# questions

tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$5$</span> </div>
source_db: db\MarksBatch0.db
title:
A compound is formed by substitution of two chlorine for two hydrogens in propane. The number of possible isomeric compounds is

slug:
a-compound-is-formed-by-substitution-of-two-chlorine-for-two-hydrogens-in-propane-the-number-of-pos-24777

question:
<div class="question">A compound is formed by substitution of two chlorine for two hydrogens in propane. The number of possible isomeric compounds is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$5$</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">$2$</span> </li> </ul>

answer:
<div class="solution">$\mathrm{C}_3 \mathrm{H}_8 \stackrel{-2 \mathrm{H}}{+2 \mathrm{Cl}}&gt;\mathrm{C}_3 \mathrm{H}_6 \mathrm{Cl}_2$, following isomers of $\mathrm{C}_3 \mathrm{H}_6 \mathrm{Cl}_2$ is possible<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ATaMaL-hnh7UIsga39A5Icz77DnU2LEmQOryYtrM2lE.original.fullsize.png"/><br/><br/>Due to presence of chiral carbon compound (IV) is optically active and forms an enantiomer. So total no of isomers $=5$</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch0.db
title:
A compressive force is applied to a uniform rod of rectangular cross-section so that its length decreases by 1% . If the Poisson's ratio for the material of the rod be 0.2 , which of the following statements is correct? "The volume approximately ...........

slug:
a-compressive-force-is-applied-to-a-uniform-rod-of-rectangular-cross-section-so-that-its-length-decr-24793

question:
<div class="question">A compressive force is applied to a uniform rod of rectangular cross-section so that its length decreases by $1 \%$. If the Poisson's ratio for the material of the rod be $0.2,$ which of the following statements is correct? "The volume approximately ...........</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">decreases by $1 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">decreases by $0.8 \%$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">decreases by $0.6 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">increases by $0.2 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">decreases by $0.6 \%$</span> </div>

answer:
<div class="solution">Given, Decrement in the length $=1 \%$ Poisson's ratio for material of the rod, $\sigma=0.2$
<br/>As we know that, Volume, $v=\pi r^{2} l,[$ where $, l$ is the length of the rod $]$
<br/>$$
<br/>\frac{\Delta V}{V}=\frac{2 \Delta r}{r}+\frac{\Delta l}{l}...(i)
<br/>$$
<br/>since, Poisson's ratio, $\sigma=\frac{-\Delta D / D}{\Delta L / L}$
<br/>$$
<br/>=\frac{-\Delta r / r}{\Delta L / L}
<br/>$$
<br/>So, Eq. (i) can be written as,
<br/>$$
<br/>\begin{aligned}
<br/>\frac{\Delta V}{V}=\frac{\Delta l}{l}(1-2 \sigma) \\
<br/>\frac{\Delta V}{V} \times 100 &amp;=\left(\frac{\Delta l}{l} \times 100\right) \times(1-2 \sigma) \\
<br/>&amp;=-1 \times[1-2 \times(0.2)] \\
<br/>&amp;=-1 \times[1-0.4] \\
<br/>&amp;=-0.6 \%
<br/>\end{aligned}
<br/>$$
<br/>Here, negative sign shows the decrement in the volume.</div>


slug: a-conducting-circular-loop-of-resistance-20-and-cross-sectional-area-20-1-0-2-m-2-is-placed-83511
tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A conducting circular loop of resistance 20Î© and cross-sectional area 20 Ã 1 0 â 2 m 2 is placed perpendicular to a spatially uniform magnetic field B, which varies with time tas B = 2 sin ( 50 Ï t ) T. Find the net charge flowing through the loop in 20 ms starting from t = 0

question:
<div class="question">A conducting circular loop of resistance $20 \Omega$ and cross-sectional area $20 \times 10^{-2} \mathrm{~m}^{2}$ is placed perpendicular to a spatially uniform magnetic field B, which varies with time tas $\mathrm{B}=2 \sin (50 \pi t)$ T. Find the net charge flowing through the loop in $20 \mathrm{~ms}$ starting from $\mathrm{t}=0$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.5 \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.2 \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0 \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.14 \mathrm{C}$</span> </li> </ul>

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

answer:
<div class="solution">Hint : $\Delta \mathrm{q}=\frac{\Delta \varphi}{\mathrm{R}}=\frac{\varphi_{2}-\varphi_{1}}{\mathrm{R}}=0, \Delta \mathrm{q}=0\left[\varphi_{1}=\varphi_{2}=0\right.$ at $\mathrm{t}=0$ and $\left.\mathrm{t}=20 \mathrm{~ms}\right]$</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A conductivity cell has been calibrated with 0.01 M 1 : 1 electrolyte solution (specific conductance, k = 1.25 Ã 1 0 â 3 S cm â 1 ) in the cell and the measured resistance was 800Î© at 2 5 â C . The cell constant will be

slug:
a-conductivity-cell-has-been-calibrated-with-0-01-m-1-1-electrolyte-solution-specific-conductance-70952

question:
<div class="question">A conductivity cell has been calibrated with
<br/>$0.01 \mathrm{M} \quad 1: 1$ electrolyte solution (specific conductance, $k=1.25 \times 10^{-3} \mathrm{S} \mathrm{cm}^{-1}$ ) in the cell and the measured resistance was $800 \Omega$ at $25^{\circ} \mathrm{C} .$ The cell constant will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.02 \mathrm{cm}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.102 \mathrm{cm}^{-1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1.00 \mathrm{cm}^{-1}$</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">$0.5 \mathrm{cm}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.00 \mathrm{cm}^{-1}$</span> </div>

answer:
<div class="solution">Given, $\mathrm{K}=1.25 \times 10^{-3} \mathrm{S} \mathrm{cm}^{-1}$
<br/>$$
<br/>\rho=\frac{1}{\mathrm{x}}=\frac{1}{1.25 \times 10^{-3}} \mathrm{S}^{-1} \mathrm{cm}
<br/>$$
<br/>From
<br/>$$
<br/>\begin{aligned}
<br/>R &amp;=\rho \frac{1}{A} \\
<br/>800 &amp;=\frac{1}{1.25 \times 10^{-3}} \times \frac{1}{A}
<br/>\end{aligned}
<br/>$$
<br/>(where, $\frac{1}{A}=$ cell constant)
<br/>$\therefore$
<br/>$$
<br/>\begin{aligned}
<br/>\frac{1}{A} &amp;=800 \times 1.25 \times 10^{-3} \\
<br/>&amp;=1 \mathrm{cm}^{-1}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A container AB in the shape of a rectangular parallelopiped of length 5 m is divided internally by a movable partition P as shown in the figure. The left compartment is filled with a given mass of an ideal gas of molar mass 32 while the right compartment is filled with an equal mass of another ideal gas of molar mass 18 at same temperature. What will be the distance of P from the left wall A when equilibrium is established?

slug:
a-container-ab-in-the-shape-of-a-rectangular-parallelopiped-of-length-5-m-is-divided-internally-by-a-95682

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ihBhW-ftYZrNUtC7-pi7SvSMLa_natyC5TPx-z6cS78.original.fullsize.png"/><br/> <br/>A container $\mathrm{AB}$ in the shape of a rectangular parallelopiped of length $5 \mathrm{~m}$ is divided internally by a movable partition $\mathrm{P}$ as shown in the figure. The left compartment is filled with a given mass of an ideal gas of molar mass 32 while the right compartment is filled with an equal mass of another ideal gas of molar mass 18 at same temperature. What will be the distance of P from the left wall A when equilibrium is established?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.8 \mathrm{~m}$</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">$3.2 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.1 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.8 \mathrm{~m}$</span> </div>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mtbqhtHF_9V-df3MiMeSOOX9X280hYUh98PyIUkmMQs.original.fullsize.png"/><br/> <br/>At equilibrium $\quad P_{1}=P_{2}, T_{1}=T_{2}$ (assuming wall to be conducting)
<br/>$\frac{P_{1} V_{1}}{\mu_{1}}=\frac{P_{2} V_{2}}{\mu_{2}}=R T$
<br/>$\frac{(5-x) 32}{m}=\frac{x \times 18}{m}$
<br/>So, $x=3.2 \mathrm{~m}$
<br/>Hence the distance of the piston from $\mathrm{A}=5-3.2=1.8 \mathrm{~m}$</div>


slug: a-convex-lens-of-focal-length-30-mathrm-cm-produces-5-times-magnified-real-image-of-an-object-92214
tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
title:
A convex lens of focal length \(30 \mathrm{~cm}\) produces 5 times magnified real image of an object. What is the object distance ?

question:
<div class="question">A convex lens of focal length \(30 \mathrm{~cm}\) produces 5 times magnified real image of an object. What is the object distance ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(36 \mathrm{~cm}\)</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">B</span> <span class="option-data">\(25 \mathrm{~cm}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(30 \mathrm{~cm}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(150 \mathrm{~cm}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(36 \mathrm{~cm}\)</span> </div>

answer:
<div class="solution">Hints: \(\frac{1}{5 \mathrm{u}}-\left(\frac{1}{-\mathrm{u}}\right)=\frac{1}{30}\)<br/>\(\begin{aligned}<br/>&amp; \frac{1}{5 \mathrm{u}}+\frac{1}{\mathrm{u}}=\frac{1}{30}, \frac{5+1}{5 \mathrm{u}}=\frac{1}{30} \\<br/>&amp; \mathrm{u}=36 \mathrm{~cm} .<br/>\end{aligned}\)</div>


tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">golden</span> </div>
source_db: db\MarksBatch0.db
title:
A copper coin was electroplated with Zn and then heated at high temperature until there is a change in colour. What will be the resulting colour?

slug:
a-copper-coin-was-electroplated-with-zn-and-then-heated-at-high-temperature-until-there-is-a-change-18836

question:
<div class="question">A copper coin was electroplated with $\mathrm{Zn}$ and then heated at high temperature until there is a change in colour. What will be the resulting colour?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">white</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">black</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">silver</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">golden</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">A copper coin was electroplated with zinc
<br/>(Zn) and then heated at high temperature until there is a change in colour. The resulting colour will be golden. The golden colour is due to the zinc migrating through the copper to form the alpha-form of brass alloy (percentage of $\mathrm{Cu}&gt;65 \%$ and that of $\mathrm{Zn}    &lt; $ $35 \%)$
<br/>$$
<br/>\mathrm{Zn}+\mathrm{Cu} \stackrel{\Delta}{\longrightarrow} \text { Brass }
<br/>$$
<br/>Thus, the option (d) is correct answer.</div>


slug: a-cricket-ball-of-mass-0-25-mathrm-kg-with-speed-10-mathrm-m-mathrm-s-collides-wit-23257
tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
title:
A cricket ball of mass \(0.25 \mathrm{~kg}\) with speed \(10 \mathrm{~m} / \mathrm{s}\) collides with a bat and returns with same speed within \(0.01 \mathrm{~S}\). The force acted on bat is

question:
<div class="question">A cricket ball of mass \(0.25 \mathrm{~kg}\) with speed \(10 \mathrm{~m} / \mathrm{s}\) collides with a bat and returns with same speed within \(0.01 \mathrm{~S}\). The force acted on bat is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(25 \mathrm{~N}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(50 \mathrm{~N}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(250 \mathrm{~N}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(500 \mathrm{~N}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Hints: \(\Delta \mathrm{P}=2 \mathrm{mV}=2 \times 0.25 \times 10=5 \frac{\mathrm{kgm}}{\mathrm{s}}\)<br/>\(\mathrm{F}=\frac{\Delta \mathrm{P}}{\Delta \mathrm{t}}=\frac{5}{0.01}=500 \mathrm{~N}\)</div>


slug: a-cricket-ball-thrown-across-a-field-is-at-heights-h-1-and-h-2-from-the-point-of-projection-41968
tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
title:
A cricket ball thrown across a field is at heights h 1 â and h 2 â from the point of projection at times t 1 â and t 2 â respectively after the throw. The ball is caught by a fielder at the same height as that of projection. The time of flight of the ball in this journey is

question:
<div class="question">A cricket ball thrown across a field is at heights $h_{1}$ and $h_{2}$ from the point of projection at times $t_{1}$ and $t_{2}$ respectively after the throw. The ball is caught by a fielder at the same height as that of projection. The time of flight of the ball in this journey is</div>

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

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

answer:
<div class="solution">For vertically moment (for $\left.h_{1}\right)$
<br/>or
<br/>$\begin{array}{l}
<br/>h_{1}=u \sin \theta t_{1}-\frac{1}{2} g t_{1}^{2} \\
<br/>t_{1}=\frac{h_{1}+\frac{1}{2} g t_{1}^{2}}{u \sin \theta}
<br/>\end{array}$
<br/>$h_{2}=u \sin \theta t_{2}-\frac{1}{2} g t_{2}^{2}$
<br/>or $t_{2}=\frac{h_{2}+\frac{1}{2} g t_{2}^{2}}{u \sin \theta}$
<br/>Divide Eq. (i) by Eq. (ia)
<br/>$\begin{array}{l}
<br/>\frac{t_{1}}{t_{2}}=\frac{h_{1}+\frac{1}{2} g t_{1}^{2} / u \sin \theta}{h_{2}+\frac{1}{2} g t_{2}^{2} / u \sin \theta} \\
<br/>\Rightarrow h_{1} t_{2}-h_{2} t_{1}=\frac{g}{2}\left(t_{1} t_{2}^{2}-t_{1}^{2} t_{2}\right)
<br/>\end{array}$
<br/>The time of flight of the ball $T=\frac{2 u \sin \theta}{g}=\frac{2}{g}(u \sin \theta) \quad$ 
<br/>$=\frac{2}{g}\left[\frac{h_{1}+\frac{1}{2} g t_{1}^{2}}{t_{1}}\right]=\frac{2}{t_{1}}\left[\frac{h_{1}}{g}+\frac{t_{1}^{2}}{2}\right]$
<br/>$=\frac{h_{1}}{t_{1}} \times \frac{2}{g}+t_{1}=\frac{h_{1}}{t_{1}} \times\left(\frac{t_{1} t_{2}^{2}-t_{1}^{2} t_{2}}{h_{1} t_{2}-h_{2} t_{1}}\right)+t_{1}$
<br/>$=\frac{h_{1} t_{1} t_{2}^{2}-h_{1} t_{1}^{2} t_{2}+h_{1} t_{1}^{2} t_{2}-h_{2} t_{1}^{3}}{t_{1}\left(h_{1} t_{2}-h_{2} t_{1}\right)}$
<br/>$=\left(\frac{h_{1} t_{2}^{2}-h_{2} t_{1}^{2}}{h_{1} t_{2}-h_{2} t_{1}}\right)$</div>


slug: a-cubical-vessel-of-height-1-mathrm-m-is-full-of-water-what-is-the-amount-of-work-done-in-pu-17401
tags: ['Physics', 'Work Power Energy', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A cubical vessel of height \(1 \mathrm{~m}\) is full of water. What is the amount of work done in pumping water out of the vessel? (Take \(\mathrm{g}=\) \(10 \mathrm{~m} \mathrm{~s}^{-2}\) )

question:
<div class="question">A cubical vessel of height \(1 \mathrm{~m}\) is full of water. What is the amount of work done in pumping water out of the vessel? (Take \(\mathrm{g}=\) \(10 \mathrm{~m} \mathrm{~s}^{-2}\) )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(1250 \mathrm{~J}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(5000 \mathrm{~J}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1000 \mathrm{~J}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2500 \mathrm{~J}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(5000 \mathrm{~J}\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/wrNc70r-dZaR8oVtsiEIeUV1lcelHc_2xQHCzEWYdEk.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \mathrm{V}=\ell^3=1 \mathrm{~m}^3 \\<br/>&amp; \mathrm{~m}=1 \times 1000=1000 \mathrm{~kg} ; \mathrm{W}=\mathrm{mgh}=1000 \times 10 \times \frac{1}{2}=5000 \mathrm{~J}<br/>\end{aligned}\)</div>


slug: a-current-i-i-0-e-t-is-flowing-in-a-circuit-consisting-of-a-parallel-combination-of-res-32790
tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
A current I = I 0 â e â Î» t is flowing in a circuit consisting of a parallel combination of resistance R and capacitance C . The total charge over the entire pulse period is

question:
<div class="question">A current $I=I_{0} e^{-\lambda t}$ is flowing in a circuit consisting of a parallel combination of resistance $R$ and capacitance $C$. The total charge over the entire pulse period is</div>

