# questions

tags: ['Physics', 'Mathematics in Physics', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-moving-with-a-uniform-speed-v-in-a-circular-path-of-radius-r-with-the-centre-at-o-when-the-particle-moves-from-a-point-p-to-q-on-the-cir-1

slug:
a-particle-is-moving-with-a-uniform-speed-v-in-a-circular-path-of-radius-r-with-the-centre-at-o-when-the-particle-moves-from-a-point-p-to-q-on-the-cir-1-43984

question:
<div class="question">A particle is moving with a uniform speed $v$ in a circular path of radius $r$ with the centre at $O$. When the particle moves from a point $P$ to $Q$ on the circle such that $\angle P O Q=\theta$, then the magnitude of the change in velocity is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a23.png"/> <br/> <br/>$\therefore$ Change in magnitude of velocity
<br/>$\begin{aligned}
<br/>|\Delta v| &amp;=\sqrt{v^{2}+v^{2}-2 v^{2} \cos \theta} \\
<br/>&amp;=2 v \sin \frac{\theta}{2}
<br/>\end{aligned}$</div>


tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-moving-with-a-uniform-speed-v-in-a-circular-path-of-radius-r-with-the-centre-at-o-when-the-particle-moves-from-a-point-p-to-q-on-the-cir

slug:
a-particle-is-moving-with-a-uniform-speed-v-in-a-circular-path-of-radius-r-with-the-centre-at-o-when-the-particle-moves-from-a-point-p-to-q-on-the-cir-57285

question:
<div class="question">A particle is moving with a uniform speed $v$ in a circular path of radius $r$ with the centre at $O$. When the particle moves from a point $P$ to $Q$ on the circle such that $\angle P O Q=\theta$, then the magnitude of the change in velocity is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a23.png"/> <br/> <br/>$\therefore$ Change in magnitude of velocity
<br/>$\begin{aligned}
<br/>|\Delta v| &amp;=\sqrt{v^{2}+v^{2}-2 v^{2} \cos \theta} \\
<br/>&amp;=2 v \sin \frac{\theta}{2}
<br/>\end{aligned}$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-projected-at-6-0-to-the-horizontal-with-a-kinetic-energy-k-the-kinetic-energy-at-the-highest-point-is-1

slug:
a-particle-is-projected-at-6-0-to-the-horizontal-with-a-kinetic-energy-k-the-kinetic-energy-at-the-highest-point-is-1-79951

question:
<div class="question">A particle is projected at $60^{\circ}$ to the horizontal with a kinetic energy $\mathrm{K}$. The kinetic energy at the highest point is</div>

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

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

answer:
<div class="solution">At highest point kinetic energy $=1 / 2 \mathrm{~m}\left(\mathrm{v} \cos 60^{\circ}\right)^2=1 / 4 \times 1 / 2 \mathrm{~m} \mathrm{v} v^2=\mathrm{K} / 4$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-projected-from-the-ground-with-a-kinetic-energy-e-at-an-angle-of-6-0-with-the-horizontal-its-kinetic-energy-at-the-highest-point-of-its-

slug:
a-particle-is-projected-from-the-ground-with-a-kinetic-energy-e-at-an-angle-of-6-0-with-the-horizontal-its-kinetic-energy-at-the-highest-point-of-its-60646

question:
<div class="question">A particle is projected from the ground with a kinetic energy $E$ at an angle of $60^{\circ}$ with the horizontal. Its kinetic energy at the highest point of its motion will be</div>

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

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

answer:
<div class="solution">At ground, $E=\frac{1}{2} m u^{2}$ 
<br/>At highest point, $E^{\prime}=\frac{1}{2} m\left(u \cos 60^{\circ}\right)^{2}$
<br/>$$
<br/>E^{\prime}=\frac{E}{4}
<br/>$$</div>


title: a-particle-is-projected-vertically-upwards-if-it-has-to-stay-above-the-ground-for-12-seconds-then
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
slug:
a-particle-is-projected-vertically-upwards-if-it-has-to-stay-above-the-ground-for-12-seconds-then-35170

question:
<div class="question">A particle is projected vertically upwards. If it has to stay above the ground for 12 seconds, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">velocity of projection is $192 \mathrm{ft} / \mathrm{sec}$</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">B</span> <span class="option-data">greatest height attained is $600 \mathrm{ft}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">velocity of projection is $196 \mathrm{ft} / \mathrm{sec}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">greatest height attained is $576 \mathrm{ft}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">velocity of projection is $192 \mathrm{ft} / \mathrm{sec}$, greatest height attained is $576 \mathrm{ft}$</span> </div>

answer:
<div class="solution">Hint:
<br/>$V=u-g t$ at $t=6$
<br/>$u-g t=0$
<br/>$\Rightarrow u=6 g=192 f t /$ sec $\quad\left(g=32 f t / \sec ^{2}\right) \ldots \ldots . .(\mathrm{i})$
<br/>$x=u t=\frac{1}{2} g t^{2}$
<br/>$=192.6-\frac{1}{2} 32.6^{2}$
<br/>$=576 \mathrm{ft}$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-subjected-to-two-simple-harmonic-motions-in-the-same-direction-having-equal-amplitudes-and-equal-frequency-if-the-resultant-amplitude-is

slug:
a-particle-is-subjected-to-two-simple-harmonic-motions-in-the-same-direction-having-equal-amplitudes-and-equal-frequency-if-the-resultant-amplitude-is-97256

question:
<div class="question">A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motion, the phase difference $(\delta)$ between the two motions is</div>

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

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

answer:
<div class="solution">$A_{\text {res }}^2=A_1^2+A_2^2+2 A_1 A_2 \cos \delta \quad A^2=A^2+A^2+2 A^2 \cos \delta, \cos \delta=-\frac{1}{2} \delta=120^{\circ}$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-travelling-along-a-straight-line-ox-the-distance-x-in-metre-of-the-particle-from-0-at-a-time-t-is-given-by-x-37-27-t-t-3-where-t-is-time

slug:
a-particle-is-travelling-along-a-straight-line-ox-the-distance-x-in-metre-of-the-particle-from-0-at-a-time-t-is-given-by-x-37-27-t-t-3-where-t-is-time-18098

question:
<div class="question">A particle is travelling along a straight line $O X$. The distance $x$ (in metre) of the particle from 0 at a time t is given by $x=37+27 t-t^{3}$, where t is time in seconds. The distance of the particle from $O$ when it comes to rest is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$81 \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$91 \mathrm{m}$</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">$101 \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$111 \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$91 \mathrm{m}$</span> </div>

answer:
<div class="solution">Given, $x=37+27 t-t^{3}$
<br/>$$
<br/>v=\frac{d x}{d t}=27-3 t^{2}
<br/>$$
<br/>According to problem,
<br/>$$
<br/>v=0 \Rightarrow 27-3 t^{2}=0
<br/>$$
<br/>Here, $t=3 \mathrm{s}$
<br/>$$
<br/>x=37+27 \times 3-(3)^{2}=91 \mathrm{m}
<br/>$$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-particle-moves-along-x-axis-and-its-displacement-at-any-time-is-given-by-x-t-2-t-3-3-t-2-4-t-in-si-units-the-velocity-of-the-particle-when-its-accel

slug:
a-particle-moves-along-x-axis-and-its-displacement-at-any-time-is-given-by-x-t-2-t-3-3-t-2-4-t-in-si-units-the-velocity-of-the-particle-when-its-accel-92658

question:
<div class="question">A particle moves along $x$ -axis and its displacement at any time is given by $x(t)=2 t^{3}-3 t^{2}+4 t$ in SI units. The velocity of the particle when its acceleration is zero, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{ms}^{-1}$</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">B</span> <span class="option-data">$3.5 \mathrm{ms}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.54 \mathrm{ms}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$8.5 \mathrm{ms}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.5 \mathrm{ms}^{-1}$</span> </div>

answer:
<div class="solution">Given, $x(t)=2 t^{3}-3 t^{2}+4 t$
<br/>So, velocity, $v=\frac{d x}{d t}=\left(6 t^{2}-6 t+4\right)$
<br/>and acceleration $a=\frac{d v}{d t}=(12 t-6)$
<br/>when acceleration is zero, $i . e . .(12 t-6)=0$
<br/>or
<br/>$$
<br/>t=\frac{6}{12}=\frac{1}{2} \mathrm{s}
<br/>$$
<br/>$\therefore$ Velocity of the particle at zero acceleration is
<br/>$$
<br/>v=6\left(\frac{1}{2}\right)^{2}-6\left(\frac{1}{2}\right) 4=2.5 \mathrm{m} / \mathrm{s}
<br/>$$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-particle-moves-with-constant-acceleration-along-a-straight-line-starting-from-rest-the-percentage-increase-in-its-displacement-during-the-4th-second

slug:
a-particle-moves-with-constant-acceleration-along-a-straight-line-starting-from-rest-the-percentage-increase-in-its-displacement-during-the-4th-second-63207

question:
<div class="question">A particle moves with constant acceleration along a straight line starting from rest. The percentage increase in its displacement during the 4th second compared to that in the 3rd second is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$33 \%$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$40 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$66 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$77 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$40 \%$</span> </div>

answer:
<div class="solution">We know that
<br/>$S=u+\frac{1}{2} a(2 n-1)$
<br/>$S_{3rd \text { sec}}=0+\frac{1}{2} a(2 \times 3-1)=\frac{5}{2} a$ (for $n=3$ s)
<br/>$S_{4th \text { sec}}=0+\frac{1}{2} a(2 \times 4-1)=\frac{7}{2} a \quad($ for $n=4 \mathrm{s})$
<br/>So, the percentage increase
<br/>$=\frac{S_{4 t h}-S_{3 \mathrm{rd}}}{S_{3 \mathrm{rd}}} \times 100$
<br/>$\begin{array}{l} =\frac{\frac{7}{2} a-\frac{5}{2} a}{\frac{5}{2} a} \times 100 \\ =\frac{\frac{2 a}{2}}{\frac{5}{2}} \times 100=2 \times 20=40 \% \end{array}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-particle-moving-in-a-straight-line-starts-from-rest-and-the-acceleration-at-any-time-t-is-a-k-t-2-2-where-a-and-k-are-positive-constants-the-maximum

slug:
a-particle-moving-in-a-straight-line-starts-from-rest-and-the-acceleration-at-any-time-t-is-a-k-t-2-2-where-a-and-k-are-positive-constants-the-maximum-76521

question:
<div class="question">A particle moving in a straight line starts from rest and the acceleration at any time $\mathrm{t}$ is $a-k t^2{ }^2$ where a and $k$ are positive constants. The maximum velocity attained by the particle is</div>

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

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

answer:
<div class="solution">$\frac{d v}{d t}=a-k t^2=(\sqrt{a}-\sqrt{k t})(\sqrt{a}+\sqrt{k t})[\because a, k&gt;0]$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Pj31golzdsKMf-LD65bstw8wtbZ7iBHl5gB45M-9zNo.original.fullsize.png"/><br/><br/>$v$ will be $\max$ at $t=\sqrt{\frac{a}{k}}$<br/>$$<br/>\because v=a t-\frac{k}{3} t^3+C<br/>$$<br/>at $\mathrm{t}=0, \mathrm{v}=0 \Rightarrow \mathrm{C}=0$<br/>$\begin{aligned} \therefore v &amp;=a t-\frac{k}{3} t^3 \\ \therefore v_{\max } &amp;=a \sqrt{\frac{a}{k}}-\frac{k}{3} \frac{a}{k} \sqrt{\frac{a}{k}} \\ &amp;=\frac{2 a}{3} \sqrt{\frac{a}{\mathrm{k}}} \end{aligned}$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-initially-at-rest-is-accelerated-by-a-uniform-electric-field-e-through-a-distance-d-and-is-then-allowed-to-approach-

slug:
a-particle-of-mass-m-and-charge-q-initially-at-rest-is-accelerated-by-a-uniform-electric-field-e-through-a-distance-d-and-is-then-allowed-to-approach-76963

question:
<div class="question">A particle of mass $M$ and charge $q$, initially at rest, is accelerated by a uniform electric field $E$ through a distance $D$ and is then allowed to approach a fixed static charge $Q$ of the same sign. The distance of the closest approach of the charge $q$ will then be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{Q}{4 \pi \varepsilon_{0} E D}$</span> </div>

answer:
<div class="solution">At the distance of closest approach, the entire $\mathrm{KE}$ of a particle is converted into electric potential energy. i.e.. $\mathrm{KE}=\mathrm{PE}$
<br/>Here, KE of the particle = work done in moving the particle in uniform electric field (E) through a distance (D)
<br/>$$
<br/>=q E \times D
<br/>$$
<br/>PE of the particle = potential $\times$ charge
<br/>$$
<br/>=\frac{q}{4 \pi c_{0} \gamma_{0}} \times Q
<br/>$$
<br/>$\therefore \quad q E D=\frac{q Q}{4 \pi \varepsilon_{0} \gamma_{0}} \Rightarrow r_{0}=\frac{Q}{4 \pi \varepsilon_{0} E D}$</div>


tags: ['Physics', 'Oscillations', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-is-at-rest-at-the-mid-point-between-two-other-fixed-similar-charges-each-of-magnitude-q-placed-a-distance-2-d-apart--1

slug:
a-particle-of-mass-m-and-charge-q-is-at-rest-at-the-mid-point-between-two-other-fixed-similar-charges-each-of-magnitude-q-placed-a-distance-2-d-apart-1-62235

question:
<div class="question">A particle of mass $M$ and charge $q$ is at rest at the mid point between two other fixed similar charges each of magnitude $Q$ placed a distance $2 d$ apart. The system is collinear as shown in the figure. The particle is now displaced by a small amount $x(x&lt; &lt; d)$ along the joining the two charges and is left to itself. It will now oscillate about the mean position with a time period $\left(\varepsilon_{0}=\right.$ permittivity of free space)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q53.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \sqrt{\frac{\pi^{3} M \varepsilon_{0} d^{3}}{Q q}}$</span> </div>

answer:
<div class="solution">Restoring force on displacement of $x$
<br/>$\begin{aligned} F &amp;=K\left[\frac{q}{(d-x)^{2}}-\frac{Q q}{(d+x)^{2}}\right] \\ &amp;=K Q q\left[\frac{1}{(d-x)^{2}}-\frac{1}{(d+x)^{2}}\right] \\ &amp;=K Q q\left[\frac{4 d x}{\left(d^{2}-x^{2}\right)^{2}}\right] \\ &amp;=K Q q\left[\frac{4 d x}{d^{4}}\right] \text { If }(d&gt;&gt;x) \\ F &amp;=K Q q\left[\frac{4 x}{d^{3}}\right] \end{aligned}$
<br/>Acceleration, $a=\frac{F}{m}=\frac{4 K Q q x}{M d^{3}}$
<br/>or $\omega^{2}=\frac{4 K Q q}{M d^{3}}$
<br/>$\therefore \quad T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{M d^{3}}{4 K O q}}$
<br/>$=2 \sqrt{\frac{\pi^{3} M d^{3} \varepsilon_{0}}{Q q}}$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-is-at-rest-at-the-mid-point-between-two-other-fixed-similar-charges-each-of-magnitude-q-placed-a-distance-2-d-apart-

slug:
a-particle-of-mass-m-and-charge-q-is-at-rest-at-the-mid-point-between-two-other-fixed-similar-charges-each-of-magnitude-q-placed-a-distance-2-d-apart-92630

question:
<div class="question">A particle of mass $M$ and charge $q$ is at rest at the mid point between two other fixed similar charges each of magnitude $Q$ placed a distance $2 d$ apart. The system is collinear as shown in the figure. The particle is now displaced by a small amount $x(x&lt; &lt; d)$ along the joining the two charges and is left to itself. It will now oscillate about the mean position with a time period $\left(\varepsilon_{0}=\right.$ permittivity of free space)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q53.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \sqrt{\frac{\pi^{3} M \varepsilon_{0} d^{3}}{Q q}}$</span> </div>

answer:
<div class="solution">Restoring force on displacement of $x$
<br/>$\begin{aligned} F &amp;=K\left[\frac{q}{(d-x)^{2}}-\frac{Q q}{(d+x)^{2}}\right] \\ &amp;=K Q q\left[\frac{1}{(d-x)^{2}}-\frac{1}{(d+x)^{2}}\right] \\ &amp;=K Q q\left[\frac{4 d x}{\left(d^{2}-x^{2}\right)^{2}}\right] \\ &amp;=K Q q\left[\frac{4 d x}{d^{4}}\right] \text { If }(d&gt;&gt;x) \\ F &amp;=K Q q\left[\frac{4 x}{d^{3}}\right] \end{aligned}$
<br/>Acceleration, $a=\frac{F}{m}=\frac{4 K Q q x}{M d^{3}}$
<br/>or $\omega^{2}=\frac{4 K Q q}{M d^{3}}$
<br/>$\therefore \quad T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{M d^{3}}{4 K O q}}$
<br/>$=2 \sqrt{\frac{\pi^{3} M d^{3} \varepsilon_{0}}{Q q}}$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-is-released-from-rest-in-a-region-of-uniform-electric-field-magnitude-e-after-a-time-t-the-distance-travelled-by-the

slug:
a-particle-of-mass-m-and-charge-q-is-released-from-rest-in-a-region-of-uniform-electric-field-magnitude-e-after-a-time-t-the-distance-travelled-by-the-12309

question:
<div class="question">A particle of mass $M$ and charge $q$ is released from rest in a region of uniform electric field magnitude $E$. After a time $t$, the distance travelled by the charge is $S$ and the kinetic energy attained by the particle is $T$. Then, the ratio $T / S$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">remains constant with time t</span> </div>

answer:
<div class="solution">Given, mass of the particle $=M$.
<br/>Charge on the particle $=q$
<br/>Electric field $=E$
<br/>Initial velocity, $u=0$
<br/>$\therefore$ Acceleration, $a=\frac{F}{M}=\frac{q E}{M}$
<br/>:.Distance travelled in electric field, $S=u t+\frac{1}{2} a t^{2}$
<br/>$$
<br/>S=\frac{1}{2}\left(\frac{q E}{M}\right) t^{2}
<br/>$$
<br/>Also, kinetic energy $T=\frac{1}{2} M\left(\frac{q E t}{M}\right)^{2}$
<br/>So, $\quad \frac{T}{S}=\frac{\frac{1}{2} M\left(\frac{q E t}{M}\right)^{2}}{\frac{1}{2}\left(\frac{q E}{M}\right) t^{2}}=q E$
<br/>$\Rightarrow$ Ratio of $\frac{T}{S}$ remains constant with time t.</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-moving-with-velocity-v-enters-regionb-form-regiona-along-the-normal-to-the-boundary-as-shown-is-the-figure-regionb-h

slug:
a-particle-of-mass-m-and-charge-q-moving-with-velocity-v-enters-regionb-form-regiona-along-the-normal-to-the-boundary-as-shown-is-the-figure-regionb-h-20977

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/vortu4tbfcKuUfrfiJLJTx2yGEtKmdFW4vY7gY8hRfo.original.fullsize.png"/><br/> <br/>A particle of mass $\mathrm{m}$ and charge $\mathrm{q}$ moving with velocity $\mathrm{v}$ enters region-b form region-a along the normal to the boundary as shown is the figure. Region-b has a uniform magnetic field $B$ perpendicular to the plane of the paper. Also, region-b has length L. Choose the correct statements :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The particle enters region-c only if $\mathrm{v}&gt;\frac{\mathrm{qLB}}{\mathrm{m}}$</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">The particle enters region-c only if $\mathrm{v}     &lt; \frac{\mathrm{qLB}}{\mathrm{m}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Path of the particle is a circle in region-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="correct"> <span class="option-label">D</span> <span class="option-data">Time spent in region -b is independent of velocity 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 answers are:
<span class="option-value">The particle enters region-c only if $\mathrm{v}&gt;\frac{\mathrm{qLB}}{\mathrm{m}}$, Path of the particle is a circle in region-b, Time spent in region -b is independent of velocity v</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kONNlzMjqI7mEKR3f9dGf47hzwufQFI2TV7WvsmFoHI.original.fullsize.png"/><br/> <br/>Particle enter region 'c' if
<br/>Radius is greater than ' $L$ ' $\Rightarrow \frac{m V}{q B}&gt;L \Rightarrow V&gt;\frac{q B L}{m}$
<br/>$\because$ B is uniform and $\overrightarrow{\mathrm{V}} \perp \overrightarrow{\mathrm{B}}$ so.
<br/>Path is circular in region b
<br/>$$
<br/>\text { Time spent }=\frac{\text { Angle cov ered }}{\omega} \quad \omega=\frac{\mathrm{qB}}{\mathrm{m}}
<br/>$$
<br/>Now angle covered depens on whether particle goes to region c or comes back to region 'a'</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-is-attached-to-three-identical-massless-springs-of-spring-constant-k-as-shown-in-the-figure-the-time-period-of-vertical-oscillati

slug:
a-particle-of-mass-m-is-attached-to-three-identical-massless-springs-of-spring-constant-k-as-shown-in-the-figure-the-time-period-of-vertical-oscillati-81955

question:
<div class="question">A particle of mass $m$ is attached to three identical massless springs of spring constant ' $k$ ' as shown in the figure. The time period of vertical oscillation of the particle is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hA0zxfPnEdI2XfsxtV7Ur2SgrGSmOkCpvMmSIz-W3Ns.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } \mathrm{T}=2 \pi \sqrt{\frac{m}{\mathrm{~K}_{\mathrm{eq}}}} \\<br/>&amp; \mathrm{F}=\mathrm{K} x+2 \mathrm{~K} x \cos ^2 45 \\<br/>&amp; \mathrm{~K}_{\mathrm{eq}} x=\mathrm{K} x+\mathrm{K} x \\<br/>&amp; \mathrm{~K}_{\mathrm{eq}}=2 \mathrm{~K}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-m-is-located-in-a-one-dimensional-potential-field-where-potential-energy-is-given-by-mathrmvmathrmxmathrma1cos-mathrmpx-where-mathr

slug:
a-particle-of-mass-m-is-located-in-a-one-dimensional-potential-field-where-potential-energy-is-given-by-mathrmvmathrmxmathrma1cos-mathrmpx-where-mathr-62413

