# questions

title: â« 0 1 â lo g ( x 1 â â 1 ) d x is equal to
slug: 0-1-lo-g-x-1-1-d-x-is-equal-to-40417
question: <div class="question">$\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch0.db
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">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">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

answer:
<div class="solution">Let $\quad I=\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x$
<br/>$\therefore$
<br/>$=\int_{0}^{1} \log \left(\frac{1-x}{x}\right) d x$
<br/>$I=\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x=-1$
<br/>$\quad\left[\because \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right]$
<br/>$2 l=0 \Rightarrow I=0$</div>


title: â« 0 2 â [ x 2 ] dx is equal to where [] if GIF
slug: 0-2-x-2-dx-is-equal-to-where-if-gif-28449
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
question:
<div class="question">$\int_{0}^{2}\left[\mathrm{x}^{2}\right] \mathrm{dx}$ is equal to where [] if GIF</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">$5-\sqrt{2}-\sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3-\sqrt{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$8 / 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5-\sqrt{2}-\sqrt{3}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\int_{0}^{1} 0 \mathrm{dx}+\int_{1}^{\sqrt{2}} 1 \cdot \mathrm{dx}+\int_{\sqrt{2}}^{\sqrt{3}} 2 \cdot \mathrm{d} x+\int_{\sqrt{3}}^{2} 3 \mathrm{~d} x$
<br/>$=5-\sqrt{2}-\sqrt{3}$</div>


title: â« 0 100 â e x â [ x ] d x is equal to
slug: 0-100-e-x-x-d-x-is-equal-to-76873
question: <div class="question">$\int_{0}^{100} e^{x-[x]} d x$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{e^{100}-1}{100}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{e^{100}-1}{e-1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$100(e-1)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{e-1}{100}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$100(e-1)$</span> </div>

answer:
<div class="solution">Let $I=\int_{0}^{100} e^{x-[x]} d x$
<br/>$=100 \int_{0}^{1} e^{x-[x]} d x$
<br/>$[\because x-[x]$ is a periodic function of period 1 and $\int_{0}^{mT} f(x) d x=m \int_{0}^{T} f(x) d x,$ where $T$ is period of $f(x)]$ $=100 \int_{0}^{1} e^{x} d x[\because x-[x]=x$ for $0    &lt; x &lt; 1]$
<br/>$=100\left[e^{x}\right]_{0}^{1}$
<br/>$=100\left[e^{1}-e^{0}\right]$
<br/>$=100(e-1)$</div>


title: â« 0 1000 â e x â [ x ] is equal to
slug: 0-1000-e-x-x-is-equal-to-36694
question: <div class="question">$\int_0^{1000} e^{x-[x]}$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{e^{1000}-1}{e-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{e^{1000}-1}{1000}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{e-1}{1000}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1000(\mathrm{e}-1)$</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">$1000(\mathrm{e}-1)$</span> </div>

answer:
<div class="solution">Hins : $I=1000 \int_0^1 \mathrm{e}^{x-[x]}$<br/>$=1000 \int_0^1 e^x d x=1000\left(e^x\right)_0^1=100(e-1)$<br/>Period of function is 1</div>


slug: 1-1-0-3-mole-of-hcl-is-added-to-a-buffer-solution-made-up-of-0-01-m-acetic-acid-and-0-01-m-so-86890
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4.66</span> </div>
source_db: db\MarksBatch0.db
title:
1 Ã 1 0 â 3 mole of HCl is added to a buffer solution made up of 0.01 M acetic acid and 0.01 M sodium acetate. The final pH of the buffer will be (given, p K a â of acetic acid is 4.75 at 2 5 â C )

question:
<div class="question">$1 \times 10^{-3}$ mole of $\mathrm{HCl}$ is added to a buffer solution made up of $0.01 \mathrm{M}$ acetic acid and 0.01 M sodium acetate. The final pH of the buffer will be (given, $\mathrm{p} K_{a}$ of acetic acid is 4.75 at $\left.25^{\circ} \mathrm{C}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4.6</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4.66</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">4.75</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4.8</span> </li> </ul>

answer:
<div class="solution">$\mathrm{CH}_{3} \mathrm{COO}^{-}+\mathrm{H}^{+} \longrightarrow \mathrm{CH}_{3} \mathrm{COOH}$
<br/>$$
<br/>\begin{aligned}
<br/>\mathrm{pH}=&amp; \mathrm{pK}_{a}+\log \frac{(\text { salt })}{(\text { acid })}=4.75+\log \frac{0.009}{0.011} \\
<br/>&amp;=4.66
<br/>\end{aligned}
<br/>$$</div>


title: 1. 2. 3. 4. The dipeptides which may be obtained from the aminoacids glycine and alanine are
slug: 1-2-3-4-the-dipeptides-which-may-be-obtained-from-the-aminoacids-glycine-and-alanine-are-82230
tags: ['Chemistry', 'Biomolecules', 'JEE Main']
source_db: db\MarksBatch0.db
question:
<div class="question">1.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ZYDtnMm7KkZRGUmsS3RyASs50NmcBatE2j8LEOk4Ds0.original.fullsize.png"/><br/> <br/>2.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5ghW8gVQ7y8yVaXOhfXYoXj-te40axycl3Vh-SyxNHo.original.fullsize.png"/><br/> <br/>3.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XDSzhQMUqz3wM9vVlj64GtqXwga0k7cQN57nrbgnvOU.original.fullsize.png"/><br/> <br/>4.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/jXs0aSTlPS2ZujB2EwL5HkF4ozTa4T2mQBGdrZoIcB4.original.fullsize.png"/><br/> <br/>The dipeptides which may be obtained from the aminoacids glycine and alanine are</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/NHz1bNAs2Vf5ChQ3UoCvzTrN866B4-Ka6i6SDvWKxUk.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/v3w8jzEm8bjQFn4sC5xYYZ--sZZqVb406VbM3qZZ6UU.original.fullsize.png"/></div>


title: â« 1 3 â â£ x â 2â£ + â£ x â 3â£ â£ x â 1â£ â d x =
slug: 1-3-x-2-x-3-x-1-d-x-20448
question: <div class="question">$\int_{1}^{3} \frac{|x-1|}{|x-2|+|x-3|} d x=$</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+\frac{4}{3} \log _{e} 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1+\frac{3}{4} \log _{e} 3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1-\frac{4}{3} \log _{e} 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1-\frac{3}{4} \log _{e} 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1+\frac{3}{4} \log _{e} 3$</span> </div>

answer:
<div class="solution">$\int_{1}^{3} \frac{|x-1|}{|x-2|+|x-3|} d x=\int_{1}^{2} \frac{x-1}{-x+2-x+3} d x+\int_{2}^{3} \frac{x-1}{x-2-x+3} d x=\int_{1}^{2} \frac{x-1}{-2 x+5}+\int_{2}^{3} \frac{x-1}{1} d x=1+\frac{3}{4} \ln 3$</div>


title: â« 1 + cos x â d x is equal to
slug: 1-cos-x-d-x-is-equal-to-96748
question: <div class="question">$\int \sqrt{1+\cos x} d x$ is equal to</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \sqrt{2} \cos \frac{\mathrm{x}}{2}+\mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{2} \sin \frac{\mathrm{x}}{2}+\mathrm{C}$</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">$\sqrt{2} \cos \frac{x}{2}+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{2} \sin \frac{x}{2}+C$</span> </li> </ul>

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

answer:
<div class="solution">Hints: $\int \sqrt{1+\cos x} d x=\sqrt{2} \int \cos \left(\frac{x}{2}\right) d x=2 \sqrt{2} \sin \left(\frac{x}{2}\right)+c$</div>


slug: 0-1-m-hci-and-0-1-m-h-2-so-4-each-of-volume-2-ml-are-mixed-and-the-volume-is-made-up-15139
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.0</span> </div>
source_db: db\MarksBatch0.db
title:
0.1 ( M ) HCI and 0.1 ( M ) H 2 â SO 4 â each of volume 2 ml are mixed and the volume is made up to 6 ml by adding 2 ml of 0.01 ( N ) NaCl solution. The pH of the resulting mixture is

question:
<div class="question">$0.1(\mathrm{M}) \mathrm{HCI}$ and $0.1(\mathrm{M}) \mathrm{H}_2 \mathrm{SO}_4$ each of volume $2 \mathrm{ml}$ are mixed and the volume is made up to $6 \mathrm{ml}$ by adding $2 \mathrm{ml}$ of $0.01(\mathrm{~N}) \mathrm{NaCl}$ solution. The $\mathrm{pH}$ of the resulting mixture is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.17</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1.0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log 2-\log 3$</span> </li> </ul>

answer:
<div class="solution">Hints: Mili moles of $\mathrm{H}^{+}=0.1 \times 2+0.1 \times 2 \times 2=0.6$<br/>Total volume in $\mathrm{ml}=6$<br/>$$<br/>\mathrm{pH}=-\log _{10}\left[\mathrm{H}^{+}\right]=-\log \left(\frac{0.6}{6}\right)=-\log 0.1=1<br/>$$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
1 mL of water has 25 drops. Let N 0 â be the Avogadro number. What is the number of molecules present in 1 drop of water ? (Density of water = 1 g / mL )

slug:
1-ml-of-water-has-25-drops-let-n-0-be-the-avogadro-number-what-is-the-number-of-molecules-pres-99783

question:
<div class="question">$1 \mathrm{~mL}$ of water has 25 drops. Let $\mathrm{N}_0$ be the Avogadro number. What is the number of molecules present in 1 drop of water ? (Density of water $=1 \mathrm{~g} / \mathrm{mL}$ )</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{0.02}{9} \mathrm{~N}_0$</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">$\frac{18}{25} \mathrm{~N}_0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{25}{18} \mathrm{~N}_0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{0.04}{25} \mathrm{~N}_0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{0.02}{9} \mathrm{~N}_0$</span> </div>

answer:
<div class="solution">Volume of one drop $=\left(\frac{1}{25}\right) \mathrm{mL}$<br/>$\therefore$ Mass of $1 \mathrm{drop}=\mathrm{V} \times \mathrm{d}$<br/>$=\left(\frac{1}{25} \mathrm{~mL}\right)(1 \mathrm{~g} / \mathrm{mL})$<br/>$=\frac{1}{25} \mathrm{~g}$<br/>Number of moles of $\mathrm{H}_2 \mathrm{O}=\frac{\text { Massof water inonedrop }}{\text { Molarmassof water }}=\frac{\frac{1}{25}}{18}=\frac{1}{25 \times 18}$<br/>$\therefore$ Number of $\mathrm{H}_2 \mathrm{O}$ Molecule $=\frac{1}{25 \times 18} \mathrm{~N}_0=\frac{1}{50 \times 9} \mathrm{~N}_0=\frac{0.02}{9} \mathrm{~N}_0$</div>


title: 1 mole of photon, each of frequency 2500 S â 1 , would have approximately a total energy of:
slug: 1-mole-of-photon-each-of-frequency-2500-s-1-would-have-approximately-a-total-energy-of-33015
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
question:
<div class="question">1 mole of photon, each of frequency $2500 \mathrm{~S}^{-1}$, would have approximately a total energy of:</div>

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

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

answer:
<div class="solution">Total Energy $=\mathrm{Nhv}=6.022 \times 10^{23} \times 6.626 \times 10^{-34} \mathrm{~J} . \mathrm{S} . \times 2500 \mathrm{~s}^{-1}=9.9 \mathrm{erg} \approx 10 \mathrm{erg}$ In (A) option, it should be 10 erg instead of $1 \mathrm{erg}$.</div>


title: 1 + n C 1 â cos Î¸ + n C 2 â cos 2 Î¸ + â¦ + n C n â cos n Î¸ equals
slug: 1-n-c-1-cos-n-c-2-cos-2-n-c-n-cos-n-equals-93448
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
question:
<div class="question">$1+{ }^{n} C_{1} \cos \theta+{ }^{n} C_{2} \cos 2 \theta+\ldots+{ }^{n} C_{n} \cos n \theta$ equals</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(2 \cos \frac{\theta}{2}\right)^{n} \cos \frac{n \theta}{2}$</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">$2 \cos ^{2} \frac{n \theta}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \cos ^{2 n} \frac{\theta}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(2 \cos ^{2} \frac{\theta}{2}\right)^{n}$</span> </li> </ul>

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

answer:
<div class="solution">Given, $1+{ }^{n} C_{1} \cos \theta+{ }^{n} C_{2} \cos 2 \theta+\ldots+{ }^{n} C_{n} \cos n \theta$
<br/>Which is real part of complex number. $\left({ }^{n} C_{0}+{ }^{n} C_{1} e^{i\theta}+\ldots\right)$
<br/>ie. $\operatorname{Re}\left({ }^{n} \mathrm{C}_{0}+{ }^{n} \mathrm{C}_{1} e^{i\theta}+\ldots\right)$
<br/>$=\operatorname{Re}\left(1+e^{\theta i}\right)^{n}=\operatorname{Ro}(1+\cos \theta+i \sin \theta)^{n}$
<br/>$=$ Re $\left(2 \cos ^{2} n \frac{\theta}{2}+i 2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}\right)$
<br/>$\left[\begin{array}{c}\because 1+\cos \theta=2 \cos ^{2} \frac{\theta}{2} \text { and } \\ \sin \theta=2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}\end{array}\right]$
<br/>$=\left(2 \cos \frac{\theta}{2}\right)^{n} \operatorname{Re}\left(\cos \frac{\theta}{2}+i \sin \frac{\theta}{2}\right)^{n}$
<br/>$=\left(2 \cos \frac{\theta}{2}\right)^{n} \operatorname{Re}\left(\cos \frac{n \theta}{2}+i \sin \frac{\theta}{2}\right)$
<br/>[by De moivre's theorem] 
<br/>$=\left(2 \cos \frac{\theta}{2}\right)^{n} \cdot \cos \frac{n \theta}{2}$</div>


title: â« 1 â x 2 â s i n â 1 x â d x is equal to where c is an arbitrary constant
slug: 1-x-2-s-i-n-1-x-d-x-is-equal-to-where-c-is-an-arbitrary-constant-96554
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
question:
<div class="question">$\int \frac{\sin ^{-1} x}{\sqrt{1-x^2}} d x$ is equal to<br/>where $c$ is an arbitrary constant</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\log \left(\sin ^{-1} \mathrm{x}\right)+\mathrm{c}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}\left(\sin ^{-1} x\right)^2+c$</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">$\log \left(\sqrt{1-x^2}\right)+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sin \left(\cos ^{-1} \mathrm{x}\right)+\mathrm{c}$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\text { Hints : } \begin{array}{rlr} <br/>&amp; I=\int t d t &amp; \sin ^{-1} x=t \\<br/>&amp; =\frac{1}{2} t^2+c &amp; \frac{1}{\sqrt{1-x^2}} d x=d t \\<br/>&amp; =\frac{1}{2}\left(\sin ^{-1} x\right)^2+c &amp;<br/>\end{array}<br/>$$</div>


title: ( 2 3 n â 1 ) will be divisible by ( â n â N )
slug: 2-3-n-1-will-be-divisible-by-n-n-16944
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\MarksBatch0.db
question:
<div class="question">$\left(2^{3 \mathrm{n}}-1\right)$ will be divisible by $(\forall \mathrm{n} \in \mathrm{N})$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">25</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } 2^{3 \mathrm{n}}=(8)^{\mathrm{n}}=(1+7)^{\mathrm{n}}=={ }^{\mathrm{n}} \mathrm{C}_0+{ }^{\mathrm{n}} \mathrm{C}_1 7+{ }^{\mathrm{n}} \mathrm{C}_2 7^2+\ldots \ldots .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} 7^{\mathrm{n}} \\<br/>&amp; \Rightarrow 2^{3 \mathrm{n}}-1=7\left[{ }^{\mathrm{n}} \mathrm{C}_1+{ }^{\mathrm{n}} \mathrm{C}_2 7+\ldots \ldots \ldots \ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} 7^{\mathrm{n}-1}\right] \\<br/>&amp; \therefore \text { divisible by } 7<br/>\end{aligned}<br/>$$</div>


slug: 2-gm-of-metal-carbonate-is-neutralized-completely-by-100-mathrm-ml-of-0-1-mathrm-n-math-28804
tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">200</span> </div>
source_db: db\MarksBatch0.db
title:
2 gm of metal carbonate is neutralized completely by \(100 \mathrm{ml}\) of \(0.1(\mathrm{~N}) \mathrm{HCl}\). The equivalent weight of metal carbonate is

question:
<div class="question">2 gm of metal carbonate is neutralized completely by \(100 \mathrm{ml}\) of \(0.1(\mathrm{~N}) \mathrm{HCl}\). The equivalent weight of metal carbonate is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">50</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">100</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">150</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">200</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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: Number of gram equivalents of \(\mathrm{HCl}\)<br/>\(=\frac{100 \times 0.1}{1000}=0.01\)<br/>Number of gram equivalents of metal carbonate required for neutralisation must also be 0.01 . Thus, mass of 1 gram eqivalent of carbonate salt \(\frac{2}{0.01}=200 \mathrm{~g}\)<br/>\(\therefore\) Equivalent mass of carbonate salt \(=200\)</div>


slug: 2-moles-of-an-ideal-monoatomic-gas-is-carried-from-a-state-p-0-v-0-to-state-2-p-0-63198
tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch0.db
title:
2 moles of an ideal monoatomic gas is carried from a state ( p 0 â , V 0 â ) to state ( 2 p 0 â , 2 V 0 â ) along a straight line path in a p â V diagram. The amount of heat absorbed by the gas in the process is given by

question:
<div class="question">2 moles of an ideal monoatomic gas is carried from a state $\left(p_{0}, V_{0}\right)$ to state $\left(2 p_{0}, 2 V_{0}\right)$ along a straight line path in a $p-V$ diagram. The amount of heat absorbed by the gas in the process is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 p_{0} V_{0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{2} p_{0} V_{0}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$6 p_{0} V_{0}$</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">$\frac{3}{2} p_{0} V_{0}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6 p_{0} V_{0}$</span> </div>

answer:
<div class="solution">The internal energy $\Delta U=n C_{r} \Delta T$
<br/>$C_{v}=$ Specific heat of gas at constant volume
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow \quad \Delta U &amp;=n \cdot \frac{3 R}{2}\left(\frac{4 p_{0} V_{0}}{n R}-\frac{p_{0} V_{0}}{n R}\right) \\
<br/>&amp;=n \cdot \frac{3 R}{2}\left(\frac{4 p_{0} V_{0}-p_{0} V_{0}}{n R}\right) \\
<br/>&amp;=n \cdot \frac{3 R}{2} \cdot \frac{3 p_{0} V_{0}}{n R} \\
<br/>&amp;=\frac{9}{2} p_{0} V_{0}
<br/>\end{aligned}
<br/>$$
<br/>Work done by the gas $W=\left(2 p_{0}+p_{0}\right) \frac{V_{0}}{2}=\frac{3 p_{0} V_{0}}{2}$
<br/>From first law of thermodynamics,
<br/>$$
<br/>\begin{aligned}
<br/>\Delta Q &amp;=d W+d U \\
<br/>&amp;=\frac{3 p_{0} V_{0}}{2}+\frac{9}{2} p_{0} V_{0}
<br/>\end{aligned}
<br/>$$
<br/>[from Eqs. (i) and (ii)]
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{3 p_{0} V_{0}}{2}+\frac{9}{2} p_{0} V_{0} \\
<br/>=\frac{12 p_{0} V_{0}}{2}=6 p_{0} V_{0}
<br/>\end{array}
<br/>$$</div>


title: â« 2 x [ f â² ( x ) + f ( x ) lo g 2 ] d x is equal to
slug: 2-x-f-x-f-x-lo-g-2-d-x-is-equal-to-15798
question: <div class="question">$\int 2^{x}\left[f^{\prime}(x)+f(x) \log 2\right] d x$ is equal to</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{x} f^{\prime}(x)+C$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2^{x} \log 2+C$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2^{x} f(x)+C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2^{x}+C$</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int 2^{n}\left[f^{\prime}(x)+f(x) \log 2\right] d x$
<br/>$\begin{array}{ll}\text { Consider } &amp; g(x)=2^{x} f(x) \\ \Rightarrow &amp; g^{\prime}(x)=2^{x} f^{\prime}(x)+2^{x} f(x) \log 2 \\ \Rightarrow &amp; g^{\prime}(x)=2^{x}\left[f^{\prime}(x)+f(x) \log 2\right] \\ \therefore \quad &amp; \quad I=\int g^{\prime}(x) d x=g(x)+C=2^{x} f(x)+C\end{array}$</div>


slug: 2-chloro-2-phenylethane-in-toluene-racemises-slowly-in-the-presence-of-small-amount-of-sbcl-5-29084
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
(+)-2-chloro-2-phenylethane in toluene racemises slowly in the presence of small amount of SbCl 5 â , due to the formation of

question:
<div class="question">(+)-2-chloro-2-phenylethane in toluene racemises slowly in the presence of small amount of $\mathrm{SbCl}_{5},$ due to the formation of</div>

