or $\beta=3 \alpha$</span> </div>

answer:
<div class="solution">Given that, $(\alpha, \beta)$ are roots of the quadratic
<br/>equation
<br/>and $\begin{aligned} a x^{2}+b x+c &amp;=0 \\ 3 b^{2} &amp;=16 a c \end{aligned}$
<br/>$\therefore$
<br/>$\alpha+\beta=\frac{-b}{a}$ and $\alpha \beta=\frac{c}{a}$
<br/>From Eq. (i), we get $\quad 3 b \cdot b=16 \cdot a \cdot c$
<br/>$\Rightarrow \quad 3\left(\frac{b}{a}\right)^{2}=16\left(\frac{c}{a}\right)$ (divide by $\left.a^{2}\right)$
<br/>$\Rightarrow \quad 3(\alpha+\beta)^{2}=16 \alpha \beta[$ from Eq. $(i i)]$
<br/>$\Rightarrow 3 \alpha^{2}+3 \beta^{2}+6 \alpha \beta=16 \alpha \beta$
<br/>$\Rightarrow \quad 3 \alpha^{2}+3 \beta^{2}=10 \alpha \beta$
<br/>$\Rightarrow \quad 3\left(\frac{\alpha}{\beta}\right)+3\left(\frac{\beta}{\alpha}\right)=10$
<br/>Let $\quad \frac{\alpha}{\beta}=x$
<br/>$\begin{array}{lr}\text { Then, } &amp; 3 x+\frac{3}{x}=10 \\ \Rightarrow &amp; 3 x^{2}-10 x+3=0 \\ \Rightarrow &amp; 3 x^{2}-9 x-x+3=0 \\ \Rightarrow &amp; 3 x(x-3)-1(x-3)=0 \\ \Rightarrow &amp; (x-3)(3 x-1)=0 \\ \Rightarrow &amp; x=\frac{1}{3}, 3 \Rightarrow \frac{\alpha}{\beta}=\frac{1}{3}, 3\end{array}$
<br/>$\therefore \quad \beta=3 \alpha$ or $\alpha=3 \beta$</div>


title: if-and-are-the-roots-of-x-2-p-x-1-0-and-is-a-root-of-x-2-p-x-1-0-then-is
slug: if-and-are-the-roots-of-x-2-p-x-1-0-and-is-a-root-of-x-2-p-x-1-0-then-is-59808
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ and $\beta$ are the roots of $x^{2}-p x+1=0$ and $\gamma$ is a root of $x^{2}+p x+1=0,$ then $(\alpha+\gamma)(\beta+\gamma)$ is</div>

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

answer:
<div class="solution">Since, $\alpha$ and $\beta$ are the roots of $x^{2}-p x+1=0$.
<br/>$\therefore$
<br/>$\alpha+\beta=p$ and $\alpha \beta=1$
<br/>Also, $\gamma$ is the root of $x^{2}+p x+1=0$ $\therefore \quad \gamma^{2}+p{\gamma}+1=0 \Rightarrow \gamma^{2}=-p \gamma-1$
<br/>Now, $(\alpha+\gamma)(\beta+\gamma)=\alpha \beta+\alpha \gamma+\beta \gamma+\gamma^{2}$
<br/>$=1+\gamma(\alpha+\beta)-p \gamma-1=\gamma(\alpha+\beta-p)$
<br/>$=\gamma \times 0=0$
<br/>$$
<br/>[\because \alpha+\beta=p]
<br/>$$
<br/>Alternate Method
<br/>Since, $\gamma=-\alpha$ or $-\beta$
<br/>$\therefore \quad(\alpha+\gamma)(\beta+\gamma)=0$</div>


title: if-angles-a-b-and-c-are-in-ap-then-b-a-c-is-equal-to
slug: if-angles-a-b-and-c-are-in-ap-then-b-a-c-is-equal-to-82420
tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are in A.P., then $\frac{\mathrm{a}+\mathrm{c}}{\mathrm{b}}$ is equal to</div>

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

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

answer:
<div class="solution">Hints: $2 \mathrm{~B}=\mathrm{A}+\mathrm{C}$<br/>$$<br/>=\frac{\sin A+\sin C}{\sin B}=\frac{2 \sin \left(\frac{A+C}{2}\right) \cos \left(\frac{A-C}{2}\right)}{\sin B}=\frac{2 \sin B}{\sin B} \cos \left(\frac{A-C}{2}\right)=2 \cos \left(\frac{A-C}{2}\right)<br/>$$</div>


title: if-aniline-is-treated-with-conc-h-2-so-4-and-heated-at-20-0-c-the-product-is
slug: if-aniline-is-treated-with-conc-h-2-so-4-and-heated-at-20-0-c-the-product-is-29277
tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If aniline is treated with conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$ and heated at $200^{\circ} \mathrm{C}$, the product is</div>

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

answer:
<div class="solution">When aniline is treated with conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$ followed by heating at $200^{\circ} \mathrm{C}$, the product obtained is sulphanilic acid i.e. $p$ -aminobenzene sulphonic acid
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a51.png"/></div>


title: if-ar-g-z-2-then-ar-g-z-ar-g-z-is
slug: if-ar-g-z-2-then-ar-g-z-ar-g-z-is-12859
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $-\pi &lt; \arg (z) &lt; -\frac{\pi}{2}$ then $\arg \bar{z}-\arg (-\bar{z})$ is</div>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/xHjyKu7U4m6kDG8h9te7-4KJTE2WbGz8ldaOGT_X5fI.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; \text { if } \arg (z)=-\pi+\theta \\<br/>&amp; \Rightarrow \arg (\bar{z})=\pi-\theta \\<br/>&amp; \arg (-\bar{z})=-\theta \\<br/>&amp; \arg (\bar{z})-\arg (-\bar{z})=\pi-\theta-(-\theta)=\pi-\theta+\theta=\pi<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-are-the-roots-of-a-x-2-b-x-c-0-a-0-and-h-h-are-the-roots-of-p-x-2-q-x-r-0-p-0-then-the-ratio-of-the-squares-of-their-discriminants-is

slug:
if-are-the-roots-of-a-x-2-b-x-c-0-a-0-and-h-h-are-the-roots-of-p-x-2-q-x-r-0-p-0-then-the-ratio-of-the-squares-of-their-discriminants-is-21489

question:
<div class="question">If $\alpha, \beta$ are the roots of $a x^{2}+b x+c=0(a \neq 0)$ and $\alpha+h, \beta+h$ are the roots of $p x^{2}+q x+r=0$ $(p \neq 0),$ then the ratio of the squares of their discriminants is</div>

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

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

answer:
<div class="solution">Given, $\alpha, \beta$ are the roots of $a x^{2}+b x+c=0$ and
<br/>$a+h, \beta+h$ are the roots of $p x^{2}+q x+r=0$
<br/>$\therefore \quad \alpha+\beta=-\frac{b}{a}, \alpha \beta=\frac{c}{a}$
<br/>and $\alpha+h+\beta+h=-\frac{q}{p},(\alpha+h)(\beta+h)=\frac{r}{p}$
<br/>Now, $\quad(\alpha+h)-(\beta+h)=\alpha-\beta$
<br/>$\Rightarrow \quad[(\alpha+h)-(\beta+h)]^{2}=(\alpha-\beta)^{2}$
<br/>$\Rightarrow[(\alpha+h)+(\beta+h)]^{2}-4(\alpha+h)(\beta+h)$
<br/>$\Rightarrow \quad \frac{q^{2}-4 p r}{p^{2}}=\frac{b^{2}-4 a c}{a^{2}}$
<br/>$\therefore$
<br/>$\frac{b^{2}-4 a c}{q^{2}-4 p r}=\frac{a^{2}}{p^{2}}$
<br/>Hence, the ratio of the square of their discriminants is $a^{2}: p^{2}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
if-are-the-roots-of-the-quadratic-equation-x-2-a-x-b-0-b-0-then-the-quadratic-equation-whose-roots-are-1-1-is

slug:
if-are-the-roots-of-the-quadratic-equation-x-2-a-x-b-0-b-0-then-the-quadratic-equation-whose-roots-are-1-1-is-99077

question:
<div class="question">If $\alpha, \beta$ are the roots of the quadratic equation $x^{2}+a x+b=0,(b \neq 0),$ then the quadratic equation whose roots are $\alpha-\frac{1}{\beta}, \beta-\frac{1}{\alpha},$ is</div>

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

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

answer:
<div class="solution">Given equation is, $x^{2}+a x+b=0,(b \neq 0)$
<br/>its roots are $\alpha$ and $\beta$. Then, sum of roots $=\alpha+\beta=-a$
<br/>Product of roots $=\alpha \cdot \beta=b$
<br/>Now,
<br/>$$
<br/>\left(\alpha-\frac{1}{\beta}\right)+\left(\beta-\frac{1}{\alpha}\right)=(\alpha+\beta)-\left(\frac{\alpha+\beta}{\alpha \beta}\right)
<br/>$$
<br/>$=-a-\frac{(-a)}{b}[$ from Eqs. (0) and $(i i)]$
<br/>$=-a+\frac{a}{b}=\frac{a}{b}(1-b)$
<br/>and $\left(\alpha-\frac{1}{\beta}\right)\left(\beta-\frac{1}{\alpha}\right)=\alpha \beta-1-1+\frac{1}{\alpha \beta}$
<br/>$=b+\frac{1}{b}-2[$ from $E q \cdot(i i)]$
<br/>$=\frac{1}{b}\left(b^{2}-2 b+1\right)=\frac{1}{b}(b-1)^{2}$
<br/>Required of quadratic equation whose roots are $\left(\alpha-\frac{1}{\beta}\right)$ and $\left(\beta-\frac{1}{\alpha}\right)$ is
<br/>$$
<br/>\begin{array}{l}
<br/>x^{2}-\left\{\left(\alpha-\frac{1}{\beta}\right)+\left(\beta-\frac{1}{\alpha}\right)\right\} x \\
<br/>+\left\{\left(\alpha-\frac{1}{\beta}\right)\left(\beta-\frac{1}{\alpha}\right)\right\}=0
<br/>\end{array}
<br/>$$
<br/>On putting the values from Eqs. (i) and (ii). we get $x^{2}-\frac{a}{b}(1-b) x+\frac{1}{b}(b-1)^{2}=0$
<br/>$\Rightarrow \quad b x^{2}+a(b-1) x+(b-1)^{2}=0, b \neq 0$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-are-the-roots-of-the-quadratic-equation-x-2-p-x-q-0-then-the-values-of-3-3-and-4-3-2-4-are-respectively

slug:
if-are-the-roots-of-the-quadratic-equation-x-2-p-x-q-0-then-the-values-of-3-3-and-4-3-2-4-are-respectively-96584

question:
<div class="question">If $\alpha, \beta$ are the roots of the quadratic equation $x^{2}+p x+q=0,$ then the values of $\alpha^{3}+\beta^{3}$ and $\alpha^{4}+\alpha^{3} \beta^{2}+\beta^{4}$ are respectively</div>

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

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

answer:
<div class="solution">$\because$ Sum of roots, $\alpha+\beta=-p$ and $\alpha \beta=q$ $\therefore \quad\left(a^{3}+\beta^{3}\right)=(\alpha+\beta)^{3}-3 \alpha \beta(\alpha+\beta)$
<br/>$$
<br/>\begin{aligned}
<br/>=(-p)^{3}-3 q(-p) \\
<br/>=&amp;-p^{3}+3 p q \\
<br/>&amp;=\left(\alpha^{4}+\beta^{4}\right)+(\alpha \beta)^{2} \\
<br/>&amp;=\left(\alpha^{2}+\beta^{2}\right)^{2}-(\alpha \beta)^{2} \\
<br/>&amp;=\left[(\alpha+\beta)^{2}-2 \alpha \beta\right]^{2}-(\alpha \beta)^{2} \\
<br/>&amp;=\left[(-p)^{2}-2 q\right]^{2}-3(q)^{2} \\
<br/>&amp;=\left[p^{2}-2 q\right]^{2}-3 q^{2} \\
<br/>&amp;=p^{4}-4 p^{2} q+4 q^{2}-q^{2} \\
<br/>&amp;=p^{4}-4 p^{2} q+3 q^{2} \\
<br/>&amp;=p^{4}-3 p^{2} q-p^{2} q+3 q^{2} \\
<br/>&amp;=p^{2}\left(p^{2}-3 q\right)-q\left(p^{2}-3 q\right) \\
<br/>&amp;=\left(p^{2}-q\right)\left(p^{2}-3 q\right)
<br/>\end{aligned}
<br/>$$</div>


title: if-b-0-1-t-1-e-t-d-t-then-a-1-a-t-a-1-e-1-is
slug: if-b-0-1-t-1-e-t-d-t-then-a-1-a-t-a-1-e-1-is-41462
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{b}=\int_{0}^{1} \frac{\mathrm{e}^{t}}{\mathrm{t}+1} d t$, then $\int_{\mathrm{a}-1}^{\mathrm{a}} \frac{\mathrm{e}^{-1}}{\mathrm{t}-\mathrm{a}-1}$ is</div>

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

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

answer:
<div class="solution">$\int_{a-1}^{a} \frac{e^{-t}}{t-a-1} d t$
<br/>$=\int_{\mathrm{a}-1}^{\mathrm{a}} \frac{\mathrm{e}^{-(2 \mathrm{a}-1-\mathrm{t})}}{2 \mathrm{a}-1-\mathrm{t}-\mathrm{a}-1} \mathrm{dt}$
<br/>$=\int_{\mathrm{a}-1}^{\mathrm{a}} \frac{\mathrm{e}^{\mathrm{t}-2 \mathrm{a}+1}}{\mathrm{a}-2-\mathrm{t}} \mathrm{dt}$
<br/>$t-(a-1)=x \quad \Rightarrow d t=d x$
<br/>$=\int_{0}^{1} \frac{e^{x} \cdot e^{-a}}{-x-1} d x$
<br/>$=-e^{-a} \int_{0}^{1} \frac{e^{x}}{x+1} d x=-b e^{-a}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-b-1-b-2-2-c-1-c-2-and-b-1-b-2-c-1-c-2-are-all-real-numbers-then-at-least-one-of-the-equations-x-2-h-1-x-c-1-0-and-x-2-b-2-x-c-2-0-has

slug:
if-b-1-b-2-2-c-1-c-2-and-b-1-b-2-c-1-c-2-are-all-real-numbers-then-at-least-one-of-the-equations-x-2-h-1-x-c-1-0-and-x-2-b-2-x-c-2-0-has-10538

question:
<div class="question">If $b_{1} b_{2}=2\left(c_{1}+c_{2}\right)$ and $b_{1}, b_{2}, c_{1}, c_{2}$ are all real numbers, then at least one of the equations $\bar{x}^{2}+h_{1} x+c_{1}=0$ and $x^{2}+b_{2} x+c_{2}=0$ has</div>

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

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

answer:
<div class="solution">We have equations
<br/>$$
<br/>\begin{aligned}
<br/>x^{2}+b_{1} x+c_{1} &amp;=0 \\
<br/>D_{1} &amp;=b_{1}^{2}-4 c_{1} \\
<br/>x^{2}+b_{2} x+c_{2} &amp;=0
<br/>\end{aligned}
<br/>$$
<br/>$$
<br/>\begin{array}{c}
<br/>D_{2}=b_{2}^{2}-4 c_{2} \\
<br/>\text { Now, } D_{1}+D_{2}=b_{1}^{2}+b_{2}^{2}-4\left(c_{1}+c_{2}\right) \\
<br/>=b_{1}^{2}+b_{2}^{2}-2 b_{1} b_{2} \quad\left[\because b_{1} b_{2}=2\left(c_{1}+c_{2}\right)\right] \\
<br/>=\left(b_{1}-b_{2}\right)^{2} \geq 0
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow$ At least one of $D_{1}$ and $D_{2}$ are non-negative
<br/>real roots.</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
if-b-and-b-are-the-ends-of-minor-axis-and-s-and-s-are-the-foci-of-the-ellipse-25-x-2-9-y-2-1-then-the-area-of-the-rhombus-sbs-b-will-be

slug:
if-b-and-b-are-the-ends-of-minor-axis-and-s-and-s-are-the-foci-of-the-ellipse-25-x-2-9-y-2-1-then-the-area-of-the-rhombus-sbs-b-will-be-42725

question:
<div class="question">If $\mathrm{B}$ and $\mathrm{B}^{\prime}$ are the ends of minor axis and $\mathrm{S}$ and $\mathrm{S}^{\prime}$ are the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, then the area of the rhombus
<br/>$\mathrm{SBS}^{\prime} \mathrm{B}^{\prime}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12 sq. unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48 sq. unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">24 sq. unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">36 sq. unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">24 sq. unit</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hLAeCZaoSZTVL4WYmy8kTp8PsOjxIsKCHHroFqdd7-4.original.fullsize.png"/><br/> <br/>Hint:
<br/>$\mathrm{B}=(0,3), \mathrm{B}^{\prime}=(0,-3)$
<br/>$\mathrm{S} \equiv(\mathrm{ae}, 0) \equiv\left(5 \times \sqrt{1-\frac{9}{25}}, 0\right) \equiv(4,0) \quad \therefore$ area $\mathrm{SBS}^{\prime} \mathrm{B}^{\prime}=2 \times$ area $\triangle \mathrm{BSB}^{\prime}=2 \times \frac{1}{2} \times 4 \times 6=24$ sq. unit</div>


title: if-be-the-roots-of-the-quadratic-equation-x-2-x-1-0-then-the-equation-whose-roots-are-19-7-is
slug: if-be-the-roots-of-the-quadratic-equation-x-2-x-1-0-then-the-equation-whose-roots-are-19-7-is-81769
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\alpha, \beta$ be the roots of the quadratic equation $\mathrm{x}^2+\mathrm{x}+1=0$ then the equation whose roots are $\alpha^{19}, \beta^7$ is</div>

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

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

answer:
<div class="solution">Hints : Roots are $\omega, \omega^2$<br/>Let $\alpha=\omega, \beta=\omega^2$ $\alpha^{19}=\omega, \beta^7=\omega^2$<br/>$\therefore$ Equation remains same i.e. $\mathrm{x}^2+\mathrm{x}+1=0$</div>


title: if-be-the-roots-of-x-2-a-x-1-b-0-then-the-value-of-2-a-1-2-a-1-a-b-2
slug: if-be-the-roots-of-x-2-a-x-1-b-0-then-the-value-of-2-a-1-2-a-1-a-b-2-74924
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\alpha, \beta$ be the roots of $x^2-a(x-1)+b=0$, then the value of $\frac{1}{\alpha^2-a \alpha}+\frac{1}{\beta^2-a \beta}+\frac{2}{a+b}$</div>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints }: x^2-a x=a+3 \quad \alpha \beta=a+b \\<br/>&amp; \alpha+\beta=a \\<br/>&amp; \alpha^2-a \alpha=-(a+b) \\<br/>&amp; \beta^2-a \alpha=-(a+b) \\<br/>&amp;-\frac{1}{a+b}-\frac{1}{a+b}+\frac{2}{a+b}=0<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">120</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-c-0-c-1-c-2-c-15-are-the-binomial-coefficients-in-the-expansion-of-1-x-15-then-the-value-of-c-0-c-1-2-c-1-c-2-3-c-2-c-3-15-c-14-c-15-is

slug:
if-c-0-c-1-c-2-c-15-are-the-binomial-coefficients-in-the-expansion-of-1-x-15-then-the-value-of-c-0-c-1-2-c-1-c-2-3-c-2-c-3-15-c-14-c-15-is-45493

question:
<div class="question">If $\mathrm{c}_{0}, \mathrm{c}_{1}, \mathrm{c}_{2}, \ldots \ldots \mathrm{c}{15}$ are the Binomial co-efficients in the expansion of $(1+\mathrm{x})^{15}$, then the value of $\frac{\mathrm{C}{1}}{\mathrm{c}{0}}+2 \frac{\mathrm{C}{2}}{\mathrm{c}{1}}+3 \frac{\mathrm{c}{3}}{\mathrm{c}{2}}+\ldots \ldots+15 \frac{\mathrm{c}{15}}{\mathrm{c}{14}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1240</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">120</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">124</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">140</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>$S_{n}=\sum_{r=1}^{15} r \frac{{ }^{15} C_{r}}{{ }^{15} c_{r-1}}=\sum_{r=1}^{15}(15-r+1)=16 \times 15-\frac{15 \times 16}{2}=120$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
title:
if-c-0-c-1-c-2-c-n-denote-the-coefficients-in-the-expansion-of-1-x-n-then-the-value-of-c-1-2-c-2-3-c-3-nc-n-is

slug:
if-c-0-c-1-c-2-c-n-denote-the-coefficients-in-the-expansion-of-1-x-n-then-the-value-of-c-1-2-c-2-3-c-3-nc-n-is-94657

question:
<div class="question">If $\mathrm{c}_0, \mathrm{c}_1, \mathrm{c}_2, \ldots \ldots \ldots \ldots \ldots . \mathrm{c}_{\mathrm{n}}$ denote the co-efficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$ then the value of $\mathrm{c}_1+2 \mathrm{c}_2+3 \mathrm{c}_3+\ldots . .+\mathrm{nc}_{\mathrm{n}}$ is</div>