question:
<div class="question">A particle of mass \(m\) is located in a one dimensional potential field where potential energy is given by : \(\mathrm{V}(\mathrm{x})=\mathrm{A}(1-\cos \mathrm{px})\), where \(\mathrm{A}\) and \(\mathrm{p}\) are constants. The period of small oscillations of the particle is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2 \pi \sqrt{\frac{m}{(A p)}}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(2 \pi \sqrt{\frac{m}{\left(A p^2\right)}}\)</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">\(2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{A}}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{2 \pi} \sqrt{\frac{A p}{m}}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \pi \sqrt{\frac{m}{\left(A p^2\right)}}\)</span> </div>

answer:
<div class="solution">Hints: \(v_x=A(1-\cos p x)\)<br/>\(F=-\frac{d u}{d x}=-A p \sin p x\)<br/>For small (x)<br/>\(\mathrm{F}=-\mathrm{AP}^2 \mathrm{x}\)<br/>\(a=-\frac{A p^2}{m} x \quad a=-\omega^2 x\)<br/>\(\omega=\sqrt{\frac{A P^2}{\mathrm{~m}}} \quad \therefore \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{m}}{A p^2}}\)</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-particle-of-mass-mathrmm-is-projected-at-a-velocity-mathrmu-making-an-angle-theta-with-the-horizontal-mathrmxaxis-if-the-angle-of-projection-theta-i

slug:
a-particle-of-mass-mathrmm-is-projected-at-a-velocity-mathrmu-making-an-angle-theta-with-the-horizontal-mathrmxaxis-if-the-angle-of-projection-theta-i-34857

question:
<div class="question">A particle of mass \(\mathrm{m}\) is projected at a velocity \(\mathrm{u}\), making an angle \(\theta\) with the horizontal ( \(\mathrm{x}\)-axis). If the angle of projection \(\theta\) is varied keeping all other parameters same, then magnitude of angular momentum (L) at its maximum height about the point of projection varies with \(\theta\) as,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/qI8HkP-sb45Pn2D2jDo55UMQmkHLpmGHyXHV8XWmWCE.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/chh5DDzepNwJFGx5KCz4EC3gvk5cZFXuUqIeQQUJCrY.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/rg0lWMp9RGWRxsyQIJXFMoUddqtnjVeRtL6kHdav-a0.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/k0rA1j27dXtlDD4BRopUZUTTNySXZZ5Efbqt-6dUHWo.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/k0rA1j27dXtlDD4BRopUZUTTNySXZZ5Efbqt-6dUHWo.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/synuolsRR_2m4zlCNmaAHBxYanZHcPneR14VlhQJ0vQ.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \mathrm{L}_0=m u \cos \theta \mathrm{H} \\<br/>&amp; =m u \cos \theta \frac{\mathrm{u}^2 \sin ^2 \theta}{2 \mathrm{~g}} \\<br/>&amp; =\frac{m u^3 \sin ^2 \theta \cos \theta}{2 \mathrm{~g}}<br/>\end{aligned}\)<br/>\(L\) is zero for \(\theta=0^{\circ}\) and \(\theta=\frac{\pi}{2}\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-particle-starts-at-the-origin-and-moves-l-unit-horizontally-to-the-right-and-reaches-p-1-then-it-moves-2-1-unit-vertically-up-and-2-reaches-p-2-then

slug:
a-particle-starts-at-the-origin-and-moves-l-unit-horizontally-to-the-right-and-reaches-p-1-then-it-moves-2-1-unit-vertically-up-and-2-reaches-p-2-then-46743

question:
<div class="question">A particle starts at the origin and moves l unit horizontally to the right and reaches $P_{1}$, then it moves $\frac{1}{2}$ unit vertically up and 2 reaches $P_{2}$, then it moves $\frac{1}{4}$ unit horizontally to right and reaches $P_{3}$, then it moves $\frac{1}{8}$ unit vertically down and reaches $P_{4}$, then it moves $\frac{1}{16}$ unit horizontally to right and reaches $P_{\mathrm{S}}$  and so on. Let $P_{n}=\left(x_{n}, y_{n}\right)$ and $\lim _{n \rightarrow \infty} x_{n}=\alpha$ and $\lim _{n \rightarrow \infty} y_{n}=\beta .$ Then, $(\alpha, \beta)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(2,3)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\frac{4}{3}, \frac{2}{5}\right)$</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">$\left(\frac{2}{5}, 1\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{4}{3}, 3\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{4}{3}, \frac{2}{5}\right)$</span> </div>

answer:
<div class="solution">According to the given information in question, we can draw the situation of particle at different stages as following
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a55.png"/> <br/>Here, $x_{1}=1, x_{2}=1, x_{3}=\frac{5}{4}, x_{4}=\frac{5}{4}$ and $x_{5}=\frac{21}{16}$
<br/>$y_{1}=0, y_{2}=\frac{1}{2}, y_{3}=\frac{1}{2}, y_{4}=\frac{3}{8}$ and $y_{5}=\frac{3}{8}$
<br/>$\therefore \quad x_{n}=1+\frac{1}{4}+\frac{1}{4^{2}}+\ldots \infty=\frac{1}{1-\frac{1}{4}}=\frac{1}{\frac{3}{4}}=\frac{4}{3}$
<br/>$\begin{aligned} \text { and } y_{n}=&amp; 0+\frac{1}{2}-\frac{1}{8}+\ldots \infty \\ &amp;=\frac{\frac{1}{2}}{1-\left(-\frac{1}{4}\right)}=\frac{\frac{1}{2}}{\left(1+\frac{1}{4}\right)}=\frac{\frac{1}{2}}{\frac{5}{4}}=\frac{2}{5} \\ \therefore \quad(\alpha, \beta)=\left(\frac{4}{3}, \frac{2}{5}\right) \end{aligned}$</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-10.33</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-particle-starts-moving-from-rest-from-a-fixed-point-in-a-fixed-direction-the-distance-s-from-the-fixed-point-at-a-timet-is-given-by-s-t-2-a-t-b-17-w

slug:
a-particle-starts-moving-from-rest-from-a-fixed-point-in-a-fixed-direction-the-distance-s-from-the-fixed-point-at-a-timet-is-given-by-s-t-2-a-t-b-17-w-25033

question:
<div class="question">A particle starts moving from rest from a fixed point in a fixed direction. The distance s from the fixed point at a timet is given by $s=t^{2}+a t-b+17,$ where $a$ and $b$ are real numbers. If the particle comes to rest after 5s at a distance of $s=25$ units from the fixed point, then values of $a$ and $b$ are. respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10, -33</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-10,-30</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-8.33 .</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">-10.33</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, $s=t^{2}+a t-b+17$ 
<br/>After 5 s. $\left[\frac{d s}{d t}\right]_{t=5}=[2 t+a]_{t=5}=0$
<br/>$\Rightarrow \quad a=-10$
<br/>At $t=5 \mathrm{s}, s=25$ units
<br/>$\therefore \quad 25=5^{2}+5 a-b+17$
<br/>$$
<br/>\mid \because s=25 \text { and } t=5 \mid
<br/>$$
<br/>$\Rightarrow \quad 25=25+5(-10)-b+17$
<br/>$\Rightarrow \quad 50-17=-b$
<br/>$\Rightarrow \quad b=-33$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-particle-vibrating-simple-harmonically-hat-an-acceleration-of-16-cms-2-when-it-is-at-distance-of-4-cm-from-the-mean-position-lt-time-period-is

slug:
a-particle-vibrating-simple-harmonically-hat-an-acceleration-of-16-cms-2-when-it-is-at-distance-of-4-cm-from-the-mean-position-lt-time-period-is-56672

question:
<div class="question">A particle vibrating simple harmonically hat an acceleration of $16 \mathrm{cms}^{-2}$ when it is at distance of $4 \mathrm{cm}$ from the mean position, lt time period is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 \mathrm{s}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.572 \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3.142 \mathrm{s}$</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">D</span> <span class="option-data">$6.028 \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.142 \mathrm{s}$</span> </div>

answer:
<div class="solution">In SHM, the acceleration of vibrating particle is proportional to displacement
<br/>$$
<br/>\begin{aligned}
<br/>|\mathbf{a}| &amp;=\omega^{2} x \\
<br/>\text { Here } a=16 \mathrm{cm} / \mathrm{s}^{2} \\
<br/>&amp;=16 \times 10^{-2} \mathrm{m} / \mathrm{s}^{2}
<br/>\end{aligned}
<br/>$$
<br/>Displacement, $x=4 \mathrm{cm}=4 \times 10^{-2} \mathrm{m}$
<br/>$\therefore \quad 16 \times 10^{-2}=\omega^{2}\left(4 \times 10^{-2}\right)$
<br/>$$
<br/>\begin{aligned}
<br/>\omega^{2} &amp;=4 \\
<br/>\omega &amp;=\pm 2 \mathrm{rad} / \mathrm{s} \\
<br/>\omega &amp;=\frac{2 \pi}{T} \\
<br/>T &amp;=\frac{2 \pi}{\omega}=\frac{2 \pi}{2}=\pi=3.142 \mathrm{s}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-particle-with-charge-e-and-mass-m-moving-along-the-x-axis-with-a-uniform-speed-u-enters-a-region-where-a-uniform-electric-field-e-is-acting-along-th

slug:
a-particle-with-charge-e-and-mass-m-moving-along-the-x-axis-with-a-uniform-speed-u-enters-a-region-where-a-uniform-electric-field-e-is-acting-along-th-26004

question:
<div class="question">A particle with charge $e$ and mass $m$, moving along the $X$ -axis with a uniform speed $u$, enters a region where a uniform electric field $E$ is acting along the $Y$ -axis. The particle starts to move in a parabola. Its focal length (neglecting any effect of gravity) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2 m u^{2}}{e E}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{e E}{2 m u^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{m u}{2 e E}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{m u^{2}}{2 e E}$</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">$\frac{m u^{2}}{2 e E}$</span> </div>

answer:
<div class="solution">From parabola
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a33.png"/> <br/> <br/>$y=\frac{1}{2} \times\left[\frac{E e}{m}\right] \times \frac{x^{2}}{u^{2}}$
<br/>$=\frac{E \cdot e}{2 m u^{2}} \cdot X^{2}$
<br/>$\begin{array}{ll}\text { As } x=4 a y &amp; \text { (for parabola) }\end{array}$
<br/>$\therefore \quad X^{2}=\frac{2 m u^{2}}{E \cdot e} \cdot y$
<br/>$a=\frac{2 m u^{2}}{4 E \cdot e}=\frac{m u^{2}}{2 E \cdot e}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-particle-with-charge-q-coulomb-tied-at-the-end-of-an-inextensible-string-of-length-r-metre-revolves-in-a-vertical-plane-at-the-centre-of-the-circula

slug:
a-particle-with-charge-q-coulomb-tied-at-the-end-of-an-inextensible-string-of-length-r-metre-revolves-in-a-vertical-plane-at-the-centre-of-the-circula-63362

question:
<div class="question">A particle with charge $Q$ coulomb, tied at the end of an inextensible string of length $R$ metre, revolves in a vertical plane. At the centre of the circular trajectory, there is a fixed charge of magnitude $Q$ coulomb. The mass of the moving charge $M$ is such that $M g=\frac{Q^{2}}{4 \pi \in_{0} R^{2}}$. If at the highest position of the particle, the tension of the string just vanishes, the horizontal velocity at the lowest point has to be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{g R}$</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">C</span> <span class="option-data">$\sqrt{2 g R}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{5 g R}$</span> </li> </ul>

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

answer:
<div class="solution">According to the question, when tension in the string is zero.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a31.png"/> <br/> <br/>As $T=0$ $M g-\frac{K Q^{2}}{R^{2}}=\frac{m v^{2}}{R} \quad\left(k=\frac{1}{4 \pi \varepsilon_{0}}\right)$
<br/>$\frac{m v^{2}}{R}=$ centripetal force required to keep the body in a circular path
<br/>$\Rightarrow \quad v=0$$\quad\left[\because M g=\frac{k Q^{2}}{R^{2}}\right]$
<br/>So the required work done to keep charge in vertical motion
<br/>$\therefore \quad W_{g}=\Delta K E=$ change in $\mathrm{K.E.}$
<br/>$\Rightarrow \quad m g(2 R)=\frac{1}{2} m v_{0}^{2}$
<br/>\begin{array}{l}
<br/>v_{0}^{2}=2 g \cdot 2 R=4 g R \\
<br/>v_{0}=\sqrt{4 g R}=2 \sqrt{g R}
<br/>\end{array}</div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-particle-with-charge-q-moves-with-a-velocity-v-in-a-direction-perpendicular-to-the-directions-of-uniform-clectric-and-magnetic-ficlds-e-and-b-respec-1

slug:
a-particle-with-charge-q-moves-with-a-velocity-v-in-a-direction-perpendicular-to-the-directions-of-uniform-clectric-and-magnetic-ficlds-e-and-b-respec-1-53011

question:
<div class="question">A particle with charge $q$ moves with a velocity $v$ in a direction perpendicular to the directions of uniform clectric and magnetic ficlds, $E$ and $B$ respectively, which are mutually perpendicular to each other. Which one of the following gives the condition for which the particle moves undeflected in its original trajectory?  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q34.png"/></div>

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

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

answer:
<div class="solution">According to the question,
<br/>Charge on particle is $=q$ 
<br/>Velocity of particle $=v$ 
<br/>Due to uniform electric field,
<br/>electric force on particle
<br/>$F_{\text {electric }}=q E$ 
<br/>Due to a uniform magnetic field, magnetic force on particle is given by,
<br/>$$
<br/>F_{\text {magnetic }}=q(v \times B)
<br/>$$
<br/>When $v$ is perpendicular to $E$ and $B,$ which are mutually perpendicular to each other, $F_{\text {electric }}=F_{\text {magnetic }}$ 
<br/>From Eqs. (i) and (ii)
<br/>$$
<br/>q E=q v B
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>v=\frac{E}{B}
<br/>$$
<br/>So, $v=\frac{E}{B}$ is the condition for which the particle moves undeflected in its original trajectory.</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-particle-with-charge-q-moves-with-a-velocity-v-in-a-direction-perpendicular-to-the-directions-of-uniform-clectric-and-magnetic-ficlds-e-and-b-respec

slug:
a-particle-with-charge-q-moves-with-a-velocity-v-in-a-direction-perpendicular-to-the-directions-of-uniform-clectric-and-magnetic-ficlds-e-and-b-respec-64242

question:
<div class="question">A particle with charge $q$ moves with a velocity $v$ in a direction perpendicular to the directions of uniform clectric and magnetic ficlds, $E$ and $B$ respectively, which are mutually perpendicular to each other. Which one of the following gives the condition for which the particle moves undeflected in its original trajectory?  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q34.png"/></div>

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

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

answer:
<div class="solution">According to the question,
<br/>Charge on particle is $=q$ 
<br/>Velocity of particle $=v$ 
<br/>Due to uniform electric field,
<br/>electric force on particle
<br/>$F_{\text {electric }}=q E$ 
<br/>Due to a uniform magnetic field, magnetic force on particle is given by,
<br/>$$
<br/>F_{\text {magnetic }}=q(v \times B)
<br/>$$
<br/>When $v$ is perpendicular to $E$ and $B,$ which are mutually perpendicular to each other, $F_{\text {electric }}=F_{\text {magnetic }}$ 
<br/>From Eqs. (i) and (ii)
<br/>$$
<br/>q E=q v B
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>v=\frac{E}{B}
<br/>$$
<br/>So, $v=\frac{E}{B}$ is the condition for which the particle moves undeflected in its original trajectory.</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-person-goes-to-office-by-car-scooter-bus-and-train-probability-of-which-are-17-37-27-and-17-respectively-probability-that-he-reaches-office-late-if-

slug:
a-person-goes-to-office-by-car-scooter-bus-and-train-probability-of-which-are-17-37-27-and-17-respectively-probability-that-he-reaches-office-late-if-48905

question:
<div class="question">A person goes to office by car, scooter, bus and train, probability of which are $1 / 7,3 / 7$. $2 / 7$ and $1 / 7,$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $2 / 9,1 / 9,4 / 9,$ and $1 / 9,$ respectively. Given that he reached office in time, the probability that he travelled by a car, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 7$</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">$2 / 7$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 / 7$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 / 7$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 7$</span> </div>

answer:
<div class="solution">Let the event of person goes to office by a car, scooter, bus or train be $A, B, C$ and $D,$ respectively. We have, $P(A)=\frac{1}{7}, P(B)=\frac{3}{7}, P(C)=\frac{2}{7}$
<br/>and $P(D)=\frac{1}{7}$
<br/>Let $E=$ He reached office in time
<br/>We have
<br/>$P\left(\frac{\bar{E}}{A}\right)=\frac{2}{9}, P\left(\frac{\bar{E}}{B}\right)=\frac{1}{9}, P\left(\frac{\bar{E}}{C}\right)=\frac{4}{9}$
<br/>and $P\left(\frac{\bar{E}}{D}\right)=\frac{1}{9}$
<br/>$\begin{aligned} P\left(\frac{A}{E}\right)=&amp; P(A) \cdot P\left(\frac{E}{A}\right) \\ &amp;+P(C) \cdot P\left(\frac{E}{C}\right)+P(D) \cdot P\left(\frac{E}{D}\right) \\ &amp;=\frac{\frac{1}{7} \cdot \frac{7}{9}}{P(A) \cdot P\left(\frac{E}{A}\right)+P(B) \cdot P\left(\frac{E}{B}\right)} \\ &amp;=\frac{1}{7} \cdot \frac{7}{9}+\frac{3}{7} \cdot \frac{8}{9}+\frac{2}{7} \cdot \frac{5}{9}+\frac{1}{7} \cdot \frac{8}{9} \\=&amp; \frac{7}{7+24+10+8}=\frac{7}{49}=\frac{1}{7} \end{aligned}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-photon-of-wavelength-300-nm-interacts-with-a-stationary-hydrogen-atom-in-ground-state-during-the-interaction-whole-energy-the-photon-is-transferred-

slug:
a-photon-of-wavelength-300-nm-interacts-with-a-stationary-hydrogen-atom-in-ground-state-during-the-interaction-whole-energy-the-photon-is-transferred-71376

question:
<div class="question">A photon of wavelength 300 nm interacts with a stationary hydrogen atom in ground state. During the interaction, whole energy the photon is transferred to the electron of the atom. State which possibility is correst. (Consider, Plank constant $=4 \times 10^{-15} eV_{s},$ velocity of light $=3 \times 10^{8} \mathrm{m} / \mathrm{s},$ ionisation energy of hydrogen $=13.6 \mathrm{eV}$ )</div>

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

answer:
<div class="solution">The energy of the photon
<br/>$$
<br/>\begin{aligned}
<br/>E &amp;=\frac{h C}{\lambda} \\
<br/>&amp;=\frac{4 \times 10^{-15} \mathrm{eVs} \times 3 \times 10^{8} \mathrm{m} / \mathrm{s}}{300 \times 10^{-9} \mathrm{m}} \\
<br/>&amp;=\frac{4 \times 10^{2}}{300} \mathrm{eV}=\frac{4}{3} \mathrm{eV}=1.33 \mathrm{eV}
<br/>\end{aligned}
<br/>$$
<br/>The ionisation energy is $13.6 \mathrm{eV}$ which is greater than energy of photon, so atom can not come into excited state and will remain in ground state</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-piece-of-wood-from-an-archaeological-sample-has-50-counts-min-1-per-gram-of-c-14-while-a-fresh-sample-of-wood-has-a-count-of-150-min-1-g-1-if-halfli

slug:
a-piece-of-wood-from-an-archaeological-sample-has-50-counts-min-1-per-gram-of-c-14-while-a-fresh-sample-of-wood-has-a-count-of-150-min-1-g-1-if-halfli-79210

question:
<div class="question">A piece of wood from an archaeological sample has 5.0 counts min $^{-1}$ per gram of $\mathrm{C} - 14,$ while a fresh sample of wood has a count of $15.0 \mathrm{min}^{-1} \mathrm{g}^{-1}$. If half-life of $\mathrm{C}-14$ is $5770 \mathrm{yr}$ the age of the archaeological sample is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8.500 \mathrm{yr}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$9,200 \mathrm{yr}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10,000 \mathrm{yr}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$11.000 \mathrm{yr}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9,200 \mathrm{yr}$</span> </div>

answer:
<div class="solution">Age of archaeological sample.
<br/>$$
<br/>t=\frac{2.303}{k} \log \left(\frac{N_{0}}{N}\right)
<br/>$$
<br/>(Here, $N_{0}=$ initial activity. $\mathrm{N}=$ remaining activity)
<br/>$$
<br/>\begin{aligned}
<br/>k &amp;=\frac{0.693}{t_{1 / 2}} \\
<br/>\quad \text { (where } t_{1 / 2} &amp;=\text { half-life) } \\
<br/>&amp;=\frac{0.693}{5770} \mathrm{yr}^{-1}
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>t=\frac{2.303 \times 5770}{0.693} \log \left(\frac{15}{5}\right)
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>=19175.0 \log 3 \mathrm{yr} \\
<br/>=9148 \mathrm{yr} \approx 9200 \mathrm{yr}
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-plane-meets-the-coordinate-axes-at-the-points-a-b-c-respectively-in-such-a-way-that-the-centroid-of-a-bc-is-1-r-r-2-for-some-real-r-if-the-plane-pas

slug:
a-plane-meets-the-coordinate-axes-at-the-points-a-b-c-respectively-in-such-a-way-that-the-centroid-of-a-bc-is-1-r-r-2-for-some-real-r-if-the-plane-pas-93916

question:
<div class="question">A plane meets the co-ordinate axes at the points $A, B, C$ respectively in such a way that the centroid of $\triangle A B C$ is $\left(1, r, r^{2}\right)$ for some real $r$. If the plane passes through the point $(5,5,-12)$ then $r=$</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{3}{2}$, $-4$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/niBK8sUh3saAhtML2CYz-tUeXjKMcXVal-I26gX0ze8.original.fullsize.png"/><br/> <br/>Centroid is $\mathrm{G}(\mathrm{a} / 3, \mathrm{~b} / 3, \mathrm{c} / 3) \equiv\left(1, \mathrm{r}, \mathrm{r}^{2}\right)$
<br/>$a / 3=1 \quad \Rightarrow a=3, b=3 r, c=3 r^{2}$
<br/>Now,
<br/>$\begin{array}{l}
<br/>\frac{5}{a}+\frac{5}{b}-\frac{12}{c}=1 \\
<br/>\Rightarrow \frac{5}{3}+\frac{5}{3 r}-\frac{12}{3 r^{2}}=1 \\
<br/>\Rightarrow \frac{5 r^{2}+5 r-12}{3 r^{2}}=1 \\
<br/>\Rightarrow 5 r^{2}+5 r-12=3 r^{2} \\
<br/>\Rightarrow 2 r^{2}+5 r-12=0 \\
<br/>\Rightarrow r=\frac{-5 \pm \sqrt{25+96}}{4} \\
<br/>\Rightarrow r=\frac{-5 \pm 11}{4}=-4,3 / 2
<br/>\end{array}$</div>


title: a-plane-progressive-wave-is-given-by-y2-cos-6284330-mathrmtmathrmx-what-is-period-of-the-wave
slug: a-plane-progressive-wave-is-given-by-y2-cos-6284330-mathrmtmathrmx-what-is-period-of-the-wave-99882
tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
question:
<div class="question">A plane progressive wave is given by \(y=2 \cos 6.284(330 \mathrm{t}-\mathrm{x})\). What is period of the wave?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{330} \mathrm{~S}\)</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">B</span> <span class="option-data">\(2 \pi \times 330 \mathrm{~S}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((2 \pi \times 330)^{-1} \mathrm{~S}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{6.284}{330} \mathrm{~S}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{330} \mathrm{~S}\)</span> </div>

answer:
<div class="solution">Hints: \(y=2 \cos 2 \pi(330 t-x)\)<br/>\(\omega=2 \pi \times 330\)<br/>\(\therefore \mathrm{T}=\frac{1}{330} \mathrm{~s}\)</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-planet-moves-around-the-sun-in-an-elliptical-orbit-with-the-sun-at-one-of-its-foci-the-physical-quantity-associated-with-the-motion-of-the-planet-th-1

slug:
a-planet-moves-around-the-sun-in-an-elliptical-orbit-with-the-sun-at-one-of-its-foci-the-physical-quantity-associated-with-the-motion-of-the-planet-th-1-59865

question:
<div class="question">A planet moves around the sun in an elliptical orbit with the sun at one of its foci. The physical quantity associated with the motion of the planet that remains constant with time is</div>