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

answer:
<div class="solution">A planar carbocation is generated when $\mathrm{SbCl}_{5}$ remove $\mathrm{Cl}^{-}$ from the substrate. This carbocation is subsequently attacked by $\mathrm{Cl}^{-}$ (nucleophile) from both the sides (j.e. from top and bottom) to produce a racemic mixture.</div>


title: 2-methylpropane on monochlorination under photochemical condition give
slug: 2-methylpropane-on-monochlorination-under-photochemical-condition-give-75313
question: <div class="question">2-methylpropane on monochlorination under photochemical condition give</div>
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2 -chloro-2-methyipropane as major product</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(1: 1)$ mixture of 1 -chioro- 2 -methylpropane and 2 -chloro-2-methylpropane</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1-chioro-2-methylpropane as a major product</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$(1: 9)$ mixture of 1 -chioro- 2 -methylpropane and 2-chloro-2-methylpropane</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1-chioro-2-methylpropane as a major product</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a16.png"/> <br/> <br/>There are nine primary hydrogens and one tertiary hydrogen in 2 -methyl propane. Tertiary hydrogen atoms react with Cl about 5.5 times as fast as primary. Even though the primary hydrogens are less reactive, there are so many of them that the primary product is the major product.</div>


slug: 3-moles-of-mono-atomic-gas-5-3-is-mixed-with-1-mole-of-a-diatomic-gas-7-3-the-va-13108
tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
3 moles of mono-atomic gas ( Î³ = 5/3 ) is mixed with 1 mole of a diatomic gas ( Î³ = 7/3 ) . The value of Î³ for the mixture will be

question:
<div class="question">3 moles of mono-atomic gas $(\gamma=5 / 3)$ is mixed with 1 mole of a diatomic gas $(\gamma=7 / 3) .$ The value of $\gamma$ for the mixture will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{9}{11}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{11}{7}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{12}{7}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{15}{7}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{11}{7}$</span> </div>

answer:
<div class="solution">The number of degrees of ireedom for the mixture is
<br/>$\therefore$$\begin{aligned} f_{\text {mix }} &amp;=\frac{n_{1} f_{1}+n_{2} f_{2}}{n_{1}+n_{2}} \\ &amp;=\frac{3 \times 3+1 \times 5}{4}=\frac{7}{2} \\ \gamma &amp;=1+\frac{2}{f} \end{aligned}$
<br/>$\Rightarrow \quad \gamma=1+\frac{4}{7}=\frac{11}{7}$</div>


slug: 3-the-sum-of-the-series-1-2-1-n-c-1-3-1-n-c-2-n-1-1-n-82128
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
title:
3. The sum â of â the â series â 1 + 2 1 â n C 1 â + 3 1 â n C 2 â + + n + 1 1 â n C n â is equal to

question:
<div class="question">$\begin{array}{llll}\text { 3. The sum } &amp; \text { of } &amp; \text { the } &amp; \text { series }\end{array}$ $1+\frac{1}{2}{ }^{n} C_{1}+\frac{1}{3}{ }^{n} C_{2}+\quad+\frac{1}{n+1}{ }^{n} C_{n}$ is equal to</div>

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

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

answer:
<div class="solution">$1+\frac{1}{2}{ }^{n} C_{1}+\frac{1}{3}{ }^{n} C_{2}+\quad+\frac{1}{n+1}{ }^{n} C_{n}$
<br/>$=\frac{1}{n+1}\left[(n+1)+\frac{(n+1) n}{21}\right.$
<br/>$+\frac{(n+1) n(n-1)}{3 !}+(1]$
<br/>$=\frac{1}{n+1}\left[^{n+1} C_{1}+{ }^{n+1} C_{2}+\quad+{ }^{n+1} C_{n+1}\right]$
<br/>\(=\frac{2^{n+1}-1}{n+1}\)</div>


title: â« 3 x l o g x â â d x is equal to
slug: 3-x-l-o-g-x-d-x-is-equal-to-60868
question: <div class="question">$\int \frac{\log \sqrt{x}}{3 x} d x$ is equal to</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}(\log \sqrt{x})^{2}+C$</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">B</span> <span class="option-data">$\frac{2}{3}(\log \sqrt{x})^{2}+C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}(\log x)^{2}+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{3}(\log x)^{2}+C$</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int \frac{\log \sqrt{x}}{3 x} d x$
<br/>Aqain, let $\log \sqrt{x}=z \Rightarrow \frac{1}{2 x} d x=d z$
<br/>$I=\int \frac{2 z}{3} d z=\frac{2}{3} \int z d z$
<br/>$=\frac{2}{3} \cdot \frac{z^{2}}{2}+C=\frac{1}{3} \left(\log \sqrt{x}\right)^{2}+C$</div>


title: â« 3 x l o g x â â dx is equal to
slug: 3-x-l-o-g-x-dx-is-equal-to-52558
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
question:
<div class="question">$\int \frac{\log \sqrt{\mathrm{x}}}{3 \mathrm{x}} \mathrm{dx}$ is equal to</div>

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

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

answer:
<div class="solution">Hints : $\mathrm{x}=\mathrm{t}^2 \Rightarrow \int \frac{\ell \mathrm{nt}}{3 \mathrm{t}^2}(2 \mathrm{tdt})=\frac{2}{3} \int \frac{\ell \mathrm{nt}}{\mathrm{t}} \mathrm{dt}=\frac{2}{3} \frac{(\ell \mathrm{nt})^2}{2}+\mathrm{c}=\frac{(\ln \sqrt{\mathrm{x}})^2}{3}+\mathrm{c}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
title:
4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is

slug:
4-boys-and-2-girls-occupy-seats-in-a-row-at-random-then-the-probability-that-the-two-girls-occupy-s-51796

question:
<div class="question">4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is</div>

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

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

answer:
<div class="solution">\(\begin{gathered} \text {Hints: } <br/>\mathrm{n}(\mathrm{e})=\lfloor 5 \cdot\lfloor 2 \\<br/>\mathrm{n}(\mathrm{s})=\lfloor 6 \\<br/>\mathrm{p}=\frac{\lfloor 5 \cdot \lfloor 2}{\lfloor 6}=\frac{2}{6}=\frac{1}{3}<br/>\end{gathered}\)</div>


title: \(4 f^2\) electronic configuration is found in
slug: 4-f-2-electronic-configuration-is-found-in-80242
question: <div class="question">\(4 f^2\) electronic configuration is found in</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{Pr}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{Pr}^{3+}\)</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">\(\mathrm{Nd}^{3+}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{Pm}^{3+}\)</span> </li> </ul>

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

answer:
<div class="solution">Hint : \(\operatorname{Pr}(59)=[X e] 4 f^3 6 s^2\)<br/>\(\therefore \mathrm{Pr}^{+3}=[\mathrm{Xe}] 4 \mathrm{f}^2\)</div>


slug: 5-m-of-0-1-m-pb-no-3-2-is-mixed-with-10-ml-of-0-02-m-ki-the-amount-of-pbl-2-precip-90337
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Main']
source_db: db\MarksBatch0.db
title:
5 m of 0.1 M Pb ( NO 3 â ) 2 â is mixed with 10 ml of 0.02 M KI . The amount of Pbl 2 â precipitated will be about

question:
<div class="question">$5 \mathrm{~m}$ of $0.1 \mathrm{M} ~ \mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}$ is mixed with $10 \mathrm{ml}$ of $0.02 \mathrm{M} ~ \mathrm{KI}$. The amount of $\mathrm{Pbl}_{2}$ precipitated will be about</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10^{-4} \mathrm{~mol}$</span> </div>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GLCRFiJeS68MwVSqK_Ce853Qlw2hl6Q1ClgaQuvplvI.original.fullsize.png"/><br/> <br/>mole of $\mathrm{Pbl}_{2} \quad \mathrm{PPt}=\frac{0.1}{1000}=1 \times 10^{-4}$</div>


title: 7 2 n + 16 n â 1 ( n â N ) is divisible by
slug: 7-2-n-16-n-1-n-n-is-divisible-by-24474
question: <div class="question">$7^{2 n}+16 n-1(n \in N)$ is divisible by</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">64</span> </div>
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">65</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">63</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">61</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">64</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">We have,
<br/>$7^{2 n}+16 n-1$
<br/>Put $n=1,$ we get
<br/>$\begin{aligned} 7^{2}+16-1 &amp; \\ =&amp; 49+15 \\ =&amp; 64 \end{aligned}$
<br/>which is divisible by 64.</div>


title: 11 â N a ^{24}$ is radioactive and it decays to
slug: 11-n-a-24-is-radioactive-and-it-decays-to-37818
question: <div class="question">${ }_{11} Na $^{24}$ is radioactive and it decays to</div>
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }_{9} \mathrm{F}^{20}$ and $\alpha$ -particles</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">${ }_{13} \mathrm{Al}^{24}$ and positron</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">${ }_{11} \mathrm{Na}^{23}$ and neutron</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">${ }_{12} \mathrm{Mg}^{24}$ and $\beta$ - particles</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">${ }_{12} \mathrm{Mg}^{24}$ and $\beta$ - particles</span> </div>

answer:
<div class="solution">$_{11} \mathrm{Na}^{24} \longrightarrow _{12} \mathrm{Mg}^{24}+_{-1}\beta^{0}$</div>


slug: 12-mathrm-mc-and-6-mathrm-mc-charges-are-given-to-the-two-conducting-plates-having-same-57929
tags: ['Physics', 'Capacitance', 'JEE Main']
source_db: db\MarksBatch0.db
title:
\(12 \mathrm{mC}\) and \(6 \mathrm{mC}\) charges are given to the two conducting plates having same cross-sectional area and placed face to face close to each other as shown in the figure. The resulting charge distribution in \(\mathrm{mC}\) on surface \(A, B, C\) and \(D\) are respectively,

question:
<div class="question">\(12 \mathrm{mC}\) and \(6 \mathrm{mC}\) charges are given to the two conducting plates having same cross-sectional area and placed face to face close to each other as shown in the figure. The resulting charge distribution in \(\mathrm{mC}\) on surface \(A, B, C\) and \(D\) are respectively,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/IbF1t8aZ_WKbL0YYJVTzTzQReb8DMPXpXolSDmxn5Dc.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(9,3,-3,9\)</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/FoqjfiIrpXtofiAiUs8939guxpFYL0osYZEIlr9EN1U.original.fullsize.png"/><br/></div>


title: 1,4 -dimethylbenzene on heating with anhydrous AlCl 3 â and HCl produces
slug: 1-4-dimethylbenzene-on-heating-with-anhydrous-alcl-3-and-hcl-produces-97558
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
question:
<div class="question">1,4 -dimethylbenzene on heating with anhydrous $\mathrm{AlCl}_{3}$ and HCl produces</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a60.png"/></div>


title: \({ }^{15} \mathrm{C}_3+{ }^{15} \mathrm{C}_5+\ldots \ldots+{ }^{15} \mathrm{C}_{15}=\)
slug: 15-mathrm-c-3-15-mathrm-c-5-ldots-ldots-15-mathrm-c-15-74422
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
question:
<div class="question">\({ }^{15} \mathrm{C}_3+{ }^{15} \mathrm{C}_5+\ldots \ldots+{ }^{15} \mathrm{C}_{15}=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2^{14}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(2^{14}-15\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2^{14}+15\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2^{14}-1\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2^{14}-15\)</span> </div>

answer:
<div class="solution">Hints: \({ }^{15} \mathrm{C}_3+{ }^{15} \mathrm{C}_5+ \ldots\ldots+{ }^{15} \mathrm{C}_{15}=2^{14}-{ }^{15} \mathrm{C}_1=2^{14}-15\)</div>


slug: 19-g-of-water-at-3-0-c-and-5-g-of-ice-at-2-0-c-are-mixed-together-in-a-calorimeter-what-19477
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
title:
19 g of water at 3 0 â C and 5 g of ice at â 2 0 â C are mixed together in a calorimeter. What is the final temperature of the mixture? Given specific heat of ice = 0.5 cal g â 1 ( â C ) â 1 and latent heat of fusion of ice = 80 cal g â 1

question:
<div class="question">$19 \mathrm{~g}$ of water at $30^{\circ} \mathrm{C}$ and $5 \mathrm{~g}$ of ice at $-20^{\circ} \mathrm{C}$ are mixed together in a calorimeter. What is the final temperature of the mixture? Given specific heat of ice $=0.5 \mathrm{cal} \mathrm{g}^{-1}\left({ }^{\circ} \mathrm{C}\right)^{-1}$ and latent heat of fusion of ice $=80 \mathrm{cal} \mathrm{g}^{-1}$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">$5 \times .5 \times 20+5 \times 80+5 t=19 \times 1 \times(30-t)$<br/>$\mathrm{t}=5^{\circ} \mathrm{C}$</div>


slug: 20-ml-0-1-n-acetic-acid-is-mixed-with-10-ml-0-1-n-solution-of-naoh-the-ph-of-the-resulting-so-50702
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4.74</span> </div>
source_db: db\MarksBatch0.db
title:
20 mL 0.1 (N) acetic acid is mixed with 10 mL 0.1 ( N ) solution of NaOH. The pH of the resulting solution is ( p K a â of acetic acid is 4.74 )

question:
<div class="question">$20 \mathrm{mL~} 0 .1$ (N) acetic acid is mixed with $10 \mathrm{mL}$ $0.1(\mathrm{N})$ solution of NaOH. The pH of the resulting solution is $\left(p K_{a}\right.$ of acetic acid is $\left.4.74\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3.74</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4.74</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">5.74</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6.74</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a31.png"/> <br/> <br/>From Henderson's equation,
<br/>$$
<br/>\begin{aligned}
<br/>p H &amp;=p K_{a}+\log \frac{\left.\mid C H_{j} C O O N a\right]}{\left|C H_{3} C O O H\right|} \\
<br/>&amp;=4.74+\log \frac{1}{1}=4.74
<br/>\end{aligned}
<br/>$$</div>


slug: 2-5-ml-of-0-4-m-weak-monoacidic-base-k-b-1-1-0-12-at-2-5-c-is-titrated-with-22463
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
2.5 ml of 0.4 ( M ) weak monoacidic base ( k b â = 1 Ã 1 0 â 12 at 2 5 â C ) is titrated with 15 2 â ( M ) HCl in water at 2 5 â C . The concentration of H + at equivalence point is ( k w â = 1 Ã 1 0 â 14 , at 2 5 â C ) ,

question:
<div class="question">$2.5 \mathrm{ml}$ of $0.4(\mathrm{M})$ weak monoacidic base $\left(\mathrm{k}_{\mathrm{b}}=1 \times 10^{-12}\right.$ at $\left.25^{\circ} \mathrm{C}\right)$ is titrated with $\frac{2}{15}(\mathrm{M}) \mathrm{HCl}$ in water at $25^{\circ} \mathrm{C}$. The concentration of $\mathrm{H}^{+}$at equivalence point is $\left(\mathrm{k}_{\mathrm{w}}=1 \times 10^{-14}\right.$, at $\left.25^{\circ} \mathrm{C}\right)$,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3.7 \times 10^{-13}$ (M)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.2 \times 10^{-7}(\mathrm{M})$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3.2 \times 10^{-2}(\mathrm{M})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.7 \times 10^{-2}(\mathrm{M})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.2 \times 10^{-2}(\mathrm{M})$</span> </div>

answer:
<div class="solution">$\mathrm{N}_{1} \mathrm{~V}_{1}=\mathrm{N}_{2} \mathrm{~V}_{2}$ (At equivalence point)
<br/>$0.4 \times 2.5=\frac{2}{15} \times \mathrm{V}_{2}$ 
<br/>$\mathrm{BOH}(\mathrm{aq})+\mathrm{HCl}(\mathrm{aq}) \rightarrow \mathrm{B} \mathrm{Cl}^{-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(I)$
<br/>$\therefore \mathrm{V}_{2}=\frac{0.4 \times 2.5}{\frac{2}{15}}=\frac{0.4 \times 2.5}{2} \times 15=7.5 \mathrm{ml}$
<br/>$\therefore$ Concentration of salt $=\frac{0.4 \times 2.5}{2.5+7.5}$
<br/>or, Salt of weak base and strong acid = $0.1$
<br/>$\therefore\left[\mathrm{H}^{+}\right]=\mathrm{Ch}=\sqrt{\frac{\mathrm{K}_{\mathrm{w}} \mathrm{C}}{\mathrm{K}_{\mathrm{b}}}}=\sqrt{\frac{10^{-14} \times 0.1}{10^{-12}}}=\sqrt{10^{-3}}=3.2 \times 10^{-2} \mathrm{M}$
<br/>as $C h^{2}=\frac{K_{w}}{K_{b}} \quad \therefore h=\sqrt{\frac{K_{w}}{K_{b} C}}$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
27 drops of mercury coalesce to form a bigger drop. What is the relative increase in surface energy?

slug:
27-drops-of-mercury-coalesce-to-form-a-bigger-drop-what-is-the-relative-increase-in-surface-energy-54502

question:
<div class="question">27 drops of mercury coalesce to form a bigger drop. What is the relative increase in surface energy?</div>

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

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

answer:
<div class="solution">Let the initial radius $r$ and finial radius $R . \frac{4}{3} \pi R^3=27 \times \frac{4}{3} \pi r^3 \Rightarrow R=3 r$<br/>$$<br/>U_i=\left(S 4 \pi r^2\right) \times 27 \quad U_f=S 4 \pi 9 r^2<br/>$$<br/>Relative increacase $=\left(\frac{U_f-U_i}{U_i}\right)=\frac{9-27}{27}=-\frac{18}{27}=-\frac{2}{3}$</div>


title: ( 32 â Ge 76 , 34 â Se 76 ) and ( 14 â Si 30 , 16 â S 32 ) are examples of
slug: 32-ge-76-34-se-76-and-14-si-30-16-s-32-are-examples-of-26861
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
question:
<div class="question">$\left({ }_{32} \mathrm{Ge}^{76},{ }_{34} \mathrm{Se}^{76}\right) \text { and } \left.{(}_{14} \mathrm{Si}^{30},{ }_{16} \mathrm{S}^{32}\right) \text { are examples of }$</div>