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

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

answer:
<div class="solution">Hints : $(1+\mathrm{x})^{\mathrm{n}}=\dot{c}_0+\mathrm{xc}_1+\mathrm{x}^2 \mathrm{c}_2+\ldots \ldots . \mathrm{x}^{\mathrm{n}} \mathrm{c}_{\mathrm{n}}$<br/>$n(1+x)^{n-1}=c_1+2 x c_2+\ldots \ldots . . n x^{n-1} c_n$<br/>Put $x=1$<br/>$$<br/>\mathrm{n}(2)^{\mathrm{n}-1}=\mathrm{c}_1+2 \mathrm{c}_2+3 \mathrm{c}_2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots+\mathrm{nc}_{\mathrm{n}}<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
title:
if-c-is-a-point-on-the-line-segment-joining-a-3-4-and-b-2-1-such-that-a-c-2-bc-then-the-coordinate-of-c-is

slug:
if-c-is-a-point-on-the-line-segment-joining-a-3-4-and-b-2-1-such-that-a-c-2-bc-then-the-coordinate-of-c-is-45202

question:
<div class="question">If $C$ is a point on the line segment joining $A(-3,4)$ and $B(2,1)$ such that $A C=2 B C$, then the coordinate of $C$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\frac{1}{3}, 2\right)$</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">$\left(2, \frac{1}{3}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(2,7)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(7,2)$</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/zQumDaJF5ElTFySJqGBu01dadP8lDy5KJdzNhpM6gZA.original.fullsize.png"/><br/><br/>Hints :<br/>$$<br/>\begin{aligned}<br/>&amp; C\left(\frac{4-3}{3}, \frac{2+4}{3}\right) \\<br/>&amp; C\left(\frac{1}{3}, 2\right)<br/>\end{aligned}<br/>$$</div>


title: if-c-is-the-reflecton-of-a-2-4-in-x-axis-and-b-is-the-reflection-of-c-in-y-axis-then-ab-is-1
slug: if-c-is-the-reflecton-of-a-2-4-in-x-axis-and-b-is-the-reflection-of-c-in-y-axis-then-ab-is-1-51136
tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{C}$ is the reflecton of $\mathrm{A}(2,4)$ in $\mathrm{x}$-axis and $\mathrm{B}$ is the reflection of $\mathrm{C}$ in $\mathrm{y}$-axis, then $|\mathrm{AB}|$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{5}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4 \sqrt{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \mathrm{A} \equiv(2,4) ; \mathrm{C} \equiv(2,-4) ; \mathrm{B} \equiv(-2,-4) \\<br/>&amp; |A B|=\sqrt{(2-(-2))^2+(4-(-4))^2}=\sqrt{4^2+8^2} \\<br/>&amp; =\sqrt{16+64}=\sqrt{80}=\sqrt{16 \times 5}=4 \sqrt{5}<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/tKXdNA_L8PJwc1FW0iuBG2zvzwotcMOsEHe4K-5wZBg.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-cl-2-is-passed-through-hot-aqueous-naoh-the-products-formed-have-cl-in-different-oxidation-states-these-are-indicated-as

slug:
if-cl-2-is-passed-through-hot-aqueous-naoh-the-products-formed-have-cl-in-different-oxidation-states-these-are-indicated-as-90278

question:
<div class="question">If $\mathrm{Cl}_{2}$ is passed through hot aqueous $\mathrm{NaOH}$, the products formed have Cl in different oxidation states. These are indicated as</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">-1 and +5</span> </div>

answer:
<div class="solution">If $\mathrm{Cl}_{2}$ is passed through hot aqueous $\mathrm{NaOH}$, it results in the formation of sodium chloride and sodium chlorate as
<br/>$$
<br/>6 \mathrm{NaOH}+3 \mathrm{Cl}_{2} \longrightarrow 5 \mathrm{NaCl}+\mathrm{NaClO}_{3}+3 \mathrm{H}_{2} \mathrm{O}
<br/>$$
<br/>In $\mathrm{NaCl}$, let oxidation state of $\mathrm{Cl}$ be $\mathrm{x}$
<br/>$$
<br/>+1+x=0 \text { or } \quad x=-1
<br/>$$
<br/>In $\operatorname{NaClO}_{3}$, let oxidation state of $\mathrm{Cl}$ be $y$
<br/>$+1+y+(-2) 3=0$ 
<br/>$y-5=0$
<br/>$\therefore y=+5$
<br/>Thus, the obtained compounds have Cl in -1 and +5 oxidation states.</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
if-cos-x-and-sin-x-are-solutions-of-the-differential-equation-a-0-d-x-2-d-2-y-a-1-d-x-d-y-a-2-y-0-where-a-0-a-1-and-a-2-are-real-constants-then-which-

slug:
if-cos-x-and-sin-x-are-solutions-of-the-differential-equation-a-0-d-x-2-d-2-y-a-1-d-x-d-y-a-2-y-0-where-a-0-a-1-and-a-2-are-real-constants-then-which-45658

question:
<div class="question">If $\cos x$ and $\sin x$ are solutions of the differential equation $a_{0} \frac{d^{2} y}{d x^{2}}+a_{1} \frac{d y}{d x}+a_{2} y=0$ where $a_{0}, a_{1}$ and $a_{2}$ are real constants, then which of the following is/are always true?</div>

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

answer:
<div class="solution">Let $f(x)=\cos x$ and $g(x)=\sin x$ Consider the Wronskian of $f(x)$ and $g(x)$.
<br/>$$
<br/>\begin{aligned}
<br/>W &amp;=\left|\begin{array}{ll}
<br/>f(x) &amp; g(x) \\
<br/>f^{\prime}(x) &amp; g^{\prime}(x)
<br/>\end{array}\right| \\
<br/>&amp;=\left|\begin{array}{ll}
<br/>\cos x &amp; \sin x \\
<br/>-\sin x &amp; \cos x
<br/>\end{array}\right| \\
<br/>&amp;=\cos ^{2} x+\sin ^{2} x=1 \neq 0
<br/>\end{aligned}
<br/>$$
<br/>Thus, the functions are linearly independent. So, the general solution of given differential equation is given by $y=A \cos x+B \sin x,$ where $A$ and $B$ are real constants.
<br/>[$\because$ if $y_{1}$ and $y_{2}$ are linearly independent solutions of the differential equation $a y^{\prime \prime}+b y^{\prime}+c=0,$ then the general solution is $y=c_{1} y_{1}+c_{2} y_{2},$ where $c_{1}$ and $c_{2}$ are constants] Hence, option (a) is true.
<br/>(b) Let $y=A \cos \left(x+\frac{\pi}{4}\right)$
<br/>$$
<br/>\begin{array}{l}
<br/>=A\left(\cos x \cdot \cos \frac{\pi}{4}-\sin x \cdot \sin \frac{\pi}{4}\right) \\
<br/>=\frac{A}{\sqrt{2}}(\cos x-\sin x) \\
<br/>=\frac{A}{\sqrt{2}} \cos x+\left(-\frac{A}{\sqrt{2}}\right) \sin x
<br/>\end{array}
<br/>$$
<br/>which is in the form of general solution.
<br/>Hence, option (b) is true.
<br/>(c) Let $y=A \cos x \sin x$
<br/>which cannot be
<br/>expressed in the form of general solution.
<br/>(d) Let $y=A \cos \left(x+\frac{\pi}{4}\right)+B \sin \left(x-\frac{\pi}{4}\right)$
<br/>$$
<br/>\begin{array}{l}
<br/>=A\left(\cos x \cdot \frac{1}{\sqrt{2}}-\sin x \cdot \frac{1}{\sqrt{2}}\right) \\
<br/>\quad+B\left(\sin x \cdot \frac{1}{\sqrt{2}}-\cos x \cdot \frac{1}{\sqrt{2}}\right) \\
<br/>=\cos x\left(\frac{A}{\sqrt{2}}-\frac{B}{\sqrt{2}}\right)+\sin x\left(\frac{B}{\sqrt{2}}-\frac{A}{\sqrt{2}}\right) \text { which is }
<br/>\end{array}
<br/>$$
<br/>in the form of general solution. Hence, option (d) is true.</div>


tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$c-x$</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-cos-x-lo-g-tan-2-x-d-x-sin-x-lo-g-tan-2-x-f-x-then-f-x-is-equal-to-assuming-c-is-a-arbitrary-real-constant

slug:
if-cos-x-lo-g-tan-2-x-d-x-sin-x-lo-g-tan-2-x-f-x-then-f-x-is-equal-to-assuming-c-is-a-arbitrary-real-constant-60678

question:
<div class="question">If $\int \cos x \log \left(\tan \frac{x}{2}\right) d x$$=\sin x \log \left(\tan \frac{x}{2}\right)+f(x),$ then $f(x)$ is equal to (assuming $c$ is a arbitrary real constant)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$c-x$</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">$c+x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x+c$</span> </li> </ul>

answer:
<div class="solution">Let $I=\int \cos x \log \left(\tan \frac{x}{2}\right)$
<br/>$=\log \left(\tan \frac{x}{2}\right) \cdot \sin x-\int \sin x \cdot \frac{1}{\tan \frac{x}{2}} \cdot \sec ^{2} \frac{x}{2} \cdot \frac{1}{2} d x$
<br/>$=\sin x \log \left(\tan \frac{x}{2}\right)-\int \sin x \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} d x$
<br/>$=\sin x \cdot \log \left(\tan \frac{x}{2}\right)-\int \frac{\sin x}{\sin x} d x$
<br/>$=\sin x \cdot \log \left(\tan \frac{x}{2}\right)-\int 1 d x$
<br/>$=\sin x \log \left(\tan \frac{x}{2}\right)-x+c$
<br/>$\therefore f(x)=c-x$</div>


title: if-cot-1-cot-2-cot-n-1-0-1-2-n-2-then-the-maximum-value-of-cos-a-1-cos-2-cos-n-is-given-by
slug: if-cot-1-cot-2-cot-n-1-0-1-2-n-2-then-the-maximum-value-of-cos-a-1-cos-2-cos-n-is-given-by-24493
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\left(\cot \alpha_1\right)\left(\cot \alpha_2\right) \ldots \ldots .\left(\cot \alpha_n\right)=1,0  &lt; \alpha_1, \alpha_2, \ldots . . \alpha_n &lt; \pi / 2$, then the maximum value of $\left(\cos a_1\right)\left(\cos \alpha_2\right) \ldots \ldots$ $\left(\cos \alpha_n\right)$ is given by</div>

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

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

answer:
<div class="solution">$\frac{1}{\left(\cos \alpha_1 \cos \alpha_2 \ldots . \cos \alpha_n\right)^2}$<br/>$\begin{aligned}<br/>&amp;=\left(1+\tan ^2 \alpha_1\right)\left(1+\tan ^2 \alpha_2\right) \ldots \ldots\left(1+\tan ^2 \alpha_n\right) \\<br/>&amp;\geq\left(2 \tan \alpha_1\right)\left(2 \tan \alpha_2\right) \ldots \ldots .\left(2 \tan \alpha_n\right)[\text { AM }-\text { GM inequality } \\<br/>&amp;=2^n \times \frac{1}{\cot \alpha_1 \cot \alpha_2 \ldots . . \cot \alpha_n}=2^n \\<br/>&amp;\therefore \cos \alpha_1 \cos \alpha_2 \ldots . . \cos \alpha_n \leq \frac{1}{2^{n / 2}}<br/>\end{aligned}$</div>


title: if-cot-3-2-x-tan-3-x-cosec-3-k-x-then-the-value-of-k-is
slug: if-cot-3-2-x-tan-3-x-cosec-3-k-x-then-the-value-of-k-is-98556
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3},$ then the value of $k$ 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">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">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1</span> </li> </ul>

answer:
<div class="solution">Given, $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3}$
<br/>Let $\theta=\frac{x}{3},$ then
<br/>$\therefore \quad \cot 2 \theta+\tan \theta=\operatorname{cosec} k \theta$
<br/>$\Rightarrow \quad \frac{\cos 2 \theta}{\sin 2 \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$
<br/>$\Rightarrow \frac{2 \cos ^{2} \theta-1}{2 \sin \theta \cos \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$
<br/>$\Rightarrow \frac{2 \cos ^{2} \theta-1+2 \sin ^{2} \theta}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$
<br/>$\Rightarrow \quad \frac{1}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$
<br/>$\Rightarrow \quad \operatorname{cosec} 2 \theta=\operatorname{cosec} k \theta$
<br/>$\Rightarrow \quad k=2$</div>


title: if-dx-d-f-x-g-x-then-a-b-f-x-g-x-dx-is-equal-to
slug: if-dx-d-f-x-g-x-then-a-b-f-x-g-x-dx-is-equal-to-84506
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\frac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{f}(\mathrm{x})\}=\mathrm{g}(\mathrm{x})$, then $\int_a^b \mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{dx}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}\left[\mathrm{f}^2(\mathrm{~b})-\mathrm{f}^2(\mathrm{a})\right]$</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">$\frac{1}{2}\left[\mathrm{~g}^2(\mathrm{~b})-\mathrm{g}^2(\mathrm{a})\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(b)-f(a)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}\left[\mathrm{f}\left(\mathrm{b}^2\right)-\mathrm{f}\left(\mathrm{a}^2\right)\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2}\left[\mathrm{f}^2(\mathrm{~b})-\mathrm{f}^2(\mathrm{a})\right]$</span> </div>

answer:
<div class="solution">Hints $: f(x)=\int g(x) d x$<br/>$$<br/>\begin{aligned}<br/>&amp; \int_a^b f(x) \cdot g(x) \cdot d x=(f(x) f(x))_a^b-\int_a^b g(x) f(x) d x \\<br/>&amp; I=f^2(b)-f^n(a)^{-1} \\<br/>&amp; I=\frac{1}{2}\left(f^2(b)-f^2(a)\right)<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
if-e-and-b-are-the-magnitudes-of-electric-and-magnetic-fields-respectively-in-some-region-of-space-then-the-possibilities-for-which-a-charged-particle

slug:
if-e-and-b-are-the-magnitudes-of-electric-and-magnetic-fields-respectively-in-some-region-of-space-then-the-possibilities-for-which-a-charged-particle-42425

question:
<div class="question">If $E$ and $B$ are the magnitudes of electric and magnetic fields respectively in some region of space, then the possibilities for which a charged particle may move in that space with a uniform velocity of magnitude $v$ are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$E=v B$</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">B</span> <span class="option-data">$E \neq 0, B=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$E=0, B \neq 0$</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">D</span> <span class="option-data">$E \neq 0, B \neq 0$</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">$E=v B$, $E=0, B \neq 0$, $E \neq 0, B \neq 0$</span> </div>

answer:
<div class="solution">The charged particle will move with uniform velocity in that space if $E=0$ and $B \neq 0,$ since a moving charge in a magnetic field experience a force. Also if $E = 0, B \neq 0,$ the charged particle will move with uniform velocity in that space.
<br/>If $E=v B$, then particle again moves with uniform velocity.</div>


tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-e-s-i-n-x-c-o-s-2-x-x-c-o-s-3-x-s-i-n-x-d-x-e-s-i-n-x-f-x-c-where-c-is-constant-of-integration-then-f-x-is-equal-to

slug:
if-e-s-i-n-x-c-o-s-2-x-x-c-o-s-3-x-s-i-n-x-d-x-e-s-i-n-x-f-x-c-where-c-is-constant-of-integration-then-f-x-is-equal-to-90430

question:
<div class="question">If $\int e^{\sin x} \cdot\left[\frac{x \cos ^{3} x-\sin x}{\cos ^{2} x}\right] d x=e^{\sin x} f(x)+c$
<br/>where c is constant of integration, then $f(x)$ is
<br/>equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sec x-x$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x-\sec x$</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">$\tan x-x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-\tan x$</span> </li> </ul>

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

answer:
<div class="solution">We have. $\int e^{\operatorname{cin} x}\left(\frac{x \cos ^{3} x-\sin x}{\cos ^{2} x}\right) d x=e^{\sin x} f(x)+c$
<br/>$\int e^{\operatorname{sin} x}(x \cos x-\sec x \tan x) d x=e^{\sin x} f(x)+c$
<br/>$\int e^{\operatorname{sin} x}(x \cos x-1+1-\sec x \tan x) d x$
<br/>$=e^{\sin x} f(x)+c$
<br/>$\left.\int[ e^{\sin x} \cos x(x-\sec x)+e^{\sin x}(1-\sec x \tan x)\right] d x$
<br/>$=e^{\sin x} f(x)+c$
<br/>$\int \frac{d}{d x}\left\{e^{\sin x}(x-\sec x)\right\} d x=e^{\sin x} f(x)+c$
<br/>$e^{\operatorname{cin} x}(x-\sec x)=e^{\sin x} f(x)+c$
<br/>$f(x)=x-\sec x$</div>


title: if-e-s-i-n-x-e-s-i-n-x-4-0-then-the-number-ofreal-values-of-x-is
slug: if-e-s-i-n-x-e-s-i-n-x-4-0-then-the-number-ofreal-values-of-x-is-43047
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $e^{\sin x}-e^{-\sin x}-4=0,$ then the number ofreal values of $x$ is</div>

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

answer:
<div class="solution">$$
<br/>\text { Given that, } e^{\sin x}-e^{-\sin x}-4=0
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;\begin{array}{l}
<br/>\text { Let } e^{\sin x}=t \\
<br/>\therefore \quad t-t^{-1}-4=0 \\
<br/>\Rightarrow \quad t-\frac{1}{t}-4=0 \\
<br/>\Rightarrow \quad t^{2}-1-4 t=0 \\
<br/>\quad t=\frac{4 \pm 2 \sqrt{5}}{2}=2 \pm \sqrt{5} \\
<br/>\Rightarrow \quad e^{\sin x}=2 \pm \sqrt{5} \\
<br/>\Rightarrow \quad \quad \quad e^{\sin x}=2+\sqrt{5} \text { and } e^{\sin x}=2-\sqrt{5} \\
<br/>\Rightarrow \quad e^{\sin x}=2 + \sqrt{5} \\
<br/>\text { Also, }-1 \leq \sin x \leq 1 \\
<br/>\therefore \frac{1}{e} \leq e^{\sin x} \leq e \\
<br/>\therefore e^{\sin x} \neq 2+\sqrt{5}
<br/>\end{array}\\
<br/>&amp;\text { Hence, there is no value of } x \text { . }
<br/>\end{aligned}
<br/>$$</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
if-electrolysis-of-aqueous-cuso-4-solution-is-carried-out-using-cuelectrodes-the-reaction-taking-place-at-the-anode-is

slug:
if-electrolysis-of-aqueous-cuso-4-solution-is-carried-out-using-cuelectrodes-the-reaction-taking-place-at-the-anode-is-82901

question:
<div class="question">If electrolysis of aqueous CuSO $_{4}$ solution is carried out using Cu-electrodes, the reaction taking place at the anode is</div>

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

answer:
<div class="solution">Electrolysis of copper sulphate solution using Cu-electrodes, 
<br/>Ions present : $\mathrm{Cu}^{2+}, \mathrm{H}^{+}, \mathrm{SO}_{4}^{2-}, \mathrm{OH}^{-}$
<br/>At cathode $\mathrm{Cu}^{2+}(a q)+2 e^{-} \rightarrow \mathrm{Cu}(s)$
<br/>At anode $\mathrm{Cu}(s)-2 e^{-} \rightarrow \mathrm{Cu}^{2+}(a q)$
<br/>So, the option (d) is correct answer.</div>


title: if-f-0-2-r-is-defined-as-f-1-tan-1-tan-1-tan-1-tan-1-then-the-range-of-f-is
slug: if-f-0-2-r-is-defined-as-f-1-tan-1-tan-1-tan-1-tan-1-then-the-range-of-f-is-69842
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f:[0, \pi / 2) \rightarrow R$ is defined as
<br/>$f(\theta)=\left|\begin{array}{ccc}1 &amp; \tan \theta &amp; 1 \\ -\tan \theta &amp; 1 &amp; \tan \theta \\ -1 &amp; -\tan \theta &amp; 1\end{array}\right|$ Then, the range of $f$ is</div>