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

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

answer:
<div class="solution">The current $1=1_{0} e^{-x}$ is of exponential nature.
<br/>$I=\frac{d Q}{d t}=I_{0} e^{-\lambda t}$
<br/>$d Q=I_{0} e^{-\lambda t} d t$
<br/>$\int_{0}^{Q} d Q=I_{0} \int_{t=0}^{t=-\infty} e^{-\lambda t} \cdot d t$
<br/>$Q=I_{0} \times \frac{1}{\lambda}=\frac{I_{0}}{\lambda}$</div>


slug: a-current-i-is-flowing-along-an-infinite-straight-wire-in-the-positive-z-direction-and-the-16298
tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A current ' I â² is flowing along an infinite, straight wire, in the positive Z -direction and the same current is flowing along a similar parallel wire 5 m apart, in the negative Z-direction. A point P is at a perpendicular distance 3 m from the first wire and 4 m from the second. What will be magnitude of the magnetic field B of P ?

question:
<div class="question">A current ' $I^{\prime}$ is flowing along an infinite, straight wire, in the positive $Z$ -direction and the same current is flowing along a similar parallel wire $5 \mathrm{m}$ apart, in the negative Z-direction. A point $P$ is at a perpendicular distance $3 \mathrm{m}$ from the first wire and $4 \mathrm{m}$ from the second. What will be magnitude of the magnetic field B of $P ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{12}\left(u_{0} I\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7}{24}\left(\mu_{0} I\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5}{24}\left(\mu_{0} I\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{25}{288}\left(\mu_{0} I\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{24}\left(\mu_{0} I\right)$</span> </div>

answer:
<div class="solution">According to the question. 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a23.png"/> <br/>Magnetic field due to first wire $(A)$ is $B_{1}=\frac{\mu_{0} \times I}{2 \pi \times 3}$
<br/>Magnetic field due to second wire $(B)$ is $B_{2}=\frac{\mu_{0} \times 1}{2 \pi \times 4}$
<br/>Net magnitude of magnetic field $B=\sqrt{B_{1}^{2}+B_{2}^{2}}$
<br/>$$
<br/>B=\frac{\mu_{0} I}{2 \pi} \sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{\mu_{0} I \times 5}{2 \pi \times 3 \times 4}
<br/>$$
<br/>Magnetic field $B=\frac{5}{24} \times \frac{\mu_{0} I}{\pi}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A current of 1 A is flowing along positive x -axis through a straight wire of length 0.5 m placed in a region of a magnetic field given by B = ( 2 i + 4 j ) T . The magnitude and the direction of the force experienced by the wire respectively are

slug:
a-current-of-1-a-is-flowing-along-positive-x-axis-through-a-straight-wire-of-length-0-5-m-placed-in-47642

question:
<div class="question">A current of 1 A is flowing along positive $x$ -axis through a straight wire of length $0.5 \mathrm{m}$ placed in a region of a magnetic field given by $\mathbf{B}=(2 \mathbf{i}+4 \mathbf{j}) \mathrm{T} .$ The magnitude and the direction of the force experienced by the wire respectively are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{18} \mathrm{N},$ along positive $z$ -axis</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{20} \mathrm{N}$, along positive $x$ -axis</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{N}$, along positive $z$ -axis</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{N},$ along positive $y$ -axis</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \mathrm{N}$, along positive $z$ -axis</span> </div>

answer:
<div class="solution">The force experienced by the wire placed in magnetic field, $F=B il$
<br/>$$
<br/>\begin{array}{l}
<br/>=(2 \hat{i}+4 \hat{j})\left(1 \hat{i} \times \frac{1}{2}\right) \\
<br/>=\hat{i} \times \hat{i}+4 \times \frac{1}{2}(\hat{i} \times \hat{j})=2 \hat{k} N
<br/>\end{array}
<br/>$$
<br/>Direction of this force can be find out by Fleming's left hand rule which is along pasitive z-axis</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A curve passes through the point ( 3 , 2 ) for which the segment of the tangent line contained between the co-ordinate axes is bisected at the point of contact. The equation of the curve is

slug:
a-curve-passes-through-the-point-3-2-for-which-the-segment-of-the-tangent-line-contained-betwe-73129

question:
<div class="question">A curve passes through the point $(3,2)$ for which the segment of the tangent line contained between the co-ordinate axes is bisected at the point of contact. The equation of the curve is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=x^2-7$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=\frac{y^2}{2}+2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x y=6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^2+y^2-5 x+7 y+11=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x y=6$</span> </div>

answer:
<div class="solution">$\frac{x}{x_1}+\frac{y}{y_1}=2$<br/>$\because\left(\frac{d y}{d x}\right)_{\left(x_1, y_1\right)}=-\frac{y_1}{x_1}$<br/>$\therefore \frac{d y}{d x}=-\frac{y}{x} \Rightarrow \frac{d y}{y}=-\frac{d x}{x} \Rightarrow \ln y=-\ln x+k \Rightarrow \ln x y=k \Rightarrow x y=e^k=C \Rightarrow x y=C$ passes through $(3,2)$<br/>$\therefore C=3 \times 2=6 \therefore x y=6$</div>


slug: a-cyclic-process-is-shown-in-mathrm-p-mathrm-v-diagram-and-mathrm-t-mathrm-s-diagram-37002
tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint :</div>
source_db: db\MarksBatch0.db
title:
A cyclic process is shown in \(\mathrm{p}-\mathrm{v}\) diagram and \(\mathrm{T}-\mathrm{S}\) diagram. Which of the following statement(s) is/are true?

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GkIqPuyS-vzfGpzgu60Tu3O7KOZRs68di4z0h4Ql680.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GVRcD2hfPGF7egbd1W0k4cmeJ7CMs1cQNrH_M0T7wT0.original.fullsize.png"/><br/><br/>A cyclic process is shown in \(\mathrm{p}-\mathrm{v}\) diagram and \(\mathrm{T}-\mathrm{S}\) diagram. Which of the following statement(s) is/are true?</div>

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


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch0.db
title:
A cylinder of height his filled with wator and is kept on a block of height h /2 The level of water in the cylinder is kept constant. Four holes numbered 1,2,3 and 4 are at the side of the cylinder and at heights 0 , h /4 , h /2 and 3 h /4 , respectively. When all four holes are openned together, the hole from which water will reach farthest distance on the plane PQ is the hole number.

slug:
a-cylinder-of-height-his-filled-with-wator-and-is-kept-on-a-block-of-height-h-2-the-level-of-water-13890

question:
<div class="question">A cylinder of height his filled with wator and is kept on a block of height $h / 2$ The level of water in the cylinder is kept constant. Four holes numbered 1,2,3 and 4 are at the side of the cylinder and at heights $0, h / 4, h / 2$ and $3 h / 4,$ respectively. When all four holes are openned together, the hole from which water will reach farthest distance on the plane $P Q$ is the hole number.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q31.png"/></div>

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

answer:
<div class="solution">Height of the cylinder from the level $P O$ is $H=h+\frac{h}{2}=\frac{3 h}{2}$
<br/>To cover maximum distance along the plane $P Q$, height of the hole is $=\frac{3 h}{2} \times \frac{1}{2}=\frac{3 h}{4}$
<br/>This height corresponds to hole (2).</div>


slug: a-cylindrical-block-floats-vertically-in-a-liquid-of-density-1-kept-in-a-container-such-that-51016
tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A cylindrical block floats vertically in a liquid of density Ï 1 â kept in a container such that the fraction of volume of the cylinder inside the liquid is x 1 â . Then some amount of another immiscible liquid of density Ï 2 â ( Ï 2 â Ï 1 â ) is added to the liquid in the container so that the cylinder now floats just fully immersed in the liquids with x 2 â , fraction of volume of the cylinder inside the liquid of density Ï 1 â . The ratio Ï 1 â / Ï 2 â will be

question:
<div class="question">A cylindrical block floats vertically in a liquid of density $\rho_{1}$ kept in a container such that the fraction of volume of the cylinder inside the liquid is $x_{1}$. Then some amount of another immiscible liquid of density $\rho_{2}\left(\rho_{2}     &lt; \rho_{1}\right)$ is added to the liquid in the container so that the cylinder now floats just fully immersed in the liquids with $x_{2}$, fraction of volume of the cylinder inside the liquid of density $\rho_{1}$. The ratio $\rho_{1} / \rho_{2}$ will be</div>

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

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

answer:
<div class="solution">The given situation is shown as <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a52.png"/> <br/> <br/>As the Bouyant force in both the cases are same
<br/>$\therefore$
<br/>$$
<br/>\rho_{1} x_{1} g=\rho_{1} x_{2} g+\rho_{2}\left(1-x_{2}\right) g
<br/>$$
<br/>On solving $\quad \frac{\rho_{1}}{\rho_{2}}=\frac{\left(1-x_{2}\right)}{\left(x_{1}-x_{2}\right)}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is

slug:
a-determinant-is-chosen-at-random-from-the-set-of-all-determinants-of-order-2-with-elements-0-or-1-o-84200

question:
<div class="question">A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{16}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{3}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5}{8}$</span> </li> </ul>

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

answer:
<div class="solution">$\left|\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right|=a d-b c=0$<br/>Case $1: a d=b c=0 \rightarrow 3 \times 3=9$ cases<br/>Case $2:$ ad $=\mathrm{bc}=1 \rightarrow 1$ case only<br/>$\therefore$ Probability $=1-\frac{9+1}{2^4}=\frac{3}{8}$</div>


slug: a-diver-at-a-depth-of-12-mathrm-m-in-water-left-mu-frac-4-3-right-sees-the-sky-in-a-11595
tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch0.db
title:
A diver at a depth of \(12 \mathrm{~m}\) in water \(\left(\mu=\frac{4}{3}\right)\) sees the sky in a cone of semivertical angle :

question:
<div class="question">A diver at a depth of \(12 \mathrm{~m}\) in water \(\left(\mu=\frac{4}{3}\right)\) sees the sky in a cone of semivertical angle :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\sin ^{-1}\left(\frac{4}{3}\right)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\tan ^{-1}\left(\frac{4}{3}\right)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\sin ^{-1}\left(\frac{3}{4}\right)\)</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">\(90^{\circ}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\sin ^{-1}\left(\frac{3}{4}\right)\)</span> </div>

answer:
<div class="solution">Hints: \(c=\sin ^{-1}\left(\frac{1}{\mu}\right)\)<br/>\(=\sin ^{-1}\left(\frac{3}{4}\right)\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_XVVcqnFFSfcKAlqBYNtXbfWBd7u9ESDxGCP9CNdxVI.original.fullsize.png"/><br/></div>


slug: a-double-ordinate-pq-of-the-hyperbola-a-2-x-2-b-2-y-2-1-is-such-that-opq-is-equilat-20018
tags: ['Mathematics', 'Hyperbola', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A double ordinate PQ of the hyperbola a 2 x 2 â â b 2 y 2 â = 1 is such that Î OPQ is equilateral, O being the centre of the hyperbola. Then the eccentricity e satisfies the relation

question:
<div class="question">A double ordinate $\mathrm{PQ}$ of the hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ is such that $\Delta \mathrm{OPQ}$ is equilateral, $\mathrm{O}$ being the centre of the hyperbola. Then the eccentricity e satisfies the relation</div>

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

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

answer:
<div class="solution">Hint:
<br/>$P \equiv(a \sec \theta, b \tan \theta) \angle \mathrm{POM}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{\mathrm{b}}{\mathrm{a}} \sin \theta$
<br/>$\Rightarrow \sin \theta=\frac{a}{b \sqrt{3}}$
<br/>$\Rightarrow \frac{b^{2}}{a^{2}}&gt;\frac{1}{3} \quad \Rightarrow 1+\frac{b^{2}}{a^{2}}&gt;\frac{4}{3} \quad \Rightarrow e&gt;\frac{2}{\sqrt{3}}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/H-XxXaiYOpHr4b0lJSHbErhFp7oQb9i_-rAnH5NYmaI.original.fullsize.png"/></div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">80</span> </div>
source_db: db\MarksBatch0.db
title:
A drop of some liquid of volume 0.04 cm 3 is placed on the surface of a glass slide. Then another glass slide is placed on it in such a way that the liquid forms a thin layer of area 20 cm 2 between the surfaces of the two slides. To separate the slides a force of 16 Ã 1 0 5 dyne has to be applied normal to the surfaces. The surface tension of the liquid is (in dyne-cm â 1 )

slug:
a-drop-of-some-liquid-of-volume-0-04-cm-3-is-placed-on-the-surface-of-a-glass-slide-then-another-gl-18310

question:
<div class="question">A drop of some liquid of volume $0.04 \mathrm{cm}^{3}$ is placed on the surface of a glass slide. Then another glass slide is placed on it in such a way that the liquid forms a thin layer of area $20 \mathrm{cm}^{2}$ between the surfaces of the two slides. To separate the slides a force of $16 \times 10^{5}$ dyne has to be applied normal to the surfaces. The surface tension of the liquid is (in dyne-cm $^{-1}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">60</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">70</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">80</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">90</span> </li> </ul>

answer:
<div class="solution">Let, thickness of layer be $x$
<br/>So, volume $V=$ Area $\times x$
<br/>$$
<br/>\begin{array}{ll}
<br/>V=A \times x &amp; (\because)=2 r) \\
<br/>x=V / A
<br/>\end{array}
<br/>$$
<br/>$2 r=\frac{V}{A} \Rightarrow r=\frac{V}{2 A}$
<br/>(i)
<br/>and $\quad \Delta P=\frac{T}{r}$
<br/>We know that $F=\Delta P \times A$
<br/>$$
<br/>=\frac{T}{r} \times A
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>F=\frac{T}{\left(\frac{V}{2 A}\right)} \times A \\
<br/>T=\frac{F \times V}{2 A^{2}}
<br/>\end{array}
<br/>$$
<br/>[from
<br/>where $F=16 \times 10^{5}$ dyne $, V=0.04 \mathrm{cm}^{3}$
<br/>$$
<br/>\begin{aligned}
<br/>A=20 \mathrm{cm}^{2}=\frac{16 \times 10^{5} \times 0.04}{2 \times 20^{2}} \\
<br/>=\frac{8 \times 10^{5} \times 4}{20^{2} \times 100}=\frac{8 \times 10^{5} \times 4}{400 \times 100} \\
<br/>=8 \times 10^{5} \times 10^{-4} &amp;=80 \mathrm{dyne} / \mathrm{cm} \\
<br/>&amp;=80 \text { dyne } \mathrm{cm}^{-1}
<br/>\end{aligned}
<br/>$$</div>


slug: a-drop-of-water-detaches-itself-from-the-exit-of-a-tap-when-surface-tension-of-water-de-45516
tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
A drop of water detaches itself from the exit of a tap when ( Ï = surface tension of water, Ï = density of water, R = radius of the tap exit, r = radius of the drop )

question:
<div class="question">A drop of water detaches itself from the exit of a tap when $(\sigma=$ surface tension of water, $\rho=$ density of water, $R=$ radius of the tap exit, $r=$ radius of the drop $)$</div>

