# questions

title: the-reagents-to-carry-out-the-following-conversion-are
slug: the-reagents-to-carry-out-the-following-conversion-are-89695
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reagents to carry out the following conversion are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q7.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{NaNH}_{2} / \mathrm{CH}_{3} \mathrm{I} ; \mathrm{HgSO}_{4} / \mathrm{dil} \cdot \mathrm{H}_{2} \mathrm{SO}_{4}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a7.png"/> <br/> <br/>(Here, Me means methyl group) Thus, the correct set of reagents is $\mathrm{NaNH}_{2} / \mathrm{CH}_{3} \mathrm{I} ; \mathrm{HgSO}_{4} / \mathrm{dil} \mathrm{H}_{2} \mathrm{SO}_{4}$</div>


title: the-reduction-of-benzenediazonium-chloride-to-phenyl-hydrazine-can-be-accomplished-by
slug: the-reduction-of-benzenediazonium-chloride-to-phenyl-hydrazine-can-be-accomplished-by-84292
tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reduction of benzenediazonium chloride to phenyl hydrazine can be accomplished by</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{SnCl}_{2}, \mathrm{HCl}$, $\mathrm{Na}_{2} \mathrm{SO}_{3}$</span> </div>

answer:
<div class="solution">Reduction of benzene diazoniums chloride occurs as follows:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a78.png"/> <br/>Hence, (a) and (b) are the correct answers.</div>


title: the-reduction-potential-of-hydrogen-halfcell-will-be-negative-if
slug: the-reduction-potential-of-hydrogen-halfcell-will-be-negative-if-12657
question: <div class="question">The reduction potential of hydrogen half-cell will be negative if</div>
tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{p}\left(\mathrm{H}_{2}\right)=1 \mathrm{~atm}$ and $\left[\mathrm{H}^{+}\right]=1.0 \mathrm{M}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}\left(\mathrm{H}_{2}\right)=1 \mathrm{~atm}$ and $\left[\mathrm{H}^{+}\right]=2.0 \mathrm{M}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{p}\left(\mathrm{H}_{2}\right)=2 \mathrm{~atm}$ and $\left[\mathrm{H}^{+}\right]=1.0 \mathrm{M}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{p}\left(\mathrm{H}_{2}\right)=2 \mathrm{~atm}$ and $\left[\mathrm{H}^{+}\right]=2.0 \mathrm{M}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{p}\left(\mathrm{H}_{2}\right)=2 \mathrm{~atm}$ and $\left[\mathrm{H}^{+}\right]=1.0 \mathrm{M}$</span> </div>

answer:
<div class="solution">Reaction $2 \mathrm{H}^{+}+2 \mathrm{e} \rightarrow \mathrm{H}_{2}$
<br/>$\therefore \mathrm{n}=2, \quad \mathrm{Q}=\frac{\mathrm{P}_{\mathrm{H}_{2}}}{\left[\mathrm{H}^{+}\right]^{2}}$
<br/>As $\mathrm{E}_{\mathrm{H}^{+} / \mathrm{H}_{2}}=\mathrm{E}_{\mathrm{H}^{+} / \mathrm{H}_{2}}^{0} \frac{-0.059}{\mathrm{n}} \log \mathrm{Q}=-\frac{0.059}{2} \log \mathrm{Q}$
<br/>$\therefore$ if $\mathrm{Q}&gt;1 \quad \mathrm{E}_{\mathrm{H}^{+} / \mathrm{H}_{2}}     &lt; 0$
<br/>(A) $Q=\frac{1}{1}=1 \quad X$
<br/>(B) $Q=\frac{1}{2^{2}} &lt; 1 \quad X$
<br/>(C) $Q=\frac{2}{12}&gt;1$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6WW-xxLPFc8hj2BD0jCtyGtYnKUa_N3qO2fnRCJJJ1o.original.fullsize.png"/><br/> <br/>(D) $Q=\frac{2}{2^{2}} &lt; 1 \quad X$</div>