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

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

answer:
<div class="solution">We have, $f(\theta)=\left|\begin{array}{ccc}1 &amp; \tan \theta &amp; 1 \\ -\tan \theta &amp; 1 &amp; \tan \theta \\ -1 &amp; -\tan \theta &amp; 1\end{array}\right|$
<br/>$=1\left(1+\tan ^{2} \theta\right)-\tan \theta$
<br/>$(-\tan \theta+\tan \theta)+1\left(\tan ^{2} \theta+1\right)$
<br/>$=2\left(1+\tan ^{2} \theta\right)-0=2 \sec ^{2} \theta$
<br/>$\therefore$ Range of $f=(2, \infty)$</div>


title: if-f-0-k-k-0-then-the-value-of-lim-x-0-x-2-2-f-x-3-f-2-x-f-4-x-is
slug: if-f-0-k-k-0-then-the-value-of-lim-x-0-x-2-2-f-x-3-f-2-x-f-4-x-is-24481
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3 k$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f^{\prime \prime}(0)=k, k \neq 0,$ then the value of $\lim _{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^{2}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$k$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 k$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3 k$</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">$4 k$</span> </li> </ul>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^{2}}$
<br/>$=\lim _{x \rightarrow 0} \frac{2 f^{\prime}(x)-3 f^{\prime}(2 x) \cdot 2+f^{\prime}(4 x) \cdot 4}{2 x}$
<br/>$=\lim _{x \rightarrow 0} \frac{f^{\prime}(x)-3 f^{\prime}(2 x)+2 f^{\prime}(4 x)}{x}$
<br/>$=\lim _{x \rightarrow 0} \frac{f^{\prime \prime}(x)-3 f''{(2 x)} \cdot 2+2 f^{\prime \prime}(4 x) \cdot 4}{1}$
<br/>$=\lim _{x \rightarrow 0} f^{\prime \prime}(x)-6 f''{(2 x)}+8 f''(4 x)$
<br/>$=f^{\prime \prime}(0)-6 f^{\prime \prime}(0)+8 f''{(0)}$
<br/>$=k-6 k+8 k$
<br/>$=3 k$</div>


title: if-f-5-7-and-f-5-7-then-x-5-x-5-x-f-5-5-f-x-is-given-by
slug: if-f-5-7-and-f-5-7-then-x-5-x-5-x-f-5-5-f-x-is-given-by-52895
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-28</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(5)=7$ and $\mathrm{f}^{\prime}(5)=7$ then $\underset{x \rightarrow 5}{ } \frac{x f(5)-5 f(x)}{x-5}$ is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">35</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-35</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">28</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Hints: $\operatorname{Lt}_{x \rightarrow 5} \frac{x f(5)-\operatorname{tf}(x)}{x-5}=\operatorname{Lt}_{x \rightarrow 5} \frac{f(5)-5 f^{\prime}(x)}{1}=f(5)-5 f^{\prime}(5)=7-5 \times 7=-28$</div>


title: if-f-is-a-realvalued-differentiable-function-such-that-f-x-f-x-0-for-all-real-x-then
slug: if-f-is-a-realvalued-differentiable-function-such-that-f-x-f-x-0-for-all-real-x-then-88055
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f$ is a real-valued differentiable function such that $f(x) f^{\prime}(x)     &lt; 0$ for all real $x,$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)$ must be an increasing function</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(x)$ must be a decreasing function</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$|f(x)|$ must be an increasing function</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$|f(x)|$ must be a decreasing 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|f(x)|$ must be a decreasing function</span> </div>

answer:
<div class="solution">Given, $f(x) f^{\prime}(x)     &lt; 0$ $\Rightarrow f(x)$ and $f^{\prime}(x)$ must be of opposite sign.
<br/>(i) Let $f(x)=e^{-x}$
<br/>$\therefore \quad f^{\prime}(x)=-e^{-x}$
<br/>$\Rightarrow f(x)&gt;0$ and $f^{\prime}(x) &lt; 0, \forall x \in R$
<br/>(ii) Let $f(x)=-e^{-x}$
<br/>$\therefore \quad f^{\prime}(x)=e^{-x}$
<br/>$\Rightarrow f(x) &lt; 0$ and $f^{\prime}(x)&gt;0, \forall x \in R$
<br/>But $|f(x)|=\left|\pm e^{-x}\right|=e^{-x}$ in both cases
<br/>$\therefore \frac{d}{d x}|f(x)|=-e^{-x} &lt; 0$ in both case, $\forall x \in R$
<br/>$\Rightarrow|f(x)|$ must be a decreasing function</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-f-r-r-be-defined-by-f-x-e-x-and-g-r-r-be-defined-by-g-x-x-2-the-mapping-gof-r-r-be-defined-by-gof-x-g-f-x-x-r-then

slug:
if-f-r-r-be-defined-by-f-x-e-x-and-g-r-r-be-defined-by-g-x-x-2-the-mapping-gof-r-r-be-defined-by-gof-x-g-f-x-x-r-then-99908

question:
<div class="question">If $f: R \rightarrow R$ be defined by $f(x)=e^{x}$ and $g: R \rightarrow R$ be defined by $g(x)=x^{2}$. The mapping gof : $R \rightarrow R$ be defined by (gof) $(x)$ $=g f(x) \mid \forall x \in R .$ Then.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">gof is injective but $g$ is not bijective</span> </div>

answer:
<div class="solution">We have, $f: R \rightarrow R$, defined by $f(x)=e^{x}$ and $g: R \rightarrow R$ defined by $g(x)=x^{2}$
<br/>Now, We have
<br/>$$
<br/>\begin{aligned}
<br/>(g v f)(x) &amp;=g(f(x)) \\
<br/>&amp;=g\left(e^{x}\right) \\
<br/>&amp;=\left(e^{x}\right)^{2} \\
<br/>&amp;=e^{2 x}, \forall x \in R
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow$ gof is injective and $g$ is neither injective nor surjective. $\Rightarrow g o f$ is injective but $g(x)$ is not bijective.</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
if-f-s-r-where-s-is-the-set-of-all-nonsingular-matrices-of-order-2-over-r-and-f-a-c-b-d-a-d-b-c-then

slug:
if-f-s-r-where-s-is-the-set-of-all-nonsingular-matrices-of-order-2-over-r-and-f-a-c-b-d-a-d-b-c-then-72726

question:
<div class="question">If $\mathrm{f}: \mathrm{S} \rightarrow \mathbb{R}$ where $S$ is the set of all non-singular matrices of order 2 over $\mathbb{R}$ and $f\left[\left(\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right)\right]=a d-b c$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{f}$ is bijective mapping</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}$ is one-one but not onto</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}$ is onto but not one-one</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}$ is neither one-one nor 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}$ is neither one-one nor onto</span> </div>

answer:
<div class="solution">Hint : 
<br/>$\mathrm{f}\left[\left(\begin{array}{ll}2 &amp; 0 \\ 0 &amp; 2\end{array}\right)\right]=4=\mathrm{f}\left[\left(\begin{array}{ll}4 &amp; 0 \\ 0 &amp; 1\end{array}\right)\right]$
<br/>$\Rightarrow$ not one-one
<br/>As $0 \in \mathbb{R}$ but S does not contain any singular matrix so, $f$ is not onto</div>


title: if-f-x-0-x-1-t-2-c-o-s-t-d-t-0-x-2-then
slug: if-f-x-0-x-1-t-2-c-o-s-t-d-t-0-x-2-then-98618
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $F(x)=\int_{0}^{x} \frac{\cos t}{\left(1+t^{2}\right)} d t, 0 \leq x \leq 2 \pi .$ Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$F$ is increasing in $\left(0, \frac{\pi}{2}\right)$ and $\left(\frac{3 \pi}{2}, 2 \pi\right)$ and decreasing in $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$</span> </div>

answer:
<div class="solution">Given function is
<br/>$$
<br/>F(x)=\int_{0}^{x} \frac{\cos t}{\left(1+t^{2}\right)} d t, 0 \leq x \leq 2 \pi
<br/>$$
<br/>On differentiation w.r.t. $x$. (apply Leibnitz rule) $F(x)=\frac{\cos x}{1+x^{2}} \times 1=\frac{\cos x}{1+x^{2}},$ where $\left(1+x^{2}\right)&gt;0$
<br/>Here, $\cos x&gt;0 \Rightarrow x \in\left(0, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$
<br/>and $\cos x     &lt; 0 \Rightarrow x \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
<br/>So, $F$ is increasing in $\left(0, \frac{\pi}{2}\right)$ and $\left(\frac{3 \pi}{2}, 2 \pi\right)$ and decreasing in $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$</div>


title: if-f-x-1-2-x-3-x-x-1-x-x-x-1-x-x-1-x-2-x-1-x-1-x-x-1-x-x-1-t-h-e-n-f100-is-equal-to
slug: if-f-x-1-2-x-3-x-x-1-x-x-x-1-x-x-1-x-2-x-1-x-1-x-x-1-x-x-1-t-h-e-n-f100-is-equal-to-18322
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">$$
<br/>\text { If } f(x)=\begin{array}{|ccc|}
<br/>1 &amp; x &amp; x+1\\
<br/>2 x &amp; x(x-1) &amp; (x+1)x\\
<br/>3 x(x-1) &amp; x(x-1)(x-2) &amp; (x+1)x(x-1)
<br/>\end{array}
<br/>$$ Then, $f(100)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">100</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">We have, $f(x)= \begin{array}{|ccc|}1 &amp; x &amp;x+1 \\ 2 x &amp; x(x-1) &amp; (x+1)x\\ 3 x(x-1) &amp; x(x-1)(x-2) &amp;(x+1)x(x-1)\end{array}$ 
<br/>Taking common $x,(x-1)$ from $R_{2}$ and $R_{3}$ respectively, we ge $f(x)=x(x-1)\left|\begin{array}{ccc}1 &amp; x &amp; x+1 \\ 2 &amp; x-1 &amp; x+1 \\ 3 x &amp; x(x-2) &amp; (x+1) x\end{array}\right|$
<br/>Taking commen $(x+1)$ from $C_{2}$, we get
<br/>$f(x)=x(x-1)(x+1)\left|\begin{array}{ccc}1 &amp; x &amp; 1 \\ 2 &amp; x-1 &amp; 1 \\ 3 x &amp; x(x-2) &amp; x\end{array}\right|$
<br/>Taking common $x$ from $R_{3}$, we get $f(x)=x^{2}(x-1)(x+1)\left|\begin{array}{ccc}1 &amp; x &amp; 1 \\ 2 &amp; x-1 &amp; 1 \\ 3 &amp; x-2 &amp; 1\end{array}\right|$
<br/>Applying $R_{1} \rightarrow R_{1}-R_{2}, R_{2}$ we get
<br/>$f(x)=x^2(x-1)(x+1)\left|\begin{array}{ccc}-1 &amp; 1 &amp; 0 \\ -1 &amp; 1 &amp; 0 \\ 3 &amp; (x-2) &amp; 1\end{array}\right|$
<br/>$\Rightarrow f(100)=0$</div>


title: if-f-x-1-x-t-dt-then-for-any-x-0-f-x-is-equal-to
slug: if-f-x-1-x-t-dt-then-for-any-x-0-f-x-is-equal-to-77173
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\int_{-1}^{x}|t|$ dt, then for any $x \geq 0, f(x)$ is equal to</div>

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

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

answer:
<div class="solution">We have, 
<br/>$\begin{aligned} f(x) &amp;=\int_{-1}^{x} t\left|d t=\int_{-1}^{0}\right| t\left|d t+\int_{0}^{x}\right| t \mid d t \\ &amp;=-\int_{-1}^{0} t d t+\int_{0}^{x} t d t \\ &amp;=\left[-\frac{t^{2}}{2}\right]_{-1}^{0}+\left[\frac{t^{2}}{2}\right]_{0}^{x} \\ &amp;=-\frac{1}{2}\left(0^{2}-(-1)^{2}\right)+\frac{1}{2}\left(x^{2}-0^{2}\right) \\ &amp;=-\frac{1}{2}(-1)+\frac{1}{2}\left(x^{2}\right)=\frac{1}{2}\left(1+x^{2}\right) \end{aligned}$</div>


title: if-f-x-2-100-x-1-g-x-3-100-x-1-then-the-set-of-real-numbers-x-such-that-f-g-x-x-is
slug: if-f-x-2-100-x-1-g-x-3-100-x-1-then-the-set-of-real-numbers-x-such-that-f-g-x-x-is-10881
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=2^{100} x+1, g(x)=3^{100} x+1$, then the set of real numbers $x$ such that $f[g(x)\}=x$ is</div>

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

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

answer:
<div class="solution">Given, $\quad f(x)=2^{100} \cdot x+1$
<br/>$g(x)=3^{300} \cdot x+1$
<br/>Now, $f\{g(x)\}=x$
<br/>$\Rightarrow \quad f\left(3^{100}. x+1\right)=x$
<br/>$\Rightarrow \quad 2^{100}\left\{3^{100} \cdot x+1\right\}+1=x$
<br/>$\Rightarrow \quad 6^{100} \cdot x+2^{100}+1=x$
<br/>$\Rightarrow \quad x\left(1-6^{100}\right)=\left(1+2^{100}\right)$
<br/>$\Rightarrow \quad x=\frac{1+2^{100}}{1-6^{100}}$
<br/>Hence, fog $(x)=x$ represent a singleton set.</div>


title: if-f-x-2-x-2-1-4-x-3-1-x-1-x-1-then-0-2-f-x-d-x-is
slug: if-f-x-2-x-2-1-4-x-3-1-x-1-x-1-then-0-2-f-x-d-x-is-62478
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\left\{\begin{array}{ll}2 x^{2}+1, &amp; x \leq 1 \\ 4 x^{3}-1, &amp; x&gt;1\end{array},\right.$ then $\int_{0}^{2} f(x) d x$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$47 / 3$</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">$50 / 3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$47 / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$47 / 3$</span> </div>

answer:
<div class="solution">Given, $f(x)=\left\{\begin{array}{ll}2 x^{2}+1, &amp; x \leq 1 \\ 4 x^{3}-1, &amp; x&gt;1\end{array}\right.$
<br/>$\begin{aligned} \therefore \int_{0}^{2} f(x) d x &amp;=\int_{0}^{1} f(x) d x+\int_{1}^{2} f(x) d x \\ &amp;=\int_{0}^{1}\left(2 x^{2}+1\right) d x+\int_{1}^{2}\left(4 x^{3}-1\right) d x \\ &amp;=\left[\frac{2 x^{3}}{3}+x\right]_{0}^{1}+\left[\frac{4 x^{4}}{4}-x\right]_{1}^{2} \\ &amp;=\frac{2}{3}(1)^{3}+1-(0+0) \\ &amp;+\left[(2)^{4}-2-\left\{(1)^{4}-1\right)\right.\end{aligned}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{2}{3}+1+[16-2-0] \\
<br/>=\frac{2}{3}+15=\frac{2+45}{3} \\
<br/>=\frac{47}{3} \mathrm{sq} \text { units }
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-f-x-and-g-x-are-twice-differentiable-functions-on-03-satisfying-f-x-g-x-f-1-4-g-1-6-f-2-3-g-2-9-then-f-1-g-1-is

slug:
if-f-x-and-g-x-are-twice-differentiable-functions-on-03-satisfying-f-x-g-x-f-1-4-g-1-6-f-2-3-g-2-9-then-f-1-g-1-is-92867

question:
<div class="question">If $f(x)$ and $g(x)$ are twice differentiable. functions on (0,3)
<br/>satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x), f^{\prime}(1)=4, g^{\prime}(1)=6, \quad f(2)=3$
<br/>$g(2)=9,$ then $f(1)-g(1)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">0</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, $f^{\prime \prime}(x)=g^{\prime \prime}(x)$
<br/>Integrating w.r.t. $x$, we get $\begin{array}{ll} &amp; f^{\prime}(x)=g^{\prime}(x)+c_{1} \\ \text { Put } &amp; x=1 \Rightarrow f^{\prime}(1)=g^{\prime}(1)+C_{1} \\ \Rightarrow &amp; 4=6+C_{1} \\ \therefore &amp; C_{1}=-2 \\ \therefore \quad &amp; f^{\prime}(x)=g^{\prime}(x)-2\end{array}$
<br/>Again, integrating w.r.t. $x$, we get $f(x)=g(x)-2 x+C_{2}$
<br/>$\Rightarrow \quad 3=9-4+C_{2} \Rightarrow C_{2}=-2$
<br/>$\therefore \quad f(x)=g(x)-2 x-2$
<br/>Put $x=1,$ we get $f(1)-g(1)=-2(1)-2=-4$</div>


title: if-f-x-e-x-x-2-2-then
slug: if-f-x-e-x-x-2-2-then-81424
question: <div class="question">If $f(x)=e^{x}(x-2)^{2},$ then</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f$ is increasing in $(-\infty, 0)$ and $(2, \infty)$ and decreasing $\operatorname{lin}(0,2)$</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">$t$ is increasing in $(-\infty, 0)$ and decreasing in $(0, \infty)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f$ is increasing in $(2, \infty)$ and decreasing in $(-\infty, 0)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f$ is increasing in (0,2) and decreasing in $(-\infty, 0)$ and $(2, \infty)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ is increasing in $(-\infty, 0)$ and $(2, \infty)$ and decreasing $\operatorname{lin}(0,2)$</span> </div>

answer:
<div class="solution">Given function is, $f(x)=e^{x}(x-2)^{2}$
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow f^{\prime}(x) &amp;=e^{x}(x-2)^{2}+2(x-2) e^{x} \\
<br/>&amp;=e^{x}(x-2)(x-2+2)=x(x-2) e^{x}
<br/>\end{aligned}
<br/>$$
<br/>Now, sign scheme of $f^{\prime}(x)$ is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a48.png"/> <br/> <br/>So, $t$ is increasing in $(-\infty, 0)$ and $(2, \infty)$ and decreasing in (0,2)</div>


title: if-f-x-f-a-x-then-0-a-x-f-x-d-x-is-equal-to
slug: if-f-x-f-a-x-then-0-a-x-f-x-d-x-is-equal-to-24637
question: <div class="question">If $f(x)=f(a-x)$ then $\int_0^a x f(x) d x$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\int_0^a f(x) d x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{a^2}{2} \int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{a}{2} \int_0^a f(x) d x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{a}{2} \int_0^{\mathrm{a}} f(x) d x$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a}{2} \int_0^a f(x) d x$</span> </div>

answer:
<div class="solution">Hints : $f(x)=f(a-x), I=\int_0^a x f(x) d x=\int_0^a(a-x) f(a-x) d x$<br/>$$<br/>\begin{aligned}<br/>&amp; =\int_0^a(a-x) f(x) d x=a \int_0^a f(x) d x-I \\<br/>&amp; \therefore 2 I=a \int_0^a f(x) d x \Rightarrow I=\frac{a}{2} \int_0^a f(x) d x<br/>\end{aligned}<br/>$$</div>


title: if-f-x-is-a-function-such-that-f-x-x-1-2-4-x-then
slug: if-f-x-is-a-function-such-that-f-x-x-1-2-4-x-then-49493
question: <div class="question">If $f(x)$ is a function such that $f^{\prime}(x)=(x-1)^{2}(4-x),$ then</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(0)=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f(x)$ is increasing in (0,3)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x=4$ is a critical point of $f(x)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(x)$ is decreasing in (3,5)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f(x)$ is increasing in (0,3), $x=4$ is a critical point of $f(x)$</span> </div>

answer:
<div class="solution">Given
<br/>$$
<br/>f^{\prime}(x)=(x-1)^{2}(4-x)
<br/>$$
<br/>The sign scheme of $f^{\prime}(x)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a75.png"/> <br/> <br/>Clearly. $f(x)$ is increasing, for $x \in(-\infty, 4)$ and decreasing, for $n \in(4, \infty)$ 
<br/>Since, $f^{\prime}(x)=0$ at $x=4$
<br/>$\mathrm{SO}, \mathrm{x}=4$ is a critical point.</div>


title: if-f-x-is-an-odd-differentiable-function-defined-on-such-that-f-3-2-then-f-3-is-equal-to
slug: if-f-x-is-an-odd-differentiable-function-defined-on-such-that-f-3-2-then-f-3-is-equal-to-65342
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)$ is an odd differentiable function defined on $(-\infty, \infty)$ such that $f^{\prime}(3)=2,$ then $f^{\prime}(-3)$ 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">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">4</span> </li> </ul>

answer:
<div class="solution">Given that $f(x)$ is an odd differentiable function.
<br/>Then, $f(-x)=-f(x)$
<br/>$\begin{array}{lr}\Rightarrow &amp; -f^{\prime}(-x)=-f^{\prime}(x) \\ \Rightarrow &amp; f^{\prime}(-x)=f^{\prime}(x)\end{array}$
<br/>Put $x=3$ in Eq. (i), we get
<br/>$\therefore$
<br/>$$
<br/>\begin{array}{l}
<br/>f^{\prime}(-3)=f^{\prime}(3) \\
<br/>f^{\prime}(-3)=2
<br/>\end{array}
<br/>$$</div>


title: if-f-x-lo-g-5-lo-g-3-x-then-f-e-is-equal-to
slug: if-f-x-lo-g-5-lo-g-3-x-then-f-e-is-equal-to-98246
question: <div class="question">If $f(x)=\log _{5} \log _{3} x,$ then $f^{\prime}(e)$ is equal to</div>
tags: ['Mathematics', 'Differentiation', '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">$e \log _{e} 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$e \log _{4} 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{e \log _{e} 5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{e \log _{e} 3}$</span> </li> </ul>