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

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

answer:
<div class="solution">${ }_{32} \mathrm{Ge}^{76}$ and $\mathrm{sq} \mathrm{Se}^{76}$ have the same mass number but different atomic number, so they are isobars. Number of neutrons,
<br/>in  $_{14} e^{\mathrm{Si}^{30}}=30-14=16$
<br/>in $_{16} e^{\mathrm{Si}^{32}}=32-16=16$
<br/>Because of the presence of same number of neutrons, $_{14} \mathrm{Si}^{30}$ and ${ }_{16} \mathrm{S}^{32}$ are called isotones.</div>


slug: 63-mathrm-g-of-a-compound-mol-wt-126-was-dissolved-in-500-mathrm-g-distilled-26604
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
\(63 \mathrm{~g}\) of a compound (Mol. Wt. \(=126)\) was dissolved in \(500 \mathrm{~g}\) distilled water. The density of the resultant solution as \(1.126 \mathrm{~g} / \mathrm{ml}\). The molarity of the solution is

question:
<div class="question">\(63 \mathrm{~g}\) of a compound (Mol. Wt. \(=126)\) was dissolved in \(500 \mathrm{~g}\) distilled water. The density of the resultant solution as \(1.126 \mathrm{~g} / \mathrm{ml}\). The molarity of the solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(1.25 \mathrm{~M}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(1.0 \mathrm{~M}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(0.75 \mathrm{~M}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(1.1 \mathrm{~M}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(1.0 \mathrm{~M}\)</span> </div>

answer:
<div class="solution">Hint : Mass of compound (solute) \(=63 \mathrm{~g}\)<br/>Mole of compound \(=\frac{63}{126}=\frac{1}{2}\) mole<br/>\(\begin{aligned}<br/>\text{Mass of solution}&amp; =\text { Mass of solute }+ \text { Mass of solvent } \\<br/>&amp; =63+500 \\<br/>&amp; =563 \mathrm{~g}<br/>\end{aligned}\)<br/>Volume of solution \(=\frac{\text { Mass }}{\text { Density }}=\frac{563}{1.126} \mathrm{ml}\)<br/>Molarity \(=\frac{\text { mole of compound }}{\text { volume of solution }(\text { inL })}\)<br/>\(\begin{aligned} &amp; =\frac{1 / 2 \times 1000}{563 / 1.126} \\ &amp; =\frac{1.126 \times 1000}{2 \times 563}=1\end{aligned}\)</div>


slug: 98-mathrm-cf-246-was-formed-along-with-a-neutron-when-an-unknown-radioactive-substance-11528
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
title:
\({ }_{98} \mathrm{Cf}^{246}\) was formed along with a neutron when an unknown radioactive substance was bombarded with \({ }_{6} \mathrm{C}^{12}\). The unknown substance was :

question:
<div class="question">\({ }_{98} \mathrm{Cf}^{246}\) was formed along with a neutron when an unknown radioactive substance was bombarded with \({ }_{6} \mathrm{C}^{12}\). The unknown substance was :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\({ }_{91} \mathrm{~Pa}^{234}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\({ }_{90} \mathrm{Th}^{234}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\({ }_{92} \mathrm{U}^{235}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\({ }_{92} \mathrm{U}^{238}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\({ }_{92} \mathrm{U}^{235}\)</span> </div>

answer:
<div class="solution">Let, the unknown radioactive substance be ${ }_{x} A^{y}$.
<br/>$x A^{y}+{ }_{6} C^{12} \longrightarrow{ }_{98} C f^{246}+{ }_{0} n^{1}$
<br/>On comparing atomic numbers, we get
<br/>$$
<br/>\begin{array}{r}
<br/>x+6=98+0 \\
<br/>x=98-6=92
<br/>\end{array}
<br/>$$
<br/>On comparing mass numbers, we get
<br/>$$
<br/>y+12=246+1
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>y &amp;=247-12 \\
<br/>&amp;=235
<br/>\end{aligned}
<br/>$$
<br/>Since, uranium has the atomic number 92 , thus, the given unknown substance was ${ }_{92} \mathrm{U}^{235}$.</div>


tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">63</span> </div>
source_db: db\MarksBatch0.db
title:
0.126 g of an acid is needed to completely neutralise 20 mL 0.1 (N) NaOH solution. The equivalent weight of the acid is

slug:
0-126-g-of-an-acid-is-needed-to-completely-neutralise-20-ml-0-1-n-naoh-solution-the-equivalent-we-65444

question:
<div class="question">$0.126 \mathrm{g}$ of an acid is needed to completely neutralise $20 \mathrm{mL} 0.1$ (N) NaOH solution. The equivalent weight of the acid is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">53</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">40</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">45</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">63</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\because$ Number of milli equivalents of NaOH<br/>$\left(n^{\prime}\right)=N V=0.1 \times 20=2$<br/>$\therefore$ Number of equivalent $=0.002$<br/>For Neutralisation Number of equivalent of base (NaOH) Number of equivalent of acid<br/>$0.002=\frac{W}{E}=\frac{0.126}{E}$<br/>$\therefore \quad E=\frac{0.126}{0.002}=63$</div>


slug: 1-56-times-10-5-mathrm-j-of-heat-is-conducted-through-a-2-mathrm-m-2-wall-of-12-ma-35039
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.13</span> </div>
source_db: db\MarksBatch0.db
title:
\(1.56 \times 10^5 \mathrm{~J}\) of heat is conducted through a \(2 \mathrm{~m}^2\) wall of \(12 \mathrm{~cm}\) thick in one hour. Temperature difference between the two sides of the wall is \(20^{\circ} \mathrm{C}\). The thermal conductivity of the material of the wall is (in \(\mathrm{W} \mathrm{m}^{-1} \mathrm{~K}^{-1}\) )

question:
<div class="question">\(1.56 \times 10^5 \mathrm{~J}\) of heat is conducted through a \(2 \mathrm{~m}^2\) wall of \(12 \mathrm{~cm}\) thick in one hour. Temperature difference between the two sides of the wall is \(20^{\circ} \mathrm{C}\). The thermal conductivity of the material of the wall is (in \(\mathrm{W} \mathrm{m}^{-1} \mathrm{~K}^{-1}\) )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.11</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0.13</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.15</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.2</span> </li> </ul>

answer:
<div class="solution">Hints: \(\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{KA} \Delta \mathrm{T}}{\mathrm{x}}\)<br/>\(\begin{aligned}<br/>&amp; \frac{1.56 \times 10^5}{3600}=\frac{\mathrm{K} \times 2 \times 20}{12 \times 10^{-2}} \\<br/>&amp; \mathrm{~K}=\frac{1.56 \times 10^5 \times 12 \times 10^{-2}}{3600 \times 2 \times 20} \\<br/>&amp; =\frac{1.56}{12}=0.13<br/>\end{aligned}\)</div>


slug: 300-gm-of-water-at-2-5-c-is-added-to-100-gm-of-ice-at-0-c-the-final-temperature-of-the-mix-12111
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
300 gm of water at 2 5 â C is added to 100 gm of ice at 0 â C . The final temperature of the mixture is

question:
<div class="question">$300 \mathrm{gm}$ of water at $25^{\circ} \mathrm{C}$ is added to $100 \mathrm{gm}$ of ice at $0^{\circ} \mathrm{C}$. The final temperature of the mixture is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$12.5^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0^{\circ} \mathrm{C}$</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">$25^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$50^{\circ} \mathrm{C}$</span> </li> </ul>

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

answer:
<div class="solution">Ref. at $0^{\circ} \mathrm{C}$ water
<br/>$(\Delta \mathrm{H})$ liberated $=300 \times 1 \times 25=7500 \mathrm{cal}$
<br/>$(\Delta \mathrm{H})$ required $=\mathrm{m}_{\text {ice }} \mathrm{L}_{\mathrm{f}}$
<br/>$=100 \times 80 \mathrm{cal}=8000 \mathrm{cal}$
<br/>as $(\Delta \mathrm{H})_{\text {req. }}&gt;(\Delta \mathrm{H})_{\text {iib }}$ the mixture remains at $0^{\circ} \mathrm{C}$
<br/>$\therefore$ Final temperature of mixture will be $0^{\circ} \mathrm{C}$.</div>


tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
5.75 mg of sodium vapour is converted to sodium ion. If the ionisation energy of sodium is 490 kJ mol â 1 and atomic weight is 23 units, the amount of energy needed for this conversion will be

slug:
5-75-mg-of-sodium-vapour-is-converted-to-sodium-ion-if-the-ionisation-energy-of-sodium-is-490-kj-mo-77009

question:
<div class="question">$5.75 \mathrm{mg}$ of sodium vapour is converted to sodium ion. If the ionisation energy of sodium is $490 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and atomic weight is 23 units, the amount of energy needed for this conversion will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.96 \mathrm{~kJ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1960 \mathrm{~kJ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$122.5 \mathrm{~kJ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.1225 \mathrm{~kJ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.1225 \mathrm{~kJ}$</span> </div>

answer:
<div class="solution">$\mathrm{Na}(\mathrm{g}) \longrightarrow \mathrm{Na}^{+}(\mathrm{g})+\mathrm{e}^{-}$
<br/>$1 \mathrm{~mol} \mathrm{Na}(\mathrm{g})$
<br/>requires $490 \mathrm{~kJ} /$ mol for ionization.
<br/>$\therefore \quad \frac{5.75 \times 10^{-3}}{23}$ mol $\quad$ requires $490 \mathrm{~kJ} \times \frac{5.75 \times 10^{-3}}{23}=0.1225 \mathrm{~kJ}=122.5 \mathrm{~J}$</div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
title:
58.4 g of NaCl and 180 g of glucose were separately dissolved in 1000 mL of water. Identify the correct statement regarding the elevation of boiling point (b.p) of the resulting solutions.

slug:
58-4-g-of-nacl-and-180-g-of-glucose-were-separately-dissolved-in-1000-ml-of-water-identify-the-corr-33914

question:
<div class="question">$58.4 \mathrm{g}$ of $\mathrm{NaCl}$ and $180 \mathrm{g}$ of glucose were separately dissolved in $1000 \mathrm{mL}$ of water. Identify the correct statement regarding the elevation of boiling point (b.p) of the resulting solutions.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">NaCl solution will show higher elevation of boiling point</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Glucose solution will show higher elevation of boiling point</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both the solutions will show equal elevation of boiling point</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The boiling point elevation will be shown by neither of the solutions</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">NaCl solution will show higher elevation of boiling point</span> </div>

answer:
<div class="solution">Elevation in boiling point, $\Delta T_{b}=i \times K_{b} \times m$ 
<br/>Molality of NaCl solution $=\frac{n}{W} \times 1000$
<br/>$=\frac{58.5}{W_{\mathrm{H}_{2} \mathrm{O}}} \times 1000=\frac{1000}{\mathrm{W}_{\mathrm{M}_{2}} \mathrm{O}}$
<br/>Molality of $f \mathrm{C}_{6} \mathrm{H}_{1} \not \mathrm{O}_{6}$ solution = $\frac{\frac{180}{180} \times 1000}{w_{\mathrm{H}_{2}} \mathrm{O}}$
<br/>$\frac{1000}{W_{M_{2}} O}$. Both the solutions have same molarity but the values for $\mathrm{NaCl}$ and glucose are ${2}$ and 1respectively. $\left.\therefore \quad \Delta T_{b(\mathrm{NaG})}=2 \times \Delta T_{b C C_{6} H_{1} 2} \mathrm{O}_{6}\right)$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
1000 droplets of water having 2 mm diameter each coalesce to form a single drop. Given the surface tension of water is 0.072 Nm â 1 . The energy loss in the process is

slug:
1000-droplets-of-water-having-2-mm-diameter-each-coalesce-to-form-a-single-drop-given-the-surface-t-34434

question:
<div class="question">1000 droplets of water having $2 \mathrm{mm}$ diameter each coalesce to form a single drop. Given the surface tension of water is $0.072 \mathrm{Nm}^{-1}$. The energy loss in the process is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$8.146 \times 10^{-4} \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4.4 \times 10^{-4} \mathrm{J}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2108 \times 10^{-5} \mathrm{J}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4.7 \times 10^{-1} \mathrm{J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8.146 \times 10^{-4} \mathrm{J}$</span> </div>

answer:
<div class="solution">Let the radius of single drop = $r$
<br/>Radius of small drop $=\frac{2 m m}{2}$
<br/>$$
<br/>\begin{array}{l}
<br/>=1 \mathrm{mm} \\
<br/>=1 \times 10^{-3} \mathrm{m}
<br/>\end{array}
<br/>$$
<br/>Surface tension of water $=0.072 \mathrm{N} / \mathrm{m}$
<br/>The volume of large drop must be equal volume of all small drops.
<br/>$$
<br/>\frac{4}{3} \pi R^{3}=1000\left(\frac{4}{3} \pi r^{3}\right)
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>A=(1000)^{\frac{1}{3}} \cdot r=10 r
<br/>$$
<br/>The initial potential energy of drops system $U_{i}=s \times n \times 4 \pi r^{2}$
<br/>$$
<br/>\begin{array}{l}
<br/>=s \times 1000 \times 4 \pi^{2} \\
<br/>=1000 \mathrm{s} \cdot 4 \pi^{2}
<br/>\end{array}
<br/>$$
<br/>When the droplets coalesce, the final potential energy of the system $U_{1}=s \cdot 4 \pi R^{2}$
<br/>$$
<br/>=s \cdot 4 \pi \times 100 r^{2}
<br/>$$
<br/>Energy loss in the formation of large drop
<br/>$$
<br/>\begin{aligned}
<br/>\Delta E &amp;=U_{t}-U_{i} \\
<br/>&amp;=s \cdot 4 \pi 100 r^{2}-1000 \mathrm{s} \cdot 4 \pi r^{2} \\
<br/>&amp;=s \cdot \pi r^{2}(400-4000) \\
<br/>&amp;=-3600 \mathrm{s} \cdot \pi^{2} \\
<br/>&amp;=-3600 \times 0.72 \times 3.14 \times\left(1 \times 10^{-3}\right)^{2}
<br/>\end{aligned}
<br/>$$
<br/>From catenation
<br/>$$
<br/>\begin{array}{l}
<br/>=813.888 \times 10^{-6} \\
<br/>=8.1388 \times 10^{-4} \mathrm{J} \\
<br/>=8.146 \times 10^{-4} \mathrm{J}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
title:
22320 cal of heat is supplied to 100 g of ice at 0 â C . If the latent heat of fusion of ice is 80 cal g â 1 and latent heat of vaporization of water is 540 cal g â 1 , the final amount of water thus obtained and its temperature respectively are

slug:
22320-cal-of-heat-is-supplied-to-100-g-of-ice-at-0-c-if-the-latent-heat-of-fusion-of-ice-is-80-84242

question:
<div class="question">22320 cal of heat is supplied to 100 g of ice at $0^{\circ} \mathrm{C}$. If the latent heat of fusion of ice is 80 cal $\mathrm{g}^{-1}$ and latent heat of vaporization of water is 540 cal $\mathrm{g}^{-1},$ the final amount of water thus obtained and its temperature respectively
<br/>are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$8 \mathrm{g}, 100^{\circ} \mathrm{C}$</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">B</span> <span class="option-data">$100 \mathrm{g} ,90^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$92 \mathrm{g}, 100^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$82 \mathrm{g}, 100^{\circ} \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 \mathrm{g}, 100^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">Heat required to convert ice to water at $100^{\circ} \mathrm{C}$
<br/>$$
<br/>Q=m \times L+m s \Delta T=18000 \mathrm{cal}
<br/>$$
<br/>Amount of heat left $=4320 \mathrm{cal}$
<br/>$$
<br/>\begin{array}{l}
<br/>m \times L=4320 \\
<br/>m=8 g \text { steam }
<br/>\end{array}
<br/>$$</div>


slug: a-1-2-3-4-b-1-2-3-4-5-6-are-two-sets-and-function-f-a-b-is-defi-74594
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
title:
A = { 1 , 2 , 3 , 4 } , B = { 1 , 2 , 3 , 4 , 5 , 6 } are two sets, and function f : A â B is defined by f ( x ) = x + 2â x â A , then the function f is

question:
<div class="question">$\mathrm{A}=\{1,2,3,4\}, \mathrm{B}=\{1,2,3,4,5,6\}$ are two sets, and function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}+2 \forall \mathrm{x} \in \mathrm{A}$, then the function $\mathrm{f}$ is</div>

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

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

answer:
<div class="solution">Hints: $\mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{y}) \Rightarrow \mathrm{x}+2=\mathrm{y}+2 \Rightarrow \mathrm{x}=\mathrm{y} \quad \therefore$ one-one</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
A 1 Î¼ F capacitor C is connected to a battery of 10 V through a resistance 1 Mega Ohm. The voltage across C after 1 s is approximately

slug:
a-1-f-capacitor-c-is-connected-to-a-battery-of-10-v-through-a-resistance-1-mega-ohm-the-voltage-94001

question:
<div class="question">A $1 \mu \mathrm{F}$ capacitor $C$ is connected to a battery of $10 \mathrm{V}$ through a resistance 1 Mega Ohm. The voltage across $C$ after 1 s is approximately</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5.6 \mathrm{V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$7.8 \mathrm{V}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$6.3 \mathrm{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10 \mathrm{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6.3 \mathrm{V}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a28.png"/> <br/> <br/>The time constant of the circuit
<br/>Here,
<br/>$$
<br/>\begin{aligned}
<br/>\tau &amp;=C \cdot R \\
<br/>C &amp;=1 \mu F=1 \times 10^{-6}
<br/>\end{aligned}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>R=1 \mathrm{M\Omega}=1 \times 10^{6} \Omega \\
<br/>\tau=1 \times 10^{-6} \times 1 \times 10^{6}=1 \mathrm{s}
<br/>\end{array}
<br/>$$
<br/>We know that in one (1) time constant $63 \%$ charging is done. $\therefore \quad \frac{63}{100} \times q_{\max }$
<br/>$=\frac{63}{100} \times 1 \times 10^{-6} \times 10$
<br/>$=\frac{63}{100} \times 1 \mu \mathrm{F} \times 10=6.3 \mu \mathrm{C}$
<br/>$V=\frac{\mathrm{q}}{\mathrm{C}}=\frac{6.3 \times 10^{-6} \mathrm{C}}{1 \times 10^{-6} \mathrm{F}}=6.3 \mathrm{V}$</div>


slug: a-5-f-capacitor-is-connected-in-series-with-10-f-capacitor-when-a-300-v-potentid-dillerence-i-74263
tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A 5 Î¼ F capacitor is connected in series with 10 Î¼ F capacitor. When a 300 v potentid dillerence is applied across this combination, the total energy stored in the capacitors is

question:
<div class="question">A $5 \mu F$ capacitor is connected in series with $10 \mathrm{\mu F}$ capacitor. When a $300 \mathrm{v}$ potentid dillerence is applied across this combination, the total energy stored in the capacitors is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$18 \mathrm{J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.5 \mathrm{J}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.15 \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.10 \mathrm{J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.15 \mathrm{J}$</span> </div>

answer:
<div class="solution">According to question the figure can be drawn as below
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a30.png"/> <br/> <br/>The equivalent capacitance,
<br/>$$
<br/>\begin{aligned}
<br/>\frac{1}{C_{e q}} &amp;=\frac{1}{C_{1}}+\frac{1}{C_{2}}=\frac{1}{5}+\frac{1}{10} \\
<br/>&amp;=\frac{2+1}{10}=\frac{3}{10} \\
<br/>\Rightarrow \quad C_{e q} &amp;=\frac{10}{3} \mu \mathrm{F}
<br/>\end{aligned}
<br/>$$
<br/>Now, the energy stored in the capacitor is
<br/>$$
<br/>\begin{aligned}
<br/>U &amp;=\frac{1}{2} C V^{2} \\
<br/>&amp;=\frac{1}{2} \times \frac{10}{3} \times 10^{-6} \times 300 \times 300 \\
<br/>&amp;=\frac{3}{10 \times 2}=\frac{3}{20}=0.15 \mathrm{J}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
title:
A 10 W electric heater is used to heat a container filled with 0.5 kg of water. It is found that the temperature of water and the container rises by 3 â K in 15 min . The container is then emptied, dried and filled with 2 kg of oil. The same heater now raises the temperature of container-oil system by 2 â K in 20 min . Assuming that there is no heat loss in the process and the specific heat of water is 4200 d kg â 1 K â 1 , the specific heat of oil in the same unit is equal to

slug:
a-10-w-electric-heater-is-used-to-heat-a-container-filled-with-0-5-kg-of-water-it-is-found-that-the-78909

question:
<div class="question">$A$ 10 W electric heater is used to heat a container filled with $0.5 \mathrm{kg}$ of water. It is found that the temperature of water and the container rises by $3^{\circ} \mathrm{K}$ in $15 \mathrm{min}$. The container is then emptied, dried and filled with $2 \mathrm{kg}$ of oil. The same heater now raises the temperature of container-oil system by $2^{\circ} \mathrm{K}$ in $20 \mathrm{min}$. Assuming that there is no heat loss in the process and the specific heat of water is $4200 \mathrm{d} \mathrm{kg}^{-1} \mathrm{K}^{-1}$, the specific heat of oil in the same unit is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.50 \times 10^{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2.55 \times 10^{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.00 \times 10^{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5.10 \times 10^{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.55 \times 10^{3}$</span> </div>

answer:
<div class="solution">$m_{1} s_{1} \Delta t+m_{2} s_{2} \Delta t=$ Work done 
<br/> <br/>$m_{1} s_{1} \Delta t+m_{2} s_{2} \Delta t=P_{1} t_{1}$
<br/>where $m_{1}=0.5 \mathrm{kg}$
<br/>Specific heat $s_{1}=4200 \mathrm{J} / \mathrm{kg}-\mathrm{K}$
<br/>$$
<br/>\begin{array}{l}
<br/>\Delta t=\Delta t_{1}=\Delta t_{2}=3 \mathrm{K} \\
<br/>P_{1}=P_{2}=10 \mathrm{W} \\
<br/>t_{1}=15 \times 60=900 \mathrm{s}
<br/>\end{array}
<br/>$$
<br/>$s_{2}=$ Specific heat capacity of container
<br/>$$
<br/>0.5 \times 4200 \times(3-0)+m_{2} \mathrm{s}_{2} \times(3-0)
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=10 \times 15 \times 60 \\
<br/>2100 \times 3+m_{2} s_{2} \times 3 &amp;=9000 \\
<br/>m_{2} s_{2} &amp;=\frac{9000-6300}{3} \\
<br/>m_{2} s_{2} &amp;=900
<br/>\end{aligned}
<br/>$$
<br/>Similarly, in case of oil
<br/>$$
<br/>m_{1} s_{0} \Delta t+m_{2} s_{2} \Delta t=P_{2} t_{2}
<br/>$$
<br/>where $s_{0}=$ specific heat capacity of oil
<br/>$$
<br/>\begin{aligned}
<br/>P_{1} &amp;=P_{2}=10 \mathrm{W} \\
<br/>2 \times s_{0} \times 2+900 \times 2 &amp;=10 \times 20 \times 60 \\
<br/>4 s_{0}+1800 &amp;=12000 \\
<br/>4 s_{0} &amp;=12000-1800 \\
<br/>s_{0} &amp;=\frac{10200}{4}=2550 \\
<br/>&amp;=2.55 \times 10^{3} \mathrm{J} \mathrm{kg}^{-1} \mathrm{k}^{-1}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A 20 cm long capillary tube is dipped vertically in water and the liquid rises upto 10 cm . If the entire system is kept is a freely falling platform, the length of the water column in the tube will be

slug:
a-20-cm-long-capillary-tube-is-dipped-vertically-in-water-and-the-liquid-rises-upto-10-cm-if-the-e-93936

question:
<div class="question">A $20 \mathrm{cm}$ long capillary tube is dipped vertically in water and the liquid rises upto $10 \mathrm{cm} .$ If the entire system is kept is a freely falling platform, the length of the water column in the tube will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15 \mathrm{cm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$20 \mathrm{cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$20 \mathrm{cm}$</span> </div>

answer:
<div class="solution">The height raised by liquid in capillary tube
<br/>$$
<br/>h=\frac{2 I \cos \theta}{\rho g h}
<br/>$$
<br/>As in freely falling platform, a body experience weightlessness.
<br/> <br/>So, the liquid will rise upto to length of the capillary.
<br/>i.e. height raised by the liquid will be $20 \mathrm{cm}$.</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
title:
A 2.0 cm object is placed 15 cm in front of a concave mirror of focal length 10 cm . What is the size and nature of the image?