title: the-reduction-product-of-ethyl-3oxobutanoate-by-nabh-4-in-methanol-is
slug: the-reduction-product-of-ethyl-3oxobutanoate-by-nabh-4-in-methanol-is-19424
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reduction product of ethyl 3-oxobutanoate by $\mathrm{NaBH}_{4}$ in methanol is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/c3CGDYJo13t0dE35a8DZYnoqewcQhE1W_xQIXQ3wq7o.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/QTqfsjfVr8yfVlfljWmJCad2nYePZ4n0tNwTXDshbtk.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/4IdnviHW-wSTjdjDkNBSRV0VWUoMOaFKEVlwGWFy5KQ.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/3llgC3MA3UxNBvUGSf-8vvRo7JnR_psVUx5s3FhnCpA.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/4IdnviHW-wSTjdjDkNBSRV0VWUoMOaFKEVlwGWFy5KQ.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Hint:
<br/>$\mathrm{NaBH}_{4}$ reduces Ketone but not ester
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/vSjzZEQ_tQjVbmfSVi93UuX0Z9P83kGoS6R-yo29_SE.original.fullsize.png"/></div>


title: the-relationship-between-the-pair-of-compounds-shown-above-are-respectively-1
slug: the-relationship-between-the-pair-of-compounds-shown-above-are-respectively-1-66293
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/UR93AqaeijC-N-W4Fup8KBSN1n_5uhZMrj1oRSQ4Q2k.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/X2wBV1YcmKAAR0P-TVDbJTefeNtLOskP6-GbxJ9pDqs.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/s8NdadrbSaE3c21NF4mA6RyzpaEwb0VpQDI69jMwFFM.original.fullsize.png"/><br/><br/>The relationship between the pair of compounds shown above are respectively,</div>

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

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Hon4bbF8sUxKvGWb_OiZAXkjn4abqmGb6rppTVMu0mo.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hZNpmVAyBWANNYvjExgRzAUGoGFGhVnTzSujtZ-3Js8.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ZweCLfI5aH4DBPEC2PvARSRt9pabUoPQd-Lcd5q1M4A.original.fullsize.png"/><br/></div>


title: the-relationship-between-the-pair-of-compounds-shown-above-are-respectively
slug: the-relationship-between-the-pair-of-compounds-shown-above-are-respectively-30639
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/TUKIvcMPt48kJP6YIQQCxF7zGA0ga9xgpjuIzeiI-YA.original.fullsize.png"/><br/> <br/>The relationship between the pair of compounds shown above are respectively</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Homomer (identical), homomer (identical) and constitutional isomer</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/bhOkubS_tzHFAIVanjho2UPjIuhxI01LT1uQ22DmgSU.original.fullsize.png"/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/j9Jb_uMKEJOh6SPrR_i3o8dFVqenGnG1iZ1p_EGK2pc.original.fullsize.png"/></div>


title: the-remainder-obtained-when-1-2-3-11-is-divided-by-12-is
slug: the-remainder-obtained-when-1-2-3-11-is-divided-by-12-is-11546
tags: ['Mathematics', 'Mathematical Induction', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The remainder obtained when $1 !+2 !+3 !+\ldots+11 !$ is divided by 12 is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">Let $S=1!+2 !+3 !+41+\ldots+11 !$
<br/>Here, we see that from $4 !$ to $111,$ we alwars get 12 factor, so it is always divisible by 12
<br/>Now. $1 !+2 !+3 !=1+2+6=9$
<br/>Hence, when $S$ is divided by $12,$ the remainder is 9.</div>


title: the-remainder-obtained-when-11-2-95-is-divided-by-15-is
slug: the-remainder-obtained-when-11-2-95-is-divided-by-15-is-41683
question: <div class="question">The remainder obtained when $11+2 !+95 !$ is divided by 15 is</div>
tags: ['Mathematics', 'Mathematical Induction', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">14</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">since, all terms from 5 lonwards are divisible by 15 and $1!+2!+3!+4 !=33$
<br/>The required remainder after dividing by 15 will be 3 .</div>


