# questions

tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-capacitor-of-capacitance-c-is-connected-in-series-with-a-resistance-r-and-d-c-source-of-emf-e-through-a-key-the-capacitor-starts-charging-when-the-k

slug:
a-capacitor-of-capacitance-c-is-connected-in-series-with-a-resistance-r-and-d-c-source-of-emf-e-through-a-key-the-capacitor-starts-charging-when-the-k-80593

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

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

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

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


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-car-is-moving-with-a-speed-of-72-km-h-1-towards-a-roadside-source-that-emits-sound-at-a-frequency-of-850-hz-the-car-driver-listens-to-the-sound-whil

slug:
a-car-is-moving-with-a-speed-of-72-km-h-1-towards-a-roadside-source-that-emits-sound-at-a-frequency-of-850-hz-the-car-driver-listens-to-the-sound-whil-44231

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

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

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

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


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-car-moving-at-a-velocity-of-17-ms-1-towards-an-approaching-bus-that-blows-a-horn-at-a-frequency-of-640-hz-on-a-straight-track-the-frequency-of-this-

slug:
a-car-moving-at-a-velocity-of-17-ms-1-towards-an-approaching-bus-that-blows-a-horn-at-a-frequency-of-640-hz-on-a-straight-track-the-frequency-of-this-48940

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

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

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

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


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-carbon-resistor-with-colour-code-is-shown-in-the-figure-there-is-no-fourth-band-in-the-resistor-the-value-of-the-resistance-is-1

slug:
a-carbon-resistor-with-colour-code-is-shown-in-the-figure-there-is-no-fourth-band-in-the-resistor-the-value-of-the-resistance-is-1-10070

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

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

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

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


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-carbon-resistor-with-colour-code-is-shown-in-the-figure-there-is-no-fourth-band-in-the-resistor-the-value-of-the-resistance-is

slug:
a-carbon-resistor-with-colour-code-is-shown-in-the-figure-there-is-no-fourth-band-in-the-resistor-the-value-of-the-resistance-is-85867


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-cell-of-emf-e-is-connected-to-a-resistance-r-1-for-time-t-and-the-amount-of-heat-generated-in-it-is-h-if-the-resistance-r-1-is-replaced-by-another-r

slug:
a-cell-of-emf-e-is-connected-to-a-resistance-r-1-for-time-t-and-the-amount-of-heat-generated-in-it-is-h-if-the-resistance-r-1-is-replaced-by-another-r-98323

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

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

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

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


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-charge-of-08-coulomb-is-divided-into-two-charges-q-1-and-q-2-these-are-kept-at-a-separation-of-30-cm-the-force-on-q-1-is-maximum-when

slug:
a-charge-of-08-coulomb-is-divided-into-two-charges-q-1-and-q-2-these-are-kept-at-a-separation-of-30-cm-the-force-on-q-1-is-maximum-when-78879

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$Q_{1}=Q_{2}=0.4 \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$Q_{1}=0.8 \mathrm{C}, Q_{2}$ negligible</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$Q_{1}$ negligible, $Q_{2}=0.8 \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$Q_{1}=0.2 \mathrm{C}, Q_{2}=0.6 \mathrm{C}$</span> </li> </ul>

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

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


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-charge-q-is-placed-at-one-corner-of-a-cube-the-electric-flux-through-any-of-the-three-faces-adjacent-to-the-charge-is-zero-the-flux-through-any-one-

slug:
a-charge-q-is-placed-at-one-corner-of-a-cube-the-electric-flux-through-any-of-the-three-faces-adjacent-to-the-charge-is-zero-the-flux-through-any-one-29250

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

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$q / 24 \varepsilon_{0}$</span> </div>

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


tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-charge-q-is-placed-at-the-origin-o-of-x-y-axes-as-shown-in-the-figure-the-work-done-in-taking-a-charge-q-from-a-to-b-along-the-straight-line-a-b-is-1

slug:
a-charge-q-is-placed-at-the-origin-o-of-x-y-axes-as-shown-in-the-figure-the-work-done-in-taking-a-charge-q-from-a-to-b-along-the-straight-line-a-b-is-1-79559

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

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

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

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


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-charge-q-is-placed-at-the-origin-o-of-x-y-axes-as-shown-in-the-figure-the-work-done-in-taking-a-charge-q-from-a-to-b-along-the-straight-line-a-b-is

slug:
a-charge-q-is-placed-at-the-origin-o-of-x-y-axes-as-shown-in-the-figure-the-work-done-in-taking-a-charge-q-from-a-to-b-along-the-straight-line-a-b-is-54022


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-charged-particle-in-a-uniform-magnetic-field-bb0-hatk-starts-moving-from-the-origin-with-velocity-v3-hati4-hatk-ms-the-trajectory-of-the-particle-an

slug:
a-charged-particle-in-a-uniform-magnetic-field-bb0-hatk-starts-moving-from-the-origin-with-velocity-v3-hati4-hatk-ms-the-trajectory-of-the-particle-an-40897

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

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

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

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


tags: ['Physics', 'Electromagnetic Waves', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-charged-particle-moves-with-constant-velocity-in-a-region-where-no-effect-of-gravity-is-felt-but-an-electrostatic-field-e-together-with-a-magnetic-f

slug:
a-charged-particle-moves-with-constant-velocity-in-a-region-where-no-effect-of-gravity-is-felt-but-an-electrostatic-field-e-together-with-a-magnetic-f-97410

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

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\vec{E} \neq 0, \vec{B} \neq 0$, $\vec{E}=0, \vec{B}=0$, $\overrightarrow{\mathrm{E}}=0, \overrightarrow{\mathrm{B}} \neq 0$</span> </div>

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


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-charged-particle-of-charge-q-and-mass-m-is-placed-at-a-distance-2-r-from-the-centre-of-a-vertical-cylindrical-region-of-radius-r-where-magnetic-fiel

slug:
a-charged-particle-of-charge-q-and-mass-m-is-placed-at-a-distance-2-r-from-the-centre-of-a-vertical-cylindrical-region-of-radius-r-where-magnetic-fiel-63587

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

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

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

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


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-charged-particle-of-mass-m-1-and-charge-q-1-is-revolving-in-a-circle-of-radius-r-another-charged-particle-of-charge-q-2-and-mass-m-2-is-situated-at--1

slug:
a-charged-particle-of-mass-m-1-and-charge-q-1-is-revolving-in-a-circle-of-radius-r-another-charged-particle-of-charge-q-2-and-mass-m-2-is-situated-at-1-66616

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

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

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


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-charged-particle-of-mass-m-1-and-charge-q-1-is-revolving-in-a-circle-of-radius-r-another-charged-particle-of-charge-q-2-and-mass-m-2-is-situated-at-

slug:
a-charged-particle-of-mass-m-1-and-charge-q-1-is-revolving-in-a-circle-of-radius-r-another-charged-particle-of-charge-q-2-and-mass-m-2-is-situated-at-43457


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-chord-a-b-is-drawn-from-the-point-a-0-3-on-the-circle-x-2-4-x-y-3-2-0-and-is-extended-to-m-such-that-a-m-2-a-b-the-locus-of-m-is

slug:
a-chord-a-b-is-drawn-from-the-point-a-0-3-on-the-circle-x-2-4-x-y-3-2-0-and-is-extended-to-m-such-that-a-m-2-a-b-the-locus-of-m-is-96321

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

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

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

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


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-circle-passing-through-0026-62-cut-the-x-axis-at-the-point-p-0-0-then-the-lenght-of-op-where-o-is-the-origin-is

slug:
a-circle-passing-through-0026-62-cut-the-x-axis-at-the-point-p-0-0-then-the-lenght-of-op-where-o-is-the-origin-is-57278

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

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

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


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-circuit-consists-of-three-batteries-of-emf-e-1-1-v-e-2-2-v-and-e-3-3-v-and-internal-resistances-1-2-and-1-respectively-which-are-connected-in-parall-1

slug:
a-circuit-consists-of-three-batteries-of-emf-e-1-1-v-e-2-2-v-and-e-3-3-v-and-internal-resistances-1-2-and-1-respectively-which-are-connected-in-parall-1-66338

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

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

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

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


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-circuit-consists-of-three-batteries-of-emf-e-1-1-v-e-2-2-v-and-e-3-3-v-and-internal-resistances-1-2-and-1-respectively-which-are-connected-in-parall

slug:
a-circuit-consists-of-three-batteries-of-emf-e-1-1-v-e-2-2-v-and-e-3-3-v-and-internal-resistances-1-2-and-1-respectively-which-are-connected-in-parall-75391


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-circular-coil-is-placed-near-a-current-carrying-conductor-both-lying-on-the-plane-of-the-paper-the-current-is-flowing-through-the-conductor-in-such-

slug:
a-circular-coil-is-placed-near-a-current-carrying-conductor-both-lying-on-the-plane-of-the-paper-the-current-is-flowing-through-the-conductor-in-such-59687

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

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">time dependent and downward.</span> </div>

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


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-circular-disc-rolls-on-a-horizontal-iloor-without-slipping-and-the-centre-of-the-disc-moves-with-a-uniform-velocity-v-which-of-the-following-values-

slug:
a-circular-disc-rolls-on-a-horizontal-iloor-without-slipping-and-the-centre-of-the-disc-moves-with-a-uniform-velocity-v-which-of-the-following-values-80765

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

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$v$, $2 v$, Zero</span> </div>

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


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-circular-loop-of-radius-r-of-conducting-wire-connected-with-a-voltage-source-of-zero-internal-resistance-produces-a-magnetic-field-b-at-its-centre-i-1

slug:
a-circular-loop-of-radius-r-of-conducting-wire-connected-with-a-voltage-source-of-zero-internal-resistance-produces-a-magnetic-field-b-at-its-centre-i-1-68332

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

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

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

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


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-circular-loop-of-radius-r-of-conducting-wire-connected-with-a-voltage-source-of-zero-internal-resistance-produces-a-magnetic-field-b-at-its-centre-i

slug:
a-circular-loop-of-radius-r-of-conducting-wire-connected-with-a-voltage-source-of-zero-internal-resistance-produces-a-magnetic-field-b-at-its-centre-i-19842


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints: \(p=1.1 .1 \cdot \frac{1}{2}=\frac{1}{2}\)</div>
source_db: db\MarksBatch1_P1.db
title:
a-coin-is-tossed-again-and-again-if-tail-appears-on-first-three-tosses-then-the-chance-that-head-appears-on-fourth-toss-is

slug:
a-coin-is-tossed-again-and-again-if-tail-appears-on-first-three-tosses-then-the-chance-that-head-appears-on-fourth-toss-is-86839