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

answer:
<div class="solution">We have,
<br/>$\therefore$
<br/>$\begin{aligned} f(x) &amp;=\log _{5} \log _{3} x \\ &amp;=\frac{\log \log _{3} x}{\log 5} \\ &amp;=\frac{\log \left(\frac{\log x}{\log 3}\right)}{\log 5} \\ &amp;=\frac{\log \log x-\log \log 3}{\log 5} \\ f^{\prime}(x)=\frac{1}{\log 5} \cdot \frac{1}{\log x} \cdot \frac{1}{x} \end{aligned}$
<br/>$\begin{aligned} f^{\prime}(e) &amp;=\frac{1}{e \log 5 \cdot \log e} \\ &amp;=\frac{1}{e \log 5} \quad[\because \log e=1] \\ &amp;=\frac{1}{e \log _{n} 5} \end{aligned}$</div>


title: if-f-x-sin-x-2-cos-2-x-4-x-4-3-then-f-attains-its
slug: if-f-x-sin-x-2-cos-2-x-4-x-4-3-then-f-attains-its-66862
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\sin x+2 \cos ^{2} x, \frac{\pi}{4} \leq x \leq \frac{3 \pi}{4} .$ Then, $f$ attains its</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">minimum at $x=\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">maximum at $x=\frac{\pi}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">minimum $x=\frac{\pi}{2}$</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">maximum at $x=\sin ^{-1}\left(\frac{1}{4}\right)$</span> </li> </ul>

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

answer:
<div class="solution">Given, $f(x)=\sin x+2 \cos ^{2} x, x \in\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$
<br/>$\therefore \quad f^{\prime}(x)=\cos x-4 \cos x \cdot \sin x$
<br/>and $f^{\prime}(x)=-\sin x-4 \cos 2 x$
<br/>For maximum or minimum of $f(x)$ 
<br/>Put $\quad f^{\prime}(x)=0$
<br/>$\Rightarrow \quad \cos x-4 \cos x \cdot \sin x=0$
<br/>$\Rightarrow \quad \cos x(1-4 \sin x)=0$
<br/>$\Rightarrow \quad \cos x=0=\cos \frac{\pi}{2}$ and $\sin x \neq \frac{1}{4}$
<br/>$\because x \in\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right] \Rightarrow x=\frac{\pi}{2}$
<br/>Now, $\quad f^{''}\left(\frac{\pi}{2}\right)=-\sin \frac{\pi}{2}-4 \cos \pi$
<br/>$$
<br/>=-1+4=3&gt;0(\mathrm{min})
<br/>$$
<br/>So, $f(x)$ is minimum at $x=\frac{\pi}{2}$
<br/>and its minimum value is $f\left(\frac{\pi}{2}\right)=\sin \frac{\pi}{2}+2 \cos ^{2} \frac{\pi}{2}=1-2 \times 0=1$</div>


tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-f-x-sin-x-cos-x-d-x-2-b-2-a-2-1-lo-g-f-x-c-where-c-is-the-constant-of-integration-then-f-x-is-equal-to

slug:
if-f-x-sin-x-cos-x-d-x-2-b-2-a-2-1-lo-g-f-x-c-where-c-is-the-constant-of-integration-then-f-x-is-equal-to-71494

question:
<div class="question">If $\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log f(x)+c$
<br/>where $c$ is the constant of integration, then $f(x)$ is equal to</div>

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

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

answer:
<div class="solution">We have. $\int f(x) \sin x \cos x d x=\frac{1}{2 b^{2}-a^{2}} \log (f(x))+c$
<br/>$\Rightarrow f(x) \sin x \cos x=\frac{1}{\left.2 b^{2}-a^{2}\right)} \cdot \frac{1}{f(x)} \cdot f'(x)$
<br/>$\Rightarrow \quad f(x) \sin 2 x=\frac{1}{b^{2}-a^{2}} \cdot \frac{f'(x)}{f(x)}$
<br/>$\Rightarrow \quad \sin 2 x=\frac{1}{b^{2}-a^{2}} \frac{f(x)}{(f(x))^{2}}$
<br/>$\Rightarrow \quad \int \sin 2 x d x=\frac{1}{b^{2}-a^{2}} \int \frac{f(x)}{(f(x))^{2}} d x$
<br/>$\Rightarrow \quad \frac{-\cos 2 x}{2}=\frac{1}{b^{2}-a^{2}} \cdot\left(\frac{-1}{f(x)}\right)$
<br/>$\Rightarrow \quad \quad \frac{\cos 2 x\left(b^{2}-a^{2}\right)}{2}=\frac{1}{f(x)}$
<br/>$\Rightarrow \quad f(x)=\frac{2}{\left(b^{2}-a^{2}\right) \cos 2 x}$</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-f-x-tan-1-l-o-g-e-x-2-l-o-g-x-2-e-tan-1-1-6-l-o-g-x-3-2-l-o-g-x-then-the-value-of-f-x-is-equal-to

slug:
if-f-x-tan-1-l-o-g-e-x-2-l-o-g-x-2-e-tan-1-1-6-l-o-g-x-3-2-l-o-g-x-then-the-value-of-f-x-is-equal-to-50603

question:
<div class="question">If $f(x)=\tan ^{-1}\left[\frac{\log \left(\frac{e}{x^{2}}\right)}{\log \left(e x^{2}\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log x}{1-6 \log x}\right]$ then the value of $f^{\prime \prime}(x)$ is equal to</div>

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

answer:
<div class="solution">Given, $f(x)=\tan ^{-1}\left[\frac{\log \left(\frac{e}{x^{2}}\right)}{\log \left(e x^{2}\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log x}{1-6 \log x}\right]$
<br/>$=\tan ^{-1}\left[\frac{\log e-2 \log x}{\log e+2 \log x}\right]+\tan ^{-1}\left[\frac{3+2 \log _{x} x}{1-6 \log x}\right]$
<br/>Let $2 \log x=\tan \theta,$ we get
<br/>$=\tan ^{-1}\left[\frac{1-\tan \theta}{1+\tan \theta}\right]+\tan ^{-1}\left[\frac{3+\tan \theta}{1-3 \tan \theta}\right]$
<br/>$\left.=\tan ^{-1}\left[\tan \left(\frac{\pi}{4}-\theta\right)\right]+[\tan ^{-1} (\tan ((\tan ^{-1} 3)+\theta))\right]$
<br/>$=\frac{\pi}{4}-\theta+\tan ^{-1} 3+\theta=\frac{\pi}{4}+\tan ^{-1} 3$
<br/>$\therefore \quad f(x)=$ constant
<br/>$\therefore f^{\prime}(x)=0$</div>


title: if-f-x-x-3-3-x-2-x-3-6-x-2-9-x-2-x-2-x-2-then
slug: if-f-x-x-3-3-x-2-x-3-6-x-2-9-x-2-x-2-x-2-then-86982
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\left\{\begin{array}{ll}x^{3}-3 x+2, &amp; x     &lt; 2 \\ x^{3}-6 x^{2}+9 x+2, &amp; x \geq 2\end{array}\right.$ then</div>

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

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

answer:
<div class="solution">Given. $f(x)=\left\{\begin{aligned} x^{3}-3 x+2, &amp; x     &lt; 2 \\ x^{3}-6 x^{2}+9 x+2, &amp; x \geq 2 \end{aligned}\right.$
<br/>LHL
<br/>$$
<br/>\begin{array}{l}
<br/>=f(2-0)=\lim _{h \rightarrow 0}(2-h)^{3}-3(2-h)+2 \\
<br/>=(2)^{3}-6+2=8-6+2=4 \\
<br/>\begin{aligned}
<br/>LH L=f(2+0) &amp;=\lim _{h \rightarrow 0}(2+h)^{3}-6(2+h)^{2} \\
<br/>&amp;=(2)^{3}-6(2)^{2}+9(2)+2 \\
<br/>&amp;=8-24+18+2=4
<br/>\end{aligned}
<br/>\end{array}
<br/>$$
<br/>$\because \mathrm{LHL}=\mathrm{RHL}$
<br/>$\therefore \lim _{x \rightarrow 2} f(x)$ exist
<br/>and $\begin{aligned} f(2)=(2)^{3} &amp;-6(2)^{2}+9(2)+2 \\ &amp;=8-24+18+2=4 \end{aligned}$
<br/>$\therefore \quad \mathrm{LHL}=\mathrm{RHL}=f(2)$
<br/>So, $f(x)$ is continuous at $x=2$
<br/>Now, $\quad f^{\prime}(x)=\left\{\begin{array}{cc}3 x^{2}-3, &amp; x &lt; 2 \\ 3 x^{2}-12 x+9, &amp; x \geq 2\end{array}\right.$
<br/>$\begin{array}{ll}\therefore \quad &amp; L f^{\prime}(2)=3(2)^{2}-3=12-3=9 \\ \text { and } \quad &amp; R f^{\prime}(2)=3(2)^{2}-12(2)+9 \\ &amp; =12-24+9=-3\end{array}$
<br/>$\because \quad L f^{\prime}(2) \neq R f^{\prime}(2)$
<br/>$\therefore f(x)$ is not differentiable at $x=2$
<br/>Hence. $f$ is continuous but not differentiable at $x=2$</div>


tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
if-f-x-x-23-x-0-then-the-area-of-the-region-enclosed-by-the-curve-y-f-x-and-the-three-lines-y-x-x-1-and-x-8-is

slug:
if-f-x-x-23-x-0-then-the-area-of-the-region-enclosed-by-the-curve-y-f-x-and-the-three-lines-y-x-x-1-and-x-8-is-74656

question:
<div class="question">If $f(x)=x^{2 / 3}, x \geq 0 .$ Then, the area of the region enclosed by the curve $y=f(x)$ and the three lines $y=x, x=1$ and $x=8$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{63}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{93}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{105}{7}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{129}{10}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{129}{10}$</span> </div>

answer:
<div class="solution">Given, $f(x)=x^{2 / 3}, x \geq 0$ and line $y=x$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a72.png"/> <br/> <br/>$\therefore$ Required area $A=\int_{x=1}^{8}\left(x-x^{2 / 3}\right) d x$
<br/>$=\left[\frac{x^{2}}{2}-\frac{3}{5} x^{5 / 3}\right]_{1}^{9}=\left(32-\frac{3}{5} \times 32\right)-\left(\frac{1}{2}-\frac{3}{5}\right)$
<br/>$=32 \times \frac{2}{5}-\frac{(5-6)}{10}=\frac{64}{5}+\frac{1}{10}$
<br/>$=\frac{128+1}{10}=\frac{129}{10}$</div>


title: if-f-x-x-n-n-being-a-nonnegative-integer-then-the-values-of-n-for-which-f-f-f-for-all-0-is
slug: if-f-x-x-n-n-being-a-nonnegative-integer-then-the-values-of-n-for-which-f-f-f-for-all-0-is-29349
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answers are: <span class="option-value">2, 0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=x^{n}, n$ being a non-negative integer, then the values of $n$ for which
<br/>$f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)$ for all $\alpha, \beta&gt;0$ 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">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">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

answer:
<div class="solution">We have,
<br/>$\begin{aligned} f(x) &amp;=x^{n} \\ f^{\prime}(x) &amp;=n x^{n-1} \end{aligned}$
<br/>Now, $f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)$
<br/>$\Rightarrow \quad n(\alpha+\beta)^{n-1}=n \alpha^{n-1}+n \beta^{n-1}$
<br/>$\Rightarrow(a+\beta)^{n-1}=a^{n-1}+\beta^{n-1}$
<br/>From options we see that $n=2$, satisfy the above equation.
<br/>$\therefore$
<br/>$$
<br/>n=2
<br/>$$</div>


title: if-f-x-x-x-1-1-x-1-x-1-1-x-1-then
slug: if-f-x-x-x-1-1-x-1-x-1-1-x-1-then-75140
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $f(x)=x\left(\frac{1}{x-1}+\frac{1}{x}+\frac{1}{x+1}\right), x&gt;1$. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)&gt;3$</span> </div>

answer:
<div class="solution">Given function is
<br/>$$
<br/>\begin{aligned}
<br/>f(x) &amp;=x\left\{\frac{1}{x-1}+\frac{1}{x}+\frac{1}{x+1}\right\}, x&gt;1 \\
<br/>&amp;=x\left\{\frac{2 x}{x^{2}-1}+\frac{1}{x}\right\}=\frac{2 x^{2}}{x^{2}-1}+1 \\
<br/>&amp;=\left\{\frac{2}{\left(1-\frac{1}{x^{2}}\right)}+1\right\}&gt;3
<br/>\end{aligned}
<br/>$$
<br/>$\therefore \quad f(x)&gt;3$</div>


title: if-four-distinct-points-2-k-3-k-2-0-0-3-00-lie-on-a-circle-then
slug: if-four-distinct-points-2-k-3-k-2-0-0-3-00-lie-on-a-circle-then-12200
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$k=1$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If four distinct points $(2 k, 3 k),(2,0),(0,3)$ (0,0) lie on a circle, then</div>

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

answer:
<div class="solution">Since, join of (2,0) and (0,3) subtends $90^{\circ}$ at (0,0)
<br/>$\Rightarrow$ It is a diameter.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a18.png"/> <br/> <br/>$\therefore$ Equation is $(x-2)(x-0)+(y-0)(y-3)=0$
<br/>$x^{2}+y^{2}-2 x-3 y=0$
<br/>$(2 k, 3 k)$ lies on it $\Rightarrow \quad 4 k^{2}+9 k^{2}-4 k-9 k=0$
<br/>$\Rightarrow$
<br/>. $13 k^{2}=13 k$
<br/>$\Rightarrow$
<br/>$k=1$
<br/>since, $k \neq 0$ otherwise $(2 k, 3 k)$ will be (0,0)</div>


title: if-frac16-sin-theta-cos-theta-tan-theta-are-in-gp-then-the-solution-set-of-theta-is
slug: if-frac16-sin-theta-cos-theta-tan-theta-are-in-gp-then-the-solution-set-of-theta-is-96624
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(\frac{1}{6} \sin \theta, \cos \theta, \tan \theta\) are in G.P., then the solution set of \(\theta\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \mathrm{n} \pi \pm \frac{\pi}{3}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint : } 6 \cos ^3 \theta+\cos ^2 \theta-1=0 \\<br/>&amp; \Rightarrow \quad(2 \cos \theta-1)\left(3 \cos ^2 \theta+2 \cos \theta+1\right)=0 \\<br/>&amp; \Rightarrow \quad \cos \theta=\frac{1}{2}\left(3 \cos ^2 \theta+2 \cos \theta+1 \neq 0\right)<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Main']
source_db: db\MarksBatch2_P1.db
title:
if-from-a-point-p-a-b-c-perpendiculars-pa-and-pb-are-drawn-to-yz-and-zx-planes-respectively-then-the-equation-of-the-plane-oab-is-1

slug:
if-from-a-point-p-a-b-c-perpendiculars-pa-and-pb-are-drawn-to-yz-and-zx-planes-respectively-then-the-equation-of-the-plane-oab-is-1-76986

question:
<div class="question">If from a point $\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})$, perpendiculars $\mathrm{PA}$ and $\mathrm{PB}$ are drawn to $\mathrm{YZ}$ and $\mathrm{ZX}$ planes respectively, then the equation of the plane $\mathrm{OAB}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">bcx $+$ cay $-a b z=0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/03qtq8qh-D3LItAAz8SVGsetZIl7mhD7y7jVDzG4Lx0.original.fullsize.png"/><br/> <br/>$\mathrm{A}(\mathrm{o}, \mathrm{b}, \mathrm{c}) \mathrm{B}(\mathrm{a}, \mathrm{o}, \mathrm{c}), \mathrm{O}(\mathrm{o}, \mathrm{o}, \mathrm{o})$ Perpendicular vector is
<br/>$\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}$
<br/>$\left|\begin{array}{ccc}\hat{\mathbf{i}} &amp; \hat{j} &amp; \hat{k} \\ 0 &amp; b &amp; c \\ a &amp; 0 &amp; c\end{array}\right|=b c \hat{i}+a c \hat{j}-a b \hat{k}$
<br/>$\therefore$ Equation of plane is $b c x+a c y-a b z=0$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
if-from-a-point-p-a-b-c-perpendiculars-pa-and-pb-are-drawn-to-yz-and-zx-planes-respectively-then-the-equation-of-the-plane-oab-is

slug:
if-from-a-point-p-a-b-c-perpendiculars-pa-and-pb-are-drawn-to-yz-and-zx-planes-respectively-then-the-equation-of-the-plane-oab-is-18035

question:
<div class="question">If from a point $\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})$, perpendiculars $\mathrm{PA}$ and $\mathrm{PB}$ are drawn to $\mathrm{YZ}$ and $\mathrm{ZX}$ planes respectively, then the equation of the plane $\mathrm{OAB}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">bcx $+$ cay $-a b z=0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/03qtq8qh-D3LItAAz8SVGsetZIl7mhD7y7jVDzG4Lx0.original.fullsize.png"/><br/> <br/>$\mathrm{A}(\mathrm{o}, \mathrm{b}, \mathrm{c}) \mathrm{B}(\mathrm{a}, \mathrm{o}, \mathrm{c}), \mathrm{O}(\mathrm{o}, \mathrm{o}, \mathrm{o})$ Perpendicular vector is
<br/>$\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}$
<br/>$\left|\begin{array}{ccc}\hat{\mathbf{i}} &amp; \hat{j} &amp; \hat{k} \\ 0 &amp; b &amp; c \\ a &amp; 0 &amp; c\end{array}\right|=b c \hat{i}+a c \hat{j}-a b \hat{k}$
<br/>$\therefore$ Equation of plane is $b c x+a c y-a b z=0$</div>


title: if-fx2-y-x2-yx-y-then-fx-y-is-equal-to
slug: if-fx2-y-x2-yx-y-then-fx-y-is-equal-to-25584
question: <div class="question">If \(f(x+2 y, x-2 y)=x y\), then \(f(x, y)\) is equal to</div>
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{4} x y\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{4}\left(x^2-y^2\right)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{1}{8}\left(x^2-y^2\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{2}\left(x^2+y^2\right)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{8}\left(x^2-y^2\right)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}\text{Hint: }<br/>&amp; x+2 y=a \\<br/>&amp; \frac{x-2 y=b}{2 x=a+b} \\<br/>&amp; 4 y=a-b<br/>\end{aligned}\)<br/>\(f(a, b)=\left(\frac{a+b}{2}\right)\left(\frac{a-b}{4}\right)=\frac{a^2-b^2}{8}\)</div>


title: if-fx3-sqrt3x2x2-then
slug: if-fx3-sqrt3x2x2-then-32411
question: <div class="question">If \(f(x)=3 \sqrt[3]{x^2}-x^2\), then</div>
tags: ['Mathematics', 'Application of Derivatives', '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 has no extrema</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(f\) is maximum at two points \(x=1\) and \(x=-1\)</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="correct"> <span class="option-label">C</span> <span class="option-data">\(f\) is minimum at \(x=0\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">f has maximum at \(x=1\) only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">\(f\) is maximum at two points \(x=1\) and \(x=-1\), \(f\) is minimum at \(x=0\)</span> </div>

answer:
<div class="solution">Hint : \(f(x)=3(x)^{\frac{2}{3}}-x^2\)<br/>Differentiate both sides wrt. \(x\)<br/>\(=f^{\prime}(x)=3 \cdot \frac{2}{3} x^{-1 / 3}-2 x=2 x^{-1 / 3}-2 x=2\left(x^{-1 / 3}-x\right)=2\left(\frac{1-x^{4 / 3}}{x^{1 / 3}}\right)\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/oAnN5fWzxsCSngTk_4Gsfkzbhg3PIgMCaeGxhDMJnBU.original.fullsize.png"/><br/><br/>\(\therefore f\) has maximum at two points \(x=-1 x=1\)<br/>\(f\) has minimum at \(x=0\)</div>


title: if-i-0-2-e-x-4-x-a-d-x-0-then-lies-in-the-interval
slug: if-i-0-2-e-x-4-x-a-d-x-0-then-lies-in-the-interval-59742
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(0,2)</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $I=\int_{0}^{2} e^{x^{4}}(x-a) d x=0$, then $\alpha$ lies in the interval</div>