title: the-remainder-when-7-7-7-7-22-times-7-is-divided-by-48-is
slug: the-remainder-when-7-7-7-7-22-times-7-is-divided-by-48-is-45427
question: <div class="question">The remainder when $7^{7^{7^{...7}}}(22$ times 7$)$ is divided by 48 is</div>
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answers are: <span class="option-value">7</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">21</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">47</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$7^{7^{7-7}}=7^{2 \mathrm{~m}+1}=7(49)^{\mathrm{m}}=7(1+48)^{\mathrm{m}}=48 \mathrm{k}+7$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
the-representation-of-the-ground-state-electronic-configuration-of-he-by-box-diagram-as-uparrow-uparrow-is-wrong-because-it-violates

slug:
the-representation-of-the-ground-state-electronic-configuration-of-he-by-box-diagram-as-uparrow-uparrow-is-wrong-because-it-violates-41797

question:
<div class="question">The representation of the ground state electronic configuration of He by box - diagram as \(\uparrow \uparrow\) is wrong because it violates</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Hysenberg's Uncertainty Principle</span> </div>

answer:
<div class="solution">Hints : According to Pauli Exclusion Principle, In any orbital, maximum two electrons can exist, having opposite spin.</div>


title: the-resistance-across-a-and-b-in-the-figure-below-will-be
slug: the-resistance-across-a-and-b-in-the-figure-below-will-be-75499
tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The resistance across \(A\) and \(B\) in the figure below will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/DPYUFDHt7gh0JZrTt1nuXEAjlsXUW3LYrmhuWuHrz98.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(3 \mathrm{R}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{R}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{R}{3}\)</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">None of these</span> </li> </ul>

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

answer:
<div class="solution">Hints: Resistance are in parallel \(\therefore \operatorname{Req}=\frac{\mathrm{R}}{3}\)</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2v$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-rms-speed-of-oxygen-is-v-at-a-particular-temperature-if-the-temperature-is-doubled-and-oxygen-molecules-dissociate-into-oxygen-atoms-the-rms-speed

slug:
the-rms-speed-of-oxygen-is-v-at-a-particular-temperature-if-the-temperature-is-doubled-and-oxygen-molecules-dissociate-into-oxygen-atoms-the-rms-speed-11803

question:
<div class="question">The rms speed of oxygen is $v$ at a particular temperature. If the temperature is doubled and oxygen molecules dissociate into oxygen atoms, the rms speed becomes</div>

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

answer:
<div class="solution">As the rms speed is given by
<br/>$$
<br/>\begin{aligned}
<br/>&amp; v_{\text {rms }}=\sqrt{\frac{3 R T}{M}}{V_{\text {ms }}} \propto \sqrt{\frac{T}{M}}
<br/>\end{aligned}
<br/>$$
<br/>When temperature is doubled and molecules dissociates into atoms, then
<br/>$$
<br/>\begin{aligned}
<br/>\frac{v_{\mathrm{rms}_{1}}}{v_{\mathrm{rms}_{2}}} &amp;=\frac{\sqrt{\frac{T}{M}}}{\sqrt{\frac{2 T}{M / 2}}}=\frac{\sqrt{\frac{T}{M}}}{\sqrt{\frac{4 T}{M}}} \\
<br/>&amp;=\frac{\sqrt{T}}{\sqrt{4 T}}=\frac{1}{2}
<br/>\end{aligned}
<br/>$$
<br/>If $v_{\mathrm{rms}},$ is $v,$ then $v_{\mathrm{rms}_{2}}$ will be $2 v$.</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-rms-speed-of-the-molecules-of-a-gas-at-10-0-c-is-v-the-temperature-at-which-the-rms-speed-will-be-3-v-is

slug:
the-rms-speed-of-the-molecules-of-a-gas-at-10-0-c-is-v-the-temperature-at-which-the-rms-speed-will-be-3-v-is-19230

