# questions

tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-correct-stability-order-of-the-following-carbocations-is-undersettextimathrmh2-stackreloplusmathrmcmathrmchmathrmchmathrmch3-undersettextiistackre

slug:
the-correct-stability-order-of-the-following-carbocations-is-undersettextimathrmh2-stackreloplusmathrmcmathrmchmathrmchmathrmch3-undersettextiistackre-40826

question:
<div class="question">The correct stability order of the following carbocations is<br/>\(\underset{\text{I}}{\mathrm{H}_2 \stackrel{\oplus}{\mathrm{C}}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}_3}\)<br/>\(\underset{\text{II}}{\stackrel{\oplus}{\mathrm{C}} \mathrm{H}_2-\mathrm{CH}=\mathrm{CH}-\mathrm{BMe}_2}\)<br/>\(\underset{\text{III}}{\mathrm{H}_2 \stackrel{\oplus}{\mathrm{C}}-\mathrm{CH}=\mathrm{CH}-\mathrm{NMe}_2}\)<br/>\(\underset{\text{Iv}}{\mathrm{H}_2 \stackrel{\oplus}{\mathrm{C}}-\mathrm{CH}=\mathrm{CH}-\mathrm{OMe}}\)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">III \( &gt; \) IV \( &gt; \) I \( &gt; \) II</span> </div>

answer:
<div class="solution">Hint: \(\quad\) III \( &gt; \) IV \( &gt; | &gt; | \mid\)<br/>\(\left.+\mathrm{R} \text { effect : }-\ddot{\mathrm{N}}\mathrm{Me}_2\right\rangle-\underset{\cdot\cdot}{\ddot{\mathrm{O}}} \mathrm{CH}_3\)<br/>\(-\mathrm{BMe}_2\) can show \(-\mathrm{R}\) effect</div>


tags: ['Chemistry', 'Coordination Compounds', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-correct-statement-about-the-magnetic-properties-of-leftmathrmfemathrmcn6right3-and-leftmathrmfef6right3-is

slug:
the-correct-statement-about-the-magnetic-properties-of-leftmathrmfemathrmcn6right3-and-leftmathrmfef6right3-is-68177

question:
<div class="question">The correct statement about the magnetic properties of \(\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}\) and \(\left[\mathrm{FeF}_6\right]^{3-}\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Both are paramagnetic</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Both are diamagnetic</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}\) is diamagnetic, \(\left[\mathrm{FeF}_6\right]^{3-}\) is paramagnetic</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}\) is paramagnetic, \(\left[\mathrm{FeF}_6\right]^{3-}\) is diamagnetic</span> </li> </ul>

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

answer:
<div class="solution">Hint : \(\mathrm{Fe}^{+3}=[\mathrm{Ar}] 3 \mathrm{~d}^5 4 \mathrm{~s}^0\)<br/>For \(\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mcTaiB2qxFfovS6oaQnnQ56ArFZEecESrtnfCaH7qtY.original.fullsize.png"/><br/><br/>Pairing of \(\mathrm{e}^{-}\)takes place as \(\mathrm{CN}^{-}\)is strong field ligand but has one unpaired electron thus paramagnetic.<br/>For \(\left[\mathrm{FeF}_6\right]^{-3}\)<br/>As \(\mathrm{F}^{\text {- }}\) is weak field ligand, so no pairing of electron, thus it has five unpaired electron. Therefore paramagnetic.</div>


title: the-correct-statement-regarding-the-following-compounds-is
slug: the-correct-statement-regarding-the-following-compounds-is-77226
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct statement regarding the following compounds is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q20.png"/></div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a20.png"/> <br/> <br/>$2-R, 3-R$ configuration, s0 it is a chiral compound<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a20a.png"/> <br/> <br/>$2-R, 3-S$ configuration has plane of symmetry or in other words, rotation of hall part is cancelled by the other half, so it is not a chiral compound.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a20b.png"/> <br/> <br/>$2-S, 3-S$ configuration, so it is also a chiral compound.</div>


title: the-correct-statement-regarding-the-following-energy-diagrams-is
slug: the-correct-statement-regarding-the-following-energy-diagrams-is-28295
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct statement regarding the following energy diagrams is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q21.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Reaction $M$ is faster and more exothermic than reaction $N$</span> </div>

answer:
<div class="solution">In reaction $M$ there is only an intermediate step. $\$ 0$ it is a fast reaction. Further, in reaction $M$ energy of product is much lesser than that of the reactant, as compared to as shown in reaction $N$. Thus reaction $M$ is faster and more exothermic than reaction $N$.</div>


title: the-correct-statements-about-b-2-h-6-is-are
slug: the-correct-statements-about-b-2-h-6-is-are-82331
question: <div class="question">The correct statement(s) about $\mathrm{B}_2 \mathrm{H}_6$ is / are :</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">All B atoms are $\mathrm{sp}^3$ hybridised</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">It is paramagnetic</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">It contains 3C - 4e bonding</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">There are two types of $\mathrm{H}$ present</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">All B atoms are $\mathrm{sp}^3$ hybridised, There are two types of $\mathrm{H}$ present</span> </div>

answer:
<div class="solution">In diborane, all boron atoms are $\mathrm{sp}^3$ hybridised. It is diamagnetic and contains $3 \mathrm{C}-2 \mathrm{e}$ bonds. There are two types of hydrogens $-$ terminal and bridging.</div>


title: the-correct-structure-of-the-dipeptide-glyala-is-1
slug: the-correct-structure-of-the-dipeptide-glyala-is-1-18479
question: <div class="question">The correct structure of the dipeptide gly-ala is</div>
tags: ['Chemistry', 'Biomolecules', 'JEE Main']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q43_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q43_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q43_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q43_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q43_optc.png"/></span> </div>

answer:
<div class="solution">The structure of dipeptide gly-ala is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a43.png"/></div>


title: the-correct-structure-of-the-drug-paracetamol-is-1
slug: the-correct-structure-of-the-drug-paracetamol-is-1-29043
question: <div class="question">The correct structure of the drug paracetamol is</div>
tags: ['Chemistry', 'Chemistry in Everyday Life', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q8_opta.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q8_optb.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q8_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q8_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q8_optb.png"/></span> </div>

answer:
<div class="solution">Paracetamol is 4 -acetamidophenol, i.e. N-(4-hydroxy phenyl) ethanamide (IUPAC Name).
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_a8.png"/></div>


title: the-cosine-of-the-angle-between-any-two-diagonals-of-a-cube-is
slug: the-cosine-of-the-angle-between-any-two-diagonals-of-a-cube-is-74062
question: <div class="question">The cosine of the angle between any two diagonals of a cube is</div>
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{\sqrt{3}}$</span> </li> </ul>

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

answer:
<div class="solution">Let the direction ratio's of two diagonals of a cube are
<br/>(1,1,1) and (-1,1,1) $\begin{aligned} \therefore \cos \theta &amp;=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{f} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} \\ &amp;=\frac{(1) \times(-1)+(1) \times(1)+(1) \times(1)}{\sqrt{(1)^{2}+(1)^{2}+(1)^{2}} \sqrt{(-1)^{2}+(1)^{2}+(1)^{2}}} \\ &amp;=\frac{-1+1+1}{\sqrt{3} \sqrt{3}}=\frac{1}{3} \end{aligned}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
the-crosssection-of-a-reflecting-surface-is-represented-by-the-equation-x-2-y-2-r-2-as-shown-in-the-figure-a-ray-travelling-in-the-positive-x-directio

slug:
the-crosssection-of-a-reflecting-surface-is-represented-by-the-equation-x-2-y-2-r-2-as-shown-in-the-figure-a-ray-travelling-in-the-positive-x-directio-32064

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8aouvsiaeCfGr0IjLBnm3rywBWz34qJ7Wt7Du53oAuo.original.fullsize.png"/><br/> <br/>The cross-section of a reflecting surface is represented by the equation $x^{2}+y^{2}=R^{2}$ as shown in the figure. A ray travelling in the positive $x$ direction is directed toward positive y direction after reflection from the surface at point $M$. The coordinate of the point $\mathrm{M}$ on the reflecting surface is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fgD1Exzuy93lV8IMDTH8m3XsIeZaAx747bo-QqtXg9Y.original.fullsize.png"/> <br/>$x^{2}+y^{2}=R^{2}$
<br/>$2 x+2 y \frac{d y}{d x}=0$
<br/>$\frac{d y}{d x}=-\frac{x}{y}=\tan 45^{\circ}=1$
<br/>$\therefore x=-\frac{R}{\sqrt{2}}$
<br/>$y=+\frac{R}{\sqrt{2}}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">$Q=\int_0^2\left(3 t^2+2 t+5\right) d t=22 C$</div>
source_db: db\MarksBatch2_P2.db
title:
the-current-flowing-through-a-wire-depends-on-time-as-i-3-t-2-2-t-5-the-charge-flowing-through-the-cross-section-of-the-wire-in-time-from-t-0-to-t-2-s

slug:
the-current-flowing-through-a-wire-depends-on-time-as-i-3-t-2-2-t-5-the-charge-flowing-through-the-cross-section-of-the-wire-in-time-from-t-0-to-t-2-s-89282

question:
<div class="question">The current flowing through a wire depends on time as $I=3 t^2+2 t+5$. The charge flowing through the cross section of the wire in time from $t=0$ to $t=2 \mathrm{sec}$. is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$22 \mathrm{C}$</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">$20 \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$18 \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$22 \mathrm{C}$</span> </div>


title: the-current-i-is-in-the-circuit-shown-is-1
slug: the-current-i-is-in-the-circuit-shown-is-1-23718
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The current $I$ is in the circuit shown is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q8.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1.33 \mathrm{A}$</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">B</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.00 \mathrm{A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.00 \mathrm{A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.33 \mathrm{A}$</span> </div>

answer:
<div class="solution">The circuit diagram can be redrawn as 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a8.png"/> <br/> <br/>For the loop $A B C D A$,
<br/>$$
<br/>+2-2 I^{\prime}+2-2 I=0
<br/>$$
<br/>For the loop $A B F E A$,
<br/>$$
<br/>\begin{array}{r}
<br/>2-2 I+2-2\left(I-I^{\prime}\right)=0 \\
<br/>4-2 I-2 I+2 I^{\prime}=0
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>4-4 I+2 I^{\prime} &amp;=0 \\
<br/>2 &amp;=2 I-I^{\prime}
<br/>\end{aligned}
<br/>$$
<br/>From Eqs (i) and (in), we get
<br/>$$
<br/>2=2 I-I^{\prime} \cdot 2=I+I^{\prime}
<br/>$$
<br/>$$
<br/>4=3 I, I=4 / 3=1.33 \mathrm{A}
<br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-current-through-a-coil-of-self-inductance-l-2-m-h-is-given-by-i-t-2-e-t-at-time-t-how-long-it-will-take-to-make-the-e-m-f-zero

slug:
the-current-through-a-coil-of-self-inductance-l-2-m-h-is-given-by-i-t-2-e-t-at-time-t-how-long-it-will-take-to-make-the-e-m-f-zero-35078

question:
<div class="question">The current through a coil of self inductance $\mathrm{L}=2 m \mathrm{H}$ is given by $\mathrm{I}=t^2 e^{-t}$ at time $t$. How long it will take to make the e $\mathrm{m} . f$ zero?</div>

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

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

answer:
<div class="solution">Hints: $\mathrm{I}=t^2 e^{-t}$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{d \mathrm{I}}{d t}=2 t e^{-t}-e^{-t} t^2=e^{-t} t(2-t) \\<br/>&amp; e=-\mathrm{L} \frac{d \mathrm{I}}{d t} \\<br/>&amp; \Rightarrow \frac{d \mathrm{I}}{d t}=0 \Rightarrow e^{-t} t(2-t)=0 \\<br/>&amp; t=2 \mathrm{sec}<br/>\end{aligned}<br/>$$</div>


title: the-curve-y-cos-x-y-12-satisfies-the-differential-equation
slug: the-curve-y-cos-x-y-12-satisfies-the-differential-equation-37030
question: <div class="question">The curve $y=(\cos x+y)^{1 / 2}$ satisfies the differential equation</div>
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(2 y-1) \frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}+\cos x=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}-2 y\left(\frac{d y}{d x}\right)^{2}+\cos x=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(2 y-1) \frac{d^{2} y}{d x^{2}}-2\left(\frac{d y}{d x}\right)^{2}+\cos x=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(2 y-1) \frac{d^{2} y}{d x^{2}}-\left(\frac{d y}{d x}\right)^{2}+\cos x=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(2 y-1) \frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}+\cos x=0$</span> </div>

answer:
<div class="solution">Given curve is
<br/>$$
<br/>y=(\cos x+y)^{1 / 2}
<br/>$$
<br/>On differentiating both sides w.r.t. $x,$ we get
<br/>$$
<br/>\begin{aligned}
<br/>\frac{d y}{d x} &amp;=\frac{1}{2(\cos x+y)^{1 / 2}} \times\left(-\sin x+\frac{d y}{d x}\right) \\
<br/>&amp;=\frac{\left(-\sin x+\frac{d y}{d x}\right)}{2 y} \\
<br/>\Rightarrow \quad &amp; 2 y \frac{d y}{d x}=-\sin x+\frac{d y}{d x} \\
<br/>\Rightarrow \quad(2 y-1) \frac{d y}{d x} &amp;=-\sin x
<br/>\end{aligned}
<br/>$$
<br/>Again, differentiating both sides w.r.t. $x$, we get
<br/>$$
<br/>\begin{array}{r}
<br/>(2 y-1) \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(2 \frac{d y}{d x}\right)=-\cos x \\
<br/>\Rightarrow \quad(2 y-1) \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(2 \frac{d y}{d x}\right)+\cos x=0
<br/>\end{array}
<br/>$$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
the-debroglie-wavelength-for-electron-e-proton-p-and-he-2-ion-are-in-the-following-order-speed-of-e-p-and-are-the-same

slug:
the-debroglie-wavelength-for-electron-e-proton-p-and-he-2-ion-are-in-the-following-order-speed-of-e-p-and-are-the-same-67372

question:
<div class="question">The de-Broglie wavelength $(\lambda)$ for electron (e), proton $(p)$ and $\mathrm{He}^{2+}$ ion $(\alpha)$ are in the following order. (Speed of e, p and $\alpha$ are the same)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$e&gt;p&gt;\alpha$</span> </div>

answer:
<div class="solution">$\lambda=\frac{\mathrm{h}}{\mathrm{mv}}$<br/>$\because$ speed are same<br/>$\begin{aligned}<br/>&amp;\therefore \lambda \propto \frac{1}{\mathrm{~m}} \\<br/>&amp;\therefore \lambda_e&gt;\lambda_p&gt;\lambda_\alpha \text { as } \quad \mathrm{m}_\alpha&gt;\mathrm{m}_{\mathrm{p}}&gt;\mathrm{m}_e<br/>\end{aligned}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
the-debroglie-wavelength-of-an-electron-is-04-1-0-10-m-when-its-kinetic-energy-is-10-kev-its-wavelength-will-be-10-1-0-10-m-when-its-kinetic-energy-is

slug:
the-debroglie-wavelength-of-an-electron-is-04-1-0-10-m-when-its-kinetic-energy-is-10-kev-its-wavelength-will-be-10-1-0-10-m-when-its-kinetic-energy-is-71327

question:
<div class="question">The de-Broglie wavelength of an electron is $0.4 \times 10^{-10} \mathrm{m}$ when its kinetic energy is $1.0 \mathrm{keV} .$ Its wavelength will be $1.0 \times 10^{-10} \mathrm{m}$
<br/>when its kinetic energy is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.2 \mathrm{keV}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.8 \mathrm{keV}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.63 \mathrm{keV}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.16 \mathrm{keV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.16 \mathrm{keV}$</span> </div>

answer:
<div class="solution">De-Broglie wavelength, $\lambda=\frac{h}{\sqrt{2 m E}}$
<br/>$\therefore \quad \frac{\lambda_{1}}{\lambda_{2}}=\sqrt{\frac{E_{2}}{E_{1}}}$
<br/>$\lambda_{1}=$ Wave length of electron
<br/>$\begin{aligned} &amp;=0.4 \times 10^{-10} \mathrm{m} \\ E_{1} &amp;=1 \cdot \text { OkeV } \\ \lambda_{2} &amp;=1 \cdot 0 \times 10^{-10} \mathrm{m} \\ E_{2} &amp;=? \\ \frac{0 \cdot 4 \times 10^{-10}}{1 \times 10^{-10}} &amp;=\sqrt{\frac{E_{2}}{1 \cdot 0 \mathrm{keV}}} \\ \frac{4}{10}=&amp; \sqrt{\frac{E_{2}}{1}} \\ \frac{16}{100}=&amp; \frac{E_{2}}{1} \\ E_{2}=&amp; 0 \cdot 16 \mathrm{keV} \end{aligned}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20: 1</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-debroglie-wavelength-of-an-electron-is-the-same-as-that-of-a-50-kev-xray-photon-the-ratio-of-the-energy-of-the-photon-to-the-kinetic-energy-of-the

slug:
the-debroglie-wavelength-of-an-electron-is-the-same-as-that-of-a-50-kev-xray-photon-the-ratio-of-the-energy-of-the-photon-to-the-kinetic-energy-of-the-95816

question:
<div class="question">The de-Broglie wavelength of an electron is the same as that of a $50 \mathrm{keV}$ X-ray photon. The ratio of the energy of the photon to the kinetic energy of the electron is (the energy equivalent of electron mass is $0.5 \mathrm{MeV}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">01:50</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">01:20</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">20: 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">50: 1</span> </li> </ul>

answer:
<div class="solution">de-Broglie wavelength $\lambda=\frac{h}{\sqrt{2 m K}}$
<br/>The kinetic energy of the electron
<br/>$$
<br/>K_{\text {electron }}=\frac{1}{2 m} \cdot \frac{h^{2}}{\lambda^{2}}
<br/>$$
<br/>Where $h=$ Planck constant $\lambda=$ wavelength
<br/>The photon energy
<br/>$$
<br/>E_{\text {phocton }}=\frac{h c}{\lambda}
<br/>$$
<br/>From Eqs. (ii) and (i), we get
<br/>$\frac{E_{\text {photon }}}{K_{\text {electron }}}=\frac{h c / \lambda}{h^{2} / 2 m \cdot \lambda^{2}}=\frac{h c \cdot \lambda^{2} \times 2 m}{h^{2} \cdot \lambda}$
<br/>where $m=0.5 \mathrm{MeV}=5 \times 10^{5} \mathrm{eV}$
<br/>$\begin{aligned} \frac{h}{\lambda c}=50 \times 10^{3} \mathrm{eV}=\frac{2 \mathrm{m} \lambda c}{h} \\=\frac{2 \mathrm{m}}{h / \lambda c} &amp;=\frac{2 \times 5 \times 10^{5}}{50 \times 10^{3}} \quad\left(\because \mathrm{m}=\frac{h}{c \lambda}\right) \\ &amp;=20: 1 \end{aligned}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-debroglie-wavelength-of-an-electron-mass-1-1-0-30-kg-charge-16-1-0-19-c-with-a-kinetic-energy-of-200-ev-is-plancks-constant-66-1-0-34-js

slug:
the-debroglie-wavelength-of-an-electron-mass-1-1-0-30-kg-charge-16-1-0-19-c-with-a-kinetic-energy-of-200-ev-is-plancks-constant-66-1-0-34-js-19763

question:
<div class="question">The de-Broglie wavelength of an electron (mass $=1 \times 10^{-30} \mathrm{kg},$ charge $\left.=16 \times 10^{-19} \mathrm{C}\right)$ with a kinetic energy of $200 \mathrm{eV}$ is (Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$9.60 \times 10^{-11} \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$8.25 \times 10^{-11} \mathrm{m}$</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">$6.25 \times 10^{-11} \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5.00 \times 10^{-11} \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8.25 \times 10^{-11} \mathrm{m}$</span> </div>

answer:
<div class="solution">The de-Broglie wavelength, $\lambda=\frac{h}{\sqrt{2 m k}}$
<br/>Given,
<br/>$$
<br/>h=6.6 \times 10^{-34} \mathrm{J}-\mathrm{s}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>m &amp;=1 \times 10^{-30} \mathrm{kg} \\
<br/>K=200 \mathrm{eV} &amp;=200 \times 1.6 \times 10^{-19} \mathrm{J}
<br/>\end{aligned}
<br/>$$
<br/>Substituting all these values
<br/>$$
<br/>\begin{aligned}
<br/>\lambda &amp;=\frac{6.6 \times 10^{-34}}{\sqrt{2 \times 1 \times 10^{-30} \times 200 \times 1.6 \times 10^{-19}}} \\
<br/>&amp;=0.825 \times 10^{-10}=8.25 \times 10^{-11} \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 2$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-debroglie-wavelength-of-an-electron-moving-with-a-velocity-c-2-c-velocity-of-light-in-vacuum-is-equal-to-the-wavelength-of-a-photon-the-ratio-of-t

slug:
the-debroglie-wavelength-of-an-electron-moving-with-a-velocity-c-2-c-velocity-of-light-in-vacuum-is-equal-to-the-wavelength-of-a-photon-the-ratio-of-t-94125

question:
<div class="question">The de-Broglie wavelength of an electron moving with a velocity $c / 2$ ( $c=$ velocity of light in vacuum) is equal to the wavelength of a photon. The ratio of the kinetic energies of electron and photon is</div>