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

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

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


tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">720</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-common-emitter-transistor-amplifier-is-connected-with-a-load-resistance-of-6-k-when-a-small-ac-signal-of-15-mv-is-added-to-the-base-emitter-voltage-

slug:
a-common-emitter-transistor-amplifier-is-connected-with-a-load-resistance-of-6-k-when-a-small-ac-signal-of-15-mv-is-added-to-the-base-emitter-voltage-57788

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">90</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">640</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">900</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">720</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>$R_{C}=6 \times 10^{3} \Omega$
<br/>$\Delta \mathrm{l}_{\mathrm{B}} \mathrm{R}_{\mathrm{B}}=15 \times 10^{-3} \mathrm{~V}$
<br/>$\Delta \mathrm{I}_{\mathrm{B}}=20 \times 10^{-6} \mathrm{~A}$
<br/>$\Delta \mathrm{l}_{\mathrm{C}}=1.8 \times 10^{-3} \mathrm{~A}$
<br/>$\mathrm{~A}_{\mathrm{v}}=\beta \times \frac{\mathrm{R}_{\mathrm{C}}}{\mathrm{R}_{\mathrm{B}}}=\frac{\Delta \mathrm{l}_{\mathrm{C}}}{\Delta_{\mathrm{B}}} \times \frac{\mathrm{R}_{\mathrm{C}}}{\mathrm{R}_{\mathrm{B}}}=\frac{1.8 \times 10^{-3}}{20 \times 10^{-6}} \times \frac{6 \times 10^{3} \times 20 \times 10^{-6}}{15 \times 10^{-3}}=720$</div>


tags: ['Chemistry', 'Solid State', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$XY$</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-compound-formed-by-elements-x-and-y-crystallises-in-the-cubic-structure-where-x-atoms-are-at-the-corners-of-a-cube-and-y-atoms-are-at-the-centre-of-

slug:
a-compound-formed-by-elements-x-and-y-crystallises-in-the-cubic-structure-where-x-atoms-are-at-the-corners-of-a-cube-and-y-atoms-are-at-the-centre-of-81111

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

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

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


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\MarksBatch1_P1.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-possible-isomeric-compounds-is-33134

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\MarksBatch1_P1.db
title:
a-compressive-force-is-applied-to-a-uniform-rod-of-rectangular-crosssection-so-that-its-length-decreases-by-1-if-the-poissons-ratio-for-the-material-o

slug:
a-compressive-force-is-applied-to-a-uniform-rod-of-rectangular-crosssection-so-that-its-length-decreases-by-1-if-the-poissons-ratio-for-the-material-o-81834

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>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-conducting-circular-loop-of-resistance-20-and-crosssectional-area-20-1-0-2-m-2-is-placed-perpendicular-to-a-spatially-uniform-magnetic-field-b-which

slug:
a-conducting-circular-loop-of-resistance-20-and-crosssectional-area-20-1-0-2-m-2-is-placed-perpendicular-to-a-spatially-uniform-magnetic-field-b-which-84421

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\MarksBatch1_P1.db
title:
a-conductivity-cell-has-been-calibrated-with-001-m-1-1-electrolyte-solution-specific-conductance-k-125-1-0-3-s-cm-1-in-the-cell-and-the-measured-resis

slug:
a-conductivity-cell-has-been-calibrated-with-001-m-1-1-electrolyte-solution-specific-conductance-k-125-1-0-3-s-cm-1-in-the-cell-and-the-measured-resis-26366

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\MarksBatch1_P1.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-le

slug:
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-le-25311

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>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-convex-lens-of-focal-length-30-mathrmcm-produces-5-times-magnified-real-image-of-an-object-what-is-the-object-distance

slug:
a-convex-lens-of-focal-length-30-mathrmcm-produces-5-times-magnified-real-image-of-an-object-what-is-the-object-distance-14048

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\MarksBatch1_P1.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-in-colour-what-will-be-the-resulting-colour-78690

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>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-cricket-ball-of-mass-025-mathrmkg-with-speed-10-mathrmm-mathrms-collides-with-a-bat-and-returns-with-same-speed-within-001-mathrms-the-force-acted-o

slug:
a-cricket-ball-of-mass-025-mathrmkg-with-speed-10-mathrmm-mathrms-collides-with-a-bat-and-returns-with-same-speed-within-001-mathrms-the-force-acted-o-84555

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>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
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

slug:
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-54311

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>


tags: ['Physics', 'Work Power Energy', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-cubical-vessel-of-height-1-mathrmm-is-full-of-water-what-is-the-amount-of-work-done-in-pumping-water-out-of-the-vessel-take-mathrmg-10-mathrmm-mathr-1

slug:
a-cubical-vessel-of-height-1-mathrmm-is-full-of-water-what-is-the-amount-of-work-done-in-pumping-water-out-of-the-vessel-take-mathrmg-10-mathrmm-mathr-1-73813

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>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-cubical-vessel-of-height-1-mathrmm-is-full-of-water-what-is-the-amount-of-work-done-in-pumping-water-out-of-the-vessel-take-mathrmg-10-mathrmm-mathr

slug:
a-cubical-vessel-of-height-1-mathrmm-is-full-of-water-what-is-the-amount-of-work-done-in-pumping-water-out-of-the-vessel-take-mathrmg-10-mathrmm-mathr-86523


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-current-i-i-0-e-t-is-flowing-in-a-circuit-consisting-of-a-parallel-combination-of-resistance-r-and-capacitance-c-the-total-charge-over-the-entire-pu

slug:
a-current-i-i-0-e-t-is-flowing-in-a-circuit-consisting-of-a-parallel-combination-of-resistance-r-and-capacitance-c-the-total-charge-over-the-entire-pu-19469

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>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-current-i-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-a-1

slug:
a-current-i-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-a-1-89280

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 2019']
source_db: db\MarksBatch1_P1.db
title:
a-current-i-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-a

slug:
a-current-i-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-a-78917


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-current-of-1-a-is-flowing-along-positive-x-axis-through-a-straight-wire-of-length-05-m-placed-in-a-region-of-a-magnetic-field-given-by-b-2-i-4-j-t-t

slug:
a-current-of-1-a-is-flowing-along-positive-x-axis-through-a-straight-wire-of-length-05-m-placed-in-a-region-of-a-magnetic-field-given-by-b-2-i-4-j-t-t-79441

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\MarksBatch1_P1.db
title:
a-curve-passes-through-the-point-3-2-for-which-the-segment-of-the-tangent-line-contained-between-the-coordinate-axes-is-bisected-at-the-point-of-conta

slug:
a-curve-passes-through-the-point-3-2-for-which-the-segment-of-the-tangent-line-contained-between-the-coordinate-axes-is-bisected-at-the-point-of-conta-40075

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>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint :</div>
source_db: db\MarksBatch1_P1.db
title:
a-cyclic-process-is-shown-in-mathrmpmathrmv-diagram-and-mathrmtmathrms-diagram-which-of-the-following-statements-isare-true

slug:
a-cyclic-process-is-shown-in-mathrmpmathrmv-diagram-and-mathrmtmathrms-diagram-which-of-the-following-statements-isare-true-59688

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\MarksBatch1_P1.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-numbere

slug:
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-numbere-77293

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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-cylindrical-block-floats-vertically-in-a-liquid-of-density-1-kept-in-a-container-such-that-the-fraction-of-volume-of-the-cylinder-inside-the-liquid-

slug:
a-cylindrical-block-floats-vertically-in-a-liquid-of-density-1-kept-in-a-container-such-that-the-fraction-of-volume-of-the-cylinder-inside-the-liquid-58060

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\MarksBatch1_P1.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

slug:
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-13620

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>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-diver-at-a-depth-of-12-mathrmm-in-water-leftmufrac43right-sees-the-sky-in-a-cone-of-semivertical-angle

slug:
a-diver-at-a-depth-of-12-mathrmm-in-water-leftmufrac43right-sees-the-sky-in-a-cone-of-semivertical-angle-36905

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>


tags: ['Mathematics', 'Hyperbola', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-double-ordinate-pq-of-the-hyperbola-a-2-x-2-b-2-y-2-1-is-such-that-opq-is-equilateral-o-being-the-centre-of-the-hyperbola-then-the-eccentricity-e-sa-1

slug:
a-double-ordinate-pq-of-the-hyperbola-a-2-x-2-b-2-y-2-1-is-such-that-opq-is-equilateral-o-being-the-centre-of-the-hyperbola-then-the-eccentricity-e-sa-1-12826

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: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-double-ordinate-pq-of-the-hyperbola-a-2-x-2-b-2-y-2-1-is-such-that-opq-is-equilateral-o-being-the-centre-of-the-hyperbola-then-the-eccentricity-e-sa

slug:
a-double-ordinate-pq-of-the-hyperbola-a-2-x-2-b-2-y-2-1-is-such-that-opq-is-equilateral-o-being-the-centre-of-the-hyperbola-then-the-eccentricity-e-sa-87219


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\MarksBatch1_P1.db
title:
a-drop-of-some-liquid-of-volume-004-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-liq

slug:
a-drop-of-some-liquid-of-volume-004-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-liq-38879

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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-drop-of-water-detaches-itself-from-the-exit-of-a-tap-when-surface-tension-of-water-density-of-water-r-radius-of-the-tap-exit-r-radius-of-the-drop

slug:
a-drop-of-water-detaches-itself-from-the-exit-of-a-tap-when-surface-tension-of-water-density-of-water-r-radius-of-the-tap-exit-r-radius-of-the-drop-60265

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>

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\MarksBatch1_P1.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-

slug:
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-97222

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-sixfaced-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\MarksBatch1_P1.db
slug:
a-fair-sixfaced-die-is-rolled-12-times-the-probability-that-each-face-turns-up-twice-is-equal-to-71797

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\MarksBatch1_P1.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-

slug:
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-33635

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\MarksBatch1_P1.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-bom

slug:
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-bom-45199

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\MarksBatch1_P1.db
title:
a-frictionless-pistoncylinder-based-enclosure-contains-some-amount-of-gas-at-a-pressure-of-400-kpa-then-heat-is-transferred-fo-the-gas-at-constant-pre

slug:
a-frictionless-pistoncylinder-based-enclosure-contains-some-amount-of-gas-at-a-pressure-of-400-kpa-then-heat-is-transferred-fo-the-gas-at-constant-pre-11646