slug:
a-2-0-cm-object-is-placed-15-cm-in-front-of-a-concave-mirror-of-focal-length-10-cm-what-is-the-siz-67549

question:
<div class="question">A $2.0 \mathrm{~cm}$ object is placed $15 \mathrm{~cm}$ in front of a concave mirror of focal length $10 \mathrm{~cm}$. What is the size and nature of the image?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$4 \mathrm{~cm}$. real</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \mathrm{~cm}$, virtual</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.0 \mathrm{~cm}$, real</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \mathrm{~cm}$. real</span> </div>

answer:
<div class="solution">$\frac{1}{\mathrm{v}}-\frac{1}{15}=\frac{1}{-10} \Rightarrow \mathrm{v}=-30 \mathrm{~cm}$<br/>$m=\frac{-30}{-15}=2$, image size $=4 \mathrm{~cm}$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch0.db
title:
A 100 mL, 0.1 = M solution of ammonium acetate is diluted by adding 100 mL of water. The pH of the resulting solution will be ( p K a â of acetic acid is nearly equal to pK b â of NH 4 â OH )

slug:
a-100-ml-0-1-m-solution-of-ammonium-acetate-is-diluted-by-adding-100-ml-of-water-the-ph-of-the-r-20028

question:
<div class="question">A $100$ mL, $0.1=\mathrm{M}$ solution of ammonium acetate is diluted by adding $100 \mathrm{mL}$ of water. The pH of the resulting solution will be $\left(p K_{a}\right.$ of acetic acid is nearly equal to $\mathrm{pK}_{b}$ of $\mathrm{NH}_{4} \mathrm{OH}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4.9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">Ammonium acetate is a salt of weak acid and weak base. When solutions of such salts diluted resulting pH of solution is calculated from following relationship
<br/>$$
<br/>\mathrm{pH}=7+\frac{1}{2} \mathrm{p} \mathrm{K}_{a}-\frac{1}{2} \mathrm{p} K_{\mathrm{b}}
<br/>$$
<br/>$\because p K_{a}=p K_{b}$
<br/>$\therefore \quad \mathrm{pH}$ of diluted solution $=7$</div>


tags: ['Physics', 'Atomic Physics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch0.db
title:
A 12.5 ev electron beam is used to bombard gaseous hydrogen at ground state. The energy level upto which the hydrogen atoms would be excited is

slug:
a-12-5-ev-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-ground-state-the-energy-level-upto-w-39136

question:
<div class="question">A $12.5$ ev electron beam is used to bombard gaseous hydrogen at ground state. The energy level upto which the hydrogen atoms would be excited 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">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">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$12.5     &lt; 12.75$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/QqVM8xlAaRejHB6J11oWTZ1jwqbLjbzU2hDr_Ypp5Pg.original.fullsize.png"/><br/> <br/>$\therefore$ 3rd state</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch0.db
title:
A 12.5 ev electron beam is used to bombard gaseous hydrogen at ground state. The energy level upto which the hydrogen atoms would be excited is

slug:
a-12-5-ev-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-ground-state-the-energy-level-upto-w-11311

question:
<div class="question">A $12.5$ ev electron beam is used to bombard gaseous hydrogen at ground state. The energy level upto which the hydrogen atoms would be excited 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">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">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$12.5     &lt; 12.75$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/QqVM8xlAaRejHB6J11oWTZ1jwqbLjbzU2hDr_Ypp5Pg.original.fullsize.png"/><br/> <br/>$\therefore$ 3rd state</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A 400Î© resistor, a 250 mH inductor and a 2.5 Î¼ F capacitor are connected in series with an AC source of peak voltage 5 V and angular frequency 2 rad/sec . What is the peak value of the electrostatic energy of the capacitor?

slug:
a-400-resistor-a-250-mh-inductor-and-a-2-5-f-capacitor-are-connected-in-series-with-an-ac-sour-53318

question:
<div class="question">A $400 \Omega$ resistor, a $250 \mathrm{mH}$ inductor and a $2.5 \mu \mathrm{F}$ capacitor are connected in series with an AC source of peak voltage $5 \mathrm{~V}$ and angular frequency $2 \mathrm{rad/sec}$. What is the peak value of the electrostatic energy of the capacitor?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mu \mathrm{J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.5 \mu \mathrm{J}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.33 \mu \mathrm{J}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$5 \mu \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$5 \mu \mathrm{J}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>The angular frequency is $2 \mathrm{KHz}$. The unit given is incorrect Assuming it to be in radian / sec $X_{L}=2 \times 10^{3} \times 250 \times 10^{-3}=500 \Omega$
<br/>$\mathrm{X}_{\mathrm{C}}=\frac{1}{2.5 \times 10^{-6} \times 2 \times 10^{3}}=200 \Omega$
<br/>$Z=\sqrt{R^{2}+\left(X_{L}-X_{c}\right)^{2}}=500 \Omega$
<br/>$\left(\mathrm{V}_{\mathrm{C}}\right)_{\text {peak }}=\mathrm{i}_{\text {peak }} \mathrm{X}_{\mathrm{c}}=\frac{\left(\mathrm{V}_{\mathrm{s}}\right)_{\text {peak }}}{\mathrm{Z}} \mathrm{X}_{\mathrm{C}}=\frac{5}{500} \times 200=2 \mathrm{~V}$
<br/>$\left(\mathrm{U}_{\mathrm{C}}\right)_{\max }=\frac{1}{2} \times 2.5 \times 10^{-6} \times(2)^{2}=5 \mu \mathrm{J}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A and B are independent events. The probability that both A and B occur is 20 1 â and the probability that neither of them occurs is 5 3 â . The probability of occurrence of A is

slug:
a-and-b-are-independent-events-the-probability-that-both-a-and-b-occur-is-20-1-and-the-probabil-52090

question:
<div class="question">A and $\mathrm{B}$ are independent events. The probability that both A and $\mathrm{B}$ occur is $\frac{1}{20}$ and the probability that neither of them occurs is $\frac{3}{5}$. The probability of occurrence of $\mathrm{A}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{1}{4}$, $\frac{1}{5}$</span> </div>

answer:
<div class="solution">\(P(A \cap B)=P(A) P(B)=\frac{1}{20} \Rightarrow P(B)=\frac{1}{2 O P(A)}\)<br/>\(P(\bar{A} \cap \bar{B})=\frac{3}{5}=1-P(A \cup B)\)<br/>\(\Rightarrow \frac{3}{5}=1-P(A)-P(B)+P(A \cap B)\)<br/>\(\Rightarrow \frac{3}{5}=1-P(A)-\frac{1}{20 P(A)}+\frac{1}{20}\)<br/>\(\Rightarrow \frac{3}{5}=\frac{21}{20}-P(A)-\frac{1}{20 P(A)}\)<br/>\(\Rightarrow \frac{12-21}{20}=-P(A)-\frac{1}{20 P(A)}\)<br/>Let \(P(A)=x\)<br/>\(\Rightarrow \frac{-9}{20}=\frac{-20 x^2-1}{20 x}\)<br/>\(\Rightarrow 20 x^2-9 x+1=0\)<br/>\(\Rightarrow(4 x-1)(5 x-1)=0\)<br/>\(\Rightarrow x=\frac{1}{4} \frac{1}{5}\)<br/>i. e. \(P(A)=\frac{1}{4} P(A)=\frac{1}{5}\)</div>


title: A and B are two independent events such that P ( A âª B ) = 0.8 and P ( A ) = 0.3 . The P ( B ) is
slug: a-and-b-are-two-independent-events-such-that-p-a-b-0-8-and-p-a-0-3-the-p-b-is-58800
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
question:
<div class="question">$A$ and $B$ are two independent events such that $P(A \cup B)=0.8$ and $P(A)=0.3$. The $P(B)$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{7}$</span> </div>

answer:
<div class="solution">Hints : Let $\mathrm{P}(\mathrm{B})=x$<br/>$$<br/>\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.3+(1-x)-0.3(1-x)<br/>$$<br/>or $\quad 0.8=1-x+0.3 x$<br/>or $\quad 1-0.7 x=0.8$<br/>or $\quad 0.7 x=0.2$<br/>or $\quad x=\frac{2}{7}$</div>


slug: a-and-b-are-two-vectors-given-by-a-2-i-3-j-and-b-i-j-the-magnitude-of-the-87350
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
title:
A and B are two vectors given by A = 2 i ^ + 3 j ^ â and B = i ^ + j ^ â . The magnitude of the component of A along B is

question:
<div class="question">$\vec{A}$ and $\vec{B}$ are two vectors given by $\vec{A}=2 \hat{i}+3 \hat{j}$ and $\vec{B}=\hat{i}+\hat{j}$. The magnitude of the component of $\vec{A}$ along $\vec{B}$ is</div>

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

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

answer:
<div class="solution">Magnitude of components of $\vec{A}$ along $\vec{B}=\frac{\vec{A} \cdot \vec{B}}{|\vec{B}|}=\frac{(2 \hat{i}+3 \hat{j})(\hat{i}+\hat{j})}{\sqrt{2}}=\frac{5}{\sqrt{2}}$</div>


slug: a-b-are-fixed-points-with-coordinates-0-a-and-0-b-a-0-b-0-p-is-variabl-97749
tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
\(A, B\) are fixed points with coordinates \((0, a)\) and \((0, b)(a > 0, b > 0)\). \(P\) is variable point \((x, 0)\) referred to rectangular axis. If the angle \(\angle \mathrm{APB}\) is maximum, then

question:
<div class="question">\(A, B\) are fixed points with coordinates \((0, a)\) and \((0, b)(a &gt; 0, b &gt; 0)\). \(P\) is variable point \((x, 0)\) referred to rectangular axis. If the angle \(\angle \mathrm{APB}\) is maximum, then</div>

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

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

answer:
<div class="solution">Hint : \(\angle A P B=\theta=\cos ^1\left(\frac{x^2+a^2+x^2+b^2-(a-b)^2}{2 \sqrt{x^2+a^2} \sqrt{x^2+b^2}}\right)\)<br/>For Max; \(\frac{\mathrm{d} \theta}{\mathrm{dx}}=0, \mathrm{x}^2=\mathrm{ab}\)</div>


slug: a-b-c-are-mutually-exclusive-events-such-that-p-a-3-3-x-1-p-b-4-1-x-43031
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A , B , C are mutually exclusive events such that P ( A ) = 3 3 x + 1 â , P ( B ) = 4 1 â x â and P ( C ) = 2 1 â 2 x â . Then the set of possible values of x are in

question:
<div class="question">$A, B, C$ are mutually exclusive events such that $P(A)=\frac{3 x+1}{3}, P(B)=\frac{1-x}{4}$ and $P(C)=\frac{1-2 x}{2}$. Then the set of possible values of $\mathrm{x}$ are in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[0,1]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\frac{1}{3}, \frac{1}{2}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\frac{1}{3}, \frac{2}{3}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\frac{1}{3}, \frac{13}{3}\right]$</span> </li> </ul>

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

answer:
<div class="solution">$0 \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{C}) \leq 1$ and $0 \leq \mathrm{P}(\mathrm{A}) \leq 1,0 \leq \mathrm{P}(\mathrm{B}) \leq 1,0 \leq \mathrm{P}(\mathrm{C}) \leq 1 \Rightarrow \frac{1}{3} \leq \mathrm{x} \leq \frac{1}{2}$</div>


slug: a-b-c-is-an-isosceles-triangle-with-an-inscribed-circle-with-centre-o-let-p-be-the-midp-92195
tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
\(A B C\) is an isosceles triangle with an inscribed circle with centre \(O\). Let \(P\) be the midpoint of \(B C\). If \(A B=A C=15\) and \(B C=10\), then \(\mathrm{OP}\) equals

question:
<div class="question">\(A B C\) is an isosceles triangle with an inscribed circle with centre \(O\). Let \(P\) be the midpoint of \(B C\). If \(A B=A C=15\) and \(B C=10\), then \(\mathrm{OP}\) equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\sqrt{5}}{\sqrt{2}}\) unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{5}{\sqrt{2}}\) 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">C</span> <span class="option-data">\(2 \sqrt{5}\) unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(5 \sqrt{2}\) unit</span> </li> </ul>

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/qg5mbamKNxxHoRAl1txrB3BjZRIJICEuPO_3KQ-w7d0.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; A P=\sqrt{15^2-5^2}=10 \sqrt{2} \\ &amp; \Delta=\frac{1}{2} \cdot B C \cdot A P=50 \sqrt{2} \\ &amp; s=\frac{15+15+10}{2}=20 \\ &amp; \because r=\frac{\Delta}{s}=\frac{50 \sqrt{2}}{20}=\frac{5}{\sqrt{2}}\end{aligned}\)</div>


slug: a-b-is-a-variable-chord-of-the-ellipse-a-2-x-2-b-2-y-2-1-if-a-b-subtends-a-right-angle-14445
tags: ['Mathematics', 'Ellipse', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A B is a variable chord of the ellipse a 2 x 2 â + b 2 y 2 â = 1 . If A B subtends a right angle at the origin O , then O A 2 1 â + O B 2 1 â equals to

question:
<div class="question">$A B$ is a variable chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. If $A B$ subtends a right angle at the origin $O$, then $\frac{1}{OA^2}+\frac{1}{OB^2}$ equals to</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/zF9yd1J0Kq837hlF3KLoMtyPnBqZjpJCS_C2MDUdxZA.original.fullsize.png"/></div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
title:
A ball is projected horizontally with a velocity of 5 m / s from the top of a building 19.6 m high. How long will the ball take of hit the ground?

slug:
a-ball-is-projected-horizontally-with-a-velocity-of-5-m-s-from-the-top-of-a-building-19-6-m-high-69144

question:
<div class="question">A ball is projected horizontally with a velocity of $5 \mathrm{~m} / \mathrm{s}$ from the top of a building $19.6 \mathrm{~m}$ high. How long will the ball take of hit the ground?</div>

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

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

answer:
<div class="solution">Hints: $\mathrm{T}=\sqrt{\frac{2 \mathrm{H}}{g}}=\sqrt{\frac{2 \times 19.6}{9.8}}=2 \mathrm{sec}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mhYHAXS9PRHqYz-F0AP7R6P5UwP9sj-T_l5uKHLZZW8.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Heights and Distances', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A balloon starting from rest is ascending from ground with uniform acceleration of \(4 \mathrm{ft} / \mathrm{sec}^2\). At the end of \(5 \mathrm{sec}\), a stone is dropped from it. If \(\mathrm{T}\) be the time to reach the stone to the ground and \(\mathrm{H}\) be the height of the balloon when the stone reaches the ground, then

slug:
a-balloon-starting-from-rest-is-ascending-from-ground-with-uniform-acceleration-of-4-mathrm-ft-86947

question:
<div class="question">A balloon starting from rest is ascending from ground with uniform acceleration of \(4 \mathrm{ft} / \mathrm{sec}^2\). At the end of \(5 \mathrm{sec}\), a stone is dropped from it. If \(\mathrm{T}\) be the time to reach the stone to the ground and \(\mathrm{H}\) be the height of the balloon when the stone reaches the ground, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{T}=6 \mathrm{sec}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{H}=112.5 \mathrm{ft}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(T=5 / 2 \mathrm{sec}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\(225 \mathrm{ft}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">\(T=5 / 2 \mathrm{sec}\)</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fsb-ZG0jfxNJBmGPuFMHIs0pRgfo51CY4i11LAmTWpc.original.fullsize.png"/><br/><br/>Let after \(5 \mathrm{sec}\) ballon is at \(B\)<br/>Given: \(a=4 \mathrm{ft} / \sec ^2 u(\) Initial velocity \()=0\)<br/>\(\mathrm{t}=5 \mathrm{sec}\)<br/>Distance covered by ballon in \(5 \mathrm{sec}\);<br/>\(S=u t+\frac{1}{2} \mathrm{at}^2=\frac{1}{2} \times 4 \times 5^2=50 \mathrm{ft}\)<br/>Let Speed of balloon at \(B\) be \(\mathrm{V}\)<br/>\(V=u+\) at \(=0+4 \times 5=20 \mathrm{ft} / \mathrm{sec}\)<br/>So, speed of ball when it's dropped<br/>at point \(B\) is \(v_1=20 \mathrm{ft} / \mathrm{sec}\).<br/>\(S=v_1 t+\frac{1}{2} a_1 t^2\)<br/>\(\Rightarrow-50=20 t+\frac{1}{2}(-32) t^2 \Rightarrow t=\frac{5}{2} \mathrm{sec}\)</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint :</div>
source_db: db\MarksBatch0.db
title:
A bar magnet falls from rest under gravity through the centre of a horizontal ring of conducting wire as shown in figure. Which of the following graph best represents the speed \((v) v s\). time (t) graph of the bar magnet?

slug:
a-bar-magnet-falls-from-rest-under-gravity-through-the-centre-of-a-horizontal-ring-of-conducting-wir-18425

question:
<div class="question">A bar magnet falls from rest under gravity through the centre of a horizontal ring of conducting wire as shown in figure. Which of the following graph best represents the speed \((v) v s\). time (t) graph of the bar magnet?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/FnUsB81yzAsRQ4_jVgUC_kdbJHkPxxgzMeXycLzc3es.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><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/76XRxSePejIgWNge779CPKHci8ws2GgZry0iLotLVaw.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/U-g2Pjz06T2Ghjtlt4G1TCfOFYg0NIz_ShroY6Y9C_o.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/IZ0r8hgbfLJmDCbaJpTs0cRJJUCRfKi1euvdZT3V6zQ.original.fullsize.png"/><br/></span> </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/FnUsB81yzAsRQ4_jVgUC_kdbJHkPxxgzMeXycLzc3es.original.fullsize.png"/><br/></span> </div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A bar magnet has a magnetic moment of 200 Am 2 . The magnet is suspended in a magnetic field of 0.30 NA â 1 m â 1 . The torque required to rotate the magnet from its equilibrium position through an angle of 3 0 â , will be

slug:
a-bar-magnet-has-a-magnetic-moment-of-200-am-2-the-magnet-is-suspended-in-a-magnetic-field-of-0-30-46956

question:
<div class="question">A bar magnet has a magnetic moment of $200 \mathrm{Am}^{2}$. The magnet is suspended in a magnetic field of $0.30 \mathrm{NA}^{-1} \mathrm{m}^{-1}$. The torque required to rotate the magnet from its equilibrium position through an angle of $30^{\circ},$ will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$30 \mathrm{N}$ -m</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$30 \sqrt{3} \mathrm{N}-\mathrm{m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$60 \mathrm{N}$ -m</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$60 \sqrt{3} \mathrm{N} \cdot \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$30 \mathrm{N}$ -m</span> </div>

answer:
<div class="solution">Given, $\mathrm{M}=200 \mathrm{A} \mathrm{m}^{2}$
<br/>$\mathrm{B}=0.30 \mathrm{NA}^{-} \mathrm{M}^{-1}$
<br/>and $\theta=30^{\circ}$
<br/>We know that the Torque,
<br/>$$
<br/>\begin{aligned}
<br/>\tau &amp;=M \times B \\
<br/>\Rightarrow \quad|t| &amp;=M B \sin \theta=200 \times 0.3 \times \frac{1}{2} \\
<br/>&amp;=100 \times 0.3=30 \mathrm{N} \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A bar of length L carrying a small mass m at one of its ends rotates with a uniform angular speed Ï in a vertical plane about the mid point of the bar. During the rotation, at some instant of time when the bar is horizontal, the mass is detached from the bar but the bar continues to rotate with some Ï . The mass moves vertically up, comes back and reaches the bar at the same point. At that place, the acceleration due to gravity is g

slug:
a-bar-of-length-l-carrying-a-small-mass-m-at-one-of-its-ends-rotates-with-a-uniform-angular-speed-59232

question:
<div class="question">A bar of length L carrying a small mass $m$ at one of its ends rotates with a uniform angular speed $\omega$ in a vertical plane about the mid point of the bar. During the rotation, at some instant of time when the bar is horizontal, the mass is detached from the bar but the bar continues to rotate with some $\omega .$ The mass moves vertically up, comes back and reaches the bar at the same point. At that place, the acceleration due to gravity is $g$</div>