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

answer:
<div class="solution">Given, $I=\int_{0}^{2} e^{x^{4}}(x-\alpha) d x=0$
<br/>$\Rightarrow \quad \int_{0}^{2} e^{x^{4}} x d x=\int_{0}^{2} e^{x^{4}} \alpha d x$
<br/>Here, we see that $\int_{0}^{2} e^{x^{4}} x d x$ gives us an area between two curves $e^{x^{4}}$ and $x$ from $x=0$ to $x=2$
<br/>Similarly. $\int_{0}^{2} e^{x^{4}} \alpha d x$ gives us a same area, so $a$ should lies between 0 to $2, i . e ., \alpha \in(0,2)$</div>


title: if-i-0-i-1-x-2-dx-then
slug: if-i-0-i-1-x-2-dx-then-69402
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{I}=\int_0^{\mathrm{I}} \frac{\mathrm{dx}}{1+\mathrm{x}^{\pi / 2}}$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\log _{\mathrm{e}} 2 &lt; 1 &lt; \pi / 4$</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">$\log _{\mathrm{e}} 2&gt;1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{I}=\pi / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{I}=\log _{\mathrm{e}} 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{\mathrm{e}} 2 &lt; 1 &lt; \pi / 4$</span> </div>

answer:
<div class="solution">Hints: $\mathrm{x}^2 &lt; \mathrm{x}^{\frac{\pi}{2}} &lt; \mathrm{x}, \quad 1+\mathrm{x}^2 &lt; 1+\mathrm{x}^{\frac{\pi}{2}} &lt; 1+\mathrm{x}$<br/>$\frac{1}{1+x^2}&gt;\frac{1}{1+x^{\frac{\pi}{2}}}&gt;\frac{1}{1+x}$<br/>$\frac{\pi}{4}&gt;\mathrm{I}&gt;(\log (1+\mathrm{x})), \quad \frac{\pi}{4}&gt;\mathrm{I}&gt;\log 2$</div>


title: if-i-1-0-0-0-1-0-0-0-1-and-p-1-0-0-0-1-0-0-0-2-then-the-matrix-p-3-2-p-2-is-equal-to
slug: if-i-1-0-0-0-1-0-0-0-1-and-p-1-0-0-0-1-0-0-0-2-then-the-matrix-p-3-2-p-2-is-equal-to-58481
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2I+P$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $I=\left(\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right)$ and $P=\left(\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; -1 &amp; 0 \\ 0 &amp; 0 &amp; -2\end{array}\right)$ Then, the matrix $P^{3}+2 P^{2}$ is equal to</div>

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

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/>and $P=\left(\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; -1 &amp; 0 \\ 0 &amp; 0 &amp; -2\end{array}\right)$
<br/>The characteristic equation of $P$ is
<br/>$$
<br/>|P-\lambda|=0
<br/>$$
<br/>$\Rightarrow\left|\begin{array}{ccc}1-\lambda &amp; 0 &amp; 0 \\ 0 &amp; -1-\lambda &amp; 0 \\ 0 &amp; 0 &amp; -2-\lambda\end{array}\right|=0$
<br/>$\Rightarrow \quad(1-\lambda)\{(1+\lambda)(2+\lambda)\}=0$
<br/>$\Rightarrow \quad\left(1-\lambda^{2}\right)(2+\lambda)=0$
<br/>$\Rightarrow \quad 2-2 \lambda^{2}+\lambda-\lambda^{3}=0$
<br/>$\Rightarrow \quad \lambda^{3}+2 \lambda^{2}-\lambda-2=0$
<br/>We know that, Caylay Hamilton theorem states that Every square matrix satisfy its characteristic equation'.
<br/>$\therefore$
<br/>$$
<br/>P^{3}+2 P^{2}-P-2 I=0
<br/>$$
<br/>$\Rightarrow$\(p^{3}+2 p^{2}=2 I+p\)</div>


title: if-i-1-0-3-f-cos-2-x-dx-and-i-2-0-f-cos-2-x-dx-then
slug: if-i-1-0-3-f-cos-2-x-dx-and-i-2-0-f-cos-2-x-dx-then-21954
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{I}_1=\int_0^{3 \pi} \mathrm{f}\left(\cos ^2 \mathrm{x}\right) \mathrm{dx}$ and $\mathrm{I}_2=\int_0^\pi \mathrm{f}\left(\cos ^2 \mathrm{x}\right) \mathrm{dx}$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{I}_1=\mathrm{I}_2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \mathrm{I}_1=\mathrm{I}_2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{I}_1=3 \mathrm{I}_2$</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">$\mathrm{I}_1=5 \mathrm{I}_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{I}_1=3 \mathrm{I}_2$</span> </div>

answer:
<div class="solution">Hints : $I_1=3 \int_0^\pi \mathrm{f}\left(\cos ^2 x\right) d x=3 I \quad$ [period is $\pi$ ]</div>


title: if-i-1-0-4-sin-2-x-d-x-and-i-2-0-4-cos-2-x-d-x-then
slug: if-i-1-0-4-sin-2-x-d-x-and-i-2-0-4-cos-2-x-d-x-then-53033
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{I}_1=\int_0^{\pi / 4} \sin ^2 \mathrm{x} d \mathrm{x}$ and $\mathrm{I}_2=\int_0^{\pi / 4} \cos ^2 \mathrm{x} d \mathrm{x}$, then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{I}_1=\mathrm{I}_2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">I $_1 &lt; \mathrm{I}_2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{I}_1&gt;\mathrm{I}_2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{I}_2=\mathrm{I}_1+\pi / 4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">I $_1 &lt; \mathrm{I}_2$</span> </div>

answer:
<div class="solution">Hints : $I_1=\int_0^{\pi / 4} \sin ^2 x d x ; I_2=\int_0^{\pi / 4} \cos ^2 x d x$<br/>$$<br/>\begin{aligned}<br/>&amp; \text { In }\left(0, \frac{\pi}{4}\right), \cos ^2 x&gt;\sin ^2 x \therefore \int_0^{\pi / 4} \cos ^2 x d x&gt;\int_0^{\pi / 4} \sin ^2 x d x \\<br/>&amp; I_2&gt;I_1 \text { i.e. } I_1 &lt; I_2<br/>\end{aligned}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/eCvvd4uuK3MvIMt4OGmCJUm2H8vZKmIRTBhLAeCBemI.original.fullsize.png"/><br/></div>


title: if-i-1-and-n-is-a-positive-integer-then-i-n-i-n-1-i-n-2-i-n-3-is-euqal-to
slug: if-i-1-and-n-is-a-positive-integer-then-i-n-i-n-1-i-n-2-i-n-3-is-euqal-to-38669
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $i=\sqrt{-1}$ and $n$ is a positive integer, then $i^n+i^{n+1}+i^{n+2}+i^{n+3}$ is euqal to</div>

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

answer:
<div class="solution">$$<br/>\text { Hints : } \dot{i}^n\left(1+\dot{i}+\dot{i}^2+\dot{i}^3\right)=\dot{i}^n(1+\dot{i}-1-\hat{i})=0<br/>$$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-i-is-the-greatest-of-i-1-0-1-e-x-cos-2-x-d-x-i-2-0-1-e-x-2-cos-2-x-d-x-i-3-0-1-e-x-2-d-x-i-4-0-1-e-x-2-2-d-x-then

slug:
if-i-is-the-greatest-of-i-1-0-1-e-x-cos-2-x-d-x-i-2-0-1-e-x-2-cos-2-x-d-x-i-3-0-1-e-x-2-d-x-i-4-0-1-e-x-2-2-d-x-then-59079

question:
<div class="question">If $\mathrm{I}$ is the greatest of $I_1=\int_0^1 e^{-x} \cos ^2 x d x, I_2=\int_0^1 e^{-x^2} \cos ^2 x d x, I_3=\int_0^1 e^{-x^2} d x, I_4=\int_0^1 e^{-x^2 / 2} d x$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{I}=\mathrm{I}_1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{I}=\mathrm{I}_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$I=I_3$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{I}=\mathrm{I}_4$</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">$\mathrm{I}=\mathrm{I}_4$</span> </div>

answer:
<div class="solution">$\mathrm{e}^{-\mathrm{x}} \cdot \cos ^2 \mathrm{x}  &lt; \mathrm{e}^{-\mathrm{x}^2} \cos ^2 \mathrm{x} &lt; \mathrm{e}^{-\mathrm{x}^2} &lt; \mathrm{e}^{-\frac{\mathrm{x}^2}{2}}$ for $0 &lt; \mathrm{x} &lt; 1$<br/>$\therefore$ By domination Law, $\mathrm{I}_4$ is maximum<br/>$\therefore \mathrm{I}=\mathrm{I}_4$</div>


title: if-i-lim-x-0-sin-x-2-e-x-x-1-2-x-2-then-limit
slug: if-i-lim-x-0-sin-x-2-e-x-x-1-2-x-2-then-limit-81230
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{I}=\lim _{\mathrm{x} \rightarrow 0} \sin \left(\frac{\mathrm{e}^{\mathrm{x}}-\mathrm{x}-1-\frac{\mathrm{x}^{2}}{2}}{\mathrm{x}^{2}}\right)$, then limit</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">exists and equals 1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">exists and equals 0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">exists and equals $\frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">exists and equals 0</span> </div>

answer:
<div class="solution">$I=\lim _{x \rightarrow 0} \sin \left(\frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}\right)=\lim _{x \rightarrow 0} \frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}=\lim _{x \rightarrow 0} \frac{e^{x}-1-0-x}{2 x}=\lim _{x \rightarrow 0} \frac{e^{x}-1}{2}=0$</div>


title: if-iint-fracx2-d-xx-sin-xcos-x2fxtan-xc-then-fx-is
slug: if-iint-fracx2-d-xx-sin-xcos-x2fxtan-xc-then-fx-is-61027
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(I=\int \frac{x^2 d x}{(x \sin x+\cos x)^2}=f(x)+\tan x+c\), then \(f(x)\) is</div>

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

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

answer:
<div class="solution">Hint : \(I=\int \frac{x^2}{(x \sin x+\cos x)^2} d x=\int \frac{x}{(x \sin x+\cos x)^2} \times \frac{x}{\cos x} d x\)<br/>\(I=-\frac{1}{(x \sin x+\cos x)} \cdot \frac{x}{\cos x}+\int \frac{1}{(x \sin x+\cos x)} \cdot \frac{(\cos x+x \sin x)}{\cos ^2 x} d x\)<br/>\(\begin{aligned} &amp; I=-\frac{1}{(x \sin x+\cos x)} \cdot \frac{x}{\cos x}+\int \sec ^2 x d x \\ &amp; I=-\frac{x}{(x \sin x+\cos x) \cos x}+\tan x+C \\ &amp; \therefore f(x)=\frac{-x}{\cos x(x \sin x+C)}\end{aligned}\)</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-in-a-a-bc-a-d-be-and-cf-are-the-altitudes-and-r-is-the-circumradius-then-the-radius-of-the-circumcircle-of-d-ef-is

slug:
if-in-a-a-bc-a-d-be-and-cf-are-the-altitudes-and-r-is-the-circumradius-then-the-radius-of-the-circumcircle-of-d-ef-is-95755

question:
<div class="question">If in a $\triangle A B C, A D, B E$ and $C F$ are the altitudes and $R$ is the circumradius, then the radius of the circumcircle of $\Delta D E F$ is</div>

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

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

answer:
<div class="solution">Let circumradius of $\Delta$ DEF be $R'$ We know. $\angle F D E=180^{\circ}-2 A$ and $F E=R \sin 2 A$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a30.png"/> <br/> <br/>Now, by sine rule in $\triangle D E F$ $2 R^{\prime}=\frac{E F}{\sin \angle F D E}=\frac{A \sin 2 A}{\sin \left(180^{\circ}-2 A\right)}$
<br/>$\therefore$
<br/>$R^{\prime}=\frac{R}{2}$</div>


title: if-in-a-triangle-a-b-c-sin-a-sin-b-sin-c-are-in-ap-then
slug: if-in-a-triangle-a-b-c-sin-a-sin-b-sin-c-are-in-ap-then-30542
question: <div class="question">If in a triangle \(A B C, \sin A, \sin B, \sin C\) are in A.P., then</div>
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">the altitudes are in A.P.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">the altitudes are in H.P.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the angles are in A.P.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the angles are in H.P.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the altitudes are in H.P.</span> </div>

answer:
<div class="solution">Hints: \(\frac{1}{2} \mathrm{ap}_1=\frac{1}{2} \mathrm{bp}_2=\frac{1}{2} \mathrm{cp}_3=\Delta\)<br/>\(\mathrm{a}=\frac{2 \Delta}{\mathrm{p}_1}\left|\mathrm{~b}=\frac{2 \Delta}{\mathrm{p}_2}\right| \mathrm{c}=\frac{2 \Delta}{\mathrm{p} 3}\)<br/>H.P.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-in-case-of-a-radio-isotope-the-value-of-halflife-leftt12right-and-decay-constant-lambda-are-identical-in-magnitude-then-their-value-should-be

slug:
if-in-case-of-a-radio-isotope-the-value-of-halflife-leftt12right-and-decay-constant-lambda-are-identical-in-magnitude-then-their-value-should-be-65893

question:
<div class="question">If in case of a radio isotope the value of half-life \(\left(T_{1/2}\right)\) and decay constant \((\lambda)\) are identical in magnitude, then their value should be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(0.693 / 2\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\((0.693)^{1 / 2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((0.693)^2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.693</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((0.693)^{1 / 2}\)</span> </div>

answer:
<div class="solution">Hint: For a radio decay \(T_{1 / 2}=\frac{0.693}{\lambda}\)<br/>If \(T_{1 / 2}=\lambda=x\) then \(\mathrm{x}=\frac{0.693}{\mathrm{x}}\)<br/>\(\Rightarrow \mathrm{x}^2=0.693, \quad \Rightarrow \mathrm{x}=\mathrm{T}_{1 / 2}=\lambda=(0.693)^{1 / 2}\)</div>


title: if-in-the-expansion-of-a-2-b-n-the-sum-of-the-5-th-and-6-th-term-is-zero-then-the-value-of-b-a-is
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
slug:
if-in-the-expansion-of-a-2-b-n-the-sum-of-the-5-th-and-6-th-term-is-zero-then-the-value-of-b-a-is-53583

question:
<div class="question">If in the expansion of $(a-2 b)^{\mathrm{n}}$, the sum of the 5 th and 6 th term is zero, then the value of $\frac{\mathrm{a}}{\mathrm{b}}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2(n-4)}{5}$</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints: } \begin{gathered}<br/>(a-2 b)^{\mathrm{n}}=\sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}(a)^{\mathrm{n}-\mathrm{r}}(-2 \mathrm{~b})^{\mathrm{r}} \\<br/>\mathrm{t}_5+\mathrm{t}_6=0<br/>\end{gathered}<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow{ }^{\mathrm{n}} \mathrm{C}_4(\mathrm{a})^{\mathrm{n}-4}(-2 \mathrm{~b})^4+{ }^{\mathrm{n}} \mathrm{C}_5(\mathrm{a})^{\mathrm{n}-5}(-2 \mathrm{~b})^5=0 \Rightarrow \frac{\mathrm{n} !}{4 !(\mathrm{n}-4) !} \mathrm{a}^{\mathrm{n}-4}(-2 \mathrm{~b})^4=-\frac{\mathrm{n} !}{5 !(\mathrm{n}-5) !}(\mathrm{a})^{\mathrm{n}-5}(-2 \mathrm{~b})^5 \\<br/>&amp; \Rightarrow \frac{1}{(\mathrm{n}-4)} \times \mathrm{a}=\frac{-1}{5}(-2 \mathrm{~b}) \Rightarrow \frac{\mathrm{a}}{\mathrm{b}}=\frac{2(\mathrm{n}-4)}{5}<br/>\end{aligned}<br/>$$</div>


title: if-inint0fracpi2-cos-n-x-cos-n-x-d-x-then-i1-i2-i3-ldots-are-in
slug: if-inint0fracpi2-cos-n-x-cos-n-x-d-x-then-i1-i2-i3-ldots-are-in-47820
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2023']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">G.P.</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(I_n=\int_0^{\frac{\pi}{2}} \cos ^n x \cos n x d x\), then \(I_1, I_2, I_3 \ldots\) are in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">A.P.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">G.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">H.P</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">no such relation</span> </li> </ul>

answer:
<div class="solution">Hint : \(I_n=\int_0^{\pi / 2} \cos ^n x \cdot \cos n x d x\)<br/>\(I_1=\int_0^{\pi / 2} \cos x \cdot \cos x d x=\int_0^{\pi / 2}\left(\frac{1+\cos 2 x}{2}\right) d x=\frac{\pi}{4}\)<br/>\(I_2=\int_0^{\pi / 2} \cos ^2 x \cdot \cos 2 x d x=\int_0^{\pi / 2}-\sin ^2 x \cdot \cos 2 x d x\)<br/>\(2 I_2=\int_0^{\pi / 2} \cos ^2 2 x=\int_0^{\pi / 2} \frac{1+\cos 4 x}{2} d x=\frac{\pi}{4}\)<br/>\(\therefore \mathrm{I}_2=\frac{\pi}{8}\)<br/>\(I_3=\int_0^{\pi / 2} \cos ^3 x \cdot \cos 3 x d x=\int_0^{\pi / 2}\left(\frac{3 \cos x+\cos 3 x}{4}\right) \cos 3 x d x\)<br/>\(\begin{aligned}<br/>&amp; =\frac{3}{4} \int_0^{\pi / 2} \cos x \cdot \cos 3 x d x+\frac{1}{4} \int_0^{\pi / 2} \cos ^2 3 x d x \\<br/>&amp; =\frac{3}{8} \int_0^{\pi / 2}(\cos 2 x+\cos 4 x) d x+\frac{1}{4} \int_0^{\pi / 2} \frac{1+\cos 6 x}{2} d x \\<br/>&amp; =\frac{\pi}{16}<br/>\end{aligned}\)<br/>\(\therefore \mathrm{I}_1, \mathrm{I}_2, \mathrm{I}_3\) are in G.P.</div>


title: if-int-fracd-xx1x2x3frac1k-log-eleftfracx33x1x24rightc-then-the-value-of-k-is
slug: if-int-fracd-xx1x2x3frac1k-log-eleftfracx33x1x24rightc-then-the-value-of-k-is-36784
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2023']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(\int \frac{d x}{(x+1)(x-2)(x-3)}=\frac{1}{k} \log _e\left\{\frac{|x-3|^3|x+1|}{(x-2)^4}\right\}+c\), then the value of \(k\) is</div>

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

answer:
<div class="solution">Hint: \(\int \frac{d x}{(x+1)(x-2)(x-3)}=\int\left(\frac{A}{x+1}+\frac{B}{(x-2)}+\frac{C}{x-3}\right) d x\)<br/>\(\begin{aligned}<br/>&amp; \frac{1}{(x+1)(x-2)(x-3)}=\frac{A(x-2)(x-3)+B(x+1)(x-3)+C(x+1)(x-2)}{(x+1)(x-2)(x-3)} \\<br/>&amp; \Rightarrow 1=A(x-2)(x-3)+B(x+1)(x-3)+C(x+1)(x-2) \\<br/>&amp; \text { for } x=-1 \\<br/>&amp; 1=A(-3)(-4) \Rightarrow A=\frac{1}{12} \\<br/>&amp; \text { for } x=2 \\<br/>&amp; 1=B(3)(-1) \Rightarrow B=-\frac{1}{3}<br/>\end{aligned}\)<br/>\(\text { for } x=3\)<br/>\(\begin{aligned} &amp; 1=C(4)(1) \Rightarrow C=\frac{1}{4} \\ &amp; \therefore I=\frac{1}{12} \int \frac{1}{x+1} d x-\frac{1}{3} \int \frac{1}{x-2} d x+\frac{1}{4} \int \frac{1}{x-3} d x \\ &amp; =\frac{1}{12} \ln |x+1|-\frac{1}{3} \ln |x-2|+\frac{1}{4} \ln |x-3|+C \\ &amp; =\frac{1}{12}\left[\ln |x+1|-\ln |x-2|^4+\ln |x-3|^3\right]+C \\ &amp; =\frac{1}{12} \ln \frac{(|x+1|)(|x-3|)^3}{(x-2)^4}-C \\ &amp; \therefore k=12\end{aligned}\)</div>


title: if-is-a-unit-vector-i-j-k-i-k-then-the-maximum-value-of-is
slug: if-is-a-unit-vector-i-j-k-i-k-then-the-maximum-value-of-is-65116
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\vec{\alpha}$ is a unit vector, $\vec{\beta}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{\gamma}=\hat{\mathrm{i}}+\hat{\mathrm{k}}$, then the maximum value of $[\vec{\alpha} \vec{\beta} \vec{\gamma}]$ is</div>