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

answer:
<div class="solution">de-Broglie wavelength
<br/>$$
<br/>\lambda=\frac{h}{m v}
<br/>$$
<br/>Here, $\lambda_{e}=\frac{h}{m_{e} \frac{c}{2}}$ and $\lambda_{p}=\frac{h}{m_{p} c}$
<br/>Given, $\quad \lambda_{e}=\lambda_{B}$
<br/>So, $\quad \frac{h}{m_{e} \frac{c}{2}}=\frac{h}{m_{p} c}$
<br/>$\frac{m_{e}}{m_{p}}=2$
<br/>Ratio of KE
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{K_{e}}{K_{p}}=\frac{\frac{1}{2} m_{e} v_{e}^{2}}{\frac{1}{2} m_{p} v_{p}^{2}} \\
<br/>\frac{K_{e}}{K_{p}}=\frac{1}{2}
<br/>\end{array}
<br/>$$</div>


title: the-decimal-equivalent-of-the-binary-number-11010101-2-is
slug: the-decimal-equivalent-of-the-binary-number-11010101-2-is-99979
question: <div class="question">The decimal equivalent of the binary number $(11010.101)_2$ is</div>
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">26.625</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9.625</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">25.265</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">26.625</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">26.265</span> </li> </ul>

answer:
<div class="solution">$$<br/>\text { Hints : }(11010.101)=0 \times 2^0+1 \times 2^1+0 \times 2^2+1 \times 2^3+1 \times 2^4+1 \times 2^{-1}+0 \times 2^{-2}+1 \times 2^{-3}=2+8+16+\frac{1}{2}+\frac{1}{8}=26.625<br/>$$</div>


title: the-decimal-number-equivalent-to-a-binary-number-1011001-is
slug: the-decimal-number-equivalent-to-a-binary-number-1011001-is-20606
question: <div class="question">The decimal number equivalent to a binary number 1011001 is</div>
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">89</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">13</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">17</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">89</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">178</span> </li> </ul>

answer:
<div class="solution">Given, binary number $=1011001$ Its equivalent decimal number
<br/>$\begin{array}{l}
<br/>=2^{\circ}+0+0+2^{3}+2^{4}+2^{6} \\
<br/>=1+8+16+64=89
<br/>\end{array}$</div>


title: the-decreasing-order-of-basic-character-of-k-2-o-bao-cao-and-mgo-is
slug: the-decreasing-order-of-basic-character-of-k-2-o-bao-cao-and-mgo-is-66409
tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$$
<br/>\begin{aligned}
<br/>&amp;\text { The decreasing order of basic character of }\\
<br/>&amp;\mathrm{K}_{2} \mathrm{O}, \mathrm{BaO}, \mathrm{CaO} \text { and } \mathrm{MgO} \text { is }
<br/>\end{aligned}
<br/>$$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{K}_{2} \mathrm{O}&gt;\mathrm{BaO}&gt;\mathrm{CaO}&gt;\mathrm{Mg} \mathrm{O}$</span> </div>

answer:
<div class="solution">On the basis of electropositivity of metal, the order of basic character is
<br/>$$
<br/>\mathrm{K}_{2} \mathrm{O}&gt;\mathrm{BaO}&gt;\mathrm{CaO}&gt;\mathrm{MgO}
<br/>$$</div>


title: the-degree-and-order-of-the-differential-equation-yxleftfracd-yd-xright2fracd-yd-x-are-respectively
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4,1</span> </div>
source_db: db\MarksBatch2_P2.db
slug:
the-degree-and-order-of-the-differential-equation-yxleftfracd-yd-xright2fracd-yd-x-are-respectively-54583

question:
<div class="question">The degree and order of the differential equation \(y=x\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}\) are respectively</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">2,1</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">Hints: \(y\left(\frac{d y}{d x}\right)^2=x\left(\frac{d y}{d x}\right)^4+1\)</div>


title: the-degree-of-the-differential-equation-x-1-d-x-d-y-2-1-d-x-d-y-2-3-1-d-x-d-y-3
slug: the-degree-of-the-differential-equation-x-1-d-x-d-y-2-1-d-x-d-y-2-3-1-d-x-d-y-3-87202
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
answer: <div class="solution">Hints: $x=e \frac{d y}{d x}, \frac{d y}{d x}=\log _e x$</div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The degree of the differential equation $x=1+\left(\frac{d y}{d x}\right)+\frac{1}{2 !}\left(\frac{d y}{d x}\right)^2+\frac{1}{3 !}\left(\frac{d y}{d x}\right)^3+\ldots \ldots \ldots$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">not defined</span> </li> </ul>


tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-density-of-the-material-of-a-cube-can-be-estimated-by-measuring-its-mass-and-the-length-of-one-of-its-sides-if-the-maximum-error-in-the-measuremen

slug:
the-density-of-the-material-of-a-cube-can-be-estimated-by-measuring-its-mass-and-the-length-of-one-of-its-sides-if-the-maximum-error-in-the-measuremen-17414

question:
<div class="question">The density of the material of a cube can be estimated by measuring its mass and the length of one of its sides. If the maximum error in the measurement of mass and length are $0.3 \%$ and $0.2 \%$ respectively, the maximum error in the estimation of the density of the cube is approximately.</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">$0.5 \%$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.9 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.7 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.9 \%$</span> </div>

answer:
<div class="solution">Given,<br/>Maximum error in the measurement of mass $=0.3 \%$<br/>Maximum error in the measurement of length $=0.2 \%$<br/>We know that,<br/>Error in density is given as,<br/>$\text {Density, } \rho=\frac{\text { mass }(m)}{\text { volume }(V)}=\frac{m}{L^{3}}$<br/>where, $L=$ side of cube<br/>Error in density is given as,<br/>$\left(\frac{\Delta \rho}{\rho}\right)=\frac{\Delta m}{m}+\frac{3 \Delta L}{L}$<br/>or $\left(\frac{\Delta \rho}{\rho}\right) \times 100=\left(\frac{\Delta m}{m}+\frac{3 \Delta L}{L}\right) \times 100$<br/>Substituting the given values, we get<br/>$\left(\frac{\Delta \rho}{\rho}\right)_{\max }=(0.3 \%+3(0.2) \%=0.3 \%+0.6 \%$<br/>$\therefore$ Maximum percentage error measurement of density $\left(\frac{\Delta \rho}{\rho}\right)_{\max }=0.9 \%$</div>


title: the-determinant-a-2-10-ab-a-c-ab-b-2-10-b-c-a-c-b-c-c-2-10-is
slug: the-determinant-a-2-10-ab-a-c-ab-b-2-10-b-c-a-c-b-c-c-2-10-is-24968
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The determinant $\left|\begin{array}{ccc}a^{2}+10 &amp; a b &amp; a c \\ a b &amp; b^{2}+10 &amp; b c \\ a c &amp; b c &amp; c^{2}+10\end{array}\right|$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">divisible by 10 but not by 100</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">divisible by 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">not divisible by 100</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">not divisible by 10</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">divisible by 100</span> </div>

answer:
<div class="solution">$=\frac{1}{a b c}\left|\begin{array}{ccc}a\left(a^{2}+10\right) &amp; a^{2} b &amp; a^{2} c \\ a b^{2} &amp; b\left(b^{2}+10\right) &amp; b^{2} c \\ a c^{2} &amp; b c^{2} &amp; c\left(c^{2}+10\right)\end{array}\right|=\frac{a b c}{a b c}\left|\begin{array}{ccc}a^{2}+10 &amp; a^{2} &amp; a^{2} \\ b^{2} &amp; b^{2}+10 &amp; b^{2} \\ c^{2} &amp; c^{2} &amp; c^{2}+10\end{array}\right|$
<br/>$\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}$
<br/>$=\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}+10\right)\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ \mathrm{~b}^{2} &amp; \mathrm{~b}^{2}+10 &amp; \mathrm{~b}^{2} \\ \mathrm{c}^{2} &amp; \mathrm{c}^{2} &amp; \mathrm{c}^{2}+10\end{array}\right|$
<br/>$=\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}+10\right)\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ \mathrm{~b}^{2} &amp; 10 &amp; 0 \\ \mathrm{c}^{2} &amp; 0 &amp; 10\end{array}\right|\left[\begin{array}{c}\mathrm{C}_{2}-\mathrm{C}_{1} \\ \mathrm{C}_{3}-\mathrm{C} 1\end{array}\right]$
<br/>$=\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}+10\right) 100$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-difference-between-orbital-angular-momentum-of-an-electron-in-a-4-f-orbital-and-another-electron-in-a-4-s-orbital-is

slug:
the-difference-between-orbital-angular-momentum-of-an-electron-in-a-4-f-orbital-and-another-electron-in-a-4-s-orbital-is-69646

question:
<div class="question">The difference between orbital angular momentum of an electron in a 4 f orbital and another electron in a 4 s orbital is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \sqrt{3}$ $\times \frac{h}{(2 \times 3.14)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \sqrt{2}$ $\times \frac{h}{(2 \times 3.14)}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{3}$ $\times \frac{h}{(2 \times 3.14)}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2 $\times \frac{h}{(2 \times 3.14)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \sqrt{3}$ $\times \frac{h}{(2 \times 3.14)}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>Orbital angular momentum $=\sqrt{\ell(\ell+1)} \frac{\mathrm{h}}{2 \pi}$
<br/>For 4 f orbital; $\ell=3 \therefore$ Orbital angular momentum $=\sqrt{3(3+1)} \frac{\mathrm{h}}{2 \pi}=2 \sqrt{3} \frac{\mathrm{h}}{2 \pi}$
<br/>For 4 s orbital; $\ell=0 \therefore$ Orbital angular momentum $=0$
<br/>$\therefore$ Difference $=(2 \sqrt{3}) \frac{\mathrm{h}}{2 \pi}$</div>


title: the-different-colours-of-litmus-in-acidic-neutral-and-basic-solutions-are-respectively
slug: the-different-colours-of-litmus-in-acidic-neutral-and-basic-solutions-are-respectively-25262
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The different colours of litmus in acidic, neutral and basic solutions are, respectively.</div>

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

answer:
<div class="solution">\begin{array}{c|c}
<br/>\hline Medium &amp; Colour of Litmus Paper \\
<br/>\hline Acidic &amp; Red \\
<br/>Neutral &amp; Violet \\
<br/>Basic &amp; Blue \\
<br/>\hline
<br/>\end{array}</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
the-differential-equation-of-all-the-ellipses-centred-at-the-origin-and-have-axes-as-the-coordinate-axes-is

slug:
the-differential-equation-of-all-the-ellipses-centred-at-the-origin-and-have-axes-as-the-coordinate-axes-is-84002

question:
<div class="question">The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is</div>

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

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

answer:
<div class="solution">equation of ellipse centred at origin and have axes as the coordinate axes is
<br/>$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
<br/>$\Rightarrow \frac{x}{a^{2}}+\frac{y y^{\prime}}{b^{2}}=0$
<br/>$\Rightarrow \quad \frac{-b^{2}}{a^{2}}=\frac{y y^{\prime}}{x}$
<br/>$\Rightarrow \quad 0=\frac{x\left(y^{\prime \prime} y^{\prime}+y^{\prime 2}\right)-y y^{\prime}}{x^{2}}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/TRL9q82T8mG6OMzCByQdhN0DjP90b8FoX1C9VsCvDZM.original.fullsize.png"/></div>


title: the-differential-equation-of-mathrmymathrmaemathrmbx-a-mathrmb-are-parameters-is
slug: the-differential-equation-of-mathrmymathrmaemathrmbx-a-mathrmb-are-parameters-is-84863
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The differential equation of \(\mathrm{y}=\mathrm{ae}^{\mathrm{bx}}\) (a &amp; \(\mathrm{b}\) are parameters) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{yy}_1=\mathrm{y}_2^2\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{yy}_2=\mathrm{y}_1^2\)</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">\(\mathrm{yy}_1^2=\mathrm{y}_2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{yy}_2^2=\mathrm{y}_1\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{yy}_2=\mathrm{y}_1^2\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hints: } y=a \cdot e^{b x} \ldots \ldots \ldots \ldots \text { (i) } \\<br/>&amp; y_1=a b e^{b x} \\<br/>&amp; y_1=b y \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \text { (ii) }  \\<br/>&amp; y_2=b_1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \text { (iii) } \\<br/>&amp; \text { Dividing (ii) &amp; (iii) } \frac{y_1}{\mathrm{y}_2}=\frac{\mathrm{y}}{\mathrm{y}_1} \Rightarrow \mathrm{y}_1{ }^2=\mathrm{yy}_2<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-differential-equation-of-the-family-of-curves-y-e-x-a-cos-x-b-sin-x-where-a-b-are-arbitrary-constants-is

slug:
the-differential-equation-of-the-family-of-curves-y-e-x-a-cos-x-b-sin-x-where-a-b-are-arbitrary-constants-is-75260

question:
<div class="question">The differential equation of the family of curves $y=e^{x}(A \cos x+B \sin x)$ where $A, B$ are arbitrary constants is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}-9 x=13$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$</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">C</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}+3 y=4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}-x y=0$</span> </li> </ul>

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

answer:
<div class="solution">Hint: 
<br/>$y=e^{x}(A \cos x+B \sin x)$
<br/>Differentiating w.r.t. x:-
<br/>$y^{\prime}=y+e^{x}(-A \sin x+B \cos x)$
<br/>Differentiating w.r.t. $x$ once again:-
<br/>$\begin{array}{c}
<br/>y^{\prime \prime}=y^{\prime}+\left(y^{\prime}-y\right)+e^{x}(-A \cos x-B \sin x) \\
<br/>=2 y^{\prime}-y-y \quad \Rightarrow y^{\prime \prime}-2 y^{\prime}+2 y=0
<br/>\end{array}$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-differential-equation-representing-the-family-of-curves-y-2-2-d-x-d-where-d-is-a-parameter-is-of

slug:
the-differential-equation-representing-the-family-of-curves-y-2-2-d-x-d-where-d-is-a-parameter-is-of-53696

question:
<div class="question">The differential equation representing the family of curves $y^{2}=2 d(x+\sqrt{d})$ where $d$ is a parameter, is of</div>

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

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

answer:
<div class="solution">Given, $\quad y^{2}=2 d(x+\sqrt{d})$
<br/>$\Rightarrow$
<br/>2y $y_{1}=2 d \Rightarrow d=y y_{1}$
<br/>From Eq. (i),
<br/>$$
<br/>y^{2}=2 y y_{1}\left(x+\sqrt{y y_{1}}\right)
<br/>$$
<br/>$\Rightarrow \quad y^{2}-2 y y_{1} x=\sqrt{y y_{1}} \cdot 2 y y_{1}$
<br/>$\Rightarrow\left(y^{2}-2 y y_{1} x\right)^{2}=4\left(y y_{1}\right)^{3}$
<br/>So, degree of above equation is 3.</div>


title: the-differential-of-f-x-lo-g-e-1-e-10-x-tan-1-e-5-x-at-x-0-and-for-d-x-02-is
slug: the-differential-of-f-x-lo-g-e-1-e-10-x-tan-1-e-5-x-at-x-0-and-for-d-x-02-is-78082
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.5$</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The differential of $f(x)=\log _{e}\left(1+e^{10 x}\right)-\tan ^{-1}\left(e^{5 x}\right)$ at $x=0$ and for $d x=0.2$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-0.2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-0.5$</span> </li> </ul>

answer:
<div class="solution">$\begin{array}{l}
<br/>\frac{f(x+d x)-f(x)}{d x}=f^{\prime}(x) \\
<br/>f(x+d x)-f(x)=f^{\prime}(x) d x
<br/>\end{array}$
<br/>Now,
<br/>$\begin{array}{l}
<br/>f^{\prime}(x)=\frac{10 e^{10 x}}{1+e^{10 x}}-\frac{5 \cdot e^{5 x}}{1+e^{10 x}}=\frac{10 e^{10 x}-5 e^{5 x}}{1+e^{10 x}} \\
<br/>\left(f^{\prime}(x)\right)_{x-0}=\frac{5}{2} \\
<br/>\Delta f(x)=0.2 \times \frac{5}{2}=0.5
<br/>\end{array}$</div>


title: the-digit-in-the-units-place-of-the-number-1-2-3-99-is
slug: the-digit-in-the-units-place-of-the-number-1-2-3-99-is-57257
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The digit in the unit's place of the number $1 !+2 !+3 !+\ldots \ldots \ldots+99 !$ is</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9WCJCWc_Njt9hB4dCF6YArczjZSbBZjfIzbXUBDmkww.original.fullsize.png"/><br/> <br/>$1+2+6+24$ $\quad\quad\quad$unit digit is zero
<br/>$=33$
<br/>$\therefore$ Unit digit = 3</div>


title: the-dimension-of-angular-momentum-is
slug: the-dimension-of-angular-momentum-is-51778
question: <div class="question">The dimension of angular momentum is</div>
tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{-1}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{1}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{M}^{2} \mathrm{L}^{1} \mathrm{T}^{-2}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{1}\right]$</span> </div>

answer:
<div class="solution">Dimension of angular momentum
<br/>$\mathbf{L}=\mathbf{r} \times \mathbf{p}$
<br/>$=[\mathrm{L}]\left[\mathrm{MLT}^{-1}\right]=\left[\mathrm{ML}^{2} \mathrm{T}^{-1}\right]$</div>


title: the-dimension-of-the-universal-constant-of-gravitation-g-is
slug: the-dimension-of-the-universal-constant-of-gravitation-g-is-14383
question: <div class="question">The dimension of the universal constant of gravitation, $G$ is</div>
tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{ML}^{2} \mathrm{T}^{-1}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{M}^{-1} \mathrm{L}^{3} \mathrm{T}^{-2}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{M}^{-1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{ML}^{3} \mathrm{T}^{-2}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{M}^{-1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$</span> </div>