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

answer:
<div class="solution">We have, $v=\frac{1}{2} \omega l$ <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a59.png"/> <br/> <br/>$$
<br/>T=\frac{2 v}{g}=\frac{\omega l}{g} \text { or } n \frac{2 \pi}{\omega }=\frac{\omega l}{g}
<br/>$$
<br/>(as completes $n$ rotations with in $I$ ) $\therefore \quad n=\frac{/ \omega^{2}}{2 \pi g}$
<br/>Distance travelled $=2 h=\frac{2 v^{2}}{2 g}=\frac{l^{2} \omega^{2}}{4 g}$</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A battery of emf E and internal resistance r is connected with an external resistance R as shown in the figure. The battery will act as a constant voltage source if

slug:
a-battery-of-emf-e-and-internal-resistance-r-is-connected-with-an-external-resistance-r-as-shown-in-31947

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hKwr_NlVIukTqoL3rhXpMxhgE2YrVc9xfMhyWY6p15E.original.fullsize.png"/><br/><br/>A battery of emf $\mathrm{E}$ and internal resistance $\mathrm{r}$ is connected with an external resistance $\mathrm{R}$ as shown in the figure. The battery will act as a constant voltage source if</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$r&lt; &lt; R$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$r&gt;&gt;R$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$r=R$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It will never act as a constant voltage source</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/yRt1sQcTHwY8CaznkhNloTbx_BjlDHnxPmU5BeXKhNY.original.fullsize.png"/><br/><br/>$V_R=\frac{E}{(R+r)} \times R, \quad V_R=\frac{E}{(1+r / R)}$<br/>$r &lt; R, V_R \approx E$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A battery of emf E and internal resistance r is connected with an external resistance R as shown in the figure. The battery will act as a constant voltage source if

slug:
a-battery-of-emf-e-and-internal-resistance-r-is-connected-with-an-external-resistance-r-as-shown-in-71239

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hKwr_NlVIukTqoL3rhXpMxhgE2YrVc9xfMhyWY6p15E.original.fullsize.png"/><br/><br/>A battery of emf $\mathrm{E}$ and internal resistance $\mathrm{r}$ is connected with an external resistance $\mathrm{R}$ as shown in the figure. The battery will act as a constant voltage source if</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$r&lt; &lt; R$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$r&gt;&gt;R$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$r=R$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It will never act as a constant voltage source</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/yRt1sQcTHwY8CaznkhNloTbx_BjlDHnxPmU5BeXKhNY.original.fullsize.png"/><br/><br/>$V_R=\frac{E}{(R+r)} \times R, \quad V_R=\frac{E}{(1+r / R)}$<br/>$r &lt; R, V_R \approx E$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">In water $\lambda=\frac{4200}{\frac{4}{3}}=3150 Ã $</div>
source_db: db\MarksBatch0.db
title:
A beam of monochromatic blue light of wavelength 4200 A Ë in air travels in water of refractive index 4/3 . Its wavelength in water will be:

slug:
a-beam-of-monochromatic-blue-light-of-wavelength-4200-a-in-air-travels-in-water-of-refractive-ind-71019

question:
<div class="question">A beam of monochromatic blue light of wavelength $4200 Ã$ in air travels in water of refractive index $4 / 3$. Its wavelength in water will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4200 Ã$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5800 Ã$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4150 Ã$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3150 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$3150 Ã$</span> </div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A biconvex lens of focal length f and radii of curvature of both the surfaces R is made of a material of refractive index n 1 â . This lens is placed in a liquid of refractive index n 2 â . Now this lens will behave like

slug:
a-biconvex-lens-of-focal-length-f-and-radii-of-curvature-of-both-the-surfaces-r-is-made-of-a-materia-86519

question:
<div class="question">A biconvex lens of focal length $f$ and radii of curvature of both the surfaces $R$ is made of a material of refractive index $n_{1}$. This lens is placed in a liquid of refractive index $n_{2}$. Now this lens will behave like</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">either as a convex or as a concave lens depending solely on $\mathrm{R}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">a convex lens depending on $n_{1}$ and $n_{2}$</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="correct"> <span class="option-label">C</span> <span class="option-data">a concave lens depending on $n_{1}$ and $n_{2}$</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">D</span> <span class="option-data">a convex lens of same focal length irrespective of $R, n_{1}$ and $n_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">a convex lens depending on $n_{1}$ and $n_{2}$, a concave lens depending on $n_{1}$ and $n_{2}$</span> </div>

answer:
<div class="solution">The lens will behave like a convex lens or a concave lens depending upon the value of $n_{1}$ and $n_{2}$.</div>


slug: a-block-of-ice-at-temperature-2-0-c-is-slowly-heated-and-converted-to-steam-at-10-0-c-57875
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
title:
A block of ice at temperature â 2 0 â C is slowly heated and converted to steam at 10 0 â C . Which of the following diagram is most appropriate?

question:
<div class="question">A block of ice at temperature $-20^{\circ} \mathrm{C}$ is slowly heated and converted to steam at $100^{\circ} \mathrm{C}$. Which of the following diagram is most appropriate?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/G02zQPahwC3O9sgAwy3uzDBUguLt8OCNfGJybHOeUjE.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><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9TTx3oiBiHnaXeTAXLdAeSsO40oWlUqd1U4FrHjJOMk.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/2dTIxymIX8Eym0DFjeh3QIjXyASxJAV0ii5UBruqWbQ.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/O6gyFTKGB9QVYskX5lxlXSFEO96GExh6P3ipR9Qc42k.original.fullsize.png"/><br/></span> </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/G02zQPahwC3O9sgAwy3uzDBUguLt8OCNfGJybHOeUjE.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/HicUGppXhfdLg9YfZ1vWrYDYfyWxpnROBuNW3Dbwa9w.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch0.db
title:
A block of mass 1 kg starts from rest at x = 0 and moves along the X -axis under the action of a force F = k t , where t is time and k = 1 Ns â 1 . The distance the block will trave] in 6 seconds is

slug:
a-block-of-mass-1-kg-starts-from-rest-at-x-0-and-moves-along-the-x-axis-under-the-action-of-a-for-49550

question:
<div class="question">A block of mass $1 \mathrm{kg}$ starts from rest at $x=0$ and moves along the $X$ -axis under the action of a force $F=k t,$ where $t$ is time and
<br/>$k=1 \mathrm{Ns}^{-1}$. The distance the block will trave] in 6 seconds is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$36 \mathrm{m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$72 \mathrm{m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$108 \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$18 \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$36 \mathrm{m}$</span> </div>

answer:
<div class="solution">From Newton's second law
<br/>$$
<br/>\begin{array}{l}
<br/>F=m a \\
<br/>m=1 \mathrm{kg} \\
<br/>F=1 \times a=\frac{d v}{d t}
<br/>\end{array}
<br/>$$
<br/>According to the question,
<br/>$$
<br/>F=k t=\frac{d v}{d t}
<br/>$$
<br/>$$
<br/>d v=k t \cdot d t
<br/>$$
<br/>Integration gives
<br/>$$
<br/>\begin{aligned}
<br/>v &amp;=\frac{k t^{2}}{2}=\frac{d x}{d t} \\
<br/>x &amp;=\frac{k t^{3}}{6}=\frac{1 \times 6 \times 6 \times 6}{6} \\
<br/>&amp;=36 \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


slug: a-block-of-mass-m-0-1-kg-is-hanging-over-a-frictionless-light-fixed-pulley-by-an-inextensible-46635
tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A block of mass m ( = 0.1 kg ) is hanging over a frictionless light fixed pulley by an inextensible string of negligible mass. The other end of the string is pulled by a constant force F in the vertically downward direction. The linear momentum of the block increases by 2 kgms â 1 in 1 s after the block starts from rest. Then, (given g = 10 ms â 2 )

question:
<div class="question">A block of mass $m(=0.1 \mathrm{kg})$ is hanging over a frictionless light fixed pulley by an inextensible string of negligible mass. The other end of the string is pulled by a constant force $F$ in the vertically downward direction. The linear momentum of the block increases by $2 \mathrm{kgms}^{-1}$ in $1 \mathrm{s}$ after the block starts from rest. Then, (given $g=10 \mathrm{ms}^{-2}$ ) <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q57.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The tension in the string is $F$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">The tension in the string is $3 \mathrm{N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The work done by the tension on the block is $20 \mathrm{J}$ during this 1 s</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">The work done against the force of gravity is $10 \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">The tension in the string is $F$, The tension in the string is $3 \mathrm{N}$, The work done against the force of gravity is $10 \mathrm{J}$</span> </div>

answer:
<div class="solution">The free body diagram <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a57.png"/> <br/> <br/>$\Rightarrow \quad F=2+m g=3 N$
<br/>also, $a=\frac{\text { unbalanced force }}{\text { mass }}=\frac{2}{0.1}=20 \mathrm{m} / \mathrm{s}^{2}$
<br/>$S=\frac{1}{2} a t^{2}=\frac{1}{2} \times 20 \times 1=10 \mathrm{m}$
<br/>Honce, work done by tension
<br/>$$
<br/>=F \times 10=3 \times 10=30 \mathrm{J}
<br/>$$
<br/>So, the work done against gravity
<br/>$$
<br/>=m g \times S=1 \times 10=10 \mathrm{J}
<br/>$$</div>


slug: a-block-of-mass-m-2-is-placed-on-a-horizontal-table-and-another-block-of-mass-m-1-is-placed-33345
tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A block of mass m 2 â is placed on a horizontal table and another block of mass m 1 â is placed on top of it. An increasing horizontal force \(F=\alpha t\) is exerted on the upper block but the lower block never moves as a result. If the coefficient of friction between the blocks is Î¼ i â and that between the lower block and the table is Î¼ 2 â , then what is the maximum possible value of Î¼ 1 â / Î¼ 2 â ?

question:
<div class="question">A block of mass $m_{2}$ is placed on a horizontal table and another block of mass $m_{1}$ is placed on top of it. An increasing horizontal force \(F=\alpha t\) is exerted on the upper block but the lower block never moves as a result. If the coefficient of friction between the blocks is $\mu_{i}$ and that between the lower block and the table is $\mu_{2}$, then what is the maximum possible value of $\mu_{1} / \mu_{2} ?$</div>

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

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

answer:
<div class="solution">According to the question.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a13.png"/> <br/> <br/>$\therefore \quad N_{2}=m_{2} g+N_{1}=\left(m_{1}+m_{2}\right) g$
<br/>: Lower block never moves.
<br/>$\therefore \quad f_{r_{2}} \geq f_{n}$
<br/>$\quad \mu_{2} N_{2} \geq \mu_{1} N_{1} \Rightarrow \mu_{2}\left(m_{1}+m_{2}\right) g \geq \mu_{1} m_{1} g$
<br/>$\Rightarrow \quad \frac{m_{1}+m_{2}}{m_{1}} \geq \frac{\mu_{1}}{\mu_{2}}$
<br/>$\therefore \quad \frac{\mu_{1}}{\mu_{2}} \leq 1+\frac{m_{2}}{m_{1}}$
<br/>$\left.\therefore \quad \frac{\mu_{1}}{\mu_{2}}\right|_{\operatorname{man}}=1+\frac{m_{2}}{m_{1}}$</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A block of mass m rests on a horizontal table with a co-efficient of static friction Î¼ . What minimum force must be applied on the block to drag it on the table?

slug:
a-block-of-mass-m-rests-on-a-horizontal-table-with-a-co-efficient-of-static-friction-what-minim-98779

question:
<div class="question">A block of mass m rests on a horizontal table with a co-efficient of static friction $\mu .$ What minimum force must be applied on the block to drag it on the table?</div>

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

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/E-Q3dXzUCPuAJnJDKUZD7UVE7YWcYpaYWWmKj_E9P-w.original.fullsize.png"/><br/> <br/>$F \sin \theta+N=m g \quad N=m g-F \sin \theta$
<br/>$F \cos \theta=\mu(m g-F \sin \theta) \quad F=\frac{\mu m g}{\cos \theta+\mu \sin \theta}$,
<br/>for $F_{\min }, \frac{d}{d \theta}[\cos \theta+\mu \sin \theta]=0$
<br/>$\therefore \tan \theta=\mu$
<br/>$\therefore F_{\min }=\frac{\mu \mathrm{mg}}{\sqrt{1+\mu^{2}}}$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A block of mass m rests on a horizontal table with a co-efficient of static friction Î¼ . What minimum force must be applied on the block to drag it on the table?

slug:
a-block-of-mass-m-rests-on-a-horizontal-table-with-a-co-efficient-of-static-friction-what-minim-53716

question:
<div class="question">A block of mass m rests on a horizontal table with a co-efficient of static friction $\mu .$ What minimum force must be applied on the block to drag it on the table?</div>

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

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/E-Q3dXzUCPuAJnJDKUZD7UVE7YWcYpaYWWmKj_E9P-w.original.fullsize.png"/><br/> <br/>$F \sin \theta+N=m g \quad N=m g-F \sin \theta$
<br/>$F \cos \theta=\mu(m g-F \sin \theta) \quad F=\frac{\mu m g}{\cos \theta+\mu \sin \theta}$,
<br/>for $F_{\min }, \frac{d}{d \theta}[\cos \theta+\mu \sin \theta]=0$
<br/>$\therefore \tan \theta=\mu$
<br/>$\therefore F_{\min }=\frac{\mu \mathrm{mg}}{\sqrt{1+\mu^{2}}}$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
A block of mass m slides with speed v on a frictionless table towards another stationary block of mass m . A massless spring with spring constant k is attached to the second block as shown in figure. The maximum distance the spring gets compress through is

slug:
a-block-of-mass-m-slides-with-speed-v-on-a-frictionless-table-towards-another-stationary-block-of-ma-71461

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fnzNC8mLQ5MK_3Mmkm6R5i7b9p7Ns0WbFhv_XQ0-5DI.original.fullsize.png"/><br/> <br/>A block of mass $m$ slides with speed $v$ on a frictionless table towards another stationary block of mass $m$. A massless spring with spring constant $\mathrm{k}$ is attached to the second block as shown in figure. The maximum distance the spring gets compress through is</div>

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

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

answer:
<div class="solution">$\frac{1}{2} \mathrm{kx}^{2}=\frac{1}{2} \times \frac{\mathrm{m} \times \mathrm{m}}{2 \mathrm{~m}} \mathrm{v}^{2} \Rightarrow \mathrm{x}=\sqrt{\frac{\mathrm{m}}{2 \mathrm{k}}} \mathrm{v}$</div>


slug: a-body-floats-in-water-with-40-of-its-volume-outside-water-when-the-same-body-floats-in-an-o-93813
tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.5</span> </div>
source_db: db\MarksBatch0.db
title:
A body floats in water with \(40 \%\) of its volume outside water. When the same body floats in an oil, \(60 \%\) of its volume remains outside oil. The relative density of oil is

question:
<div class="question">A body floats in water with \(40 \%\) of its volume outside water. When the same body floats in an oil, \(60 \%\) of its volume remains outside oil. The relative density of oil is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.9</span> </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">1.2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1.5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Hints: \(\mathrm{V} \sigma \mathrm{g}=0.6 \mathrm{~V} \sigma_1 \mathrm{~g}\) ....(1)<br/>\(\mathrm{~V} \sigma \mathrm{g}=0.4 \mathrm{~V} \sigma_2 \mathrm{~g}\) ....(2)<br/>\(\mathrm{V \sigma} \mathrm{g}=0.4 \mathrm{V \sigma}_2 \mathrm{~g}\)<br/>Dividing (1) and (2) \(1=\frac{6}{4} \frac{\sigma_1}{\sigma_2} \therefore \frac{\sigma_2}{\sigma_1}=\frac{3}{2}\)</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.5</span> </div>
source_db: db\MarksBatch0.db
title:
A body floats in water with 40% of its volume outside water. When the same body floats in oil, 60% of its volume remains outside oil. The relative density of the oil is

slug:
a-body-floats-in-water-with-40-of-its-volume-outside-water-when-the-same-body-floats-in-oil-60-o-83045

question:
<div class="question">A body floats in water with $40 \%$ of its volume outside water. When the same body floats in oil, $60 \%$ of its volume remains outside oil. The relative density of the oil is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1.5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.8</span> </li> </ul>

answer:
<div class="solution">Hints : Fraction of immersed part $f=\frac{d}{\rho}$<br/>Case-1,<br/>$f=1-0.4=0.6$<br/>$0.6=\frac{d}{1}$<br/>$d=0.6$<br/>Case-2,<br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; f=1-0.6=0.4 \\<br/>&amp; f=\frac{d}{\rho_{\text {oil }}} \\<br/>&amp; 0.4=\frac{0.6}{\rho_{\text {oil }}} \\<br/>&amp; \rho_{\text {oil }}=1.5<br/>\end{aligned}<br/>$$</div>


slug: a-body-is-projected-from-the-ground-with-a-velocity-v-3-i-10-j-ms-1-the-maximu-50216
tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A body is projected from the ground with a velocity v = ( 3 i ^ + 10 j ^ â ) ms â 1 . The maximum height attained and the range of the body respectively are (given g = 10 ms â 2 )

question:
<div class="question">A body is projected from the ground with a velocity $\mathbf{v}=(3 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}) \mathrm{ms}^{-1} .$ The maximum height attained and the range of the body respectively are (given $g=10 \mathrm{ms}^{-2}$ )</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 \mathrm{m}$ and $6 \mathrm{m}$</span> </div>

answer:
<div class="solution">(a) Given, $v=(3 i+10 j) m / s$
<br/>$\Rightarrow \quad v_{x}=3$ and $v_{y}=10$
<br/>$\therefore$ Maximum height attained, $H=\frac{v_{y}^{2}}{2 g}$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=\frac{10 \times 10}{2 \times 10}=5 \mathrm{m} \\
<br/>\text { Range }=v_{x} \times T=V_{x} \times \frac{2 v_{y}}{g} &amp;=\frac{3 \times 2 \times 10}{10}=6 \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
title:
A body is projected with a speed u m / s at an angle Î² with the horizontal. The kinetic energy at the highest point is 3/4 th of the initial kinetic energy. The value of Î² is :

slug:
a-body-is-projected-with-a-speed-u-m-s-at-an-angle-with-the-horizontal-the-kinetic-energy-at-t-51137

question:
<div class="question">A body is projected with a speed $u \mathrm{~m} / \mathrm{s}$ at an angle $\beta$ with the horizontal. The kinetic energy at the highest point is $3 / 4$ th of the initial kinetic energy. The value of $\beta$ is :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$60^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$120^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$30^{\circ}$</span> </div>

answer:
<div class="solution">Hints: (K.E.) at maximum height $=\frac{1}{2} m\left(u^2 \cos ^2 \beta\right)$<br/>$\mathrm{K} . \mathrm{E} .=\mathrm{K} \cos ^2 \beta$<br/>Here, $K \cos ^2 \beta=\frac{3}{4} \mathrm{~K}$<br/>$$<br/>\begin{aligned}<br/>&amp; \cos \beta=\frac{\sqrt{3}}{2} \\<br/>&amp; \beta=30^{\circ}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
A body of density 1.2 Ã 1 0 3 kg / m 3 is dropped from rest from a height 1 m into a liquid of density 2.4 Ã 1 0 3 kg / m 3 . Neglecting all dissipative effects, the maximum depth to which the body sinks before returning to float on the surface is