question:
<div class="question">The r.m.s speed of the molecules of a gas at $100^{\circ} \mathrm{C}$ is $v .$ The temperature at which the r.m.s. speed will be $\sqrt{3} v$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$546^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$646^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$746^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$846^{\circ} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$846^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">$v_{\text {tms }}=\sqrt{\frac{3 R T}{M}}$
<br/>Given, $\quad T=100^{\circ} \mathrm{C}=373 \mathrm{K}$
<br/>$\therefore \quad v=\sqrt{\frac{3 R \times 373}{M}}$
<br/>For $\quad \sqrt{3} v=\sqrt{\frac{3 R T}{M}}$
<br/>$\therefore \quad \frac{\sqrt{3} v}{v}=\frac{\sqrt{3 R T / M}}{\sqrt{3 R \times 373 / M}}$
<br/>$\Rightarrow \quad \sqrt{3}=\sqrt{\frac{T}{373}}$
<br/>$\Rightarrow \quad 3=\frac{T}{373}$
<br/>$\Rightarrow \quad T=1119 \mathrm{K}$
<br/>or $\quad T=846^{\circ} \mathrm{C}$</div>


title: the-rms-value-of-potential-difference-v-shown-in-the-figure-is
slug: the-rms-value-of-potential-difference-v-shown-in-the-figure-is-49674
tags: ['Physics', 'Alternating Current', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/U69wMoPBQrbiOmQ35KDUClaiH9e9EYMFdlwd0lfCXxg.original.fullsize.png"/><br/> <br/>The rms value of potential difference $v$ shown in the figure is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{v}_{0}}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{v}_{0}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{v_{0}}{\sqrt{3}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{v_{0}}{\sqrt{2}}$</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> </ul>

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

answer:
<div class="solution">$\mathrm{V}_{\mathrm{rms}}^{2} \times \mathrm{T}=\mathrm{V}_{0}^{2} \times \frac{\mathrm{T}}{2}+0$
<br/>$\mathrm{~V}_{\mathrm{rms}}^{2}=\frac{\mathrm{V}_{0}^{2}}{2} \quad \mathrm{~V}_{\mathrm{rms}}=\frac{\mathrm{V}_{0}}{\sqrt{2}}$</div>


tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
the-rms-velocity-of-co-gas-molecules-at-2-7-c-is-approximately-1000-m-s-for-molecules-at-600-k-the-rms-velocity-is-approximately

slug:
the-rms-velocity-of-co-gas-molecules-at-2-7-c-is-approximately-1000-m-s-for-molecules-at-600-k-the-rms-velocity-is-approximately-67527

question:
<div class="question">The RMS velocity of CO gas molecules at $27^{\circ} \mathrm{C}$ is approximately $1000 \mathrm{m} / \mathrm{s}$. For molecules at $600 \mathrm{K}$, the RMS velocity is approximately</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2000 \mathrm{m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1414 \mathrm{m} / \mathrm{s}$</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">C</span> <span class="option-data">$1000 \mathrm{m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1500 \mathrm{m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1414 \mathrm{m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">As trom the formula, $v_{\text {ms }}=\sqrt{\frac{3 R T}{M}}$
<br/>Given that $\left(v_{\text {rms }}\right)_{CO}=1000 \mathrm{m} / \mathrm{s}$
<br/>$(Temp )_{CO}=27^{\circ} \mathrm{C}=300 \mathrm{K}$
<br/>$(Temp.) _{\mathrm{N}_{\mathrm{2}}}=600 \mathrm{K}$
<br/>Now putting the values, we get
<br/>$\frac{\left(V_{\operatorname{mot}}\right)_{CO}}{\left(v_{m}\right)_{N_{2}}}=\sqrt{\frac{3 R}{3 R} \times \frac{T_{CO}}{M_{CO}} \times \frac{M_{N_{2}}}{T_{N_{2}}}}$
<br/>$\frac{1000}{\left(v_{\text {rad }}\right)_{N_{2}}}=\sqrt{\frac{300}{28} \times \frac{28}{600}}=\frac{1}{\sqrt{2}}$
<br/>or. $\left(v_{\text {rms }}\right)_{N_{2}}=1000 \times 1414$ $=1414 \mathrm{m} / \mathrm{s}$</div>


tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'WBJEE', 'WBJEE 2017']
answer: <div class="solution">Among the given statements (b) and (c) are correct these are the facts.</div>
source_db: db\MarksBatch2_P3.db
title:
the-role-of-fluorspar-which-is-added-in-small-quantities-in-the-electrolytic-reduction-of-alumina-dissolved-in-fused-cryolite-is

slug:
the-role-of-fluorspar-which-is-added-in-small-quantities-in-the-electrolytic-reduction-of-alumina-dissolved-in-fused-cryolite-is-18801

question:
<div class="question">The role of fluorspar, which is added in small quantities in the electrolytic reduction of alumina dissolved in fused cryolite is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">to make fused mixture conducting, to lower the melting temperature of the mixture</span> </div>


title: the-rolles-theorem-is-applicable-in-the-interval-1-x-1-for-the-function
slug: the-rolles-theorem-is-applicable-in-the-interval-1-x-1-for-the-function-18357
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The Rolle's theorem is applicable in the interval $-1 \leq \mathrm{x} \leq 1$ for the function</div>

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

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

answer:
<div class="solution">Hints : $f(x)=x^2 \mid$ and $\mathrm{f}(1)=\mathrm{f}(-1)$ for $\mathrm{f}(\mathrm{x})=|\mathrm{x}|$ but at $\mathrm{x}=0, \mathrm{f}(\mathrm{x})=|\mathrm{x}|$ is not differentiable hence (2) is the correct option.<br/>$$<br/>f(1)=1=f(-1)<br/>$$</div>


tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-root-mean-square-rms-speed-of-x2-gas-is-mathrmx-mathrmms-at-a-given-temperature-when-the-temperature-is-doubled-the-mathrmx2-molecules-dissociated

slug:
the-root-mean-square-rms-speed-of-x2-gas-is-mathrmx-mathrmms-at-a-given-temperature-when-the-temperature-is-doubled-the-mathrmx2-molecules-dissociated-50268

question:
<div class="question">The root mean square (rms) speed of \(X_2\) gas is \(\mathrm{x} \mathrm{m/s}\) at a given temperature. When the temperature is doubled, the \(\mathrm{X}_2\) molecules dissociated completely into atoms. The root mean square speed of the sample of gas then becomes (in \(\mathrm{m} / \mathrm{s}\) )</div>

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

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

answer:
<div class="solution">Hint : \(C_{m m s}=\sqrt{\frac{3 R T}{M}}\)<br/>\(\begin{array}{ll}<br/>\mathrm{T}_1=\mathrm{T} &amp; \mathrm{T}_2=2 \mathrm{~T} \\<br/>\mathrm{M}_1=\mathrm{M} &amp; \mathrm{M}_2=\mathrm{M} / 2 \\<br/>\mathrm{C}_1=\mathrm{X} &amp; \mathrm{C}_2=?<br/>\end{array}\)<br/>\(\frac{C_1}{C_2}=\sqrt{\frac{T_1}{M_1} \times \frac{\mathrm{M}_2}{\mathrm{~T}_2}}=\sqrt{\frac{\mathrm{T}}{\mathrm{M}} \times \frac{\mathrm{M} / 2}{2 T}}=\frac{1}{2}\)<br/>\(\therefore \frac{\mathrm{x}}{\mathrm{C}_2}=\frac{1}{2}\), Hence \(\mathrm{C}_2=2 \mathrm{x~} \mathrm{m} / \mathrm{s}\)</div>


title: the-roots-of-the-quadratic-equation-x-2-2-3-x-22-0-are
slug: the-roots-of-the-quadratic-equation-x-2-2-3-x-22-0-are-61507
question: <div class="question">The roots of the quadratic equation $x^2-2 \sqrt{3} x-22=0$ are :</div>
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">imaginry</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">real, rational and equal</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">real, irrational and unequal</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">real, rational and unequal</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">real, irrational and unequal</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints: } \begin{aligned}<br/>&amp; x^2-2 \sqrt{3}-22=0 \\<br/>&amp; D=12+(4 \times 22)&gt;0<br/>\end{aligned}<br/>$$<br/>$\because$ coeffs are irrational,<br/>$$<br/>x=\frac{2 \sqrt{3} \pm \sqrt{12+88}}{2}<br/>$$<br/>$\therefore$ Roots are irrational, real, unequal.</div>