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

answer:
<div class="solution">A planet revolves around the sun, is an elliptical orbit under the effect of gravitational pull on the planet. <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a16.png"/> <br/> <br/>So, torque, $C=r \times F=r F \sin 180^{\circ} n=0$
<br/>$$
<br/>\text{As } \mathrm{C}=\frac{\mathrm{dL}}{d t} ; \mathrm{soL}=\mathrm{a} \text { constant }
<br/>$$
<br/>$\Rightarrow$ Angular momentum is constant.</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-planet-moves-around-the-sun-in-an-elliptical-orbit-with-the-sun-at-one-of-its-foci-the-physical-quantity-associated-with-the-motion-of-the-planet-th

slug:
a-planet-moves-around-the-sun-in-an-elliptical-orbit-with-the-sun-at-one-of-its-foci-the-physical-quantity-associated-with-the-motion-of-the-planet-th-71059

question:
<div class="question">A planet moves around the sun in an elliptical orbit with the sun at one of its foci. The physical quantity associated with the motion of the planet that remains constant with time is</div>

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

answer:
<div class="solution">A planet revolves around the sun, is an elliptical orbit under the effect of gravitational pull on the planet. <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a16.png"/> <br/> <br/>So, torque, $C=r \times F=r F \sin 180^{\circ} n=0$
<br/>$$
<br/>\text{As } \mathrm{C}=\frac{\mathrm{dL}}{d t} ; \mathrm{soL}=\mathrm{a} \text { constant }
<br/>$$
<br/>$\Rightarrow$ Angular momentum is constant.</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-planoconcave-lens-is-made-of-glass-of-refractive-index-15-and-the-radius-of-curvature-of-its-curved-face-is-100-mathrmcm-what-is-the-power-of-the-le

slug:
a-planoconcave-lens-is-made-of-glass-of-refractive-index-15-and-the-radius-of-curvature-of-its-curved-face-is-100-mathrmcm-what-is-the-power-of-the-le-93139

question:
<div class="question">A plano-concave lens is made of glass of refractive index 1.5 and the radius of curvature of its curved face is \(100 \mathrm{~cm}\). What is the power of the lens?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(+0.5 \mathrm{D}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(-0.5 \mathrm{D}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\(-2 \mathrm{D}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(+2 \mathrm{D}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-0.5 \mathrm{D}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints: } P=\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right) \\<br/>&amp; =(1.5-1)\left(\frac{1}{\infty}-\frac{1}{1 \mathrm{~m}}\right) \\<br/>&amp; =0.5(-1) \\<br/>&amp; P=-0.5 \mathrm{D}<br/>\end{aligned}\)</div>


title: a-planoconvex-lens-f-20-cm-is-silvered-at-plane-surface-now-focal-length-will-be
slug: a-planoconvex-lens-f-20-cm-is-silvered-at-plane-surface-now-focal-length-will-be-66883
tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
question:
<div class="question">A plano-convex lens $(f=20 \mathrm{~cm})$ is silvered at plane surface. Now focal length will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$40 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } \mathrm{P}=2 \mathrm{P}_{\mathrm{L}}+\mathrm{P}_{\mathrm{M}} \\<br/>&amp; \mathrm{P}_{\mathrm{M}}=0 \\<br/>&amp; \mathrm{P}=\frac{1}{f} \times 2=\frac{2}{f} \\<br/>&amp; -\frac{1}{\mathrm{~F}}=\frac{2}{f} \\<br/>&amp; \mathrm{~F}=-\frac{f}{2}<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/rygb0J1g4hQUq9dfyWb0Hi0e2bAfGZ82kwxlIn9TCfY.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-plot-of-in-mathrmk-against-frac1mathrmt-abscissa-is-expected-to-be-a-straight-line-with-intercept-on-ordinate-axis-equal-to

slug:
a-plot-of-in-mathrmk-against-frac1mathrmt-abscissa-is-expected-to-be-a-straight-line-with-intercept-on-ordinate-axis-equal-to-90189

question:
<div class="question">A plot of In \(\mathrm{k}\) against \(\frac{1}{\mathrm{~T}}\) (abscissa) is expected to be a straight line with intercept on ordinate axis equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\Delta \mathrm{S}^{\circ}}{2.303 \mathrm{R}}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{\Delta \mathrm{S}^{\circ}}{\mathrm{R}}\)</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">C</span> <span class="option-data">\(-\frac{\Delta S^{\circ}}{R}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{R} \times \Delta \mathrm{S}^{\circ}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\Delta \mathrm{S}^{\circ}}{\mathrm{R}}\)</span> </div>

answer:
<div class="solution">Hints: \(\Delta \mathrm{G}^{\circ}=-\mathrm{RT} \operatorname{InK}\)<br/>or, \(\Delta \mathrm{H}^{\circ}-\mathrm{T} \Delta \mathrm{S}^{\circ}=-\mathrm{RT} \operatorname{InK}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/zGxcZa5a1eHSd8Tu-FQ_OF9yL_WHunsuW4gNtcGPI0o.original.fullsize.png"/><br/><br/>or, \(\operatorname{InK}=\frac{-\Delta \mathrm{H}^{\circ}}{\mathrm{RT}}+\frac{\Delta \mathrm{S}^{\circ}}{\mathrm{R}}\) comparing with \(\mathrm{y}=\mathrm{m} \cdot \mathrm{x}+\mathrm{c}\)<br/>\(\therefore \mathrm{y}\) intercept is \(\frac{\Delta \mathrm{S}^{\circ}}{\mathrm{R}}\)</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-point-charge-q-is-carried-from-a-point-a-to-another-point-b-on-the-axis-of-a-charged-ring-of-radius-r-carrying-a-charge-q-if-the-point-a-is-at-a-dis

slug:
a-point-charge-q-is-carried-from-a-point-a-to-another-point-b-on-the-axis-of-a-charged-ring-of-radius-r-carrying-a-charge-q-if-the-point-a-is-at-a-dis-98837

question:
<div class="question">A point charge $-q$ is carried from a point $A$ to another point $B$ on the axis of a charged ring of radius $r$ carrying a charge $+q$. If the point $A$ is at a distance $\frac{4}{3} r$ from the centre of the
<br/>ring and the point $B$ is $\frac{3}{4} r$ from the centre but
<br/>on the opposite side, what is the net work that need to be done for this?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{1}{5} \cdot \frac{q^{2}}{4 \pi \varepsilon_{0} I}$</span> </div>

answer:
<div class="solution">A charged circular ring of radius $R$ is shown in figure.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a23.png"/> <br/>From the figure, $d=\sqrt{x^{2}+r^{2}}$
<br/>$\therefore$ Potential due to a uniform ring of positive charge $q$ at point $P$
<br/>$$
<br/>\begin{aligned}
<br/>V &amp;=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{d} \\
<br/>&amp;=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{\sqrt{x^{2}+r^{2}}} \quad[\text { from Eq. }
<br/>\end{aligned}
<br/>$$
<br/>Now,<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a23a.png"/> <br/>$\therefore C B=\sqrt{O C^{2}+O B^{2}}=\sqrt{r^{2}+\left(\frac{3}{4}\right)^{2} r^{2}}=\frac{5}{4} r$
<br/>Similarly, $A C=\frac{5}{3} r$
<br/>$\therefore$ Work done in bringing a point charge $-q$ from $A$
<br/>to $B$,
<br/>$\begin{aligned} W &amp;=-q\left(V_{B}-V_{A}\right) \\ &amp;=-q\left[\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{C B}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{C A}\right] \\=&amp; \frac{-1}{4 \pi \varepsilon_{0}} \cdot q^{2}\left[\frac{1}{\frac{5}{4} r}-\frac{1}{5}\right] \\=&amp; \frac{-1}{4 \pi \varepsilon_{0}} \cdot \frac{q^{2}}{5 r}[4-3]=-\frac{1}{5} \cdot \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q^{2}}{r} \end{aligned}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">No Solution Available</div>
source_db: db\MarksBatch1_P1.db
title:
a-point-charge-q-is-placed-at-the-centre-of-a-cube-of-side-l-the-electric-flux-emerging-from-the-cube-is

slug:
a-point-charge-q-is-placed-at-the-centre-of-a-cube-of-side-l-the-electric-flux-emerging-from-the-cube-is-80705

question:
<div class="question">A point charge $+q$ is placed at the centre of a cube of side $\mathrm{L}$. The electric flux emerging from the cube is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{q}{\varepsilon_0}$</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><li class=""> <span class="option-label">B</span> <span class="option-data">Zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{6 q L^2}{\varepsilon_0}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{q}{6 L^2 \varepsilon_0}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{q}{\varepsilon_0}$</span> </div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-point-is-in-motion-along-a-hyperbola-y-x-10-so-that-its-abscissa-x-increases-uniformly-at-a-rate-of-1-unit-per-second-then-the-rate-of-change-of-its

slug:
a-point-is-in-motion-along-a-hyperbola-y-x-10-so-that-its-abscissa-x-increases-uniformly-at-a-rate-of-1-unit-per-second-then-the-rate-of-change-of-its-41350

question:
<div class="question">A point is in motion along a hyperbola $y=\frac{10}{x}$ so that its abscissa $x$ increases uniformly at a rate of 1 unit per second. Then, the rate of change of its ordinate when the point passes through (5,2)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">decreases at the rate of $\frac{2}{5}$ unit per second</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>\begin{array}{r}
<br/>\frac{d x}{d t}=\operatorname{l and} y=\frac{10}{x} \\
<br/>\text { Now, } y=\frac{10}{x}
<br/>\end{array}
<br/>$$
<br/>On differentiating both the sides w.r.t. $t$, we get
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow \quad \frac{d y}{d t} &amp;=\frac{-10}{x^{2}} \times \frac{d x}{d t} \\
<br/>&amp;=\frac{-10}{(5)^{2}} \times 1
<br/>\end{aligned}
<br/>$$
<br/>$$
<br/>=\frac{-10}{25}=\frac{-2}{5} \quad\left[x=5, \frac{d x}{d t}=1\right]
<br/>$$
<br/>$\therefore y$ decreases at the rate of $\frac{2}{5}$ unit per second.</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-point-moves-in-such-a-way-that-the-difference-of-its-distance-from-two-points-80-and-80-always-remains-4-then-the-locus-of-the-point-is

slug:
a-point-moves-in-such-a-way-that-the-difference-of-its-distance-from-two-points-80-and-80-always-remains-4-then-the-locus-of-the-point-is-26894

question:
<div class="question">A point moves in such a way that the difference of its distance from two points (8,0) and (-8,0) always remains $4 .$ Then, the locus of the point is</div>

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

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

answer:
<div class="solution">We know by the definition of hyperbola, "A hyperbola is the locus of a point which moves in such a way that the difference between two fixed points remains constant, ie., $|P S-P S|=2 a$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-point-moves-so-that-the-sum-of-squares-of-its-distance-from-the-points-12-and-21-is-always-6-then-its-locus-is

slug:
a-point-moves-so-that-the-sum-of-squares-of-its-distance-from-the-points-12-and-21-is-always-6-then-its-locus-is-32139

question:
<div class="question">A point moves, so that the sum of squares of its distance from the points (1,2) and (-2,1) is always $6 .$ Then, its locus is</div>

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

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

answer:
<div class="solution">Let $P$ be any point, whose coordinate is $(h, k)$
<br/>Given.
<br/> <br/>P moves, so that the sum of squares of its distances from the points $A(1,2)$ and $B(-2,1)$ is $6 .$
<br/>1.e., $(P A)^{2}+(P B)^{2}=6$
<br/>$\Rightarrow \quad(h-1)^{2}+(k-2)^{2}+(h+2)^{2}+(k-1)^{2}=6$
<br/>$\Rightarrow \quad h^{2}+1-2 h+k^{2}+4-4 k+h^{2}+4+4 h$
<br/>$$
<br/>+k^{2}+1-2 k=6
<br/>$$
<br/>$\Rightarrow \quad 2 h^{2}+2 k^{2}+2 h-6 k+4=0$
<br/>$\Rightarrow \quad h^{2}+k^{2}+h-3 k+2=0$
<br/>$\therefore$ Required locus is
<br/>$$
<br/>x^{2}+y^{2}+x-3 y+2=0
<br/>$$
<br/>Which represent a circle. Whose centre is $\left(\frac{-1}{2}, \frac{3}{2}\right)$ and radius $=\sqrt{\frac{1}{4}+\frac{9}{4}-2}=\sqrt{\frac{5}{2}-2}=\frac{1}{\sqrt{2}}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-point-object-is-held-above-a-thin-equiconvex-lens-at-its-focus-the-focal-length-is-01-m-and-the-lens-rests-on-a-horizontal-thin-plane-mirror-the-fin

slug:
a-point-object-is-held-above-a-thin-equiconvex-lens-at-its-focus-the-focal-length-is-01-m-and-the-lens-rests-on-a-horizontal-thin-plane-mirror-the-fin-42088

question:
<div class="question">A point object is held above a thin equiconvex lens at its focus. The focal length is $0.1 \mathrm{m}$ and the lens rests on a horizontal thin plane mirror. The final image will be formed at</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.1 \mathrm{m}$ above the center of the lens</span> </div>

answer:
<div class="solution">As the lens is placed on a horizontal thin plane mirror.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a20.png"/> <br/> <br/>So due to role of plane mirror the image will be formed at the same point $O$ above $0.1 \mathrm{m}$ distance above the center of the lens.</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-point-object-is-placed-on-the-axis-of-a-thin-convex-lens-of-focal-length-005-m-at-a-distance-of-02-m-from-the-lens-and-its-image-is-formed-on-the-ax

slug:
a-point-object-is-placed-on-the-axis-of-a-thin-convex-lens-of-focal-length-005-m-at-a-distance-of-02-m-from-the-lens-and-its-image-is-formed-on-the-ax-27330

question:
<div class="question">A point object is placed on the axis of a thin convex lens of focal length $0.05 \mathrm{m}$ at a distance of $0.2 \mathrm{m}$ from the lens and its image is formed on the axis. If the object is now made to oscillate along the axis with a small amplitude of $A \mathrm{cm},$ then what is the amplitude of oscillation of the image? $\left[\right.$ you may assume, $\frac{1}{1+x} \approx 1-x,$ where $\left.x&lt; &lt; 1\right]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{4 A}{9} \times 10^{-2} m$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5 A}{9} \times 10^{-2} m$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{A}{3} \times 10^{-2} m$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{A}{9} \times 10^{-2} \mathrm{m}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{A}{9} \times 10^{-2} \mathrm{m}$</span> </div>

answer:
<div class="solution">According to the question, we can draw the following diagram
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a28.png"/> <br/>Given, $u=-0.2 \mathrm{m}$ and $f=0.05 \mathrm{m}$
<br/>As we know that,
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{1}{f}=\frac{1}{v}-\frac{1}{u} \\
<br/>\frac{1}{v}=\frac{1}{f}+\frac{1}{u} \\
<br/>\frac{1}{v}=\frac{1}{0.05}-\frac{1}{0.2} \\
<br/>\frac{1}{v}=\frac{100}{5}-\frac{10}{2} \\
<br/>\frac{1}{v}=20-5 \Rightarrow v=\frac{1}{15} \mathrm{m}
<br/>\end{array}
<br/>$$
<br/>Now, differentiating eq. (i), we get
<br/>$$
<br/>\begin{array}{l}
<br/>-\frac{d v}{v^{2}}=-\frac{d u}{u^{2}} \quad \therefore \quad d v=d u \frac{v^{2}}{u^{2}} \\
<br/>A_{\max }=A \times\left(\frac{1}{15}\right)^{2} \times\left(\frac{1}{(-0.2)}\right)^{2} \\
<br/>A_{\max }=A \times \frac{1}{225} \times 25 \\
<br/>A_{\max }=\frac{A}{9}
<br/>\end{array}
<br/>$$
<br/>Here $A$ is in $\mathrm{cm} .$
<br/>Hence, $A_{\max }=\frac{A}{9} \times 10^{-2} \mathrm{m}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-point-p-lies-on-a-line-through-q-1-2-3-and-is-parallel-to-the-line-1-x-4-y-5-z-if-p-lies-on-the-plane-2-x-3-y-4-z-22-0-then-segment-pq-equals

slug:
a-point-p-lies-on-a-line-through-q-1-2-3-and-is-parallel-to-the-line-1-x-4-y-5-z-if-p-lies-on-the-plane-2-x-3-y-4-z-22-0-then-segment-pq-equals-26927

question:
<div class="question">A point $P$ lies on a line through $Q(1,-2,3)$ and is parallel to the line $\frac{x}{1}=\frac{y}{4}=\frac{z}{5},$ If $P$ lies on the plane $2 x+3 y-4 z+22=0,$ then segment PQ equals</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{42}$ units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{32}$ units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4 units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5 units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{42}$ units</span> </div>

answer:
<div class="solution">Equation of line through $Q(1,-2,3)$ and
<br/>parallel to the line $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is $\frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}=\lambda \quad$ (say)
<br/>since, point $P$ lies on above line.
<br/>$\therefore$
<br/>$$
<br/>P(\lambda+1,4 \lambda-2,5 \lambda+3)
<br/>$$
<br/>since, $P$ lies on the given plane. $\therefore \quad 2(\lambda+1)+3(4 \lambda-2)-4(5 \lambda+3)+22=0$
<br/>$\Rightarrow \quad 2 \lambda+2+12 \lambda-6-20 \lambda-12+22=0$
<br/>$$
<br/>\Rightarrow \quad-6 \lambda+6=0
<br/>$$
<br/>$\Rightarrow$
<br/>$\lambda=1$
<br/>$\therefore$
<br/>$P(2,2,8)$
<br/>$\therefore \quad P Q=\sqrt{(2-1)^{2}+(2+2)^{2}+(3-8)^{2}}$
<br/>$\Rightarrow \quad P Q=\sqrt{1+16+25}=\sqrt{42}$</div>


title: a-point-p-lies-on-the-circle-x-2-y-2-169-if-q-5-12-and-r-12-5-then-the-qpr-is
slug: a-point-p-lies-on-the-circle-x-2-y-2-169-if-q-5-12-and-r-12-5-then-the-qpr-is-20131
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
question:
<div class="question">A point $P$ lies on the circle $x^{2}+y^{2}=169$. If $Q=(5,12)$ and $R=(-12,5),$ then the $\angle Q P R$ is</div>