slug:
a-body-of-density-1-2-1-0-3-kg-m-3-is-dropped-from-rest-from-a-height-1-m-into-a-liquid-of-dens-22854

question:
<div class="question">A body of density $1.2 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ is dropped from rest from a height $1 \mathrm{~m}$ into a liquid of density $2.4 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. Neglecting all dissipative effects, the maximum depth to which the body sinks before returning to float on the surface is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.1 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 \mathrm{~m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.01 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \mathrm{~m}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{d}=1.2 \times 10^{3}, \mathrm{~h}=1 \mathrm{~m}, \rho=2.4 \times 10^{3}$
<br/>By work Energy Theorem,
<br/>$$
<br/>\begin{array}{l}
<br/>\mathrm{Mg}(\mathrm{h}+\mathrm{d})=\mathrm{Bd} \\
<br/>\mathrm{V}\left(1.2 \times 10^{-3}\right) \mathrm{g}(1+\mathrm{d})=\mathrm{V}\left(2.4 \times 10^{-3}\right) \mathrm{g} \mathrm{d} \\
<br/>\therefore \mathrm{d}=1 \mathrm{~m}
<br/>\end{array}
<br/>$$</div>


slug: a-body-of-mass-2-mathrm-kg-moves-in-a-horizontal-circular-path-of-radius-5-mathrm-m-a-21026
tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A body of mass \(2 \mathrm{~kg}\) moves in a horizontal circular path of radius \(5 \mathrm{~m}\). At an instant, its speed is \(2 \sqrt{5} \mathrm{~m} / \mathrm{s}\) and is increasing at the rate of \(3 \mathrm{~m} / \mathrm{s}^2\). The magnitude of force acting on the body at the instant is,

question:
<div class="question">A body of mass \(2 \mathrm{~kg}\) moves in a horizontal circular path of radius \(5 \mathrm{~m}\). At an instant, its speed is \(2 \sqrt{5} \mathrm{~m} / \mathrm{s}\) and is increasing at the rate of \(3 \mathrm{~m} / \mathrm{s}^2\). The magnitude of force acting on the body at the instant is,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(6 \mathrm{~N}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(8 \mathrm{~N}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(14 \mathrm{~N}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(10 \mathrm{~N}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(10 \mathrm{~N}\)</span> </div>

answer:
<div class="solution">Hint: \(F=\) ma<br/>\(\begin{aligned}<br/>&amp; =m \sqrt{a_c^2+a_T^2} \\<br/>&amp; =m \sqrt{\left(\frac{20}{5}\right)^2+9} \\<br/>&amp; =2 \sqrt{16+9}=2 \times 5=10 \mathrm{~N}<br/>\end{aligned}\)</div>


slug: a-body-of-mass-6-mathrm-kg-is-acted-upon-by-a-force-which-causes-a-displacement-in-it-given-b-78450
tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
title:
A body of mass \(6 \mathrm{~kg}\) is acted upon by a force which causes a displacement in it given by \(x=\frac{t^2}{4}\) metre where \(t\) is the time in second. The work done by the force in 2 seconds is

question:
<div class="question">A body of mass \(6 \mathrm{~kg}\) is acted upon by a force which causes a displacement in it given by \(x=\frac{t^2}{4}\) metre where \(t\) is the time in second. The work done by the force in 2 seconds is</div>

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints : } \mathrm{m}=6 \mathrm{~kg} \quad \mathrm{x}=\frac{\mathrm{t}^2}{4} \\<br/>&amp; \frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{v}=\frac{\mathrm{t}}{2} \quad \mathrm{v}(0)=0 ; \mathrm{v}(2)=\frac{2}{2}=1 \\<br/>&amp; \mathrm{~K}_{\mathrm{i}}=\frac{1}{2} \mathrm{~m}(0)^2=0 ; \mathrm{K}_{\mathrm{f}}=\frac{1}{2} \mathrm{~m}(1)^2=\frac{1}{2} \times 6 \times 1=3 ; \mathrm{W}=\mathrm{K}_{\mathrm{f}}-\mathrm{K}_{\mathrm{i}}=3-0=3 \mathrm{~J}<br/>\end{aligned}\)</div>


slug: a-body-of-mass-m-is-thrown-vertically-upward-with-speed-3-v-e-where-v-e-is-the-escape-63548
tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A body of mass m is thrown vertically upward with speed 3 â v e â , where v e â is the escape velocity of a body from earth surface. The final velocity of the body is

question:
<div class="question">A body of mass $m$ is thrown vertically upward with speed $\sqrt{3} v_e$, where $v_e$ is the escape velocity of a body from earth surface. The final velocity of the body is</div>

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

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

answer:
<div class="solution">$-\frac{\mathrm{GMm}}{\mathrm{R}}+\frac{1}{2} \mathrm{~m}\left(\sqrt{3} \mathrm{v}_{\mathrm{e}}\right)^2=0+\frac{1}{2} \mathrm{mv}^2, \quad \mathrm{v}=\sqrt{2} \mathrm{v}_{\mathrm{e}}$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A body of mass m is thrown with velocity u from the origin of a co-ordinate axes at an angle Î¸ with the horizon. The magnitude of the angular momentum of the particle about the origin at time t when it is at the maximum height of the trajectory is proportional to

slug:
a-body-of-mass-m-is-thrown-with-velocity-u-from-the-origin-of-a-co-ordinate-axes-at-an-angle-with-55283

question:
<div class="question">A body of mass $m$ is thrown with velocity u from the origin of a co-ordinate axes at an angle $\theta$ with the horizon. The magnitude of the angular momentum of the particle about the origin at time $\mathrm{t}$ when it is at the maximum height of the trajectory is proportional to</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/tEeo08p6qilPBu21NB9VZpSasp39TcyruSUhZeeuuAA.original.fullsize.png"/><br/><br/>$\vec{L}=m u \cos \theta h_{\max }=m u \cos \theta \frac{u^2 \sin ^2 \theta}{2 g} \Rightarrow L=\frac{m u^3 \sin ^2 \theta \cos \theta}{2 g} \quad L \propto u^3$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch0.db
title:
A body starts from rest, under the action of an engine working at a constant power and moves along a straight line. The displacement s is given as a function of time ( f as

slug:
a-body-starts-from-rest-under-the-action-of-an-engine-working-at-a-constant-power-and-moves-along-a-51266

question:
<div class="question">A body starts from rest, under the action of an engine working at a constant power and moves along a straight line. The displacement s is given as a function of time ( $f$ as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$s=a t+b t^{2}, a$ and $b$ are constants</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$s=b t^{2}, b$ is a constant</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$s=a t^{3 / 2},$ a is a 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">D</span> <span class="option-data">$s=a t,$ a is a constant</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$s=a t^{3 / 2},$ a is a constant</span> </div>

answer:
<div class="solution">Given, Power $(P)=$ constant
<br/>Kinetic Energy (KE) $=\frac{1}{2} m v^{2}$
<br/>We know that, $P=\frac{K E}{\Delta t} \Rightarrow P=\frac{m v^{2}}{\Delta t}$
<br/>$\because P=$ constant. 
<br/>Hence, velocity of the body $v \propto \sqrt{t}$...(i)
<br/>As, Velocity $v=\frac{d s}{d t}$...(ii)
<br/>From Eqs. (i) and (ii), we get
<br/>So, $\quad \frac{d s}{d t} \propto \sqrt{t}$
<br/>Integrating the above cquation w.r.t. time $(t)$
<br/>$$
<br/>\int \frac{d s}{d t} \propto \int \sqrt{t}
<br/>$$
<br/>we get, displacement of the body $s \propto t^{3 / 2}$
<br/>Displacement $s=a t^{3 / 2}$, where $a$ is constant.</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
title:
A body when fully immersed in a liquid of specific gravity 1.2 weighs 44 gwt. The same body when fully immersed in water weighs 50 gwt. The mass of the body is

slug:
a-body-when-fully-immersed-in-a-liquid-of-specific-gravity-1-2-weighs-44-gwt-the-same-body-when-ful-64084

question:
<div class="question">A body when fully immersed in a liquid of specific gravity 1.2 weighs 44 gwt. The same body when fully immersed in water weighs 50 gwt. The mass of the body is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">36 \mathrm{g}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48 \mathrm{g}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">64 \mathrm{g}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$80 \mathrm{g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$80 \mathrm{g}$</span> </div>

answer:
<div class="solution">Balancing equation
<br/>$$
<br/>W=m g-V \rho g
<br/>$$
<br/>Hence,
<br/>$$
<br/>\begin{array}{l}
<br/>44=m-1.2 \mathrm{V} \\
<br/>50=m-V
<br/>\end{array}
<br/>$$
<br/>On solving Eqs. (i) and (ii) we get
<br/>$$
<br/>m=80 \mathrm{g}
<br/>$$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
title:
A box is moved along a straight line by a machine delivering constant power. The distance moved by the body in time \(\mathrm{t}\) is proportional to

slug:
a-box-is-moved-along-a-straight-line-by-a-machine-delivering-constant-power-the-distance-moved-by-t-23150

question:
<div class="question">A box is moved along a straight line by a machine delivering constant power. The distance moved by the body in time \(\mathrm{t}\) is proportional to</div>

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

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints: } \mathrm{P}=\mathrm{Fv}=\mathrm{m} \cdot \frac{\mathrm{dv}}{\mathrm{dt}} \cdot \mathrm{v} \\<br/>&amp; \int \mathrm{vdv}=\int \mathrm{P} / \mathrm{mdt} ; \quad \frac{\mathrm{v}^2}{2}=\frac{\mathrm{Pt}}{\mathrm{m}} \\<br/>&amp; V=\sqrt{\frac{2 p}{m}} \mathrm{t}^{\frac{1}{2}} ; \frac{\mathrm{dx}}{\mathrm{dt}}=\sqrt{\frac{2 \mathrm{p}}{\mathrm{m}}} \mathrm{t}^{\frac{1}{2}} \\<br/>&amp; \int d x=\sqrt{\frac{2 p}{m}} \int t^{\frac{1}{2}} d t ; \quad x=\sqrt{\frac{2 p}{m}} \frac{t^{\frac{3}{2}}}{\frac{3}{2}}=\frac{2}{3} \sqrt{\frac{2 p}{m}} \mathrm{t}^{\frac{3}{2}} \\<br/>&amp; \mathrm{x} \alpha \mathrm{t}^{\frac{3}{2}} \\<br/>&amp;<br/>\end{aligned}\)</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
title:
A box of mass 2 kg is placed on the roof of a car. The box would remain stationary until the car attains a maximum acceleration. Coefficient of static friction between the box and the roof of the car is 0.2 and g = 10 ms â 2 This maximum acceleration of the car, for the box to remain stationary, is

slug:
a-box-of-mass-2-kg-is-placed-on-the-roof-of-a-car-the-box-would-remain-stationary-until-the-car-att-76649

question:
<div class="question">A box of mass $2 \mathrm{kg}$ is placed on the roof of a car. The box would remain stationary until the car attains a maximum acceleration. Coefficient of static friction between the box and the roof of the car is 0.2 and $g=10 \mathrm{ms}^{-2}$ This maximum acceleration of the car, for the box to remain stationary, is</div>

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

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

answer:
<div class="solution">Given, $m=2 \mathrm{kg}, \mu=0.2$ and $g=10 \mathrm{ms}^{-2}$
<br/>Here, $\quad m a=\mu m g$
<br/>$$
<br/>\begin{array}{l}
<br/>a=\mu g \\
<br/>a=0.2 \times 10 \\
<br/>a=2 \mathrm{m} / \mathrm{s}^{2}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch0.db
title:
A boy of mass 40 kg is climbing a vertical pole at a constant speed. If the coefficient of friction between his palms and the pole is 0.8 and g = 10 m / s 2 , the horizontal force that he is applying on the pole is

slug:
a-boy-of-mass-40-kg-is-climbing-a-vertical-pole-at-a-constant-speed-if-the-coefficient-of-friction-90054

question:
<div class="question">A boy of mass $40 \mathrm{~kg}$ is climbing a vertical pole at a constant speed. If the coefficient of friction between his palms and the pole is 0.8 and $g=10 \mathrm{~m} / \mathrm{s}^2$, the horizontal force that he is applying on the pole is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$300 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$400 \mathrm{~N}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$500 \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$600 \mathrm{~N}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$500 \mathrm{~N}$</span> </div>

answer:
<div class="solution">Hints: Here $\mu=0.8$<br/>Frictional force $=\mu \mathrm{N}_1=m g$<br/>$$<br/>\mathrm{N}_1=\frac{m g}{\mu}=\frac{400}{0.8}=500 \mathrm{~N}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
A bulb is placed at the centre of a circular track of radius 10 m . A vertical wall is erected touching the track at a point P. A man is running along the track with a speed of 10 m / sec . Starting from P the speed with which his shadow is running along the wall when he is at an angular distance of 6 0 â from P is

slug:
a-bulb-is-placed-at-the-centre-of-a-circular-track-of-radius-10-m-a-vertical-wall-is-erected-touch-14057

question:
<div class="question">A bulb is placed at the centre of a circular track of radius $10 \mathrm{~m}$. A vertical wall is erected touching the track at a point P. A man is running along the track with a speed of $10 \mathrm{~m} / \mathrm{sec}$. Starting from $\mathrm{P}$ the speed with which his shadow is running along the wall when he is at an angular distance of $60^{\circ}$ from $P$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30 \mathrm{~m} / \mathrm{sec}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$40 \mathrm{~m} / \mathrm{sec}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$60 \mathrm{~m} / \mathrm{sec}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$80 \mathrm{~m} / \mathrm{sec}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$40 \mathrm{~m} / \mathrm{sec}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kiZM8RNMXifCLEl-q3p1D65TnuCyCTTdPNZ3MUt3-pA.original.fullsize.png"/><br/> <br/>$\overrightarrow{\mathrm{v}}=\mathrm{r} \frac{\mathrm{d} \theta}{\mathrm{dt}}=10$
<br/>$\tan \theta=\frac{\mathrm{y}}{\mathrm{r}}$ or, $\mathrm{y}=\mathrm{r} \tan \theta$ or, $\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{r} \sec ^{2} \theta \cdot \frac{\mathrm{d} \theta}{\mathrm{dt}}=10 \times \sec ^{2}\left(60^{\circ}\right)=40 \mathrm{~m} / \mathrm{sec}$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
title:
A bullet is fired with a velocity u making an angle of 6 0 â with the horizontal plane. The horizontal component o the velocity of the bullet when it reaches the maximum height is

slug:
a-bullet-is-fired-with-a-velocity-u-making-an-angle-of-6-0-with-the-horizontal-plane-the-horizo-49524

question:
<div class="question">A bullet is fired with a velocity u making an angle of $60^{\circ}$ with the horizontal plane. The horizontal component o the velocity of the bullet when it reaches the maximum height is</div>

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

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

answer:
<div class="solution">Horizontal velocity would be constant so the value of velocity at the highest point will be $u / 2$</div>


slug: a-bullet-of-mass-4-2-1-0-2-kg-moving-at-a-speed-of-300-ms-1-gets-stuck-into-a-block-w-62732
tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch0.db
title:
A bullet of mass 4.2 Ã 1 0 â 2 kg , moving at a speed of 300 ms â 1 , gets stuck into a block with a mass 9 times that of the bullet. If the block is free to move without any kind of friction, the heat generated in the process will be

question:
<div class="question">A bullet of mass $4.2 \times 10^{-2} \mathrm{kg},$ moving at a speed of $300 \mathrm{ms}^{-1}$, gets stuck into a block with a mass 9 times that of the bullet. If the block is free to move without any kind of friction, the heat generated in the process will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$45 \mathrm{cal}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$405 \mathrm{cal}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$450 \mathrm{cal}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1701 \mathrm{cal}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$405 \mathrm{cal}$</span> </div>

answer:
<div class="solution">Let mass of bullet = $m$
<br/>Mass of block $=\mathrm{M}$ 
<br/>Velocity of bullet $=v=300 \mathrm{m} / \mathrm{s}$
<br/>Velocity of combined system $M+m=V$
<br/>Here, fhom momentum conservation $\frac{M+m}{m} V=v$
<br/>$v=\frac{300 \times 42 \times 10^{-4}}{4.2 \times 10^{-4}+9\left(4.2 \times 10^{-4}\right)}$
<br/>$30 \mathrm{ms}$
<br/>Now, heat produced = Loss in kinetic energy of bullet
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{1}{2} m^{2}-\frac{1}{2}(M+m) V^{2} \\
<br/>=\frac{1}{2} \times 4.2 \times 10^{-4}(300)^{2}-\frac{1}{2}\left(4.2 \times 10^{-2}\right. \\
<br/>+9 \times 4.2 \times 1)(30)^{2} \\
<br/>=63 \times 270 \\
<br/>=1701 \mathrm{J} \\
<br/>=\frac{1701}{4.2} \mathrm{Cal}
<br/>\end{array}
<br/>$$
<br/>$=405 \mathrm{Cal}$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A bullet of mass m travelling with a speed v hits a block of mass M initially at rest and gets embedded in it. The combined system is free to move and there is no other force acting on the system. The heat generated in the process will be

slug:
a-bullet-of-mass-m-travelling-with-a-speed-v-hits-a-block-of-mass-m-initially-at-rest-and-gets-embed-67204

question:
<div class="question">A bullet of mass $m$ travelling with a speed $v$ hits a block of mass $M$ initially at rest and gets embedded in it. The combined system is free to move and there is no other force acting on the system. The heat generated in the process will be</div>

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

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

answer:
<div class="solution">Mass of bullet $=m_{1}=m$
<br/>Initial speed of bullet $=u_{1}=v$
<br/>Mass of block $=m_{2}=M$ 
<br/>Initial speed of block $=u_{2}=0$
<br/>Let the common velocity of the bodies after collision $=V$ According to conservation of linear momentum
<br/>$$
<br/>\begin{aligned}
<br/>m \times v+M \times 0 &amp;=(m+M) V \\
<br/>V &amp;=\frac{m v}{(m+M)}
<br/>\end{aligned}
<br/>$$
<br/>: Heat generated = loss in $\mathrm{KE}$
<br/>$=$ Initial KE - Final KE
<br/>$=\frac{1}{2} m v^{2}-\frac{1}{2}(m+M) v^{2}$
<br/>$=\frac{1}{2} m v^{2}-\frac{1}{2}(m+M)\left(\frac{m v}{m+M}\right)^{2}$
<br/>$=\frac{1}{2} m v^{2}-\frac{1}{2} \frac{m^{2} v^{2}}{(m+M)}$
<br/>$=\frac{1}{2} m v^{2}\left(1-\frac{m}{m+M}\right)$
<br/>$=\frac{1}{2} m v^{2}\left(\frac{m+M-m}{m+M}\right)$
<br/>$=\frac{1}{2} \frac{m M}{(m+M)} v^{2}$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch0.db
title:
A bullet on penetrating 30 cm into its target loses its velocity by 50% . What additional distance will it penetrate into the target before it comes to rest?

slug:
a-bullet-on-penetrating-30-cm-into-its-target-loses-its-velocity-by-50-what-additional-distance-w-75104

question:
<div class="question">A bullet on penetrating $30 \mathrm{cm}$ into its target loses its velocity by $50 \%$. What additional distance will it penetrate into the target before it comes to rest?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$20 \mathrm{cm}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$5 \mathrm{cm}$</span> </li> </ul>

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

answer:
<div class="solution">Given $S=30 \mathrm{cm}=30 \times 10^{-2} \mathrm{m}$ and loss in
<br/>velocity is $50 \%$ Equation of motion
<br/>$$
<br/>v^{2}=u^{2}+2 a s
<br/>$$
<br/>Here, $\frac{u^{2}}{4}=u^{2}+2 a \times 30 \times 10^{-2}$
<br/>$$
<br/>0=\frac{u^{2}}{4}+3 a \times x
<br/>$$
<br/>On solving Eqs. (i) and (ii), we get
<br/>$$
<br/>x=10 \mathrm{cm}
<br/>$$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">850</span> </div>
source_db: db\MarksBatch0.db
title:
A candidate is required to answer 6 out of 12 questions which are divided into two parts A and B , each containing 6 questions and he/she is not permitted to attempt more than 4 questions from any part. In how many different ways can he/she make up hisher choice of 6 questions?