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\MarksBatch1_P1.db
title:
a-galvanometer-can-be-converted-to-a-voltmeter-of-fullscale-deflection-v-0-by-connecting-a-series-resistance-r-1-and-can-be-converted-to-an-ammeter-of

slug:
a-galvanometer-can-be-converted-to-a-voltmeter-of-fullscale-deflection-v-0-by-connecting-a-series-resistance-r-1-and-can-be-converted-to-an-ammeter-of-41509

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\MarksBatch1_P1.db
title:
a-galvanometer-having-internal-resistance-10-requires-001-a-for-a-full-scale-deflection-to-convert-this-galvanometer-to-a-voltmeter-of-fullscale-defle

slug:
a-galvanometer-having-internal-resistance-10-requires-001-a-for-a-full-scale-deflection-to-convert-this-galvanometer-to-a-voltmeter-of-fullscale-defle-67795

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\MarksBatch1_P1.db
title:
a-gas-bubble-of-2-cm-diameter-rises-througha-liquid-of-density-175-g-cm-3-with-a-fixed-speed-of-035-cms-1-neglect-the-density-of-the-gas-the-coefficie

slug:
a-gas-bubble-of-2-cm-diameter-rises-througha-liquid-of-density-175-g-cm-3-with-a-fixed-speed-of-035-cms-1-neglect-the-density-of-the-gas-the-coefficie-59628

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>


tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-given-amount-of-fe-2-is-oxidized-by-x-mol-of-mno-4-in-acidic-medium-the-number-of-moles-of-cr-2-o-7-2-required-to-oxidize-the-same-amount-of-fe-2-in

slug:
a-given-amount-of-fe-2-is-oxidized-by-x-mol-of-mno-4-in-acidic-medium-the-number-of-moles-of-cr-2-o-7-2-required-to-oxidize-the-same-amount-of-fe-2-in-37764

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>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.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-a

slug:
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-a-95382

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\MarksBatch1_P1.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-m-here-and-

slug:
a-glass-slab-consists-of-thin-uniform-layers-of-progressively-decreasing-refractive-indices-ri-see-figure-such-that-the-ri-of-any-layer-is-m-here-and-27597

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\MarksBatch1_P1.db
title:
a-golf-ball-of-mass-50-gm-placed-on-a-tee-is-struck-by-a-golfclub-the-speed-of-the-golf-ball-as-it-leaves-the-tee-is-100-m-s-the-time-of-contact-on-th

slug:
a-golf-ball-of-mass-50-gm-placed-on-a-tee-is-struck-by-a-golfclub-the-speed-of-the-golf-ball-as-it-leaves-the-tee-is-100-m-s-the-time-of-contact-on-th-95812

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>

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\MarksBatch1_P1.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

slug:
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-94612

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\MarksBatch1_P1.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-statements-is-are-tr

slug:
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-statements-is-are-tr-90013

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\MarksBatch1_P1.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-inside-the-sphere-are-respectively-98338

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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P1.db
title:
a-hollow-sphere-of-external-radius-r-and-thickness-t-r-is-made-of-a-metal-of-density-the-sphere-will-float-in-water-if

slug:
a-hollow-sphere-of-external-radius-r-and-thickness-t-r-is-made-of-a-metal-of-density-the-sphere-will-float-in-water-if-55105

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\MarksBatch1_P1.db
title:
a-homonuclear-diatomic-gas-molecule-shows-2electron-magnetic-moment-the-oneelectron-and-twoelectron-reduced-species-obtained-from-above-gas-molecule-c

slug:
a-homonuclear-diatomic-gas-molecule-shows-2electron-magnetic-moment-the-oneelectron-and-twoelectron-reduced-species-obtained-from-above-gas-molecule-c-35611

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>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-horizontal-fire-hose-with-a-nozzle-of-crosssectional-arca-21-5-1-0-3-m-2-delivers-a-cubic-metre-of-water-in-10-s-what-will-be-the-maximum-possible-i

slug:
a-horizontal-fire-hose-with-a-nozzle-of-crosssectional-arca-21-5-1-0-3-m-2-delivers-a-cubic-metre-of-water-in-10-s-what-will-be-the-maximum-possible-i-11025

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\MarksBatch1_P1.db
title:
a-horizontal-semicircular-wire-of-radius-r-is-connected-to-a-battery-through-two-similar-springs-x-and-y-to-an-electric-cell-which-sends-current-i-thr

slug:
a-horizontal-semicircular-wire-of-radius-r-is-connected-to-a-battery-through-two-similar-springs-x-and-y-to-an-electric-cell-which-sends-current-i-thr-65790

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>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-hyperbola-having-the-transverse-axis-of-length-2-sin-is-confocal-with-the-ellipse-3-x-2-4-y-2-12-it-equation-is

slug:
a-hyperbola-having-the-transverse-axis-of-length-2-sin-is-confocal-with-the-ellipse-3-x-2-4-y-2-12-it-equation-is-90119

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\MarksBatch1_P1.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-q-is

slug:
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-q-is-83224

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\MarksBatch1_P1.db
title:
a-junction-diode-has-a-resistance-of-25-when-forward-biased-and-2500-when-reverse-biased-the-current-in-the-diode-for-the-arrangement-shown-will-be-1

slug:
a-junction-diode-has-a-resistance-of-25-when-forward-biased-and-2500-when-reverse-biased-the-current-in-the-diode-for-the-arrangement-shown-will-be-1-55915

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\MarksBatch1_P1.db
title:
a-junction-diode-has-a-resistance-of-25-when-forward-biased-and-2500-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-current-in-the-diode-for-the-arrangement-shown-will-be-75715


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.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

slug:
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-95527

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\MarksBatch1_P1.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-th

slug:
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-th-72413

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>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-letter-lock-consists-of-three-rings-with-15-different-letters-if-mathrmn-denotes-the-number-of-ways-in-which-it-is-possible-to-make-unsuccessful-att

slug:
a-letter-lock-consists-of-three-rings-with-15-different-letters-if-mathrmn-denotes-the-number-of-ways-in-which-it-is-possible-to-make-unsuccessful-att-79311

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\MarksBatch1_P1.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

slug:
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-88049

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>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-line-cuts-the-x-axis-at-a-5-0-and-the-y-axis-at-b-0-3-a-variable-line-pq-is-drawn-perpendicular-to-a-b-cutting-the-x-axis-at-p-and-the-y-axis-at-q-i

slug:
a-line-cuts-the-x-axis-at-a-5-0-and-the-y-axis-at-b-0-3-a-variable-line-pq-is-drawn-perpendicular-to-a-b-cutting-the-x-axis-at-p-and-the-y-axis-at-q-i-49776

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>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-line-passes-through-the-point-1-1-and-makes-an-angle-sin-1-5-3-in-the-positive-direction-of-x-axis-if-this-line-meets-the-curve-x-2-4-y-9-at-a-and-b

slug:
a-line-passes-through-the-point-1-1-and-makes-an-angle-sin-1-5-3-in-the-positive-direction-of-x-axis-if-this-line-meets-the-curve-x-2-4-y-9-at-a-and-b-62821

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>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch1_P1.db
title:
a-line-passing-through-the-point-of-intersection-of-x-y-4-and-x-y-2-makes-an-angle-tan-1-4-3-with-the-x-axis-it-intersects-the-parabola-y-2-4-x-3-at-p-1

slug:
a-line-passing-through-the-point-of-intersection-of-x-y-4-and-x-y-2-makes-an-angle-tan-1-4-3-with-the-x-axis-it-intersects-the-parabola-y-2-4-x-3-at-p-1-31835

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>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-line-passing-through-the-point-of-intersection-of-x-y-4-and-x-y-2-makes-an-angle-tan-1-4-3-with-the-x-axis-it-intersects-the-parabola-y-2-4-x-3-at-p

slug:
a-line-passing-through-the-point-of-intersection-of-x-y-4-and-x-y-2-makes-an-angle-tan-1-4-3-with-the-x-axis-it-intersects-the-parabola-y-2-4-x-3-at-p-24409

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>

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>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
title:
a-line-through-the-point-a-2-0-which-makes-an-angle-of-3-0-with-the-positive-direction-of-x-axis-is-rotated-about-a-in-clockwise-direction-through-an-

slug:
a-line-through-the-point-a-2-0-which-makes-an-angle-of-3-0-with-the-positive-direction-of-x-axis-is-rotated-about-a-in-clockwise-direction-through-an-14704

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>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-line-with-positive-direction-cosines-passes-through-the-point-p-2-1-2-and-makes-equal-angle-with-coordinate-axes-the-line-meets-the-plane-2-x-y-z-9-

slug:
a-line-with-positive-direction-cosines-passes-through-the-point-p-2-1-2-and-makes-equal-angle-with-coordinate-axes-the-line-meets-the-plane-2-x-y-z-9-97611

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\MarksBatch1_P1.db
title:
a-liquid-of-bulk-modulus-k-is-compressed-by-applying-an-external-pressure-such-that-its-density-increases-by-001-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-increases-by-001-the-pressure-applied-on-the-liquid-is-13750

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>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P1.db
title:
a-long-conducting-wire-carrying-a-current-i-is-bent-at-12-0-see-figure-the-magnetic-field-b-at-a-point-p-on-the-right-bisector-of-bending-angle-at-a-d-1

slug:
a-long-conducting-wire-carrying-a-current-i-is-bent-at-12-0-see-figure-the-magnetic-field-b-at-a-point-p-on-the-right-bisector-of-bending-angle-at-a-d-1-43904

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: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-long-conducting-wire-carrying-a-current-i-is-bent-at-12-0-see-figure-the-magnetic-field-b-at-a-point-p-on-the-right-bisector-of-bending-angle-at-a-d

slug:
a-long-conducting-wire-carrying-a-current-i-is-bent-at-12-0-see-figure-the-magnetic-field-b-at-a-point-p-on-the-right-bisector-of-bending-angle-at-a-d-11906