title: the-second-ionisation-energy-of-the-following-elements-follows-the-order
slug: the-second-ionisation-energy-of-the-following-elements-follows-the-order-71506
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The second ionisation energy of the following elements follows the order</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Zn}&gt;\mathrm{Cd}    &lt; \mathrm{Hg}$</span> </div>

answer:
<div class="solution">Key Point For similar electronic configuration of outer most shell, size will decide the value of ionisation energy. More the size, lesser is the value of ionisation energy.
<br/>Electronic configuration - IInd I.E
<br/>$\mathrm{Zn}^{+}$ ion $(Z=30)=[\mathrm{Ar}] 3 \mathrm{d}^{10} 4 \mathrm{s}^{1}, - \quad(1734 \mathrm{kJ} / \mathrm{mol})$
<br/>$\mathrm{Cd}^{+}$ ion $(Z=48)=[\mathrm{Kr}] 4 d^{10} 5 \mathrm{s}^{1} - (1631 \mathrm{kJ} / \mathrm{mol})$
<br/>$\mathrm{Hg}^{+}$ ion $(Z=80)=[\mathrm{Xe}] 4 f^{14} \cdot 5 d^{10} 6 \mathrm{s}^{1} - (1809 \mathrm{kJ} / \mathrm{mol})$
<br/>$\because$ Size of $\mathrm{Cd}^{+}&gt;\mathrm{Zn}^{+}$, thus it has lower IInd ionisation energy while due to lanthanoid effect (i.e., poor screening by $4 f$ and $5 d$ electrons, $\mathrm{Hg}^{+}$ has higher IInd ionisation-energy. Hence, correct order is,
<br/>$\mathrm{Zn} &gt; \mathrm{Cd} &lt; \mathrm{Hg}$ and option (a) is the correct answer.</div>


title: the-second-law-of-thermodynamics-says-that-in-a-cyclic-process
slug: the-second-law-of-thermodynamics-says-that-in-a-cyclic-process-52408
question: <div class="question">The second law of thermodynamics says that in a cyclic process :</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">Because $0 \mathrm{~K}$ temperature is unattainable.</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">work cannot be converted into heat</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">heat cannot be converted into work</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">work cannot be completely converted into heat</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">heat cannot be completely converted into work</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">heat cannot be completely converted into work</span> </div>


title: the-second-order-derivative-of-a-sin-3-t-with-respect-to-a-cos-3-t-at-t-4-is
slug: the-second-order-derivative-of-a-sin-3-t-with-respect-to-a-cos-3-t-at-t-4-is-23527
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The second order derivative of a $\sin ^3 t$ with respect to a $\cos ^3 t$ at $t=\frac{\pi}{4}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{12 a}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{4 \sqrt{2}}{3 a}$</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">D</span> <span class="option-data">$\frac{3 a}{4 \sqrt{2}}$</span> </li> </ul>