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

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

answer:
<div class="solution">Let mass of the drop $=m$ 
<br/>Weight of the drop = mg 
<br/>It will act downward. The force due to surface tension on the drop $=\sigma2 \pi R$ 
<br/>Where, $R=$ radius of tap If will act in upward direction. The drop of water will detach when
<br/>$$
<br/>m g&gt;\sigma 2 \pi R
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>\text { But, } \quad m=v \times \rho=\frac{4}{3} \pi^{3} p g \\
<br/>\therefore \quad \frac{4}{3} \pi^{3} \rho g&gt;\sigma \times 2 \pi A
<br/>\end{array}
<br/>$$
<br/>$$
<br/>r&gt;\left[\frac{3}{2} \frac{\sigma \cdot R}{\rho g}\right]^{1 / 3}
<br/>$$
<br/>Hence, none of the given options is correct.</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A fair coin is tossed at a fixed number of times. If the probability of getting exactly 3 heads equals the probability of getting exactly 5 heads, then the probability of getting exactly one head is

slug:
a-fair-coin-is-tossed-at-a-fixed-number-of-times-if-the-probability-of-getting-exactly-3-heads-equa-89521

question:
<div class="question">A fair coin is tossed at a fixed number of times. If the probability of getting exactly 3 heads equals the probability of getting exactly 5 heads, then the probability of getting exactly one head is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 64$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 / 32$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$1 / 16$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 32$</span> </div>

answer:
<div class="solution">Let the coin be tossed $n$ times 
<br/>Let getting head is consider to be success $\therefore \quad p=\frac{1}{2}, q=1-p=1-\frac{1}{2}=\frac{1}{2}$
<br/>It is given that,
<br/>$P(X=3)=P(X=5)$
<br/>$\Rightarrow \quad{ }^{n} C_{3}\left(\frac{1}{2}\right)^{3}\left(\frac{1}{2}\right)^{n-3}={ }^{n} C_{5}\left(\frac{1}{2}\right)^{5}\left(\frac{1}{2}\right)^{n-5}$
<br/>$\Rightarrow \quad{ }^{n} C_{3}={ }^{n} C_{5}$
<br/>$\Rightarrow \quad n=3+5 \quad\left[\because^{n} C_{x}={ }^{n} C_{y} \Rightarrow x+y=n \mid\right.$
<br/>$\Rightarrow \quad n=8$
<br/>Now, $P(X=1)={ }^{8} C_{1}\left(\frac{1}{2}\right)^{1}\left(\frac{1}{2}\right)^{8-1}$
<br/>$={ }^{8} C_{1} \times\left(\frac{1}{2}\right)^{8}=\frac{1}{32}$</div>


title: A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
slug:
a-fair-six-faced-die-is-rolled-12-times-the-probability-that-each-face-turns-up-twice-is-equal-to-12012

question:
<div class="question">A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{12 !}{2^{6} 6^{12}}$</span> </div>

answer:
<div class="solution">Required probability
<br/>$$
<br/>\begin{aligned}={ }^{12} C_{2} \times{ }^{10} C_{2} \times{ }^{8} C_{2} \times{ }^{6} C_{2} \times{ }^{4} C_{2} \times{ }^{2} C_{2} \times\left(\frac{1}{6}\right)^{12} \\=\frac{12 !}{10 ! \times 2 !} \times \frac{10 !}{8 ! \times 2 !} \times \frac{8 !}{6 ! \times 2 !} \times \frac{6 !}{4 ! \times 2 !} \\ \times \frac{4 !}{2 ! \times 2 !} \times \frac{2 !}{2 ! \times 1 !} \times\left(\frac{1}{6}\right)^{12} \end{aligned}
<br/>$$
<br/>$=\frac{12 !}{2^{6} \times 6^{12}}$</div>


tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A family of curves is such that the length intercepted on the y -axis between the origin and the tangent at a point is three times the ordinate of the point of contact. The family of curves is

slug:
a-family-of-curves-is-such-that-the-length-intercepted-on-the-y-axis-between-the-origin-and-the-tan-80337

question:
<div class="question">A family of curves is such that the length intercepted on the $y$ -axis between the origin and the tangent at a point is three times the ordinate of the point of contact. The family of curves is</div>

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

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

answer:
<div class="solution">Let the general equation of tangent.
<br/>Which passes through the point $(x, y)$ is 
<br/>$(Y-y)=\frac{d y}{d x}(x-x)$
<br/>$\Rightarrow \quad Y-y=X \frac{d y}{d x}-x \frac{d y}{d x}$
<br/>for length of $Y$ -intercept, put $X=0$ in Eq. $(1),$ we get $Y-y=-x \frac{d y}{d x}$
<br/>$\Rightarrow \quad Y=y-x \frac{d y}{d x}$
<br/>Now,according to the guestion Y-intercept $=3 \times$ ordinate of the point of contact $\Rightarrow \quad y-x \frac{d y}{d x}=3 y$
<br/>$\Rightarrow \quad-x \frac{d y}{d x}=2 y$
<br/>$\Rightarrow$
<br/>$\int \frac{d y}{y}=-\int \frac{2 d x}{x}$ (on integrating)
<br/>$\Rightarrow \quad \log y=-2 \log x+\log C$
<br/>$\Rightarrow \quad \log y+\log x^{2}=\log C$
<br/>$\Rightarrow y x^{2}=C$. where $C$ is a constant.</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A fighter plane, flying horizontally with a speed 360 kmph at an altitude of 500 m drops a bomb for a target straight ahead of it on the ground. The bomb .should be dropped at what approximate distance ahead of the target ? Assume that acceleration due to gravity ( g ) is 10 ms â 2 . Also neglect air drag

slug:
a-fighter-plane-flying-horizontally-with-a-speed-360-kmph-at-an-altitude-of-500-m-drops-a-bomb-for-58687

question:
<div class="question">A fighter plane, flying horizontally with a speed $360 \mathrm{kmph}$ at an altitude of $500 \mathrm{~m}$ drops a bomb for a target straight ahead of it on the ground. The bomb .should be dropped at what approximate distance ahead of the target ? Assume that acceleration due to gravity $(\mathrm{g})$ is $10 \mathrm{~ms}^{-2}$. Also neglect air drag</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1000 \mathrm{~m}$</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">$50 \sqrt{5} \mathrm{~m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$500 \sqrt{5} \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$866 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1000 \mathrm{~m}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$v=360 \mathrm{~km} / \mathrm{h}=100 \mathrm{~m} / \mathrm{s} .$
<br/>$\mathrm{h}=500 \mathrm{~m}$
<br/>$\mathrm{R}=\mathrm{u} \sqrt{\frac{2 \mathrm{~h}}{\mathrm{~g}}}=100 \times \sqrt{\frac{2 \times 500}{10}}=1000 \mathrm{~m}$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A frictionless piston-cylinder based enclosure contains some amount of gas at a pressure of 400 kPa . Then heat is transferred fo the gas at constant pressure in a quasi-static process. The piston moves up slowly through a height of 10 cm . If the piston has a cross-sectional area of 0.3 m 2 , the work done by the gas in this process is

slug:
a-frictionless-piston-cylinder-based-enclosure-contains-some-amount-of-gas-at-a-pressure-of-400-kpa-19704

question:
<div class="question">A frictionless piston-cylinder based enclosure contains some amount of gas at a pressure of $400 \mathrm{kPa}$. Then heat is transferred fo the gas at constant pressure in a quasi-static process. The piston moves up slowly through a height of $10 \mathrm{cm} .$ If the piston has a cross-sectional area of $0.3 \mathrm{m}^{2},$ the work done by the gas in this process is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 \mathrm{kJ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$12 \mathrm{kJ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$7.5 \mathrm{kJ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$24 \mathrm{kJ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$12 \mathrm{kJ}$</span> </div>

answer:
<div class="solution">$W_{\text {constant pressure }}=p \times \Delta V$ $=400 \times 10^{3} \times 0.3 \times 10 \times 10^{-2}$
<br/>$=400 \times 10 \times 3$
<br/>$=12000=12 \mathrm{kJ}$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A galvanometer can be converted to a voltmeter of full-scale deflection V 0 â by connecting a series resistance R 1 â and can be converted to an ammeter of full-scale deflection I 0 â by connecting a shunt resistance R 2 â . What is the current flowing through the galvanometer at its full-scale deflection?

slug:
a-galvanometer-can-be-converted-to-a-voltmeter-of-full-scale-deflection-v-0-by-connecting-a-seri-66599

question:
<div class="question">A galvanometer can be converted to a voltmeter of full-scale deflection $\mathrm{V}_{0}$ by connecting a series resistance $\mathrm{R}_{1}$ and can be converted to an ammeter of full-scale deflection $I_{0}$ by connecting a shunt resistance $\mathrm{R}_{2}$. What is the current flowing through the galvanometer at its full-scale deflection?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{V}_{0}-\mathrm{I}_{0} \mathrm{R}_{2}}{\mathrm{R}_{1}-\mathrm{R}_{2}}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$R_{1}=\frac{V_{0}}{I_{g}}-G_{\ldots} \ldots \ldots . .(1)$
<br/>$\mathrm{R}_{2}=\frac{\mathrm{I}_{\mathrm{g}} \mathrm{G}}{\mathrm{I}_{0}-\mathrm{I}_{\mathrm{g}}} \ldots \ldots \ldots . .(2)$
<br/>Eliminating G we get
<br/>$I_{g}=\frac{V_{0}-I_{0} R_{2}}{R_{1}-R_{2}}$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
title:
A galvanometer having internal resistance 10Î© requires 0.01 A for a full scale deflection. To convert this galvanometer to a voltmeter of full-scale deflection at 120 V , we need to connect a resistance of

slug:
a-galvanometer-having-internal-resistance-10-requires-0-01-a-for-a-full-scale-deflection-to-conve-86255

question:
<div class="question">A galvanometer having internal resistance $10 \Omega$ requires 0.01 A for a full scale deflection. To convert this galvanometer to a voltmeter of full-scale deflection at $120 \mathrm{V}$, we need to connect a resistance of</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$11990 \Omega$ in series</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$11990 \Omega$ in parallel</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$12010 \Omega$ in series</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12010 \Omega$ in parallel</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$11990 \Omega$ in series</span> </div>

answer:
<div class="solution">Resistance $R=\frac{V}{i_{g}}-G=\frac{120}{0.01}-10$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=12000-10 \\
<br/>R=11990 \Omega
<br/>\end{aligned}
<br/>$$
<br/>To convert a galvanometer into a voltmeter, high resistance should connect in series.</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
A gas bubble of 2 cm diameter rises througha liquid of density 1.75 g cm â 3 with a fixed speed of 0.35 cms â 1 . Neglect the density of the gas. The coefficient of viscosity of the liquid is

slug:
a-gas-bubble-of-2-cm-diameter-rises-througha-liquid-of-density-1-75-g-cm-3-with-a-fixed-speed-of-78995

question:
<div class="question">A gas bubble of $2 \mathrm{cm}$ diameter rises througha liquid of density $1.75 \mathrm{g} \mathrm{cm}^{-3}$ with a fixed speed of $0.35 \mathrm{cms}^{-1}$. Neglect the density of the gas. The coefficient of viscosity of the liquid is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">870 poise</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1120 poise</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">982 poise</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1089 poise</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">1089 poise</span> </div>

answer:
<div class="solution">The gas bubble is rising through the liquid of densily $1.75 \mathrm{g} / \mathrm{cm}^{3}$ with constant speed. The acting forces on bubble are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a21.png"/> <br/> <br/>Buoyancy force of liquid $=F_{0}$ (in upward direction) Viscous force due to liquid = $F_{V}$, (in downward direction)
<br/>$$
<br/>\therefore \quad F_{V}=F_{b}
<br/>$$
<br/>For free falling in a viscous medium
<br/>$$
<br/>F_{V}=6 \pi \eta {V_{T}}
<br/>$$
<br/>$\eta$ = viscosity coefficient of the liquid 
<br/>$r=$ radius of bubble
<br/>$$
<br/>\begin{aligned}
<br/>F_{D} &amp;=m g=v \times \rho g \\
<br/>&amp;=\frac{4}{3} \pi r^{3} \rho_{e} \cdot g
<br/>\end{aligned}
<br/>$$
<br/>$\begin{aligned} 6 \pi \eta r V_{t} &amp;=\frac{4}{3} \pi r^{3} \rho_{e} \cdot g \\ \eta &amp;=\frac{24 \pi \cdot r^{3} \rho_{e} \cdot g}{3 \times 6 \pi rV_{T}}=\frac{2}{9} \cdot \frac{r^{2} \rho_{e} \cdot g}{v_{T}} \end{aligned}$
<br/>$\begin{aligned} D &amp;=2 \mathrm{cm}, r=1 \mathrm{cm}=1 \times 10^{-2} \mathrm{m} \\ \rho_{e} &amp;=1.75 \mathrm{g} / \mathrm{cm}^{3}=1.75 \times 10^{* 3} \mathrm{kg} / \mathrm{m}^{3} \\ g &amp;=10 \mathrm{m} / \mathrm{s}^{2} \\ \mathrm{v}_{T} &amp;=\text { terminal velocity } \\ &amp;=0.35 \times 10^{-2} \mathrm{m} / \mathrm{s} \\ \eta &amp;=\frac{2}{9} \times \frac{\left(10^{-2}\right)^{2} \times 1.75 \times 10^{+3} \times 10}{0.35 \times 10^{-2}} \\ &amp;=\frac{2}{9} \times \frac{10^{-4} \times 1.75 \times 10^{3} \times 10}{0.35 \times 10^{-2}} \\ &amp;=\frac{2 \times 175 \times 10 \times 10}{9 \times 35} \times 10 \text { poise } \\ &amp;=1089 \text { poise } \end{aligned}$</div>