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-longitudinal-wave-is-represented-by-x-x-0-sin-2-n-t-x-the-maximum-particle-velocity-will-be-four-times-the-wave-velocity-if

slug:
a-longitudinal-wave-is-represented-by-x-x-0-sin-2-n-t-x-the-maximum-particle-velocity-will-be-four-times-the-wave-velocity-if-73245

question:
<div class="question">A longitudinal wave is represented by $x=x_0 \sin 2 \pi(n t-x / \lambda)$. The maximum particle velocity will be four times the wave velocity if:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\lambda=\frac{\pi x_0}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\lambda=2 \pi x_0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\lambda=\frac{\pi x_0}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\lambda=4 \pi x_0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\lambda=\frac{\pi x_0}{2}$</span> </div>

answer:
<div class="solution">Hints: Maximum particle velocity $\left(\mathrm{V}_{\mathrm{p}}\right)=\mathrm{A} \omega=2 \pi n x_0$<br/>Wave velocity $\left(\mathrm{V}_\omega\right)=n \lambda$<br/>Here, $\mathrm{V}_{\mathrm{P}}=4 \mathrm{~V}_\omega$<br/>$$<br/>2 \pi n x_0=4 n \lambda<br/>$$<br/>$$<br/>\lambda=\frac{\pi}{2} x_0<br/>$$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-luminous-object-is-separated-from-a-screen-by-distance-d-a-convex-lens-is-placed-between-the-object-and-the-screen-such-that-it-forms-a-distinct-ima

slug:
a-luminous-object-is-separated-from-a-screen-by-distance-d-a-convex-lens-is-placed-between-the-object-and-the-screen-such-that-it-forms-a-distinct-ima-35523

question:
<div class="question">A luminous object is separated from a screen by distance $d$. A convex lens is placed between the object and the screen such that it forms a distinct image on the screen. The maximum possible focal length of this convex lens is</div>

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

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

answer:
<div class="solution">Focal length of a convex lens by displacement method
<br/>Focal length of the convex lens
<br/>$$
<br/>f=\frac{a^{2}-b^{2}}{4 a}
<br/>$$
<br/>where $a$ = distance between the image and object and $b=$ distance between two positions of lens
<br/>Fact The distance bemeen the ho points should be 4 times greater than the focal iength of the convex lens, ie, $d&gt;4 f$.
<br/>So, the maximum possible focal length of this convex lens is $\frac{d}{4}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-magnetic-field-b-2-t-4-t-2-where-t-x-time-is-applied-perpendicular-to-the-plany-of-a-circular-wire-of-radius-r-and-resistance-r-if-all-the-units-are

slug:
a-magnetic-field-b-2-t-4-t-2-where-t-x-time-is-applied-perpendicular-to-the-plany-of-a-circular-wire-of-radius-r-and-resistance-r-if-all-the-units-are-43000

question:
<div class="question">A magnetic field $B=2 t+4 t^{2}$ (where, $t_{x}$ time) is applied perpendicular to the plany of a circular wire of radius $r$ and resistance
<br/>R. If all the units are in SI the electric charge that flows through the circular wire during $t=0 \mathrm{s}$ to $t=2 \mathrm{s}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{6 \pi^{2}}{R}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{20 \pi r^{2}}{R}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{32 \pi^{2}}{R}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{48 \pi^{2}}{R}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{20 \pi r^{2}}{R}$</span> </div>

answer:
<div class="solution">Given, $B=2 t+4 t^{2}$
<br/>at $\quad t=0, B_{1}=0$
<br/>and at $t=2, B_{2}=2 \times 2+4(2)^{2}$
<br/>$$
<br/>=4+16=20 \mathrm{Wb} / \mathrm{m}^{2}
<br/>$$
<br/>We have, $\Delta Q=\frac{\Delta \phi}{R}=\frac{\pi^{2}\left(B_{2}-B_{1}\right)}{R}$
<br/>$$
<br/>=\frac{\pi r^{2} \left[20-0\right]}{R}=\frac{20 \pi r^{2}}{R}
<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P1.db
title:
a-magnetic-needle-is-placed-in-a-uniform-magnetic-field-and-is-aligned-with-the-field-the-needle-is-now-rotated-by-an-angle-of-6-0-and-the-work-done-i

slug:
a-magnetic-needle-is-placed-in-a-uniform-magnetic-field-and-is-aligned-with-the-field-the-needle-is-now-rotated-by-an-angle-of-6-0-and-the-work-done-i-59243

question:
<div class="question">A magnetic needle is placed in a uniform magnetic field and is aligned with the field. The needle is now rotated by an angle of $60^{\circ}$ and the work done is $W$. The torque on the magnetic needle at this position is</div>

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

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

answer:
<div class="solution">Given, work done $=W$ and $\theta=60^{\circ}$ We know that
<br/>Hence,
<br/>$$
<br/>\begin{array}{l}
<br/>W=M B(1-\cos \theta) \\
<br/>W=M B\left(1-\cos 60^{\circ}\right) \\
<br/>W=\frac{M B}{2} \\
<br/>|\tau|=M B \sin 60^{\circ}=\sqrt{3} W
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
title:
a-mapping-from-in-to-in-is-defined-as-follows-f-in-in-f-n-n-5-2-n-in-in-is-the-set-of-natural-numbers-then

slug:
a-mapping-from-in-to-in-is-defined-as-follows-f-in-in-f-n-n-5-2-n-in-in-is-the-set-of-natural-numbers-then-37380

question:
<div class="question">A Mapping from IN to IN is defined as follows :<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{f}: \mathrm{IN} \rightarrow \mathrm{IN} \\<br/>&amp; \mathrm{f}(\mathrm{n})=(\mathrm{n}+5)^2, \mathrm{n} \in \mathrm{IN}<br/>\end{aligned}<br/>$$<br/>(IN is the set of natural numbers). Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}$ is one-to-one but not onto</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \mathrm{f}: \mathrm{IN} \rightarrow \mathrm{IN} ; \mathrm{f}(\mathrm{n})=(\mathrm{n}+5)^2 \\<br/>&amp; \left(\mathrm{n}_1+5\right)^2=\left(\mathrm{n}_2+5\right)^2 \\<br/>&amp; \Rightarrow\left(\mathrm{n}_1-\mathrm{n}_2\right)\left(\mathrm{n}_1+\mathrm{n}_2+10\right)=0 \\<br/>&amp; \Rightarrow \mathrm{n}_1=\mathrm{n}_2 \rightarrow \text { one-to-one }<br/>\end{aligned}<br/>$$<br/>There does not exist $n \in$ IN such that $(n+5)^2=1$<br/>Hence $\mathrm{f}$ is not onto</div>


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1: 1</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-mass-m-at-rest-is-broken-into-two-pieces-having-masses-m-and-m-m-the-two-masses-are-then-separated-by-a-distance-r-the-gravitational-force-between-t

slug:
a-mass-m-at-rest-is-broken-into-two-pieces-having-masses-m-and-m-m-the-two-masses-are-then-separated-by-a-distance-r-the-gravitational-force-between-t-54934

question:
<div class="question">A mass $M$ at rest is broken into two pieces having masses $m$ and $(M-m) .$ The two masses are then separated by a distance $r$ The gravitational force between them will be the maximum when the ratio of the masses $[m:(M-m)]$ of the two parts is</div>

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

answer:
<div class="solution">The gravitation force between two masses,
<br/>$$
<br/>F=\frac{G m_{1} m_{2}}{r}
<br/>$$
<br/>here, $\quad m_{1}=m$
<br/>and
<br/>$$
<br/>\begin{array}{l}
<br/>m_{2}=(M-m) \\
<br/>F=\frac{G m(M-m)}{r^{2}}
<br/>\end{array}
<br/>$$
<br/>$\therefore$
<br/>For maximum gravitational force, $\frac{d F}{d m}=0$
<br/>$\therefore$
<br/>$$
<br/>\frac{d}{d m}[m(M-m)]=0
<br/>$$
<br/>By solving, we get $m=\frac{M}{2}$
<br/>So, $\quad \frac{m}{(M-m)}=\frac{1}{1}$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3: 1</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-mass-of-1-kg-is-suspended-by-means-of-a-thread-the-system-is-i-lifted-up-with-an-acceleration-of-49-ms-2-ii-lowered-with-an-acceleration-of-49-ms-2-

slug:
a-mass-of-1-kg-is-suspended-by-means-of-a-thread-the-system-is-i-lifted-up-with-an-acceleration-of-49-ms-2-ii-lowered-with-an-acceleration-of-49-ms-2-37611

question:
<div class="question">A mass of $1 \mathrm{kg}$ is suspended by means of a thread. The system is
<br/>(i) lifted up with an acceleration of $4.9 \mathrm{ms}^{-2}$
<br/>(ii) lowered with an acceleration of $4.9 \mathrm{ms}^{-2}$. The ratio of tension in the first and second case is</div>

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

answer:
<div class="solution">When the body of mass m lifted up the forces acting on the body
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a34.png"/> <br/> <br/>$$
<br/>\begin{aligned}
<br/>T_{1}-m g &amp;=\frac{m g}{2} \\
<br/>T_{1} &amp;=m g+\frac{m g}{2}=\frac{3 m g}{2}
<br/>\end{aligned}
<br/>$$
<br/>(ii) When the body lowered the acting forces<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a34a.png"/> <br/> <br/>$$
<br/>\left(m g-T_{2}\right)=\frac{m g}{2}
<br/>$$
<br/>$$
<br/>T_{2}=m g-\frac{m g}{2}=\frac{m q}{2}
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>\begin{aligned}
<br/>\frac{T_{1}}{T_{2}} &amp;=\frac{3 m g / 2}{m g / 2}=\frac{3}{1} \\
<br/>&amp;=3: 1
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">zero</span> </div>
answer: <div class="solution">Hints : Poisson's ratio is 0.5 so there is no change in volume</div>
source_db: db\MarksBatch1_P1.db
title:
a-material-has-poissons-ratio-050-if-a-uniform-rod-of-it-suffers-a-longitudinal-strain-of-2-times-103-then-the-percentage-change-in-volume-is

slug:
a-material-has-poissons-ratio-050-if-a-uniform-rod-of-it-suffers-a-longitudinal-strain-of-2-times-103-then-the-percentage-change-in-volume-is-58886

question:
<div class="question">A material has Poisson's ratio 0.50. If a uniform rod of it suffers a longitudinal strain of \(2 \times 10^{-3}\), then the percentage change in volume is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">100</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-metal-m-forms-two-oxides-the-ratio-m-o-by-weight-in-the-two-oxides-are-25-4-and-25-6-the-minimum-value-of-atomic-mass-of-m-is

slug:
a-metal-m-forms-two-oxides-the-ratio-m-o-by-weight-in-the-two-oxides-are-25-4-and-25-6-the-minimum-value-of-atomic-mass-of-m-is-91440

question:
<div class="question">A metal (M) forms two oxides. The ratio $\mathrm{M}: \mathrm{O}$ (by weight) in the two oxides are $25: 4$ and $25: 6$. The minimum value of atomic mass of $M$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">50</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">100</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">150</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">200</span> </li> </ul>

answer:
<div class="solution">Let two oxide be<br/>$\mathrm{M}_2 \mathrm{O}_x$ and $\mathrm{M}_2 \mathrm{Oy}$<br/>$\underline{\text{As per question}}$<br/>$\frac{2 a}{16 x}=\frac{25}{4}$ and $\frac{2 a}{16 y}=\frac{25}{6}$<br/>$x=\frac{a}{50}, y=\frac{3 a}{100}$ where $a=$ atomic mass of Metal<br/>As $x$ and $y$ to be an integer,<br/>If we take $a=50$, then $x=1, y=1.5$ (not possible)<br/>If we take $a=100$ then $x=2, y=3$ (possible)<br/>$\therefore$ Minimum Atomic Mass $=100 \mathrm{u}$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-metal-m-specific-heat-016-forms-a-metal-chloride-with-65-chlorine-present-in-it-the-formula-of-the-metal-chloride-will-be

slug:
a-metal-m-specific-heat-016-forms-a-metal-chloride-with-65-chlorine-present-in-it-the-formula-of-the-metal-chloride-will-be-80526

question:
<div class="question">A metal $M$ (specific heat 0.16 ) forms a metal chloride with $65 \%$ chlorine present in it. The formula of the metal chloride will be</div>