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

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

answer:
<div class="solution">Given equation of circle is
<br/>$$
<br/>x^{2}+y^{2}=169
<br/>$$
<br/>Its centre $=(0,0)$ and radius $=13$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a7.png"/> <br/> <br/>Now, slope of $O R=\frac{-5}{12}=m_{1}\left(\because\right.$ slope $\left.=\frac{y_{2}-y}{x_{2}-x}\right)$
<br/>and slope of $O Q=\frac{12}{5}=m_{2}$
<br/>$\because m_{1} \cdot m_{2}=-1 \Rightarrow \angle R O Q=\frac{\pi}{2}$
<br/>We know that, angle made by the chord of circle at circumference is equal to the half of the angle made by the same chord at the centre of circle.
<br/>$\therefore \quad \angle Q P R=\frac{1}{2} \cdot \angle R O Q=\frac{1}{2} \times \frac{\pi}{2}=\frac{\pi}{4}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-point-source-is-placed-at-coordinates-01-in-x-y-plane-a-ray-of-light-from-the-source-is-reflected-on-a-plane-mirror-placed-along-the-x-axis-and-perp

slug:
a-point-source-is-placed-at-coordinates-01-in-x-y-plane-a-ray-of-light-from-the-source-is-reflected-on-a-plane-mirror-placed-along-the-x-axis-and-perp-72585

question:
<div class="question">A point source is placed at coordinates (0,1) in $x y$ -plane. A ray of light from the source is reflected on a plane mirror placed along the
<br/>$X$ -axis and perpendicular to the $x y$ -plane. The reflected ray passes through the point $(3,3) .$ What is the path length of the ray from (0,1) to (3,3)$?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">$\sqrt{13}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1+2 \sqrt{3}$</span> </li> </ul>

answer:
<div class="solution">Ray diagram for the question.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a3.png"/> <br/> <br/>$I$ is image of source $S$ by the plane mirror placed perpendicularly along $X$ -axis.
<br/>$S M=I M$
<br/>$\therefore P M+M S=P M+M I=P I$
<br/>$\therefore \quad P I=\sqrt{(3-0)^{2}+(3+1)^{2}}$
<br/>$=\sqrt{9+16}=\sqrt{25}=5$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-point-source-of-light-is-used-in-an-experiment-of-photoelectric-effects-if-the-distance-between-the-source-and-the-photoelectric-surface-is-doubled-

slug:
a-point-source-of-light-is-used-in-an-experiment-of-photoelectric-effects-if-the-distance-between-the-source-and-the-photoelectric-surface-is-doubled-77697

question:
<div class="question">A point source of light is used in an experiment of photo-electric effects. If the distance between the source and the photo-electric surface is doubled, which of the following may result?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">photo-electric current will decrease</span> </div>

answer:
<div class="solution">Hint: 
<br/>Intensity $\mathrm{I} \propto \frac{1}{\text { d }}$
<br/>and photo-electric current $\propto$ intensity
<br/>$\therefore$ Current will decrease</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-poker-hand-consists-of-5-cards-drawn-at-random-from-a-wellshuffled-pack-of-52-cards-then-the-probability-that-a-poker-hand-consists-of-a-pair-and-a-

slug:
a-poker-hand-consists-of-5-cards-drawn-at-random-from-a-wellshuffled-pack-of-52-cards-then-the-probability-that-a-poker-hand-consists-of-a-pair-and-a-62232

question:
<div class="question">A poker hand consists of 5 cards drawn at random from a well-shuffled pack of 52 cards. Then, the probability that a poker hand consists of a pair and a triple of equal face values (for example, 2 sevens and 3 kings or 2 aces and 3 queens, etc. ) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{6}{4165}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{23}{4165}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1797}{4165}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{4165}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{6}{4165}$</span> </div>

answer:
<div class="solution">Required probability
<br/>\(\frac{{ }^{13} c_1 \times{ }^4 c_2 \times{ }^{12} c_1 \times{ }^4 c_3}{{ }^{52} c_5}\)
<br/>\(\frac{13 \times 6 \times 12 \times 4}{\frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1}}\)
<br/>$=\frac{13 \times 6 \times 12 \times 4}{52 \times 51 \times 10 \times 49 \times 2}$
<br/>$=\frac{6}{4165}$</div>


title: a-polygon-has-44-diagonals-the-number-of-its-sides-is-1
slug: a-polygon-has-44-diagonals-the-number-of-its-sides-is-1-81664
question: <div class="question">A polygon has 44 diagonals. The number of its sides is</div>
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: db\MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">13</span> </li> </ul>

answer:
<div class="solution">Hints: ${ }^n \mathrm{C}_2-n=44$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{n(n-1)}{2}-n=44 \\<br/>&amp; n\left[\frac{n-1}{2}-1\right]=44 \\<br/>&amp; n(n-3)=88 \\<br/>&amp; n(n-3)=11 \times 8 \\<br/>&amp; n=11<br/>\end{aligned}<br/>$$</div>


title: a-positive-acute-angle-is-divided-into-two-parts-whose-tangents-are-2-1-and-3-1-then-the-angle-is
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
slug:
a-positive-acute-angle-is-divided-into-two-parts-whose-tangents-are-2-1-and-3-1-then-the-angle-is-87937

question:
<div class="question">A positive acute angle is divided into two parts whose tangents are $\frac{1}{2}$ and $\frac{1}{3}$. Then the angle is</div>

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

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

answer:
<div class="solution">Hints : Angle $\theta=\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=\tan ^{-1}\left(\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \cdot \frac{1}{3}}\right)$<br/>$=\tan ^{-1}\left(\frac{5 / 6}{5 / 6}\right)=\tan ^{-1}(1)=\pi / 4$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-positive-charge-q-is-situated-at-the-centre-of-a-cube-the-electric-flux-through-any-face-of-the-cube-is-in-si-units

slug:
a-positive-charge-q-is-situated-at-the-centre-of-a-cube-the-electric-flux-through-any-face-of-the-cube-is-in-si-units-29886

question:
<div class="question">A positive charge $Q$ is situated at the centre of a cube. The electric flux through any face of the cube is (in SI units)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{Q}{6 \varepsilon_{0}}$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \pi Q$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{Q}{4 \pi \varepsilon_{0}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{Q}{6 \pi \varepsilon_{0}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{Q}{6 \varepsilon_{0}}$</span> </div>

answer:
<div class="solution">The positive charge $Q$ is situated at the centre of a cube
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a8.png"/> <br/>According to Gauss theorem, the flux from any closed surface
<br/>$$
<br/>\phi_{E}=\frac{Q}{\varepsilon_{0}}
<br/>$$
<br/>Cube has 6 faces, so electric flux from any face $\phi_{z}=\frac{Q}{6 \varepsilon_{0}}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-problem-in-mathematics-is-given-to-4-students-whose-chances-of-solving-individually-are-2-1-3-1-4-1-and-5-1-the-probability-that-the-problem-will-be

slug:
a-problem-in-mathematics-is-given-to-4-students-whose-chances-of-solving-individually-are-2-1-3-1-4-1-and-5-1-the-probability-that-the-problem-will-be-28495

question:
<div class="question">A problem in mathematics is given to 4 students whose chances of solving individually are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5},$ The probability that the problem will be solved at least by one student is</div>

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

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

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { (c) Let } P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\\
<br/>&amp;\begin{aligned}
<br/>P(C) &amp;=\frac{1}{4}, P(D)=\frac{1}{5} \\
<br/>\text { Now, } P(A&amp;\cup B \cup C \cup D) \\
<br/>&amp;=1-P(\bar{A} \cap \bar{B} \cap \bar{C} \cap \bar{D}) \\
<br/>&amp;=1-P(\bar{A}) P(\bar{B}) P(\bar{C}) P(\bar{D}) \\
<br/>&amp;=1-\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \\
<br/>&amp;=1-\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)=1-\frac{1}{5}=\frac{4}{5}
<br/>\end{aligned}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-progressive-wave-moving-along-x-axis-is-represented-by-y-a-sin-2-v-t-x-the-wavelength-at-which-the-maximum-particle-velocity-is-3-times-the-wave-vel

slug:
a-progressive-wave-moving-along-x-axis-is-represented-by-y-a-sin-2-v-t-x-the-wavelength-at-which-the-maximum-particle-velocity-is-3-times-the-wave-vel-55298

question:
<div class="question">A progressive wave moving along $x$ -axis is represented by $y=A \sin \left[\frac{2 \pi}{\lambda}(v t-x)\right]$ The
<br/>wavelength $(\lambda)$ at which the maximum particle velocity is 3 times the wave velocity is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(2 / 3) \pi A$</span> </div>

answer:
<div class="solution">$y=A \sin \left[\frac{2 \pi}{\lambda}(v t-x)\right]$
<br/>$v=\frac{2 \pi}{\lambda} V A$
<br/>$$
<br/>\begin{aligned}
<br/>\left(v_{p}\right)_{\max } &amp;=3 v \\
<br/>\lambda &amp;=\frac{2 \pi}{3} A
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-projectile-thrown-with-an-initial-velocity-of-10-ms-1-at-an-angle-with-the-horizontal-has-a-range-of-5-m-taking-g-10-ms-2-and-neglecting-air-resista

slug:
a-projectile-thrown-with-an-initial-velocity-of-10-ms-1-at-an-angle-with-the-horizontal-has-a-range-of-5-m-taking-g-10-ms-2-and-neglecting-air-resista-69602

question:
<div class="question">A projectile thrown with an initial velocity of $10 \mathrm{ms}^{-1}$ at an angle $\alpha$ with the horizontal. has a range of $5 \mathrm{m}$. Taking $g=10 \mathrm{ms}^{-2}$ and neglecting air resistance, what will he the estimated value of $\alpha$?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$15^{\circ}$</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">$30^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$45^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$75^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$15^{\circ}$, $75^{\circ}$</span> </div>

answer:
<div class="solution">Given, initial velocity of particle,
<br/>$$
<br/>u=10 \mathrm{ms}^{-1}
<br/>$$
<br/>Range, $R=5 \mathrm{m}$ Gravitational acceleration,
<br/>$$
<br/>g=10 \mathrm{ms}^{-2}
<br/>$$
<br/>As we known that, 
<br/>Range of projectile, $R=\frac{u^{2} \sin 2 \theta}{g}$
<br/>or
<br/>$$
<br/>\begin{aligned}
<br/>5 &amp;=\frac{(10)^{2} \sin 2 \alpha}{10} \\
<br/>\sin 2 \alpha &amp;=\frac{5}{10} \\
<br/>\sin 2 \alpha &amp;=\frac{1}{2} \\
<br/>\alpha &amp;=15^{\circ} \\
<br/>\left(90^{\circ}-15^{\circ}\right.&amp;\left.=75^{\circ}\right)
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-proton-and-an-electron-initially-at-rest-are-accelerated-by-the-same-potential-difference-assuming-that-a-proton-is-2000-times-heavier-than-an-elect

slug:
a-proton-and-an-electron-initially-at-rest-are-accelerated-by-the-same-potential-difference-assuming-that-a-proton-is-2000-times-heavier-than-an-elect-78344

question:
<div class="question">A proton and an electron initially at rest are accelerated by the same potential difference. Assuming that a proton is 2000 times heavier than an electron, what will be the relation between the de Broglie wavelength of the proton $\left(\lambda_{p}\right)$ and that of electron $\left(\lambda_{c}\right) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\lambda_{p}=\frac{\lambda_{0}}{20 \sqrt{5}}$</span> </div>

answer:
<div class="solution">As we know that de-Broglie wavelength is given as $\lambda=\frac{h}{p}=\frac{h}{m v}$
<br/>where, $h=$ Planck constant
<br/>$$
<br/>\begin{array}{l}
<br/>p=\text { momentum of particle } \\
<br/>v=\text { velocity of particle }
<br/>\end{array}
<br/>$$
<br/>and $m=$ mass of the particle.
<br/>Eq.(i) can be written as,
<br/>$$
<br/>\lambda=\frac{h}{2 m(\mathrm{KE})}=\frac{h}{\sqrt{2 m q v}} \quad[\because \mathrm{KE}=q v]
<br/>$$
<br/>where, $\mathrm{KE}=$ Kinetic energy of particle
<br/>Hence, $\lambda \propto \frac{1}{\sqrt{m}}$
<br/>$$
<br/>\text { Now, } \frac{\lambda_{\text {proton }}}{\lambda_{\text {electron }}}=\sqrt{\frac{m_{\text {electron }}}{m_{\text {proton }}}}
<br/>$$
<br/>Given, mass of proton, $m_{\text {proton }}=2000 \mathrm{m}_{\text {electron }}$
<br/>$\frac{\lambda_{\text {proton }}}{\lambda_{\text {electron }}}=\sqrt{\frac{1}{2000}}$
<br/>$\frac{\lambda_{\text {proton }}}{\lambda_{\text {electron }}}=\frac{1}{20 \sqrt{5}} \Rightarrow \lambda_{p}=\frac{\lambda_{e}}{20 \sqrt{5}}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-proton-is-moving-with-a-uniform-velocity-of-1-0-6-ms-1-along-the-y-axis-under-the-joint-action-of-a-magnetic-field-along-z-axis-and-an-electric-fiel

slug:
a-proton-is-moving-with-a-uniform-velocity-of-1-0-6-ms-1-along-the-y-axis-under-the-joint-action-of-a-magnetic-field-along-z-axis-and-an-electric-fiel-95368

question:
<div class="question">A proton is moving with a uniform velocity of $10^{6} \mathrm{ms}^{-1}$ along the $Y$ -axis, under the joint action of a magnetic field along $Z$ -axis and an electric field of magnitude $2 \times 10^{4} \mathrm{Vm}^{-1}$ along the negative $X$ -axis. If the electric field is switched off, the proton starts moving in a circle. The radius of the circle is nearly given: $\frac{e}{m}$ ratio for proton $=10^{4} \mathrm{Ckg}^{-1}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.5 \mathrm{m}$</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">$0.2 \mathrm{m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.1 \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.05 \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.5 \mathrm{m}$</span> </div>

answer:
<div class="solution">Velocity of proton $=10^{6} \mathrm{m} / \mathrm{s}$ along $y$ -direction
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a13.png"/> <br/> <br/>$$
<br/>\text { Electric field }=2 \times 10^{4} \mathrm{V} / \mathrm{m}
<br/>$$
<br/>$\frac{e}{=}$ for proton $=10^{-8} \mathrm{C} / \mathrm{kg}$
<br/>it
<br/>The magnetic field,
<br/>$$
<br/>\begin{aligned}
<br/>B &amp;=\frac{E}{v} \quad\left(\because q, v B=q_{\rho} E\right) \\
<br/>&amp;=\frac{2 \times 10^{4}}{10^{6}}=2 \times 10^{-2} \mathrm{T}
<br/>\end{aligned}
<br/>$$
<br/>The radius of circular path
<br/>$$
<br/>\begin{aligned}
<br/>\gamma &amp;=\frac{m v}{q_{p} \cdot B}=\frac{10^{-6} \times 10^{6}}{2 \times 10^{-2}} \\
<br/>&amp;=\frac{1}{2}=0 \cdot 5 \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-proton-of-mass-m-and-charge-q-is-moving-in-a-plane-with-kinetic-energy-e-if-there-exists-a-uniform-magnetic-field-b-perpendicular-to-the-plane-of-th

slug:
a-proton-of-mass-m-and-charge-q-is-moving-in-a-plane-with-kinetic-energy-e-if-there-exists-a-uniform-magnetic-field-b-perpendicular-to-the-plane-of-th-72083

question:
<div class="question">A proton of mass $m$ and charge $q$ is moving in a plane with kinetic energy $E$. If there exists a uniform magnetic field $B$, perpendicular to the plane of the motion, the proton will move in a circular path of radius</div>

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

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

answer:
<div class="solution">Given, Kinetic energy $=E$
<br/>Mass $=m$
<br/>Magnetic field $=B$
<br/>Charge $=q$ We know that
<br/>$$
<br/>F=q v B \sin \theta
<br/>$$
<br/>(motion of a charged particle in a uniform magnetic field) If
<br/>$$
<br/>\theta=90^{\circ}
<br/>$$
<br/>Then
<br/>$$
<br/>F=q v B
<br/>$$
<br/>We know that also (centripetal force)
<br/>$$
<br/>F=\frac{m v^{2}}{r}
<br/>$$
<br/>From Eqs. (i) and (ii), we get
<br/>$$
<br/>\begin{aligned}
<br/>q v B &amp;=\frac{m v^{2}}{r}, r=\frac{m v}{q B}\left[\because E=\frac{1}{2} m v^{2}, v=\sqrt{\frac{2 E}{m}}\right] \\
<br/>\therefore \quad r &amp;=\frac{m \sqrt{\frac{2 E}{m}}}{q B} \Rightarrow r=\frac{\sqrt{2 E m}}{q B}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$T$</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-proton-of-mass-m-moving-with-a-speed-v-c-velocity-of-light-in-vacuum-completes-circular-orbit-in-time-t-in-a-uniform-whetic-ficld-if-the-speed-of-th

slug:
a-proton-of-mass-m-moving-with-a-speed-v-c-velocity-of-light-in-vacuum-completes-circular-orbit-in-time-t-in-a-uniform-whetic-ficld-if-the-speed-of-th-34750

question:
<div class="question">A proton of mass $m$ moving with a speed $v$ ( $&lt; &lt; $ c. velocity of light in vacuum) completes circular orbit in time $T$ in a uniform Whetic ficld. If the speed of the proton is increased to $\sqrt{2}$ v, what will be time needed to plete the circular orbit?</div>

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

answer:
<div class="solution">$\because$ When a charged particle moves at a circular path in uniform magnetic field, it time period is independent from its speed.
<br/>Time period of proton will not change with change in its speed.</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-radioactive-atom-mathrmxmathrmy-emits-two-alpha-particles-and-one-beta-particle-successively-the-number-of-neutrons-in-the-nucleus-of-the-product-wi

slug:
a-radioactive-atom-mathrmxmathrmy-emits-two-alpha-particles-and-one-beta-particle-successively-the-number-of-neutrons-in-the-nucleus-of-the-product-wi-57419

question:
<div class="question">A radioactive atom \(\mathrm{X}_{\mathrm{Y}}\) emits two \(\alpha\) particles and one \(\beta\) particle successively. The number of neutrons in the nucleus of the product will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{X}-\mathrm{Y}-5\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9altCu2MQk-eikmdUk79yhUPIkGkkIJhrDoT4k4_PbE.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>\text { Number of neutrons } &amp; =\text { Mass no. }- \text { Atomic no. } \\<br/>&amp; =\mathrm{X}-8-\mathrm{Y}+3 \\<br/>&amp; =\mathrm{X}-\mathrm{Y}-5<br/>\end{aligned}\)</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-radioactive-nucleus-of-mass-number-a-initially-at-rest-emits-an-alphaparticle-with-a-speed-v-the-recoil-speed-of-the-daughter-nucleus-will-be

slug:
a-radioactive-nucleus-of-mass-number-a-initially-at-rest-emits-an-alphaparticle-with-a-speed-v-the-recoil-speed-of-the-daughter-nucleus-will-be-52653

question:
<div class="question">A radioactive nucleus of mass number A, initially at rest, emits an \(\alpha\)-particle with a speed \(v\). The recoil speed of the daughter nucleus will be</div>

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

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

answer:
<div class="solution">Hints : From conservation of momentum \(4 . \mathrm{V}=(\mathrm{A}-4) \mathrm{V}_1 ; \mathrm{V}=\frac{4 \mathrm{~V}}{\mathrm{~A}-4}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-ray-of-light-along-x-3-y-3-gets-reflected-upon-reaching-x-axis-the-equation-of-the-reflected-ray-is-1

slug:
a-ray-of-light-along-x-3-y-3-gets-reflected-upon-reaching-x-axis-the-equation-of-the-reflected-ray-is-1-71434

question:
<div class="question">A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching $x$-axis, the equation of the reflected ray is</div>

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

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

answer:
<div class="solution">$\mathrm{A} \equiv(\sqrt{3}, 0)$
<br/>Let, B $(0,1)$
<br/>$\therefore \mathrm{B}^{\prime} \equiv(0,-1)$
<br/>$\therefore$ Equation of $\mathrm{AB}^{\prime}: \frac{\mathrm{y}-0}{0+1}=\frac{\mathrm{x}-\sqrt{3}}{\sqrt{3}-0} \Rightarrow \sqrt{3} \mathrm{y}=\mathrm{x}-\sqrt{3}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6ToocRo59C-9cnQSlrto454exHR8Pf2dJPUh6dRb8qE.original.fullsize.png"/></div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-ray-of-light-is-incident-at-an-angle-i-on-a-glass-slab-of-refractive-index-the-angla-between-reflected-and-refracted-light-is-9-0-then-the-relations

slug:
a-ray-of-light-is-incident-at-an-angle-i-on-a-glass-slab-of-refractive-index-the-angla-between-reflected-and-refracted-light-is-9-0-then-the-relations-93228

question:
<div class="question">A ray of light is incident at an angle $i$ on a glass slab of refractive index $\mu$. The angla between reflected and refracted light is $90^{\circ}$ Then, the relationship between $i$ and $\mu$ is</div>

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

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

answer:
<div class="solution">As situation can be diagrammatically as below
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a16.png"/> <br/> <br/>From law of reflection
<br/>$$
<br/>i=\theta
<br/>$$
<br/>Now, $\quad \theta+r+90^{\circ}=180^{\circ}$
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow \quad i+r+90^{\circ} &amp;=180^{\circ} \\
<br/>r &amp;=90^{\circ}-i
<br/>\end{aligned}
<br/>$$
<br/>Also, from Snell's law $\frac{\sin i}{\sin r}=\mu \Rightarrow \frac{\sin i}{\sin \left(90^{\circ}-i\right)}=\frac{\sin i}{\cos i}=\mu$
<br/>$\Rightarrow \tan i=\mu$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-ray-of-light-is-incident-on-a-right-angled-isosceles-prism-parallel-to-its-base-as-shown-in-the-figure-refractive-index-of-the-material-of-the-prism

slug:
a-ray-of-light-is-incident-on-a-right-angled-isosceles-prism-parallel-to-its-base-as-shown-in-the-figure-refractive-index-of-the-material-of-the-prism-74201

question:
<div class="question">A ray of light is incident on a right angled isosceles prism parallel to its base as shown in the figure. Refractive index of the material of the prism is $\sqrt{2}$. Then. which of the following statement(s) is/are true? <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q37.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">The refiection at $P$ is total internal., The ray emerging at $R$ is parallel to the ray incident at $S .$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a37.png"/> <br/> <br/>Critical angle $\theta_{C}=\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)=45^{\circ}$
<br/>Snell's law at points, $\left(\because n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}\right)$
<br/>$1 \sin 45^{\circ}=\sqrt{2} \sin \theta$
<br/>$\therefore \quad \theta=30^{\circ}$
<br/>At $P \because$ incidence angle at $P=75^{\circ}&gt;\theta_{C}$
<br/>$\therefore$ TIR at point $P$
<br/>At $Q \because$ incidence angle at $Q=15^{\circ} &lt; \theta_{C}$
<br/>$\therefore$ Partial reflection and refraction at $Q .$ As $S P \| Q R$, emergent ray at $R$ is parallel to incid,
<br/>ray at $S$.
<br/>$\therefore$ Net deviation $=180^{\circ}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-ray-of-light-is-reflected-by-a-plane-mirror-e-0-e-and-n-be-the-unit-vectors-along-the-incident-ray-reflected-ray-and-the-normal-to-the-reflecting-su

slug:
a-ray-of-light-is-reflected-by-a-plane-mirror-e-0-e-and-n-be-the-unit-vectors-along-the-incident-ray-reflected-ray-and-the-normal-to-the-reflecting-su-37856

question:
<div class="question">A ray of light is reflected by a plane mirror. $\hat{e}_{0}, \hat{e}$ and $\hat{n}$ be the unit vectors along the incident ray, reflected ray and the normal to the reflecting surface respectively. Which of the following gives an expression for Ãª? <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q1.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\hat{e}_{0}+2\left(\hat{e}_{0} \cdot \hat{n}\right) \hat{n}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\hat{\boldsymbol{e}}_{0}-2\left(\hat{\boldsymbol{e}}_{0} \cdot \hat{\mathbf{n}}\right) \hat{\mathbf{n}}$</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">$\hat{\mathbf{e}}_{0}-\left(\hat{\boldsymbol{\theta}}_{0} \cdot \hat{\mathbf{n}}\right) \hat{\mathbf{n}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{d}) \hat{\mathrm{e}}_{0}+\left(\hat{e}_{0} \cdot \hat{n}\right) \hat{\mathrm{n}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\hat{\boldsymbol{e}}_{0}-2\left(\hat{\boldsymbol{e}}_{0} \cdot \hat{\mathbf{n}}\right) \hat{\mathbf{n}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a1.png"/> <br/>$\text { (b) According to the question, }$
<br/>We know that incident ray, reflected ray and normal lie in the same plane. and angle of incidence $=$ angle of reflection
<br/>Therefore $\hat{\mathrm{n}}$ will be along the angle bisector of e and $-\hat{e}_{0}$
<br/>i.e. $\hat{\mathbf{n}}=\frac{\hat{e}+\left(-\hat{\mathbf{e}}_{0}\right)}{\left|\hat{e}-\hat{\mathbf{e}}_{0}\right|}$
<br/>[$\because$ Bisector will along a vector dividing in same ratio as the ratio of sides forming that angle ]
<br/>But $\hat{\mathbf{n}}$ is a unit vector where $\left|\mathbf{e}-\hat{\mathbf{e}}_{0}\right|=O C$
<br/>$\begin{aligned}
<br/>&amp;=20 P=2|\hat{e}| \\
<br/>\cos \theta &amp;=2 \cos \theta
<br/>\end{aligned}$
<br/>Substituting this value in Eq. (i), we get
<br/>$\hat{\mathbf{n}}=\frac{\hat{\mathbf{e}}-\hat{\mathbf{e}}_{0}}{2 \cos \theta}$
<br/>$\begin{array}{l}
<br/>\hat{e}=\hat{e}_{0}+(2 \cos \theta) \hat{n} \\
<br/>\left.\hat{e}=\hat{e}_{0}-2 \hat{n} \cdot \hat{e}_{0}\right) \hat{n}\left[\because \hat{n} \cdot \hat{e}_{0}=-\cos \theta\right]
<br/>\end{array}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-ray-of-light-strikes-a-glass-plate-at-an-angle-of-6-0-if-the-reflected-and-refracted-rays-are-perpendicular-to-each-other-the-refractive-index-of-gl

slug:
a-ray-of-light-strikes-a-glass-plate-at-an-angle-of-6-0-if-the-reflected-and-refracted-rays-are-perpendicular-to-each-other-the-refractive-index-of-gl-42580

question:
<div class="question">A ray of light strikes a glass plate at an angle of $60^{\circ}$. If the reflected and refracted rays are perpendicular to each other, the refractive index of glass is</div>