title: A gas can be liquefied at temperature T and pressure p provided
slug: a-gas-can-be-liquefied-at-temperature-t-and-pressure-p-provided-36863
question: <div class="question">A gas can be liquefied at temperature T and pressure $p$ provided</div>
tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$T=T_{c}$ and $p    &lt; p_{c}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$T    &lt; T_{c}$ and $p&gt;p_{c}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$T&gt;T_{c}$ and $p&gt;p_{c}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$T&gt;T_{c}$ and $p    &lt; p_{c}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$T    &lt; T_{c}$ and $p&gt;p_{c}$</span> </div>

answer:
<div class="solution">A gas can be liquefied at temperature $T$ when it is lower than critical temperature, $T_{c}$ and pressure $p$ is greater than cntical pressure.
<br/>Hence. $T    &lt; T_{c}$ and $p&gt;p_{c}$</div>


slug: a-given-amount-of-fe-2-is-oxidized-by-x-mol-of-mno-4-in-acidic-medium-the-number-of-moles-of-82877
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
A given amount of Fe 2 + is oxidized by x mol of MnO 4 â in acidic medium. The number of moles of Cr 2 â O 7 â 2 â required to oxidize the same amount of Fe 2 + in acidic medium is

question:
<div class="question">A given amount of $\mathrm{Fe}^{2+}$ is oxidized by $\mathrm{x}$ mol of $\mathrm{MnO}_{4}$ in acidic medium. The number of moles of $\mathrm{Cr}_{2} \mathrm{O}_{7}{ }^{2-}$ required to oxidize the same amount of $\mathrm{Fe}^{2+}$ in acidic medium is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.83 \mathrm{x}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.0 \mathrm{x}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.2 \mathrm{x}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.83 \mathrm{x}$</span> </div>

answer:
<div class="solution">As per Law of Equivalence
<br/>no. of eq. of $\mathrm{Fe}^{+2}=$ no. of eq. of $\mathrm{KMnO}_{4}=$ no. of eq. of $\mathrm{Cr}_{2} \mathrm{O}_{7}^{-2}$
<br/>$\begin{array}{l}
<br/>\mathrm{MnO}_{4}^{-}(\mathrm{O} \cdot \text { Agent })+5 \mathrm{e}^{-}+8 \mathrm{H}^{+} \rightarrow \mathrm{Mn}^{+2}+4\mathrm{H}_{2} \mathrm{O} \\
<br/>(\mathrm{n} . \mathrm{f}=5) \\
<br/>\mathrm{Cr}_{2} \mathrm{O}_{7}^{-2}(\mathrm{O} \cdot \mathrm{A})+14 \mathrm{H}^{+}+6 \mathrm{e}^{-} \rightarrow 2 \mathrm{Cr}^{+3}+7 \mathrm{H}_{2} \mathrm{O} \\
<br/>(\mathrm{n} . \mathrm{f}=6)
<br/>\end{array}$
<br/>no. of eq. of $\mathrm{KMnO}_{4}=$ no. of moles $\times \mathrm{n}$. factor $=\mathrm{x} \times 5$
<br/>no. of eq. of $\mathrm{Cr}_{2} \mathrm{O}_{7}^{-2}=$ no. of moles $\times n . \mathrm{f}=\mathrm{y} \times 6$
<br/>From Law of equivalence
<br/>$\begin{array}{l}
<br/>6 y=5 x \\
<br/>\text { or } y=5 / 6 x=0.83 x
<br/>\end{array}$</div>


slug: a-given-quantity-of-gas-is-taken-from-a-to-c-in-two-ways-a-directly-from-a-rightar-52375
tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A given quantity of gas is taken from \(A\) to \(C\) in two ways; \(a\) ) directly from \(A \rightarrow C\) along a straight line and \(b\) ) in two steps, from \(A \rightarrow B\) and then from \(B \rightarrow C\). Work done and heat absorbed along the direct path \(A \rightarrow C\) is \(200 \mathrm{~J}\) and \(280 \mathrm{~J}\) respectively

question:
<div class="question">A given quantity of gas is taken from \(A\) to \(C\) in two ways; \(a\) ) directly from \(A \rightarrow C\) along a straight line and \(b\) ) in two steps, from \(A \rightarrow B\) and then from \(B \rightarrow C\). Work done and heat absorbed along the direct path \(A \rightarrow C\) is \(200 \mathrm{~J}\) and \(280 \mathrm{~J}\) respectively<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sBNmmkeIhzP07obOXXnsHcihrZpL6GJXB9gRixsq-4g.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(80 \mathrm{~J}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(160 \mathrm{~J}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(120 \mathrm{~J}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(160 \mathrm{~J}\)</span> </div>

answer:
<div class="solution">Hint : \(\Delta W=200 \mathrm{~J} \quad \Delta Q=280 \mathrm{~J} \quad\) for path \(A C\)<br/>\(\therefore \quad \Delta U=\Delta Q-\Delta W=280-200=80 \mathrm{~J}\)<br/>\((\Delta U)\) is same for both paths.<br/>\(\therefore \Delta Q=\Delta W+\Delta U=80+80=160 \mathrm{~J}\)</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">50</span> </div>
source_db: db\MarksBatch0.db
title:
A glass slab consists of thin uniform layers of progressively decreasing refractive indices RI (see figure) such that the RI of any layer is Î¼ â m Î Î¼ . Here, Î¼ and Î Î¼ denote the RI of 0 th layer and the difference in RI between any two consecutive layers, respectively. The integer m = 0 , 1 , 2 , 3 , â¦ denotes the numbers of the successive layers. A ray of light from the oth layer enters the 1st layer at an angle of incidence of 3 0 â . After undergoing the mth refraction, the ray emerges parallel to the interface. If Î¼ = 1.5 and Î Î¼ = 0.015 , the value of m is

slug:
a-glass-slab-consists-of-thin-uniform-layers-of-progressively-decreasing-refractive-indices-ri-see-89354

question:
<div class="question">A glass slab consists of thin uniform layers of progressively decreasing refractive indices RI (see figure) such that the RI of any layer is $\mu-m \Delta \mu$. Here, $\mu$ and $\Delta \mu$ denote the $\mathrm{RI}$ of 0 th layer and the difference in RI between any two consecutive layers, respectively. The integer $m=0,1,2,3, \ldots$ denotes the numbers of the successive layers. A ray of light from the oth layer enters the 1st layer at an angle of incidence of $30^{\circ}$. After undergoing the mth refraction, the ray emerges parallel to the interface. If $\mu=1.5$ and $\Delta \mu=0.015,$ the value of $m$ is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q48.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">30</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">40</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">50</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">By Snell's law. $\mu \sin i=$ constant
<br/>$$
<br/>\begin{aligned}
<br/>\mu \sin i &amp;=(\mu-m \Delta \mu) \sin r \\
<br/>\text { where } \mu=1.5, i &amp;=30^{\circ}, r=90^{\circ}, \Delta \mu=0.015 \\
<br/>1.5 \sin 30^{\circ} &amp;=(1.5-m \times 0.15) \sin 90^{\circ} \\
<br/>\frac{3}{2} \times \frac{1}{2} &amp;=(1.5-m \times 0.015) \times 1 \\
<br/>\frac{3}{4} &amp;=\frac{3}{2}-\frac{15 m}{1000} \\
<br/>15 \mathrm{m} &amp;=\left(\frac{3}{2}-\frac{3}{4}\right) \times 1000 \\
<br/>15 \mathrm{m} &amp;=\frac{6-3}{4} \times 1000 \Rightarrow m=\frac{3}{4} \times \frac{1000}{15} \\
<br/>m &amp;=\frac{3000}{60} \Rightarrow m=50
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A golf ball of mass 50 gm placed on a tee, is struck by a golf-club. The speed of the golf ball as it leaves the tee is 100 m / s , the time of contact on the ball is 0.02 s . If the force decreases to zero linearly with time, then the force at the beginning of the contact is

slug:
a-golf-ball-of-mass-50-gm-placed-on-a-tee-is-struck-by-a-golf-club-the-speed-of-the-golf-ball-as-i-57676

question:
<div class="question">A golf ball of mass $50 \mathrm{gm}$ placed on a tee, is struck by a golf-club. The speed of the golf ball as it leaves the tee is $100 \mathrm{~m} / \mathrm{s}$, the time of contact on the ball is $0.02 \mathrm{~s}$. If the force decreases to zero linearly with time, then the force at the beginning of the contact is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$200 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$250 \mathrm{~N}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$500 \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/FqdiqrllOar_9_sY0_E-IN-BcpUkGiXrfPvHcf0F0Fs.original.fullsize.png"/><br/><br/>Impulse $=$ Area under F-t graph $=$ change in linear momentum<br/>$$<br/>=\frac{1}{2} \times F \times 0.02=\frac{50}{1000}(100-0) \Rightarrow F=500 \mathrm{~N}<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch0.db
title:
A heating element of resistance r is fitted inside an adiabatic cylinder which carries a frictionless piston of mass m and cross section A as shown in diagram. The cylinder contains one mole diatomic current the element temperatures time t as (a and Î² are constants), while pressure remains constant. The atmospheric pressure above the piston is P 0 â . Then

slug:
a-heating-element-of-resistance-r-is-fitted-inside-an-adiabatic-cylinder-which-carries-a-frictionles-97371

question:
<div class="question">$A$ heating element of resistance $r$ is fitted inside an adiabatic cylinder which carries a frictionless piston of mass $m$ and cross section A as shown in diagram. The cylinder contains one mole diatomic current the element temperatures time $t$ as (a and $\beta$ are constants), while pressure remains constant. The atmospheric pressure above the piston is $P_{0}$. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">the rate of increase in internal energy is $\frac{5}{2} R(\alpha+\beta t)$</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">the current flowing in the element is $\sqrt{\frac{5}{2 r} R(\alpha+\beta t)}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the piston moves upwards with constant acceleration</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the piston moves upwards with constant speed</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">the rate of increase in internal energy is $\frac{5}{2} R(\alpha+\beta t)$</span> </div>

answer:
<div class="solution">We know that Internal energy, $U=\frac{n f R T}{2}=\frac{5 R}{2}\left(\alpha t+\frac{1}{2} \beta t^{2}\right)$
<br/>Differentiate with respect to $t$
<br/>$$
<br/>\frac{d U}{d t}=\frac{5 R}{2}[\alpha+\beta t]
<br/>$$
<br/>But, $d Q=n C_{p} d T$
<br/>$$
<br/>\begin{array}{l}
<br/>\therefore \quad \frac{d Q}{d t}=n C_{p} \times \frac{d T}{d t} \Rightarrow i^{2} r=\frac{7}{2} R \times[\alpha+\beta t] \\
<br/>\quad i=\sqrt{\frac{7 R}{2 r}(\alpha+\beta t)} \\
<br/>p V=n R T
<br/>\end{array}
<br/>$$
<br/>or $\quad V=\frac{n R T}{p}=\frac{n R}{p}\left(\alpha t+\frac{1}{2} \beta t^{2}\right)$
<br/>$X=\frac{n R}{p A}\left(\alpha t+\frac{1}{2} \beta t^{2}\right)$
<br/>$$
<br/>v=\frac{n R}{p A}(\alpha+\beta t)
<br/>$$
<br/>and acceleration $=\frac{n R}{p A} \times \beta$
<br/>So, the rate of increase in internal energy is $\frac{5}{2} R(\alpha+\beta t)$ and the piston moves upwards with constant acceleration.</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A hemisphere of radius R is placed in a uniform electric field E so that its axis is parallel to the field. Which of the following statement(s) is / are true?

slug:
a-hemisphere-of-radius-r-is-placed-in-a-uniform-electric-field-e-so-that-its-axis-is-parallel-to-the-80442

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/YjlNWB0OFl0uBSn6kQVmeb7E816oGWA7tPoONEKwO1M.original.fullsize.png"/><br/><br/>A hemisphere of radius $\mathrm{R}$ is placed in a uniform electric field $\mathrm{E}$ so that its axis is parallel to the field. 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">Flux through the curved surface of hemisphere is $\pi R^2 E$.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Flux through the circular surface of hemisphere is $\pi R^2 E$.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Total flux enclosed is 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">Work done in moving a point charge $q$ from $A$ to $B$ via the path $A C B$ depends upon $R$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Flux through the curved surface of hemisphere is $\pi R^2 E$., Total flux enclosed is zero</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp;\quad \text { as } \phi_{\text {in }}=0, \quad \phi_{\text {total }}=0 \\<br/>&amp;\phi_{\text {curved }}+\phi_{\text {flat }}=0 \\<br/>&amp;\phi_{\text {flat }}=-E \times \pi R^2 \\<br/>&amp;\therefore \phi_{\text {curved }}=E \times \pi R^2<br/>\end{aligned}<br/>$$<br/>Also $\Delta \mathrm{V}=-\overrightarrow{\mathrm{E}} \cdot \Delta \overrightarrow{\mathrm{r}}$ for uniform eletrified.<br/>$$<br/>\begin{aligned}<br/>&amp;\text { as } \vec{E} \perp \Delta \vec{r} \\<br/>&amp;\Delta V=0 \\<br/>&amp;W=q \Delta V=0<br/>\end{aligned}<br/>$$<br/>so work is independent of $R$ in moving charge $q$ from $A+B$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
A hollow metal sphere of radius R is charged with a charge Q . The electric potential and intensity inside the sphere are respectively

slug:
a-hollow-metal-sphere-of-radius-r-is-charged-with-a-charge-q-the-electric-potential-and-intensity-95261

question:
<div class="question">A hollow metal sphere of radius $R$ is charged with a charge $Q .$ The electric potential and intensity inside the sphere are respectively</div>

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

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

answer:
<div class="solution">Inside a hollow charged sphere, the electric field intensity is always zero.
<br/>$\therefore$ $E=0$ but potential is constant.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a3.png"/> <br/>Potential on the surtace of the hollow sphere
<br/>$$
<br/>V=\frac{K Q}{R}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R}=\frac{Q}{4 \pi \varepsilon_{0} R}
<br/>$$
<br/>This is the potential $(\mathrm{V})$ inside the charged hollow meta sphere.</div>


slug: a-hollow-sphere-of-external-radius-r-and-thickness-t-r-is-made-of-a-metal-of-density-the-sp-60477
tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A hollow sphere of external radius R and thickness t ( R ) is made of a metal of density Ï . The sphere will float in water, if

question:
<div class="question">$A$ hollow sphere of external radius $R$ and thickness $t(&lt; &lt; R)$ is made of a metal of density $\rho$. The sphere will float in water, if</div>