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

answer:
<div class="solution">Hints : $y=a \sin ^3 \mathrm{t} ; \mathrm{x}=\mathrm{a} \cos ^3 \mathrm{t}$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\mathrm{dy}}{\mathrm{dt}}=3 \sin ^2 \mathrm{t} \cos \mathrm{t} ; \frac{\mathrm{dx}}{\mathrm{dt}}=-3 \mathrm{a} \cos ^2 \mathrm{t} \sin \mathrm{t} \\<br/>&amp; \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{3 a \sin ^2 t \cos t}{-3 a \cos ^2 t \operatorname{sint}}=-\frac{\operatorname{sint}}{\cos t}=-\operatorname{tant} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx} \mathrm{x}^2}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\frac{\mathrm{d}}{\mathrm{dx}}(-\tan t)=\frac{\mathrm{d}}{\mathrm{dt}}(-\operatorname{tant}) \cdot \frac{\mathrm{dt}}{\mathrm{dx}} \\<br/>&amp; =\left(-\sec ^2 \mathrm{t}\right) \frac{1}{-3 \cos ^2 \mathrm{t} \sin t}=\frac{1}{+3 \cos ^4 \mathrm{t} \sin t} \\<br/>&amp; \left.\frac{d^2 y}{d x^2}\right|_{t=\pi / 4}=\frac{1}{3 a\left(\frac{1}{\sqrt{2}}\right)^4 \cdot\left(\frac{1}{\sqrt{2}}\right)}=\frac{(\sqrt{2})^5}{3 a}=\frac{4 \sqrt{2}}{3 a}<br/>\end{aligned}<br/>$$</div>


title: the-sequence-log-mathrma-log-fracmathrma2mathrmb-log-fracmathrma3mathrmb2-ldots-ldots-ldots-is
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
slug:
the-sequence-log-mathrma-log-fracmathrma2mathrmb-log-fracmathrma3mathrmb2-ldots-ldots-ldots-is-87841

question:
<div class="question">The sequence \(\log \mathrm{a}, \log \frac{\mathrm{a}^2}{\mathrm{~b}}, \log \frac{\mathrm{a}^3}{\mathrm{~b}^2}, \ldots \ldots \ldots\). is</div>

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

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

answer:
<div class="solution">\(\begin{aligned}\text {Hints: } <br/>&amp; \log \mathrm{a} \cdot(2 \log \mathrm{a}-\log \mathrm{b})(3 \log \mathrm{a}-2 \log \mathrm{b}) \\<br/>&amp; =\mathrm{T}_2-\mathrm{T}_1=\log \mathrm{a}-\log \mathrm{b} \\<br/>= &amp; \mathrm{T}_3-\mathrm{T}_2=\log \mathrm{a}-\log \mathrm{b}<br/>\end{aligned}\)</div>


title: the-shape-of-x-e-f-5-will-be
slug: the-shape-of-x-e-f-5-will-be-80426
question: <div class="question">The shape of $XeF_{5}^{-}$ will be</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">planar</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">square pyramid</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">trigonal bipyramidal</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">planar</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">pentagonal bipyramid</span> </li> </ul>

answer:
<div class="solution">Shape of XeF $\rightarrow$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a50.png"/> <br/> <br/>The Xe show $s p^{3} d^{3}$ -hybridisation in $XeF^{-}_{5}$, Thus, its geometry is Pentagonal-bipyramidal. But, number of electron pair $=\frac{8+5+1}{2}=7$ Number of bond pair $=5^{2},$ number of lone pair $=2$ A lot of electrons are present at axil position and all bonds are in same plane. Hence, the shape is planar.</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
the-side-a-b-of-a-bc-is-fixed-and-is-of-length-2-a-unit-the-vertex-moves-in-the-plane-such-that-the-vertical-angle-is-always-constant-and-is-let-x-axi

slug:
the-side-a-b-of-a-bc-is-fixed-and-is-of-length-2-a-unit-the-vertex-moves-in-the-plane-such-that-the-vertical-angle-is-always-constant-and-is-let-x-axi-93159

question:
<div class="question">The side $A B$ of $\triangle A B C$ is fixed and is of length $2 a$ unit. The vertex moves in the plane such that the vertical angle is always constant and is $\alpha$. Let $x$-axis be along $A B$ and the origin be at $A$. Then the locus of the vertex is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^2+y^2+2 a x \sin \alpha+a^2 \cos \alpha=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^2+y^2-2 a x-2 a y \cot \alpha=0$</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">$x^2+y^2-2 a x \cos \alpha-a^2=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^2+y^2-a x \sin \alpha-$ ay $\cos \alpha=0$</span> </li> </ul>