answer:
<div class="solution">The dimensions of universal constant $G$, is obtained by Newton's law of gravitation<br/>$F=\frac{G m_{1} m_{2}}{r^{2}}$<br/>$\begin{array}{l}<br/>G=\frac{F \times r^{2}}{m_{1} m_{2}} \\<br/>F=\text { Force }<br/>\end{array}$<br/>$r=$ Distance $m_{1}$ and $m_{2}$ are masses<br/>$\begin{aligned}<br/>G &amp;=\frac{\left[M^{1} L^{1} T^{-2}\right]\left[L^{2}\right]}{[M][M]} \\<br/>&amp;=\frac{\left[M^{\prime} L^{2} T^{-2}\right]}{\left[M^{2}\right]}=\left[M^{-1} L^{2} T^{-2}\right]<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-direction-ratios-of-the-normal-to-the-plane-passing-through-the-points-123-121-and-parallel-to-2-x-2-3-y-1-4-z-is

slug:
the-direction-ratios-of-the-normal-to-the-plane-passing-through-the-points-123-121-and-parallel-to-2-x-2-3-y-1-4-z-is-39319

question:
<div class="question">The direction ratios of the normal to the plane passing through the points (1,2,-3) ,(-1,-2,1) and parallel to $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z}{4}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(2, 3, 4)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(14,-8,-1)$</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">$(-2,0,-3)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(1,-2,-3)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(14,-8,-1)$</span> </div>

answer:
<div class="solution">Direction ratios of line joining the points (1,2,-3) and (-1,-2,1) is $(-1-1,-2-2,1+3)$
<br/>i.e. (-2,-4,4)
<br/>Let the direction ratios of the normal to the plane is $(a, b, c),$ then
<br/>$$
<br/>\begin{array}{r}
<br/>2 a+3 b+4 c=0 ....(i)\\
<br/>-2 a-4 b+4 c=0 ....(ii)
<br/>\end{array}
<br/>$$
<br/>and
<br/>By Eqs.
<br/>(i) and (ii), we get
<br/>$\begin{aligned} \frac{a}{3(4)-(4)(-4)} &amp;=\frac{b}{(-2)(4)-(2)(4)} \\ &amp;=\frac{c}{(2)(-4)-(-2)(3)} \\ \Rightarrow \quad \frac{a}{12+16} &amp;=\frac{b}{-8-8}=\frac{c}{-8+6} \\ \Rightarrow \quad \frac{a}{28} &amp;=\frac{b}{-16}=\frac{c}{-2} \\ \Rightarrow \quad \frac{a}{14} &amp;=\frac{b}{-8}=\frac{c}{-1} \end{aligned}$
<br/>Hence, direction ratios are (14,-8,-1)</div>


title: the-dispersed-phase-and-dispersion-medium-of-fog-respectively-are
slug: the-dispersed-phase-and-dispersion-medium-of-fog-respectively-are-54289
question: <div class="question">The dispersed phase and dispersion medium of fog respectively are</div>
tags: ['Chemistry', 'Surface Chemistry', 'WBJEE', 'WBJEE 2015']
answer: <div class="solution">The dispersed phase is liquid and dispersion medium is gas in the fog</div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">solid, liquid</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">liquid, liquid</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">liquid, gas</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">gas, liquid</span> </li> </ul>

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


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-displacement-of-a-particle-at-time-t-is-x-where-x-t-4-k-t-3-if-the-velocity-of-the-particle-at-time-t-2-is-minimum-then

slug:
the-displacement-of-a-particle-at-time-t-is-x-where-x-t-4-k-t-3-if-the-velocity-of-the-particle-at-time-t-2-is-minimum-then-52385

question:
<div class="question">The displacement of a particle at time $t$ is $x$, where $x=t^4-k t^3$. If the velocity of the particle at time $t=2$ is minimum, then</div>

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

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

answer:
<div class="solution">Hints : $\frac{d x}{d t}=4 t^3-3 k t^2$<br/>$\frac{d v}{d t}=12 t^2-6 \mathrm{kt}$ at $\mathrm{t}=2$<br/>$\Rightarrow \frac{\mathrm{dv}}{\mathrm{dt}}=0,48-12 \mathrm{k}=0 \quad ; \mathrm{k}=4$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-displacement-of-a-particle-in-a-periodic-motion-is-given-by-y-4-cos-2-2-t-sin-1000-t-this-displacement-may-be-considered-as-the-result-of-superpos

slug:
the-displacement-of-a-particle-in-a-periodic-motion-is-given-by-y-4-cos-2-2-t-sin-1000-t-this-displacement-may-be-considered-as-the-result-of-superpos-48148

question:
<div class="question">The displacement of a particle in a periodic motion is given by $y=4 \cos ^{2}\left(\frac{t}{2}\right) \sin (1000 t)$. This displacement may be considered as the result of superposition of $n$ independent harmonic oscillations. Here $n$ is</div>

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

answer:
<div class="solution">Given, $Y=4 \cos ^{2}\left(\frac{t}{2}\right) \cdot \sin (1000 t)$
<br/>$=2 \times 2 \cos ^{2}\left(\frac{t}{2}\right) \cdot \sin (1000 t)$
<br/>$\quad\left[\because 2 \cos ^{2} \frac{t}{2}=(1+\cos t)\right]$
<br/>$=2(1+\cos t) \sin (1000 t)$
<br/>$=2 \sin (1000 t)+2 \cos t \cdot \sin (1000 t)$
<br/>$=2 \sin (1000 t)+2 \sin (1000 t) \cdot \cos t$
<br/>$=2 \sin (1000 t)+\sin (1000 t+t)$
<br/>$$
<br/>\begin{aligned}[\because-2 \sin A \cdot \cos B=\sin (A+B)+\sin (A-B)] \\=2 \sin (1000 t)+\sin (1001 t)+\sin (999 t) \end{aligned}
<br/>$$
<br/>So, $n=3$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-displacement-of-a-particle-in-shm-varies-according-to-the-relation-x4cos-pi-tsin-pi-mathrmt-the-amplitude-of-the-particle-is

slug:
the-displacement-of-a-particle-in-shm-varies-according-to-the-relation-x4cos-pi-tsin-pi-mathrmt-the-amplitude-of-the-particle-is-78895

question:
<div class="question">The displacement of a particle in S.H.M. varies according to the relation \(x=4(\cos \pi t+\sin \pi \mathrm{t})\). The amplitude of the particle 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">4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(4 \sqrt{2}\)</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">8</span> </li> </ul>

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

answer:
<div class="solution">Hints: \(\mathrm{R} \sin \delta=4\)<br/>\(\mathrm{R} \cos \delta=4\)<br/>\(R=4 \sqrt{2}\)</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-displacement-of-a-plane-progressive-wave-in-a-medium-travelling-towards-positive-xaxis-with-velocity-4-mathrmm-mathrms-at-t0-is-given-by-y3-sin-2-

slug:
the-displacement-of-a-plane-progressive-wave-in-a-medium-travelling-towards-positive-xaxis-with-velocity-4-mathrmm-mathrms-at-t0-is-given-by-y3-sin-2-36413

question:
<div class="question">The displacement of a plane progressive wave in a medium, travelling towards positive \(x\)-axis with velocity \(4 \mathrm{~m} / \mathrm{s}\) at \(t=0\) is given by \(y=3 \sin 2 \pi\left(-\frac{x}{3}\right)\). Then the expression for the displacement at a later time \(t=4\) sec will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(y=3 \sin 2 \pi\left(-\frac{x-16}{3}\right)\)</span> </div>

answer:
<div class="solution">Hint : Let \(y=3 \sin [\omega t-\mathrm{kx}]\)<br/>\(\frac{\text { at } t=0}{y=3 \sin (-k x)}\)<br/>\(k=\frac{2 \pi}{3}\)<br/>\(v=\frac{\omega}{k}\)<br/>\(4=\frac{\omega}{2 \pi / 3} \quad \Rightarrow \quad \omega=\frac{8 \pi}{3}\)<br/>\(\begin{gathered}\therefore \quad y=3 \sin \left[\frac{8 \pi}{3} t-\frac{2 \pi}{3} x\right] \\ \therefore t=4 \\ y=3 \sin \left[\frac{8 \pi}{3} \times 4-\frac{2 \pi}{3} x\right] \\ =3 \sin 2 \pi\left[\frac{-x+16}{3}\right] \\ y=3 \sin \left[2 \pi\left(-\frac{x-16}{3}\right)\right]\end{gathered}\)</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-displacement-x-of-a-particle-at-time-t-is-given-by-x-at-2-bt-c-where-a-b-c-are-constants-and-v-is-velocity-of-a-particle-then-the-value-of-4-ax-v-

slug:
the-displacement-x-of-a-particle-at-time-t-is-given-by-x-at-2-bt-c-where-a-b-c-are-constants-and-v-is-velocity-of-a-particle-then-the-value-of-4-ax-v-16577

question:
<div class="question">The displacement $\mathrm{x}$ of a particle at time $\mathrm{t}$ is given by $\mathrm{x}=\mathrm{At}^2+\mathrm{Bt}+\mathrm{C}$ where $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are constants and $\mathrm{v}$ is velocity of a particle, then the value of $4 \mathrm{Ax}-\mathrm{v}^2$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \mathrm{AC}-\mathrm{B}^2$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \mathrm{x}=\mathrm{At}^2+\mathrm{Bt}+\mathrm{c} \\<br/>&amp; \mathrm{v}=2 \mathrm{At}+\mathrm{B} \Rightarrow \mathrm{v}^2=4 \mathrm{~A}^2 \mathrm{t}^2+4 \mathrm{AB} \mathrm{t}+\mathrm{B}^2 \\<br/>&amp; 4 \mathrm{Ax}=4 \mathrm{~A}^2 \mathrm{t}^2+4 \mathrm{AB} \mathrm{t}+4 \mathrm{AC} \\<br/>&amp; \Rightarrow \mathrm{v}^2-4 \mathrm{ax}=\mathrm{B}^2-4 \mathrm{AC} \\<br/>&amp; \Rightarrow 4 A \mathrm{Ax}-\mathrm{v}^2=4 A C-\mathrm{B}^2<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
the-distance-between-a-light-source-and-photoelectric-cell-is-d-if-the-distance-is-decreased-to-2-d-then

slug:
the-distance-between-a-light-source-and-photoelectric-cell-is-d-if-the-distance-is-decreased-to-2-d-then-25173

question:
<div class="question">The distance between a light source and photoelectric cell is d. If the distance is decreased to $\frac{d}{2},$ then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">the emission of electron per second will be four times, stopping potential will remain same</span> </div>

answer:
<div class="solution">The intensity of emitted electrons is
<br/>$$
<br/>\begin{array}{l}
<br/>I=\frac{1}{r^{2}} \\
<br/>I_{1}=\frac{1}{d^{2}} \\
<br/>I_{2}=\frac{1}{(d / 2)^{2}}=4 \cdot \frac{1}{d^{2}}
<br/>\end{array}
<br/>$$
<br/>and I $\propto$ Number of photons per second $=N$
<br/>$\therefore$
<br/>$$
<br/>N=\frac{1}{r^{2}}
<br/>$$
<br/>$\therefore$ Number of emithed electrons will become 4 times From Enstein's photo-electric equation
<br/>$$
<br/>K E_{max}=h v-\phi
<br/>$$
<br/>where, $\phi=$ work function of surtace
<br/>$\mathrm{v}=$ frequency of incident photon
<br/>Since, v remains unchanged, thus $KE_{max}$ as well as stopping potential remans unchanged
<br/>$\because \mathrm{KE}_{max}=\mathrm{eV}_{s}=\mathrm{v}$
<br/>where. $V_{s}$ = stopping potential</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-distance-between-an-object-and-a-divergent-lens-is-m-times-the-focal-length-of-the-lens-the-linear-magnification-produced-by-the-lens-is

slug:
the-distance-between-an-object-and-a-divergent-lens-is-m-times-the-focal-length-of-the-lens-the-linear-magnification-produced-by-the-lens-is-67893

question:
<div class="question">The distance between an object and a divergent lens is $\mathrm{m}$ times the focal length of the lens. The linear magnification produced by the lens is</div>

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

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

answer:
<div class="solution">$\mathrm{u}=-\mathrm{mf}$<br/>$\frac{1}{\mathrm{v}}-\frac{1}{(-m f)}=-\frac{1}{f} \Rightarrow \frac{1}{\mathrm{v}}=-\frac{1}{f}\left(1+\frac{1}{m}\right) \Rightarrow-\frac{\mathrm{v}}{\mathrm{u}}=\left(\frac{1}{1+m}\right)$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-distance-covered-by-a-particle-in-t-seconds-is-given-by-x-3-8-t-4-t-2-after-1-second-velocity-will-be

slug:
the-distance-covered-by-a-particle-in-t-seconds-is-given-by-x-3-8-t-4-t-2-after-1-second-velocity-will-be-37183

question:
<div class="question">The distance covered by a particle in $t$ seconds is given by $x=3+8 t-4 t^2$. After 1 second velocity will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0 unit/second</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3 units/second</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4 units/second</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7 units/second</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">0 unit/second</span> </div>

answer:
<div class="solution">$$<br/>\begin{array}{r}<br/>\text { Hints : } v=\frac{d x}{d t}=8-8 t \\<br/>t=1, v=8-8=0<br/>\end{array}<br/>$$</div>


title: the-domain-of-definition-of-f-x-2-x-1-x-is-here-a-b-x-a-x-b-and-a-b-x-a-x-b
slug: the-domain-of-definition-of-f-x-2-x-1-x-is-here-a-b-x-a-x-b-and-a-b-x-a-x-b-16492
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The domain of definition of $f(x)=\sqrt{\frac{1-|x|}{2-|x|}}$ is Here $(a, b)=\{x: a    &lt; x &lt; b\}$ and
<br/>$[a, b]=\{x: a \leq x \leq b\}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(-\infty,-1) \cup(2, \infty)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">[-1,1]$\cup(2, \infty) \cup(-\infty,-2)$</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">$(-\infty, 1) \cup(2, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">[-1,1]$\cup (2, \infty)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">[-1,1]$\cup(2, \infty) \cup(-\infty,-2)$</span> </div>

answer:
<div class="solution">We have, $f(x)=\sqrt{\frac{1-|x|}{2-|x|}}$
<br/>$f(x)$ is defined for all $x$ satisfying $\frac{1-|x|}{2-|x|} \geq 0 \Rightarrow \frac{|x|-1}{|x|-2} \geq 0$
<br/>$\Rightarrow \quad|x| \leq$ lor $|x|&gt;2$
<br/>$\Rightarrow \quad x \in[-1,1]$ or $x \in(-\infty,-2) \cup(2, \infty)$
<br/>$\Rightarrow \quad x \in(-\infty,-2) \cup[-1,1] \cup(2, \infty)$</div>


title: the-domain-of-definition-of-the-function-f-x-1-lo-g-e-1-x-is
slug: the-domain-of-definition-of-the-function-f-x-1-lo-g-e-1-x-is-82231
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The domain of definition of the function $f(x)=\sqrt{1+\log _{\mathrm{e}}(1-x)}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\infty &lt; x \leq 0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\infty &lt; x \leq \frac{e-1}{e}$</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">C</span> <span class="option-data">$-\infty &lt; x \leq 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x \geq 1-e$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\infty &lt; x \leq \frac{e-1}{e}$</span> </div>

answer:
<div class="solution">Hints : $1-x&gt;0 \Rightarrow X &lt; 1$<br/>$$<br/>\begin{aligned}<br/>&amp; 1+\log _e(1-x) \geq 0 \\<br/>&amp; \log _e(1-x) \geq-1 \Rightarrow 1-x \geq e^{-1} \\<br/>&amp; x \leq 1-\frac{1}{e} \\<br/>&amp; x \leq \frac{e-1}{e}<br/>\end{aligned}<br/>$$</div>


title: the-domain-of-f-x-x-1-x-1-is
slug: the-domain-of-f-x-x-1-x-1-is-19729
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The domain of $f(x)=\sqrt{\left(\frac{1}{\sqrt{x}}-\sqrt{x+1}\right)}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(0, \frac{\sqrt{5}-1}{2}\right]$</span> </div>

answer:
<div class="solution">Hint:
<br/>$x^{2}+x \leq 1$ and $x&gt;0$
<br/>$x \in\left[\frac{-1-\sqrt{5}}{2}, \frac{\sqrt{5}-1}{2}\right]$
<br/>$x \in\left(0, \frac{\sqrt{5}-1}{2}\right]$</div>


title: the-domain-of-the-function-f-x-cos-1-2-1-x-is
slug: the-domain-of-the-function-f-x-cos-1-2-1-x-is-68975
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The domain of the function $f(x)=\sqrt{\cos ^{-1}\left(\frac{1-|x|}{2}\right)}$ is</div>

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

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

answer:
<div class="solution">Hints: $f(x)=\sqrt{\cos ^{-1}\left(\frac{1-|x|}{2}\right)}$<br/>$$<br/>-1 \leq \frac{1-|\mathrm{x}|}{2} \leq 1 \Rightarrow-2-1 \leq-|\mathrm{x}| \leq 2-1 \Rightarrow-3 \leq-|\mathrm{x}| \leq 1 \Rightarrow-1 \leq \mathrm{x} \mid \leq 3 \Rightarrow \mathrm{x} \in[-3,3]<br/>$$</div>


title: the-ease-of-dehydrohalogenation-of-alkyl-halide-with-alcoholic-mathrmkoh-is
slug: the-ease-of-dehydrohalogenation-of-alkyl-halide-with-alcoholic-mathrmkoh-is-22406
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The ease of dehydrohalogenation of alkyl halide with alcoholic \(\mathrm{KOH}\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(3^{\circ} &gt; 2^{\circ} &gt; 1^{\circ}\)</span> </div>

answer:
<div class="solution">Hints : Such dehydrohalogenation follows \(\mathrm{E}_2\) mechanism. The driving force of such reactions is the stability of alkene produced. Since tetriary alkyl halide can give more substituted alkene, it reacts fastest followed by secondary and primary i.e. \(3^{\circ} &gt; 2^{\circ} &gt; 1^{\circ}\).</div>


tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-ease-of-hydrolysis-in-the-compounds-ch-3-cocl-i-ch-3-co-o-coch-3-ii-ch-3-cooc-2-h-5-iii-and-ch-3-conh-2-iv-is-of-the-order

slug:
the-ease-of-hydrolysis-in-the-compounds-ch-3-cocl-i-ch-3-co-o-coch-3-ii-ch-3-cooc-2-h-5-iii-and-ch-3-conh-2-iv-is-of-the-order-17067

question:
<div class="question">The ease of hydrolysis in the compounds $\mathrm{CH}_{3} \mathrm{COCl}(\mathrm{I}), \mathrm{CH}_{3}-\mathrm{CO}-\mathrm{O}-\mathrm{COCH}_{3}(\mathrm{II})$
<br/>$\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(\mathrm{III})$ and $\mathrm{CH}_{3} \mathrm{CONH}_{2}$ (IV) is of
<br/>the order</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$I&gt;II&gt;III&gt;IV$</span> </div>

answer:
<div class="solution">The ease of hydrolysis of carbonyl compounds depend on the group attached to the carbonyl carbon. Among the given options, chlorine $(-$ Cl) group is clectron withdrawing group which makes the carbonyl group electrophilic. Hence, hydrolysis can occur most easily. The order for electron withdrawing tendency among given options is as follows:
<br/>$$
<br/>\begin{array}{l}
<br/>-\mathrm{Cl}&gt;-\mathrm{OCOCH}_{3}&gt;-\mathrm{OC}_{2} \mathrm{H}_{5}&gt;-\mathrm{NH}_{2} \\\end{array}$$
<br/>So, the correct order for the ease of hydrolysis in the given compounds is as follows:
<br/>$\mathrm{CH}_{3} \mathrm{COCl}&gt;\mathrm{CH}_{3} \mathrm{COOCOCH}_{3}$
<br/>$&gt;\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}&gt;\mathrm{CH}_{3}{\mathrm{IV}} \mathrm{CONH}_{2}$</div>


title: the-ease-of-nitration-of-the-following-three-hydrocarbons-follows-the-order
slug: the-ease-of-nitration-of-the-following-three-hydrocarbons-follows-the-order-60941
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The ease of Nitration of the following three hydrocarbons follows the order<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/DtqUyAACPmYbjbe7YfgqFmGw5-y7jL_roHofMfQUf60.original.fullsize.png"/><br/></div>