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

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

answer:
<div class="solution">The density of material is $\rho$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a9.png"/> <br/> <br/>The hollow sphere will float if its weight is less than the weight of the water displaced by the volume of the sphere This implies mass of the sphere is less than that for the same volume of water. Now, mass of spherical cell
<br/>$$
<br/>m_{1}=4 \pi R^{2} \times t \times \rho
<br/>$$
<br/>While the mass of water having same volume
<br/>$$
<br/>m_{2}=\frac{4}{3} \pi R^{3} \times \rho_{g}=\frac{4}{3} \pi R^{3}
<br/>$$
<br/>where, $\rho_{g}=$ density of water $=\frac{1 \mathrm{kg}}{\mathrm{m}^{3}}$
<br/>For the floatation of sphere,
<br/>$$
<br/>m_{1} \leq m_{2}
<br/>$$
<br/>$4 \pi R^{2} \times t \times \rho \leq \frac{4}{3} \pi R^{3}$
<br/>$\Rightarrow \quad t \rho \leq \frac{R}{3} \Rightarrow t \leq \frac{R}{3 \rho}$</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch0.db
title:
A homonuclear diatomic gas molecule shows 2-electron magnetic moment. The one-electron and two-electron reduced species obtained from above gas molecule can act as both oxidizing and reducing agents. When the gas molecule is one-electron oxidized the bond length decreases compared to the neutral molecule. The gas molecule is

slug:
a-homonuclear-diatomic-gas-molecule-shows-2-electron-magnetic-moment-the-one-electron-and-two-elect-94204

question:
<div class="question">A homonuclear diatomic gas molecule shows 2-electron magnetic moment. The one-electron and two-electron reduced species obtained from above gas molecule can act as both oxidizing and reducing agents. When the gas molecule is one-electron oxidized the bond length decreases compared to the neutral molecule. The gas molecule is</div>

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

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/yXyj0VLjn-zT47T1-QBngt58kJs2NVrGqS1Fe5qS9Cs.original.fullsize.png"/></div>


slug: a-horizontal-fire-hose-with-a-nozzle-of-cross-sectional-arca-21-5-1-0-3-m-2-delivers-97940
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch0.db
title:
A horizontal fire hose with a nozzle of cross-sectional arca 21 â 5 â Ã 1 0 â 3 m 2 delivers a cubic metre of water in 10 s. What will be the maximum possible increase in the temperature of water while it hits a rigid wall (neglecting the effect of gravity)?

question:
<div class="question">A horizontal fire hose with a nozzle of cross-sectional arca $\frac{5}{\sqrt{21}} \times 10^{-3} \mathrm{m}^{2}$ delivers a cubic metre of water in 10 s. What will be the maximum possible increase in the temperature of water while it hits a rigid wall (neglecting the effect of gravity)?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1^{\circ} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.1^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.01^{\circ} \mathrm{C}$</span> </li> </ul>

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

answer:
<div class="solution">Given, cross sectional area of nozzle
<br/>$$
<br/>A=\frac{5}{\sqrt{21}} \times 10^{-3} \mathrm{m}^{2}
<br/>$$
<br/>and rate of heat transfer $Q=\frac{1}{10}=10^{-1} \mathrm{m}^{3} / \mathrm{s}$
<br/>Heat transfer $Q=A V$ $\therefore V=\frac{Q}{A}=\frac{10^{-1} \mathrm{m}^{3} / \mathrm{s}}{\frac{5}{\sqrt{21}} \times 10^{-3} \mathrm{m}^{2}}=2 \sqrt{21} \times 10 \mathrm{m} / \mathrm{s}$
<br/>When it hits a rigid wall then maximum possible increase in temperature of water can be expressed as
<br/>$$
<br/>\frac{1}{2} m v^{2}=m s \Delta T
<br/>$$
<br/>where, $m=$ mass, $s=$ specific heat of water
<br/>$$
<br/>=1 \mathrm{cal} / \mathrm{g}=4.2 \times 10^{3} \mathrm{J} / \mathrm{kg}
<br/>$$
<br/>and $\Delta T=$ increase in temperature
<br/>$$
<br/>\begin{aligned}
<br/>\text { From Eq. }(i), \Delta T=&amp; \frac{1}{2} \frac{v^{2}}{s} \\
<br/>&amp;=\frac{\left(2 \sqrt{21} \times 10^{1}\right)^{2}}{2 \times 4.2 \times 10^{3}} \\
<br/>&amp;=\frac{84 \times 10^{2}}{8.4 \times 10^{3}} \\
<br/>&amp;=1^{\circ} \mathrm{C}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A horizontal semi-circular wire of radius r is connected to a battery through two similar springs X and Y to an electric cell, which sends current I through it. A vertically downward uniform magnetic field B is applied on the wire, as shown in the figure. What is the force acting on each spring?

slug:
a-horizontal-semi-circular-wire-of-radius-r-is-connected-to-a-battery-through-two-similar-springs-x-14005

question:
<div class="question">A horizontal semi-circular wire of radius $\mathrm{r}$ is connected to a battery through two similar springs $\mathrm{X}$ and $\mathrm{Y}$ to an electric cell, which sends current I through it. A vertically downward uniform magnetic field B is applied on the wire, as shown in the figure. What is the force acting on each spring?<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ZxajP_msPkg4t3zAonun29cvusg0wH9BzVqAaN9Sw-U.original.fullsize.png"/></div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/qSyQNlA375Bb9yomgju8sm5wW9ZUtB8RflIvACqOJVk.original.fullsize.png"/><br/><br/>$F_B=i \times 2 r \times B$<br/>$2 F=F_B$<br/>$F=i r B$</div>


slug: a-hyperbola-having-the-transverse-axis-of-length-2-sin-is-confocal-with-the-ellipse-3-x-2-4-y-46185
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A hyperbola, having the transverse axis of length 2 sin Î¸ is confocal with the ellipse 3 x 2 + 4 y 2 = 12 . It equation is

question:
<div class="question">A hyperbola, having the transverse axis of length $2 \sin \theta$ is confocal with the ellipse $3 x^{2}+4 y^{2}=12$. It equation is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$x^{2} \operatorname{cosec}^{2} \theta-y^{2} \sec ^{2} \theta=1$</span> </div>

answer:
<div class="solution">Given. $2 a_{1}=2 \sin \theta$
<br/>$\Rightarrow \quad a_{1}=\sin \theta$
<br/>and $3 x^{2}+4 y^{2}=12$
<br/>$\Rightarrow \quad \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$
<br/>Here, $a^{2}=4$ and $b^{2}=3$
<br/>$\therefore$
<br/>$b^{2}=a^{2}\left(1-e^{2}\right)$
<br/>$\Rightarrow$
<br/>$3=4\left(1-e^{2}\right)$
<br/>$\Rightarrow$
<br/>$e^{2}=1-\frac{3}{4}=\frac{1}{4}$
<br/>$e=\frac{1}{2}$
<br/>Focus, $F(a e, 0)=F\left(2 \times \frac{1}{2}, 0\right)$
<br/>$=F(1,0)$
<br/>For hyperbola foci are same
<br/>$a_{1} e_{1}=a e=1$
<br/>$\therefore$
<br/>$(\sin \theta) e_{1}=1$
<br/>$\Rightarrow$
<br/>$e_{1}=\operatorname{cosec} \theta$
<br/>and $b_{1}^{2}=a_{1}^{2}\left(e_{1}^{2}-1\right)=a_{1}^{2} e_{1}^{2}-a_{1}^{2}$
<br/>$\Rightarrow \quad b_{1}^{2}=1-\sin ^{2} \theta=\cos ^{2} \theta$
<br/>$\frac{x^{2}}{a_{1}^{2}}-\frac{y^{2}}{b_{1}^{2}}=1$
<br/>$\Rightarrow \quad \frac{x^{2}}{\sin ^{2} \theta}-\frac{y^{2}}{\cos ^{2} \theta}=1$
<br/>$\Rightarrow x^{2} \operatorname{cosec}^{2} \theta-y^{2} \sec ^{2} \theta=1$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3^n</span> </div>
source_db: db\MarksBatch0.db
title:
A is a set containing n elements. P and Q are two subsets of A . Then the number of ways of choosing P and Q so that P â© Q = Ï is

slug:
a-is-a-set-containing-n-elements-p-and-q-are-two-subsets-of-a-then-the-number-of-ways-of-choosing-41270

question:
<div class="question">A is a set containing n elements. $P$ and $Q$ are two subsets of $A$. Then the number of ways of choosing $P$ and $Q$ so that $P \cap Q=\varphi$ is</div>

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

answer:
<div class="solution">$\sum_{r=0}^n{ }^n C_r \cdot 2^{n-r}=(2+1)^n=3^n$ (Assuming $P$ contains r elements then $Q$ can be formed with $n-r$ elements in $2^{n-r}$ ways)</div>


tags: ['Physics', 'Semiconductors', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A junction diode has a resistance of 25Î© when forward biased and 2500Î© when reverse biased. The current in the diode, for the arrangement shown will be

slug:
a-junction-diode-has-a-resistance-of-25-when-forward-biased-and-2500-when-reverse-biased-the-cu-58963

question:
<div class="question">A junction diode has a resistance of $25 \Omega$ when forward biased and $2500 \Omega$ when reverse biased. The current in the diode, for the arrangement shown will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OWBV48beSVcoKVMHfHYHaPhh3mK3e_IpE3Pi4kL084g.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{15} \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{7} \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{25} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{180} \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{7} \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\mathrm{R}_{\mathrm{eq}}=25+10=35 \Omega$<br/>Because diode is forward biased. So $I=\frac{V}{R_{e q}}=\frac{5}{35}=\frac{1}{7} \mathrm{~A}$</div>


tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
title:
A junction diode has a resistance of 25Î© when forward biased and 2500Î© when reverse biased. The current in the diode, for the arrangement shown will be

slug:
a-junction-diode-has-a-resistance-of-25-when-forward-biased-and-2500-when-reverse-biased-the-cu-87291

question:
<div class="question">A junction diode has a resistance of $25 \Omega$ when forward biased and $2500 \Omega$ when reverse biased. The current in the diode, for the arrangement shown will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OWBV48beSVcoKVMHfHYHaPhh3mK3e_IpE3Pi4kL084g.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{15} \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{7} \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{25} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{180} \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{7} \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\mathrm{R}_{\mathrm{eq}}=25+10=35 \Omega$<br/>Because diode is forward biased. So $I=\frac{V}{R_{e q}}=\frac{5}{35}=\frac{1}{7} \mathrm{~A}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A ladder 20 ft long leans against a vertical wall. The top end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is

slug:
a-ladder-20-ft-long-leans-against-a-vertical-wall-the-top-end-slides-downwards-at-the-rate-of-2-ft-71435

question:
<div class="question">A ladder $20 \mathrm{ft}$ long leans against a vertical wall. The top end slides downwards at the rate of $2 \mathrm{ft}$ per second. The rate at which the lower end moves on a horizontal floor when it is $12 \mathrm{ft}$ from the wall is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{-8}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{6}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{17}{4}$</span> </li> </ul>

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

answer:
<div class="solution">$$
<br/>x^{2}+y^{2}=(20)^{2}=400
<br/>$$
<br/>We have. $\frac{d y}{d t}=2 \mathrm{ft} / \mathrm{sec}$
<br/>When
<br/>$$
<br/>x=12
<br/>$$
<br/>then $\left(13^{2}+y^{2}=400\right.$
<br/>$\Rightarrow \quad 144+y^{2}=400$
<br/>$\Rightarrow \quad y^{2}=400-144=256$
<br/>$\Rightarrow \quad y=16$
<br/>Now, $2 x \frac{d x}{d t}+2 y \frac{d y}{d t}=0$
<br/>$\Rightarrow \quad x \frac{d x}{d t}=-y \frac{d y}{d t}$
<br/>$\Rightarrow \quad 12\left(\frac{d x}{d t}\right)=-16(2)$
<br/>$\Rightarrow \quad \frac{d x}{d t}=\frac{-8}{3}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a51.png"/></div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch0.db
title:
A large number of particles are placed around the origin, each at a distance R from the origin. The distance of the centro of mass of the system from the origin is

slug:
a-large-number-of-particles-are-placed-around-the-origin-each-at-a-distance-r-from-the-origin-the-28871

question:
<div class="question">A large number of particles are placed around the origin, each at a distance $R$ from the origin. The distance of the centro of mass of the system from the origin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">equal to $R$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">less than equal to $R$</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">greater than $R$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">greater than equal to $R$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">less than equal to $R$</span> </div>

answer:
<div class="solution">As large number of particles is situated at a distance $R$ from the ongin If particles are uniformly distubuted and make a circulat boundary atound the ongm, then centre of mass will be at the ongin While if the patticles are not uniformly distnbuted, then centre of mass will lie between particle and origin. This implies distance between centre of mass and ongin is always less than equal to $R$.</div>


slug: a-letter-lock-consists-of-three-rings-with-15-different-letters-if-mathrm-n-denotes-the-numbe-88111
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch0.db
title:
A letter lock consists of three rings with 15 different letters. If \(\mathrm{N}\) denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then

question:
<div class="question">A letter lock consists of three rings with 15 different letters. If \(\mathrm{N}\) denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">482 divides \(\mathrm{N}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">N is the product of two distinct prime numbers.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\mathrm{N}\) is the product of three distinct prime numbers.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">16 divides \(\mathrm{N}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">482 divides \(\mathrm{N}\), \(\mathrm{N}\) is the product of three distinct prime numbers.</span> </div>

answer:
<div class="solution">Hint : Number of unsuccessful attempts \(=15^3-1=3374=2 \times 1687=2 \times 7 \times 241\)<br/>Divisible by 482<br/>\(\mathrm{N}\) is the product of three distinct prime number</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A light charged particle is revolving in a circle of radius r in electrostatic attraction of a static heavy particle with opposite charge How does the magnetic field vary at the centre of the circle due to the moving charge depend on r ?