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

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

answer:
<div class="solution">$[\vec{\alpha} \vec{\beta} \vec{\gamma}]=\vec{\alpha} \cdot(\vec{\beta} \times \vec{\gamma})=\vec{\alpha} \cdot\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; k \\ 1 &amp; 1 &amp; -1 \\ 1 &amp; 0 &amp; 1\end{array}\right|$<br/>$=\vec{\alpha} \cdot(\hat{i}+2 \hat{j}-\hat{k})$ is maximum $\Rightarrow$ angle between $\vec{\alpha} \&amp; \hat{i}+2 \hat{j}-\hat{k}$ will be 0<br/>$\therefore[\vec{\alpha} \vec{\beta} \vec{\gamma}]=|\vec{\alpha}||\hat{i}+2 \hat{j}-\hat{k}|=\sqrt{6}$</div>


title: if-is-an-imaginary-cube-root-of-unity-then-the-value-of-2-2-2-2-3-3-2-n-1-n-n-2-is
slug: if-is-an-imaginary-cube-root-of-unity-then-the-value-of-2-2-2-2-3-3-2-n-1-n-n-2-is-86235
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is an imaginary cube root of unity, then the value of $(2-\omega)\left(2-\omega^{2}\right)+2(3-\omega)\left(3-\omega^{2}\right)$ $+\ldots+(n-1)(n-\omega)\left(n-\omega^{2}\right)$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{n^{2}}{4}(n+1)^{2}-n$</span> </div>

answer:
<div class="solution">Given,
<br/>$(2-\omega)\left(2-\omega^{2}\right)+2(3-\omega)\left(3 \omega^{2}\right)+\ldots+$ $\quad(n-1)(n-\omega)\left(n \omega^{2}\right)$
<br/>Now, $\begin{aligned} T_{n} &amp;=(n-1)(n-\omega)\left(n-\omega^{2}\right) \\ &amp;=(n-1)\left(n^{2}-n \omega-n \omega^{2}+\omega^{3}\right) \\ &amp;=(n-1)\left(n^{2}+n+1\right)=n^{3}-1 \\ \therefore \quad S_{n} &amp;=\Sigma T_{n} \\ &amp;=\Sigma n^{3}-\Sigma 1=\frac{n^{2}(n+1)^{2}}{4}-n \end{aligned}$</div>


title: if-is-an-imaginary-cube-root-of-unity-then-the-value-of-the-determinant-1-1-2-2-0-2-2-is
slug: if-is-an-imaginary-cube-root-of-unity-then-the-value-of-the-determinant-1-1-2-2-0-2-2-is-26997
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is an imaginary cube root of unity, then the value of the determinant $\left|\begin{array}{ccc}1+\omega &amp; 0 &amp; -\omega \\ 1+\omega^{2} &amp; \omega &amp; -\omega^{2} \\ \omega+\omega^{2} &amp; \omega &amp; -\omega^{2}\end{array}\right|$ is</div>

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

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

answer:
<div class="solution">$\left|\begin{array}{ccc}1+\omega &amp; \omega^{2} &amp; -\omega \\ 1+\omega^{2} &amp; \omega &amp; -\omega^{2} \\ \omega+\omega^{2} &amp; \omega &amp; -\omega^{2}\end{array}\right|$
<br/>$\left|\begin{array}{ccc}0 &amp; \omega^{2} &amp; -\omega \\ 0 &amp; \omega &amp; -\omega^{2} \\ -1+\omega &amp; \omega &amp; -\omega^{2}\end{array}\right|$
<br/>$=(-1+\omega)\left(-\omega^{4}+\omega^{2}\right)=(\omega-1)\left(\omega^{2}-c\right.$
<br/>$=\omega^{3}-\omega^{2}-\omega^{2}+\omega=-3 \omega^{2}$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
title:
if-it-is-assumed-that-92-235-u-decays-only-by-emitting-and-particles-the-possible-product-of-the-decay-is

slug:
if-it-is-assumed-that-92-235-u-decays-only-by-emitting-and-particles-the-possible-product-of-the-decay-is-22789

question:
<div class="question">If it is assumed that ${ }_{92}^{235} U$ decays only by emitting $\alpha$ and $\beta$ particles, the possible product of the decay is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }_{89}^{225} A c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">${ }_{89}^{227} A c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">${ }_{89}^{230} A c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${ }_{89}^{231} A c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }_{89}^{227} A c$</span> </div>

answer:
<div class="solution">New mass no. $=235-2 \times 4=227$<br/>New at. no. $=92-2 \times 2+1=92-4+1=89$</div>


title: if-l-o-g-e-2-x-e-x-1-1-d-x-lo-g-e-2-3-then-the-value-of-x-is
slug: if-l-o-g-e-2-x-e-x-1-1-d-x-lo-g-e-2-3-then-the-value-of-x-is-40373
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\int_{\log _{e} 2}^{x}\left(e^{x}-1\right)^{-1} d x=\log _{e} \frac{3}{2}$ then the value of $x$ 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">$\mathrm{e}^{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\log 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{\mathrm{e}}$</span> </li> </ul>

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

answer:
<div class="solution">$\int_{\log _{e}{ }^{2}}^{x}\left(e^{x}-1\right)^{-1} d x=\log _{e}^{\frac{3}{2}} \int_{\log _{e}{ }^{2}}^{x} \frac{e^{-x}}{1-e^{-x}} d x=\log _{e}^{\frac{3}{2}}$
<br/>$e^{-x}=\frac{1}{4}$
<br/>$x=\ln 4$</div>


title: if-lim-x-0-1-c-x-1-c-x-1-x-4-then-lim-x-0-1-2-c-x-1-2-c-x-1-x-is
slug: if-lim-x-0-1-c-x-1-c-x-1-x-4-then-lim-x-0-1-2-c-x-1-2-c-x-1-x-is-99139
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow 0}\left(\frac{1+c x}{1-c x}\right)^{1 / x}=4$, then $\lim _{x \rightarrow 0}\left(\frac{1+2 c x}{1-2 c x}\right)^{1 / x}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">D</span> <span class="option-data">64</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>$\mathrm{e}^{2 \mathrm{c}}=4 \Rightarrow \mathrm{e}^{4 \mathrm{c}}=16$</div>


title: if-lim-x-0-t-a-n-3-x-2-a-s-i-n-x-s-i-n-2-x-exists-and-is-equal-to-1-then-the-value-of-a-is
slug: if-lim-x-0-t-a-n-3-x-2-a-s-i-n-x-s-i-n-2-x-exists-and-is-equal-to-1-then-the-value-of-a-is-64201
tags: ['Mathematics', 'Limits', '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
question:
<div class="question">If $\lim _{x \rightarrow 0} \frac{2 a \sin x-\sin 2 x}{\tan ^{3} x}$ exists and is equal to 1 then the value of $a$ is</div>

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

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{2 a \sin x-\sin 2 x}{\tan ^{3} x}$
<br/>$$
<br/>=\lim _{x \rightarrow 0} \frac{2 a\left(x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots\right)-\left(2 x-\frac{(2 x)^{3}}{3 !}+\frac{(2 x)^{5}}{5 !}-\ldots\right)}{\left(x+\frac{x^{3}}{3}+\frac{2}{15} x^{5}+\ldots\right)^{3}}
<br/>$$
<br/>$$
<br/>=\lim _{x \rightarrow 0} \frac{\left(2 a-2 x+\left(-\frac{2 a}{3 !}+\frac{2^{3}}{3 !}\right) x^{3}+\left(\frac{2 a}{5 !}-\frac{2^{5}}{5 !}\right) x^{5}+\ldots\right.}{x^{3}\left(1+\frac{x^{2}}{3}+\frac{2}{15} x^{4}+\ldots\right)^{3}}
<br/>$$
<br/>since, it is given that given limit is exist, then
<br/>$$
<br/>2 a-2=0 \Rightarrow \alpha=1
<br/>$$</div>


title: if-lim-x-0-x-2-a-x-e-x-b-l-o-g-1-x-3-then-the-values-of-a-and-b-are-respectively
slug: if-lim-x-0-x-2-a-x-e-x-b-l-o-g-1-x-3-then-the-values-of-a-and-b-are-respectively-18485
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2,2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow 0} \frac{a x e^{x}-b \log (1+x)}{x^{2}}=3$ then the values of $a$ and $b$ are, respectively</div>

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

answer:
<div class="solution">We have. $\lim _{x \rightarrow 0} \frac{\operatorname{ax} e^{x}-b \log (1+x)}{x^{2}}=3\left[\frac{0}{0}\right.$ form $]$
<br/>Using L' Hospital's rule, we get $\lim _{x \rightarrow 0} \frac{a e^{x}+a x e^{x}-\frac{b}{1+x}}{2 x}=3 \quad\left[\frac{0}{0}\right.$ form $]$
<br/>$\Rightarrow a-b=0 \Rightarrow a=b$
<br/>Using L' Hospital's rule, we get
<br/>$$
<br/>\lim _{x \rightarrow 0} \frac{a e^{x}+a e^{x}+a x e^{x}+\frac{b}{(1+x)^{2}}}{2}=3
<br/>$$
<br/>$\Rightarrow \quad \lim _{x \rightarrow 0} 2 a e^{x}+a x e^{x}+\frac{b}{(1+x)^{2}}=6$
<br/>$\Rightarrow$
<br/>$2 a+b=6$
<br/>$\Rightarrow \quad 3 a=6 \Rightarrow a=2$
<br/>On putting the value of $a$ we get $b=2$</div>


title: if-lo-g-2-6-2-x-1-lo-g-2-2-x-1-8-then-the-values-of-x-are
slug: if-lo-g-2-6-2-x-1-lo-g-2-2-x-1-8-then-the-values-of-x-are-28468
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\log _{2}^{6}+\frac{1}{2 x}=\log _{2}\left(2^{\frac{1}{x}}+8\right)$, then the values of $x$ are</div>

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

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

answer:
<div class="solution">We have,
<br/>$\log _{2}^{6}+\frac{1}{2 x}=\log _{2}\left(2^{\frac{1}{x}}+8\right)$
<br/>$\Rightarrow \quad 1+\log _{2}^{3}+\frac{1}{2 x}=\log _{2}\left(2^{\frac{1}{x}}+8\right)$
<br/>$\Rightarrow \quad \frac{2^{\frac{1}{x}}+8}{3}=2^{1+\frac{1}{2 x}}$
<br/>$\Rightarrow 2^{\frac{1}{x}}+8=3 \cdot 2 \cdot 2^{\frac{1}{2 x}}$
<br/>$\Rightarrow 2^{\frac{1}{x}}+8=6 \cdot 2^{\frac{1}{2 x}}$
<br/>Let $y=2^{\frac{1}{2 x}}$
<br/>$\Rightarrow \quad y^{2}+8=6 y$
<br/>$\Rightarrow \quad y^{2}-6 y+8=0$
<br/>$\Rightarrow \quad(y-4)(y-2)=0$
<br/>$y=4,2$
<br/>$2^{\frac{1}{2 x}}=4$ and $2^{\frac{1}{2 x}}=2$
<br/>$\frac{1}{2 x}=2$ and $\frac{1}{2 x}=1$
<br/>$x=\frac{1}{4}, \frac{1}{2}$</div>


title: if-lo-g-02-x-1-lo-g-004-x-5-then
slug: if-lo-g-02-x-1-lo-g-004-x-5-then-18030
question: <div class="question">$\text {If } \log _{0.2}(x-1)&gt;\log _{0.04}(x+5), \text { then }$</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-1    &lt; x &lt; 4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2    &lt; x &lt; 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1    &lt; x &lt; 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1    &lt; x &lt; 3$</span> </li> </ul>

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

answer:
<div class="solution">We have, $\log _{0.2}(x-1)&gt;\log _{0.04}(x+5)$
<br/>$\Rightarrow \quad \log _{0.2}(x-1)&gt;\log _{0.2^{2}}(x+5)$
<br/>$\Rightarrow \quad \log _{0.2}(x-1)&gt;\frac{1}{2} \log _{0.2}(x+5)$
<br/>$\Rightarrow \quad 2 \log _{0.2}(x-1)&gt;\log _{0.2}(x+5)$
<br/>$\Rightarrow \quad \log _{0.2}(x-1)^{2}&gt;\log _{0.2}(x+5)$
<br/>$\Rightarrow \quad(x-1)^{2}    &lt; x+5$
<br/>$\left[\because \log _{a} x&gt;\log _{a} y \Rightarrow x &lt; y,\right.$ if $\left.0 &lt; a &lt; 1\right]$
<br/>$\Rightarrow \quad x^{2}-2 x+1 &lt; x+5$
<br/>$\Rightarrow \quad x^{2}-3 x-4 &lt; 0$
<br/>$\Rightarrow \quad x^{2}-4 x+x-4 &lt; 0$
<br/>$\Rightarrow x(x-4)+1(x-4) &lt; 0$
<br/>$\Rightarrow \quad(x-4)(x+1) &lt; 0$
<br/>$\Rightarrow \quad x \in(-1,4)$
<br/>But $\quad x&gt;1$
<br/>$\Rightarrow \quad x \in(1,4)$</div>


title: if-lo-g-03-x-1-lo-g-009-x-1-then-x-lies-in-the-interval
slug: if-lo-g-03-x-1-lo-g-009-x-1-then-x-lies-in-the-interval-93583
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\log _{0.3}(x-1)    &lt; \log _{0.09}(x-1),$ then $x$ lies in
<br/>the interval</div>

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

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

answer:
<div class="solution">Given, $\quad \log _{0.3}(x-1)    &lt; \log _{0.09}(x-1)$
<br/>$\Rightarrow \quad \log _{0.3}(x-1) &lt; \log _{0.3^{2}}(x-1)$
<br/>$\Rightarrow \quad \log _{0.3}(x-1)^{2} &lt; \log _{0.3}(x-1)$
<br/>$\Rightarrow \quad(x-1)^{2}&gt;x-1 \quad[\because 0.3 &lt; 1]$
<br/>$\Rightarrow x^{2}+1-2 x-x+1&gt;0$
<br/>$\Rightarrow \quad x^{2}-3 x+2&gt;0$
<br/>$\Rightarrow \quad(x-1)(x-2)&gt;0$
<br/>$\Rightarrow \quad x &lt; 1, x&gt;2 \Rightarrow x&gt;2 \quad[\because x \neq 1]$
<br/>$\therefore \quad(2, \infty)$
<br/>Hence, x les in the interval $(2, \infty)$</div>


title: if-lo-g-5-x-lo-g-x-3-x-lo-g-3-x-y-lo-g-x-x-3-then-y-equals
slug: if-lo-g-5-x-lo-g-x-3-x-lo-g-3-x-y-lo-g-x-x-3-then-y-equals-79674
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">125</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\left(\log _{5} x\right)\left(\log _{x} 3 x\right)\left(\log _{3 x} y\right)=\log _{x} x^{3},$ then $y$ equals</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">125</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">25</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 / 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">243</span> </li> </ul>

answer:
<div class="solution">We have, 
<br/>$\begin{aligned} &amp; \log _{5} x \cdot \log _{x} 3 x \cdot \log _{3 x} y=\log _{x} x^{3} \\ \Rightarrow &amp; \frac{\log x}{\log 5} \times \frac{\log 3 x}{\log x} \times \frac{\log y}{\log 3 x}=3 \log _{x} x \end{aligned}$
<br/>$\left[\because \log _{b} a=\frac{\log a}{\log b}\right.$ and $\left.\log a^{m}=m \log a\right]$
<br/>$\frac{\log y}{\log 5}=3$
<br/>$\left[\because \log _{a} a=1\right]$
<br/>$\begin{array}{ll}\Rightarrow &amp; \log y=3 \log 5 \\ \Rightarrow &amp; \log y=\log 5^{3}\end{array}$
<br/>$y=5^{3}=125$</div>


title: if-lo-g-e-x-2-16-lo-g-e-4-x-11-then
slug: if-lo-g-e-x-2-16-lo-g-e-4-x-11-then-23017
question: <div class="question">If $\log _{e}\left(x^{2}-16\right) \leq \log _{e}(4 x-11)$, then</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4     &lt; x \leq 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x     &lt; -4$ or $x&gt;4$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-1 \leq x \leq 5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x     &lt; -1$ or $x&gt;5$</span> </li> </ul>

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

answer:
<div class="solution">Given, \(\log _{e}\left(x^{2}-16\right) \leq \log _{e}(4 x-11)\) if and only if
<br/>\(\begin{array}{l}
<br/>x^{2}-16 \leq 4 x-11 \\
<br/>\Rightarrow x^{2}-4 x-5 \leq 0 \\
<br/>x^{2}-5 x+x-5 \leq 0 \\
<br/>\Rightarrow x(x-5)+1(x-5) \leq 0 \\
<br/>(x-5)(x+1) \leq 0
<br/>\end{array}\)
<br/>Sign scheme of \(x^{2}-4 x-5 \leq 0\)
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a80.png"/> <br/>\(-1 \leq x \leq 5\)</div>


title: if-log-3-xlog-3-y2log-3-2-and-log-3xy2-then
slug: if-log-3-xlog-3-y2log-3-2-and-log-3xy2-then-66223
question: <div class="question">If \(\log _3 x+\log _3 y=2+\log _3 2\) and \(\log _3(x+y)=2\), then</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(x=1, y=8\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(x=8, y=1\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(x=3, y=6\)</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">\(x=9, y=3\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x=3, y=6\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}\text {Hints: } <br/>&amp; \log _3 \mathrm{x}+\log _3 \mathrm{y}=2+\log _3 2 \\<br/>&amp; \Rightarrow \mathrm{x} \cdot \mathrm{y}=18 \\<br/>&amp; \log (\mathrm{x}+\mathrm{y})=2 \Rightarrow \mathrm{x}+\mathrm{y}=9<br/>\end{aligned}\)<br/>we will get \(x=3\) and \(y=6\)</div>


title: if-log-7-2lambda-then-the-value-of-log-4928-is
slug: if-log-7-2lambda-then-the-value-of-log-4928-is-92452
question: <div class="question">If \(\log _7 2=\lambda\), then the value of \(\log _{49}(28)\) is</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((2 \lambda+1)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((2 \lambda+3)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{1}{2}(2 \lambda+1)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2(2 \lambda+1)\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: \(\log _{49} 28=\log _{72} 4 \times 7\)<br/>\(=\frac{1}{2}\left[2 \log _7 2+\log _7 7\right]=\frac{1}{2}[2 \lambda+1]\)</div>


title: if-m-0-n-2-x-2-c-o-s-x-d-x-n-0-4-x-1-2-s-i-n-x-c-o-s-x-d-x-then-the-value-of-m-n-is
slug: if-m-0-n-2-x-2-c-o-s-x-d-x-n-0-4-x-1-2-s-i-n-x-c-o-s-x-d-x-then-the-value-of-m-n-is-13963
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $M=\int_{0}^{n / 2} \frac{\cos x}{x+2} d x, N=\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{(x+1)^{2}} d x,$ then
<br/>the value of $M-N$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{\pi+4}$</span> </div>

answer:
<div class="solution">Given, $M=\int_{0}^{\frac{\pi}{2}} \frac{\cos x}{(x+2)} d x$
<br/>$\begin{aligned} \text { and } \quad N &amp;=\int_{0}^{\frac{\pi}{4}}  \frac{\sin x \cos x}{(x+1)^{2}} d x \\ &amp;=\int_{0}^{\pi / 4} \frac{1}{2} \cdot \frac{\sin 2 x}{(x+1)^{2}} d x \end{aligned}$
<br/>Put $2 x=t \Rightarrow d x=\frac{a}{2}$
<br/>$\begin{aligned} \therefore \quad N &amp;=\int_{0}^{\frac{\pi}{2}}  \frac{\sin t}{4(t / 2+1)^{2}} d t \\ &amp;=\int_{0}^{\frac{\pi}{2}} \frac{\sin t}{(t+2)^{2}} d t \end{aligned}$
<br/>$\therefore M-N=\int_{0}^{\frac{\pi}{2}}(\cos x) \cdot \frac{1}{(x+2)} d x$
<br/>$-\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{(x+2)^{2}} d x$
<br/>$=\left(\frac{\sin x}{x+2}\right)_{0}^{\pi 2}-\int_{0}^{\pi 2}-\frac{\sin x}{(x+2)^{2}} d x$
<br/>$-\frac{\sin \pi / 2}{\pi / 2+2}=\frac{1}{\frac{\pi+4}{2}}=\frac{2}{\pi+4}$</div>


title: if-m-is-a-3-3-matrix-such-that-0-1-2-m-1-0-0-3-4-5-m-0-1-0-then-6-7-8-m-is-equal-to
slug: if-m-is-a-3-3-matrix-such-that-0-1-2-m-1-0-0-3-4-5-m-0-1-0-then-6-7-8-m-is-equal-to-92458
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $M$ is a $3 \times 3$ matrix such that $(0~1~2) M=(1~0~0),(3~4~5) M=(0~1~0)$, then $(6~7~8) M$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(\left.\begin{array}{llll}2 &amp; 1 &amp; -2\end{array}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(\left.\begin{array}{llll}0 &amp; 0 &amp; 1\end{array}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(\left.\begin{array}{llll}-1 &amp; 2 &amp; 0\end{array}\right)$</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">$(\left.\begin{array}{llll}9 &amp; 10 &amp; 8\end{array}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\left.\begin{array}{llll}-1 &amp; 2 &amp; 0\end{array}\right)$</span> </div>

answer:
<div class="solution">$\left(\begin{array}{lll}0 &amp; 1 &amp; 2\end{array}\right) \mathrm{m}=\left(\begin{array}{lll}1 &amp; 0 &amp; 0\end{array}\right)$ ....(i)
<br/>$\left(\begin{array}{lll}3 &amp; 4 &amp; 5\end{array}\right) \mathrm{m}=\left(\begin{array}{lll}0 &amp; 1 &amp; 0\end{array}\right)$
<br/>$\left(\begin{array}{lll}6 &amp; 8 &amp; 10\end{array}\right) \mathrm{m}=\left(\begin{array}{lll}0 &amp; 2 &amp; 0\end{array}\right)$ ...(ii)
<br/>(ii) $-$ (i)
<br/>$\left(\begin{array}{lll}6 &amp; 8 &amp; 10\end{array}\right) \mathrm{m}-\left(\begin{array}{lll}0 &amp; 1 &amp; 2\end{array}\right) \mathrm{m}$ = $\left(\begin{array}{lll}0 &amp; 2 &amp; 0\end{array}\right)-\left(\begin{array}{lll}1 &amp; 0 &amp; 0\end{array}\right)$
<br/>$\Rightarrow$ $\left(\begin{array}{lll}6 &amp; 7 &amp; 8\end{array}\right) \mathrm{m}=\left(\begin{array}{lll}-1 &amp; 2 &amp; 0\end{array}\right)$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
if-m-is-any-square-matrix-of-order-3-over-r-and-if-m-be-the-transpose-of-m-then-a-d-j-m-adj-m-is-equal-to

slug:
if-m-is-any-square-matrix-of-order-3-over-r-and-if-m-be-the-transpose-of-m-then-a-d-j-m-adj-m-is-equal-to-46045

question:
<div class="question">If $M$ is any square matrix of order 3 over $R$. and if $M^{\prime}$ be the transpose of $M$, then $adj\left(M^{\prime}\right)-(\text { adj } M)^{\prime}$ is equal to</div>