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

answer:
<div class="solution">Hints: Stability order of Arenium ion<br/>\(\text { II } &gt; \text { III } &gt; \text { I }\)</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
the-eccentric-angle-in-the-first-quadrant-of-a-point-on-the-ellipse-10-x-2-8-y-2-1-at-a-distance-3-units-from-the-centre-of-the-ellipse-is

slug:
the-eccentric-angle-in-the-first-quadrant-of-a-point-on-the-ellipse-10-x-2-8-y-2-1-at-a-distance-3-units-from-the-centre-of-the-ellipse-is-72820

question:
<div class="question">The eccentric angle in the first quadrant of a point on the ellipse $\frac{x^{2}}{10}+\frac{y^{2}}{8}=1$ at a distance 3 units from the centre of the ellipse is</div>

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

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

answer:
<div class="solution">Let $P(\sqrt{10} \cos \theta, \sqrt{8} \sin \theta)$ be the required point
<br/>on $\frac{x^{2}}{10}+\frac{y^{2}}{8}=1$
<br/>Whose distance from centre (0,0) is 3 units. $\therefore 10 \cos ^{2} \theta+8 \sin ^{2} \theta=9=9\left(\sin ^{2} \theta+\cos ^{2} \theta\right)$
<br/>$$
<br/>\Rightarrow \quad \tan ^{2} \theta=1 \Rightarrow \theta=\frac{\pi}{4}
<br/>$$</div>


title: the-eccentricity-of-the-hyperbola-4-x29-y236-is
slug: the-eccentricity-of-the-hyperbola-4-x29-y236-is-85147
question: <div class="question">The eccentricity of the hyperbola \(4 x^2-9 y^2=36\) is</div>
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\sqrt{11}}{3}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\sqrt{15}}{3}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{\sqrt{13}}{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{\sqrt{14}}{3}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\sqrt{13}}{3}\)</span> </div>

answer:
<div class="solution">Hints: \(\frac{x^2}{9}-\frac{y^2}{4}=1\)<br/>\(\begin{aligned}<br/>&amp; a=3, b=2 \\<br/>&amp; \therefore e=\sqrt{\frac{a^2+b^2}{a^2}}=\sqrt{\frac{13}{9}}=\frac{\sqrt{13}}{3}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
the-effective-resistance-between-a-and-b-in-the-figure-is-12-7-if-each-side-of-the-cube-has-1-resistance-the-effective-resistance-between-the-same-two

slug:
the-effective-resistance-between-a-and-b-in-the-figure-is-12-7-if-each-side-of-the-cube-has-1-resistance-the-effective-resistance-between-the-same-two-42284

question:
<div class="question">The effective resistance between $A$ and $B$ in the figure is $\frac{7}{12} \Omega$ if each side of the cube has $1 \Omega$ resistance. The effective resistance between the same two points, when the link $A B$ is removed, is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a35.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{12} \Omega$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5}{12} \Omega$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{7}{5} \Omega$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5}{7} \Omega$</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a35.png"/> <br/> <br/>Let when link AB is removed, $x$ is an equivalent of remaining part of cube without link.
<br/> <br/>AB and rest part will be in paralel, so according to question
<br/>$$
<br/>\begin{array}{c}
<br/>\frac{7}{12}=\frac{R_{1} R_{2}}{R_{1}+R}=\frac{1(x)}{1+x}=\frac{x}{1+x} \\
<br/>\frac{7}{12}=\frac{x}{1+x} \\
<br/>12 x=7+7 x \\
<br/>12 x-7 x=7
<br/>\end{array}
<br/>5 x=7
<br/>$$
<br/>$$
<br/>x=\frac{7}{5} \Omega
<br/>$$</div>


tags: ['Physics', 'Electromagnetic Waves', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint: \(\vec{E} \cdot \vec{B}=0\) and in the plane of \(X Y\)</div>
source_db: db\MarksBatch2_P2.db
title:
the-electric-field-of-a-plane-electromagnetic-wave-of-wave-number-k-and-angular-frequency-omega-is-given-by-vecee0hatihatj-sin-k-zomega-t-which-of-the

slug:
the-electric-field-of-a-plane-electromagnetic-wave-of-wave-number-k-and-angular-frequency-omega-is-given-by-vecee0hatihatj-sin-k-zomega-t-which-of-the-87134

question:
<div class="question">The electric field of a plane electromagnetic wave of wave number \(k\) and angular frequency \(\omega\) is given by \(\vec{E}=E_0(\hat{i}+\hat{j}) \sin (k z-\omega t)\). Which of the following gives the direction of the associated magnetic field \(\vec{B}\) ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\hat{k}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(-\hat{i}+\hat{j}\)</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">\(-\hat{i}-\hat{j}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\hat{\mathrm{i}}-\hat{\mathrm{k}}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-\hat{i}+\hat{j}\)</span> </div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
the-electric-potential-for-an-electric-field-directed-parallel-to-x-axis-is-shown-in-the-figure-choose-the-correct-plot-of-electric-field-strength

slug:
the-electric-potential-for-an-electric-field-directed-parallel-to-x-axis-is-shown-in-the-figure-choose-the-correct-plot-of-electric-field-strength-23856

question:
<div class="question">The electric potential for an electric field directed parallel to $\mathrm{X}$-axis is shown in the figure. Choose the correct plot of electric field strength.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Jg921ZiO-LFyT3ZakaG1oo7VtQrKiUHl7OkIjRMRtw4.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/eyPFGx2Bh8KlDXyUYz1qc7Rq1iP2jpgNf3xylkHz1UI.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Nf3P5thlen06Xd8jYnR8o3wY4oIPPcRiwBh7ePH1O8w.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Xk3uQJ6jTHejiG4FXVVcwuo-skgmG1L9nnlbaW9U3-Y.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/E5l2LIs1KhOIbWDK8vkMexErtCguUdb4EzwCD8ZtNDs.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/eyPFGx2Bh8KlDXyUYz1qc7Rq1iP2jpgNf3xylkHz1UI.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Ww7mBwuShly3LLWTgoAmf9-0CCkuCZETQZpXdErLSTY.original.fullsize.png"/></div>


title: the-electronic-configuration-of-cu-is
slug: the-electronic-configuration-of-cu-is-87630
question: <div class="question">The electronic configuration of $\mathrm{Cu}$ is</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">[Ne] $3 s^{2}, 3 p^{6}, 3 d^{9}, 4 s^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">[Ne] $3 s^{2}, 3 p^{6}, 3 d^{10}, 4 s^{1}$</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">[Ne] $3 s^{2}, 3 p^{6}, 3 d^{3}, 4 s^{2}, 4 p^{6}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$[\mathrm{Ne}] 3 \mathrm{s}^{2}, 3 p^{6}, 3 d^{5} \cdot 4 s^{2} \cdot 4 p^{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">[Ne] $3 s^{2}, 3 p^{6}, 3 d^{10}, 4 s^{1}$</span> </div>

answer:
<div class="solution">Atomic number of Cu is 29 .
<br/>Thus, its electronic configuration can be written as
<br/>$$
<br/>\mathrm{Cu}_{29}=1s^{2}, 2 s^{2}, 2 p^{6}, 3 s^{2}, 3 p^{6}, 3 d^{10}, 4 s^{1}
<br/>$$
<br/>(Since, fully filled orbitals are more stable, so one electron from 4 s orbital transfers into $3 d$ orbital.
<br/>The electronic configuration of neon is
<br/>$$
<br/>\mathrm{Ne}_{10}=1 \mathrm{s}^{2}, 2 s^{2}, 2 p^{6}
<br/>$$
<br/>Thus, using inert gas, the configuration of Cu can also be represented as
<br/>$$
<br/>\mathrm{Cu}_{29}=[\mathrm{Ne}] 3 \mathrm{s}^{2}, 3 p^{6}, 3 d^{10}, 4 \mathrm{s}^{1}
<br/>$$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-electronic-transitions-from-mathrmn2-to-mathrmn1-will-produce-shortest-wavelength-in-where-mathrmn-principal-quantum-state

slug:
the-electronic-transitions-from-mathrmn2-to-mathrmn1-will-produce-shortest-wavelength-in-where-mathrmn-principal-quantum-state-63745

question:
<div class="question">The electronic transitions from \(\mathrm{n}=2\) to \(\mathrm{n}=1\) will produce shortest wavelength in (where \(\mathrm{n}=\) principal quantum state)</div>

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

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

answer:
<div class="solution">Hints: \(\frac{1}{\lambda}=\mathrm{z}^2 \cdot \mathrm{R}_{\mathrm{H}}\left[1 / \mathrm{n}_1^2-1 / \mathrm{n}_2^2\right]\)<br/>\(\begin{aligned}<br/>&amp; \Rightarrow \frac{1}{\lambda}=(\mathrm{z})^2 \cdot \mathrm{R}_{\mathrm{H}}\left\{\frac{1}{1}-\frac{1}{4}\right\}=\frac{3}{4} \mathrm{R}_{\mathrm{H}} \mathrm{z}^2 \\<br/>&amp; \therefore \lambda \propto 1 / \mathrm{z}^2<br/>\end{aligned}\)<br/>Hence, for shortest \(\lambda, \mathrm{z}\) must be maximum, which is for \(\mathrm{Li}^{+2}\).</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-emission-spectrum-of-hydrogen-discovered-first-and-the-region-of-the-electromagnetic-spectrum-in-which-it-belongs-respectively-are

slug:
the-emission-spectrum-of-hydrogen-discovered-first-and-the-region-of-the-electromagnetic-spectrum-in-which-it-belongs-respectively-are-45669

question:
<div class="question">The emission spectrum of hydrogen discovered first and the region of the electromagnetic spectrum in which it belongs, respectively are</div>

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

answer:
<div class="solution">In 1885 , Balmer for the first time showed that the wave numbers of spectral lines present in the visible region in hydrogen spectrum are given by
<br/>$$
<br/>\bar{v}\left(\mathrm{cm}^{-1}\right)=109677\left(\frac{1}{(2)^{2}}-\frac{1}{n^{2}}\right)
<br/>$$
<br/>Here, $n=3,4,5, \ldots$
<br/>Thus, Balmer spectrum of hydrogen was discovered first and it lies in the visible region.</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-energy-of-an-electron-in-first-bohr-orbit-of-mathrmhatom-is-136-mathrmev-the-possible-energy-value-of-electron-in-the-excited-state-of-mathrmli2-i

slug:
the-energy-of-an-electron-in-first-bohr-orbit-of-mathrmhatom-is-136-mathrmev-the-possible-energy-value-of-electron-in-the-excited-state-of-mathrmli2-i-60677

question:
<div class="question">The energy of an electron in first Bohr orbit of \(\mathrm{H}\)-atom is \(-13.6 \mathrm{eV}\). The possible energy value of electron in the excited state of \(\mathrm{Li}^{2+}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(-122.4 \mathrm{eV}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(-30.6 \mathrm{eV}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(-30.6 \mathrm{eV}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\(13.6 \mathrm{eV}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-30.6 \mathrm{eV}\)</span> </div>

answer:
<div class="solution">Hints: \(E_n=\frac{E_1}{n^2} \times z^2\)<br/>\(=\frac{-13.6}{4} \times 9=-30.6 \mathrm{eV}\)<br/>For the excited state, \(\mathrm{n}=2\) and for \(\mathrm{Li}^{++}\)ion, \(\mathrm{z}=3\)</div>


tags: ['Physics', 'Electromagnetic Waves', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-energy-of-gamma-ray-photon-is-e-and-that-of-an-xray-photon-is-e-x-if-the-visible-light-photon-has-an-energy-of-e-v-then-we-can-say-that

slug:
the-energy-of-gamma-ray-photon-is-e-and-that-of-an-xray-photon-is-e-x-if-the-visible-light-photon-has-an-energy-of-e-v-then-we-can-say-that-93357

question:
<div class="question">The energy of gamma $(\gamma)$ ray photon is $E$, and that of an X-ray photon is $E_{X}$. If the visible light photon has an energy of $E_{v}$, then we can say that</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$E_{\gamma}&gt;E_{x}&gt;E_{v}$</span> </div>

answer:
<div class="solution">$E_{y} \geq 100 \mathrm{keV}$<br/>$\begin{aligned}<br/>E_{X} &amp;=100 \mathrm{eV} \text { to } 100 \mathrm{keV} \\<br/>E_{v} &amp;=2.48 \mathrm{eV}<br/>\end{aligned}$<br/>So, we can say that<br/>$E_{y}&gt;E_{X}&gt;E_{v}$</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-energy-released-by-the-fission-of-one-uranium-atom-is-200-mev-the-number-of-fissions-per-second-required-to-produce-32-w-of-power-is-take-1-ev-16-

slug:
the-energy-released-by-the-fission-of-one-uranium-atom-is-200-mev-the-number-of-fissions-per-second-required-to-produce-32-w-of-power-is-take-1-ev-16-59258

question:
<div class="question">The energy released by the fission of one uranium atom is $200 \mathrm{MeV}$. The number of fissions per second required to produce $3.2 \mathrm{~W}$ of power is (Take $1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10^7$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10^{10}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10^{15}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$10^{11}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10^{11}$</span> </div>

answer:
<div class="solution">Hints: $u=200 \mathrm{MeV}=200 \times 10^6 \mathrm{eV}=200 \times 10^6 \times 1.6 \times 10^{-19} \mathrm{~J}$<br/>$\mathrm{E}=3.2 \mathrm{~J}$<br/>No of fissions $=\frac{3.2}{2 \times 1.6 \times 10^{-11}}=10^{11}$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
the-energy-required-to-break-one-mole-of-hydrogenhydrogen-bonds-in-h-2-is-436-kj-what-is-the-longest-wavelength-of-light-required-to-break-a-single-hy

slug:
the-energy-required-to-break-one-mole-of-hydrogenhydrogen-bonds-in-h-2-is-436-kj-what-is-the-longest-wavelength-of-light-required-to-break-a-single-hy-28389

question:
<div class="question">The energy required to break one mole of hydrogen-hydrogen bonds in $\mathrm{H}_{2}$ is $436 \mathrm{kJ}$. What is the longest wavelength of light required to break a single hydrogenhydrogen bond?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$68.5 \mathrm{nm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$137 \mathrm{nm}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$274 \mathrm{nm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$548 \mathrm{nm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$274 \mathrm{nm}$</span> </div>

answer:
<div class="solution">Given, energy of one mole of $\mathrm{H}_{2}$ bonds $=436 \mathrm{k} \mathrm{J}$
<br/>$$
<br/>\begin{array}{l}
<br/>\because E=\frac{n \cdot h c}{\lambda}\\
<br/>\therefore \quad \lambda=\frac{n.h c}{E} \\
<br/>\lambda=\frac{6.634 \times 10^{-} 34 \mathrm{s} \times 3 \times 10^{3} \mathrm{ms}^{-1} \times 6.023 \times 10^{23}}{436 \times 10^{3} \mathrm{J}} \mathrm{m} \\
<br/>=0.2736 \times 10^{-34+8+23-3} \\
<br/>=0.2736 \times 10^{-6} \mathrm{m} \\
<br/>=273.6 \mathrm{nm}=274 \mathrm{nm}
<br/>\end{array}
<br/>$$</div>


title: the-enol-form-in-which-ethyle3oxobutanoate-exists-is
slug: the-enol-form-in-which-ethyle3oxobutanoate-exists-is-64978
question: <div class="question">The enol form in which ethyle-3-oxobutanoate exists is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XHK0QSTuIkBCOLPijlFtMNljFcQ8m10-EwGRKXtVUEw.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/r7qmT5h77whHJlEIoDpQGWhaLTM6DI7Os25N5ncWzj8.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sZYZFU7OwqNryWcffWvICDFuj3E4xZFHPyVrCTwbCt8.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/3vLpiJGyGR5lUTTGm8HzaXJRVwSSslfZCRjLFNTDGco.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sZYZFU7OwqNryWcffWvICDFuj3E4xZFHPyVrCTwbCt8.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/CqLZonEDx3JUgHP38TYW01AUvjLO3mRqnbngCBCn_Qk.original.fullsize.png"/><br/><br/>Stablised by H-bonding and Ketonic group enolises relatively more than ester group.</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-enthalpy-of-vaporisation-of-a-certain-liquid-at-its-boiling-point-of-3-5-c-is-2464-kj-mol-1-the-value-of-change-in-entropy-for-the-process-is

slug:
the-enthalpy-of-vaporisation-of-a-certain-liquid-at-its-boiling-point-of-3-5-c-is-2464-kj-mol-1-the-value-of-change-in-entropy-for-the-process-is-41436

question:
<div class="question">The enthalpy of vaporisation of a certain liquid at its boiling point of $35^{\circ} \mathrm{C}$ is $24.64 \mathrm{kJ} \mathrm{mol}^{-1}$. The value of change in entropy for the process is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$704 \mathrm{K}\mathrm{mol}^{-1} \mathrm{mol}^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$80 \mathrm{K}\mathrm{mol}y^{-1} \mathrm{mol}^{-1}$</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">$24.64 \mathrm{K}\mathrm{mol}^{-1} \mathrm{mol}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7.04 \mathrm{K}\mathrm{mol}^{-1} \mathrm{mol}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$80 \mathrm{K}\mathrm{mol}y^{-1} \mathrm{mol}^{-1}$</span> </div>

answer:
<div class="solution">Entropy of vaporisation ( $\Delta$ S)
<br/>$$
<br/>=\frac{\text { enthalpy of vaporisation }(\Delta H)}{\text { boiling point (in K)} }
<br/>$$
<br/>Given, $\Delta H=24.64 \mathrm{kJ} \mathrm{mol}^{-1}$ and
<br/>$\text { boiling point }=35+273=308 \mathrm{k}$
<br/>$\therefore$
<br/>$$
<br/>\begin{aligned}
<br/>\Delta S &amp;=\frac{24.64 \times 10^{3} \mathrm{J} \mathrm{mol}^{-1}}{308 \mathrm{K}} \\
<br/>&amp;=80 \mathrm{JK}^{-1} \mathrm{mol}^{-1}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
the-entropy-s-of-a-black-hole-can-be-written-as-s-k-b-a-where-k-b-is-the-boltzmann-constant-and-a-is-the-area-of-the-black-hole-then-has-dimension-of

slug:
the-entropy-s-of-a-black-hole-can-be-written-as-s-k-b-a-where-k-b-is-the-boltzmann-constant-and-a-is-the-area-of-the-black-hole-then-has-dimension-of-47402

question:
<div class="question">The Entropy (S) of a black hole can be written as $S=\beta k_B A$, where $k_B$ is the Boltzmann constant and $A$ is the area of the black hole. Then $\beta$ has dimension of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{L}^2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{ML}^2 \mathrm{~T}^{-1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{L}^{-2}$</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">D</span> <span class="option-data">dimensionless</span> </li> </ul>