slug:
a-light-charged-particle-is-revolving-in-a-circle-of-radius-r-in-electrostatic-attraction-of-a-stati-32412

question:
<div class="question">A light charged particle is revolving in a circle of radius $r$ in electrostatic attraction of a static heavy particle with opposite charge How does the magnetic field vary at the centre of the circle due to the moving charge depend on $r ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$B \propto \frac{1}{r}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$B \propto \frac{1}{r^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$B \propto \frac{1}{r^{\frac{3}{2}}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$B \propto \frac{1}{r^{\frac{5}{2}}}$</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">$B \propto \frac{1}{r^{\frac{5}{2}}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a31.png"/> <br/> <br/>$$
<br/>\begin{aligned}
<br/>&amp;\begin{array}{l}
<br/>F_{\text {centripetal }}=m r \omega^{2}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{1} q_{2}}{r^{2}} \\
<br/>\therefore \quad \omega^{2}=\frac{1}{4 \pi \varepsilon_{0} m} \cdot \frac{q_{1} q_{2}}{r^{3}} \Rightarrow \omega=\sqrt{\frac{q_{1} q_{2}}{4 \pi \varepsilon_{0} m r^{3}}}
<br/>\end{array}\\
<br/>&amp;\omega \propto \frac{1}{r^{3 / 2}}
<br/>\end{aligned}
<br/>$$
<br/>Current produced due to moving $q_{1}$
<br/>$$
<br/>i=\frac{\omega}{2 \pi} q_{1}
<br/>$$
<br/>Thus, magnetic field due to motion of $q_{1}$ in circular path at point $\underline{O}$ $B=\frac{\mu_{0}}{2} \cdot \frac{i}{r}=\frac{\mu_{0}}{2} \cdot \frac{\omega q_{1}}{2 \pi r} \quad[$ using $\mathrm{Eq} \cdot(\mathrm{ii})]$
<br/>Hence, $B \propto \frac{\omega}{r}$
<br/>Magnetic field $B \propto \frac{1}{r^{3 / 2}}$ $[$ using Eq. (i) ]</div>


slug: a-line-cuts-the-x-axis-at-a-5-0-and-the-y-axis-at-b-0-3-a-variable-line-pq-is-dr-30472
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch0.db
title:
A line cuts the X -axis at A ( 5 , 0 ) and the Y -axis at B ( 0 , â 3 ) . A variable line PQ is drawn perpendicular to A B cutting the X -axis at P and the Y -axis at Q . If A Q and BP meet at R , then the locus of R is

question:
<div class="question">A line cuts the $X$ -axis at $A(5,0)$ and the $Y$ -axis at $B(0,-3)$. A variable line $P Q$ is drawn perpendicular to $A B$ cutting the $X$ -axis at $P$ and the $Y$ -axis at $Q$. If $A Q$ and $B P$ meet at $R$, then the locus of $R$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}-5 x+3 y=0$</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">$x^{2}+y^{2}+5 x+3 y=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}+5 x-3 y=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}-5 x-3 y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}-5 x+3 y=0$</span> </div>

answer:
<div class="solution">Equation of line $A B$ is
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{x}{5}+\frac{y}{-3}=1 \\
<br/>3 x-5 y=15
<br/>\end{array}
<br/>$$
<br/>Perpendicular line to $A B$ is
<br/>$$
<br/>5 x+3 y=\lambda
<br/>$$
<br/>Coordinate of $P$ is $\left(\frac{\lambda}{5}, 0\right)$ and coordinate of $Q$ is $(0, \lambda / 3)$ 
<br/>Now, equation of line $A Q$ is
<br/>$x / 5+\frac{y}{\lambda / 3}=1$
<br/>$\Rightarrow \quad \frac{x}{5}+\frac{3 y}{\lambda}=1$
<br/>$\Rightarrow \quad \frac{3 y}{\lambda}=1-\frac{x}{5}$
<br/>$\Rightarrow \quad \frac{1}{\lambda}=\frac{1}{3 y}\left(1-\frac{x}{5}\right)$
<br/>and equation of line $B P$ is
<br/>$\frac{x}{\lambda / 5}+\frac{y}{-3}=1$
<br/>$\Rightarrow \quad \frac{5 x}{\lambda}-\frac{y}{3}=1$
<br/>$\Rightarrow \quad \frac{1}{\lambda}=\frac{1}{5 x}\left(\frac{y}{3}+1\right)$
<br/>From Eqs. (i) and (ii), 
<br/>$\frac{1}{3 y}\left(1-\frac{x}{5}\right)=\frac{1}{5 x}\left(\frac{y}{3}+1\right)$
<br/>$\Rightarrow \quad 5 x\left(1-\frac{x}{5}\right)=3 y\left(\frac{y}{3}+1\right)$
<br/>$\Rightarrow \quad 5 x-x^{2}=y^{2}+3 y$
<br/>$\Rightarrow x^{2}+y^{2}-5 x+3 y=0$
<br/>which is a circle.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a60.png"/></div>


slug: a-line-passes-through-the-point-1-1-and-makes-an-angle-sin-1-5-3-in-the-positi-33484
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch0.db
title:
A line passes through the point ( â 1 , 1 ) and makes an angle sin â 1 ( 5 3 â ) in the positive direction of x -axis. If this line meets the curve x 2 = 4 y â 9 at A and B , then â£ A B â£ is equal to

question:
<div class="question">A line passes through the point $(-1,1)$ and makes an angle $\sin ^{-1}\left(\frac{3}{5}\right)$ in the positive direction of $x$-axis. If this line meets the curve $x^2=4 y-9$ at $A$ and $B$, then $|A B|$ is equal to</div>

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

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

answer:
<div class="solution">$4 y-3 x=7$ and $x^2=4 y-9$<br/>On solving, $x^2-3 x+2=0 \Rightarrow x=1,2$ and $y=\frac{5}{2}, \frac{13}{4}$<br/>$A\left(1, \frac{5}{2}\right)$ and $B\left(2, \frac{13}{4}\right)$<br/>$$<br/>|A B|=\frac{5}{4}<br/>$$</div>


slug: a-line-passing-through-the-point-of-intersection-of-x-y-4-and-x-y-2-makes-an-angle-tan-46302
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch0.db
title:
A line passing through the point of intersection of x + y = 4 and x â y = 2 makes an angle tan â 1 ( 4 3 â ) with the X -axis. It intersects the parabola y 2 = 4 ( x â 3 ) at points ( x 1 â , y 1 â ) and ( x 2 â , y 2 â ) , respectively. Then â£ x 1 â â x 2 â â£ is equal to

question:
<div class="question">A line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan ^{-1}\left(\frac{3}{4}\right)$ with the $X$ -axis. It intersects the parabola $y^{2}=4(x-3)$ at points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right),$ respectively. Then $\left|x_{1}-x_{2}\right|$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{16}{9}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{32}{9}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{40}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{80}{9}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{32}{9}$</span> </div>

answer:
<div class="solution">Given equations are
<br/>$x+y=4  ...(i)$
<br/>and $\quad x-y=2$ ...(ii)
<br/>From Eqs. (i) and $(ii),$ we get
<br/>$x=3$ and $y=1$
<br/>The line through this point making an angle tan $^{-1} \frac{3}{4}$ with the $X$ -axis is 
<br/>$(y-1)=\frac{3}{4}(x-3) \quad\left[\because m=\frac{3}{4}\right]$
<br/>$\Rightarrow \quad y=\frac{3 x}{4}-\frac{5}{4}=\frac{3 x-5}{4}$
<br/>Since, this line intersects the parabola
<br/>$y^{2}=4(x-3)$ at points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$, respectively.
<br/>$\therefore$ Putting $y=\frac{3 x-5}{4}$ in equation of parabola, we get
<br/>$$
<br/>\left(\frac{3 x-5}{4}\right)^{2}=4(x-3)
<br/>$$
<br/>$\Rightarrow \quad 9 x^{2}-94 x+217=0$
<br/>$\Rightarrow \quad x_{1}+x_{2}=\frac{94}{9}$ and $x_{1} x_{2}=\frac{217}{9}$
<br/>$\begin{aligned} \therefore \quad\left|x_{1}-x_{2}\right| &amp;=\sqrt{\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}} \\ &amp;=\sqrt{\left(\frac{94}{9}\right)^{2}-4 \times \frac{217}{9}} \\ &amp;=\frac{32}{9} \end{aligned}$</div>


slug: a-line-passing-through-the-point-of-intersection-of-x-y-4-and-x-y-2-makes-an-angle-tan-15703
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch0.db
title:
A line passing through the point of intersection of x + y = 4 and x â y = 2 makes an angle tan â 1 ( 4 3 â ) with the x -axis. It intersects the parabola y 2 = 4 ( x â 3 ) at points ( x 1 â , y 1 â ) and ( x 2 â , y 2 â ) respectively. Then, â£ x 1 â â x 2 â â£ is equal to

question:
<div class="question">A line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan ^{-1}\left(\frac{3}{4}\right)$ with the $x$ -axis. It intersects the parabola $y^{2}=4(x-3)$ at points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ respectively. Then, $\left|x_{1}-x_{2}\right|$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{16}{9}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{32}{9}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{40}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{80}{9}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{32}{9}$</span> </div>

answer:
<div class="solution">Given lines are
<br/>$$
<br/>x+y=4 \text { and } x-y=2
<br/>$$
<br/>On solving these lines, we get
<br/>$x=3$ and $y=1$ 
<br/>Now, the equation of line which passes through the intersection point (3, 1) having slope
<br/>$\theta=\tan ^{-1}\left(\frac{3}{4}\right)$ is $(y-1)=\frac{3}{4}(x-3)$
<br/>$\Rightarrow \quad 4 y-4=3 x-9$
<br/>$\Rightarrow \quad 3 x-4 y=5$
<br/>Now for the intersection point of the line (i) with parabola $y^{2}=4(x-3)$. Put $y=\left(\frac{3 x-5}{4}\right)$
<br/>then we get $\quad \frac{(3 x-5)^{2}}{16}=4(x-3)$
<br/>$\Rightarrow \quad 9 x^{2}+25-30 x=64 x-192$
<br/>$\Rightarrow \quad 9 x^{2}-94 x+217=0$
<br/>$\Rightarrow \quad x=\frac{94 \pm \sqrt{8836-7812}}{18}=\frac{94 \pm \sqrt{1024}}{18}$
<br/>$\Rightarrow \quad x=\frac{94 \pm 32}{18}=\frac{126}{18}$ or $\frac{62}{18}$
<br/>$\Rightarrow \quad x_{1}=\frac{21}{3}=7$
<br/>and $\quad x_{2}=\frac{31}{9}$
<br/>$\therefore \quad\left|x_{1}-x_{2}\right|=\left|7-\frac{31}{9}\right|=\left|\frac{32}{9}\right|=\frac{32}{9}$</div>


slug: a-line-through-the-point-a-2-0-which-makes-an-angle-of-3-0-with-the-positive-direction-of-86578
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch0.db
title:
A line through the point A ( 2 , 0 ) which makes an angle of 3 0 â with the positive direction of x -axis is rotated about A in clockwise direction through an angle 1 5 â . Then the equation of the straight line in the new position is

question:
<div class="question">A line through the point $\mathrm{A}(2,0)$ which makes an angle of $30^{\circ}$ with the positive direction of $x$-axis is rotated about A in clockwise direction through an angle $15^{\circ}$. Then the equation of the straight line in the new position is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(2-\sqrt{3}) x+y-4+2 \sqrt{3}=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(2-\sqrt{3}) x-y-4+2 \sqrt{3}=0$</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">$(2-\sqrt{3}) x-y+4+2 \sqrt{3}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(2-\sqrt{3})_x+y+4+2 \sqrt{3}=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(2-\sqrt{3}) x-y-4+2 \sqrt{3}=0$</span> </div>

answer:
<div class="solution">Hints : Equation of line in new position :<br/>$$<br/>\begin{aligned}<br/>&amp; y-0=\tan 15^{\circ}(x-2) \\<br/>&amp; \Rightarrow y=\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)(x-2) \\<br/>&amp; \Rightarrow y=\frac{(\sqrt{3}-1)^2}{2}(x-2)<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow 2 y=(4-2 \sqrt{3})(x-2) \\<br/>&amp; \Rightarrow y=(2-\sqrt{3})(x-2) \\<br/>&amp; \Rightarrow(2-\sqrt{3}) x-y-4+2 \sqrt{3}=0<br/>\end{aligned}<br/>$$</div>


slug: a-line-with-positive-direction-cosines-passes-through-the-point-p-2-1-2-and-makes-equal-13293
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch0.db
title:
A line with positive direction cosines passes through the point P ( 2 , â 1 , 2 ) and makes equal angle with co-ordinate axes. The line meets the plane 2 x + y + z = 9 at point Q . The length of the line segment PQ equals.

question:
<div class="question">A line with positive direction cosines passes through the point $\mathrm{P}(2,-1,2)$ and makes equal angle with co-ordinate axes. The line meets the plane $2 x+y+z=9$ at point $Q$. The length of the line segment $P Q$ equals.</div>

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

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

answer:
<div class="solution">Let d.c. of line is $(\mathrm{k}, \mathrm{k}, \mathrm{k})$
<br/>$\begin{array}{l}
<br/>\Rightarrow 3 \mathrm{k}^{2}=1 \Rightarrow \mathrm{k}=\frac{1}{\sqrt{3}} \\
<br/>\Rightarrow \text { d.c. of line is }\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)
<br/>\end{array}$
<br/>Equation of a line passing through point $p$ is
<br/>$\begin{array}{l}
<br/>\frac{x-2}{\frac{1}{\sqrt{3}}}=\frac{y+1}{\frac{1}{\sqrt{3}}}=\frac{z-2}{\frac{1}{\sqrt{3}}}=k \\
<br/>\Rightarrow x=k+2, y=k-1, z=k+2 \\
<br/>\therefore 2(k+2)+(k-1)+(k+2)=9 \\
<br/>4 k=4 \Rightarrow k=1
<br/>\end{array}$
<br/>$\therefore$ Point $\mathrm{Q}$ is $(3,0,3)$
<br/>$P Q=\sqrt{1^{2}+1^{2}+1^{2}}=\sqrt{3}$ units.</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch0.db
title:
A liquid of bulk modulus k is compressed by applying an external pressure such that its density increases by 0.01%. The pressure applied on the liquid is

slug:
a-liquid-of-bulk-modulus-k-is-compressed-by-applying-an-external-pressure-such-that-its-density-incr-31644

question:
<div class="question">A liquid of bulk modulus $k$ is compressed by applying an external pressure such that its density increases by $0.01 \% .$ The pressure applied on the liquid is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{k}{10000}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{k}{1000}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1000 \mathrm{k}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.01 k$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{k}{10000}$</span> </div>

answer:
<div class="solution">The Bulk modulus $k$
<br/>$$
<br/>k=-\frac{p}{\Delta V / V}
<br/>$$
<br/>Where, $p=$ Pressure
<br/>$$
<br/>\begin{aligned}
<br/>\Delta V &amp;=\text { Change in volume } \\
<br/>V &amp;=\text { Volume of liquid }
<br/>\end{aligned}
<br/>$$
<br/>From (i)
<br/>$$
<br/>\frac{p}{k}=-\frac{\Delta V}{V}=\frac{\Delta p}{p}
<br/>$$
<br/>$\Rightarrow \quad p=\frac{k \Delta p}{p}$
<br/>$$
<br/>\begin{aligned}
<br/>\Delta p &amp;=0 \cdot 01 \%=0 \cdot 01 / 100 \\
<br/>p &amp;=\frac{k}{10000}
<br/>\end{aligned}
<br/>$$</div>


slug: a-long-conducting-wire-carrying-a-current-i-is-bent-at-12-0-see-figure-the-magnetic-field-b-a-95524
tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch0.db
title:
A long conducting wire carrying a current I is bent at 12 0 â (see figure). The magnetic field B at a point P on the right bisector of bending angle at a distance d from the bend is ( Î¼ 0 â is the permeability of free space)

question:
<div class="question">A long conducting wire carrying a current $I$ is bent at $120^{\circ}$ (see figure). The magnetic field $B$ at a point $P$ on the right bisector of bending angle at a distance $d$ from the bend is $\left(\mu_{0}\right.$ is the permeability of free space) <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q49.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3 \mu_{0} I}{2 \pi d}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mu_{0} I}{2 \pi d}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mu_{0} I}{\sqrt{3} n d}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{3} \mu_{0} I}{2 \pi d}$</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> </ul>