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

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

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\begin{aligned}
<br/>\text { Since, } \operatorname{adj}\left(M^{\prime}\right)=&amp;(\operatorname{adj} M)^{\prime} \\
<br/>=&amp; \operatorname{adj}\left(M^{\prime}\right)-(\operatorname{adj} M)^{\prime} \\
<br/>&amp;\left[\because \operatorname{adj}\left(A^{\prime}\right)=(\operatorname{adj} A)^{\prime}\right]
<br/>\end{aligned}\\
<br/>&amp;=0 \text { , a null matrix. }
<br/>\end{aligned}
<br/>$$</div>


title: if-mathrma-and-mathrmb-are-two-matrices-such-that-mathrmamathrmb-and-mathrmab-are-both-defined-then
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
slug:
if-mathrma-and-mathrmb-are-two-matrices-such-that-mathrmamathrmb-and-mathrmab-are-both-defined-then-12365

question:
<div class="question">If \(\mathrm{A}\) and \(\mathrm{B}\) are two matrices such that \(\mathrm{A}+\mathrm{B}\) and \(\mathrm{AB}\) are both defined, then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A, B are square matrices of the same order</span> </div>

answer:
<div class="solution">Hints : Addition is defined if order of \(A\) is equal to order of \(B\)<br/>A B<br/>\(\mathrm{nxm} \mathrm{nxm}\) is defined if \(\mathrm{m}=\mathrm{n}\)<br/>\(\Rightarrow A, B\) are square matrices of same order</div>


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-mathrmg-is-the-acceleration-due-to-gravity-on-the-surface-of-the-earth-the-gain-in-potential-energy-of-an-object-of-mass-mathrmm-raised-from-the-ea

slug:
if-mathrmg-is-the-acceleration-due-to-gravity-on-the-surface-of-the-earth-the-gain-in-potential-energy-of-an-object-of-mass-mathrmm-raised-from-the-ea-53565

question:
<div class="question">If \(\mathrm{g}\) is the acceleration due to gravity on the surface of the earth, the gain in potential energy of an object of mass \(\mathrm{m}\) raised from the earth's surface to a height equal to the radius \(R\) of the earth is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\mathrm{mg} \mathrm{R}}{4}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{m g R}{2}\)</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">\(m g R\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2 \mathrm{mg} \mathrm{R}\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: \(\Delta \mathrm{U}=\frac{\mathrm{mgh}}{1+\frac{\mathrm{h}}{\mathrm{R}}}=\frac{\mathrm{mgR}}{1+\frac{\mathrm{R}}{\mathrm{R}}}=\frac{\mathrm{mgR}}{2}\)</div>


title: if-mathrmn-1-is-an-integer-and-mathrmx-neq-0-then-1mathrmxmathrmnmathrmnx1-is-divisible-by
slug: if-mathrmn-1-is-an-integer-and-mathrmx-neq-0-then-1mathrmxmathrmnmathrmnx1-is-divisible-by-71165
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(\mathrm{n} &gt; 1\) is an integer and \(\mathrm{x} \neq 0\), then \((1+\mathrm{x})^{\mathrm{n}}-\mathrm{nx}-1\) is divisible by</div>

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

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

answer:
<div class="solution">\(\begin{array}{r}\text {Hints: } <br/>(1+\mathrm{x})^n={ }^n C_0+{ }^n C_1 x+{ }^n C_2 x^2+{ }^n C_3 \mathrm{x} 3+\ldots \ldots . \\<br/>=1+\mathrm{nx}+\mathrm{x}^2\left({ }^n C_2+{ }^n C_3 \mathrm{x}+\ldots \ldots . .\right) \\<br/>(1+\mathrm{x})^{\mathrm{n}}-\mathrm{nx}-1=\mathrm{x}^2\left({ }^n C_2+{ }^n C_3 \mathrm{x}+\ldots \ldots .\right)<br/>\end{array}\)</div>


title: if-momentum-is-increased-by-20-then-kinetic-energy-increases-by
slug: if-momentum-is-increased-by-20-then-kinetic-energy-increases-by-89271
question: <div class="question">If momentum is increased by $20 \%$, then kinetic energy increases by</div>
tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$48 \%$</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">$40 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$36 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$44 \%$</span> </div>

answer:
<div class="solution">Hints: $\mathrm{K}=\frac{\mathrm{P}^2}{2 m}$<br/>Here $\mathrm{P}^{\prime}=1.2 \mathrm{P}$<br/>Hence, $K^{\prime}=\frac{(1.2 \mathrm{P})^2}{2 m}$<br/>$$<br/>\mathrm{K}^{\prime}=1.44 \frac{\mathrm{P}^2}{2 m}<br/>$$<br/>$\mathrm{K}^{\prime}=1.44 \mathrm{~K}$ or Percentage increase in $\mathrm{K}=44 \%$</div>


title: if-n-1-c-3-n-1-c-4-n-c-3-then-n-is-just-greater-than-integer
slug: if-n-1-c-3-n-1-c-4-n-c-3-then-n-is-just-greater-than-integer-99347
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If ${ }^{\mathrm{n}-1} \mathrm{C}_3+{ }^{\mathrm{n}-1} \mathrm{C}_4&gt;{ }^{\mathrm{n}} \mathrm{C}_3$, then $\mathrm{n}$ is just greater than integer</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: }:^{n-1} C_3+{ }^{n-1} C_4&gt;{ }^n C_3 \\<br/>&amp; \qquad \Rightarrow^n C_4&gt;^n C_3 \Rightarrow \frac{n !}{4 !(n-4) !}&gt;\frac{n !}{3 !(n-3) !} \Rightarrow \frac{1}{4}&gt;\frac{1}{(n-3)} \Rightarrow n-3&gt;4 \Rightarrow n&gt;7<br/>\end{aligned}<br/>$$</div>


title: if-n-c-r-1-36-n-c-r-84-and-n-c-r-1-126-then-the-value-of-n-c-8-is
slug: if-n-c-r-1-36-n-c-r-84-and-n-c-r-1-126-then-the-value-of-n-c-8-is-57445
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If ${ }^{n} C_{r-1}=36,{ }^{n} C_{r}=84$ and ${ }^{n} C_{r+1}=126$ then the value of ${ }^{n} C_{8}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</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">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">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Given,
<br/>$$
<br/>\begin{aligned}
<br/>{ }^{n} C_{r-1} &amp;=36 \\
<br/>{ }^{n} C_{r} &amp;=84 \\
<br/>{ }^{n} C_{r+1} &amp;=126
<br/>\end{aligned}
<br/>$$
<br/>On dividing 
<br/>$$
<br/>\frac{r}{n-r+1}=\frac{36}{84}
<br/>$$
<br/>$\Rightarrow \quad 84 r=36 n-36 r+36$
<br/>$\Rightarrow \quad 120 r=36 n+36$
<br/>Also,
<br/>$$
<br/>\begin{array}{r}
<br/>\frac{r+1}{n-r}=\frac{64}{126} \\
<br/>\Rightarrow \quad 126 r+126=84 n-84 r
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow$
<br/>$$
<br/>210r=84 n-126
<br/>$$
<br/>On solving Eqs. (iv) and (v). we get $n=9$ and $r=3$
<br/>\begin{array}{l}
<br/>\hline
<br/>\end{array}
<br/>$$
<br/>{ }^{n} C_{8}={ }^{9} C_{8}=9
<br/>$$</div>


title: if-n-c4-n-c5-and-n-c6-are-in-ap-then-n-is
slug: if-n-c4-n-c5-and-n-c6-are-in-ap-then-n-is-85267
question: <div class="question">If \({ }^n C_4,{ }^n C_5\) and \({ }^n C_6\) are in A.P., then \(n\) is</div>
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">7 or 14</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">14</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">14 or 21</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">7 or 14</span> </div>

answer:
<div class="solution">Hints: \({ }^n \mathrm{C}_4,{ }^n \mathrm{C}_5,{ }^n \mathrm{C}_6\) are in \(\mathrm{AP}\)<br/>\(\begin{aligned}<br/>&amp; \text { 2. }{ }^n \mathrm{C}_5={ }^n \mathrm{C}_4+{ }^n \mathrm{C}_6 \\<br/>&amp; \frac{2}{5(n-5)}=\frac{1}{(n-4)}+\frac{1}{30} \\<br/>&amp;<br/>\end{aligned}\)<br/>by solving \(n=14\) or 7</div>


tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">area</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-n-denotes-a-positive-integer-h-the-planck-constant-q-the-charge-and-b-the-magnetic-field-then-the-quantity-2-qb-nh-has-the

slug:
if-n-denotes-a-positive-integer-h-the-planck-constant-q-the-charge-and-b-the-magnetic-field-then-the-quantity-2-qb-nh-has-the-70179

question:
<div class="question">If $n$ denotes a positive integer, $h$ the Planck constant, $q$ the charge and $B$ the magnetic field, then the quantity $\left[\frac{n h}{2 \pi q B}\right]$ has the</div>

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

answer:
<div class="solution">$\left[\frac{n h}{2 \pi q B}\right]$<br/>where $n$ and $2 \pi$ dimensionless quantity<br/>So, $[h]=[m v r]$<br/>$\begin{aligned}<br/>\mid q B] &amp;=\left[\frac{q v B}{v}\right]=\left[\frac{F}{v}\right] \\<br/>\therefore \quad\left[\frac{m v r}{F / v}\right] &amp;=\frac{[m v r][v]}{[F]} \\<br/>&amp;=\left[\frac{\left[m v^{2} r\right]}{r}\right]=\left[r^{2}\right\}=\text { Area }<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-n-is-even-posirive-integer-then-the-condition-that-the-greatest-term-in-the-expansion-of-a-x-n-may-also-have-the-greatest-coefficient-is

slug:
if-n-is-even-posirive-integer-then-the-condition-that-the-greatest-term-in-the-expansion-of-a-x-n-may-also-have-the-greatest-coefficient-is-11746

question:
<div class="question">If $n$ is even posirive integer, then the condition that the greatest term in the expansion of $(a+x)^{n}$ may also have the greatest coefficient, is</div>

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

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

answer:
<div class="solution">For greatest term of $(1+x)^{n},$ we have
<br/>$\quad \frac{n}{2}    &lt; \frac{n+1}{1+x} &lt; \frac{n}{2}+1$
<br/>$\Rightarrow \quad \frac{n}{2} &lt; \frac{n+1}{1+x}$ and $\frac{n+1}{1+x} &lt; \frac{n}{2}+1$
<br/>$\Rightarrow \quad 1+x &lt; \frac{n+1}{n / 2}$ and $\frac{n+1}{\frac{n}{2}+1} &lt; 1+x$
<br/>$\Rightarrow \quad x &lt; \frac{n+1-n / 2}{n / 2}$
<br/>and $\quad \frac{n+1-(n / 2-1)}{\frac{n+2}{2}} &lt; x$
<br/>$\Rightarrow \quad x &lt; \frac{n+2}{n}$ and $\frac{n}{n+2} &lt; x$
<br/>$\Rightarrow \quad \frac{n}{n+2} &lt; x &lt; \frac{n+2}{n}$</div>


title: if-n-t-number-of-radioatoms-are-present-at-time-t-the-following-expression-will-be-a-constant
slug: if-n-t-number-of-radioatoms-are-present-at-time-t-the-following-expression-will-be-a-constant-88717
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $n_t$ number of radioatoms are present at time $t$, the following expression will be a constant :</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{d} \operatorname{In} \mathrm{n} / \mathrm{dt}$</span> </div>

answer:
<div class="solution">$\frac{d N}{d t}=\lambda \mathrm{N} \Rightarrow-\frac{d \ln N}{d t}=\lambda<br/> Hence C</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-omega-neq-1-is-a-cube-root-of-unity-then-the-sum-of-the-series-s12-omega3-omega2ldots-ldots-ldots-3-n-omega3-n1-is

slug:
if-omega-neq-1-is-a-cube-root-of-unity-then-the-sum-of-the-series-s12-omega3-omega2ldots-ldots-ldots-3-n-omega3-n1-is-48676

question:
<div class="question">If \(\omega \neq 1\) is a cube root of unity, then the sum of the series \(S=1+2 \omega+3 \omega^2+\ldots \ldots \ldots . .+3 n \omega^{3 n-1}\) is</div>

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

answer:
<div class="solution">\(\begin{aligned}<br/>\text {Hints : } &amp; s=1+2 \omega+3 \omega^2+\ldots \ldots \ldots .+3 n \omega^{3 n-1} \\<br/>&amp; s \omega=\omega+2 \omega^2+\ldots \ldots \ldots \ldots .+(3 n-1) \omega^{3 n}+3 n \omega^{3 n} \\<br/>&amp; s(1-\omega)=1+\omega+\omega^2+\ldots \ldots \ldots .+\omega^{3 n-1}-3 n \omega^{3 \mathrm{~h}} \\<br/>&amp; =0-3 n \\<br/>&amp; s=\frac{-3 n}{1-\omega}=\frac{3 n}{\omega-1}<br/>\end{aligned}\)</div>


title: if-one-end-of-a-diameter-of-the-circle-3-x-2-3-y-2-9-x-6-y-5-0-is-1-2-then-the-other-end-is
slug: if-one-end-of-a-diameter-of-the-circle-3-x-2-3-y-2-9-x-6-y-5-0-is-1-2-then-the-other-end-is-11934
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(2,-4)</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If one end of a diameter of the circle $3 x^{2}+3 y^{2}-9 x+6 y+5=0$ is $(1,2),$ then the other end is</div>

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

answer:
<div class="solution">Given equation of circle is
<br/>$$
<br/>\begin{array}{r}
<br/>3 x^{2}+3 y^{2}-9 x+6 y+5=0 \\
<br/>\Rightarrow \quad x^{2}+y^{2}-3 x+2 y+\frac{5}{3}=0
<br/>\end{array}
<br/>Centre
<br/>=\left(\frac{3}{2},-1\right)
<br/>$$
<br/>and radius $=\sqrt{\frac{9}{4}+1-\frac{5}{3}}$
<br/>$$
<br/>=\sqrt{\frac{19}{12}}=\frac{1}{2} \sqrt{\frac{19}{3}}
<br/>$$
<br/>We know that, centre of the circle is the mid-point of the diameter.
<br/>Lives one and of point of diameter in (1,2) Let the other end point of diameter is $(h, k)$
<br/>Then, $\quad\left(\frac{3}{2},-1\right)=\left(\frac{1+h}{2}, \frac{2+k}{2}\right)$
<br/>$\Rightarrow \quad \frac{1+h}{2}=\frac{3}{2}$
<br/>$\Rightarrow 1+h=3$
<br/>$\Rightarrow h=2$
<br/>and $\frac{2+k}{2}=-1$
<br/>$\Rightarrow 2+k=-2$
<br/>$\Rightarrow k=-4$
<br/>So, the other end point is (2,-4)</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-one-of-the-cube-roots-of-1-be-omega-then-leftbeginarrayccc1-1omega2-omega2-1i-1-omega21-i-1omega-1endarrayright

slug:
if-one-of-the-cube-roots-of-1-be-omega-then-leftbeginarrayccc1-1omega2-omega2-1i-1-omega21-i-1omega-1endarrayright-55963

question:
<div class="question">If one of the cube roots of 1 be \(\omega\), then<br/>\(\left|\begin{array}{ccc}1 &amp; 1+\omega^2 &amp; \omega^2 \\ 1-i &amp; -1 &amp; \omega^2-1 \\ -i &amp; -1+\omega &amp; -1\end{array}\right|=\)</div>

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

answer:
<div class="solution">Hints: \(\mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_3\)<br/>\(\begin{aligned}<br/>&amp; \mathrm{C}_3 \rightarrow \mathrm{C}_3+\mathrm{C}_2 \\<br/>&amp; \mathrm{C}_3 \rightarrow \mathrm{C}_3+\omega \mathrm{C}_1 \\<br/>&amp; \mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_1<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: db\MarksBatch2_P1.db
title:
if-one-of-the-diameter-of-the-circle-given-by-the-equation-x-2-y-2-4-x-6-y-12-0-is-a-chord-of-a-circle-s-whose-centre-is-2-3-the-radius-of-s-is-1

slug:
if-one-of-the-diameter-of-the-circle-given-by-the-equation-x-2-y-2-4-x-6-y-12-0-is-a-chord-of-a-circle-s-whose-centre-is-2-3-the-radius-of-s-is-1-10366

question:
<div class="question">If one of the diameter of the circle, given by the equation $x^{2}+y^{2}+4 x+6 y-12=0,$ is a chord of a circle $S,$ whose centre is $(2,-3$ ), the radius of $S$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{41}$ unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \sqrt{5}$ unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \sqrt{2}$ unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sqrt{5}$ unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{41}$ unit</span> </div>

answer:
<div class="solution">Given, equation of circle is $x^{2}+y^{2}+4 x+6 y-12=0$
<br/>Centre $C(-2,-3)$ and radius $=\sqrt{(-2)^{2}+(-3)^{2}+12}=5$
<br/>$$
<br/>x^{2}+y^{2}+4 x+6 y-12=0
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>C_{1} C_{2} &amp;=\sqrt{\left(2+{2})^{2}+(-3+3)^{2}\right.} \\
<br/>&amp;=\sqrt{(4)^{2}+(0)^{2}}=4
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Radius of circle, $S$
<br/>$$
<br/>=\sqrt{(4)^{2}+({5})^{2}}=\sqrt{16+25}=\sqrt{41} \text { unit }
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a43.png"/></div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-one-of-the-diameter-of-the-circle-given-by-the-equation-x-2-y-2-4-x-6-y-12-0-is-a-chord-of-a-circle-s-whose-centre-is-2-3-the-radius-of-s-is

slug:
if-one-of-the-diameter-of-the-circle-given-by-the-equation-x-2-y-2-4-x-6-y-12-0-is-a-chord-of-a-circle-s-whose-centre-is-2-3-the-radius-of-s-is-96857

question:
<div class="question">If one of the diameter of the circle, given by the equation $x^{2}+y^{2}+4 x+6 y-12=0,$ is a chord of a circle $S,$ whose centre is $(2,-3$ ), the radius of $S$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{41}$ unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \sqrt{5}$ unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \sqrt{2}$ unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sqrt{5}$ unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{41}$ unit</span> </div>

answer:
<div class="solution">Given, equation of circle is $x^{2}+y^{2}+4 x+6 y-12=0$
<br/>Centre $C(-2,-3)$ and radius $=\sqrt{(-2)^{2}+(-3)^{2}+12}=5$
<br/>$$
<br/>x^{2}+y^{2}+4 x+6 y-12=0
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>C_{1} C_{2} &amp;=\sqrt{\left(2+{2})^{2}+(-3+3)^{2}\right.} \\
<br/>&amp;=\sqrt{(4)^{2}+(0)^{2}}=4
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Radius of circle, $S$
<br/>$$
<br/>=\sqrt{(4)^{2}+({5})^{2}}=\sqrt{16+25}=\sqrt{41} \text { unit }
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a43.png"/></div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-one-of-the-diameters-of-the-curve-x-2-y-2-4-x-6-y-9-0-is-a-chord-of-a-circle-with-centre-1-1-the-radius-of-this-circle-is

slug:
if-one-of-the-diameters-of-the-curve-x-2-y-2-4-x-6-y-9-0-is-a-chord-of-a-circle-with-centre-1-1-the-radius-of-this-circle-is-76970

question:
<div class="question">If one of the diameters of the curve $x^{2}+y^{2}-4 x-6 y+9=0$ is a chord of a circle with centre $(1,1),$ the radius of this circle is</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">$\sqrt{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Given circle
<br/>$x^{2}+y^{2}-4 x-6 y+9=0$
<br/>$\therefore$ Centre $=\left(-\frac{1}{2} \times(-4),-\frac{1}{2} \times(-6)\right)$
<br/>$=(2.3)$
<br/>$\begin{aligned} \text { Radius } &amp;=\sqrt{(-2)^{2}+(-3)^{2}-9} \\ &amp;=\sqrt{4+9-9} \\ &amp;=2 \end{aligned}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a54.png"/> <br/> <br/>$$
<br/>\begin{aligned}
<br/>\therefore \quad A B &amp;=\sqrt{(2-1)^{2}+(3-1)^{2}}=\sqrt{1+4}=\sqrt{5} \\
<br/>\therefore \quad A C &amp;=\sqrt{A B^{2}+B C^{2}}=\sqrt{(\sqrt{5})^{2}+(2)^{2}} \\
<br/>&amp;=\sqrt{5+4}=\sqrt{9}=3
<br/>\end{aligned}
<br/>$$
<br/>Radius of required circle $=3$</div>


title: if-one-root-of-x2p-xq20-p-and-q-are-real-be-less-than-2-and-other-be-greater-than-2-then
slug: if-one-root-of-x2p-xq20-p-and-q-are-real-be-less-than-2-and-other-be-greater-than-2-then-97446
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If one root of \(x^2+p x-q^2=0, p\) and \(q\) are real, be less than 2 and other be greater than 2, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(4+2 p+q^2 &gt; 0\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4+2 p+q^2  &lt; 0\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(4+2 p-q^2 &gt; 0\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(4+2 p-q^2  &lt; 0\)</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(4+2 p-q^2  &lt; 0\)</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/By49xtSDGC7gqXFkkOTZxn1AGfSuDX0uxgOv0k_9Dug.original.fullsize.png"/><br/><br/>\(\alpha &lt; 2\) and \(\beta &gt; 2\)<br/>Let, \(\mathrm{p}(\mathrm{x})=\mathrm{x}^2+\mathrm{px}-\mathrm{q}^2\)<br/>\(\begin{aligned}<br/>&amp; \therefore \mathrm{p}(2) &lt; 0 \\<br/>&amp; \Rightarrow 2^2+2 p-\mathrm{q}^2 &lt; 0 \\<br/>&amp; \Rightarrow 4+2 p-\mathrm{q}^2 &lt; 0<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
if-p-0-0-q-1-0-and-r-2-1-2-3-are-three-given-points-then-the-centre-of-the-circle-for-which-the-lines-pq-qr-and-rp-are-the-tangents-is

slug:
if-p-0-0-q-1-0-and-r-2-1-2-3-are-three-given-points-then-the-centre-of-the-circle-for-which-the-lines-pq-qr-and-rp-are-the-tangents-is-44159

question:
<div class="question">If $P(0,0), Q(1,0)$ and $R\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ are three given points, then the centre of the circle for which the lines $P Q, Q R$ and $R P$ are the tangents is</div>