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

answer:
<div class="solution">$\begin{array}{ll} \mathrm{S}= \beta \mathrm{K}_\beta \mathrm{A} \\ \frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]}{[\theta]}=[\beta]\left[\frac{\mathrm{ML}^2 \mathrm{~T}^{-2}}{\theta}\right]\left[\mathrm{L}^2\right] \quad[\beta]=\left[\mathrm{L}^{-2}\right] \end{array}$</div>


title: the-equation-2-x-2-5-x-y-12-y-2-0-represents-a
slug: the-equation-2-x-2-5-x-y-12-y-2-0-represents-a-77515
question: <div class="question">The equation $2 x^{2}+5 x y-12 y^{2}=0$ represents a</div>
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">circle</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">pair of non-perpendicular intersecting straight lines</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">pair of perpendicular straight lines</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">hyperbola</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">pair of non-perpendicular intersecting straight lines</span> </div>

answer:
<div class="solution">Given equation is
<br/>$$
<br/>\begin{array}{r}
<br/>2 x^{2}+5 x y-12 y^{2}=0 \\
<br/>\Rightarrow \quad 2 x^{2}+8 x y-3 x y-12 y^{2}=0 \\
<br/>\Rightarrow \quad 2 x(x+4 y)-3 y(x+4 y)=0 \\
<br/>\Rightarrow \quad(x+4 y)(2 x-3 y)=0 \\
<br/>\therefore \quad x+4 y=0 \text { and } 2 x-3 y=0
<br/>\end{array}
<br/>$$
<br/>which represent a pair of straight lines. 
<br/>Compair Eq. (i) with $a x^{2}+2 h x y+b y^{2}=0$
<br/>we get $a=2$ and $b=-12$
<br/>$\because a+b \neq 0$
<br/>So, lines are not perpendicular to each other. Hence, it is a pair of non-perpendicular intersecting straight lines.</div>


title: the-equation-3-sin-x-cos-x-4-has
slug: the-equation-3-sin-x-cos-x-4-has-53752
question: <div class="question">The equation $\sqrt{3} \sin x+\cos x=4$ has</div>
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">only one solution</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">two solutions</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">infinitely many solutions</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">no solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">no solution</span> </div>

answer:
<div class="solution">Hints : $\sqrt{3} \sin x+\cos x=2 \sin \left(x+\frac{\pi}{6}\right) \leq 2$. Therefore<br/>$\sqrt{3} \sin x+\cos x=4 \quad$ cannot have a solution</div>


title: the-equation-6-x-8-x-1-0-x-has
slug: the-equation-6-x-8-x-1-0-x-has-95344
question: <div class="question">The equation $6^{x}+8^{x}=10^{x}$ has</div>
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">no real root</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">infinitely many rational roots</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">exactly one real root</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">two distinct real roots</span> </li> </ul>

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

answer:
<div class="solution">$6^{x}+8^{x}=10^{x}\left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}=1 \Rightarrow\left(\frac{3}{5}\right)^{x}=1-\left(\frac{4}{5}\right)^{x}$
<br/>$\therefore \quad 6^{x}+8^{x}=10^{x}$ has exactly one real root
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XYzjtRkq5Rj3rTShp6C7YHqFdDqnEcZ5j0u66SmkjTU.original.fullsize.png"/></div>


title: the-equation-8-x212-y24-x4-y10-represents
slug: the-equation-8-x212-y24-x4-y10-represents-63940
question: <div class="question">The equation \(8 x^2+12 y^2-4 x+4 y-1=0\) represents</div>
tags: ['Mathematics', 'Pair of Lines', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">an ellipse</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a hyperbola</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a parabola</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a circle</span> </li> </ul>

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

answer:
<div class="solution">Hints: \(a x^2+b y^2+2 h x y+2 g x+2 f y+c=0\)<br/>represents ellipse if \(\mathrm{h}^2-a b  &lt; 0\)<br/>\(\begin{aligned}<br/>&amp; 3 x^2+12 y^2-4 x+4 y-1=0 \\<br/>&amp; h=0, a=3, b=12 \\<br/>&amp; h^2-a b &lt; 0<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-a-line-parallel-to-the-line-3-x-4-y-0-and-touching-the-circle-x-2-y-2-9-in-the-first-quadrant-is

slug:
the-equation-of-a-line-parallel-to-the-line-3-x-4-y-0-and-touching-the-circle-x-2-y-2-9-in-the-first-quadrant-is-20353

question:
<div class="question">The equation of a line parallel to the line $3 x+4 y=0$ and touching the circle $x^{2}+y^{2}=9$ in the first quadrant, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 x+4 y=15$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 x+4 y=45$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 x+4 y=9$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x+4 y=27$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 x+4 y=15$</span> </div>

answer:
<div class="solution">Equation of circle is $x^{2}+y^{2}=9$  ...(i)
<br/>And equation of line is $3 x+4 y=0$  ....(ii)
<br/>Equation of line parallel to the line (ii) .
<br/>$=$
<br/>$$
<br/>\begin{aligned}
<br/>3 x+4 y &amp;=k \\
<br/>y &amp;=\frac{-3 x}{4}+\frac{k}{4} ...(iii)
<br/>\end{aligned}
<br/>$$
<br/>$\because$ Circle touches the line (iii), so by the condition of tangency
<br/>$$
<br/>c^{2}=a^{2}\left(1+m^{2}\right)
<br/>$$
<br/>$\Rightarrow \quad\left(\frac{k}{4}\right)^{2}=9\left[1+\left(\frac{-3}{4}\right)^{2}\right]$
<br/>$\left[\because c=\frac{k}{4}, a=3, m=\frac{-3}{4}\right]$
<br/>$\Rightarrow$
<br/>$$
<br/>k=\pm 15
<br/>$$
<br/>Hence, $3 x+4 y=15$ touches the circle in the first quadrant.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a36.png"/></div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-a-progressive-wave-is-y-4-sin-4-t-004-x-3-where-x-is-in-meter-and-t-is-in-second-the-velocity-of-the-wave-is

slug:
the-equation-of-a-progressive-wave-is-y-4-sin-4-t-004-x-3-where-x-is-in-meter-and-t-is-in-second-the-velocity-of-the-wave-is-10386

question:
<div class="question">The equation of a progressive wave is $y=4 \sin (4 \pi t-0.04 x+\pi / 3)$ where $x$ is in meter and $t$ is in second. The velocity of the wave is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$100 \pi \mathrm{m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$50 \pi \mathrm{m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$25 \pi \mathrm{m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi \mathrm{m} / \mathrm{s}$</span> </li> </ul>

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

answer:
<div class="solution">Hints : Velocity of wave $=\frac{\omega}{\mathrm{K}}=\frac{4 \pi}{0.04}=100 \pi \mathrm{m} / \mathrm{sec}$</div>


title: the-equation-of-auxiliary-circle-of-the-ellipse-16-x-2-25-y-2-32-x-100-y-284-is
slug: the-equation-of-auxiliary-circle-of-the-ellipse-16-x-2-25-y-2-32-x-100-y-284-is-77561
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation of auxiliary circle of the ellipse $16 x^{2}+25 y^{2}+32 x-100 y=284$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+2 x-4 y-20=0$</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">$x^{2}+y^{2}+2 x-4 y=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(x+1)^{2}+(y-2)^{2}=400$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(x+1)^{2}+(y-2)^{2}=225$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+2 x-4 y-20=0$</span> </div>

answer:
<div class="solution">Given equation of ellipse is
<br/>$$
<br/>16 x^{2}+25 y^{2}+32 x-100 y=284
<br/>$$
<br/>Simplifying the given equation, we have ellipse as
<br/>$$
<br/>\frac{(x+1)^{2}}{25}+\frac{(y-2)^{2}}{16}=1
<br/>$$
<br/>So, the auxiliary equation of circle is
<br/>$$
<br/>\begin{array}{r}
<br/>(x+1)^{2}+(y-2)^{2}=25 \\
<br/>\Rightarrow \quad x^{2}+2 x+1+y^{2}-4 y+4=25
<br/>\end{array}
<br/>$$
<br/>$\therefore$
<br/>$x^{2}+y^{2}+2 x-4 y-20=0$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-circle-of-radius-17-unit-with-centre-on-the-positive-side-of-x-axis-and-through-the-point-0-1-is

slug:
the-equation-of-circle-of-radius-17-unit-with-centre-on-the-positive-side-of-x-axis-and-through-the-point-0-1-is-70986

question:
<div class="question">The equation of circle of radius $\sqrt{17}$ unit, with centre on the positive side of $x$ -axis and through the point $(0,1)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}-8 x-1=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}+8 x-1=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}-9 y+1=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x^{2}+2 y^{2}-3 x+2 y=4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}-8 x-1=0$</span> </div>

answer:
<div class="solution">Hint: 
<br/>Let centre be $\equiv(a, 0)(a&gt;0)$
<br/>So, circle : $(x-a)^{2}+y^{2}=17 ;$ as it passes through $(0,1)$, so $a^{2}+1=17 \quad \Rightarrow a=4(a \neq-4, a&gt;0)$
<br/>$\therefore$ Equation is : $x^{2}+y^{2}-8 x-1=0$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-hyperbola-whose-coordinates-of-the-foci-are-80-and-the-length-of-latusrectum-is-24-units-is

slug:
the-equation-of-hyperbola-whose-coordinates-of-the-foci-are-80-and-the-length-of-latusrectum-is-24-units-is-10818

question:
<div class="question">The equation of hyperbola whose coordinates of the foci are (Â±8,0) and the length of latusrectum is 24 units, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 x^{2}-y^{2}=48$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 x^{2}-y^{2}=48$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}-3 y^{2}=48$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}-4 y^{2}=48$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 x^{2}-y^{2}=48$</span> </div>

answer:
<div class="solution">Let equation of hyperbola be
<br/>$$
<br/>\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
<br/>$$
<br/>Given, foci, $(\pm 8,0)=(\pm a e, 0)$
<br/>$\Rightarrow \quad a e=8$
<br/>and length of latusrectum $=\frac{2 b^{2}}{a}$
<br/>$\therefore$
<br/>$24=\frac{2 b^{2}}{a}$
<br/>$\Rightarrow \quad b^{2}=12 a$
<br/>$\therefore$ From Eq. (ii).
<br/>$$
<br/>a^{2} e^{2}=64
<br/>$$
<br/>$\Rightarrow$
<br/>$a^{2}\left(\frac{a^{2}+b^{2}}{a^{2}}\right)=64$
<br/>$\Rightarrow \quad a^{2}+b^{2}=64$
<br/>$\Rightarrow \quad a^{2}+12 a=64$
<br/>$\Rightarrow \quad a^{2}+12 a-64=0$
<br/>$\Rightarrow \quad a^{2}+16 a-4 a-64=0$
<br/>$\Rightarrow \quad a(a+16)-4(a+16)=0$
<br/>$\Rightarrow \quad(a+16)(a-4)=0$
<br/>$\Rightarrow \quad a=4$
<br/>[$\because$ a cannot be negative] On putting $a=4$ in Eq. (iii), we get
<br/>$$
<br/>b^{2}=12 \times 4 \Rightarrow b^{2}=48
<br/>$$
<br/>$\therefore$ From Eq. (i). $\frac{x^{2}}{16}-\frac{y^{2}}{48}=1$
<br/>$\Rightarrow \quad 3 x^{2}-y^{2}=48$</div>


title: the-equation-of-normal-of-x-2-y-2-2-x-4-y-5-0-at-2-1-is
slug: the-equation-of-normal-of-x-2-y-2-2-x-4-y-5-0-at-2-1-is-24579
question: <div class="question">The equation of normal of $x^2+y^2-2 x+4 y-5=0$ at $(2,1)$ is</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y=3 x-5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 y=3 x-4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y=3 x+4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=x+1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=3 x-5$</span> </div>

answer:
<div class="solution">Hints: $0(1,-2) \mathrm{A}(2,1)$<br/>Slope $A \rightarrow \frac{y-1}{-2-1}=\frac{x-2}{1-2}, \quad \frac{y-1}{-3}=\frac{x-2}{-1}=1, y-1=3(x-2)$ $y=3 x-5$</div>


title: the-equation-of-one-of-the-curves-whose-slope-at-any-point-is-equal-to-y-2-x-is
slug: the-equation-of-one-of-the-curves-whose-slope-at-any-point-is-equal-to-y-2-x-is-92423
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation of one of the curves whose slope at any point is equal to $\mathrm{y}+2 \mathrm{x}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=2\left(e^x+x-1\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$y=2\left(e^x-x-1\right)$</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">$y=2\left(e^x-x+1\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=2\left(e^x+x+1\right)$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \frac{d y}{d x}=y+2 x \quad \text { Put } y+2 x=z \Rightarrow \frac{d y}{d x}+z=\frac{d z}{d x} \\<br/>&amp; \frac{d z}{d x}-2=z, \quad \frac{d z}{d x}=z+2 \Rightarrow \int \frac{d z}{z+2}=\int d x \\<br/>&amp; \log (z+2)=x+c, \log (y+2 x+2)=x+c \\<br/>&amp; y+2 x+2=x+c, y=2\left(e^x-x-1\right)<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">$\mathrm{JK}^{-1} \mathrm{~mol}^{-1}$</div>
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-state-for-n-moles-of-an-ideal-gas-is-pv-nrt-where-r-is-a-constant-the-si-unit-for-r-is

slug:
the-equation-of-state-for-n-moles-of-an-ideal-gas-is-pv-nrt-where-r-is-a-constant-the-si-unit-for-r-is-12213

question:
<div class="question">The equation of state for $\mathrm{n}$ moles of an ideal gas is $\mathrm{PV}=\mathrm{nRT}$, where $\mathrm{R}$ is a constant. The SI unit for $\mathrm{R}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{JK}^{-1}$ per molecule</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{JK}^{-1} \mathrm{~mol}^{-1}$</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">$\mathrm{J} \mathrm{Kg}^{-1} \mathrm{~K}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{JK}^{-1} \mathrm{~g}^{-1}$</span> </li> </ul>

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


tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-state-of-a-gas-is-given-by-p-v-3-a-v-b-2-c-t-where-p-v-t-are-pressure-volume-and-temperature-respectively-and-a-b-c-are-constants-the-

slug:
the-equation-of-state-of-a-gas-is-given-by-p-v-3-a-v-b-2-c-t-where-p-v-t-are-pressure-volume-and-temperature-respectively-and-a-b-c-are-constants-the-16179

question:
<div class="question">The equation of state of a gas is given by $\left(p+\frac{a}{v^{3}}\right)\left(V-b^{2}\right)=c T,$ where $p, V, T$ are pressure, volume and temperature respectively and $a, b, c$ are constants. The dimensions of $a$ and $b$ are respectively</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{ML}^{8} \mathrm{T}^{-2}\right.$ and $\left.\mathrm{L}^{3 / 2}\right]$</span> </div>

answer:
<div class="solution">$[P]=\left[\frac{a}{V^{3}}\right] \Rightarrow\left[M L^{-1} T^{-2}\right]=\frac{a}{\left[L^{3}\right]^{3}}$
<br/>$\begin{aligned} \Rightarrow \quad a &amp;=\left[M L^{8} T^{-2}\right] \\ &amp;[V]=\left[b^{2}\right] \Rightarrow\left[L^{3}\right]=b^{2} \Rightarrow b=\left[L^{3 / 2}\right] \end{aligned}$</div>


title: the-equation-of-the-chord-of-the-circle-x-2-y-2-4-x-0-whose-mid-point-is-1-0-is
slug: the-equation-of-the-chord-of-the-circle-x-2-y-2-4-x-0-whose-mid-point-is-1-0-is-47958
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=1$</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation of the chord of the circle $x^2+y^2-4 x=0$ whose mid point is $(1,0)$ is</div>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Wr--fhAEOL2cr5uxdPopCkpGFgmwnKXCyi1pfEQBqbU.original.fullsize.png"/><br/><br/><br/>$$<br/>\text { Equation: } X=1<br/>$$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-circle-passing-through-the-point-11-and-the-points-of-intersection-of-x-2-y-2-6-x-8-0-and-x-2-y-2-6-0-is

slug:
the-equation-of-the-circle-passing-through-the-point-11-and-the-points-of-intersection-of-x-2-y-2-6-x-8-0-and-x-2-y-2-6-0-is-78660

question:
<div class="question">The equation of the circle passing through the point (1,1) and the points of intersection of $x^{2}+y^{2}-6 x-8=0$ and $x^{2}+y^{2}-6=0$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+3 x-5=0$</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">$x^{2}+y^{2}-4 x+2=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}+6 x-4=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}-4 y-2=0$</span> </li> </ul>

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

answer:
<div class="solution">Let $\quad S_{1}=x^{2}+y^{2}-6 x-8=0$
<br/>and $\quad S_{2}=x^{2}+y^{2}-6=0$
<br/>Now, the equation of the circle passing through the point (1,1) and the point of intersection of $S_{1}$ and $S_{2}$ is
<br/>$$
<br/>S_{1}+\lambda S_{2}=0
<br/>$$
<br/>$\Rightarrow \quad\left(x^{2}+y^{2}-6 x-8\right)+\lambda\left(x^{2}+y^{2}-6\right)=0$
<br/>$\Rightarrow \quad(1+\lambda) x^{2}+(1+\lambda) y^{2}-6 x$
<br/>$$
<br/>+(-8-6 \lambda)=0 \ldots 0
<br/>$$
<br/>Since, Eq. (i) passes through the point (1,1)
<br/>$\because$ Put $x=1, y=1$ in Eq. (ii), we get
<br/>$$
<br/>\begin{array}{r}
<br/>(1+\lambda)+(1+\lambda)-6+(-8-6 \lambda)=0 \\
<br/>\Rightarrow \quad-4 \lambda-12=0
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad \lambda=-3$
<br/>On putting the value of ' $\lambda$ ' in Eq. ( 0 , we get
<br/>$$
<br/>\begin{array}{r}
<br/>-2 x^{2}-2 y^{2}-6 x+10=0 \\
<br/>\Rightarrow \quad x^{2}+y^{2}+3 x-5=0
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-common-tangent-with-positive-slope-to-the-parabola-y-2-8-3-x-and-the-hyperbola-4-x-2-y-2-4-is

slug:
the-equation-of-the-common-tangent-with-positive-slope-to-the-parabola-y-2-8-3-x-and-the-hyperbola-4-x-2-y-2-4-is-14029

question:
<div class="question">The equation of the common tangent with positive slope to the parabola $y^{2}=8 \sqrt{3} x$ and the hyperbola $4 x^{2}-y^{2}=4$ is</div>