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

answer:
<div class="solution">We know that
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a49.png"/> <br/> <br/>$$
<br/>\begin{aligned}
<br/>B_{\text {net }} &amp;=2\left[\frac{\mu_{0} i}{4 \pi r}\left(\sin \theta_{1}+\sin \theta_{2}\right)\right] \\
<br/>&amp;=2\left[\frac{\mu_{0}}{4 \pi} \times \frac{i}{\frac{d \sqrt{3}}{2}} \times\left(\sin 90^{\circ}+\sin 30^{\circ}\right)\right] \\
<br/>&amp;=2\left[\frac{\mu_{0}}{4 \pi} \times \frac{2 i}{d \sqrt{3}} \times\left(1+\frac{1}{2}\right)\right] \\
<br/>&amp;=2\left[\frac{\mu_{0}}{4 \pi} \times \frac{2 i}{d \sqrt{3}} \times \frac{3}{2}\right]=\frac{\sqrt{3} \mu_{0} i}{2 \pi d}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
63-mathrmg-of-a-compound-mol-wt-126-was-dissolved-in-500-mathrmg-distilled-water-the-density-of-the-resultant-solution-as-1126-mathrmg-mathrmml-the-mo

slug:
63-mathrmg-of-a-compound-mol-wt-126-was-dissolved-in-500-mathrmg-distilled-water-the-density-of-the-resultant-solution-as-1126-mathrmg-mathrmml-the-mo-33236

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

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

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

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


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
98-mathrmcf246-was-formed-along-with-a-neutron-when-an-unknown-radioactive-substance-was-bombarded-with-6-mathrmc12-the-unknown-substance-was

slug:
98-mathrmcf246-was-formed-along-with-a-neutron-when-an-unknown-radioactive-substance-was-bombarded-with-6-mathrmc12-the-unknown-substance-was-15950

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\({ }_{91} \mathrm{~Pa}^{234}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\({ }_{90} \mathrm{Th}^{234}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\({ }_{92} \mathrm{U}^{235}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\({ }_{92} \mathrm{U}^{238}\)</span> </li> </ul>

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

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


tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">63</span> </div>
source_db: db\MarksBatch1_P1.db
title:
0126-g-of-an-acid-is-needed-to-completely-neutralise-20-ml-01-n-naoh-solution-the-equivalent-weight-of-the-acid-is

slug:
0126-g-of-an-acid-is-needed-to-completely-neutralise-20-ml-01-n-naoh-solution-the-equivalent-weight-of-the-acid-is-29770

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

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

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


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.13</span> </div>
source_db: db\MarksBatch1_P1.db
title:
156-times-105-mathrmj-of-heat-is-conducted-through-a-2-mathrmm2-wall-of-12-mathrmcm-thick-in-one-hour-temperature-difference-between-the-two-sides-of-

slug:
156-times-105-mathrmj-of-heat-is-conducted-through-a-2-mathrmm2-wall-of-12-mathrmcm-thick-in-one-hour-temperature-difference-between-the-two-sides-of-63343

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

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

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


title: 300-gm-of-water-at-2-5-c-is-added-to-100-gm-of-ice-at-0-c-the-final-temperature-of-the-mixture-is
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
slug:
300-gm-of-water-at-2-5-c-is-added-to-100-gm-of-ice-at-0-c-the-final-temperature-of-the-mixture-is-34304

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$12.5^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0^{\circ} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$25^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$50^{\circ} \mathrm{C}$</span> </li> </ul>

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

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


tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
575-mg-of-sodium-vapour-is-converted-to-sodium-ion-if-the-ionisation-energy-of-sodium-is-490-kj-mol-1-and-atomic-weight-is-23-units-the-amount-of-ener

slug:
575-mg-of-sodium-vapour-is-converted-to-sodium-ion-if-the-ionisation-energy-of-sodium-is-490-kj-mol-1-and-atomic-weight-is-23-units-the-amount-of-ener-49825

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

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

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

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


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
584-g-of-nacl-and-180-g-of-glucose-were-separately-dissolved-in-1000-ml-of-water-identify-the-correct-statement-regarding-the-elevation-of-boiling-poi

slug:
584-g-of-nacl-and-180-g-of-glucose-were-separately-dissolved-in-1000-ml-of-water-identify-the-correct-statement-regarding-the-elevation-of-boiling-poi-42932

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">NaCl solution will show higher elevation of boiling point</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Glucose solution will show higher elevation of boiling point</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both the solutions will show equal elevation of boiling point</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The boiling point elevation will be shown by neither of the solutions</span> </li> </ul>

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

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


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
1000-droplets-of-water-having-2-mm-diameter-each-coalesce-to-form-a-single-drop-given-the-surface-tension-of-water-is-0072-nm-1-the-energy-loss-in-the

slug:
1000-droplets-of-water-having-2-mm-diameter-each-coalesce-to-form-a-single-drop-given-the-surface-tension-of-water-is-0072-nm-1-the-energy-loss-in-the-27109

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

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

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

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


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
22320-cal-of-heat-is-supplied-to-100-g-of-ice-at-0-c-if-the-latent-heat-of-fusion-of-ice-is-80-cal-g-1-and-latent-heat-of-vaporization-of-water-is-540

slug:
22320-cal-of-heat-is-supplied-to-100-g-of-ice-at-0-c-if-the-latent-heat-of-fusion-of-ice-is-80-cal-g-1-and-latent-heat-of-vaporization-of-water-is-540-56103

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$8 \mathrm{g}, 100^{\circ} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$100 \mathrm{g} ,90^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$92 \mathrm{g}, 100^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$82 \mathrm{g}, 100^{\circ} \mathrm{C}$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-1-2-3-4-b-1-2-3-4-5-6-are-two-sets-and-function-f-a-b-is-defined-by-f-x-x-2-x-a-then-the-function-f-is

slug:
a-1-2-3-4-b-1-2-3-4-5-6-are-two-sets-and-function-f-a-b-is-defined-by-f-x-x-2-x-a-then-the-function-f-is-13359

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

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

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

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


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-1-f-capacitor-c-is-connected-to-a-battery-of-10-v-through-a-resistance-1-mega-ohm-the-voltage-across-c-after-1-s-is-approximately

slug:
a-1-f-capacitor-c-is-connected-to-a-battery-of-10-v-through-a-resistance-1-mega-ohm-the-voltage-across-c-after-1-s-is-approximately-57158

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

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

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

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


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-5-f-capacitor-is-connected-in-series-with-10-f-capacitor-when-a-300-v-potentid-dillerence-is-applied-across-this-combination-the-total-energy-stored

slug:
a-5-f-capacitor-is-connected-in-series-with-10-f-capacitor-when-a-300-v-potentid-dillerence-is-applied-across-this-combination-the-total-energy-stored-84048

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

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

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

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


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-10-w-electric-heater-is-used-to-heat-a-container-filled-with-05-kg-of-water-it-is-found-that-the-temperature-of-water-and-the-container-rises-by-3-k

slug:
a-10-w-electric-heater-is-used-to-heat-a-container-filled-with-05-kg-of-water-it-is-found-that-the-temperature-of-water-and-the-container-rises-by-3-k-53852

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

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

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

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


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-20-cm-long-capillary-tube-is-dipped-vertically-in-water-and-the-liquid-rises-upto-10-cm-if-the-entire-system-is-kept-is-a-freely-falling-platform-th

slug:
a-20-cm-long-capillary-tube-is-dipped-vertically-in-water-and-the-liquid-rises-upto-10-cm-if-the-entire-system-is-kept-is-a-freely-falling-platform-th-23275

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15 \mathrm{cm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$20 \mathrm{cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$20 \mathrm{cm}$</span> </div>

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


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
title:
a-20-cm-object-is-placed-15-cm-in-front-of-a-concave-mirror-of-focal-length-10-cm-what-is-the-size-and-nature-of-the-image

slug:
a-20-cm-object-is-placed-15-cm-in-front-of-a-concave-mirror-of-focal-length-10-cm-what-is-the-size-and-nature-of-the-image-64522

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

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

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

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


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-100-ml-01-m-solution-of-ammonium-acetate-is-diluted-by-adding-100-ml-of-water-the-ph-of-the-resulting-solution-will-be-p-k-a-of-acetic-acid-is-nearl

slug:
a-100-ml-01-m-solution-of-ammonium-acetate-is-diluted-by-adding-100-ml-of-water-the-ph-of-the-resulting-solution-will-be-p-k-a-of-acetic-acid-is-nearl-70385

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

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

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


tags: ['Physics', 'Atomic Physics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-125-ev-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-ground-state-the-energy-level-upto-which-the-hydrogen-atoms-would-be-excited-is-1

slug:
a-125-ev-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-ground-state-the-energy-level-upto-which-the-hydrogen-atoms-would-be-excited-is-1-77768

question:
<div class="question">A $12.5$ ev electron beam is used to bombard gaseous hydrogen at ground state. The energy level upto which the hydrogen atoms would be excited is</div>

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

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


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-125-ev-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-ground-state-the-energy-level-upto-which-the-hydrogen-atoms-would-be-excited-is

slug:
a-125-ev-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-ground-state-the-energy-level-upto-which-the-hydrogen-atoms-would-be-excited-is-50462


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-400-resistor-a-250-mh-inductor-and-a-25-f-capacitor-are-connected-in-series-with-an-ac-source-of-peak-voltage-5-v-and-angular-frequency-2-radsec-wha

slug:
a-400-resistor-a-250-mh-inductor-and-a-25-f-capacitor-are-connected-in-series-with-an-ac-source-of-peak-voltage-5-v-and-angular-frequency-2-radsec-wha-66238

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mu \mathrm{J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.5 \mu \mathrm{J}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.33 \mu \mathrm{J}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$5 \mu \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$5 \mu \mathrm{J}$</span> </div>

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


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-and-b-are-independent-events-the-probability-that-both-a-and-b-occur-is-20-1-and-the-probability-that-neither-of-them-occurs-is-5-3-the-probability-

slug:
a-and-b-are-independent-events-the-probability-that-both-a-and-b-occur-is-20-1-and-the-probability-that-neither-of-them-occurs-is-5-3-the-probability-10524

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

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

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

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


title: a-and-b-are-two-vectors-given-by-a-2-i-3-j-and-b-i-j-the-magnitude-of-the-component-of-a-along-b-is
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
slug:
a-and-b-are-two-vectors-given-by-a-2-i-3-j-and-b-i-j-the-magnitude-of-the-component-of-a-along-b-is-37652

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

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

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

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


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-b-are-fixed-points-with-coordinates-0-a-and-0-ba-0-b-0-p-is-variable-point-x-0-referred-to-rectangular-axis-if-the-angle-angle-mathrmapb-is-maximum-

slug:
a-b-are-fixed-points-with-coordinates-0-a-and-0-ba-0-b-0-p-is-variable-point-x-0-referred-to-rectangular-axis-if-the-angle-angle-mathrmapb-is-maximum-47829

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

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

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

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


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-b-c-are-mutually-exclusive-events-such-that-p-a-3-3-x-1-p-b-4-1-x-and-p-c-2-1-2-x-then-the-set-of-possible-values-of-x-are-in

slug:
a-b-c-are-mutually-exclusive-events-such-that-p-a-3-3-x-1-p-b-4-1-x-and-p-c-2-1-2-x-then-the-set-of-possible-values-of-x-are-in-66541

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

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

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

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


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-b-c-is-an-isosceles-triangle-with-an-inscribed-circle-with-centre-o-let-p-be-the-midpoint-of-b-c-if-a-ba-c15-and-b-c10-then-mathrmop-equals

slug:
a-b-c-is-an-isosceles-triangle-with-an-inscribed-circle-with-centre-o-let-p-be-the-midpoint-of-b-c-if-a-ba-c15-and-b-c10-then-mathrmop-equals-69803

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

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

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

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


tags: ['Mathematics', 'Ellipse', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-b-is-a-variable-chord-of-the-ellipse-a-2-x-2-b-2-y-2-1-if-a-b-subtends-a-right-angle-at-the-origin-o-then-o-a-2-1-o-b-2-1-equals-to-1

slug:
a-b-is-a-variable-chord-of-the-ellipse-a-2-x-2-b-2-y-2-1-if-a-b-subtends-a-right-angle-at-the-origin-o-then-o-a-2-1-o-b-2-1-equals-to-1-47873

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

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

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

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


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-b-is-a-variable-chord-of-the-ellipse-a-2-x-2-b-2-y-2-1-if-a-b-subtends-a-right-angle-at-the-origin-o-then-o-a-2-1-o-b-2-1-equals-to

slug:
a-b-is-a-variable-chord-of-the-ellipse-a-2-x-2-b-2-y-2-1-if-a-b-subtends-a-right-angle-at-the-origin-o-then-o-a-2-1-o-b-2-1-equals-to-38944


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-ball-is-projected-horizontally-with-a-velocity-of-5-m-s-from-the-top-of-a-building-196-m-high-how-long-will-the-ball-take-of-hit-the-ground

slug:
a-ball-is-projected-horizontally-with-a-velocity-of-5-m-s-from-the-top-of-a-building-196-m-high-how-long-will-the-ball-take-of-hit-the-ground-57618

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

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

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

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


tags: ['Mathematics', 'Heights and Distances', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-balloon-starting-from-rest-is-ascending-from-ground-with-uniform-acceleration-of-4-mathrmft-mathrmsec2-at-the-end-of-5-mathrmsec-a-stone-is-dropped-

slug:
a-balloon-starting-from-rest-is-ascending-from-ground-with-uniform-acceleration-of-4-mathrmft-mathrmsec2-at-the-end-of-5-mathrmsec-a-stone-is-dropped-24945

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{T}=6 \mathrm{sec}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{H}=112.5 \mathrm{ft}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(T=5 / 2 \mathrm{sec}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(225 \mathrm{ft}\)</span> </li> </ul>

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

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


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint :</div>
source_db: db\MarksBatch1_P1.db
title:
a-bar-magnet-falls-from-rest-under-gravity-through-the-centre-of-a-horizontal-ring-of-conducting-wire-as-shown-in-figure-which-of-the-following-graph-

slug:
a-bar-magnet-falls-from-rest-under-gravity-through-the-centre-of-a-horizontal-ring-of-conducting-wire-as-shown-in-figure-which-of-the-following-graph-84670

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/FnUsB81yzAsRQ4_jVgUC_kdbJHkPxxgzMeXycLzc3es.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/76XRxSePejIgWNge779CPKHci8ws2GgZry0iLotLVaw.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/U-g2Pjz06T2Ghjtlt4G1TCfOFYg0NIz_ShroY6Y9C_o.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/IZ0r8hgbfLJmDCbaJpTs0cRJJUCRfKi1euvdZT3V6zQ.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/FnUsB81yzAsRQ4_jVgUC_kdbJHkPxxgzMeXycLzc3es.original.fullsize.png"/><br/></span> </div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-bar-magnet-has-a-magnetic-moment-of-200-am-2-the-magnet-is-suspended-in-a-magnetic-field-of-030-na-1-m-1-the-torque-required-to-rotate-the-magnet-fr

slug:
a-bar-magnet-has-a-magnetic-moment-of-200-am-2-the-magnet-is-suspended-in-a-magnetic-field-of-030-na-1-m-1-the-torque-required-to-rotate-the-magnet-fr-91931