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

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

answer:
<div class="solution">Here, angle of incidence is $60^{\circ}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a8.png"/> <br/> <br/>$$
<br/>\mu=\frac{\sin i}{\sin r}
<br/>$$
<br/>$\sin i=\mu \sin r$
<br/>here,
<br/>$$
<br/>\begin{aligned}
<br/>r &amp;=90-i \\
<br/>\sin i &amp;=\mu \sin \left(90^{\circ}-i\right)=\mu \cos i \\
<br/>\mu &amp;=\frac{\sin i}{\cos i}=\tan i=\tan 60^{\circ} \\
<br/>\mu &amp;=\sqrt{3}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-ray-of-monochromatic-light-is-incident-on-the-plane-surface-of-separation-between-two-media-x-and-y-with-angle-of-incidence-i-in-medium-x-and-angle-

slug:
a-ray-of-monochromatic-light-is-incident-on-the-plane-surface-of-separation-between-two-media-x-and-y-with-angle-of-incidence-i-in-medium-x-and-angle-99482

question:
<div class="question">A ray of monochromatic light is incident on the plane surface of separation between two media \(X\) and \(Y\) with angle of incidence 'i' in medium \(X\) and angle of refraction ' \(r\) ' in medium \(Y\). The given graph shows the relation between sin \(i\) and \(\sin r\). If \(V_x\) and \(V_{\mathrm{Y}}\) are the velocities of the ray in media \(X\) and \(Y\) respectively, then which of the following is true?<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dAsDIq2ls_rYexhLrwpuHK_X4L1cipk_XnZGbvdzg5M.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{V}_{\mathrm{X}}=\sqrt{3} \mathrm{~V}_Y\)</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/wgj5AwKlYqIguyrBjqKOH4t85s9GWX9QtgAXyTbvtOY.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \mu_1 \sin i=\mu_2 \sin r \\ &amp; \frac{C}{V_X} \sin i=\frac{C}{V_Y} \sin r \\ &amp; \frac{\sin i}{\sin r}=\frac{V_X}{V_Y}=\sqrt{3} \\ &amp; V_Y=\frac{V_X}{\sqrt{3}} \\ &amp; V_X=\sqrt{3} V_Y\end{aligned}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dUR9dpAbqUyP_XH0f8moX7iN45AoU5_I84xV6JhJkCA.original.fullsize.png"/><br/><br/>\(y=m x\)<br/>\(\sin r=\left(\tan 30^{\circ}\right) \sin \mathrm{i}\)<br/>\(\frac{\sin i}{\sin r}=\frac{1}{\tan 30^{\circ}}=\frac{\sqrt{3}}{1}\)</div>


tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-rectangle-a-b-c-d-has-its-side-parallel-to-the-line-y2-x-and-vertices-a-b-d-are-on-lines-y1-x1-and-x1-respectively-the-coordinate-of-c-can-be-1

slug:
a-rectangle-a-b-c-d-has-its-side-parallel-to-the-line-y2-x-and-vertices-a-b-d-are-on-lines-y1-x1-and-x1-respectively-the-coordinate-of-c-can-be-1-98611

question:
<div class="question">A rectangle \(A B C D\) has its side parallel to the line \(y=2 x\) and vertices \(A, B, D\) are on lines \(y=1, x=1\) and \(x=-1\) respectively. The coordinate of \(C\) can be</div>

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

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

answer:
<div class="solution">Hint: Case I<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/7S15ruluWKJKf4q7gz0TVtzk4FWRmq9rlUJpaE6BJ6E.original.fullsize.png"/><br/><br/>line \(y=2 x\ldots(1)\)<br/>\(A D\) parallel to \(y=2 x\)<br/>\(\Rightarrow \mathrm{M}_{\mathrm{AD}}=2 ~\&amp;~ \mathrm{M}_{\mathrm{AB}}=-\frac{1}{2}\)<br/>for \(a=3\)<br/>\(A(3,1), B(1,2), C(-3,-6), D(-1,-7)\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/SNGTmUw5DGV_rxGmmZ1zFxtWIdnL0AeToThWGu5cj2c.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \text { for } a=-3 \\ &amp; C(3,3), A(-3,1), B(1,-1), D(-1,5)\end{aligned}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/PtZaSYFJ2ZNQZ--gAKsKMdBc1tAkYbf9b6w2u5uovM0.original.fullsize.png"/><br/><br/>Case II AD perpendicular to line (1)<br/>\(\Rightarrow M_{A D}=\frac{1}{2}, M_{A B}=2 \Rightarrow b=3-2 a \Rightarrow d=\frac{a+1}{2}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ceJhDdZ9HLCuO8ldt5BLirocavCuA6WHKFv7XDYLN3Q.original.fullsize.png"/><br/><br/>for \(a=3, C(-3,-2)\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lGF7kYE2pR3NQr2VLXxNRvqPMtzXkAbjscTUNbTWNV8.original.fullsize.png"/><br/><br/>for \(a=-3, C(3,7)\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/SLzQYFuBUEUTTvLVWdKsGMUoyZ8vWdbJSGNC-dHkXaU.original.fullsize.png"/><br/><br/>\({ }^*\) In both cases if abscissa of \(C: 3\) or -3<br/>then co-ordinates of \(C\) can be \((3,3)\) or \((3,7)\) or \((-3,-6)\) or \((-3,-2)\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-rectangle-a-b-c-d-has-its-side-parallel-to-the-line-y2-x-and-vertices-a-b-d-are-on-lines-y1-x1-and-x1-respectively-the-coordinate-of-c-can-be

slug:
a-rectangle-a-b-c-d-has-its-side-parallel-to-the-line-y2-x-and-vertices-a-b-d-are-on-lines-y1-x1-and-x1-respectively-the-coordinate-of-c-can-be-41952

question:
<div class="question">A rectangle \(A B C D\) has its side parallel to the line \(y=2 x\) and vertices \(A, B, D\) are on lines \(y=1, x=1\) and \(x=-1\) respectively. The coordinate of \(C\) can be</div>

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

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

answer:
<div class="solution">Hint: Case I<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/7S15ruluWKJKf4q7gz0TVtzk4FWRmq9rlUJpaE6BJ6E.original.fullsize.png"/><br/><br/>line \(y=2 x\ldots(1)\)<br/>\(A D\) parallel to \(y=2 x\)<br/>\(\Rightarrow \mathrm{M}_{\mathrm{AD}}=2 ~\&amp;~ \mathrm{M}_{\mathrm{AB}}=-\frac{1}{2}\)<br/>for \(a=3\)<br/>\(A(3,1), B(1,2), C(-3,-6), D(-1,-7)\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/SNGTmUw5DGV_rxGmmZ1zFxtWIdnL0AeToThWGu5cj2c.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \text { for } a=-3 \\ &amp; C(3,3), A(-3,1), B(1,-1), D(-1,5)\end{aligned}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/PtZaSYFJ2ZNQZ--gAKsKMdBc1tAkYbf9b6w2u5uovM0.original.fullsize.png"/><br/><br/>Case II AD perpendicular to line (1)<br/>\(\Rightarrow M_{A D}=\frac{1}{2}, M_{A B}=2 \Rightarrow b=3-2 a \Rightarrow d=\frac{a+1}{2}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ceJhDdZ9HLCuO8ldt5BLirocavCuA6WHKFv7XDYLN3Q.original.fullsize.png"/><br/><br/>for \(a=3, C(-3,-2)\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lGF7kYE2pR3NQr2VLXxNRvqPMtzXkAbjscTUNbTWNV8.original.fullsize.png"/><br/><br/>for \(a=-3, C(3,7)\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/SLzQYFuBUEUTTvLVWdKsGMUoyZ8vWdbJSGNC-dHkXaU.original.fullsize.png"/><br/><br/>\({ }^*\) In both cases if abscissa of \(C: 3\) or -3<br/>then co-ordinates of \(C\) can be \((3,3)\) or \((3,7)\) or \((-3,-6)\) or \((-3,-2)\)</div>


title: a-rectangular-coil-carryingcurrent-is-placed-in-a-nonuniform-magnetic-field-on-that-coil-the-total
tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
slug:
a-rectangular-coil-carryingcurrent-is-placed-in-a-nonuniform-magnetic-field-on-that-coil-the-total-51945

question:
<div class="question">A rectangular coil carrying-current is placed in a non-uniform magnetic field. On that coil, the total</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">force is non-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><li class=""> <span class="option-label">B</span> <span class="option-data">force is zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">torque is zero</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">torque is non-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">force is non-zero, torque is non-zero</span> </div>

answer:
<div class="solution">Current-carrying rectangular coil is placed in a non-uniform magnetic field. So, the force on each arm of the coil will different giving the resultant force as non-zero.
<br/>Thus, option (a) is correct.
<br/>Also, the torque acting on current-carrying coil in non-uniform magnetic field will be non-zero.
<br/>Thus, option (d) is also correct.
<br/>following reason.</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-relation-on-the-set-of-real-number-r-is-defined-as-x-y-x-y-0-then-which-of-the-following-isare-true

slug:
a-relation-on-the-set-of-real-number-r-is-defined-as-x-y-x-y-0-then-which-of-the-following-isare-true-60568

question:
<div class="question">A relation $\rho$ on the set of real number $R$ is defined as $\{x \rho y: x y&gt;0\} .$ Then, which of the following is/are true?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\rho$ is symmetric but not reflexive, $\rho$ is symmetric and transitive</span> </div>

answer:
<div class="solution">We have, $x p y: x y &gt; 0$
<br/>(i) Reflexive Suppose $x \rho x \in R$
<br/>$\Rightarrow x^{2} &gt; 0$
<br/>which is not true when $x=0$. Hence, relation is not reflexive.
<br/>(ii) Symmetric $x p y \in R$ $\Rightarrow \quad x y&gt;0$
<br/>$\Rightarrow \quad y x&gt;0$
<br/>$\Rightarrow \quad y \rho x \in R$
<br/>If $(x, y) \in R,$ then, relation is symmetric.
<br/>(iii) Transitive $\mathrm{xp} \mathrm{y} \in \mathrm{R}$
<br/>$x y&gt;0$
<br/>and $\quad y \rho z \in R$ $y z&gt;0$
<br/>$\begin{array}{ll}<br/>\Rightarrow x y^{2} z &gt; 0 \\<br/>\Rightarrow x z &gt; 0 \\<br/>\Rightarrow (x, z) \in R \\<br/>\text {If }(x, y) \in R, \text { then }(y, z) \in R \\<br/>\Rightarrow (x, z) \in R<br/>\end{array}$
<br/>Hence, relation is transitive.</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-rifleman-is-firing-at-a-distant-target-and-has-only-10-chance-of-hitting-it-the-least-number-of-rounds-he-must-fire-to-have-more-than-50-chance-of-h

slug:
a-rifleman-is-firing-at-a-distant-target-and-has-only-10-chance-of-hitting-it-the-least-number-of-rounds-he-must-fire-to-have-more-than-50-chance-of-h-98821

question:
<div class="question">A rifleman is firing at a distant target and has only $10 \%$ chance of hitting it. The least number of rounds he must fire to have more than $50 \%$ chance of hitting it at least once, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">9</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">11</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>$\mathrm{P}$ (hitting a target) $=\frac{1}{10}$
<br/>$\therefore \mathrm{P}($ not hitting a target $)=\frac{9}{10}$
<br/>$\therefore$ Let number of trials $=n$
<br/>So, $\mathrm{P}($ hitting at least once $)=1-\mathrm{P}$ (missing all) $=1-\left(\frac{9}{10}\right)^{n} \geq \frac{1}{2}$
<br/>$\Rightarrow(0.9)^{n} \leq 0.5$
<br/>$(0.9)^{6}=0.531441,(0.9)^{7}=0.4782969 \Rightarrow \mathrm{n}=7$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-sample-of-hydrogen-atom-in-its-ground-state-is-radiated-with-photons-of-102-ev-energies-the-radiation-from-the-sample-is-absorbed-by-excited-ionized

slug:
a-sample-of-hydrogen-atom-in-its-ground-state-is-radiated-with-photons-of-102-ev-energies-the-radiation-from-the-sample-is-absorbed-by-excited-ionized-88758

question:
<div class="question">A sample of hydrogen atom in its ground state is radiated with photons of $10.2 \mathrm{eV}$ energies. The radiation from the sample is absorbed by excited ionized $\mathrm{He}^{+}$. 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">$\mathrm{He}^{+}$electron moves from $n=2$ to $n=4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">In the $\mathrm{He}^{+}$emission spectra, there will be 6 lines</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Smallest wavelength of $\mathrm{He}^{+}$spectrum is obtained when transition taken place from $n=4$ to $n=3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{He}^{+}$electron moves from $n=2$ to $n=3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{He}^{+}$electron moves from $n=2$ to $n=4$, Smallest wavelength of $\mathrm{He}^{+}$spectrum is obtained when transition taken place from $n=4$ to $n=3$</span> </div>

answer:
<div class="solution">$\begin{gathered}<br/>E=13.6 \times 2^2\left(\frac{1}{n^2}-\frac{1}{m^2}\right) \\<br/>n=2 \text { to } m=4 \\<br/>E=13.6 \times 2^2 \times\left(\frac{1}{2^2}-\frac{1}{4^2}\right)=10.2 \mathrm{eV}<br/>\end{gathered}$<br/>So $\mathrm{He}^{+}$electron can move from $n=2$ to $n=4$<br/>Number of spectral lines $=\frac{n(n-1)}{2}=\frac{4}{2}(4-1)=6$<br/>Smallest wavelength correspond to $n=4$ to $n=3$ transition</div>


tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-sample-of-mgco-3-is-dissolved-in-dil-hcl-and-the-solution-is-neutralized-with-ammonia-and-buffered-with-nh-cl-4-nh-4-oh-disodium-hydrogen-phosphate-

slug:
a-sample-of-mgco-3-is-dissolved-in-dil-hcl-and-the-solution-is-neutralized-with-ammonia-and-buffered-with-nh-cl-4-nh-4-oh-disodium-hydrogen-phosphate-49391

question:
<div class="question">A sample of $\mathrm{MgCO}_3$ is dissolved in dil. $\mathrm{HCl}$ and the solution is neutralized with ammonia and buffered with $\mathrm{NH} \mathrm{Cl}_4 /$ $\mathrm{NH}_4 \mathrm{OH}$. Disodium hydrogen phosphate reagent is added to the resulting solution. A white precipitate is formed. What is the formula of the precipitate?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Mg}_3\left(\mathrm{PO}_4\right)_2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{MgHPO}_4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Mg}_2 \mathrm{P}_2 \mathrm{O}_7$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/shFLyp12fYziTq3xkymuveQCCPKJ9PAmEW2NO5uXUO4.original.fullsize.png"/></div>


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-satellite-has-kinetic-energy-k-potential-energy-v-and-total-energy-e-which-of-the-following-statements-is-true

slug:
a-satellite-has-kinetic-energy-k-potential-energy-v-and-total-energy-e-which-of-the-following-statements-is-true-33382

question:
<div class="question">A satellite has kinetic energy $K$, potential energy $V$ and total energy $E .$ Which of the following statements is true?</div>

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

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

answer:
<div class="solution">As we know that for the earth and satellite system 
<br/>Kinetic energy, $\quad K=\frac{G M m}{2 a}$
<br/>Potential energy, $V=-\frac{G M m}{a}$
<br/>and total energy, $E=-\frac{G M m}{2 a}$
<br/>where, $a=$ radius of the orbit of the satellite and $m=$ mass of the satellite
<br/>On the basis of above three expressions for the energies, we have
<br/>$K=\frac{G M m}{2 a}=-\left(-\frac{G M m}{2}\right)=-\frac{V}{2}$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-saturated-solution-of-baso-4-at-2-5-c-is-4-1-0-5-m-the-solubility-of-baso-4-in-01-m-na-2-so-4-at-this-temperature-will-be

slug:
a-saturated-solution-of-baso-4-at-2-5-c-is-4-1-0-5-m-the-solubility-of-baso-4-in-01-m-na-2-so-4-at-this-temperature-will-be-68098

question:
<div class="question">A saturated solution of $\mathrm{BaSO}_{4}$ at $25^{\circ} \mathrm{C}$ is $4 \times 10^{-5} \mathrm{M}$. The solubility of $\mathrm{BaSO}_{4}$ in $0.1 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}$ at this temperature will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.6 \times 10^{-9} \mathrm{M}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.6 \times 10^{-8} \mathrm{M}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \times 10^{-6} \mathrm{M}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 \times 10^{-4} \mathrm{M}$</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> </ul>

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

answer:
<div class="solution">$\begin{array}{r}
<br/>\mathrm{BaSO}_{4} \rightleftharpoons \mathrm{Ba}^{2+}+\mathrm{SO}_{4}^{2-} \\
<br/>\mathrm{s} \quad \mathrm{s}+0.1 \\
<br/>\mathrm{Na}_{2} \mathrm{SO}_{4} \rightarrow 2 \mathrm{Na}^{+}+\mathrm{SO}_{4}^{2-} \\
<br/>0.2 \quad 0.1
<br/>\end{array}$
<br/>$\begin{array}{l}
<br/>\mathrm{K}_{\mathrm{sp}}=\left[\mathrm{Ba}^{+2}\right]\left[\mathrm{SO}_{4}^{-2}\right] \\
<br/>\therefore 4 \times 10^{-5}=\mathrm{s} \times(\mathrm{s}+0.1)
<br/>\end{array}$
<br/>or $4 \times 10^{-5} \approx \mathrm{s} \times 0.1 \quad \therefore \mathrm{s}=4 \times 10^{-4}$ as $\mathrm{s}&lt;&lt; &lt; 0.1 \quad \therefore \mathrm{s}+0.1 \approx 0.1$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-scientist-proposes-a-new-temperature-scale-in-which-the-ice-point-is-25-x-x-is-the-new-unit-of-temperature-and-the-steam-point-is-305-x-the-specific

slug:
a-scientist-proposes-a-new-temperature-scale-in-which-the-ice-point-is-25-x-x-is-the-new-unit-of-temperature-and-the-steam-point-is-305-x-the-specific-65541

question:
<div class="question">A scientist proposes a new temperature scale in which the ice point is $25 \mathrm{X}(\mathrm{X}$ is the new unit of temperature) and the steam point is $305 \mathrm{X}$. The specific heat capacity of water in this new scale is $\left(\operatorname{in} J \mathrm{kg}^{-1} \mathrm{X}^{-1}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.2 \times 10^{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.0 \times 10^{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.2 \times 10^{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1.5 \times 10^{3}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.5 \times 10^{3}$</span> </div>

answer:
<div class="solution">Given, $305 \times-25 X=100^{\circ} \mathrm{C}$
<br/>(: $\mathrm{X}$ is the new unit of temperature
<br/>$$
<br/>(305-25) \mathrm{X}=100^{\circ} \mathrm{C} \Rightarrow 280 \mathrm{X}=100^{\circ} \mathrm{C}
<br/>$$
<br/>$\therefore \quad 1^{\circ} \mathrm{C}=2.8 \mathrm{x}$
<br/>The specinc heat capacity of water
<br/>$\begin{aligned}=4200 \frac{j 0 u l e}{k g \cdot^{\circ} C} &amp;=4200 \times \frac{j \text { oule }}{k g \times 2.8 \times} \\ &amp;=1500 J / \mathrm{kg}-\mathrm{X} \\ &amp;=1.5 \times 10^{3} \mathrm{J} \mathrm{kg}^{-1} \mathrm{X}^{-1} \end{aligned}$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-shell-of-mass-5-m-acted-upon-by-no-external-force-and-initially-at-rest-bursts-into-three-fragments-of-masses-m-2-m-and-2-m-respectively-the-first-t

slug:
a-shell-of-mass-5-m-acted-upon-by-no-external-force-and-initially-at-rest-bursts-into-three-fragments-of-masses-m-2-m-and-2-m-respectively-the-first-t-37356

question:
<div class="question">A shell of mass $5 M$, acted upon by no external force and initially at rest, bursts into three fragments of masses $M, 2 M$ and $2 M$ respectively. The first two fragments move in opposite directions with velocities of magnitudes 2v and $v$ respectively. The third fragment will</div>