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

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

answer:
<div class="solution">Equation of tangent in slope form of parabola $y^{2}=8 \sqrt{3} x$ is
<br/>$$
<br/>y=m x+c
<br/>$$
<br/>where, $\quad c=\frac{a}{m}$
<br/>$\therefore$
<br/>$c=\frac{2 \sqrt{3}}{m}$
<br/>Also, tangent to the hyperbola
<br/>$$
<br/>4 x^{2}-y^{2}=4
<br/>$$
<br/>or $\frac{x^{2}}{1}-\frac{y^{2}}{4}=1$ is
<br/>$$
<br/>\begin{array}{l}
<br/>c^{2}=a^{2} m^{2}-b^{2} \\
<br/>c^{2}=1 m^{2}-4
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad\left(\frac{2 \sqrt{3}}{m}\right)^{2}=m^{2}-4$
<br/>$\Rightarrow \quad \frac{12}{m^{2}}=m^{2}-4$
<br/>$\Rightarrow \quad m^{4}-4 m^{2}-12=0$
<br/>$\Rightarrow \quad m^{4}-6 m^{2}+2 m^{2}-12=0$
<br/>$\Rightarrow \quad m^{2}\left(m^{2}-6\right)+2\left(m^{2}-6\right)=0$
<br/>$\Rightarrow \quad\left(m^{2}+2\right)\left(m^{2}-6\right)=0$
<br/>$\Rightarrow \quad m^{2}-6=0$ and $m^{2}+2 \neq 0$
<br/>$\Rightarrow \quad m^{2}=6$
<br/>$\Rightarrow \quad m=\pm \sqrt{6}$
<br/>i.e $m=\sqrt{6}$ as $m$ is positive slope.
<br/>$\therefore$ 
<br/>$y=\sqrt{6} x+\frac{2 \sqrt{3}}{\sqrt{6}}$
<br/>$\Rightarrow \quad y=\sqrt{6} x+\sqrt{2}$</div>


title: the-equation-of-the-directrices-of-the-hyperbola-3-x-2-3-y-2-18-x-12-y-2-0-is
slug: the-equation-of-the-directrices-of-the-hyperbola-3-x-2-3-y-2-18-x-12-y-2-0-is-65373
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation of the directrices of the hyperbola $3 x^{2}-3 y^{2}-18 x+12 y+2=0$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x=3 \pm \sqrt{\frac{13}{6}}$</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">B</span> <span class="option-data">$x=3 \pm \sqrt{\frac{6}{13}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x=6 \pm \sqrt{\frac{13}{3}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=6 \pm \sqrt{\frac{3}{13}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x=3 \pm \sqrt{\frac{13}{6}}$</span> </div>

answer:
<div class="solution">Given equation is
<br/>$$
<br/>3 x^{2}-3 y^{2}-18 x+12 y+2=0
<br/>$$
<br/>It can be written as
<br/>$\frac{(x-3)^{2}}{\left(\sqrt{\frac{13}{3}}\right)^{2}}-\frac{(y-2)^{2}}{\left(\sqrt{\frac{13}{3}}\right)^{2}}=1$
<br/>$\begin{aligned} \text { here, } a &amp;=b=\sqrt{\frac{13}{3}} \\ \therefore \quad e &amp;=\sqrt{1+\frac{b^{2}}{a^{2}}} \\ &amp;=\sqrt{1+1} \\ &amp;=\sqrt{2} \end{aligned}$
<br/>$\therefore$ Equation of the directrix are
<br/>$$
<br/>x=3 \pm \sqrt{\frac{13}{6}}
<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-latus-rectum-of-a-parabola-is-x-y-8-and-the-equation-of-the-tangent-at-the-vertex-is-x-y-12-then-the-length-of-the-latus-rectum-is

slug:
the-equation-of-the-latus-rectum-of-a-parabola-is-x-y-8-and-the-equation-of-the-tangent-at-the-vertex-is-x-y-12-then-the-length-of-the-latus-rectum-is-17135

question:
<div class="question">The equation of the latus rectum of a parabola is $x+y=8$ and the equation of the tangent at the vertex is $x+y=12$. Then the length of the latus rectum is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 \sqrt{2}$ unit</span> </div>

answer:
<div class="solution">Hint: 
<br/>The distance between latus rectum and equation of tangent at vertex is 'a'. Here $\mathrm{a}=\frac{4}{\sqrt{1+1}}=2 \sqrt{2}$
<br/>So, length of latus rectum $=4 \mathrm{a}=8 \sqrt{2}$ unit</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-locus-of-the-point-of-intersection-of-the-straight-lines-x-sin-theta1cos-theta-ya-sin-theta-and-x-sin-theta1-cos-theta-ya-sin-thet

slug:
the-equation-of-the-locus-of-the-point-of-intersection-of-the-straight-lines-x-sin-theta1cos-theta-ya-sin-theta-and-x-sin-theta1-cos-theta-ya-sin-thet-36041

question:
<div class="question">The equation of the locus of the point of intersection of the straight lines \(x \sin \theta+(1-\cos \theta) y=a \sin \theta\) and \(x\) sin \(\theta-(1+\) \(\cos \theta) y+a \sin \theta=0\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(y \pm a x\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(x= \pm\) ay</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(y^2=4 x\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(x^2+y^2=a^2\)</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">\(x^2+y^2=a^2\)</span> </div>

answer:
<div class="solution">Hints: \(y=a \sin \theta\)<br/>\(x=a \cos \theta \text {. }\)<br/>\(x^2+y^2=a^2\)</div>


title: the-equation-of-the-plane-through-123-and-221-and-parallel-to-x-axis-is
slug: the-equation-of-the-plane-through-123-and-221-and-parallel-to-x-axis-is-20893
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation of the plane through (1,2,-3) and (2,-2,1) and parallel to $X$ -axis is</div>

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

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

answer:
<div class="solution">Equation of the plane will be $\left|\begin{array}{ccc}x-1 &amp; y-2 &amp; z+3 \\ 2-1 &amp; -2-2 &amp; 1-(-3) \\ 1 &amp; 0 &amp; 0\end{array}\right|=0$
<br/>$\Rightarrow\left|\begin{array}{ccc}x-1 &amp; y-2 &amp; z+3 \\ 1 &amp; -4 &amp; 4 \\ 1 &amp; 0 &amp; 0\end{array}\right|=0$
<br/>$\Rightarrow \quad 1[4(y-2)+4(z+3)]=0$
<br/>[Expanding along $\left.R_{j}\right]$ 
<br/>$\Rightarrow \quad 4 y-8+4 z+12=0$
<br/>$\Rightarrow \quad 4 y+4 z+4=0$
<br/>$\Rightarrow \quad y+z+1=0$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-plane-through-the-intersection-of-the-planes-x-y-z-1-and-2-x-3-y-z-4-and-parallel-to-the-x-axis-is

slug:
the-equation-of-the-plane-through-the-intersection-of-the-planes-x-y-z-1-and-2-x-3-y-z-4-and-parallel-to-the-x-axis-is-18218

question:
<div class="question">The equation of the plane through the intersection of the planes $x+y+z=1$ and $2 x+3 y-z+4$ and parallel to the $x$-axis is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y+3 z+6=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y+3 z-6=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y-3 z+6=0$</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">$y-3 z-6=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y-3 z+6=0$</span> </div>

answer:
<div class="solution">$\quad(2 x+3 y-z+4)+\lambda(x + y+z-1)=0$<br/>$\overrightarrow{\mathrm{n}}=(2+\lambda) \hat{\mathrm{i}}+(3+\lambda) \hat{\mathrm{j}}+(-1+\lambda) \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{n}} \cdot \overrightarrow{\mathrm{i}}=0 \Rightarrow \lambda=-2$<br/>$\Rightarrow \mathrm{y}-3 z+6=0$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-plane-through-the-point-2-1-3-and-parallel-to-the-lines-3-x-1-2-y-2-4-z-and-2-x-3-y-1-2-z-2-is

slug:
the-equation-of-the-plane-through-the-point-2-1-3-and-parallel-to-the-lines-3-x-1-2-y-2-4-z-and-2-x-3-y-1-2-z-2-is-69329

question:
<div class="question">The equation of the plane through the point $(2,-1,-3)$ and parallel to the lines $\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}$ and $\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$8 x+14 y+13 z+37=0$</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">$8 x-14 y-13 z-37=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 x-14 y-13 z+37=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 x+14 y+13 z+37=0$</span> </div>

answer:
<div class="solution">The equation of plane passing through \((2,-1,-3)\) is \(a(x-2)+b(y+1)+c(z+3)=0\) Also, as the plane is parallel is the given two lines,
<br/>\(\therefore 3 a+2 b-4 c=0\) and \(2 a-3 b+2 c=0\)
<br/>\(\Rightarrow \mathrm{a}=-8, \mathrm{~b}=-14, \mathrm{c}=-13\)
<br/>\(\therefore\) The equation of the required plane is
<br/>\(-8(x-2)-14(y+1)-13(z+3)=0\)
<br/>\(\Rightarrow 8 x+14 y+13 z+37=0\)
<br/>So the answer is (a).</div>


title: the-equation-of-the-plane-which-bisects-the-line-joining-the-points-123-and-345-at-right-angles-is
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
slug:
the-equation-of-the-plane-which-bisects-the-line-joining-the-points-123-and-345-at-right-angles-is-17016

question:
<div class="question">The equation of the plane, which bisects the line joining the points (1,2,3) and (3,4,5) at right angles is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+y+z=9$</span> </div>

answer:
<div class="solution">The given points are $A(1,2,3)$ and $B(3,4,5)$ The direction ratios of line segment $A B$ is given by
<br/>$$
<br/>\begin{array}{l}
<br/>\left(x_{2}-x_{1}\right),\left(y_{2}-y_{1}\right),\left(z_{2}-z_{1}\right) \\
<br/>(3-1),(4-2),(5-3)=(2,2,2)=(1,1,1)
<br/>\end{array}
<br/>$$
<br/>Since, the plane bisects $A B$ at right angles, $\mathbf{A B}$ is the normal to the plane. Which is n Therefore, $\mathbf{n}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$
<br/>Let $c$ be the mid-point of $A B$
<br/>$$
<br/>\begin{aligned}
<br/>\left(\frac{x_{1}+x_{2}}{2},\right.&amp;\left.\frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right) \\
<br/>=&amp; c\left(\frac{1+3}{2}, \frac{2+4}{2}, \frac{3+5}{2}\right) \\
<br/>=&amp; c(2,3,4)
<br/>\end{aligned}
<br/>$$
<br/>Let this vector be $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$
<br/>Hence, the required equation of vector
<br/>$$
<br/>\{\mathbf{r}-(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \mathbf{k})\}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\mathbf{k})=0
<br/>$$
<br/>We know that, $\mathbf{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$
<br/>$$
<br/>\begin{array}{l}
<br/>\Rightarrow\{(x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}})-(2\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})\}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=0 \\
<br/>\Rightarrow \quad(x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}) \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \\
<br/>=(2\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \\
<br/>\Rightarrow x+y+z=2+3+4 \\
<br/>\Rightarrow x+y+z=9
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-equation-of-the-straight-line-passing-through-the-point-4-3-and-making-intercepts-on-the-coordinate-axes-whose-sum-is-1-is-1

slug:
the-equation-of-the-straight-line-passing-through-the-point-4-3-and-making-intercepts-on-the-coordinate-axes-whose-sum-is-1-is-1-69332

question:
<div class="question">The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{x}{2}-\frac{y}{3}=1$, $\frac{x}{-2}+\frac{y}{1}=1$</span> </div>

answer:
<div class="solution">Hint: 
<br/>Let line be $\frac{x}{a}+\frac{y}{b}=1$, then $a+b=-1$
<br/>$\frac{4}{a}-\frac{3}{1+a}=1 \Rightarrow a=\pm 2$</div>


title: the-equation-of-the-tangent-to-the-conic-x-2-y-2-8-x-2-y-11-0-at-2-1-is
slug: the-equation-of-the-tangent-to-the-conic-x-2-y-2-8-x-2-y-11-0-at-2-1-is-74162
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation of the tangent to the conic $x^2-y^2-8 x+2 y+11=0$ at $(2,1)$ is</div>

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

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

answer:
<div class="solution">Hints : Equation of tangent at $\left(x_1, y_1\right)$ is<br/>$$<br/>\begin{aligned}<br/>&amp; x x_1-y y_1-4\left(x+x_1\right)+\left(y+y_1\right)+11=0 \\<br/>&amp; x_1=2 ; y=1<br/>\end{aligned}<br/>$$<br/>$\therefore$ Equation of tangent is<br/>$$<br/>\begin{aligned}<br/>&amp; 2 x-y-4(x+2)+(y+1)+11=0 \\<br/>&amp; \text { or } \quad-2 x-8+12=0<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{array}{ll}<br/>\text { or } &amp; -2 x+4=0 \\<br/>\text { or } &amp; 2 x=4 \\<br/>\text { or } &amp; x=2 \\<br/>\text { or } &amp; x-2=0<br/>\end{array}<br/>$$</div>


title: the-equation-r-cos-3-2-represents
slug: the-equation-r-cos-3-2-represents-33539
question: <div class="question">The equation $r \cos \left(\theta-\frac{\pi}{3}\right)=2$ represents</div>
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a circle</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a parabola</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">an ellipse</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a straight line</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Hint: 
<br/>$r \cos \left(\theta-\frac{\pi}{3}\right)=2 \quad \Rightarrow r \cos \theta \times \frac{1}{2}+r \sin \theta \times \frac{\sqrt{3}}{2}=2 \Rightarrow x+\sqrt{3} y=4(x=r \cos \theta, y=r \sin \theta)$
<br/>$\therefore$ a straight line</div>


title: the-equation-r2-cos-2leftthetafracpi3right2-represents
slug: the-equation-r2-cos-2leftthetafracpi3right2-represents-21238
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation \(r^2 \cos ^2\left(\theta-\frac{\pi}{3}\right)=2\) represents</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a pair of straight lines</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint: } r^2\left(\frac{\cos \theta}{2}+\frac{\sqrt{3}}{2} \sin \theta\right)^2=2 \\<br/>&amp; \Rightarrow \quad(r \cos \theta+\sqrt{3} r \sin \theta)^2=8 \\<br/>&amp; \Rightarrow \quad(x+y \sqrt{3})^2=8 \\<br/>&amp; \Rightarrow \quad(x+y \sqrt{3}-2 \sqrt{2})(x+y \sqrt{3}+2 \sqrt{2})=0 \\<br/>&amp; \Rightarrow \quad \text { a pair of straight lines }<br/>\end{aligned}\)</div>


title: the-equation-sin-x-sin-x-cos-x-k-has-real-solutions-where-k-is-a-real-number-then
slug: the-equation-sin-x-sin-x-cos-x-k-has-real-solutions-where-k-is-a-real-number-then-17897
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation $\sin x(\sin x+\cos x)=k$ has real solutions, where k is a real number. Then,</div>

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

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

answer:
<div class="solution">Let, $f(x)=\sin x(\sin x+\cos x)$
<br/>$=\sin ^{2} x+\sin x \cos x$
<br/>$=\frac{1-\cos 2 x}{2}+\frac{2 \sin x \cos x}{2}$
<br/>$=\frac{1}{2}-\frac{1}{2} \cos 2 x+\frac{1}{2} \sin 2 x$
<br/>$=\frac{1}{2}+\frac{1}{2}(\sin 2 x-\cos 2 x)$
<br/>$\begin{aligned} \text { Now, }-\sqrt{(1)^{2}+(1)^{2}} &amp; \leq \sin 2 x-\cos 2 x \\ &amp; \leq \sqrt{(1)^{2}+(1)^{2}} \end{aligned}$
<br/>$\left[\because \sqrt{a^{2}+b^{2}} \leq a \sin x+b \cos x \leq \sqrt{a^{2}+b^{2}}\right]$
<br/>$\Rightarrow \quad-\sqrt{2} \leq \sin 2 x-\cos 2 x \leq \sqrt{2}$
<br/>$\Rightarrow \quad-\frac{\sqrt{2}}{2} \leq \frac{\sin 2 x-\cos 2 x}{2} \leq \frac{\sqrt{2}}{2}$
<br/>$\Rightarrow \frac{1}{2}-\frac{\sqrt{2}}{2} \leq \frac{1}{2}+\frac{\sin 2 x-\cos 2 x}{2} \leq \frac{1}{2}+\frac{\sqrt{2}}{2}$
<br/>$\Rightarrow \frac{1-\sqrt{2}}{2} \leq f(x) \leq \frac{1+\sqrt{2}}{2}$
<br/>$\therefore \quad \frac{1-\sqrt{2}}{2} \leq k \leq \frac{1+\sqrt{2}}{2}$</div>


title: the-equation-x-3-y-x-2-x-y-0-represents
slug: the-equation-x-3-y-x-2-x-y-0-represents-89623
question: <div class="question">The equation $x^{3}-y x^{2}+x-y=0$ represents</div>
tags: ['Mathematics', 'Pair of Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a hyperbola and two straight lines</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">a straight line</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a parabola and two straight lines</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a straight line and a circle</span> </li> </ul>

answer:
<div class="solution">$x^{3}-y x^{2}+x-y=0$
<br/>$x^{2}(x-y)+(x-y)=0$
<br/>$\left(x^{2}+1\right)(x-y)=0$
<br/>$x^{2}+1 \neq 0$
<br/>$x-y=0$
<br/>$x=y$
<br/>So, the equation represents a straight line.</div>


title: the-equation-x-l-o-g-3-x-2-2-9-l-o-g-3-x-5-3-3-has
slug: the-equation-x-l-o-g-3-x-2-2-9-l-o-g-3-x-5-3-3-has-91438
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation $x^{\left(\log _{3} x\right)^{2}-\frac{9}{2} \log _{3} x+5}=3 \sqrt{3}$ has</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">at least one real root, exactly one irrational root</span> </div>

answer:
<div class="solution">Hint $: 2 t^{3}-9 t^{2}+10 t-3=0, t=\log _{3} x$
<br/>$\Rightarrow(t-3)(t-1)\left(t-\frac{1}{2}\right)=0$
<br/>$\Rightarrow x=3^{3}, 3^{1}, 3^{1 / 2}$</div>


title: the-equation-x-lo-g-x-3-x
slug: the-equation-x-lo-g-x-3-x-97882
question: <div class="question">The equation $x \log x=3-x$</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">has no root in (1,3)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">has exactly one root in (1,3)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x \log x-(3-x)&gt;0$ in [1,3]</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x \log x-(3-x)    &lt; 0$ in [1,3]</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">has exactly one root in (1,3)</span> </div>

answer:
<div class="solution">Let $\quad f(x)=x \log x+x-3$
<br/>$\Rightarrow \quad f^{\prime}(x)=x \cdot \frac{1}{x}+\log x+1$
<br/>$\Rightarrow \quad f^{\prime}(x)=\log x+2&gt;0$
<br/>$\Rightarrow f(1)=-2$ and $f(3)=3 \log 3, f(1) \cdot f(3    &lt; 0)$
<br/>Hence, exactly one root in $x \in(1,3)$ as $f(x)&gt;0$</div>


title: the-equation-y-2-4-x-4-y-k-0-represents-a-parabola-whose-latusrectum-is
slug: the-equation-y-2-4-x-4-y-k-0-represents-a-parabola-whose-latusrectum-is-17377
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation $y^{2}+4 x+4 y+k=0$ represents a parabola whose latusrectum is</div>