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

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

answer:
<div class="solution">Given, specific heat $=0.16$ 
<br/>Let metal chloride be $M \mathrm{Cl}_{x}$ then. $\frac{6.4}{\text { specific heat }}=$ Atomic weight of metal $\frac{6.4}{0.16}=$ Atomic weight of metal
<br/>Atomic weight $=40$ 
<br/>40 is the atomic weight of calcium. According to question, metal chloride $\left(M C l_{x}\right)$ have $\approx 65 \%$ chlorine present in it. $\frac{x \times \text { Atomic weight of chlorine }}{40+x \times \text { Atomic weight of chlorine }} \times 100=65$
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{x \times 355}{40+x \times 355} \times 100=65 \\
<br/>x=209 \approx 2(a p p r o x .)
<br/>\end{array}
<br/>$$
<br/>So, the formula of metal chloride will be $M \mathrm{Cl}_{2}$</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-metal-rod-is-fixed-rigidly-at-two-ends-so-as-to-prevent-its-thermal-expansion-if-l-and-y-preventively-denote-the-length-of-the-rod-respective-of-lin

slug:
a-metal-rod-is-fixed-rigidly-at-two-ends-so-as-to-prevent-its-thermal-expansion-if-l-and-y-preventively-denote-the-length-of-the-rod-respective-of-lin-79331

question:
<div class="question">A metal rod is fixed rigidly at two ends so as to prevent its thermal expansion. If $L, \alpha$ and $Y$ preventively denote the length of the rod, respective of linear thermal expansion and coefficient of Young's modulus of its material, then for an Young's modulus of its material, then for an increase in temperature of the rod by $\Delta T$, the longitudinal stress developed in the rod is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">directly proportional to $\Delta T / Y$</span> </div>

answer:
<div class="solution">Strain $=\alpha \cdot \Delta T$
<br/>Stress $Y \propto \Delta T$
<br/>where $L=$ length of the rod $\boldsymbol{\alpha}=$ coefficient of linear thermal expansion
<br/>$Y=$ Young's modulus of its material So, the longitudinal stress developed in the rod is directly proportional to $\frac{\Delta T}{Y}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-metal-sphere-of-radius-r-carrying-charge-q-is-surrounded-by-a-thick-concentric-metal-shell-of-inner-and-outer-radii-a-and-b-respectively-the-net-cha

slug:
a-metal-sphere-of-radius-r-carrying-charge-q-is-surrounded-by-a-thick-concentric-metal-shell-of-inner-and-outer-radii-a-and-b-respectively-the-net-cha-74386

question:
<div class="question">A metal sphere of radius $R$ carrying charge $q$ is surrounded by a thick concentric metal shell of inner and outer radii a and $\mathrm{b}$ respectively. The net charge on the shell is zero. The potential at the centre of the sphere, when the outer surface of the shell is grounded will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{q}}{4 \pi \epsilon_{0}}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{a}}\right)$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/PXfLyLFKo99kt6XzIZDBzk7ziLA7K0ypia0WCpfzB6A.original.fullsize.png"/> <br/>\(V_{A}=V_{B}+V_{C}+V_{c}\)
<br/>\(V_{c}=\frac{1}{4 x \varepsilon_{0}}\left(\frac{-q}{a}\right)+\frac{1}{4 x \varepsilon_{0}}\left(\frac{q}{R}\right)+0\)
<br/>\(=\frac{1}{4 \pi \varepsilon_{0}}\left(\frac{q}{R}-\frac{q}{a}\right)\)
<br/>\(=\frac{2}{4 \pi \varepsilon 0}\left[\frac{1}{R}-\frac{1}{a}\right]\)</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$4: 1$</span> </div>
source_db: db\MarksBatch1_P1.db
title:
a-metal-wire-of-circular-crosssection-has-a-nsistance-r-1-the-wire-is-now-stretched-without-breaking-so-that-its-length-is-doubled-and-the-density-is-

slug:
a-metal-wire-of-circular-crosssection-has-a-nsistance-r-1-the-wire-is-now-stretched-without-breaking-so-that-its-length-is-doubled-and-the-density-is-95918

question:
<div class="question">$A$ metal wire of circular cross-section has a n'sistance $R_{1}$. The wire is now stretched without breaking, so that its length is doubled and the density is assumed to remain the same. If the resistance of the wire now beromes $R_{2},$ then $R_{2}: R_{1}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1: 1$</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">$4: 1$</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">$1: 4$</span> </li> </ul>

answer:
<div class="solution">As we know that, $R_{1}=\rho \frac{1}{a}=\rho \frac{l^{2}}{V}$
<br/>where, $1=$ length of wire $a=$ area of cross-section of the wire and $V=$ volume of the wire $R_{1} \propto l^{2}$
<br/>$\Rightarrow \quad \frac{R_{1}}{R_{2}}=\left(\frac{l_{1}}{I_{2}}\right)^{2}=\left(\frac{1}{2}\right)^{2} \Rightarrow R_{2}: R_{1}=4: 1$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-metallic-block-of-mass-20-kg-is-dragged-with-a-uniform-velocity-of-05-ms-1-on-a-horizontal-table-for-21-s-the-coefficient-of-static-friction-between

slug:
a-metallic-block-of-mass-20-kg-is-dragged-with-a-uniform-velocity-of-05-ms-1-on-a-horizontal-table-for-21-s-the-coefficient-of-static-friction-between-57969

question:
<div class="question">A metallic block of mass $20 \mathrm{~kg}$ is dragged with a uniform velocity of $0.5 \mathrm{~ms}^{-1}$ on a horizontal table for $2.1 \mathrm{~s}$. The co-efficient of static friction between the block and the table is $0.10$. What will be the maximum possible rise in temperature of the metal block if the specific heat of the block is $0.1$ C.G.S unit ? Assume $g=10 \mathrm{~ms}^{-2}$ uniform rise in temperature throughout the whole block. [lgnore absorption of heat by the table]</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.0025^{\circ} \mathrm{C}$</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">$0.025^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.001^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.05^{\circ} \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.0025^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>friction $f=\mu \mathrm{mg}=0.1 \times 20 \times 10=20$
<br/>$\mathrm{~W}_{\mathrm{f}}=$ heat
<br/>$\Rightarrow(\mathrm{f.u}) \mathrm{t}=\mathrm{mc} \Delta \mathrm{T}$
<br/>$\Rightarrow 20 \times 0.5 \times 2.1=20 \times 0.1 \times 4.2 \times 10^{3} . \Delta \mathrm{T}$
<br/>$\Rightarrow \Delta \mathrm{T}=0.0025^{\circ} \mathrm{C}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-metallic-loop-is-placed-in-a-uniform-magnetic-field-b-with-the-planc-of-the-loop-perpendicular-to-b-under-which-condition-given-an-emf-will-be-induc

slug:
a-metallic-loop-is-placed-in-a-uniform-magnetic-field-b-with-the-planc-of-the-loop-perpendicular-to-b-under-which-condition-given-an-emf-will-be-induc-76621

question:
<div class="question">A metallic loop is placed in a uniform magnetic field $\mathbf{B}$ with the planc of the loop perpendicular to $\mathbf{B}$. Under which condition given an emf will be induced in the loop "If the loop is ....."</div>

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

answer:
<div class="solution">According to the question.
<br/>Given, A metallic loop is placed in uniform magnetic field B. Magnetic field $\mathbf{B}$ and metallic loop are perpendicular to each other. Then, if metallic loop is moved along $\mathbf{B}$ or rotated about its own axis. the net flux associated with it remains constant. Thus, no emf will be induced in these cases. However, when the loop is squeezed to a smaller area or rotated about one of its diameters. Then its flux changes. Thus, emf is induced. So, option (b) and (d) both are correct.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a40.png"/></div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-missile-is-fired-from-the-ground-level-rises-x-meters-vertically-upwards-in-t-sec-where-x100-tfrac252-t2-the-maximum-height-reached-is

slug:
a-missile-is-fired-from-the-ground-level-rises-x-meters-vertically-upwards-in-t-sec-where-x100-tfrac252-t2-the-maximum-height-reached-is-91577

question:
<div class="question">A missile is fired from the ground level rises \(x\) meters vertically upwards in \(t \sec\), where \(x=100 t-\frac{25}{2} t^2\). The maximum height reached is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(100 \mathrm{~m}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(300 \mathrm{~m}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(200 \mathrm{~m}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(125 \mathrm{~m}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(200 \mathrm{~m}\)</span> </div>

answer:
<div class="solution">Hint : \(x=100 t-\frac{25}{2} t^2\)<br/>\(\frac{d x}{d t}=100-25 \mathrm{t}\) for maximum and minimum \(\frac{d x}{d t}=0 \Rightarrow t=4 \mathrm{sec}\)<br/>\(\frac{d^2 x}{d t^2}=-25  &lt; 0\)<br/>\(\therefore\) at \(\mathrm{x}=4\) it will have maixmum height<br/>\(x_{\max }=100 \times 4-\frac{25}{2} \times 16=400-200=200 \mathrm{~m}\)</div>