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

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

answer:
<div class="solution">Let the velocity of third fragment is $v'$. Then by the conservation of linear momentum
<br/>$$
<br/>\begin{aligned}
<br/>&amp; 5(0)=M \times 2 \mathrm{v}-2 M \times \mathrm{v}+2 \mathrm{M} \times \mathrm{v}^{\prime} \\
<br/>\Rightarrow \quad &amp; \mathrm{v}^{\prime}=0
<br/>\end{aligned}
<br/>$$
<br/>i.e., the third fragment will be at rest.</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-shown-in-the-figure-a-liquid-is-at-same-levels-in-two-arms-of-a-utube-of-uniform-crosssection-when-at-rest-if-the-utube-moves-with-an-acceleration-f

slug:
a-shown-in-the-figure-a-liquid-is-at-same-levels-in-two-arms-of-a-utube-of-uniform-crosssection-when-at-rest-if-the-utube-moves-with-an-acceleration-f-38625

question:
<div class="question">A shown in the figure, a liquid is at same levels in two arms of a U-tube of uniform cross-section when at rest. If the \(U\)-tube moves with an acceleration ' \(f\) ' towards right, the difference between liwuid height between two arms of the \(\mathrm{U}\)-tube will be, (acceleration due to gravity \(=\mathrm{g}\) )<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/DuzrTR8OzG0rtlKnvUgsBk7J7vCL70_Ie8k6lRe6P58.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0lcPdy6_s-Advht3AcfJ9YmvUT6dpFLKQajiDXKpjNo.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \tan \theta=\frac{\text { acceleration }}{\mathrm{g}} \\ &amp; \frac{\mathrm{h}}{\mathrm{a}}=\frac{\text { acceleration }}{\mathrm{g}} \\ &amp; \frac{\mathrm{h}}{\mathrm{a}}=\frac{\mathrm{f}}{\mathrm{g}} \\ &amp; \mathrm{h}=\frac{\mathrm{fa}}{\mathrm{g}}\end{aligned}\)</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-simple-pendulum-consisting-of-a-small-ball-of-mass-m-attached-to-a-massless-string-hanging-vertically-from-the-ceiling-is-oscillating-with-an-amplit

slug:
a-simple-pendulum-consisting-of-a-small-ball-of-mass-m-attached-to-a-massless-string-hanging-vertically-from-the-ceiling-is-oscillating-with-an-amplit-75206

question:
<div class="question">A simple pendulum, consisting of a small ball of mass $m$ attached to a massless string hanging vertically from the ceiling, is oscillating with an amplitude such that $T_{\max }=2 T_{\min }$ where $T_{\max }$ and $T_{\min }$ are the maximum and minimum tension in the string respectively. The value of maximum tension $\mathrm{T}_{\max }$ in the string is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/CrB_6_1zsW5yDpNjTclZgVfm7EsHO-CN-N180PPQb5o.original.fullsize.png"/><br/> <br/>$\mathrm{T}_{\min }=m g \cos \theta$
<br/>$\mathrm{T}_{\max }=\mathrm{mg}+\frac{\mathrm{mv}^{2}}{\mathrm{I}}=\mathrm{mg}+\frac{\mathrm{mv}^{2}}{\mathrm{I}}$
<br/>$\frac{1}{2} \mathrm{mv}^{2}=\mathrm{mgl}(1-\cos \theta)$
<br/>$\mathrm{T}_{\max }=2 \mathrm{~T}_{\min }$
<br/>$\mathrm{mg}+\frac{\mathrm{mv}^{2}}{\mathrm{I}}=2 \mathrm{mg} \cos \theta$
<br/>$\therefore \cos \theta=\frac{3}{4}$
<br/>$\mathrm{v}^{2}=2 \mathrm{gl}\left(1-\frac{3}{4}\right)=\frac{\mathrm{gl}}{2}$
<br/>$\mathrm{~T}_{\max }=\mathrm{mg}+\frac{\mathrm{m}}{\mathrm{I}} \times \frac{\mathrm{g} \mathrm{l}}{2}=\frac{3 \mathrm{mg}}{2}$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-simple-pendulum-of-length-is-displaced-so-that-its-taught-string-is-horizontal-and-then-released-a-uniform-bar-pivoted-at-one-end-is-simultaneously-

slug:
a-simple-pendulum-of-length-is-displaced-so-that-its-taught-string-is-horizontal-and-then-released-a-uniform-bar-pivoted-at-one-end-is-simultaneously-58145

question:
<div class="question">A simple pendulum of length $\ell$ is displaced so that its taught string is horizontal and then released. A uniform bar pivoted at one end is simultaneously released from its horizontal position. If their motions are synchronous, what is the length of the bar?</div>

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

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

answer:
<div class="solution">Hint: 
<br/>$v=\sqrt{2 g h}$
<br/>$=\sqrt{2 g \mid \sin \theta}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/UUUAYlP-JJjKgBjz3whWWkLqUmmrw_KCLudfK80Tzvk.original.fullsize.png"/><br/> <br/>$\omega_{1}=\frac{v}{l}=\sqrt{\frac{2 g \sin \theta}{1}}$
<br/>$m g h=\frac{1}{2} l \omega^{2}$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fozVfPIcLox95Lk-RH_AJ_gkVuhkCikYoTgfmI2ynVk.original.fullsize.png"/><br/> <br/>$\Rightarrow \mathrm{mg} \cdot \frac{\mathrm{l}^{\prime} \sin \theta}{2}=\frac{1}{2} \times \frac{\mathrm{ml}^{2}}{3} \times \omega_{2}^{2}$
<br/>$\omega_{2}=\sqrt{\frac{3 \mathrm{~g} \sin \theta}{\mathrm{l}}}=\omega_{1} \Rightarrow \frac{2}{\mathrm{l}}=\frac{3}{\mathrm{l}}$
<br/>$\Rightarrow \mathrm{I}^{\prime}=\frac{3}{2}I$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-simple-pendulum-of-length-l-swings-in-a-vertical-plane-the-tension-of-the-string-when-it-makes-an-angle-with-the-vertical-and-the-bob-of-mass-m-move

slug:
a-simple-pendulum-of-length-l-swings-in-a-vertical-plane-the-tension-of-the-string-when-it-makes-an-angle-with-the-vertical-and-the-bob-of-mass-m-move-12329

question:
<div class="question">A simple pendulum of length $L$ swings in a vertical plane. The tension of the string when it makes an angle $\theta$ with the vertical and the bob of mass $m$ moves with a speed $v$ is $\mathrm{fg}$ is the gravitational acceleration)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$m g \cos \theta+m v^{2} / L$</span> </div>

answer:
<div class="solution">The situation is given below
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a5.png"/> <br/> <br/>For motion along a vertical circular track, the required centripetal force is along the radius and towards the centre of the circle is given by $T-m g \cos \theta=\frac{m v^{2}}{L}$
<br/>$\Rightarrow \quad T=\frac{m v^{2}}{L}+m g \cos \theta$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-small-bar-magnet-of-dipole-moment-m-is-moving-with-speed-v-along-x-direction-towards-a-small-closed-circular-conducting-loop-of-radius-a-with-its-ce

slug:
a-small-bar-magnet-of-dipole-moment-m-is-moving-with-speed-v-along-x-direction-towards-a-small-closed-circular-conducting-loop-of-radius-a-with-its-ce-59795

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/yKrUu-icrcru9hDF-8xAJlb0dz0jJYKxcbHVCqiUUjw.original.fullsize.png"/><br/> <br/>A small bar magnet of dipole moment $M$ is moving with speed $v$ along $x$ direction towards a small closed circular conducting loop of radius ' $a$ ' with its centre $O$ at $x=0$ (see figure). Assume $x&gt;&gt;a$ and the coil has a resistance $R$. Then which of the following statemenmt(s) is/are true?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Induced EMF is proportional to $\frac{1}{x^{4}}$, The magnetic moment $\mu$ due to induced current in the coil is proportional to $\mathrm{a}^{4}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/iMP2OOkDmKfV3qBWhRGjFQbtMhICsJcIxneHBMIdAt8.original.fullsize.png"/><br/> <br/>Magnetic field at an axial point is $2\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M}{x^{3}}$
<br/>$$
<br/>\phi=2\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M}{x^{3}} \cdot \pi a^{2}
<br/>$$
<br/>$\mathrm{EMF}=\frac{-\mathrm{d} \phi}{\mathrm{dt}}=-2\left(\frac{\mu_{0}}{4 \pi}\right) \mathrm{M} \pi \mathrm{a}^{2}\left(\frac{-3}{\mathrm{x}^{4}}\right) \frac{\mathrm{dx}}{\mathrm{dt}}=6\left(\frac{\mu_{0}}{4 \pi}\right) \frac{\mathrm{M} \pi \mathrm{a}^{2}}{\mathrm{x}^{4}} \mathrm{~V} \quad \mathrm{EMF} \propto \frac{1}{\mathrm{x}^{4}}$
<br/>Current in coil $=\frac{E M F}{R} \quad i=6\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M \pi a^{2}}{x^{4}} \frac{V}{R}$
<br/>Magnetic moment $\mu=\mathrm{i} \pi \mathrm{a}^{2}, \mu \propto \mathrm{a}^{4}$
<br/>Rate at which heat is produced $=\mathrm{i}^{2} \mathrm{R} . \propto \frac{1}{\mathrm{x}^{8}}$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-small-mass-m-attached-to-one-end-of-a-spring-with-a-negligible-mass-and-an-unstretched-length-l-executes-virtual-oscillation-with-angular-frequency-

slug:
a-small-mass-m-attached-to-one-end-of-a-spring-with-a-negligible-mass-and-an-unstretched-length-l-executes-virtual-oscillation-with-angular-frequency-23525

question:
<div class="question">A small mass $m$, attached to one end of a spring with a negligible mass and an unstretched length $L$, executes virtual oscillation with angular frequency $\omega_{0}$. When the mass is rotated with an angular speed $\omega$ by holding the other end of the spring at a fixed point, the mass moves uniformly in a circular path in a horizontal plane. Then the increase in length of the spring during the rotation is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{\omega^{2} L}{\omega_{0}^{2}-\omega^{2}}\)</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=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\omega_{0}^{2} L}{\omega^{2}-\omega_{0}^{2}}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\omega^{2} L}{\omega_{0}^{2}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{\omega_{0}^{2} L}{\omega^{2}}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\omega^{2} L}{\omega_{0}^{2}-\omega^{2}}\)</span> </div>

answer:
<div class="solution">The given situation can be shown as <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a46.png"/> <br/> <br/>From figure, $K\left(x \sin \theta=m \omega^{2}(L+x) \sin \theta\right.$
<br/>$\Rightarrow \quad K x=\operatorname{m\omega}^{2}(L+x)$...(i)
<br/>Also, $\quad \sqrt{\frac{K}{m}}=\omega_{0}$
<br/>$\Rightarrow \quad K=m \omega_{0}^{2}$
<br/>Substituting the value in Eq. (i)
<br/>$$
<br/>\begin{aligned}
<br/>m \omega_{0}^{2} x &amp;=m \omega^{2}(L+x) \\
<br/>\Rightarrow \quad x &amp;=\frac{\omega^{2} L}{\omega_{0}^{2}-\omega^{2}}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-small-metal-sphere-of-radius-a-is-falling-with-a-velocity-v-through-a-vertical-column-of-a-viscous-liquid-if-the-coefficient-of-viscosity-of-the-liq

slug:
a-small-metal-sphere-of-radius-a-is-falling-with-a-velocity-v-through-a-vertical-column-of-a-viscous-liquid-if-the-coefficient-of-viscosity-of-the-liq-50518

question:
<div class="question">A small metal sphere of radius $a$ is falling with a velocity $v$ through a vertical column of a viscous liquid. If the coefficient of viscosity of the liquid is $\eta$, then the sphere encounters an opposing force of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6\pi \eta av$</span> </div>

answer:
<div class="solution">Stokes established that if a sphere of radius a moves with velocity $v$ through a fluid of viscosily
<br/>Coefficient $\mu$ the viscous force opposing the motion of thy sphere is $F=6\pi \eta a v$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Zero</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-small-spherical-body-of-radius-r-and-density-p-moves-with-the-terminal-velocity-v-in-a-fluid-of-coefficient-of-viscosity-and-density-what-will-be-th

slug:
a-small-spherical-body-of-radius-r-and-density-p-moves-with-the-terminal-velocity-v-in-a-fluid-of-coefficient-of-viscosity-and-density-what-will-be-th-67535

question:
<div class="question">A small spherical body of radius $r$ and density p moves with the terminal velocity $v$ in a fluid of coefficient of viscosity $\eta$ and density $\bar{\sigma}$. What will be the net force on the body?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{4 \pi}{3} r^{3}(p-\sigma) g$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$6 \pi \eta rv$</span> </li><li class="correct"> <span class="option-label">C</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><li class=""> <span class="option-label">D</span> <span class="option-data">Infinity</span> </li> </ul>

answer:
<div class="solution">When the spherical body falls with constant velocity, i.e. terminal velocity then the net force becomes zero, i.e the weight of body is equal to the buoyancy force. Hence, $F_{\text {net }}=0$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-small-steel-ball-bounces-on-a-steel-plate-held-horizontally-on-each-bounce-the-speed-of-the-ball-arriving-at-the-plate-is-reduced-by-a-factor-e-coef

slug:
a-small-steel-ball-bounces-on-a-steel-plate-held-horizontally-on-each-bounce-the-speed-of-the-ball-arriving-at-the-plate-is-reduced-by-a-factor-e-coef-54230

question:
<div class="question">A small steel ball bounces on a steel plate held horizontally. On each bounce the speed of the ball arriving at the plate is reduced by a factor
<br/>e (coefficient of restitution) in the rebound, so that $V_{upward}=eV_{downward}$
<br/>If the ball is initially dropped from a height of $0.4 \mathrm{m}$ above the plate and if 10 seconds later the bouncing ceases, the value of $e$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{\frac{2}{7}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{13}{18}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{17}{18}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{17}{18}$</span> </div>

answer:
<div class="solution">Given, steel ball bounces on a steel ple held horizontally, so $$
<br/>V_{\text {upward }}=e V_{\text {downward }}
<br/>$$
<br/>According to the relation,
<br/>$h_{n}=e^{2 n} h$
<br/>$\begin{aligned} \therefore \quad t &amp;=\sqrt{\frac{2 h}{g}}+2 \sqrt{\frac{2 h e^{2}}{g}}+2 \sqrt{\frac{2 h e^{4}}{g}}+\ldots \\ &amp;=\sqrt{\frac{2 h}{g}}\left[1+2 e+2 e^{2}+\ldots\right] \\ &amp;=\sqrt{\frac{2 h}{g}}\left[\frac{1+e}{1-e}\right] \\ \therefore \quad t &amp;=10 \mathrm{sec} \\ \therefore 10 &amp;=\sqrt{\frac{2(0 \cdot 4)}{10}}\left(\frac{1+e}{1-e}\right) \end{aligned}$
<br/>Solving above relation $\therefore e=\frac{25 \sqrt{2}-1}{25 \sqrt{2}+1}=\frac{17}{18}$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-smooth-massless-string-passes-over-a-smooth-fixed-pulley-two-masses-m-1-and-m-2-m-1-m-2-are-tied-at-the-two-ends-of-the-string-the-masses-are-allowe

slug:
a-smooth-massless-string-passes-over-a-smooth-fixed-pulley-two-masses-m-1-and-m-2-m-1-m-2-are-tied-at-the-two-ends-of-the-string-the-masses-are-allowe-44929

question:
<div class="question">A smooth massless string passes over a smooth fixed pulley. Two masses $m_{1}$ and $m_{2},\left(m_{1}&gt;m_{2}\right)$ are tied at the two ends of the string. The masses are allowed to move under gravity starting from rest. The total external force acting on the two masses is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(m_{1}+m_{2}\right) g$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\left(m_{1}-m_{2}\right)^{2}}{m_{1}+m_{2}} g$</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><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(m_{1}-m_{2}\right) g$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\left(m_{1}+m_{2}\right)^{2}}{m_{1}-m_{2}} g$</span> </li> </ul>

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

answer:
<div class="solution">We know that Acceleration, $a_{\mathrm{CM}}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} \times g\left(\because m_{1}&gt;m_{2}\right)$
<br/>So, resultant external force.
<br/>$$
<br/>\begin{aligned}
<br/>F &amp;=\left(m_{1}+m_{2}\right) a_{\mathrm{CM}} \\
<br/>&amp;=\left(m_{1}+m_{2}\right) \times\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} \times g \\
<br/>&amp;=\frac{\left(m_{1}-m_{2}\right)^{2}}{\left(m_{1}+m_{2}\right)} \times g
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-solid-maintained-at-t-1-c-is-kept-in-an-evacuated-chamber-at-temperature-t-2-c-t-2-t-1-the-rate-of-heat-absorbed-by-the-body-is-proportional-to

slug:
a-solid-maintained-at-t-1-c-is-kept-in-an-evacuated-chamber-at-temperature-t-2-c-t-2-t-1-the-rate-of-heat-absorbed-by-the-body-is-proportional-to-56314

question:
<div class="question">A solid maintained at $t_{1}^{\circ} \mathrm{C}$ is kept in an evacuated chamber at temperature $t_{2}^{\circ} \mathrm{C}$ $\left(t_{2} \gg t_{1}\right) .$ The rate of heat absorbed by the body is proportional to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$t_{2}^{4}-t_{1}^{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(f_{2}^{4}+273\right)-\left(t_{1}^{4}+273\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$t_{2}-t_{1}$</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">$t_{2}^{2}-t_{1}^{2}$</span> </li> </ul>

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

answer:
<div class="solution">As we know that, the rate of cooling is
<br/>$$
<br/>\frac{d \theta}{d t}=b A\left(\theta-\theta_{0}\right)
<br/>$$
<br/>where, $b A=$ constant $\Rightarrow \quad \frac{d \theta}{d t} \propto \theta-\theta_{0}$
<br/>Also $-\frac{d T}{d t} \propto T-T_{0}$
<br/>where, $T=$ temperature and $\quad t=$ time
<br/>So, the rate of heat absorbed by cooled body is proportional to $t_{2}-t_{1}$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-solid-rectangular-sheet-has-two-different-coefficients-of-linear-expansion-1-and-2-along-its-length-and-breadth-respectively-the-coefficient-of-surf

slug:
a-solid-rectangular-sheet-has-two-different-coefficients-of-linear-expansion-1-and-2-along-its-length-and-breadth-respectively-the-coefficient-of-surf-32483

question:
<div class="question">A solid rectangular sheet has two different coefficients of linear expansion $\alpha_{1}$ and $\alpha_{2}$ along its length and breadth respectively. The coefficient of surface expansion is (for $\left.\alpha_{1} t&lt; &lt; 1, \alpha_{2} t&lt; &lt; 1\right)$</div>

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

answer:
<div class="solution">The coefficient of linear expansion along its length $=\alpha_{1}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a7.png"/> <br/> <br/>The coefficient of linear expansion along its breadth $=\alpha_{2}$ Increase in length,
<br/>$$
<br/>L_{i}=l_{0}\left(1+\alpha_{1} \Delta t\right)
<br/>$$
<br/>Increase in breadth,
<br/>$$
<br/>B_{t}=b_{0}\left(1+\alpha_{2} \Delta t_{2}\right)
<br/>$$
<br/>Let coefficient of surface expansion is $\beta$
<br/>$$
<br/>\begin{aligned}
<br/>\text { Area } &amp;=\text { length } \times \text { breadth } \\
<br/>&amp;=L_{0}\left(1+\alpha_{1} \Delta t\right) \times b_{0}\left(1+\alpha_{2} \Delta t\right) \\
<br/>&amp;=L_{0} b_{0}\left(1+\alpha_{1} \Delta t\right)\left(1+\alpha_{2} \Delta t\right) \\
<br/>&amp;=S_{0}\left(1+\alpha_{1} \Delta t+\alpha_{2} \Delta t+\ldots\right) \\
<br/>\text { where, } S_{0} &amp;=l_{0} \cdot b_{0}
<br/>\end{aligned}
<br/>$$
<br/>$=$ Initial area of surface In state of expansion,
<br/>$$
<br/>\begin{aligned}
<br/>S_{t} &amp;=L_{r} \times B_{r} \\
<br/>&amp;=L_{0} b_{0}\left(1+\alpha_{1} \Delta t\right)\left(1+\alpha_{2} \Delta t\right) \\
<br/>&amp;=S_{0}\left(1+\alpha_{1} \Delta t+\alpha_{2} \Delta t+\ldots\right) \\
<br/>S_{t} &amp;=S_{0}(1+\beta \Delta t) \\
<br/>\therefore S_{0}(1+\beta \Delta t) &amp;=S_{0}\left(1+\alpha_{1} \Delta t+\alpha_{2} \Delta t+\ldots\right) \\
<br/>\beta \cdot \Delta t &amp;=\alpha_{1} \Delta t+\alpha_{2} \Delta t \\
<br/>\beta &amp;=\alpha_{1}+\alpha_{2}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-solid-spherical-ball-and-a-hollow-spherical-ball-of-two-different-materials-of-densities-1-and-2-respectively-have-same-outer-radii-and-same-mass-wh-1

slug:
a-solid-spherical-ball-and-a-hollow-spherical-ball-of-two-different-materials-of-densities-1-and-2-respectively-have-same-outer-radii-and-same-mass-wh-1-13477