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

answer:
<div class="solution">$y^{2}+4 x+4 y+k=0$ $y^{2}+2 \times 2 y+4-4+4 x+k=0$
<br/>$(y+2)^{2}=-4 x-k+4$
<br/>$(y+2)^{2}=-4\left(x-\frac{4+k}{4}\right)$
<br/>$\therefore$ Latusrectum $=4$ units</div>


title: the-equation-z-z-2-3-i-z-2-3-i-z-4-0-represents-a-circle-of-radius
slug: the-equation-z-z-2-3-i-z-2-3-i-z-4-0-represents-a-circle-of-radius-30056
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3 unit</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equation $z \bar{z}+(2-3 i) z+(2+3 i) \bar{z}+4=0$ represents a circle of radius</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2 unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">C</span> <span class="option-data">4 unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6 unit</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>Centre and radius of $z \bar{z}+\bar{a} z+a \bar{z}+b=0$ are $-a$ and $\sqrt{|a|^{2}-b} \therefore$ radius $=\sqrt{13-4}=3$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-equations-of-the-circles-which-touch-both-the-axes-and-the-line-4-x-3-y-12-and-have-centres-in-the-first-quadrant-are

slug:
the-equations-of-the-circles-which-touch-both-the-axes-and-the-line-4-x-3-y-12-and-have-centres-in-the-first-quadrant-are-99073

question:
<div class="question">The equations of the circles, which touch both the axes and the line $4 x+3 y=12$ and have centres in the first quadrant, are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+x-y+1=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}-2 x-2 y+1=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}-12 x-12 y+36=0$</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">$x^{2}+y^{2}-6 x-6 y+36=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$x^{2}+y^{2}-2 x-2 y+1=0$, $x^{2}+y^{2}-12 x-12 y+36=0$</span> </div>

answer:
<div class="solution">Radius $(r)=$ perpendicular distance on line $4 x+3 y=12$ from centre
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a76.png"/> <br/> <br/>$\Rightarrow \quad r=\frac{|4 r+3 r-12|}{\sqrt{16+9}}$
<br/>$\Rightarrow \quad|7 r-12|=5 r$
<br/>$\Rightarrow \quad 7 r-12=\pm 5 r$
<br/>$\therefore \quad 2 r=12 \Rightarrow r=6$
<br/>and $12 r=12 \Rightarrow r=1$
<br/>(1) When centre is (1,1) and radius is $1,$ then equation of circle is
<br/>$$
<br/>(x-1)^{2}+(y-1)^{2}=1
<br/>$$
<br/>$\Rightarrow \quad x^{2}+y^{2}-2 x-2 y+1=0$
<br/>(ii) When centre is (6,6) and radius is 2 , then equation of circle is
<br/>$(x-6)^{2}+(y-6)^{2}=36$
<br/>$\Rightarrow \quad x^{2}+y^{2}-12 x-12 y+36=0$</div>


title: the-equations-of-the-lines-through-1-1-and-making-angles-of-4-5-with-the-line-x-y-0-are
slug: the-equations-of-the-lines-through-1-1-and-making-angles-of-4-5-with-the-line-x-y-0-are-68296
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equations of the lines through $(1,1)$ and making angles of $45^{\circ}$ with the line $x+y=0$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{x}-1=0, \mathrm{y}-1=0$</span> </div>

answer:
<div class="solution">Hints : $m=1, y-1=\frac{m \pm \tan 45}{1 \mp m \tan 45}(x-1), y-1=\frac{(-1) \pm 1}{1 \pm 1}(x-1)$<br/>$y=1, x=1$</div>


title: the-equations-x-2-x-a-0-and-x-2-a-x-1-0-have-a-common-real-root
slug: the-equations-x-2-x-a-0-and-x-2-a-x-1-0-have-a-common-real-root-24008
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equations $x^{2}+x+a=0 \quad$ and
<br/>$x^{2}+a x+1=0$ have a common real root</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">for no value of $a$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">for exactly one value of $a$</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">for exactly two values of $a$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">for exactly three values of $a$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">for exactly one value of $a$</span> </div>

answer:
<div class="solution">Let \(\alpha\) be the common roots
<br/>\(\begin{array}{l}
<br/>\therefore \alpha^{2}+\alpha+a=0 \ldots \text { (i) } \\
<br/>\text { and } \alpha^{2}+a \alpha+1=0 \ldots \text { (ii) } \\
<br/>\frac{\alpha^{2}}{1-a^{2}}=\frac{\alpha}{a-1}=\frac{1}{a-1}[a \neq 1]
<br/>\end{array}\)
<br/>Eliminating \(\alpha\), we get
<br/>\(\begin{array}{l}
<br/>(a-1)^{2}=\left(1-a^{2}\right)(a-1) \\
<br/>\Rightarrow a-1)=1-a^{2} \\
<br/>\Rightarrow a^{2}+a-2=0 \\
<br/>\Rightarrow a^{2}+2 a-a-2=0 \\
<br/>\Rightarrow a(a+2)-1(a+2)=0 \\
<br/>\Rightarrow(a+2)(a-1)=0 \\
<br/>\Rightarrow a=-2 \cdot[\because a \neq 1]
<br/>\end{array}\)</div>


title: the-equations-y-3-x-y-1-are-the-sides-of
slug: the-equations-y-3-x-y-1-are-the-sides-of-24269
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equations $\mathrm{y}= \pm \sqrt{3 x}, \mathrm{y}=1$ are the sides of</div>

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

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

answer:
<div class="solution">Hints: $y=\tan 60^{\circ} x, y=-\tan 60^{\circ} x$<br/>$y=1$, equilateral</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-equilibrium-constant-for-the-following-reactions-are-given-at-2-5-c-2-a-b-c-k-1-10-2-b-c-d-k-2-16-2-c-d-2-p-k-3-25-the-equilibrium-constant-for-th

slug:
the-equilibrium-constant-for-the-following-reactions-are-given-at-2-5-c-2-a-b-c-k-1-10-2-b-c-d-k-2-16-2-c-d-2-p-k-3-25-the-equilibrium-constant-for-th-61639

question:
<div class="question">The equilibrium constant for the following reactions are given at $25^{\circ} \mathrm{C}$
<br/>$\begin{array}{l}
<br/>2 \mathrm{~A} \rightleftharpoons \mathrm{B}+\mathrm{C}, \mathrm{K}_{1}=1.0 \\
<br/>2 \mathrm{~B} \rightleftharpoons \mathrm{C}+\mathrm{D}, \mathrm{K}_{2}=16 \\
<br/>2 \mathrm{C}+\mathrm{D} \rightleftharpoons 2 \mathrm{P}, \mathrm{K}_{3}=25
<br/>\end{array}$
<br/>The equilibrium constant for the reaction $\mathrm{P} \rightleftharpoons \mathrm{A}+\frac{1}{2} \mathrm{~B}$ at $25^{\circ} \mathrm{C}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{20}$</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">B</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{42}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">21</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{20}$</span> </div>

answer:
<div class="solution">Hint:
<br/>We can manipulate the given equations as follows
<br/>$\frac{1}{2} \mathrm{~B}+\frac{1}{2} \mathrm{C} \rightleftharpoons \mathrm{A}, \quad \mathrm{K}_{1}^{\prime}=1$
<br/>$\frac{1}{2} \mathrm{C}+\frac{1}{2} \mathrm{D} \rightleftharpoons \mathrm{B}, \quad \mathrm{K}_{2}^{\prime}=\frac{1}{\sqrt{16}}$
<br/>$\mathrm{P} \rightleftharpoons \mathrm{C}+\frac{1}{2} \mathrm{D}, \quad \mathrm{K}_{\mathrm{3}}^{\prime}=\frac{1}{\sqrt{25}}$
<br/>$\mathrm{P} \rightleftharpoons \mathrm{A}+\frac{1}{2} \mathrm{~B}, \quad \mathrm{~K}_{\text {final }}=1 \times \frac{1}{4} \times \frac{1}{5}=\frac{1}{20}$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
the-equilibrium-constant-for-the-reaction-n-2-g-o-2-g-2-no-g-is-4-1-0-4-at-2000-k-in-presence-of-a-catalyst-the-equilibrium-is-attained-10-times-faste

slug:
the-equilibrium-constant-for-the-reaction-n-2-g-o-2-g-2-no-g-is-4-1-0-4-at-2000-k-in-presence-of-a-catalyst-the-equilibrium-is-attained-10-times-faste-10983

question:
<div class="question">The equilibrium constant for the reaction $\mathrm{N}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})$ is $4 \times 10^{-4}$ at $2000 \mathrm{~K}$. In presence of a catalyst the equilibrium is attained 10 times faster. Therefore, the equilibrium constant, in presence of the catalyst at $2000 \mathrm{~K}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$4 \times 10^{-4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \times 10^{-3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \times 10^{-5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.5 \times 10^{-4}$</span> </li> </ul>

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

answer:
<div class="solution">As per Van't Hoff Equation
<br/>$\frac{\mathrm{d}\left(\ln \mathrm{K}_{\mathrm{eq}}\right)}{\mathrm{dT}}=\frac{\Delta \mathrm{H}_{\mathrm{T}}}{\mathrm{RT}^{2}}$
<br/>As catalyst don't affect enthalpy of reaction $\left(\Delta \mathrm{H}_{\mathrm{r}}\right)$ and Temperature is constant. Therefore, $\mathrm{K}_{\mathrm{eq}}$ (equilibrium constant) will remain constant i.e. $4 \times 10^{-4}$.</div>


title: the-equilibrium-constant-k-of-a-reaction-may-be-written-as
slug: the-equilibrium-constant-k-of-a-reaction-may-be-written-as-62080
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The equilibrium constant $(\mathrm{K})$ of a reaction may be written as :</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$K=e^{-\Delta \mathrm{G}^0 / R T}$</span> </div>

answer:
<div class="solution">$\Delta \mathrm{G}^{\mathrm{o}}=-\mathrm{RT} \ln \mathrm{K}$<br/>$\Rightarrow \frac{\Delta \mathrm{G}^{\mathrm{o}}}{-\mathrm{RT}}=\ln \mathrm{K}$<br/>$\mathrm{K}=e^{-\Delta \mathrm{G}^{\circ} / \mathrm{RT}}$</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-equivalent-conductance-of-mathrmnacl-mathrmhcl-and-mathrmch3-mathrmcoona-at-infinite-dilution-are-1264542616-and-91-mathrmohm1-mathrmcm2-mathrmeq1

slug:
the-equivalent-conductance-of-mathrmnacl-mathrmhcl-and-mathrmch3-mathrmcoona-at-infinite-dilution-are-1264542616-and-91-mathrmohm1-mathrmcm2-mathrmeq1-42178

question:
<div class="question">The equivalent conductance of \(\mathrm{NaCl}, \mathrm{HCl}\) and \(\mathrm{CH}_3 \mathrm{COONa}\) at infinite dilution are \(126.45,426.16\) and 91 \(\mathrm{ohm}^{-1} \mathrm{~cm}^2 \mathrm{eq}^{-1}\) respectively at \(25^{\circ} \mathrm{C}\). The equivalent conductance of acetic acid (at infinite dilution) would be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(461.61 \mathrm{~ohm}^{-1} \mathrm{~cm}^2 \mathrm{~eq}^{-1}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(390.71 \mathrm{~ohm}^{-1} \mathrm{~cm}^2 \mathrm{~eq}^{-1}\)</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">cannot be determined from the given data</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(208.71 \mathrm{~ohm}^{-1} \mathrm{~cm}^2 \mathrm{~eq}^{-1}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(390.71 \mathrm{~ohm}^{-1} \mathrm{~cm}^2 \mathrm{~eq}^{-1}\)</span> </div>

answer:
<div class="solution">Hint : According to Kohlrausch's law<br/>\(\begin{aligned}<br/>&amp; \wedge_{\mathrm{CH}_3 \mathrm{COOH}}^0=\wedge_{\mathrm{CH}_{3}\mathrm{COONa}}^0 +\wedge_{\mathrm{HCl}}^0-\wedge_{\mathrm{NaCl}}^0 \\<br/>&amp; \wedge_{\mathrm{CH}_3 \mathrm{COOH}}^0=(91+426.16-126.45) \mathrm{~ohm}^{-1} \mathrm{~cm}^2 \mathrm{~eq}^{-1} \\<br/>&amp; \wedge_{\mathrm{CH}_3 \mathrm{COOH}}^0=390.71 \mathrm{~ohm}^{-1} \mathrm{~cm}^2 \mathrm{~eq}^{-1}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$r$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-equivalent-resistance-between-the-points-a-and-b-of-the-electrical-network-shown-in-the-figure-is-1

slug:
the-equivalent-resistance-between-the-points-a-and-b-of-the-electrical-network-shown-in-the-figure-is-1-76837

question:
<div class="question">The equivalent resistance between the points $a$ and $b$ of the electrical network shown in the figure is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q34.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 r$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 r$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 r$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$r$</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> </ul>

answer:
<div class="solution">The equlvalent circuit for the given electrical network is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a34.png"/> <br/> <br/>This is a balanced Wheatstone bridge, so the arm containing $2 r$ not in use. So, we have<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a34a.png"/> <br/> <br/>So, the equivalent resistance between a and bis
<br/>$$
<br/>\frac{1}{r^{\prime}}=\frac{1}{2 r}+\frac{1}{2 r} \Rightarrow r^{\prime}=\frac{2 r}{2}=r
<br/>$$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$r$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-equivalent-resistance-between-the-points-a-and-b-of-the-electrical-network-shown-in-the-figure-is

slug:
the-equivalent-resistance-between-the-points-a-and-b-of-the-electrical-network-shown-in-the-figure-is-55523

question:
<div class="question">The equivalent resistance between the points $a$ and $b$ of the electrical network shown in the figure is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_q34.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 r$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 r$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 r$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$r$</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> </ul>

answer:
<div class="solution">The equlvalent circuit for the given electrical network is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a34.png"/> <br/> <br/>This is a balanced Wheatstone bridge, so the arm containing $2 r$ not in use. So, we have<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a34a.png"/> <br/> <br/>So, the equivalent resistance between a and bis
<br/>$$
<br/>\frac{1}{r^{\prime}}=\frac{1}{2 r}+\frac{1}{2 r} \Rightarrow r^{\prime}=\frac{2 r}{2}=r
<br/>$$</div>


tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
the-equivalent-weight-of-k-2-cr-2-o-7-in-acidic-medium-is-expressed-in-terms-of-its-molecular-weight-m-as

slug:
the-equivalent-weight-of-k-2-cr-2-o-7-in-acidic-medium-is-expressed-in-terms-of-its-molecular-weight-m-as-31586

question:
<div class="question">The equivalent weight of $\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}$ in acidic medium is expressed in terms of its molecular weight (M) as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$M / 3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$M / 4$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$M / 6$</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">$M / 7$</span> </li> </ul>

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

answer:
<div class="solution">In acidic medium,
<br/>$$
<br/>\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \longrightarrow \mathrm{Cr}^{3+}
<br/>$$
<br/>Equivalent weight of $\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}=\frac{\text { Molecular weight }}{\text { Change in oxidation state of } \mathrm{c}}$
<br/>$$
<br/>=\frac{M}{2(6-3)}=\frac{M}{6}
<br/>$$</div>


tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-equivalent-weight-of-mathrmkio3-in-the-given-reaction-is-mathrmm-molecular-mass-2-mathrmcrmathrmoh34-mathrmohmathrmkio3-rightarrow-2-mathrmcro4-25

slug:
the-equivalent-weight-of-mathrmkio3-in-the-given-reaction-is-mathrmm-molecular-mass-2-mathrmcrmathrmoh34-mathrmohmathrmkio3-rightarrow-2-mathrmcro4-25-71903

question:
<div class="question">The equivalent weight of \(\mathrm{KIO}_3\) in the given reaction is (\(\mathrm{M}=\) molecular mass) :<br/>\(2 \mathrm{Cr}(\mathrm{OH})_3+4 \mathrm{OH}^{-}+\mathrm{KIO}_3 \rightarrow 2 \mathrm{CrO}_4{ }^{2-}+5 \mathrm{H}_2 \mathrm{O}+\mathrm{KI}\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{M}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{M} / 2\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\mathrm{M} / 6\)</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">\(\mathrm{M} / 8\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{M} / 6\)</span> </div>

answer:
<div class="solution">Hint : \(\mathrm{Cr}(\mathrm{OH})_3+4 \overline{\mathrm{O}} \mathrm{H}+\mathrm{K~} \mathrm{I}^{+5} \mathrm{O}_3 \longrightarrow 2 \mathrm{CrO}_4^{-2}+5 \mathrm{H}_2 \mathrm{O}+\mathrm{K~} \mathrm{I}^{-1}\)<br/>Change in oxidation state of iodine \(=6\)<br/>\(\therefore\) Equivalent weight of \(\mathrm{KIO}_3=\mathrm{M} / 6\)</div>


title: the-even-function-of-the-following-is
slug: the-even-function-of-the-following-is-10210
question: <div class="question">The even function of the following is</div>
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(f(x)=\frac{a^x+a^{-x}}{a^x-a^{-x}}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(f(x)=\frac{a^x+1}{a^x-1}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(f(x)=x \cdot \frac{a^x-1}{a^x+1}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{f}(\mathrm{x})=\log _2\left(\mathrm{x}+\sqrt{\mathrm{x}^2+1}\right)\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: \(f(-x)=(-x) \frac{a^{-x}-1}{a^{-x}+1}\)<br/>\(\begin{aligned}<br/>&amp; =(-n) \frac{1-a^x}{1+a^x} \\<br/>&amp; =n \frac{\left(a^x-1\right)}{\left(a^x+1\right)}=f(x)<br/>\end{aligned}\)</div>


title: the-exact-order-of-acidity-of-the-compounds-pnitrophenol-acetic-acid-acetylene-and-ethanol-is-1
tags: ['Chemistry', 'General Organic Chemistry', 'JEE Main']
source_db: db\MarksBatch2_P2.db
slug:
the-exact-order-of-acidity-of-the-compounds-pnitrophenol-acetic-acid-acetylene-and-ethanol-is-1-10423

question:
<div class="question">The exact order of acidity of the compounds p-nitrophenol, acetic acid, acetylene and ethanol is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">acetylene $     &lt; $ ethanol $ &lt; $ p-nitrophenol $ &lt; $ acetic acid</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Ewy4UEuhUhsQzNd2LMeUeZtf-lRvUZmlxyeeoAaILiE.original.fullsize.png"/><br/> <br/>Acetic acid is most acidic as its conjugte base has equivalent Resonating structure followed by para-nitrophenol. Ethanol is less acidic then para-nitrophenol as conjugate base of p-nitrophenol is resonance stablized but not ethoxide $\left(\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{O}\right)$ Also electronegativity of carbon is lesser than oxygen, hence acetylene is less acidic.</div>


title: the-exact-order-of-boiling-points-of-the-compounds-n-pentane-isopentane-butanone-and-1-butanol-is
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
slug:
the-exact-order-of-boiling-points-of-the-compounds-n-pentane-isopentane-butanone-and-1-butanol-is-41468

question:
<div class="question">The exact order of boiling points of the compounds $\mathrm{n}$-pentane, isopentane, butanone and 1 -butanol is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">isopentane $     &lt; \mathrm{n}$-pentane $ &lt; $ butanone $ &lt; 1$-butanol</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5mxerLYNWXzvzUn_ZcMi10ShyVxn0duIV9KmQtPb92o.original.fullsize.png"/><br/> <br/>As intermolecular force varies as:
<br/>H bonding &gt; dipole dipole force &gt; vander waals force
<br/>$\therefore$ boiling point order
<br/>$\mathrm{n}$-butanol $&gt;$ butanone $&gt;\mathrm{n}$-Pentane $&gt;$ iso pentane</div>


title: the-expression-1-i-n-2-1-i-n-equals
slug: the-expression-1-i-n-2-1-i-n-equals-56164
question: <div class="question">The expression $\frac{(1+i)^{n}}{(1-i)^{n-2}}$ equals</div>
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-i^{n+1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$i^{n+1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-2 i^{n+1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