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

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

answer:
<div class="solution">$\begin{aligned}
<br/>&amp;\text { } I=\left(\frac{a x_{1}+b x_{2}+c x_{3}}{a+b+c}, \frac{a y_{1}+b y_{2}+c y_{3}}{a+b+c}\right)\\
<br/>&amp;a=\sqrt{\frac{1}{4}+\frac{3}{4}}=1, b=1, c=1\\
<br/>&amp;\text { Centre of the circle. }\\
<br/>&amp;=\left(\frac{1 \times 0+1 \times 1+1 \times \frac{1}{2}}{1+1+1}, \frac{1 \times 0+1 \times 0+1 \times \sqrt{3} / 2}{1+1+1}\right)\\
<br/>&amp;=\left(\frac{0+1+\frac{1}{2}}{3} \frac{0+0+\frac{\sqrt{3}}{2}}{3}\right)\\
<br/>&amp;=\left(\frac{3}{\frac{2}{3}}, \frac{\frac{\sqrt{3}}{2}}{3}\right)
<br/>\end{aligned}$
<br/>$=\left(\frac{3}{2} \times \frac{1}{3}, \frac{\sqrt{3}}{2} \times \frac{1}{3}\right)$
<br/>$=\left(\frac{1}{2}, \frac{1}{2 \sqrt{3}}\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a38.png"/></div>


title: if-p-1-1-2-3-1-1-q-p-p-t-then-the-value-of-the-determinant-of-q-is
slug: if-p-1-1-2-3-1-1-q-p-p-t-then-the-value-of-the-determinant-of-q-is-29313
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $P=\left[\begin{array}{lll}1 &amp; 2 &amp; 1 \\ 1 &amp; 3 &amp; 1\end{array}\right] Q=P P^{T}$, then the value of the
<br/>determinant of $Q$ is</div>

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

answer:
<div class="solution">Given, $P=\left[\begin{array}{lll}1 &amp; 2 &amp; 1 \\ 1 &amp; 3 &amp; 1\end{array}\right]$
<br/>$$
<br/>\begin{array}{l}
<br/>\therefore \quad Q=P P^{T}=\left[\begin{array}{lll}
<br/>1 &amp; 2 &amp; 1 \\
<br/>1 &amp; 3 &amp; 1
<br/>\end{array}\right]\left[\begin{array}{ll}
<br/>1 &amp; 1 \\
<br/>2 &amp; 3 \\
<br/>1 &amp; 1
<br/>\end{array}\right] \\
<br/>=\left[\begin{array}{ll}
<br/>1 \times 1+2 \times 2+1 \times 1 &amp; 1 \times 1+2 \times 3+1 \times 1 \\
<br/>1 \times 1+3 \times 2+1 \times 1 &amp; 1 \times 1+3 \times 3+1 \times 1
<br/>\end{array}\right] \\
<br/>=\left[\begin{array}{ll}
<br/>1+4+1 &amp; 1+6+1 \\
<br/>1+6+1 &amp; 1+9+1
<br/>\end{array}\right]=\left[\begin{array}{lr}
<br/>6 &amp; 8 \\
<br/>8 &amp; 11
<br/>\end{array}\right] \\
<br/>\therefore \text { The determintant of } Q=\left|\begin{array}{lr}
<br/>6 &amp; 8 \\
<br/>8 &amp; 11
<br/>\end{array}\right| \\
<br/>=66-64=2
<br/>\end{array}
<br/>$$</div>


title: if-p-1-1-2-3-4-3-3-4-is-the-adjoint-of-the-3-3-matrix-a-and-det-a-4-then-is-equal-to
slug: if-p-1-1-2-3-4-3-3-4-is-the-adjoint-of-the-3-3-matrix-a-and-det-a-4-then-is-equal-to-13680
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{p}=\left[\begin{array}{lll}1 &amp; \alpha &amp; 3 \\ 1 &amp; 3 &amp; 3 \\ 2 &amp; 4 &amp; 4\end{array}\right]$ is the adjoint of the $3 \times 3$ matrix $\mathrm{A}$ and $\operatorname{det} \mathrm{A}=4$, then $\alpha$ is equal to</div>

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

answer:
<div class="solution">$|\mathrm{p}|=2 \alpha-6=(\operatorname{det} \mathrm{A})^2=16$<br/>$\Rightarrow \alpha=11$</div>


title: if-p-1-2-2-1-3-2-2-1-and-q-1-2-1-3-4-1-5-6-1-then
slug: if-p-1-2-2-1-3-2-2-1-and-q-1-2-1-3-4-1-5-6-1-then-90221
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $P=1+\frac{1}{2 \times 2}+\frac{1}{3 \times 2^{2}}+\ldots$
<br/>and $Q=\frac{1}{1 \times 2}+\frac{1}{3 \times 4}+\frac{1}{5 \times 6}+\ldots,$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$P=Q$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 P=Q$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$P=2 Q$</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">$P=4 Q$</span> </li> </ul>

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

answer:
<div class="solution">Given,
<br/>$$
<br/>P=1+\frac{1}{2 \times 2}+\frac{1}{3 \times 2^{2}}+\ldots
<br/>$$
<br/>and $\quad Q=\frac{1}{1 \times 2}+\frac{1}{3 \times 4}+\frac{1}{5 \times 6}+\ldots$
<br/>Now, $\quad \frac{P}{2}=\frac{(1 / 2)^{1}}{1}+\frac{(1 / 2)^{2}}{2}+\frac{(1 / 2)^{3}}{3}+\ldots$
<br/>$\Rightarrow \quad-p / 2=-\left(\frac{1 / 2}{1}\right)^{1}-\frac{(1 / 2)^{2}}{3}-\frac{(1 / 2)^{3}}{3} \ldots$
<br/>$\Rightarrow \quad-P / 2=\log _{e}\left(1-\frac{1}{2}\right)$
<br/>$\Rightarrow \quad-\frac{P}{2}=\log _{\theta} \frac{1}{2}$
<br/>$\Rightarrow \quad P=2 \log _{e} 2$
<br/>Now, $Q=\left(1-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)+\ldots$
<br/>$\Rightarrow \quad Q=\log _{e} 2$
<br/>Now, from Eqs. (i) and (ii), we get
<br/>$$
<br/>P=2 Q
<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-p-1-p-2-and-p-3-p-4-are-two-focal-chords-of-the-parabola-y-2-4-a-x-then-the-chords-p-1-p-3-and-p-2-p-4-intersect-on-the

slug:
if-p-1-p-2-and-p-3-p-4-are-two-focal-chords-of-the-parabola-y-2-4-a-x-then-the-chords-p-1-p-3-and-p-2-p-4-intersect-on-the-65886

question:
<div class="question">If $P_1 P_2$ and $P_3 P_4$ are two focal chords of the parabola $y^2=4 a x$ then the chords $P_1 P_3$ and $P_2 P_4$ intersect on the</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">directrix of the parabola</span> </div>

answer:
<div class="solution">Let $P_i\left(a_i{ }^2, 2 a t_i\right), i=1,2,3,4$<br/>then, $t_1 t_2=t_3 t_4=-1$<br/>equation of $P_1 P_3$ : $\left(t_1+t_3\right) y=2 x+2 a t_1 t_3$ ...(1)<br/>$P_2 P_4:\left(t_2+t_4\right) y=2 x+2 a t_2 t_4$ ...(2)<br/>Putting $x=-a$ in equation (1) gives $y=\frac{2 a t_1 t_3-2 a}{t_1+t_3}$ Putting $x=-a$ in equation (2) gives<br/>$y=\frac{2 \mathrm{at}_2 \mathrm{t}_4-2 \mathrm{a}}{\mathrm{t}_2+\mathrm{t}_4}=\frac{\frac{2 \mathrm{a}}{\mathrm{t}_4 \mathrm{t}_3}-2 \mathrm{a}}{-\frac{1}{\mathrm{t}_1}-\frac{1}{\mathrm{t}_3}}=\frac{2 \mathrm{a}\left(1-\mathrm{t}_1 \mathrm{t}_3\right)}{-\left(\mathrm{t}_1+\mathrm{t}_3\right)}=\frac{2 \mathrm{a}\left(\mathrm{t}_1 \mathrm{t}_3-1\right)}{\left(\mathrm{t}_1+\mathrm{t}_3\right)}$<br/>$\therefore$ These lines meet at $\left(-\mathrm{a}, \frac{2 \mathrm{a}\left(\mathrm{t}_1 \mathrm{t}_3-1\right)}{\mathrm{t}_1+\mathrm{t}_3}\right)$</div>


title: if-p-2-1-1-2-3-2-4-4-3-then-p-5-is-equal-to
slug: if-p-2-1-1-2-3-2-4-4-3-then-p-5-is-equal-to-86869
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$P$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $P=\left(\begin{array}{rrr}2 &amp; -2 &amp; -4 \\ -1 &amp; 3 &amp; 4 \\ 1 &amp; -2 &amp; -3\end{array}\right),$ then $P^{5}$ is equal to</div>

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

answer:
<div class="solution">Given, $P=\left(\begin{array}{ccc}2 &amp; -2 &amp; -4 \\ -1 &amp; 3 &amp; 4 \\ 1 &amp; -2 &amp; -3\end{array}\right)$
<br/>$\begin{aligned} P^{2} &amp;=P \cdot P=\left(\begin{array}{ccc}2 &amp; -2 &amp; -4 \\ -1 &amp; 3 &amp; 4 \\ 1 &amp; -2 &amp; -3\end{array}\right)\left(\begin{array}{ccc}2 &amp; -2 &amp; -4 \\ -1 &amp; 3 &amp; 4 \\ 1 &amp; -2 &amp; -3\end{array}\right) \\ &amp;=\left(\begin{array}{ccc}4+2-4 &amp; -4-6+8 &amp; -8 &amp; -8+12 \\ -2-3+4 &amp; 2+9-8 &amp; 4+12-12 \\ 2+2-3 &amp; -2-6+6 &amp; -4-8+9\end{array}\right) \\ &amp;=\left(\begin{array}{ccc}2 &amp; -2 &amp; -4 \\ -1 &amp; 3 &amp; 4 \\ 1 &amp; -2 &amp; -3\end{array}\right)=P \\ \therefore P^{4}=P^{2}=P \\ \Rightarrow P^{5}=P^{2}=P \end{aligned}$</div>


title: if-p-2-1-sin-2-3-1-cos-2-then
slug: if-p-2-1-sin-2-3-1-cos-2-then-86972
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{P}=\frac{1}{2} \sin ^2 \theta+\frac{1}{3} \cos ^2 \theta$ then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3} \leq \mathrm{P} \leq \frac{1}{2}$</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">B</span> <span class="option-data">$\mathrm{P} \geq \frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 2 \leq \mathrm{P} \leq 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{\sqrt{13}}{6} \leq \mathrm{P} \leq \frac{\sqrt{13}}{6}$</span> </li> </ul>

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

answer:
<div class="solution">Hints: $P=\frac{1}{2} \sin ^2 \theta+\frac{1}{3} \cos ^2 \theta=\frac{1}{2} \sin ^2 \theta+\frac{1}{3}\left(1-\sin ^2 \theta\right)=\frac{1}{3}+\frac{1}{6} \sin ^2 \theta$<br/>$$<br/>\begin{aligned}<br/>&amp; 0 \leq \sin ^2 \theta \leq 1 \Rightarrow \frac{1}{3} \leq \frac{1}{3}+\frac{1}{6} \sin ^2 \theta \leq \frac{1}{3}+\frac{1}{6} \\<br/>&amp; \Rightarrow \frac{1}{3} \leq \mathrm{P} \leq \frac{1}{2}<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-p-and-p-are-the-vapour-pressure-of-the-pure-solvent-and-solution-and-n-1-and-n-2-are-the-moles-of-solute-and-solvent-respectively-in-the-solution-t

slug:
if-p-and-p-are-the-vapour-pressure-of-the-pure-solvent-and-solution-and-n-1-and-n-2-are-the-moles-of-solute-and-solvent-respectively-in-the-solution-t-14256

question:
<div class="question">If $p^{\circ}$ and $p$ are the vapour pressure of the pure solvent and solution and $n_{1}$ and $n_{2}$ are the moles of solute and solvent respectively in the solution then the correct relation between $p$ and $p^{\circ}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p=p^{\circ}\left[\frac{n_{2}}{n_{1}+n_{2}}\right]$</span> </div>

answer:
<div class="solution">Given, 
<br/>$$
<br/>\begin{array}{l}
<br/>p^{\circ}= \text {vapour pressure of pure solvent}\\
<br/>p=\text { vapour pressure of solution } \\
<br/>n_{1}=\text { moles of solute } \\
<br/>n_{2}=\text { moles of solvent }
<br/>\end{array}
<br/>$$
<br/>Thus, according to Raoult's law.
<br/>$$
<br/>\frac{p^{\circ}-p}{p^{\circ}}=x_{1}=\frac{n_{1}}{n_{1}+n_{2}}
<br/>$$
<br/>$\left(x_{1}=\right.$ mole fraction of solute)
<br/>or, $\quad 1-\frac{p}{p^{\circ}}=\frac{n_{1}}{n_{1}+n_{2}}$
<br/>$$
<br/>\frac{p}{p^{\circ}}=1-\frac{n_{1}}{n_{1}+n_{2}}
<br/>$$
<br/>$\Rightarrow$
<br/>$$
<br/>\begin{array}{c}
<br/>\frac{n_{1}+n_{2}-n_{1}}{n_{1}+n_{2}} \\
<br/>\frac{p}{p_{\circ}}=\frac{n_{2}}{n_{1}+n_{2}}
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow p=p^{\circ} \left( \frac{n_{2}}{n_{1}+n_{2}} \right)$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-p-be-a-point-on-the-parabola-y-2-4-a-x-with-focus-f-let-q-denote-the-foot-of-the-perpendicular-from-p-onto-the-directrix-then-t-a-n-pfq-t-a-n-pqf-i

slug:
if-p-be-a-point-on-the-parabola-y-2-4-a-x-with-focus-f-let-q-denote-the-foot-of-the-perpendicular-from-p-onto-the-directrix-then-t-a-n-pfq-t-a-n-pqf-i-51875

question:
<div class="question">If $P$ be a point on the parabola $y^{2}=4 a x$ with focus $F$. Let $Q$ denote the foot of the perpendicular from $P$ onto the directrix. Then, $\frac{\tan \angle P Q F}{\tan \angle P F Q}$ 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">$\frac{1}{2}$</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">$\frac{1}{4}$</span> </li> </ul>

answer:
<div class="solution">Equation of parabola is
<br/>$$
<br/>y^{2}=4 a x
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a74.png"/> <br/> <br/>Let the parametric coordinate of point $P$ on the parabola is $(a, 2 a)$
<br/>Now. $Q F=2 \sqrt{2} a$
<br/>$$
<br/>P Q=2 a \text { and } P F=2 a
<br/>$$
<br/>we observe that, $O F^{2}=P Q^{2}+P F^{2}$
<br/>$\Rightarrow \quad 8 a^{2}=4 a^{2}+4 a^{2}=8 a^{2}$
<br/>So, $\Delta Q P F$ form a right angle isoceles triangle. In which, $\angle P Q F=\angle P F Q$
<br/>$\Rightarrow$
<br/>$\tan \angle P Q F=\tan \angle P F Q$
<br/>$\Rightarrow$
<br/>$\frac{\tan \angle P Q F}{\tan \angle P F Q}=1$</div>


title: if-p-cos-4-sin-4-sin-4-cos-4-and-x-2-1-2-1-then-p-3-x-is-equal-to
slug: if-p-cos-4-sin-4-sin-4-cos-4-and-x-2-1-2-1-then-p-3-x-is-equal-to-92819
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $p=\left(\begin{array}{cc}\cos \frac{\pi}{4} &amp; -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} &amp; \cos \frac{\pi}{4}\end{array}\right)$ and $X=\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)$. Then,
<br/>$P^{3} X$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\begin{array}{l}0 \\ 1\end{array}\right)$</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">$\left(\begin{array}{r}-1 \\ 0\end{array}\right)$</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">$\left(\begin{array}{r}-1 \\ 0\end{array}\right)$</span> </div>

answer:
<div class="solution">Given, $P=\left(\begin{array}{cc}\cos \frac{\pi}{4} &amp; -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} &amp; \cos \frac{\pi}{4}\end{array}\right)=\left(\begin{array}{cc}\frac{1}{\sqrt{2}} &amp; -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}}\end{array}\right)$
<br/>$\Rightarrow \quad P=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 &amp; -1 \\ 1 &amp; 1\end{array}\right)$
<br/>$\begin{aligned} \text { Now, } P^{2} &amp;=P \cdot P=\frac{1}{2}\left(\begin{array}{cc}1 &amp; -1 \\ 1 &amp; 1\end{array}\right)\left(\begin{array}{cc}1 &amp; -1 \\ 1 &amp; 1\end{array}\right) \\ &amp;=\frac{1}{2}\left(\begin{array}{cc}1-1 &amp; -1 &amp; -1 \\ 1+1 &amp; -1+1\end{array}\right) \\ &amp;=\frac{1}{2}\left(\begin{array}{cc}0 &amp; -2 \\ 2 &amp; 0\end{array}\right)=\left(\begin{array}{cc}0 &amp; -1 \\ 1 &amp; 0\end{array}\right) \end{aligned}$
<br/>$\begin{aligned} P^{3} &amp;=P \cdot P^{2}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 &amp; -1 \\ 1 &amp; 1\end{array}\right) \cdot\left(\begin{array}{cc}0 &amp; -1 \\ 1 &amp; 0\end{array}\right) \\ &amp;=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}0-1 &amp; -1-0 \\ 0+1 &amp; -1+0\end{array}\right) \\ &amp;=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1 &amp; -1 \\ 1 &amp; -1\end{array}\right) \end{aligned}$
<br/>Also, given $X=\left(\begin{array}{c}1 / \sqrt{2} \\ \frac{1}{\sqrt{2}}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{l}1 \\ 1\end{array}\right)$
<br/>$\begin{aligned} P^{3} X &amp;=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1 &amp; -1 \\ 1 &amp; -1\end{array}\right) \cdot \frac{1}{\sqrt{2}}\left(\begin{array}{l}1 \\ 1\end{array}\right) \\ &amp;=\frac{1}{2}\left(\begin{array}{c}-1-1 \\ 1-1\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}-2 \\ 0\end{array}\right)=\left(\begin{array}{c}-1 \\ 0\end{array}\right) \end{aligned}$</div>