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-modified-gravitational-potential-is-given-by-vfracg-mrfracar2-if-the-constant-a-is-expressed-in-terms-of-gravitational-constant-g-mass-m-and-velocit

slug:
a-modified-gravitational-potential-is-given-by-vfracg-mrfracar2-if-the-constant-a-is-expressed-in-terms-of-gravitational-constant-g-mass-m-and-velocit-10124

question:
<div class="question">A modified gravitational potential is given by \(V=-\frac{G M}{r}+\frac{A}{r^2}\). If the constant \(A\) is expressed in terms of gravitational constant (G), mass ( \(M\) ) and velocity of light (c), then from dimensional analysis, \(A\) is,</div>

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

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

answer:
<div class="solution">Hint : \(V=-\frac{G M}{r}+\frac{A}{r^2}\)<br/>\([A]=\frac{[G M]}{[r]}\left[r^2\right]=[G M][r]\)<br/>now, we know, \(\frac{G M}{r}\) gives dimension of \(c^2\)<br/>\(\frac{[\mathrm{GM}]}{[\mathrm{r}]}=\left[\mathrm{c}^2\right] \Rightarrow[\mathrm{r}]=\frac{[\mathrm{GM}]}{\left[\mathrm{c}^2\right]}\)<br/>\(\Rightarrow[\mathrm{A}]=\frac{[\mathrm{GM}][\mathrm{GM}]}{\left[\mathrm{c}^2\right]}\)<br/>\([\mathrm{A}]=\frac{\mathrm{G}^2 \mathrm{M}^2}{\mathrm{c}^2}\)</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch1_P1.db
title:
a-mouse-of-mass-mathrmm-jumps-on-the-outside-edge-of-a-rotating-ceiling-fan-of-moment-of-inertia-i-and-radius-mathrmr-the-fractional-loss-of-angular-v

slug:
a-mouse-of-mass-mathrmm-jumps-on-the-outside-edge-of-a-rotating-ceiling-fan-of-moment-of-inertia-i-and-radius-mathrmr-the-fractional-loss-of-angular-v-82295

question:
<div class="question">A mouse of mass \(\mathrm{m}\) jumps on the outside edge of a rotating ceiling fan of moment of inertia I and radius \(\mathrm{R}\). The fractional loss of angular velocity of the fan as a result is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{m R^2}{\mathrm{I}+\mathrm{mR}^2}\)</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">\(\frac{I}{I+m R^2}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\mathrm{I}-\mathrm{mR}^2}{\mathrm{I}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{I-m R^2}{I+m R^2}\)</span> </li> </ul>

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

answer:
<div class="solution">Hint : \(I \omega_0=\left(\mathrm{I}+\mathrm{mR}^2\right) \omega\)<br/>\(\omega_0 \rightarrow\) Initial angular velocity<br/>\(\omega=\frac{I \omega_0}{I+m R^2}\)<br/>\(\omega \rightarrow\) Final angular velocity<br/>So, \(\frac{\omega_0-\omega}{\omega_0}=I-\frac{I}{I+m R^2}=\frac{m R^2}{I+m R^2}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P1.db
title:
a-moving-line-intersects-the-lines-x-y-0-and-x-y-0-at-the-points-a-b-respectively-such-that-the-area-of-the-triangle-with-vertices-0-0-a-b-has-a-const

slug:
a-moving-line-intersects-the-lines-x-y-0-and-x-y-0-at-the-points-a-b-respectively-such-that-the-area-of-the-triangle-with-vertices-0-0-a-b-has-a-const-35081

question:
<div class="question">A moving line intersects the lines $x+y=0$ and $x-y=0$ at the points A, B respectively such that the area of the triangle with vertices $(0,0), A \&amp; B$ has a constant area $C$. The locus of the mid-point $A B$ is given by the equation</div>

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

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

answer:
<div class="solution">Let mid point of $\mathrm{AB}=(\mathrm{h}, \mathrm{k})$
<br/>Let, $A=(\alpha,-\alpha), B=(\beta, \beta)$
<br/>$\therefore \beta+\alpha=2 \mathrm{~h}, \quad \beta-\alpha=2 \mathrm{k}$
<br/>Area $\Delta \mathrm{AOB}=\frac{1}{2} \times \mathrm{OA} \times \mathrm{OB}=\frac{1}{2} \sqrt{2 \alpha^{2}} \sqrt{2 \beta^{2}}=|\alpha \beta|=\mathrm{c}$
<br/>$\begin{array}{l}
<br/>\therefore \alpha^{2} \beta^{2}=\mathrm{c}^{2} \Rightarrow 16 \alpha^{2} \beta^{2}=16 \mathrm{c}^{2} \Rightarrow\left\{(\beta+\alpha)^{2}-(\beta-\alpha)^{2}\right\}^{2}=16 \mathrm{c}^{2} \\
<br/>\Rightarrow\left(4 \mathrm{~h}^{2}-4 \mathrm{k}^{2}\right)^{2}=16 \mathrm{c}^{2} \rightarrow \text { locus }:\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right)^{2}=\mathrm{c}^{2}
<br/>\end{array}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/s3QJ_akdM5cTNYqBZDECKe_8wIzkt06oGQmxKKKMeoc.original.fullsize.png"/></div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-negative-charge-is-placed-at-the-midpoint-between-two-fixed-equal-positive-charges-separated-by-a-distance-2-d-if-the-negative-charge-is-given-a-sma

slug:
a-negative-charge-is-placed-at-the-midpoint-between-two-fixed-equal-positive-charges-separated-by-a-distance-2-d-if-the-negative-charge-is-given-a-sma-24570

question:
<div class="question">A negative charge is placed at the midpoint between two fixed equal positive charges, separated by a distance $2 d$. If the negative charge is given a small displacement $x(x&lt; &lt; a)$ perpendicular to the line joining the positive charges, how the force $(F)$ developed on it will approximately depend on $x ?$</div>

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

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

answer:
<div class="solution">According to the question,
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a20.png"/> <br/>Force experienced by the charge $-q$ due to charge $Q$
<br/>$$
<br/>\begin{array}{l}
<br/>\qquad F=-\frac{2 k Q q}{r^{2}} \cos \theta \ldots(\mathrm{i})\left[\text { where, } k=\frac{1}{4 \pi \varepsilon_{0}}\right] \\
<br/>\text { From diagram, } \cos \theta=\frac{x}{r}
<br/>\end{array}
<br/>$$
<br/>By substituting the value of $\cos \theta$ in Eq. (i)
<br/>$$
<br/>F=-\frac{2 k Q q}{r^{2}} \cdot \frac{x}{r}
<br/>$$
<br/>or
<br/>$$
<br/>F=-\frac{2 k Q x}{r^{3}}
<br/>$$
<br/>or
<br/>$$
<br/>F=-\frac{2 k Q x}{\left(x^{2}+d^{2}\right)^{3 / 2}} \quad\left[\begin{array}{l}
<br/>\because r^{2}=x^{2}+d^{2} \\
<br/>r=\sqrt{x^{2}+d^{2}}
<br/>\end{array}\right]
<br/>$$
<br/>For, $x&lt; &lt; d,$ so $x^{2}$ can be neglected
<br/>$$
<br/>F=-\frac{2 k Q x}{d^{3}}
<br/>$$
<br/>So, the force developed by negative charge $(-q)$ due to the system of the charges as shown in the figure is,
<br/>$$
<br/>F=\frac{-4 k Q q x}{d^{3}}
<br/>$$
<br/>$\Rightarrow \quad F \propto x$
<br/>So, the force developed by negative charge is directly proportional to the distance $x$.</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">Considering only interaction between charge $q_a$ and $q_b$.</div>
source_db: db\MarksBatch1_P1.db
title:
a-neutral-conducting-solid-sphere-of-radius-r-has-two-spherical-cavities-of-radius-a-and-b-as-shown-in-the-figure-centre-to-centre-distance-between-tw

slug:
a-neutral-conducting-solid-sphere-of-radius-r-has-two-spherical-cavities-of-radius-a-and-b-as-shown-in-the-figure-centre-to-centre-distance-between-tw-12557

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/39Rhm7mglnT87696mR-q4XvGv_SG9MJH-vHT3qXTJLg.original.fullsize.png"/><br/><br/>A neutral conducting solid sphere of radius $\mathrm{R}$ has two spherical cavities of radius a and $b$ as shown in the figure. Centre to centre distance between two cavities is $c . q_a$ and $q_b$ charges are placed at the centres of cavities respectively. The force between $q_a$ and $q_b$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}_{\mathrm{a}} \mathrm{q}_{\mathrm{b}}}{\mathrm{c}^2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4 \pi \varepsilon_0} q_a q_b\left(\frac{1}{a^2}+\frac{1}{b^2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">insufficient data</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}_{\mathrm{a}} \mathrm{q}_{\mathrm{b}}}{\mathrm{c}^2}$</span> </div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-nonzero-current-passes-through-the-galvanometer-g-shown-in-the-circuit-when-the-key-k-is-closed-and-its-value-does-not-change-when-the-key-is-opened

slug:
a-nonzero-current-passes-through-the-galvanometer-g-shown-in-the-circuit-when-the-key-k-is-closed-and-its-value-does-not-change-when-the-key-is-opened-64086

question:
<div class="question">A non-zero current passes through the galvanometer $G$ shown in the circuit when the key $K$ is closed and its value does not change when the key is opened. Then, which of the following statement(s) is/are true?<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q36.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The galvanometer resistance is infinite.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The current through the galvanometer is $40 \mathrm{mA}$.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">After the key is closed, the current through the $200 \Omega$ resistor is same as the current through the $300 \Omega$ resistor.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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 galvanometer resistance is 150$\Omega$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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 current through the galvanometer is $40 \mathrm{mA}$., After the key is closed, the current through the $200 \Omega$ resistor is same as the current through the $300 \Omega$ resistor., The galvanometer resistance is 150$\Omega$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a36.png"/>$\therefore I_{C}=\frac{10}{100+G}$
<br/>$300 I_{3}=\frac{10}{100+G} G$
<br/>$I_{3}=\frac{G}{30}\left[\frac{1}{100+G}\right]$
<br/>$\therefore \quad I_{3}+I_{G}=\frac{1}{(100+G)}\left[10+\frac{G}{30}\right]=\frac{(300+G)}{(100+G) \times 30}$
<br/>According to the problem, $10=\left(\frac{200}{3}+\frac{300 G}{300+G}\right)\left[\frac{(300+G)}{(100+6) 30}\right]$
<br/>$Y=R C$
<br/>$10=\frac{60000+1100 G}{3} \times \frac{1}{100+G} \times \frac{1}{30}$
<br/>$\Rightarrow 900(100+G)=60000+1100 G$
<br/>$\Rightarrow \quad 30000=200 G$
<br/>$\therefore$
<br/>$G=150 \Omega$
<br/>$I_{G}=\frac{10}{100+150}=\frac{10}{250}=40 \mathrm{mA}$
<br/>As $\quad \frac{200}{100}=\frac{300}{150}$
<br/>So, $\quad I_{200}=I_{300}$</div>


tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-nor-gate-and-a-nand-gate-are-connected-as-shown-in-the-figure-two-different-sets-of-inputs-are-given-to-this-setup-in-the-first-case-the-inputs-to-t

slug:
a-nor-gate-and-a-nand-gate-are-connected-as-shown-in-the-figure-two-different-sets-of-inputs-are-given-to-this-setup-in-the-first-case-the-inputs-to-t-54481

question:
<div class="question">A NOR gate and a NAND gate are connected as shown in the figure. Two different sets of inputs are given to this setup. In the first case, the inputs to the gates are $A=0, B=0$, $C=0 .$ In the second case, the inputs are $A=1, B=0, C=1 .$ The output $D$ in the first case and second case respectively are <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q4.png"/></div>