question:
<div class="question">A solid spherical ball and a hollow spherical ball of two different materials of densities $\rho_{1}$ and $\rho_{2}$ respectively have same outer radii and same mass. What will be the ratio, the moment of inertia (about an axis passing through the centre) of the hollow sphere to that of the solid sphere?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\rho_{2}}{\rho_{1}}\left[1-\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{5}{3}}\right]$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a33.png"/> <br/> <br/>$\therefore \quad M_{1}=\rho_{1} \frac{4}{3} \pi R^{3}$ and $M_{2}=\rho_{2} \frac{4}{3} \pi\left(R^{3}-r^{3}\right)$
<br/>$\therefore \quad \rho_{1} \frac{4}{3} \pi R^{3}=\rho_{2} \frac{4}{3} \pi\left(R^{3}-r^{3}\right)$
<br/>$\Rightarrow \quad \rho_{1} R^{3}=\rho_{2}\left(R^{3}-r^{3}\right)$
<br/>$\Rightarrow \quad \frac{\rho_{1}}{\rho_{2}}=1-\frac{r^{3}}{R^{3}}$
<br/>$\therefore \frac{r}{R}=\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{1}{3}}$
<br/>Moment of inertia, $\begin{aligned} \frac{I_{M}}{I_{S}} &amp;=\frac{\frac{2}{5} M_{2}\left[\frac{R^{5}-r^{3}}{R^{3}-r^{3}}\right]}{\frac{2}{5} M_{1} R^{2}} \\ &amp;=\frac{\rho_{2} \frac{4}{3} \pi\left(R^{3}-r^{3}\right)\left[\frac{R^{5}-r^{5}}{\left(R^{3}-r^{3}\right)}\right]}{\rho_{1} \frac{4}{3} \pi R^{3} R^{2}} \\ &amp;=\frac{\rho_{2}}{\rho_{1}}\left[\frac{R^{5}-r^{5}}{R^{5}}\right]=\frac{\rho_{2}}{\rho_{1}}\left[1-\left(\frac{r}{R}\right)^{3}\right] \end{aligned}$
<br/>$\therefore \quad \frac{I_{H}}{I_{s}}=\frac{\rho_{2}}{\rho_{1}}\left[1-\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{5}{3}}\right]$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-solid-spherical-ball-and-a-hollow-spherical-ball-of-two-different-materials-of-densities-1-and-2-respectively-have-same-outer-radii-and-same-mass-wh

slug:
a-solid-spherical-ball-and-a-hollow-spherical-ball-of-two-different-materials-of-densities-1-and-2-respectively-have-same-outer-radii-and-same-mass-wh-21840

question:
<div class="question">A solid spherical ball and a hollow spherical ball of two different materials of densities $\rho_{1}$ and $\rho_{2}$ respectively have same outer radii and same mass. What will be the ratio, the moment of inertia (about an axis passing through the centre) of the hollow sphere to that of the solid sphere?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\rho_{2}}{\rho_{1}}\left[1-\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{5}{3}}\right]$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a33.png"/> <br/> <br/>$\therefore \quad M_{1}=\rho_{1} \frac{4}{3} \pi R^{3}$ and $M_{2}=\rho_{2} \frac{4}{3} \pi\left(R^{3}-r^{3}\right)$
<br/>$\therefore \quad \rho_{1} \frac{4}{3} \pi R^{3}=\rho_{2} \frac{4}{3} \pi\left(R^{3}-r^{3}\right)$
<br/>$\Rightarrow \quad \rho_{1} R^{3}=\rho_{2}\left(R^{3}-r^{3}\right)$
<br/>$\Rightarrow \quad \frac{\rho_{1}}{\rho_{2}}=1-\frac{r^{3}}{R^{3}}$
<br/>$\therefore \frac{r}{R}=\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{1}{3}}$
<br/>Moment of inertia, $\begin{aligned} \frac{I_{M}}{I_{S}} &amp;=\frac{\frac{2}{5} M_{2}\left[\frac{R^{5}-r^{3}}{R^{3}-r^{3}}\right]}{\frac{2}{5} M_{1} R^{2}} \\ &amp;=\frac{\rho_{2} \frac{4}{3} \pi\left(R^{3}-r^{3}\right)\left[\frac{R^{5}-r^{5}}{\left(R^{3}-r^{3}\right)}\right]}{\rho_{1} \frac{4}{3} \pi R^{3} R^{2}} \\ &amp;=\frac{\rho_{2}}{\rho_{1}}\left[\frac{R^{5}-r^{5}}{R^{5}}\right]=\frac{\rho_{2}}{\rho_{1}}\left[1-\left(\frac{r}{R}\right)^{3}\right] \end{aligned}$
<br/>$\therefore \quad \frac{I_{H}}{I_{s}}=\frac{\rho_{2}}{\rho_{1}}\left[1-\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{5}{3}}\right]$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-solid-uniform-sphere-resting-on-a-rough-horizontal-plane-is-given-a-horizontal-impulse-directed-through-its-centre-so-that-it-starts-sliding-with-an

slug:
a-solid-uniform-sphere-resting-on-a-rough-horizontal-plane-is-given-a-horizontal-impulse-directed-through-its-centre-so-that-it-starts-sliding-with-an-35771

question:
<div class="question">A solid uniform sphere resting on a rough horizontal plane is given a horizontal impulse directed through its centre so that it starts sliding with an initial velocity $v_{0}$. When it finally atarta rolling without shpping the speed of its centre is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{7} v_{0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{7} v_{0}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5}{7} v_{0}$</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">$\frac{6}{7} v_{0}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{7} v_{0}$</span> </div>

answer:
<div class="solution">Let, the final velocity be V
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a46.png"/> <br/> <br/>So, angular momentum will remain conserved along point of contact By conservation of angular momentum Angular momentum will remain conserved along point of contact
<br/>$$
<br/>\begin{aligned}
<br/>I_{\omega} &amp;=\text { constant } \\
<br/>m v_{0} r &amp;=m v r+\frac{2}{5} m r^{2} \times \omega \quad\left(\because \omega=\frac{v}{r}\right) \\
<br/>m v_{0} r &amp;=m v r+\frac{2}{5} m r^{2}\left(\frac{v}{r}\right) \\
<br/>v_{0} &amp;=v+\frac{2}{5} v \\
<br/>v_{0} &amp;=\frac{7}{5} v \Rightarrow v=\frac{5}{7} v_{0}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2023']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">82000</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-solution-containing-4-mathrmg-of-polymer-in-40-litre-solution-at-27circ-mathrmc-shows-an-osmotic-pressure-of-30-times-104-mathrmatm-the-molar-mass-o

slug:
a-solution-containing-4-mathrmg-of-polymer-in-40-litre-solution-at-27circ-mathrmc-shows-an-osmotic-pressure-of-30-times-104-mathrmatm-the-molar-mass-o-50970

question:
<div class="question">A solution containing \(4 \mathrm{~g}\) of polymer in 4.0 litre solution at \(27^{\circ} \mathrm{C}\) shows an osmotic pressure of \(3.0 \times 10^{-4} \mathrm{~atm}\). The molar mass of the polymer in \(\mathrm{g} / \mathrm{mol}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">820000</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">82000</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">8200</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">820</span> </li> </ul>

answer:
<div class="solution">Hint : \(\pi=i C(M) R T\)<br/>\(3.0 \times 10^{-4}=1 \times C(M) \times 0.0821 \times 300\)<br/>\(\therefore \mathrm{C}(\mathrm{M})=1.22 \times 10^{-5} \text {, Molarity }=\frac{\text { no. of moles }}{\text { vol. of solution }(\mathrm{L})}\)<br/>\(1.22 \times 10^{-5}=\frac{4 / \mathrm{M}}{4}\). Hence \(M=81967 \approx 82000 \mathrm{~g} / \mathrm{mol}\)</div>


tags: ['Chemistry', 'Coordination Compounds', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-solution-is-made-by-a-concentrated-solution-of-co-no-3-2-with-a-concentrated-solution-of-nano-2-in-50-acetic-acid-a-solution-of-a-salt-containing-me

slug:
a-solution-is-made-by-a-concentrated-solution-of-co-no-3-2-with-a-concentrated-solution-of-nano-2-in-50-acetic-acid-a-solution-of-a-salt-containing-me-80570

question:
<div class="question">A solution is made by a concentrated solution of $\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}$ with a concentrated solution of $\mathrm{NaNO}_{2}$ in $50 \%$ acetic acid. A solution of a salt containing metal $\mathrm{M}$ is added to the mixture, when a yellow precipitate is formed. Metal ' $\mathrm{M}$ ' is</div>

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

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

answer:
<div class="solution">$\begin{array}{l}
<br/>\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{3}+\mathrm{NaNO}_{2} \longrightarrow \mathrm{Na}_{3}\left[\mathrm{Co}\left(\mathrm{NO}_{2}\right)_{6}\right] \\
<br/>\mathrm{Na}_{3}\left[\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{6}\right]+\mathrm{K}^{+} \longrightarrow \mathrm{K}_{3}\left[\mathrm{Co}\left(\mathrm{NO}_{2}\right)_{6}\right] \downarrow \text { yellow ppt }
<br/>\end{array}$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-solution-is-saturated-with-srco-3-and-sr-f-2-the-co-3-2-is-found-to-be-12-1-0-3-m-the-concentration-of-f-in-the-solution-would-be

slug:
a-solution-is-saturated-with-srco-3-and-sr-f-2-the-co-3-2-is-found-to-be-12-1-0-3-m-the-concentration-of-f-in-the-solution-would-be-13989

question:
<div class="question">A solution is saturated with $\mathrm{SrCO}_{3}$ and $\mathrm{Sr} \mathrm{F}_{2}$. The $\left[\mathrm{CO}_{3}^{2-}\right]$ is found to be $1.2 \times 10^{-3} \mathrm{M}$. The concentration of $\mathrm{F}$ - in the solution would be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3.7 \times 10^{-6} \mathrm{M}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.2 \times 10^{-3} \mathrm{M}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5.1 \times 10^{-7} \mathrm{M}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3.7 \times 10^{-2} \mathrm{M}$
<br/>Given : $\mathrm{K}_{\mathrm{sp}}\left(\mathrm{SrCO}_{3}\right)=7.0 \times 10^{-10}, \mathrm{~K}_{\mathrm{sp}}\left(\mathrm{SrF}_{2}\right)=7.9 \times 10^{-10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.7 \times 10^{-2} \mathrm{M}$
<br/>Given : $\mathrm{K}_{\mathrm{sp}}\left(\mathrm{SrCO}_{3}\right)=7.0 \times 10^{-10}, \mathrm{~K}_{\mathrm{sp}}\left(\mathrm{SrF}_{2}\right)=7.9 \times 10^{-10}$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\begin{aligned}
<br/>\mathrm{Sr} \mathrm{CO}_{3}(\mathrm{~s}) \stackrel{\mathrm{S}}{\rightleftarrows} \mathrm{Sr}^{2+}(\mathrm{aq})+\mathrm{CO}_{3}^{2-}(\mathrm{aq}) \\
<br/>\mathrm{S}_{1}+\mathrm{S}_{2} \quad \mathrm{~S}_{1} \\
<br/>\mathrm{SrF}_{2}(\mathrm{~s}) \rightleftarrows \mathrm{Sr}^{2+}(\mathrm{aq})+2 \mathrm{~F}^{-}(\mathrm{aq}) \\
<br/>\mathrm{S}_{1}+\mathrm{S}_{2} \quad 2 \mathrm{~S}_{2}
<br/>\end{aligned}$
<br/>Where $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ are solubilities of $\mathrm{SrCO}_{3}(\mathrm{~s})$ and $\mathrm{Sr} \mathrm{F}_{2}(\mathrm{~s})$ respectively.
<br/>$\begin{array}{l}
<br/>\mathrm{S}_{1}=1.2 \times 10^{-3} \mathrm{M}(\text { Given }) \\
<br/>\mathrm{K}_{\mathrm{sp}}\left(\mathrm{SrCO}_{3}\right)=\left[\mathrm{Sr}^{2+}\right]\left[\mathrm{CO}_{3}^{-2}\right] \\
<br/>7 \times 10^{-10}=\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right) \mathrm{S}_{1}-\ldots-\text { (i) } \\
<br/>\mathrm{K}_{\text {sp }}\left(\mathrm{SrF}_{2}\right)=\left[\mathrm{Sr}^{2+}\right]\left[\mathrm{F}^{-}\right]^{2} \\
<br/>7.9 \times 10^{-10}=\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\left(2 \mathrm{~S}_{2}\right)^{2}
<br/>\end{array}$
<br/>Dividing equation (ii) by equation (i)
<br/>$\begin{array}{l}
<br/>\frac{7.9 \times 10^{-10}}{7 \times 10^{-10}}=\frac{4 \mathrm{~S}_{2}^{2}}{\mathrm{~S}_{1}} \\
<br/>\begin{array}{l}
<br/>\therefore \mathrm{S}_{2}=\sqrt{\frac{7.9}{7} \times \frac{1.2 \times 10^{-3}}{4}} \\
<br/>\quad=1.84 \times 10^{-2} \\
<br/>\therefore\left[\mathrm{F}-\mathrm{]}=2 \mathrm{~S}_{2}=2 \times 1.84 \times 10^{-2}=3.68 \times 10^{-2} \mathrm{M}\right.
<br/>\end{array}
<br/>\end{array}$</div>


title: a-solution-of-nano-3-when-treated-with-a-mixture-of-zn-dust-and-a-yields-ammonia-a-can-be
slug: a-solution-of-nano-3-when-treated-with-a-mixture-of-zn-dust-and-a-yields-ammonia-a-can-be-55032
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
question:
<div class="question">A solution of $\mathrm{NaNO}_{3}$, when treated with a mixture of $\mathrm{Zn}$ dust and ' $\mathrm{A}$ ' yields ammonia. ' $\mathrm{A}$ ' can be</div>

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

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

answer:
<div class="solution">$\mathrm{NaNO}_{3}+4 \mathrm{Zn}+\frac{7 \mathrm{NaOH}}{(\mathrm{A})} \rightarrow 4 \mathrm{Na}_{2} \mathrm{ZnO}_{2}+\mathrm{NH}_{3}+2 \mathrm{H}_{2} \mathrm{O}$
<br/>A is caustic soda</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-sphere-of-radius-r-has-a-volume-density-of-charge-k-r-where-r-is-the-distance-from-the-centre-of-the-sphere-and-k-is-constant-the-magnitude-of-the-e

slug:
a-sphere-of-radius-r-has-a-volume-density-of-charge-k-r-where-r-is-the-distance-from-the-centre-of-the-sphere-and-k-is-constant-the-magnitude-of-the-e-56427

question:
<div class="question">A sphere of radius $R$ has a volume density of charge $\rho=k r,$ where $r$ is the distance from the centre of the sphere and $k$ is constant. The magnitude of the electric field which exists at the surface of the sphere is given by $\left(\varepsilon_{0}=\right.$ permittivity of free space)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{4 \pi k R^{4}}{3 \varepsilon_{0}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{k R}{3 e_{0}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4 \pi k R}{\varepsilon_{0}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{k R^{2}}{4 \varepsilon_{0}}$</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">$\frac{k R^{2}}{4 \varepsilon_{0}}$</span> </div>

answer:
<div class="solution">By Gauss's theorem
<br/>\(\begin{aligned} \mathrm{E}\left(4 \pi \mathrm{r}^{2}\right) &amp;=\frac{\int \mathrm{p} \times 4 \pi \mathrm{r}^{2} \mathrm{dr}}{\varepsilon_{0}} \\ &amp;=\frac{\int \mathrm{kr} \times 4 \pi \mathrm{r}^{2} \mathrm{dr}}{\varepsilon_{0}} \\ &amp;=\mathrm{E}=\frac{\mathrm{Kr}^{2}}{4 \varepsilon_{0}} \end{aligned}\)</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-spherical-ball-a-of-mass-4-kg-moving-along-a-straight-line-strikes-another-spherical-ball-b-of-mass-1-kg-at-rest-after-the-collision-a-and-b-move-wi-2

slug:
a-spherical-ball-a-of-mass-4-kg-moving-along-a-straight-line-strikes-another-spherical-ball-b-of-mass-1-kg-at-rest-after-the-collision-a-and-b-move-wi-2-95532

question:
<div class="question">A spherical ball $A$ of mass 4 kg, moving along a straight line strikes another spherical ball $B$ of mass $1 \mathrm{kg}$ at rest. After the collision, $A$ and $B$ move with velocities $y_{1} \mathrm{ms}^{-1}$ and $v_{2} \mathrm{ms}^{-1}$ respectively making angles of $30^{\circ}$ and $60^{\circ}$ with
<br/>respect to the original direction of motion of $A .$ The ratio $\frac{y_{1}}{v_{2}}$ will be</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_a23.png"/> <br/> <br/>Along $y$ -axis momentum remains zero. Here
<br/>$$
<br/>\begin{aligned}
<br/>4 v_{1} \sin 30^{\circ} &amp;=v_{2} \sin 60^{\circ} \\
<br/>\frac{y_{1}}{v_{2}} &amp;=\frac{\sqrt{3}}{4}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-spherical-ball-a-of-mass-4-kg-moving-along-a-straight-line-strikes-another-spherical-ball-b-of-mass-1-kg-at-rest-after-the-collision-a-and-b-move-wi-3

slug:
a-spherical-ball-a-of-mass-4-kg-moving-along-a-straight-line-strikes-another-spherical-ball-b-of-mass-1-kg-at-rest-after-the-collision-a-and-b-move-wi-3-64934

question:
<div class="question">A spherical ball $A$ of mass 4 kg, moving along a straight line strikes another spherical ball $B$ of mass $1 \mathrm{kg}$ at rest. After the collision, $A$ and $B$ move with velocities $y_{1} \mathrm{ms}^{-1}$ and $v_{2} \mathrm{ms}^{-1}$ respectively making angles of $30^{\circ}$ and $60^{\circ}$ with
<br/>respect to the original direction of motion of $A .$ The ratio $\frac{y_{1}}{v_{2}}$ will be</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_a23.png"/> <br/> <br/>Along $y$ -axis momentum remains zero. Here
<br/>$$
<br/>\begin{aligned}
<br/>4 v_{1} \sin 30^{\circ} &amp;=v_{2} \sin 60^{\circ} \\
<br/>\frac{y_{1}}{v_{2}} &amp;=\frac{\sqrt{3}}{4}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-spherical-convex-surface-of-power-5-dioptre-separates-object-and-image-space-of-refractive-indices-10-and-3-4-respectively-the-radius-of-curvature-o

slug:
a-spherical-convex-surface-of-power-5-dioptre-separates-object-and-image-space-of-refractive-indices-10-and-3-4-respectively-the-radius-of-curvature-o-33659

question:
<div class="question">A spherical convex surface of power 5 dioptre separates object and image space of refractive indices $1.0$ and $\frac{4}{3}$ respectively. The radius of curvature of the surface is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$5 \mathrm{~cm}$</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">$5 \mathrm{~cm}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fk0Z4DoJheuuneaAwY_Z6q2yi2D3XV9EHkMWRl0UXqQ.original.fullsize.png"/><br/> <br/>$P=5 D$
<br/>$u=\infty$ then
<br/>$\frac{1}{\infty}-\frac{4 / 3}{f}=\frac{1-4 / 3}{R}$
<br/>$-\frac{4}{3 f}=-\frac{1}{3 R}$
<br/>$\frac{f}{4}=R \quad \quad f=4 R$
<br/>$\therefore \frac{100}{5}=4 R \Rightarrow R=\frac{25}{5}=5 \mathrm{~cm}$</div>


tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-spherical-liquid-drop-is-placed-on-a-horizontal-plane-a-small-disturbance-causes-the-volume-of-the-drop-to-oscillate-the-time-period-of-oscillation--1

slug:
a-spherical-liquid-drop-is-placed-on-a-horizontal-plane-a-small-disturbance-causes-the-volume-of-the-drop-to-oscillate-the-time-period-of-oscillation-1-42341

question:
<div class="question">A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation ( $T$ ) of the liquid drop depends on radius $(n$ of the drop, density $(\rho)$ and surface tension (5) of the liquid. Which among the following will be a possible expression for $T$ (where, $k$ is a dimensionless constant)?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$k \sqrt{\frac{\rho r}{S}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$k \sqrt{\frac{\rho^{2} r}{S}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$k \sqrt{\frac{\rho r^{3}}{S}}$</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">D</span> <span class="option-data">$k \sqrt{\frac{\rho r^{3}}{s^{2}}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$k \sqrt{\frac{\rho r^{3}}{S}}$</span> </div>

answer:
<div class="solution">According to the question. time period. $T \alpha r^{a} p^{b} s^{c}$
<br/>$r=k r^{a} p^{b} s^{c}$
<br/>Thus, putting dimension, we get
<br/>Equating the dimensions of both sides, we get
<br/>$[T]=[\mathrm{L}]^{a}\left[\mathrm{ML}^{-3}]^ \mathrm{b}\left[\mathrm{MT}^{-2} ]^c\right.\right.$
<br/>$[T]=[M]^{b+c} \cdot[L]^{2a-3b} [T]^{-2c}$
<br/>$b+c=0, a-3 b=0$
<br/>and $-2 x=$
<br/>$\begin{array}{l}
<br/>c=-\frac{1}{2} \\
<br/>b=\frac{1}{2} \\
<br/>a=3 b=\frac{3}{2}
<br/>\end{array}$
<br/>Putting these value of $a, b$ and $c$ into Eq. (i), we get
<br/>$T=k \cdot r^{\frac{3}{2}} p^{\frac{1}{2}} \cdot s^{-\frac{1}{2}}$
<br/>$=k \sqrt{\frac{\rho}{s} r^{3}}$</div>


title: a-spring-of-force-constant-k-is-cut-into-three-equal-parts-the-force-constant-of-each-part-would-be
tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3 k$</span> </div>
answer: <div class="solution">Hints $: \mathrm{K} \propto \frac{1}{l}$</div>
source_db: db\MarksBatch1_P1.db
slug:
a-spring-of-force-constant-k-is-cut-into-three-equal-parts-the-force-constant-of-each-part-would-be-92970

question:
<div class="question">A spring of force constant $k$ is cut into three equal parts. The force constant of each part would be</div>

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


title: a-spring-of-force-constant-k-is-cut-into-two-equal-halves-the-force-constant-of-each-half-is
slug: a-spring-of-force-constant-k-is-cut-into-two-equal-halves-the-force-constant-of-each-half-is-50686
tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">As $\quad \mathrm{K} \ell=$ constant<br/>$K^{\prime}=2 K$</div>
source_db: db\MarksBatch1_P1.db
question:
<div class="question">A spring of force constant $\mathrm{k}$ is cut into two equal halves. The force constant of each half is</div>