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

answer:
<div class="solution">$(\mathrm{c}) \frac{(1+i)^{n}}{(1-i)^{n-2}}=\frac{(1+i)^{n}}{(1-i)^{n}(1-i)^{-2}}$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=\left(\frac{1+i}{1-i}\right)^{n}(1-i)^{2} \\
<br/>=&amp;\left(\frac{1+i}{1-i} \times \frac{1+i}{1+i}\right)^{n}\left(1+i^{2}-2 i\right) \\
<br/>=&amp;\left(\frac{1+2 i+i^{2}}{1-i^{2}}\right)^{n}\left(1+i^{2}-2 i\right) \\
<br/>=&amp;\left(\frac{1+2 i-1}{1-(-1)}\right)^{n}(1-1-2 i) \\
<br/>=&amp;\left(\frac{2 i}{2}\right)^{n}(-2 i) \\
<br/>=&amp; i^{n}(-2 i)=-2^{n+1}
<br/>\end{aligned}
<br/>$$</div>


title: the-expression-a-a-b-b-aa-a-b-simplifies-to
slug: the-expression-a-a-b-b-aa-a-b-simplifies-to-25871
question: <div class="question">The expression $\bar{A}(A+B)+(B+A A)(A+\bar{B})$ simplifies to</div>
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$A+B$</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$A+B$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$A B$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\overline{\mathrm{A}+\mathrm{B}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\bar{A}+\bar{B}$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \bar{A}(A+B)+(B+A A)(A+\bar{B}) \\=&amp; \bar{A} A+\bar{A} B+(B+A)(A+\bar{B}) \\=&amp; O+\bar{A} B+A B+A+B \bar{B}+A \bar{B} \\=&amp;(\bar{A}+A) B+A+O+A \bar{B} \\=&amp; 1 . B+A(1+\bar{B}) \\=&amp; A+B \end{aligned}$</div>


title: the-expression-a-x-2-b-x-c-a-b-and-c-are-real-has-the-same-sign-as-that-of-a-for-all-x-if
slug: the-expression-a-x-2-b-x-c-a-b-and-c-are-real-has-the-same-sign-as-that-of-a-for-all-x-if-98138
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The expression $a x^{2}+b x+c(a, b$ and $c$ are real $)$ has the same sign as that of a for all $x$ if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$b^{2}-4 a c&gt;0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$b^{2}-4 a c \neq 0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$b^{2}-4 a c \leq 0$</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">$\mathrm{b}$ and $\mathrm{c}$ have the same sign as that of $\mathrm{a}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$b^{2}-4 a c \leq 0$</span> </div>

answer:
<div class="solution">Hint:
<br/>C-l: If $a&gt;0, a x^{2}+b x+c&gt;0, b^{2}-4 a c    &lt; 0, \quad C-I I:$ If $a \leq 0, a x^{2}+b x+c \leq 0, D \leq 0$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-expression-fracint0nx-d-xint0nx-d-x-where-x-and-x-are-respectively-integral-and-fractional-part-of-x-and-n-in-mathbbn-is-equal-to

slug:
the-expression-fracint0nx-d-xint0nx-d-x-where-x-and-x-are-respectively-integral-and-fractional-part-of-x-and-n-in-mathbbn-is-equal-to-28359

question:
<div class="question">The expression \(\frac{\int_0^n[x] d x}{\int_0^n\{x\} d x}\), where \([x]\) and \(\{x\}\) are respectively integral and fractional part of \(x\) and \(n \in \mathbb{N}\), is equal to</div>

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

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

answer:
<div class="solution">Hint : \(\frac{\int_0^n[x] d x}{\int_0^n\{x\} d x}=\frac{I_1}{I_2}\)<br/>\(\begin{aligned} &amp; I_1=\int_0^1[x] d x+\int_1^2[x] d x+\ldots+\int_{n-1}^n[x] d x \\ &amp; =0+1+\ldots .+(n-1)=\frac{n(n-1)}{2} \\ &amp; I_2=\int_0^n\{x\} d x=n \int_0^1\{x\} d x=n\left[\frac{x^2}{2}\right]_0^1=\frac{n}{2} \\ &amp; \therefore \frac{I_1}{I_2}=\frac{n(n-1)}{n}=n-1\end{aligned}\)</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-family-of-curves-yea-sin-x-where-a-is-arbitrary-constant-is-represented-by-the-differential-equation

slug:
the-family-of-curves-yea-sin-x-where-a-is-arbitrary-constant-is-represented-by-the-differential-equation-22432

question:
<div class="question">The family of curves \(y=e^{a \sin x}\), where ' \(a\) ' is arbitrary constant, is represented by the differential equation</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(y \log y=\tan x \frac{d y}{d x}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned} &amp; \Rightarrow \log y=a \sin x \\ &amp; \Rightarrow \frac{d}{d x}\left(\frac{\log y}{\sin x}\right)=\frac{d}{d x}(a) \Rightarrow \frac{\sin x\left(\frac{d y}{d x}\right)}{y}-\log y \times \cos x=0 \\ &amp; \Rightarrow y \log y=\tan x \times \frac{d y}{d x}\end{aligned}\)</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-figure-represents-two-equipotential-lines-in-mathrmxmathrmy-plane-for-an-electric-field-the-mathrmxcomponent-mathrmemathrmx-of-the-electric-field-

slug:
the-figure-represents-two-equipotential-lines-in-mathrmxmathrmy-plane-for-an-electric-field-the-mathrmxcomponent-mathrmemathrmx-of-the-electric-field-31018

question:
<div class="question">The figure represents two equipotential lines in \(\mathrm{x}-\mathrm{y}\) plane for an electric field. The \(\mathrm{x}\)-component \(\mathrm{E}_{\mathrm{x}}\) of the electric field in space between these equipotential lines is,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0ozGyFKcoPyuKE1DcIxbU5bWdefHb9f3Qwj_yGobOeo.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(100 \mathrm{~V} / \mathrm{m}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(-100 \mathrm{~V} / \mathrm{m}\)</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">\(200 \mathrm{~V} / \mathrm{m}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-200 \mathrm{~V} / \mathrm{m}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-100 \mathrm{~V} / \mathrm{m}\)</span> </div>

answer:
<div class="solution">Hint : \(E_{\mathrm{x}}=-\frac{d v}{d x}=-\left(\frac{4-2}{2}\right)=-1 \mathrm{~V} / \mathrm{cm}=-100 \mathrm{~V} / \mathrm{m}\)</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-figure-shows-two-identical-parallel-plate-capacitors-a-and-b-of-capacitances-c-connected-to-a-battery-the-key-mathrmk-is-initially-closed-the-swit

slug:
the-figure-shows-two-identical-parallel-plate-capacitors-a-and-b-of-capacitances-c-connected-to-a-battery-the-key-mathrmk-is-initially-closed-the-swit-29193

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/WQGnONrUDMcHTPchaG4wVwZ9mHyO3-Plipl2f-FefwI.original.fullsize.png"/><br/><br/>The figure shows two identical parallel plate capacitors \(A\) and \(B\) of capacitances \(C\) connected to a battery. The key \(\mathrm{K}\) is initially closed. The switch is now opened and the free spaces between the plates of the capacitors are filled with a dielectric constant 3. 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">Which the switch is closed, total energy stored in the two capacitors is \(\mathrm{CV}^2\)</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">When the switch is opened, no charge is stored in the capacitor B</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">When the switch is opened, energy stored in the capacitor \(\mathrm{B}\) is \(\frac{3}{2} \mathrm{CV}^2\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">When the switch is opened, total energy stored in two capacitors is \(\frac{5}{2} \mathrm{CV}^2\)</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">Which the switch is closed, total energy stored in the two capacitors is \(\mathrm{CV}^2\), When the switch is opened, total energy stored in two capacitors is \(\frac{5}{2} \mathrm{CV}^2\)</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/li2RykiqUMUbWEIP0euNBQGhjI5QcGCXqCdJwHQSz0g.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \mathrm{U}=\frac{1}{2} \times 2 \mathrm{CV}^2 \\ &amp; =\mathrm{CV}^2\end{aligned}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/7GJDviDxdEl483NBNUp5HmKY2zz-lZStyG5Gk3Jlnv4.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \mathrm{U}_{\mathrm{A}}=\frac{1}{2}(\mathrm{KC}) \mathrm{V}^2=\frac{3}{2} \mathrm{CV}^2 \\ &amp; \mathrm{U}_B=\frac{\mathrm{q}^2}{2 \mathrm{KC}}=\frac{[\mathrm{CV}]^2}{2 \mathrm{KC}}=\frac{\mathrm{CV}^2}{2 \mathrm{~K}}=\frac{1}{6} \mathrm{CV}^2\end{aligned}\)<br/>\(\underline{\text{Total energy when switch is open}}\)<br/>\(\begin{aligned}<br/>&amp; \mathrm{U}=\frac{1}{2} \mathrm{KCV}^2+\frac{1}{6} \mathrm{CV}^2 \\<br/>&amp; =\frac{3}{2} \mathrm{CV}^2+\frac{1}{6} \mathrm{CV}^2 \\<br/>&amp; =\frac{10}{6} \mathrm{CV}^2=\frac{5}{3} \mathrm{CV}^2<br/>\end{aligned}\)</div>


title: the-first-electron-affinity-of-c-n-and-o-will-be-of-the-order
slug: the-first-electron-affinity-of-c-n-and-o-will-be-of-the-order-12211
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The first electron affinity of $\mathrm{C}, \mathrm{N}$ and $\mathrm{O}$ will be of the order</div>

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

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

answer:
<div class="solution">Key Point
<br/>(i) Half-filled/fully-filled configuration have (-) ve or low electron affinity i.e., (+) ve or high electron gain enthalpy.
<br/>(ii) Smaller the size, more is value of electron affinity, i.e. more $(-)$ ve will be the electron gain enthalpy.
<br/>On moving across a period, the size of atoms decreases due to increase in nuclear charge.
<br/>(i) Electron configuration of the elements are shown below. 
<br/>$C(Z=6)=1 s^{2} 2 s^{2} 2 p^{2}$
<br/>$\mathrm{N}(Z=7)=1 s^{2} 2 s^{2} 2 p^{3}$
<br/>$\mathrm{O}(Z=8)=1 \mathrm{s}^{2} 2 \mathrm{s}^{2} 2 p^{4}$
<br/>(ii) Due to half-filled electronic configuration of nitrogen (i.e. 3 electrons in $3 p$ orbitals), it is highly stable and has $(+)$ ve value of electron gain enthalpy (i.e. about $30.9 \mathrm{kJ} / \mathrm{mol}$ ).
<br/>(iii) As size of ' $\mathrm{O}^{\prime},$ is smaller than that of ${ }^{\prime} \mathrm{C}^{\prime},{ }^{\prime} \mathrm{O}^{\prime}$
<br/>atom has higher $(-)$ ve electron gain enthalpy (about $-141.1 \mathrm{kJ} / \mathrm{mol}$ ) than that of carbon (C-atom) [about $-122.3 \mathrm{kJ} / \mathrm{mol}]$.
<br/>Hence, correct order is $\mathrm{O}&gt;\mathrm{C}&gt;\mathrm{N}$ and $(\mathrm{b})$ is the correct option.</div>


title: the-focus-of-the-conic-x-2-6-x-4-y-1-0-is
slug: the-focus-of-the-conic-x-2-6-x-4-y-1-0-is-65749
question: <div class="question">The focus of the conic $x^{2}-6 x+4 y+1=0$ is</div>
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(2,3)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(3,2)</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">(1,4)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">(3,1)</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>x^{2}-6 x+4 y+1=0
<br/>$$
<br/>$\Rightarrow \quad(x-3)^{2}-9+4 y+1=0$
<br/>$\Rightarrow \quad(x-3)^{2}+4 y-8=0$
<br/>$\Rightarrow \quad(x-3)^{2}=-4(y-2)$
<br/>It represents parabola whose vertex is (3,2)
<br/>$\therefore$ Focus $=(3,-1+2)=(3,1)$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-following-equilibrium-constants-are-given-n-2-3-h-2-2-nh-3-k-1-n-2-o-2-2-no-k-2-h-2-2-1-o-2-h-2-o-k-3-the-equilibrium-constant-for-the-oxidation-o

slug:
the-following-equilibrium-constants-are-given-n-2-3-h-2-2-nh-3-k-1-n-2-o-2-2-no-k-2-h-2-2-1-o-2-h-2-o-k-3-the-equilibrium-constant-for-the-oxidation-o-43883

question:
<div class="question">The following equilibrium constants are given
<br/>$\begin{array}{l}
<br/>\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3} ; K_{1} \\
<br/>\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO} ; K_{2}
<br/>\end{array}$
<br/>$\mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{H}_{2} \mathrm{O} ; K_{3}$
<br/>The equilibrium constant for the oxidation of 2 mole of $\mathrm{NH}_{3}$ to give NO is</div>

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

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

answer:
<div class="solution">$\begin{array}{l}<br/>\text {Given, } \\<br/>\begin{aligned}<br/>\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3} ; K_{1} &amp;=\frac{\left[\mathrm{NH}_{3}\right]^{2}}{\left[\mathrm{N}_{2}\right]\left[\mathrm{H}_{2}\right]^{3}} \\<br/>\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons &amp; 2 \mathrm{NO} ; K_{2}=\frac{[\mathrm{NO}]^{2}}{\left[\mathrm{N}_{2}\right]\left[\mathrm{O}_{2}\right]} \\<br/>\mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons &amp; \mathrm{H}_{2} \mathrm{O} ; K_{3}=\frac{\left[\mathrm{H}_{2} \mathrm{O}\right]}{\left[\mathrm{H}_{2}\right]\left[\mathrm{O}_{2}\right]^{1 / 2}}<br/>\end{aligned}<br/>\end{array}$<br/>The chemical equation for the oxidation of 2 mol of<br/>$\mathrm{NH}_{3}$ to give NO is<br/>$2 \mathrm{NH}_{3}+\frac{5}{2} \mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO}+3 \mathrm{H}_{2} \mathrm{O}$<br/>(ii) and To get the equation (iv) from equation (i),<br/>(iii) following steps are followed:<br/>Reversing equation (i), we get<br/>$2 \mathrm{NH}_{3} \rightleftharpoons \mathrm{N}_{2}+3 \mathrm{H}_{2} \text { so } K^{\prime}=\frac{1}{K_{1}}$<br/>Multiplying cquation (iii) by 3, we get<br/>$3 \mathrm{H}_{2}+\frac{3}{2} \mathrm{O}_{2} \rightleftharpoons 3 \mathrm{H}_{2} \mathrm{O} \mathrm{so} \mathrm{K}^{\prime}=K_{3}^{3}$<br/>Adding equation (ii) and (vi)<br/>$\mathrm{N}_{2}+3 \mathrm{H}_{2}+\frac{5}{2} \mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO}+3 \mathrm{H}_{2} \mathrm{O}$<br/>so, $K^{\prime}=K_{2} \cdot K_{3}^{3}$<br/>On combining (v) and (vii), we get the required equation having equilibrium constant $K^{\prime}=K_{2} \cdot \frac{K_{3}^{3}}{K_{1}}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2$</span> </div>
answer: <div class="solution">$t_{1 / 2} \propto \frac{1}{a^{n-1}}$<br/>Hence (C)</div>
source_db: db\MarksBatch2_P2.db
title:
the-following-graph-shows-how-t-12-halflife-of-a-reactant-r-changes-with-the-initial-reactant-concentration-a-0-the-order-of-the-reaction-will-be-1

slug:
the-following-graph-shows-how-t-12-halflife-of-a-reactant-r-changes-with-the-initial-reactant-concentration-a-0-the-order-of-the-reaction-will-be-1-45170

question:
<div class="question">The following graph shows how $\mathrm{T}_{1 / 2}$ (half-life) of a reactant $\mathrm{R}$ changes with the initial reactant concentration $a_0$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/tZGtvEdXz_z8XAFGceIrI6XWqdC-FpXjr44QVU_wOxg.original.fullsize.png"/><br/><br/>The order of the reaction will be :</div>

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


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2$</span> </div>
answer: <div class="solution">$t_{1 / 2} \propto \frac{1}{a^{n-1}}$<br/>Hence (C)</div>
source_db: db\MarksBatch2_P2.db
title:
the-following-graph-shows-how-t-12-halflife-of-a-reactant-r-changes-with-the-initial-reactant-concentration-a-0-the-order-of-the-reaction-will-be

slug:
the-following-graph-shows-how-t-12-halflife-of-a-reactant-r-changes-with-the-initial-reactant-concentration-a-0-the-order-of-the-reaction-will-be-36945

question:
<div class="question">The following graph shows how $\mathrm{T}_{1 / 2}$ (half-life) of a reactant $\mathrm{R}$ changes with the initial reactant concentration $a_0$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/tZGtvEdXz_z8XAFGceIrI6XWqdC-FpXjr44QVU_wOxg.original.fullsize.png"/><br/><br/>The order of the reaction will be :</div>

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-foot-of-the-perpendicular-drawn-from-the-point-184-on-the-line-joining-the-point-0114-and-231-is

slug:
the-foot-of-the-perpendicular-drawn-from-the-point-184-on-the-line-joining-the-point-0114-and-231-is-29674

question:
<div class="question">The foot of the perpendicular drawn from the point (1,8,4) on the line joining the point (0,-11,4) and (2,-3,1) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(4,5,2)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(-4,5,2)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(4,-5,2)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">(4,5,-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">(4,5,-2)</span> </div>

answer:
<div class="solution">Equation of line joining points (0,-11,4) and (2,-3,1) is
<br/>$$
<br/>\frac{x-2}{2}=\frac{y+3}{8}=\frac{z-1}{-3}=\lambda
<br/>$$
<br/>Let $P$ is any point of the above line then coordinate of $P$ is $(2 \lambda+2,8 \lambda-3,-3 \lambda+1)$
<br/>$\therefore \mathrm{DR}^{\prime} \mathrm{s}$ of $P Q$ is $(2 \lambda+1,8 \lambda-11,-3 \lambda-3$
<br/>Now, $(2 \lambda+1)(4+(8 \lambda-11)(8)+(-3 \lambda-3) (-3)=0$
<br/>$\Rightarrow \quad 4 \lambda+2+64 \lambda-88+9 \lambda+9=0$
<br/>$77 \lambda-77=0$
<br/> <br/>$\therefore$ Required foot of perpendicular.
<br/>$$
<br/>P(4,5,-2)
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a50.png"/></div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-force-f-acting-on-a-particle-moving-in-a-straight-line-is-shown-below-what-is-the-work-done-by-the-force-on-the-particle-in-the-1-st-meter-of-the-

slug:
the-force-f-acting-on-a-particle-moving-in-a-straight-line-is-shown-below-what-is-the-work-done-by-the-force-on-the-particle-in-the-1-st-meter-of-the-51975

question:
<div class="question">The force $\mathrm{F}$ acting on a particle moving in a straight line is shown below. What is the work done by the force on the particle in the $1^{\text {st }}$ meter of the trajectory?<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/NB0LTrUsup820TVd0kaycpwu_ofkA98Mnz97ZQhKLiY.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \mathrm{~J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10 \mathrm{~J}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15 \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2.5 \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Work done in 1 meter $=$ area of shaded curve $=1 / 2 \times 1 \times 5=2.5 \mathrm{~J}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Gb-PglyWx0MBhyffsnhlNNhp6FZV2ZuxT1-8mGdtbZk.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-formal-potential-of-f-e-3-f-e-2-in-a-sulphuric-acid-and-phosphoric-acid-mixture-e-061-v-is-much-lower-than-the-standard-potential-e-077-v-this-is-

slug:
the-formal-potential-of-f-e-3-f-e-2-in-a-sulphuric-acid-and-phosphoric-acid-mixture-e-061-v-is-much-lower-than-the-standard-potential-e-077-v-this-is-99337

question:
<div class="question">The formal potential of $Fe^{3+}/Fe^{2+}$ in a sulphuric acid and phosphoric acid mixture $\left(E^{\circ}=+0.61 \mathrm{V}\right)$ is much lower than the standard potential $\left(E^{\circ}=+0.77 \mathrm{V}\right)$. This is due to</div>

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

answer:
<div class="solution">Since, $\mathrm{H}_{2} \mathrm{SO}_{4}$ and $\mathrm{H}_{3} \mathrm{PO}_{4}$, both are strong acids so their presence makes the medium more acidic. The $\mathrm{Fe}^{3+}$ ions form complex, IFeHPO $\left._{4}\right]^{+}$ with these acid and hence, the concentration of $\mathrm{Fe}^{3+}$ ion reduces which results in lowering in the formal reduction potential.</div>