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

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

answer:
<div class="solution">In first case $A=0, B=0$
<br/>$\therefore$ Output of NOR gate, $Y=\overline{A+B}=1$ 
<br/>This output is the input for NAND gate, $i . e ., Y=1$
<br/>and $C=0$
<br/>$\therefore \quad D=\overline{Y \cdot C}=1$
<br/>In second case $A=1, B=0$
<br/>$\therefore$ Output of NOR gate, $Y=\overline{A+B}=0$ 
<br/>This output is the input for NAND gate $i . e . . Y=0$
<br/>and $C=1$
<br/>$\therefore \quad D=\overline{Y \cdot C}=1$</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-nucleus-x-emits-a-beta-particle-to-produce-a-nucleus-y-if-their-atomic-masses-are-m-x-and-m-y-respectively-the-maximum-energy-of-the-beta-particle-e

slug:
a-nucleus-x-emits-a-beta-particle-to-produce-a-nucleus-y-if-their-atomic-masses-are-m-x-and-m-y-respectively-the-maximum-energy-of-the-beta-particle-e-15711

question:
<div class="question">A nucleus X emits a beta particle to produce a nucleus Y. If their atomic masses are $\mathrm{M}_{\mathrm{x}}$ and $\mathrm{M}_{\mathrm{y}}$ respectively, the maximum energy of the beta particle emitted is (where $m_{e}$ is the mass of an electron and $c$ is the velocity of light)</div>

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

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/gKv3Gl__KGXJ4TL8YAEkwNsi4Jya45ra2uT2_xsXiuQ.original.fullsize.png"/><br/> <br/>$\mathrm{m}_{\mathrm{x}}=\mathrm{M}_{\mathrm{x}}-\mathrm{zm}_{\mathrm{e}}$
<br/>$\frac{\mathrm{m}_{\mathrm{y}}=\mathrm{M}_{\mathrm{y}}-(\mathrm{z}+1) \mathrm{m}_{\mathrm{e}}}{\mathrm{m}_{\mathrm{x}}-\mathrm{m}_{\mathrm{y}}=\mathrm{M}_{\mathrm{x}}-\mathrm{M}_{\mathrm{y}}+\mathrm{m}_{\mathrm{e}}}$
<br/>$\mathrm{E}=\Delta \mathrm{mC}^{2}=\left(\mathrm{m}_{\mathrm{x}}-\mathrm{m}_{\mathrm{y}}-\mathrm{m}_{\mathrm{e}}\right) \mathrm{C}^{2}=\left(\mathrm{M}_{\mathrm{x}}-\mathrm{M}_{\mathrm{y}}\right) \mathrm{C}^{2}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P1.db
title:
a-pair-of-parallel-metal-plates-are-kept-with-a-separation-d-one-plate-is-at-a-potential-v-and-the-other-is-at-ground-potential-a-narrow-beam-of-elect

slug:
a-pair-of-parallel-metal-plates-are-kept-with-a-separation-d-one-plate-is-at-a-potential-v-and-the-other-is-at-ground-potential-a-narrow-beam-of-elect-54169

question:
<div class="question">A pair of parallel metal plates are kept with a separation 'd'. One plate is at a potential $+\mathrm{V}$ and the other is at ground potential. A narrow beam of electrons enters the space between the plates with a velocity $\mathrm{v}_{0}$ and in a direction parallel to the plates. What will be the angle of the beam with the plates after it travels an axial distance L?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan ^{-1}\left(\frac{\mathrm{eVL}}{\mathrm{mdv}_{0}}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\tan ^{-1}\left(\frac{\mathrm{eVL}}{\mathrm{mdv}_{0}^{2}}\right)$</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">$\sin ^{-1}\left(\frac{\mathrm{eVL}}{\mathrm{mdv}_{0}}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\cos ^{-1}\left(\frac{\mathrm{eVL}}{\mathrm{mdv}_{0}^{2}}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1}\left(\frac{\mathrm{eVL}}{\mathrm{mdv}_{0}^{2}}\right)$</span> </div>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/n6u7oGksKJ8wCw4i2_GEQ5wHinsQ04ol5HHpcE5uc90.original.fullsize.png"/><br/> <br/>$t=\frac{L}{v_{0}}$
<br/>$v_{y}=\frac{e}{m} \cdot \frac{v}{d} \times \frac{L}{v_{0}}=\frac{e v L}{m d v_{0}}$
<br/>$\tan \theta=\frac{v_{y}}{v_{x}}=\frac{e v L}{m d v_{0} \cdot v_{0}}$
<br/>$\Rightarrow \theta=\tan ^{-1}\left(\frac{e v L}{m d v_{0}^{2}}\right)$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P1.db
title:
a-parallel-beam-of-light-is-incident-on-a-glass-prism-in-the-shape-of-a-quarter-cylinder-of-radius-r-005-m-and-refractive-index-n-15-placed-on-a-horiz

slug:
a-parallel-beam-of-light-is-incident-on-a-glass-prism-in-the-shape-of-a-quarter-cylinder-of-radius-r-005-m-and-refractive-index-n-15-placed-on-a-horiz-39031

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_q21.png"/> <br/>A parallel beam of light is incident on a glass prism in the shape of a quarter cylinder of radius $R=0.05 \mathrm{m}$ and refractive index $n=1.5,$ placed on a horizontal table as shown in the figure. Beyond the cylinder, a patch of light is found whose the nearest distance $x$ from the cylinder is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(3 \sqrt{5}-5) \times 10^{-2} \mathrm{m}$</span> </div>

answer:
<div class="solution">Given, radius $R=0.05 \mathrm{m}$ 
<br/>Refractive index $n=1 \cdot 5$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a21.png"/> <br/> <br/>The critical angle, $\sin c=\frac{1}{n}=\frac{1}{1 \cdot 5}$
<br/>$$
<br/>=\frac{10}{15}=\frac{2}{3}
<br/>$$
<br/>From above figure, $\cos c=\frac{R}{R+X}$
<br/>$\sqrt{1-\sin ^{2} c}=\frac{R}{R+X}$
<br/>$\sqrt{1-\frac{4}{9}}=\frac{R}{R+X}$
<br/>$\therefore$
<br/>$$
<br/>\begin{aligned}
<br/>\frac{\sqrt{5}}{3} &amp;=\frac{R}{R+X} \\
<br/>\sqrt{5}(R+X) &amp;=3 R \\
<br/>\sqrt{5} R+\sqrt{5} X &amp;=3 R \\
<br/>X &amp;=(3 \sqrt{5}-5) \times 10^{-2} \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Alternating Current', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-parallel-plate-capacitor-in-series-with-a-resistance-of-100-an-inductor-of-20-mh-and-an-ac-voltage-source-of-variable-frequency-shows-resonance-at-a

slug:
a-parallel-plate-capacitor-in-series-with-a-resistance-of-100-an-inductor-of-20-mh-and-an-ac-voltage-source-of-variable-frequency-shows-resonance-at-a-33779

question:
<div class="question">A parallel plate capacitor in series with a resistance of $100 \Omega$, an inductor of $20 \mathrm{mH}$ and an AC voltage source of variable frequency shows resonance at a frequency of $\frac{1250}{\pi}$ Hz If this capacitor is charged by a DC voltage source to a voltage 25 v, what amount of charge will be stored in cach plate of the capacitor?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.2 \mu C$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{mC}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.2 \mathrm{mC}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$0.2 \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.2 \mathrm{mC}$</span> </div>

answer:
<div class="solution">In given, Series $R-L-C$ circuit Resistance $R=100 \Omega$
<br/>Inductance of Inductor, $L=20 \mathrm{mH}=20 \times 10^{-3} \mathrm{H}$ 
<br/>Resonance frequency $f=\frac{1250}{\pi} \mathrm{Hz}$ 
<br/>Source voltage $V_{D C}=25 \mathrm{V}$ 
<br/>According to resonant frequency.
<br/>$$
<br/>\omega_{0}=2 \pi f_{0}=\frac{1}{\sqrt{L C}}
<br/>$$
<br/>or $\left(2 \pi f_{0}\right)^{2}=\frac{1}{L C}$
<br/>or $\quad 4 \pi^{2} \frac{1250 \times 1250}{\pi \times \pi}=\frac{1}{L C}$
<br/>Or
<br/>$$
<br/>\begin{aligned}
<br/>C=&amp; \frac{1000}{1250 \times 1250 \times 4 \times 20} \\
<br/>&amp;\left(\because \mathrm{By} \text { substituting } L=20 \times 10^{-3}\right) \\
<br/>C=&amp; 8 \times 10^{-6} \mathrm{F}
<br/>\end{aligned}
<br/>$$
<br/>or
<br/>We know that, Charge $Q_{s}=C V$
<br/>$$
<br/>Q_{s}=8 \times 10^{-6} \times 25=0.2 \mathrm{mC}
<br/>$$
<br/>So, the amount of charge stored in each plate of capacitor is $0.2 \mathrm{mC}$.</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-parallel-plate-capacitor-is-charged-and-then-disconnected-from-the-charging-battery-if-the-plates-are-now-moved-farther-apart-by-pulling-at-them-by-

slug:
a-parallel-plate-capacitor-is-charged-and-then-disconnected-from-the-charging-battery-if-the-plates-are-now-moved-farther-apart-by-pulling-at-them-by-97189

question:
<div class="question">A parallel plate capacitor is charged and then disconnected from the charging battery. If the plates are now moved farther apart by pulling at them by means of insulating handles, then</div>