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

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


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-square-conducting-loop-is-placed-near-an-infinitely-long-current-carrying-wire-with-one-edge-parallel-to-the-wire-as-shown-in-the-figure-if-the-curr-1

slug:
a-square-conducting-loop-is-placed-near-an-infinitely-long-current-carrying-wire-with-one-edge-parallel-to-the-wire-as-shown-in-the-figure-if-the-curr-1-24591

question:
<div class="question">A square conducting loop is placed near an infinitely long current carrying wire with one edge parallel to the wire as shown in the figure. If the current in the straight wire is suddenly halved. which of the following statements will be true? "The loop will .......".  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q24.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">move away from the wire</span> </div>

answer:
<div class="solution">As we know that, magnetic field due to a long wire at a distance $x$ from it is given by $B=\frac{\mu_{0} I}{2 \pi x},$ where, $I$ is the current flowing through the wire $\beta \propto I$
<br/>Magnetic flux associated with the square loop,
<br/>$$
<br/>\phi \propto \beta \propto I
<br/>$$
<br/>Now, if the current increase, then $\phi$ also increases. Direction of long wire will be $\otimes$.
<br/>This means, magnetic field due to induced current will be opposite to the existing magnetic field, i.e. according to Lenz's law, The induced current in the loop will be in the anti-clockwise direction. Now,
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a24.png"/> <br/>since, wires attract each other, if current flowing through them is in same direction and repel each other, if currents are in opposite direction.
<br/>$\therefore$ Part $C D$ of the loop will experience a force of   repulsion, whereas part $A B$ will experience attraction. Parts $B C$ and $A D$ will not experience any force. Thus, the overall force will be a force of repulsion because $A B$ is closer to straight. The force between two current carrying conductors is inversely proportional to the distance between them
<br/>$\because$
<br/>$$
<br/>\begin{array}{l}
<br/>F \propto \frac{1}{r} \\
<br/>r_{1} &lt; r_{2}
<br/>\end{array}
<br/>$$
<br/>So,
<br/>$$
<br/>F_{C D}&gt;F_{A B}
<br/>$$
<br/>$$
<br/>F_{\text {net }}=F_{C D}-F_{A B}
<br/>$$
<br/>Hence, the loop will moves away from the wire.</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-square-conducting-loop-is-placed-near-an-infinitely-long-current-carrying-wire-with-one-edge-parallel-to-the-wire-as-shown-in-the-figure-if-the-curr

slug:
a-square-conducting-loop-is-placed-near-an-infinitely-long-current-carrying-wire-with-one-edge-parallel-to-the-wire-as-shown-in-the-figure-if-the-curr-28481

question:
<div class="question">A square conducting loop is placed near an infinitely long current carrying wire with one edge parallel to the wire as shown in the figure. If the current in the straight wire is suddenly halved. which of the following statements will be true? "The loop will .......".  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q24.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">move away from the wire</span> </div>

answer:
<div class="solution">As we know that, magnetic field due to a long wire at a distance $x$ from it is given by $B=\frac{\mu_{0} I}{2 \pi x},$ where, $I$ is the current flowing through the wire $\beta \propto I$
<br/>Magnetic flux associated with the square loop,
<br/>$$
<br/>\phi \propto \beta \propto I
<br/>$$
<br/>Now, if the current increase, then $\phi$ also increases. Direction of long wire will be $\otimes$.
<br/>This means, magnetic field due to induced current will be opposite to the existing magnetic field, i.e. according to Lenz's law, The induced current in the loop will be in the anti-clockwise direction. Now,
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a24.png"/> <br/>since, wires attract each other, if current flowing through them is in same direction and repel each other, if currents are in opposite direction.
<br/>$\therefore$ Part $C D$ of the loop will experience a force of   repulsion, whereas part $A B$ will experience attraction. Parts $B C$ and $A D$ will not experience any force. Thus, the overall force will be a force of repulsion because $A B$ is closer to straight. The force between two current carrying conductors is inversely proportional to the distance between them
<br/>$\because$
<br/>$$
<br/>\begin{array}{l}
<br/>F \propto \frac{1}{r} \\
<br/>r_{1} &lt; r_{2}
<br/>\end{array}
<br/>$$
<br/>So,
<br/>$$
<br/>F_{C D}&gt;F_{A B}
<br/>$$
<br/>$$
<br/>F_{\text {net }}=F_{C D}-F_{A B}
<br/>$$
<br/>Hence, the loop will moves away from the wire.</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-steel-and-a-brass-wire-each-of-length-50-cm-and-crosssectional-area-0005-cm-2-hang-from-a-ceiling-and-are-15-cm-apart-lower-ends-of-the-wires-are-at-1

slug:
a-steel-and-a-brass-wire-each-of-length-50-cm-and-crosssectional-area-0005-cm-2-hang-from-a-ceiling-and-are-15-cm-apart-lower-ends-of-the-wires-are-at-1-87129

question:
<div class="question">A steel and a brass wire, each of length $50 \mathrm{~cm}$ and cross-sectional area $0.005 \mathrm{~cm}^{2}$ hang from a ceiling and are $15 \mathrm{~cm}$ apart. Lower ends of the wires are attached to a light horizontal bar. A suitable downward load is applied to the bar so that each of the wires extends in length by $0.1 \mathrm{~cm}$. At what distance from the steel wire the load must be applied? [Young's modulus of steel is $2 \times 10^{12}$ dynes $/ \mathrm{cm}^{2}$ and that of brass is $1 \times 10^{12}$ dynes $\left./ \mathrm{cm}^{2}\right]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7.5 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{~cm}$</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">C</span> <span class="option-data">$10 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \mathrm{~cm}$</span> </li> </ul>

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/L6JRQ9ioUV-Cdjw6AI3g8poyt5w8D6GYFAjDDyNQFfc.original.fullsize.png"/><br/> <br/>At equilibrium, Taking torque about point $\mathrm{O}$
<br/>$\begin{aligned}
<br/>T_{1} x=T_{2}(15-x) &amp; \Rightarrow Y_{1} x=Y_{2}(15-x) \\
<br/>\left[Y=\frac{T}{A} \times \frac{L}{\ell} ; T \propto Y\right] &amp; 1 \times 10^{12} x=2 \times 10^{12}(15-x) ; 3 x=15, \quad \text { So, } x=5 \mathrm{~cm}
<br/>\end{aligned}$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-steel-and-a-brass-wire-each-of-length-50-cm-and-crosssectional-area-0005-cm-2-hang-from-a-ceiling-and-are-15-cm-apart-lower-ends-of-the-wires-are-at

slug:
a-steel-and-a-brass-wire-each-of-length-50-cm-and-crosssectional-area-0005-cm-2-hang-from-a-ceiling-and-are-15-cm-apart-lower-ends-of-the-wires-are-at-29205

question:
<div class="question">A steel and a brass wire, each of length $50 \mathrm{~cm}$ and cross-sectional area $0.005 \mathrm{~cm}^{2}$ hang from a ceiling and are $15 \mathrm{~cm}$ apart. Lower ends of the wires are attached to a light horizontal bar. A suitable downward load is applied to the bar so that each of the wires extends in length by $0.1 \mathrm{~cm}$. At what distance from the steel wire the load must be applied? [Young's modulus of steel is $2 \times 10^{12}$ dynes $/ \mathrm{cm}^{2}$ and that of brass is $1 \times 10^{12}$ dynes $\left./ \mathrm{cm}^{2}\right]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7.5 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{~cm}$</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">C</span> <span class="option-data">$10 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \mathrm{~cm}$</span> </li> </ul>

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/L6JRQ9ioUV-Cdjw6AI3g8poyt5w8D6GYFAjDDyNQFfc.original.fullsize.png"/><br/> <br/>At equilibrium, Taking torque about point $\mathrm{O}$
<br/>$\begin{aligned}
<br/>T_{1} x=T_{2}(15-x) &amp; \Rightarrow Y_{1} x=Y_{2}(15-x) \\
<br/>\left[Y=\frac{T}{A} \times \frac{L}{\ell} ; T \propto Y\right] &amp; 1 \times 10^{12} x=2 \times 10^{12}(15-x) ; 3 x=15, \quad \text { So, } x=5 \mathrm{~cm}
<br/>\end{aligned}$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-stone-falls-freely-from-rest-and-the-total-distance-covered-by-it-in-the-last-second-of-its-motion-equals-the-distance-covered-by-it-in-the-first-th

slug:
a-stone-falls-freely-from-rest-and-the-total-distance-covered-by-it-in-the-last-second-of-its-motion-equals-the-distance-covered-by-it-in-the-first-th-16627

question:
<div class="question">A stone falls freely from rest and the total distance covered by it in the last second of its motion equals the distance covered by it in the first three seconds of its motion. The stone remains in the air for</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 \mathrm{~s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{~s}$</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">$7 \mathrm{~s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{~s}$</span> </li> </ul>

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

answer:
<div class="solution">Hints : $u=0$<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{S}_3=0+\frac{1}{2} g t^2=\frac{1}{2} \times 10 \times 9=45 \\<br/>&amp; \mathrm{~S}_t \text { th }=u+(2 t-1) \frac{g}{2} \\<br/>&amp; \mathrm{~S}_t \text { th }=0+5(2 t-1)=45 \\<br/>&amp; 2 t-1=9 \\<br/>&amp; t=5 \mathrm{sec}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-stone-of-relative-density-mathrmk-is-released-from-rest-on-the-surface-of-a-lake-if-viscous-effects-are-ignored-the-stone-sinks-in-water-with-an-acc

slug:
a-stone-of-relative-density-mathrmk-is-released-from-rest-on-the-surface-of-a-lake-if-viscous-effects-are-ignored-the-stone-sinks-in-water-with-an-acc-95986

question:
<div class="question">A stone of relative density \(\mathrm{K}\) is released from rest on the surface of a lake. If viscous effects are ignored, the stone sinks in water with an acceleration of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{g}\left(1-\frac{1}{\mathrm{~K}}\right)\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kN8EY5H-o_TPUoQMw69wGKF9ZjYD8qLwJaCL4CtRrG0.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; F=v \sigma g-v \rho g=v \sigma g\left(1-\frac{\rho}{\sigma}\right)=m g\left(1-\frac{1}{\mathrm{k}}\right) \\<br/>&amp; a=g\left(1-\frac{1}{\mathrm{k}}\right)<br/>\end{aligned}\)</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-straight-conductor-01-m-long-moves-in-uniform-magnetic-field-01-t-the-velocity-the-conductor-is-15-m-s-and-is-directer-perpendicular-to-the-field-th

slug:
a-straight-conductor-01-m-long-moves-in-uniform-magnetic-field-01-t-the-velocity-the-conductor-is-15-m-s-and-is-directer-perpendicular-to-the-field-th-25623

question:
<div class="question">A straight conductor $0.1 \mathrm{m}$ long moves in uniform magnetic field $0.1 \mathrm{T}$. The velocity the conductor is $15 \mathrm{m} / \mathrm{s}$ and is directer perpendicular to the field. The emf inducod between the two ends of the conductor is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.10 \mathrm{V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.15 \mathrm{V}$</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">$1.50 \mathrm{V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$15.00 \mathrm{v}$</span> </li> </ul>

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

answer:
<div class="solution">Given, length of conductor $l=0.1 
<br/>\mathrm{m}$
<br/>Mangetic field. $B=0.1 \mathrm{T}$ 
<br/>Velocity of conductor, $v=15 \mathrm{m} / \mathrm{s}$
<br/>The angle between $\mathbf{v}$ and $\mathbf{B}$ is $90^{\circ}$
<br/>When $\mathbf{v}$ and $\mathbf{B}$ are mutually perpendicular, then emf (induced) is given by
<br/>$$
<br/>\varepsilon=v B l=15 \times 0.1 \times 0.1=\frac{15}{100}=0.15 \mathrm{V}
<br/>$$</div>


title: a-straight-line-joining-the-points-111-and-000-intersects-the-plane-2-x-2-y-z-10-at
slug: a-straight-line-joining-the-points-111-and-000-intersects-the-plane-2-x-2-y-z-10-at-61406
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
question:
<div class="question">A straight line joining the points (1,1,1) and (0,0,0) intersects the plane $2 x+2 y+z=10$ at</div>

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

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

answer:
<div class="solution">Equation of line joining the points (1,1,1) and (0,0,0) is
<br/>$$
<br/>\begin{aligned}
<br/>\frac{x-0}{1-0} &amp;=\frac{y-0}{1-0}=\frac{z-0}{1-0}=\lambda \\
<br/>\Rightarrow &amp; x=y=z=\lambda
<br/>\end{aligned}
<br/>$$
<br/>So, the point is $(\lambda, \lambda, \lambda)$ The point intersects the plane $2 x+2 y+z=10$
<br/>$\therefore 2(\lambda)+2(\lambda)+\lambda=10$
<br/>$\Rightarrow 5 \lambda=10$
<br/>$\Rightarrow \lambda=2$
<br/>Hence, the point is (2,2,2) .</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$m+n$</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-straight-line-meets-the-coordinate-axes-at-a-and-b-a-circle-is-circumscribed-about-the-triangle-oab-o-being-the-origin-if-m-and-n-are-the-distances--1

slug:
a-straight-line-meets-the-coordinate-axes-at-a-and-b-a-circle-is-circumscribed-about-the-triangle-oab-o-being-the-origin-if-m-and-n-are-the-distances-1-23684

question:
<div class="question">A straight line meets the co-ordinate axes at $\mathrm{A}$ and $\mathrm{B}$. A circle is circumscribed about the triangle $\mathrm{OAB}, \mathrm{O}$ being the origin. If $\mathrm{m}$ and $\mathrm{n}$ are the distances of the tangent to the circle at the origin from the points $\mathrm{A}$ and $\mathrm{B}$ respectively, the diameter of the circle is</div>

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

answer:
<div class="solution">Clearly, AB is one of diameter<br/>$\because \mathrm{AM}$ and $\mathrm{BN}$ are parallel and $\angle \mathrm{BNO}=\angle \mathrm{AMO}=\pi / 2$<br/>$\therefore$ Points $\mathrm{N}, \mathrm{O}$ and $\mathrm{M}$ are collinear.<br/>$\therefore \triangle \mathrm{BNMA}$ is a rectangle<br/>$\Rightarrow \mathrm{AB}=\mathrm{MN}=\mathrm{m}+\mathrm{n}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/i4M6OkA-OzslW88r47tJs7dF9cqM2vpzJs7kkgxP-bc.original.fullsize.png"/></div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$m+n$</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-straight-line-meets-the-coordinate-axes-at-a-and-b-a-circle-is-circumscribed-about-the-triangle-oab-o-being-the-origin-if-m-and-n-are-the-distances-

slug:
a-straight-line-meets-the-coordinate-axes-at-a-and-b-a-circle-is-circumscribed-about-the-triangle-oab-o-being-the-origin-if-m-and-n-are-the-distances-62103

question:
<div class="question">A straight line meets the co-ordinate axes at $\mathrm{A}$ and $\mathrm{B}$. A circle is circumscribed about the triangle $\mathrm{OAB}, \mathrm{O}$ being the origin. If $\mathrm{m}$ and $\mathrm{n}$ are the distances of the tangent to the circle at the origin from the points $\mathrm{A}$ and $\mathrm{B}$ respectively, the diameter of the circle is</div>

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

answer:
<div class="solution">Clearly, AB is one of diameter<br/>$\because \mathrm{AM}$ and $\mathrm{BN}$ are parallel and $\angle \mathrm{BNO}=\angle \mathrm{AMO}=\pi / 2$<br/>$\therefore$ Points $\mathrm{N}, \mathrm{O}$ and $\mathrm{M}$ are collinear.<br/>$\therefore \triangle \mathrm{BNMA}$ is a rectangle<br/>$\Rightarrow \mathrm{AB}=\mathrm{MN}=\mathrm{m}+\mathrm{n}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/i4M6OkA-OzslW88r47tJs7dF9cqM2vpzJs7kkgxP-bc.original.fullsize.png"/></div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3: 4$</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-straight-line-through-the-origin-o-meets-the-parallel-lines-4-x-2-y-9-and-2-x-y-6-0-at-p-and-q-respectively-the-point-o-divides-the-segment-pq-in-th

slug:
a-straight-line-through-the-origin-o-meets-the-parallel-lines-4-x-2-y-9-and-2-x-y-6-0-at-p-and-q-respectively-the-point-o-divides-the-segment-pq-in-th-12038

question:
<div class="question">A straight line through the origin $\mathrm{O}$ meets the parallel lines $4 \mathrm{x}+2 \mathrm{y}=9$ and $2 \mathrm{x}+\mathrm{y}+6=0$ at $\mathrm{P}$ and $\mathrm{Q}$ respectively. The point O divides the segment $\mathrm{PQ}$ in the ratio</div>

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

answer:
<div class="solution">Hint :
<br/>$\Rightarrow \frac{\mathrm{OP}}{\mathrm{OQ}}=\frac{\mathrm{OM}}{\mathrm{ON}}=\Rightarrow \frac{9 / 2}{12 / 2}=\frac{3}{4}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lSbr94n9Wpew_JvzApJH-vgkRKqkSPh6Qv0i2Tqw350.original.fullsize.png"/></div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-straight-line-through-the-point-32-is-ndined-at-an-angle-6-0-to-the-line-3-x-y-1-if-it-intersects-the-x-axis-then-its-equation-vill-be

slug:
a-straight-line-through-the-point-32-is-ndined-at-an-angle-6-0-to-the-line-3-x-y-1-if-it-intersects-the-x-axis-then-its-equation-vill-be-21147

question:
<div class="question">A straight line through the point (3,-2) is ndined at an angle $60^{\circ}$ to the line $\sqrt{3} x+y=1$. If it intersects the $X$ -axis, then its equation vill be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y+x \sqrt{3}+2+3 \sqrt{3}=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$y-x \sqrt{3}+2+3 \sqrt{3}=0$</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">$y-x \sqrt{3}-2-2 \sqrt{3}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-x \sqrt{3}+2-3 \sqrt{3}=0$</span> </li> </ul>

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

answer:
<div class="solution">Given line,
<br/>$$
<br/>\begin{array}{l}
<br/>\sqrt{3} x+y=1 \\
<br/>y=-\sqrt{3} x+c
<br/>\end{array}
<br/>$$
<br/>We know that, $m=-\sqrt{3}$
<br/>$$
<br/>\begin{aligned}
<br/>\text { that, } m &amp;=-\sqrt{3} \quad[\because y=m x+c] \\
<br/>\tan \theta &amp;=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right| \\
<br/>\tan 60^{\circ} &amp;=\left|\frac{m_{1}-(-\sqrt{3})}{1+m_{1}(-\sqrt{3})}\right| \quad\left[\because \theta=60^{\circ}\right] \\
<br/>\sqrt{3}=&amp;\left|\frac{m_{1}+\sqrt{3}}{1-\sqrt{3}} m_{1}\right| \\
<br/>\pm \sqrt{3}=&amp; \frac{m_{1}+\sqrt{3}}{1-\sqrt{3} m_{1}}
<br/>\end{aligned}
<br/>$$
<br/>\begin{array}{l|l}
<br/>\hline taking + sign &amp; taking - sign \\
<br/>\hline+\sqrt{3}=\frac{m_{1}+\sqrt{3}}{1-\sqrt{3} m_{1}} &amp; -\sqrt{3}=\frac{m_{1}+\sqrt{3}}{1-\sqrt{3} m_{1}} \\
<br/>\sqrt{3}\left(1-\sqrt{3} m_{1}\right)=m_{1}+\sqrt{3} &amp; \Rightarrow-\sqrt{3}\left(1-\sqrt{3} m_{1}\right) \\
<br/>&amp; =m_{1}+\sqrt{3} \\
<br/>\Rightarrow \sqrt{3}-3 m_{1}=m_{1}+\sqrt{3} &amp; \Rightarrow-\sqrt{3}-3 m_{1}=m_{1}+\sqrt{3} \\
<br/>\Rightarrow &amp; \Rightarrow 3 m_{1}-m_{1}=2 \sqrt{3} \\
<br/>\Rightarrow 4m_{1}=0 &amp; \Rightarrow \quad 2 m_{1}=2 \sqrt{3} \\
<br/>\Rightarrow m_{1}=0 &amp; \Rightarrow \quad m_{1}=\sqrt{3}
<br/>\end{array}
<br/>given point (3,-2) 
<br/>$y=m x$ + c
<br/>$$
<br/>\begin{aligned}
<br/>-2 &amp;=3 \sqrt{3}+c \\
<br/>c &amp;=-2-3 \sqrt{3}
<br/>\end{aligned}
<br/>$$
<br/>Hence, required equation
<br/>$$
<br/>\therefore \quad y=m x+c
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>y=\sqrt{3} x-2-3 \sqrt{3} \\
<br/>y-\sqrt{3} x+2+3 \sqrt{3}=0 \\
<br/>y-x \sqrt{3}+2+3 \sqrt{3}=0
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-straight-line-through-the-point-of-intersection-of-the-lines-x-2-y-4-and-2-x-y-4-meets-the-coordinate-axes-at-a-and-b-the-locus-of-the-midpoint-of-a

slug:
a-straight-line-through-the-point-of-intersection-of-the-lines-x-2-y-4-and-2-x-y-4-meets-the-coordinate-axes-at-a-and-b-the-locus-of-the-midpoint-of-a-28471

question:
<div class="question">A straight line through the point of intersection of the lines $x+2 y=4$ and $2 x+y=4$ meets the coordinate axes at $A$ and $B$. The locus of the mid-point of $A B$ is</div>

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

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

answer:
<div class="solution">Given lines are,
<br/>$$
<br/>x+2 y=4
<br/>$$
<br/>and $\quad 2 x+y=4$
<br/>Solving Eqs. (i) and (ii), we get
<br/>$$
<br/>x=\frac{4}{3}, y=\frac{4}{3}
<br/>$$
<br/>Let $\quad A B: \frac{x}{a}+\frac{y}{b}=1$
<br/>Meets $x$ -axis at $A$ and $y$ -axis at $B$. If the straight line (iii) passes through the point $\left(\frac{4}{3}, \frac{4}{3}\right),$ then $\frac{4}{3 a}+\frac{4}{3 b}=1$
<br/>$\Rightarrow \quad \frac{1}{a}+\frac{1}{b}=\frac{3}{4}$
<br/>Let the mid-point of $A B$ be $(h, k)$
<br/>$\therefore \quad h=\frac{a+0}{2}, k=\frac{0+b}{2}$
<br/>$\Rightarrow \quad a=2 h, b=2 k$
<br/>$\therefore$ From Eq. $(\mathrm{iv}), \frac{1}{2 h}+\frac{1}{2 k}=\frac{3}{4}$
<br/>$\Rightarrow \quad 2 h+2 k=3 h k$
<br/>Replace $x, y$ from $h, k$ respectively, we get required locus
<br/>i.e. $2(x+y)=3 x y$</div>