slug:
a-candidate-is-required-to-answer-6-out-of-12-questions-which-are-divided-into-two-parts-a-and-b-e-56875

question:
<div class="question">A candidate is required to answer 6 out of 12 questions which are divided into two parts A and $B$, each containing 6 questions and he/she is not permitted to attempt more than 4 questions from any part. In how many different ways can he/she make up hisher choice of 6 questions?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">850</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">800</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">750</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">700</span> </li> </ul>

answer:
<div class="solution">$$
<br/>\begin{array}{|c|c|}
<br/>\hline A &amp; B \\
<br/>\hline 4 &amp; 2 \\
<br/>\hline 3 &amp; 3 \\
<br/>\hline 2 &amp; 4 \\
<br/>\hline
<br/>\end{array}
<br/>$$
<br/>$\therefore$ Total number of ways $={ }^{6} C_{4} \times{ }^{6} C_{2}+{ }^{6} C_{3} \times{ }^{6} C_{3}+{ }^{6} C_{2} \times{ }^{6} C_{4}$
<br/>$=\frac{6 \times 5}{2 \times 1} \times \frac{6 \times 5}{2 \times 1}+\frac{6 \times 5 \times 4}{3 \times 2 \times 1} \times \frac{6 \times 5 \times 4}{3 \times 2 \times 1}+\frac{6 \times 5}{2 \times 1} \times \frac{6 \times 5}{2 \times 1}$
<br/>$=15 \times 15+20 \times 20+15 \times 15$
<br/>$=225+400+225$
<br/>$=850$</div>


slug: a-capacitor-of-capacitance-c-0-is-charged-to-a-potential-v-0-and-is-connected-with-another-c-72933
tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A capacitor of capacitance C 0 â is charged to a potential V 0 â and is connected with another capacitor of capacitance C as shown. After closing the switch S , the common potential across the two capacitors becomes V . The capacitance C is given by

question:
<div class="question">A capacitor of capacitance $C_{0}$ is charged to a potential $V_{0}$ and is connected with another capacitor of capacitance $C$ as shown. After closing the switch $S,$ the common potential across the two capacitors becomes $V$. The capacitance $C$ is given by <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q24.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{C_{0}\left(V_{0}-V\right)}{V}$</span> </div>

answer:
<div class="solution">When the switch $S$ is closed, the common potential across the two capacitors becomes $V$. so charge on isolated plates remains same.
<br/>$$
<br/>\begin{array}{lr}
<br/>\therefore \quad C_{0} V_{0}=\left(C_{0}+C\right)V \\
<br/>\Rightarrow \quad C V=C_{0} V_{0}-C_{0} V \\
<br/>\Rightarrow \quad C=\frac{C_{0}\left(V_{0}-V\right)}{V}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch0.db
title:
A capacitor of capacitance C is connected in series with a resistance R and D C source of emf E through a key. The capacitor starts charging when the key is closed. By the time the capacitor has been fully charged. what amount of energy is dissipated in the resistance R ?

slug:
a-capacitor-of-capacitance-c-is-connected-in-series-with-a-resistance-r-and-d-c-source-of-emf-e-thro-73076

question:
<div class="question">A capacitor of capacitance $C$ is connected in series with a resistance $R$ and $D C$ source of emf $E$ through a key. The capacitor starts charging when the key is closed. By the time the capacitor has been fully charged. what amount of energy is dissipated in the resistance $R ?$  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q31.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} C E^{2}$</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">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CE}^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{E^{2}}{R}$</span> </li> </ul>

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

answer:
<div class="solution">We know that, energy stored in the capacitor
<br/>$$
<br/>=\frac{1}{2} C E^{2}
<br/>$$
<br/>and energy supplied by the source of emf
<br/>$$
<br/>=C E^{2}
<br/>$$
<br/>$\therefore$ Energy dissipated in resistance $R$ = Energy supplied by the source of emf $E$ -  Energy stored in the capacitor
<br/>$$
<br/>\begin{array}{l}
<br/>=C E^{2}-\frac{1}{2} C E^{2} \\
<br/>=\left(1-\frac{1}{2}\right) C E^{2}=\frac{1}{2} C E^{2}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
title:
A car is moving with a speed of 72 km h â 1 towards a roadside source that emits sound at a frequency of 850 Hz . The car driver listens to the sound while approaching the source and again while moving away from the source after crossing it. If the velocity of sound is 340 ms â 1 , the difference of the two frequencies, the driver hears is

slug:
a-car-is-moving-with-a-speed-of-72-km-h-1-towards-a-roadside-source-that-emits-sound-at-a-freque-37757

question:
<div class="question">A car is moving with a speed of $72 \mathrm{km}\mathrm{h}^{-1}$ towards a roadside source that emits sound at a frequency of $850 \mathrm{Hz}$. The car driver listens to the sound while approaching the source and again while moving away from the source after crossing it. If the velocity of sound is $340 \mathrm{ms}^{-1}$, the difference of the two frequencies, the driver hears is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$50 \mathrm{Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$85 \mathrm{Hz}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$100 \mathrm{Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$150 \mathrm{Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$100 \mathrm{Hz}$</span> </div>

answer:
<div class="solution">By Doppler's Effect When observer is moving with velocity $v_{0}$ towards a source at rest then approach frequency
<br/>$$
<br/>N_{\text {Approach }}=N\left(\frac{v+v_{0}}{v}\right)
<br/>$$
<br/>where $N=850 \mathrm{Hz}, v=340 \mathrm{ms}^{-1}, v_{0}=72 \mathrm{kmh}^{-1}$
<br/>$$
<br/>=20 \mathrm{ms}^{-1}=850\left(\frac{340+20}{340}\right)
<br/>$$
<br/>Similarly. when observer is moving away from source, then
<br/>$$
<br/>N_{\text {Soparation }}=N\left(\frac{v-v_{0}}{v}\right)=850\left(\frac{340-20}{340}\right)
<br/>$$
<br/>The different of the two frequency.
<br/>$$
<br/>\begin{array}{l}
<br/>N_{\text {Approuch }}-N_{\text {Soparation }} \\
<br/>\qquad \begin{aligned}
<br/>&amp;=850\left(\frac{360}{340}\right)-850\left(\frac{320}{340}\right) \\
<br/>&amp;=\frac{850}{340} \times 40=\frac{850}{8.5}=100 \mathrm{Hz}
<br/>\end{aligned}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A car moving at a velocity of 17 ms â 1 towards an approaching bus that blows a horn at a frequency of 640 Hz on a straight track. The frequency of this horn appears to be 680 Hz to the car driver. If the velocity of sound in air is 340 ms â 1 , then the velocity of the approaching bus is

slug:
a-car-moving-at-a-velocity-of-17-ms-1-towards-an-approaching-bus-that-blows-a-horn-at-a-frequenc-20974

question:
<div class="question">A car moving at a velocity of $17 \mathrm{ms}^{-1}$ towards an approaching bus that blows a horn at a frequency of $640 \mathrm{Hz}$ on a straight track. The frequency of this horn appears to be $680 \mathrm{Hz}$ to the car driver. If the velocity of sound in air is $340 \mathrm{ms}^{-1}$, then the velocity of the approaching bus is</div>

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

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

answer:
<div class="solution">Given, velocity of sound, $v=340 \mathrm{m} / \mathrm{s}$
<br/>Velocity of listner, $v_{L}=17 \mathrm{m} / \mathrm{s}$ 
<br/>Velocity of source $=v_{s}$ 
<br/>Frequency of hom emitted
<br/>$$
<br/>v=640 \mathrm{Hz}
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a45.png"/> <br/> <br/>The apparent frequency
<br/>$$
<br/>\begin{aligned}
<br/>v^{\prime} &amp;=v \frac{\left(v+v_{l}\right)}{v-v_{s}} \\
<br/>680 &amp;=640\left(\frac{340+17}{340-v_{s}}\right)
<br/>\end{aligned}
<br/>$$
<br/>On solving we get $v_{s}=4 \mathrm{m} / \mathrm{s}$</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A carbon resistor with colour code is shown in the figure. There is no fourth band in the resistor. The value of the resistance is

slug:
a-carbon-resistor-with-colour-code-is-shown-in-the-figure-there-is-no-fourth-band-in-the-resistor-51826

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dD3w8-_kECcgRGJaXfmDtwH8W5bHcfpdyidYHonw5Lg.original.fullsize.png"/><br/> <br/>A carbon resistor with colour code is shown in the figure. There is no fourth band in the resistor. The value of the resistance is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$24 \mathrm{M} \Omega \pm 20 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$14 \mathrm{k} \Omega \pm 5 \%$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$24 \mathrm{k} \Omega \pm 20 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$34 \mathrm{k} \Omega \pm 10 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$24 \mathrm{k} \Omega \pm 20 \%$</span> </div>

answer:
<div class="solution">RYO
<br/>By colour coding of resistors.
<br/>\(R=24 \times 10^3 \Omega=24 \mathrm{k} \Omega\)</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
A carbon resistor with colour code is shown in the figure. There is no fourth band in the resistor. The value of the resistance is

slug:
a-carbon-resistor-with-colour-code-is-shown-in-the-figure-there-is-no-fourth-band-in-the-resistor-63906

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dD3w8-_kECcgRGJaXfmDtwH8W5bHcfpdyidYHonw5Lg.original.fullsize.png"/><br/> <br/>A carbon resistor with colour code is shown in the figure. There is no fourth band in the resistor. The value of the resistance is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$24 \mathrm{M} \Omega \pm 20 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$14 \mathrm{k} \Omega \pm 5 \%$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$24 \mathrm{k} \Omega \pm 20 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$34 \mathrm{k} \Omega \pm 10 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$24 \mathrm{k} \Omega \pm 20 \%$</span> </div>

answer:
<div class="solution">RYO
<br/>By colour coding of resistors.
<br/>\(R=24 \times 10^3 \Omega=24 \mathrm{k} \Omega\)</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A cell of emf E is connected to a resistance R 1 â for time t and the amount of heat generated in it is H . If the resistance R 1 â is replaced by another resistance R 2 â and is connected to the cell at the same time t , the amount of heat generated in R 2 â is 4 H . Then internal resistance of the cell is

slug:
a-cell-of-emf-e-is-connected-to-a-resistance-r-1-for-time-t-and-the-amount-of-heat-generated-in-67197

question:
<div class="question">A cell of emf $E$ is connected to a resistance $R_{1}$ for time $t$ and the amount of heat generated in it is $H$. If the resistance $R_{1}$ is replaced by another resistance $R_{2}$ and is connected to the cell at the same time $t,$ the amount of heat generated in $R_{2}$ is $4 \mathrm{H}$. Then internal resistance of the cell is</div>

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

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

answer:
<div class="solution">We have, $H=I^{2} R$ According to given condition, $I_{1}^{2} R_{1}=H$
<br/>and
<br/>$$
<br/>I_{2}^{2} R_{2}=4 H
<br/>$$
<br/>$$
<br/>\Rightarrow \quad \frac{E^{2}}{\left(R_{1}+r\right)^{2}} R_{1}=H \text { and } \frac{E^{2}}{\left(R_{2}+r\right)^{2}} R_{2}=4 H
<br/>$$
<br/>$\therefore \quad \frac{R_{2}}{\left(R_{2}+r\right)^{2}}=\frac{4 R_{1}}{\left(R_{1}+r\right)^{2}}$
<br/>$\Rightarrow \quad \sqrt{R_{2}}\left(R_{1}+r\right)=2 \sqrt{R_{1}}\left(R_{2}+r\right)$
<br/>On solving. $r=\frac{\sqrt{R_{1} R_{2}}\left[\sqrt{R_{1}}-2 \sqrt{R_{2}}\right]}{2 \sqrt{R_{1}}-\sqrt{R_{2}}}$</div>


slug: a-charge-of-0-8-coulomb-is-divided-into-two-charges-q-1-and-q-2-these-are-kept-at-a-separa-31725
tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch0.db
title:
A charge of 0.8 coulomb is divided into two charges Q 1 â and Q 2 â . These are kept at a separation of 30 cm . The force on Q 1 â is maximum when

question:
<div class="question">A charge of 0.8 coulomb is divided into two charges $Q_{1}$ and $Q_{2}$. These are kept at a separation of $30 \mathrm{cm}$. The force on $Q_{1}$ is maximum when</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$Q_{1}=Q_{2}=0.4 \mathrm{C}$</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">B</span> <span class="option-data">$Q_{1}=0.8 \mathrm{C}, Q_{2}$ negligible</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$Q_{1}$ negligible, $Q_{2}=0.8 \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$Q_{1}=0.2 \mathrm{C}, Q_{2}=0.6 \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$Q_{1}=Q_{2}=0.4 \mathrm{C}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a10.png"/> <br/> <br/>$$
<br/>\frac{Q_{1}}{30 c m} \stackrel{Q_{2}}{6}
<br/>$$
<br/>Checking the given options one by one
<br/>(i) $Q_{1}=Q_{2}=0 \cdot 4 \mathrm{C}$
<br/>The force on $Q_{1}$ due to $Q_{2}$
<br/>$$
<br/>\begin{aligned}
<br/>F &amp;=k_{p} \frac{Q_{1} \times Q_{2}}{30 \times 10^{-2}} \\
<br/>&amp;=k \frac{Q_{1} Q_{2} \times 100}{30} \\
<br/>&amp;=k \frac{0-4 \times 0.4 \times 100}{30} \\
<br/>&amp;=\frac{k \times 0 \cdot 16 \times 100}{30}
<br/>\end{aligned}
<br/>$$
<br/>(ii) When $Q_{1}=0 \cdot 8 \mathrm{C} Q_{2}=0$
<br/>$$
<br/>F=\frac{k \times Q_{1} Q_{2}}{30 \times 10^{-2}}=0
<br/>$$
<br/>(iii) when $Q_{1}=0, Q_{2}=0-8 \mathrm{C}$
<br/>$$
<br/>F=0
<br/>$$
<br/>(iv) $Q_{1}=0 \cdot 2 \mathrm{C} \mathrm{Q}_{2}=0 \cdot 6 \mathrm{C}$
<br/>$$
<br/>\begin{aligned}
<br/>F &amp;=\frac{k \cdot 0 \cdot 2 \times 0 \cdot 6}{30 \times 10^{-2}} \\
<br/>&amp;=\frac{k \times 0 \cdot 12 \times 10^{2}}{30}
<br/>\end{aligned}
<br/>$$
<br/>Thus we find that, in option(a), the force on
<br/>$Q_{1}$ will be maximum.
<br/>$\therefore$ correct answer is $Q_{1}=Q_{2}=0 \cdot 4 \mathrm{C}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A charge q is placed at one corner of a cube. The electric flux through any of the three faces adjacent to the charge is zero. The flux through any one of the other three faces is

slug:
a-charge-q-is-placed-at-one-corner-of-a-cube-the-electric-flux-through-any-of-the-three-faces-adjac-94213

question:
<div class="question">A charge $q$ is placed at one corner of a cube. The electric flux through any of the three faces adjacent to the charge is zero. The flux through any one of the other three faces is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$q / 3 \varepsilon_{0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$q / 6 \varepsilon_{0}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$q / 12 \varepsilon_{0}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$q / 24 \varepsilon_{0}$</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">$q / 24 \varepsilon_{0}$</span> </div>

answer:
<div class="solution">Consider the diagram
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a34.png"/> <br/> <br/>In above the flux coverage of three face is shown in figure. Which is like a quadrant in any plane. So, flux will be
<br/>$$
<br/>\phi=\frac{1}{4} \times \frac{q}{6 \varepsilon_{0}}=\frac{q}{24 \varepsilon_{0}}
<br/>$$</div>


slug: a-charge-q-is-placed-at-the-origin-o-of-x-y-axes-as-shown-in-the-figure-the-work-done-in-taki-65180
tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A charge + q is placed at the origin O of X â Y axes as shown in the figure. The work done in taking a charge Q from A to B along the straight line A B is

question:
<div class="question">A charge $+q$ is placed at the origin $O$ of $X-Y$ axes as shown in the figure. The work done in taking a charge $Q$ from $A$ to $B$ along the straight line $A B$ is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_q5.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{q Q}{4 \pi \varepsilon_{0}}\left(\frac{a-b}{a b}\right)$</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">$\frac{q Q}{4 \pi \varepsilon_{0}}\left(\frac{b-a}{a b}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{q Q}{4 \pi \varepsilon_{0}}\left(\frac{b}{a^{2}}-\frac{1}{b}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{q Q}{4 \pi \varepsilon_{0}}\left(\frac{a}{b^{2}}-\frac{1}{b}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{q Q}{4 \pi \varepsilon_{0}}\left(\frac{a-b}{a b}\right)$</span> </div>

answer:
<div class="solution">We know that
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_a5.png"/> <br/> <br/>and $\quad V=\frac{Q}{4 \pi \varepsilon_{0} r}$
<br/>$\begin{aligned} \text { Hence, } \quad W &amp;=q\left[\frac{Q}{4 \pi \varepsilon_{0} b}-\frac{Q}{4 \pi \varepsilon_{0} a}\right] \\ &amp;=\frac{Q q}{4 \pi \varepsilon_{0}}\left[\frac{a-b}{a b}\right] \end{aligned}$</div>


slug: a-charged-particle-in-a-uniform-magnetic-field-b-b-0-hat-k-starts-moving-from-the-origin-with-73149
tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A charged particle in a uniform magnetic field \(B=B_0 \hat{k}\) starts moving from the origin with velocity \(v=3 \hat{i}+4 \hat{k}\) m/s. The trajectory of the particle and the time \(t\) at which it reaches \(2 \mathrm{~m}\) above \(x-y\) plane are,

question:
<div class="question">A charged particle in a uniform magnetic field \(B=B_0 \hat{k}\) starts moving from the origin with velocity \(v=3 \hat{i}+4 \hat{k}\) m/s. The trajectory of the particle and the time \(t\) at which it reaches \(2 \mathrm{~m}\) above \(x-y\) plane are,</div>

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

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/74VLgV0oFFB-NczEXsGDwEhEr1y-14xdCUVw2Ng7oAI.original.fullsize.png"/><br/><br/>Velocity along z-direction will be const.<br/>\(\mathrm{t}=\frac{\mathrm{s}}{V_z}=\frac{2}{4}=\frac{1}{2} \mathrm{sec}\)<br/>Path will be helical.</div>


tags: ['Physics', 'Electromagnetic Waves', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A charged particle moves with constant velocity in a region where no effect of gravity is felt but an electrostatic field E together with a magnetic field B may be present. Then which of the following cases are possible?

slug:
a-charged-particle-moves-with-constant-velocity-in-a-region-where-no-effect-of-gravity-is-felt-but-a-27558

question:
<div class="question">A charged particle moves with constant velocity in a region where no effect of gravity is felt but an electrostatic field $\overrightarrow{\mathrm{E}}$
<br/>together with a magnetic field $\vec{B}$ may be present. Then which of the following cases are possible?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\vec{E} \neq 0, \vec{B} \neq 0$</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=""> <span class="option-label">B</span> <span class="option-data">$\vec{E} \neq 0, \vec{B}=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\vec{E}=0, \vec{B}=0$</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">$\overrightarrow{\mathrm{E}}=0, \overrightarrow{\mathrm{B}} \neq 0$</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">$\vec{E} \neq 0, \vec{B} \neq 0$, $\vec{E}=0, \vec{B}=0$, $\overrightarrow{\mathrm{E}}=0, \overrightarrow{\mathrm{B}} \neq 0$</span> </div>

answer:
<div class="solution">If a charged particle is moving in a gravity-free space without changing its velocity, then three cases possible
<br/>(i) Particle can move with constant velocity in any direction, if \(\vec{E}=0, \vec{B}=0\).
<br/>(ii) If \(\vec{E}, \vec{B}\) are having values such that \(q E=q v B\) and hence both forces (Force due to magnetic field and force due to electric field) acts in such a way that they cancel each other, in this case also particle can move with uniform velocity.
<br/>(iii) If \(\vec{E}=0, \vec{B} \neq 0\) and \(\vec{B}\) is such that it acts in the direction of velocity then magnetic force would be zero and hence velocity can remain constant.</div>


slug: a-charged-particle-of-charge-q-and-mass-m-is-placed-at-a-distance-2-r-from-the-centre-of-60594
tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A charged particle of charge \(q\) and mass \(m\) is placed at a distance \(2 R\) from the centre of a vertical cylindrical region of radius \(R\) where magnetic field varies as \(\vec{B}=\left(4 t^2-2 t+6\right) \hat{k}\) where \(t\) is time. Then which of the following statement(s) is/are true?