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

answer:
<div class="solution">We know that
<br/>The parallel plate capacitor $C=\frac{\varepsilon_{0} \cdot A}{d}$
<br/>When $d$ (distance between two parallel plate) increase, then $C$ will be decrease
<br/>$\because$
<br/>$Q=C V \quad$ where $Q=$ constant
<br/>$\therefore$ The voltage across the capacitor increases.</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P1.db
title:
a-parent-nucleus-x-undergoes-decay-with-a-halflife-of-75000-yrs-the-daughter-nucleus-y-undergoes-decay-with-a-halflife-of-9-months-in-a-particular-sam

slug:
a-parent-nucleus-x-undergoes-decay-with-a-halflife-of-75000-yrs-the-daughter-nucleus-y-undergoes-decay-with-a-halflife-of-9-months-in-a-particular-sam-75367

question:
<div class="question">A parent nucleus $X$ undergoes $\alpha$ -decay with a half-life of 75000 yrs. The daughter nucleus $Y$ undergoes $\beta$ -decay with a half-life of 9 months. In a particular sample, it is found that the rate of emission of $\beta$ -particles is nearly constant (over several months) at $10^{7} / \mathrm{h} .$ What will be the number of $\alpha$ -particles emitted in an hour?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$10^{7}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10^{12}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10^{14}$</span> </li> </ul>

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

answer:
<div class="solution">According to the question,<br/>$X \stackrel{\text { a-particle }}{\longrightarrow} Y \longrightarrow Z(\text { say) }$<br/>According to Rutherford Soddy law of radioactive decay, the rate of decay of radioactive along at any instant is proportional to the number of atoms present at that instant<br/>$\frac{d N}{d t}=-\lambda N$<br/>since, $\lambda=$ constant (decay constant) and $N=$ number of atoms<br/>Rate of decay of $Y$ particle is given as,<br/>$\lambda_{X} N_{X}-\lambda_{\gamma} N_{Y}=\frac{d N_{Y}}{d t}=0$<br/>[ $\because$ Decay rate for $\beta$ -particle become constant after some time]<br/>Given, rate of emission of $\beta$ -particle $=10^{7} \mathrm{h}$ $\lambda_{X} N_{X}=\lambda_{\gamma} N_{Y}=10^{7} / \mathrm{h}$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-acted-upon-by-a-constant-power-then-which-of-the-following-physical-quantity-remains-constant

slug:
a-particle-is-acted-upon-by-a-constant-power-then-which-of-the-following-physical-quantity-remains-constant-79249

question:
<div class="question">A particle is acted upon by a constant power. Then, which of the following physical quantity remains constant?</div>

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

answer:
<div class="solution">We have Power $=$ Rate of doing work $=\frac{d W}{d t}$
<br/>or, $\quad P=\frac{d W}{d t}=$ rate of change in $\mathrm{KE}$
<br/>i.e., $\quad P=\frac{d W}{d t}=\frac{d(\mathrm{KE})}{d t}=$ constant</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-executing-linear-simple-harmonic-motion-of-amplitude-a-at-what-displacement-is-the-energy-of-the-particle-half-potential-and-half-kineti

slug:
a-particle-is-executing-linear-simple-harmonic-motion-of-amplitude-a-at-what-displacement-is-the-energy-of-the-particle-half-potential-and-half-kineti-29021

question:
<div class="question">A particle is executing linear simple harmonic motion of amplitude A. At what displacement is the energy of the particle half potential and half kinetic?</div>

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

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

answer:
<div class="solution">Hints : Total Energy $(\mathrm{E})=\frac{1}{2} m \omega^2 \mathrm{~A}^2$<br/>P.E. $=\frac{1}{2} m \omega^2 x^2$<br/>As P.E. $=\frac{E}{2}$<br/>Then, $\frac{1}{2} m \omega^2 \mathrm{~A}^2 \times \frac{1}{2}=\frac{1}{2} m \omega^2 x^2$<br/>$$<br/>x^2=\frac{\mathrm{A}^2}{2} \Rightarrow x=\frac{\mathrm{A}}{\sqrt{2}}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-in-motion-along-a-curve-12-y-x-3-the-rate-of-change-of-its-ordinate-exceeds-that-of-abscissa-in

slug:
a-particle-is-in-motion-along-a-curve-12-y-x-3-the-rate-of-change-of-its-ordinate-exceeds-that-of-abscissa-in-49038

question:
<div class="question">A particle is in motion along a curve $12 y=x^{3}$. The rate of change of its ordinate exceeds that of abscissa in</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$x    &lt; -2$, $x&gt;2$</span> </div>

answer:
<div class="solution">Given, curve, $12 y=x^{3}$
<br/>and $\frac{d y}{d t}&gt;\frac{d x}{d t}$ ....(i)
<br/>Now, $12 \frac{d y}{d t}=3 x^{2} \frac{d x}{d t}$....(ii)
<br/>From Eqs. (i) and (ii), we get
<br/>$\Rightarrow$
<br/>$$
<br/>\begin{aligned}
<br/>3 x^{2} \frac{d x}{d t} &amp;&gt;12 \frac{d x}{d t} \\
<br/>3 x^{2} &amp;&gt;12
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow \quad x^{2}-4&gt;0$
<br/>$\begin{array}{ll}\Rightarrow &amp; (x-2)(x+2)&gt;0 \\ \Rightarrow &amp; x \in(-\infty,-2) \cup(2, \infty)\end{array}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-moving-in-a-straight-line-at-time-t-the-distance-between-the-particle-from-its-starting-point-is-given-by-x-t-6-t-z-its-acceleration-wil

slug:
a-particle-is-moving-in-a-straight-line-at-time-t-the-distance-between-the-particle-from-its-starting-point-is-given-by-x-t-6-t-z-its-acceleration-wil-65845

question:
<div class="question">A particle is moving in a straight line. At time $t$, the distance between the particle from its starting point is given by $x=t-6 t+z$. Its acceleration will be zero at</div>

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

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

answer:
<div class="solution">Hints : $x=t-6 t^2+t^3$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{d x}{d t}=1-12 t+3 t^2 \\<br/>&amp; \frac{d^2 x}{d t^2}=-12+6 t<br/>\end{aligned}<br/>$$<br/>Acceleration $=\frac{d^2 x}{d t^2}$<br/>$\therefore$ Acceleration $=0 \Rightarrow 6 t-12=0 \Rightarrow t=2$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-moving-in-x-y-plane-according-to-r-b-cos-t-i-b-sin-t-j-where-is-constant-which-of-the-following-statements-is-are-true

slug:
a-particle-is-moving-in-x-y-plane-according-to-r-b-cos-t-i-b-sin-t-j-where-is-constant-which-of-the-following-statements-is-are-true-64597

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{E}{\omega}$ is a constant where $E$ is the total energy of the particle, The trajectory of the particle in $x-y$ plane is a circle</span> </div>

answer:
<div class="solution">$\vec{r}=b \cos \omega t i+b \sin \omega t j$<br/>$\vec{v}=\frac{d \vec{r}}{d t}=-b \omega \sin \omega t+b \omega \cos \omega t j$<br/>$(\vec{v})=\sqrt{b^2 \omega^2 \sin ^2 \omega t+b^2 \omega^2 \cos ^2 \omega t}$<br/>$(\vec{v})=b \omega \rightarrow$ constant $\quad \therefore E=\frac{1}{2} m v^2=$ constant<br/>$E=\frac{1}{2} m\left[b^2 \omega^2\right] \quad \Rightarrow \frac{E}{\omega}=\frac{1}{2} m b^2 \cdot \omega=$ constant<br/>$x=b \cos \omega t$<br/>$y=b \sin \omega t$<br/>$x^2+y^2=b^2 \rightarrow$ equation of circle<br/>$\therefore \vec{a} \perp \vec{v} \quad$ So $\vec{a} \neq \omega^2 \vec{v}$</div>


tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-moving-uniformly-in-a-circular-path-of-radius-r-when-it-moves-through-an-angular-displacement-then-the-magnitude-of-the-corresponding-li

slug:
a-particle-is-moving-uniformly-in-a-circular-path-of-radius-r-when-it-moves-through-an-angular-displacement-then-the-magnitude-of-the-corresponding-li-61877

question:
<div class="question">A particle is moving uniformly in a circular path of radius $r$. When it moves through an angular displacement $\theta$, then the magnitude of the corresponding linear displacement will be</div>

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

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

answer:
<div class="solution">$\ln \Delta A O B \quad \sin \frac{\theta}{2}=\frac{A B}{A O}$<br/>$\Leftrightarrow A O=r)$<br/>$\begin{aligned}<br/>A B &amp;=A O \sin \frac{\theta}{2} \Rightarrow A B=r \sin \frac{\theta}{2} \\<br/>A C &amp;=A B+B C \quad(\because A B=B C) \\<br/>&amp;=r \sin \frac{\theta}{2}+r \sin \frac{\theta}{2} \\<br/>A C &amp;=2 r \sin \frac{\theta}{2}<br/>\end{aligned}$<br/>So, the magnitude of the corresponding linear displacement will be $2 r \sin \frac{\theta}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a21.png"/></div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P1.db
title:
a-particle-is-moving-with-a-constant-speed-v-in-a-circle-what-is-the-magnitude-of-average-velocity-after-half-rotation

slug:
a-particle-is-moving-with-a-constant-speed-v-in-a-circle-what-is-the-magnitude-of-average-velocity-after-half-rotation-11986

question:
<div class="question">A particle is moving with a constant speed \(v\) in a circle. What is the magnitude of average velocity after half rotation?</div>

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

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lYDq1PNJJAxgvcltZnUHl_YYnj2jqZlgVEgdqbfkHeM.original.fullsize.png"/><br/><br/>\(\mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{V}} ; \mathrm{t}_{\mathrm{o}}=\frac{\mathrm{T}}{2}=\frac{\pi \mathrm{r}}{\mathrm{V}} ; \quad \mathrm{V}_{\mathrm{av}}=\frac{2 \mathrm{r}}{\frac{\pi \mathrm{r}}{\mathrm{v}}}=\frac{2 \mathrm{v}}{\pi}\)</div>