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

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

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

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


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-bar-of-length-l-carrying-a-small-mass-m-at-one-of-its-ends-rotates-with-a-uniform-angular-speed-in-a-vertical-plane-about-the-mid-point-of-the-bar-d

slug:
a-bar-of-length-l-carrying-a-small-mass-m-at-one-of-its-ends-rotates-with-a-uniform-angular-speed-in-a-vertical-plane-about-the-mid-point-of-the-bar-d-82243

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

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">This possible if the quantity $\frac{\omega^{2} I}{2 \pi g}$ is an integer, The total distance travelled by the mass n in air is proportional to $\omega^{2}$, The total distance travelled by the mass in air and its total time of flight are both independent on its mass.</span> </div>

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


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-battery-of-emf-e-and-internal-resistance-r-is-connected-with-an-external-resistance-r-as-shown-in-the-figure-the-battery-will-act-as-a-constant-volt-1

slug:
a-battery-of-emf-e-and-internal-resistance-r-is-connected-with-an-external-resistance-r-as-shown-in-the-figure-the-battery-will-act-as-a-constant-volt-1-97677

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

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

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

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


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-battery-of-emf-e-and-internal-resistance-r-is-connected-with-an-external-resistance-r-as-shown-in-the-figure-the-battery-will-act-as-a-constant-volt

slug:
a-battery-of-emf-e-and-internal-resistance-r-is-connected-with-an-external-resistance-r-as-shown-in-the-figure-the-battery-will-act-as-a-constant-volt-29968


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">In water $\lambda=\frac{4200}{\frac{4}{3}}=3150 Ã $</div>
source_db: db\MarksBatch1_P1.db
title:
a-beam-of-monochromatic-blue-light-of-wavelength-4200-a-in-air-travels-in-water-of-refractive-index-43-its-wavelength-in-water-will-be

slug:
a-beam-of-monochromatic-blue-light-of-wavelength-4200-a-in-air-travels-in-water-of-refractive-index-43-its-wavelength-in-water-will-be-55944

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4200 Ã$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5800 Ã$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4150 Ã$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3150 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3150 Ã$</span> </div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-biconvex-lens-of-focal-length-f-and-radii-of-curvature-of-both-the-surfaces-r-is-made-of-a-material-of-refractive-index-n-1-this-lens-is-placed-in-a

slug:
a-biconvex-lens-of-focal-length-f-and-radii-of-curvature-of-both-the-surfaces-r-is-made-of-a-material-of-refractive-index-n-1-this-lens-is-placed-in-a-78581

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

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

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

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


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-ice-at-temperature-2-0-c-is-slowly-heated-and-converted-to-steam-at-10-0-c-which-of-the-following-diagram-is-most-appropriate

slug:
a-block-of-ice-at-temperature-2-0-c-is-slowly-heated-and-converted-to-steam-at-10-0-c-which-of-the-following-diagram-is-most-appropriate-77715

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/G02zQPahwC3O9sgAwy3uzDBUguLt8OCNfGJybHOeUjE.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9TTx3oiBiHnaXeTAXLdAeSsO40oWlUqd1U4FrHjJOMk.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2dTIxymIX8Eym0DFjeh3QIjXyASxJAV0ii5UBruqWbQ.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/O6gyFTKGB9QVYskX5lxlXSFEO96GExh6P3ipR9Qc42k.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/G02zQPahwC3O9sgAwy3uzDBUguLt8OCNfGJybHOeUjE.original.fullsize.png"/><br/></span> </div>

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-mass-1-kg-starts-from-rest-at-x-0-and-moves-along-the-x-axis-under-the-action-of-a-force-f-k-t-where-t-is-time-and-k-1-ns-1-the-distance-th

slug:
a-block-of-mass-1-kg-starts-from-rest-at-x-0-and-moves-along-the-x-axis-under-the-action-of-a-force-f-k-t-where-t-is-time-and-k-1-ns-1-the-distance-th-81510

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

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

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

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-mass-m-01-kg-is-hanging-over-a-frictionless-light-fixed-pulley-by-an-inextensible-string-of-negligible-mass-the-other-end-of-the-string-is-

slug:
a-block-of-mass-m-01-kg-is-hanging-over-a-frictionless-light-fixed-pulley-by-an-inextensible-string-of-negligible-mass-the-other-end-of-the-string-is-23430

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

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">The tension in the string is $F$, The tension in the string is $3 \mathrm{N}$, The work done against the force of gravity is $10 \mathrm{J}$</span> </div>

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-mass-m-2-is-placed-on-a-horizontal-table-and-another-block-of-mass-m-1-is-placed-on-top-of-it-an-increasing-horizontal-force-falpha-t-is-ex

slug:
a-block-of-mass-m-2-is-placed-on-a-horizontal-table-and-another-block-of-mass-m-1-is-placed-on-top-of-it-an-increasing-horizontal-force-falpha-t-is-ex-45092

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

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

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

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


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-mass-m-rests-on-a-horizontal-table-with-a-coefficient-of-static-friction-what-minimum-force-must-be-applied-on-the-block-to-drag-it-on-the--1

slug:
a-block-of-mass-m-rests-on-a-horizontal-table-with-a-coefficient-of-static-friction-what-minimum-force-must-be-applied-on-the-block-to-drag-it-on-the-1-49615

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

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

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

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-mass-m-rests-on-a-horizontal-table-with-a-coefficient-of-static-friction-what-minimum-force-must-be-applied-on-the-block-to-drag-it-on-the-

slug:
a-block-of-mass-m-rests-on-a-horizontal-table-with-a-coefficient-of-static-friction-what-minimum-force-must-be-applied-on-the-block-to-drag-it-on-the-61792


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-block-of-mass-m-slides-with-speed-v-on-a-frictionless-table-towards-another-stationary-block-of-mass-m-a-massless-spring-with-spring-constant-k-is-a

slug:
a-block-of-mass-m-slides-with-speed-v-on-a-frictionless-table-towards-another-stationary-block-of-mass-m-a-massless-spring-with-spring-constant-k-is-a-85830

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

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

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

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


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.5</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-body-floats-in-water-with-40-of-its-volume-outside-water-when-the-same-body-floats-in-an-oil-60-of-its-volume-remains-outside-oil-the-relative-densi

slug:
a-body-floats-in-water-with-40-of-its-volume-outside-water-when-the-same-body-floats-in-an-oil-60-of-its-volume-remains-outside-oil-the-relative-densi-82135

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1.5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.5</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-body-floats-in-water-with-40-of-its-volume-outside-water-when-the-same-body-floats-in-oil-60-of-its-volume-remains-outside-oil-the-relative-density-

slug:
a-body-floats-in-water-with-40-of-its-volume-outside-water-when-the-same-body-floats-in-oil-60-of-its-volume-remains-outside-oil-the-relative-density-97957

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

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

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


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-body-is-projected-from-the-ground-with-a-velocity-v-3-i-10-j-ms-1-the-maximum-height-attained-and-the-range-of-the-body-respectively-are-given-g-10-

slug:
a-body-is-projected-from-the-ground-with-a-velocity-v-3-i-10-j-ms-1-the-maximum-height-attained-and-the-range-of-the-body-respectively-are-given-g-10-27559

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

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

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

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


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-body-is-projected-with-a-speed-u-m-s-at-an-angle-with-the-horizontal-the-kinetic-energy-at-the-highest-point-is-34-th-of-the-initial-kinetic-energy-

slug:
a-body-is-projected-with-a-speed-u-m-s-at-an-angle-with-the-horizontal-the-kinetic-energy-at-the-highest-point-is-34-th-of-the-initial-kinetic-energy-91178

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

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

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

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


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-body-of-density-12-1-0-3-kg-m-3-is-dropped-from-rest-from-a-height-1-m-into-a-liquid-of-density-24-1-0-3-kg-m-3-neglecting-all-dissipative-effects-t

slug:
a-body-of-density-12-1-0-3-kg-m-3-is-dropped-from-rest-from-a-height-1-m-into-a-liquid-of-density-24-1-0-3-kg-m-3-neglecting-all-dissipative-effects-t-93800

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

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

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

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-body-of-mass-2-mathrmkg-moves-in-a-horizontal-circular-path-of-radius-5-mathrmm-at-an-instant-its-speed-is-2-sqrt5-mathrmm-mathrms-and-is-increasing

slug:
a-body-of-mass-2-mathrmkg-moves-in-a-horizontal-circular-path-of-radius-5-mathrmm-at-an-instant-its-speed-is-2-sqrt5-mathrmm-mathrms-and-is-increasing-96517

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

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

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

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


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-body-of-mass-6-mathrmkg-is-acted-upon-by-a-force-which-causes-a-displacement-in-it-given-by-xfract24-metre-where-t-is-the-time-in-second-the-work-do

slug:
a-body-of-mass-6-mathrmkg-is-acted-upon-by-a-force-which-causes-a-displacement-in-it-given-by-xfract24-metre-where-t-is-the-time-in-second-the-work-do-83742

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

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

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

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


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-body-of-mass-m-is-thrown-vertically-upward-with-speed-3-v-e-where-v-e-is-the-escape-velocity-of-a-body-from-earth-surface-the-final-velocity-of-the-

slug:
a-body-of-mass-m-is-thrown-vertically-upward-with-speed-3-v-e-where-v-e-is-the-escape-velocity-of-a-body-from-earth-surface-the-final-velocity-of-the-87453

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

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

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

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


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-body-of-mass-m-is-thrown-with-velocity-u-from-the-origin-of-a-coordinate-axes-at-an-angle-with-the-horizon-the-magnitude-of-the-angular-momentum-of-

slug:
a-body-of-mass-m-is-thrown-with-velocity-u-from-the-origin-of-a-coordinate-axes-at-an-angle-with-the-horizon-the-magnitude-of-the-angular-momentum-of-90336

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

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

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

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


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-body-starts-from-rest-under-the-action-of-an-engine-working-at-a-constant-power-and-moves-along-a-straight-line-the-displacement-s-is-given-as-a-fun

slug:
a-body-starts-from-rest-under-the-action-of-an-engine-working-at-a-constant-power-and-moves-along-a-straight-line-the-displacement-s-is-given-as-a-fun-78504

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

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

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

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


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-body-when-fully-immersed-in-a-liquid-of-specific-gravity-12-weighs-44-gwt-the-same-body-when-fully-immersed-in-water-weighs-50-gwt-the-mass-of-the-b

slug:
a-body-when-fully-immersed-in-a-liquid-of-specific-gravity-12-weighs-44-gwt-the-same-body-when-fully-immersed-in-water-weighs-50-gwt-the-mass-of-the-b-49851

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

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

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

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


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-box-is-moved-along-a-straight-line-by-a-machine-delivering-constant-power-the-distance-moved-by-the-body-in-time-mathrmt-is-proportional-to

slug:
a-box-is-moved-along-a-straight-line-by-a-machine-delivering-constant-power-the-distance-moved-by-the-body-in-time-mathrmt-is-proportional-to-49941

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

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

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

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-box-of-mass-2-kg-is-placed-on-the-roof-of-a-car-the-box-would-remain-stationary-until-the-car-attains-a-maximum-acceleration-coefficient-of-static-f

slug:
a-box-of-mass-2-kg-is-placed-on-the-roof-of-a-car-the-box-would-remain-stationary-until-the-car-attains-a-maximum-acceleration-coefficient-of-static-f-56391

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

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

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

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


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-boy-of-mass-40-kg-is-climbing-a-vertical-pole-at-a-constant-speed-if-the-coefficient-of-friction-between-his-palms-and-the-pole-is-08-and-g-10-m-s-2

slug:
a-boy-of-mass-40-kg-is-climbing-a-vertical-pole-at-a-constant-speed-if-the-coefficient-of-friction-between-his-palms-and-the-pole-is-08-and-g-10-m-s-2-83005

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

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

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

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


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-bulb-is-placed-at-the-centre-of-a-circular-track-of-radius-10-m-a-vertical-wall-is-erected-touching-the-track-at-a-point-p-a-man-is-running-along-th

slug:
a-bulb-is-placed-at-the-centre-of-a-circular-track-of-radius-10-m-a-vertical-wall-is-erected-touching-the-track-at-a-point-p-a-man-is-running-along-th-52197

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30 \mathrm{~m} / \mathrm{sec}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$40 \mathrm{~m} / \mathrm{sec}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$60 \mathrm{~m} / \mathrm{sec}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$80 \mathrm{~m} / \mathrm{sec}$</span> </li> </ul>

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

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


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
title:
a-bullet-is-fired-with-a-velocity-u-making-an-angle-of-6-0-with-the-horizontal-plane-the-horizontal-component-o-the-velocity-of-the-bullet-when-it-rea

slug:
a-bullet-is-fired-with-a-velocity-u-making-an-angle-of-6-0-with-the-horizontal-plane-the-horizontal-component-o-the-velocity-of-the-bullet-when-it-rea-35321

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

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

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

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


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-bullet-of-mass-42-1-0-2-kg-moving-at-a-speed-of-300-ms-1-gets-stuck-into-a-block-with-a-mass-9-times-that-of-the-bullet-if-the-block-is-free-to-move

slug:
a-bullet-of-mass-42-1-0-2-kg-moving-at-a-speed-of-300-ms-1-gets-stuck-into-a-block-with-a-mass-9-times-that-of-the-bullet-if-the-block-is-free-to-move-66693

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$45 \mathrm{cal}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$405 \mathrm{cal}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$450 \mathrm{cal}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1701 \mathrm{cal}$</span> </li> </ul>

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

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


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-bullet-of-mass-m-travelling-with-a-speed-v-hits-a-block-of-mass-m-initially-at-rest-and-gets-embedded-in-it-the-combined-system-is-free-to-move-and-

slug:
a-bullet-of-mass-m-travelling-with-a-speed-v-hits-a-block-of-mass-m-initially-at-rest-and-gets-embedded-in-it-the-combined-system-is-free-to-move-and-12882

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

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

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

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


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-bullet-on-penetrating-30-cm-into-its-target-loses-its-velocity-by-50-what-additional-distance-will-it-penetrate-into-the-target-before-it-comes-to-r

slug:
a-bullet-on-penetrating-30-cm-into-its-target-loses-its-velocity-by-50-what-additional-distance-will-it-penetrate-into-the-target-before-it-comes-to-r-59466

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$20 \mathrm{cm}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 \mathrm{cm}$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">850</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-candidate-is-required-to-answer-6-out-of-12-questions-which-are-divided-into-two-parts-a-and-b-each-containing-6-questions-and-heshe-is-not-permitte

slug:
a-candidate-is-required-to-answer-6-out-of-12-questions-which-are-divided-into-two-parts-a-and-b-each-containing-6-questions-and-heshe-is-not-permitte-18215

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">850</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">800</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">750</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">700</span> </li> </ul>

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


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-capacitor-of-capacitance-c-0-is-charged-to-a-potential-v-0-and-is-connected-with-another-capacitor-of-capacitance-c-as-shown-after-closing-the-switc

slug:
a-capacitor-of-capacitance-c-0-is-charged-to-a-potential-v-0-and-is-connected-with-another-capacitor-of-capacitance-c-as-shown-after-closing-the-switc-33608

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

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

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

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