question:
<div class="question">A charged particle of charge \(q\) and mass \(m\) is placed at a distance \(2 R\) from the centre of a vertical cylindrical region of radius \(R\) where magnetic field varies as \(\vec{B}=\left(4 t^2-2 t+6\right) \hat{k}\) where \(t\) is time. Then which of the following statement(s) is/are true?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Induced electric field lines form closed loops</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">Electric field varies linearly with \(r\) if \(r  &lt; R\), where \(r\) is the radial distance from the centerline of the cylinder</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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 charged particle will move in clockwise direction when viewed from top</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Acceleration of the charged particle is \(\frac{7 q}{2 m}\) when \(t=2 \mathrm{sec}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Induced electric field lines form closed loops, Electric field varies linearly with \(r\) if \(r  &lt; R\), where \(r\) is the radial distance from the centerline of the cylinder</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/pJXWPbSAI8XNf4GmKPZlR7r0NtYtbsYD_W7mIMLC1Cc.original.fullsize.png"/><br/><br/>\(r &lt; R\)<br/>\(E \times 2 \pi r=\frac{d \phi}{d t}=\frac{d}{d t}\left(4 t^2-2 t+6\right) \times \pi r^2\)<br/>\(\begin{aligned}<br/>&amp; E \times 2 \pi r=(8 t-2) \pi r^2 \\<br/>&amp; E=\frac{(8 t-2) r}{2} \\<br/>&amp; E=(4 t-1) r<br/>\end{aligned}\)<br/>\(E \propto r\)<br/>For \(r &gt; R\)<br/>\(\begin{aligned}<br/>&amp; E \times 2 \pi \cdot(2 R)=\frac{d}{d t}\left[4 t^2-2 t+6\right] \times \pi[R]^2 \\<br/>&amp; E .4 \pi R=[8 t-2] \pi R^2 \\<br/>&amp; E=\frac{[8 t-2] R}{4} \quad \text { at } t=2, E=\frac{14}{4} R=\frac{7 R}{2} \\<br/>&amp; \text { acceleration }=\frac{E q}{m}=\frac{7 R q}{2 m}=\frac{7 q R}{2 m}<br/>\end{aligned}\)</div>


slug: a-charged-particle-of-mass-m-1-and-charge-q-1-is-revolving-in-a-circle-of-radius-r-another-28761
tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A charged particle of mass m 1 â and charge q 1 â is revolving in a circle of radius r . Another charged particle of charge q 2 â and mass m 2 â is situated at the centre of the circle. If the velocity and time period of the revolving particle be v and T respectively, then

question:
<div class="question">A charged particle of mass $m_{1}$ and charge $q_{1}$ is revolving in a circle of radius $r .$ Another charged particle of charge $q_{2}$ and mass $m_{2}$ is situated at the centre of the circle. If the velocity and time period of the revolving particle be $v$ and $T$ respectively, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$v=\sqrt{\frac{q \cdot q_{2} r}{4 \pi \varepsilon_{0} m_{1}}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$v=\frac{1}{m_{1}} \sqrt{\frac{q_{1} q_{2}}{4 \pi \varepsilon_{0} r}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$T=\sqrt{\frac{16 \pi^{3} \varepsilon_{0} m_{1}^{2} r^{3}}{q_{1} q_{2}}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of the above</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">None of the above</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a36.png"/> <br/> <br/>The necessary centripetal force, to move in a circular path is provided by the Coulomb's force.
<br/>$$
<br/>\begin{array}{r}
<br/>\frac{m_{1} v^{2}}{\gamma}=\frac{1}{4 \pi \epsilon_{0}} \cdot \frac{q_{1} q_{2}}{r^{2}} \\
<br/>v=\sqrt{\frac{1}{4 \pi \epsilon_{0}} \cdot \frac{q_{1} q_{2}}{r m_{1}}}
<br/>\end{array}
<br/>$$
<br/>We know, $v=r\omega$
<br/>$$
<br/>\omega=\frac{2 \pi}{T}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>T=\frac{2 \pi}{\omega} \\
<br/>=\frac{2 \pi \cdot r}{v}
<br/>\end{array}
<br/>$$
<br/>Substitution the value of $v$ 
<br/>$T=\frac{2 \pi r}{\sqrt{\frac{q_{1} \cdot q_{2}}{4 \pi \varepsilon_{0} R \cdot m_{1}}}}$
<br/>$T=\sqrt{\frac{4 \pi^{2} r^{2} \times 4 \pi \varepsilon_{0} \cdot r \cdot m_{1}}{q q_{2}}}$
<br/>$T=\sqrt{\frac{16 \pi^{3} \varepsilon_{0} r^{3} \cdot m_{1}}{q_{1} \cdot q_{2}}}$
<br/>None of the given options is correct.
<br/>Both the charges are positive or both negative. (Sino answers have the product $q A_{2}$ inside square loy Therefore, circular motion is not possible. The statement of question is wrong. Either $q_{1}$ or should be negative.</div>


slug: a-chord-a-b-is-drawn-from-the-point-a-0-3-on-the-circle-x-2-4-x-y-3-2-0-and-is-13144
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A chord A B is drawn from the point A ( 0 , 3 ) on the circle x 2 + 4 x + ( y â 3 ) 2 = 0 , and is extended to M such that A M = 2 A B . The locus of M is

question:
<div class="question">A chord $A B$ is drawn from the point $A(0,3)$ on
<br/>the circle $x^{2}+4 x+(y-3)^{2}=0,$ and is extended to $M$ such that $A M=2 A B$. The locus
<br/>of $M$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}-8 x-6 y+9=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}+8 x+6 y+9=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}+8 x-6 y+9=0$</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">$x^{2}+y^{2}-8 x+6 y+9=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+8 x-6 y+9=0$</span> </div>

answer:
<div class="solution">Given, $A M=2 A B$
<br/>$\Rightarrow B$ is mid-point of $A M$
<br/>$\therefore$ Coordinate of $B$ is $\left(\frac{0+x_{1}}{2}, \frac{3+y_{1}}{2}\right)$
<br/>$$
<br/>=\left(\frac{x_{1}}{2}, \frac{y_{1}+3}{2}\right)
<br/>$$
<br/>since, $B$ lies on the circle $x^{2}+4 x+(y-3)^{2}=0$
<br/>$\therefore\left(\frac{x_{1}}{2}\right)^{2}+4\left(\frac{x_{1}}{2}\right)+\left(\frac{y_{1}+3}{2}-3\right)^{2}=0$
<br/>$\Rightarrow \quad \frac{x_{1}^{2}}{4}+2 x_{1}+\left(\frac{y_{1}-3}{2}\right)^{2}=0$
<br/>$\Rightarrow \quad \frac{x_{1}^{2}}{4}+2 x_{1}+\frac{y_{1}^{2}+9-6 y_{1}}{4}=0$
<br/>$\Rightarrow \quad x_{1}^{2}+y_{1}^{2}+8 x_{1}-6 y_{1}+9=0$
<br/>Hence, locus of a point is $x^{2}+y^{2}+8 x-6 y+9=0$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a44.png"/></div>


slug: a-circle-passing-through-0-0-2-6-6-2-cut-the-x-axis-at-the-point-p-0-0-then-the-54929
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: db\MarksBatch0.db
title:
A circle passing through (0,0),(2,6) (6,2) cut the x -axis at the point P î  = ( 0 , 0 ) Then, the lenght of OP , where O is the origin, is

question:
<div class="question">A circle passing through (0,0),(2,6) (6,2) cut the $x$ -axis at the point $P \neq(0,0)$ Then, the lenght of $O P,$ where $O$ is the origin, is</div>

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

answer:
<div class="solution">Let the equation of circle is
<br/>$$
<br/>x^{2}+y^{2}+2 g x+2 f y+c=0
<br/>$$
<br/>When, circle (i) passes through the onigin
<br/>Then, $\quad c=0$
<br/>When, circle (i) passes through the point (2,6). 
<br/>Then $4+36+4 g+12 f+0=0$
<br/>$\Rightarrow \quad 4 g+12 f+40=0$
<br/>$\Rightarrow g+3 f=-10$
<br/>When, circle (1) passes through the point (6,2) Then, $36+4+12 g+4 t+0=0$ [from Eq. $\Rightarrow \quad 12 g+4 f+40=0$
<br/>$\Rightarrow \quad 3 g+t+10=0$
<br/>On solving Eqs. (iii) and (iv), we get $g=\frac{-5}{2}$ and $f=\frac{-5}{2}$
<br/>$\therefore$ Equation of circle becomes
<br/>$$
<br/>x^{2}+y^{2}-5 x-5 y=0
<br/>$$
<br/>Circle cut the $x$ -axis.
<br/>So, put $y=0$ in Eq. $(v)$, we get
<br/>$$
<br/>x^{2}-5 x=0 \Rightarrow x(x-5)=0
<br/>$$
<br/>$\Rightarrow \quad x=5$
<br/>So, the circle cut the $x$ -axis at point $P(5,0)$
<br/>$\therefore$ The length of $O P=5$</div>


slug: a-circuit-consists-of-three-batteries-of-emf-e-1-1-v-e-2-2-v-and-e-3-3-v-and-int-40281
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A circuit consists of three batteries of emf E 1 â = 1 V , E 2 â = 2 V and E 3 â = 3 V and internal resistances 1Î© , 2Î© and 1Î© respectively which are connected in parallel as shown in the figure. The potential difference between points P and Q is

question:
<div class="question">A circuit consists of three batteries of emf $E_{1}=1 \mathrm{V}, E_{2}=2 \mathrm{V}$ and $E_{3}=3 \mathrm{V}$ and internal resistances $1 \Omega, 2 \Omega$ and $1 \Omega$ respectively which are connected in parallel as shown in the figure. The potential difference between points $P$ and $Q$ is  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q50.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.0 \mathrm{V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2.0 \mathrm{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.2 \mathrm{V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.0 \mathrm{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.0 \mathrm{V}$</span> </div>

answer:
<div class="solution">$1 \Omega .2 \Omega$ and $1 \Omega$ are in parallel
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a50.png"/> <br/> <br/>So, the required internal resistance
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r-{2}}+\frac{1}{r-{3}} \\
<br/>\frac{1}{r}=\frac{1}{1}+\frac{1}{2}+\frac{1}{1} \\
<br/>\frac{1}{r}=\frac{2+1+2}{2} \Rightarrow r=\frac{2}{5} \Omega
<br/>\end{array}
<br/>$$
<br/>The potential difference between points $P$ and $Q$
<br/>$\begin{aligned} E_{\text {dif }} &amp;=\frac{\frac{E_{1}}{r_{1}}+\frac{E_{2}}{5_{2}}+\frac{E_{3}}{5_{3}}}{1 / r}=\frac{1}{1}+\frac{2}{2}+\frac{3}{1} \\ &amp;=\frac{2+2+6}{5 / 2} \\=&amp; \frac{2}{5 / 2}=\frac{10 / 2}{5 / 2}=\frac{5}{5} \times 2=2 \mathrm{V} \end{aligned}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A circular coil is placed near a current carrying conductor, both lying on the plane of the paper. The current is flowing through the conductor in such a way that the induced current in the loop is clockwise as shown in the figure. The current in the wire is,

slug:
a-circular-coil-is-placed-near-a-current-carrying-conductor-both-lying-on-the-plane-of-the-paper-t-27116

question:
<div class="question">A circular coil is placed near a current carrying conductor, both lying on the plane of the paper. The current is flowing through the conductor in such a way that the induced current in the loop is clockwise as shown in the figure. The current in the wire is,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/rs1OIqN6mLeEe1nW-twqsJXSDUlQ55UR4ePCD19S514.original.fullsize.png"/><br/></div>

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

answer:
<div class="solution">Hint : If current is increasing in upward direction, so magnetic field is increasing in out of plane in order to oppose it induced current will be in clockwise direction<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/V7_WflNIFdTzwNmODCrVksKg4JXebG9Wlp4XI-sFO4M.original.fullsize.png"/><br/><br/>Similarly if current is decreasing in downward direction, so magnetic field is decreasing into the plane in order to support it induced current will be in clockwise direction.</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A circular disc rolls on a horizontal Iloor without slipping and the centre of the disc moves with a uniform velocity v . Which of the following values of the velocily at a point on the rim of the disc can have?

slug:
a-circular-disc-rolls-on-a-horizontal-iloor-without-slipping-and-the-centre-of-the-disc-moves-with-a-73378

question:
<div class="question">A circular disc rolls on a horizontal Iloor without slipping and the centre of the disc moves with a uniform velocity $v$. Which of the following values of the velocily at a point on the rim of the disc can have?</div>

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

answer:
<div class="solution">Consider the rolling motion of a circular disc as shown in the adjacent diagram. Velocity of point of contact $(P)$ is $v_{P}=v-\omega R=v-v=0$
<br/>$$
<br/>[\because v=\omega R]
<br/>$$
<br/>velocity of point ${ }^{\prime} S^{\prime}, v_{S}=v+\omega R=2 V$
<br/>velocity of centre ' $O^{\prime}, v_{0}=v-\omega(0)=V$ Velocity of a point ' $T$ at distance $R$ from $P$ is $V_{T}=\omega R=\mathrm{O}_{V}$
<br/>Thus possible velocities are $v, 2 v$ and $0 .$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a39.png"/></div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A circular loop of radius r of conducting wire connected with a voltage source of zero internal resistance produces a magnetic field B at its centre. If instead, a circular loep of radius 2 r , made of same material, having the same cross-section is connected to the same voltage source, what will be the magnetic ficld at its centre?

slug:
a-circular-loop-of-radius-r-of-conducting-wire-connected-with-a-voltage-source-of-zero-internal-resi-37007

question:
<div class="question">A circular loop of radius $r$ of conducting wire connected with a voltage source of zero internal resistance produces a magnetic field
<br/>$B$ at its centre. If instead, a circular loep of radius $2 r,$ made of same material, having the same cross-section is connected to the same voltage source, what will be the magnetic ficld at its centre?</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a28.png"/> <br/> <br/>$$
<br/>B=\frac{\mu_{0}}{2} \cdot \frac{I}{r}
<br/>$$
<br/>$\because$
<br/>$$
<br/>I_{1}=\frac{V}{R_{1}}
<br/>$$
<br/>Case (I) $l_{1}=2 \pi r$ (length of the loop).
<br/>$$
<br/>R_{1}=\rho \frac{l_{1}}{A} \text { and } B_{1}=\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{r}
<br/>$$
<br/>Case (II) $I_{2}=2 \pi \cdot 2 r=2 l_{1}$
<br/>$$
<br/>\begin{array}{l}
<br/>R_{2}=\rho \cdot \frac{l_{2}}{A}=\rho \cdot \frac{2 l_{1}}{A}=2 R_{1} \\
<br/>I_{2}=\frac{V}{R_{2}}=\frac{V}{2 R_{1}}=\frac{I_{1}}{2} \\
<br/>B_{2}=\frac{\mu_{0}}{2} \cdot \frac{I_{2}}{2 r}=\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{2 \cdot 2 r}
<br/>\end{array}
<br/>$$
<br/>From Eqs. (i) and (ii),
<br/>$$
<br/>\frac{B_{2}}{B_{1}}=\frac{\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{2 \cdot 2 r}}{\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{r}}=\frac{1}{4}
<br/>$$
<br/>Magnetic field $B_{2}=\frac{B_{1}}{4}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A circular loop of radius r of conducting wire connected with a voltage source of zero internal resistance produces a magnetic field B at its centre. If instead, a circular loep of radius 2 r , made of same material, having the same cross-section is connected to the same voltage source, what will be the magnetic ficld at its centre?

slug:
a-circular-loop-of-radius-r-of-conducting-wire-connected-with-a-voltage-source-of-zero-internal-resi-78214

question:
<div class="question">A circular loop of radius $r$ of conducting wire connected with a voltage source of zero internal resistance produces a magnetic field
<br/>$B$ at its centre. If instead, a circular loep of radius $2 r,$ made of same material, having the same cross-section is connected to the same voltage source, what will be the magnetic ficld at its centre?</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a28.png"/> <br/> <br/>$$
<br/>B=\frac{\mu_{0}}{2} \cdot \frac{I}{r}
<br/>$$
<br/>$\because$
<br/>$$
<br/>I_{1}=\frac{V}{R_{1}}
<br/>$$
<br/>Case (I) $l_{1}=2 \pi r$ (length of the loop).
<br/>$$
<br/>R_{1}=\rho \frac{l_{1}}{A} \text { and } B_{1}=\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{r}
<br/>$$
<br/>Case (II) $I_{2}=2 \pi \cdot 2 r=2 l_{1}$
<br/>$$
<br/>\begin{array}{l}
<br/>R_{2}=\rho \cdot \frac{l_{2}}{A}=\rho \cdot \frac{2 l_{1}}{A}=2 R_{1} \\
<br/>I_{2}=\frac{V}{R_{2}}=\frac{V}{2 R_{1}}=\frac{I_{1}}{2} \\
<br/>B_{2}=\frac{\mu_{0}}{2} \cdot \frac{I_{2}}{2 r}=\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{2 \cdot 2 r}
<br/>\end{array}
<br/>$$
<br/>From Eqs. (i) and (ii),
<br/>$$
<br/>\frac{B_{2}}{B_{1}}=\frac{\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{2 \cdot 2 r}}{\frac{\mu_{0}}{2} \cdot \frac{I_{1}}{r}}=\frac{1}{4}
<br/>$$
<br/>Magnetic field $B_{2}=\frac{B_{1}}{4}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints: \(p=1.1 .1 \cdot \frac{1}{2}=\frac{1}{2}\)</div>
source_db: db\MarksBatch0.db
title:
A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is

slug:
a-coin-is-tossed-again-and-again-if-tail-appears-on-first-three-tosses-then-the-chance-that-head-a-36845

question:
<div class="question">A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{16}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{1}{8}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{4}\)</span> </li> </ul>

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


tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">720</span> </div>
source_db: db\MarksBatch0.db
title:
A common emitter transistor amplifier is connected with a load resistance of 6 k Î© . When a small a.c. signal of 15 mV is added to the base emitter voltage, the alternating base current is 20 Î¼ A and the alternating collector current is 1.8 mA . What is the voltage gain of the amplifier?

slug:
a-common-emitter-transistor-amplifier-is-connected-with-a-load-resistance-of-6-k-when-a-small-a-24871

question:
<div class="question">A common emitter transistor amplifier is connected with a load resistance of $6 \mathrm{k} \Omega$. When a small a.c. signal of 15 $\mathrm{mV}$ is added to the base emitter voltage, the alternating base current is $20 \mu \mathrm{A}$ and the alternating collector current is $1.8 \mathrm{~mA}$. What is the voltage gain of the amplifier?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">90</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">640</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">900</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">720</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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: 
<br/>$R_{C}=6 \times 10^{3} \Omega$
<br/>$\Delta \mathrm{l}_{\mathrm{B}} \mathrm{R}_{\mathrm{B}}=15 \times 10^{-3} \mathrm{~V}$
<br/>$\Delta \mathrm{I}_{\mathrm{B}}=20 \times 10^{-6} \mathrm{~A}$
<br/>$\Delta \mathrm{l}_{\mathrm{C}}=1.8 \times 10^{-3} \mathrm{~A}$
<br/>$\mathrm{~A}_{\mathrm{v}}=\beta \times \frac{\mathrm{R}_{\mathrm{C}}}{\mathrm{R}_{\mathrm{B}}}=\frac{\Delta \mathrm{l}_{\mathrm{C}}}{\Delta_{\mathrm{B}}} \times \frac{\mathrm{R}_{\mathrm{C}}}{\mathrm{R}_{\mathrm{B}}}=\frac{1.8 \times 10^{-3}}{20 \times 10^{-6}} \times \frac{6 \times 10^{3} \times 20 \times 10^{-6}}{15 \times 10^{-3}}=720$</div>


tags: ['Chemistry', 'Solid State', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$XY$</span> </div>
source_db: db\MarksBatch0.db
title:
A compound formed by elements X and Y crystallises in the cubic structure, where X atoms are at the corners of a cube and Y atoms are at the centre of the body. The formula of the compounds is

slug:
a-compound-formed-by-elements-x-and-y-crystallises-in-the-cubic-structure-where-x-atoms-are-at-the-93604

question:
<div class="question">A compound formed by elements $X$ and $Y$ crystallises in the cubic structure, where $X$ atoms are at the corners of a cube and $Y$ atoms are at the centre of the body. The formula of the compounds is</div>

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

answer:
<div class="solution">Number of $X$ atoms at the corners $=8$ Number of $X$ atoms per unit cell $=8 \times \frac{1}{8}=1$ atom Number of $Y$ atoms at the centre of the body $=1$ atom Hence, the formula of the compound is $X Y$.</div>