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

answer:
<div class="solution">Let, $c=z=x+i y$<br/>$\begin{aligned}<br/>&amp;\arg \left(\frac{2 \mathrm{a}-\mathrm{z}}{0-\mathrm{z}}\right)=\alpha \\<br/>&amp;\Rightarrow \arg \left(\frac{(\mathrm{x}-2 \mathrm{a})+\mathrm{iy}}{\mathrm{x}+\mathrm{iy}}\right)=\alpha \\<br/>&amp;\Rightarrow \mathrm{x}^2+\mathrm{y}^2-2 \mathrm{ax}-2 \mathrm{ay} \cot \alpha=0<br/>\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/DDKrzp4ZD6ZWmVkW4M9jP_8W2hFGRcKMNDpViqCgnIk.original.fullsize.png"/></div>


title: the-sine-of-the-angle-between-the-straight-line-3-x-2-4-y-3-5-z-4-and-the-plane-2-x-2-y-z-5-is
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
slug:
the-sine-of-the-angle-between-the-straight-line-3-x-2-4-y-3-5-z-4-and-the-plane-2-x-2-y-z-5-is-48502

question:
<div class="question">The sine of the angle between the straight line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ and the plane $2 x-2 y+z=5$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2 \sqrt{3}}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{2}}{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{5 \sqrt{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{5}}{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{2}}{10}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>Let the angle be $\theta$. Then, $\cos \left(90^{\circ}-\theta\right)=\frac{(3 \hat{\mathbf{i}}+4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}) \cdot(2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})}{|3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}| \times|2 \hat{\mathbf{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}|} \Rightarrow \sin \theta=\frac{3}{\sqrt{50} \sqrt{9}}=\frac{\sqrt{2}}{10}$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
title:
the-slope-at-any-point-of-a-curve-y-f-x-is-given-by-d-x-d-y-3-x-2-and-it-passes-through-1-1-the-equation-of-the-curve-is

slug:
the-slope-at-any-point-of-a-curve-y-f-x-is-given-by-d-x-d-y-3-x-2-and-it-passes-through-1-1-the-equation-of-the-curve-is-31455

question:
<div class="question">The slope at any point of a curve $y=f(x)$ is given by $\frac{d y}{d x}=3 x^2$ and it passes through $(-1,1)$. The equation of the curve is</div>

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

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

answer:
<div class="solution">Hints: $\frac{d y}{d x}=3 x^2 \Rightarrow \int d y=\int 3 x^2 d x \Rightarrow y=x^3+\mathrm{C}$<br/>Curve passes through $(-1,1)$. Hence $1=-1+C \Rightarrow C=2$<br/>$$<br/>\therefore y=x^3+2<br/>$$</div>


title: the-smallest-positive-root-of-the-equation-tan-x-x-0-lies-in
slug: the-smallest-positive-root-of-the-equation-tan-x-x-0-lies-in-24918
question: <div class="question">The smallest positive root of the equation $\tan x-x=0$ lies in</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(0, \frac{\pi}{2}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{\pi}{2}, \pi\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\pi, \frac{3 \pi}{2}\right)$</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">$\left(\frac{3 \pi}{2}, 2 \pi\right)$</span> </li> </ul>

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

answer:
<div class="solution">Given equation is
<br/>$\tan x-x=0$
<br/>$\tan x=x$
<br/>Solutions are abscissae of points of intersection of the curves $y=\tan x$ and $y=x$. From the figure, it is clearty visible that solution lies in $\left(\pi, \frac{3 \pi}{2}\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a29.png"/></div>


title: the-smallest-value-of-5-cos-12-is
slug: the-smallest-value-of-5-cos-12-is-79442
question: <div class="question">The smallest value of $5 \cos \theta+12$ is</div>
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">17</span> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } 5 \cos \theta+12,-1 \leq \cos \theta \leq 1 \\<br/>&amp; \Rightarrow-5 \leq 5 \cos \theta \leq 5 \\<br/>&amp; \therefore 5 \cos \theta+12 \geq-5+12 \Rightarrow 5 \cos \theta+12 \geq 7<br/>\end{aligned}<br/>$$</div>


title: the-solubility-of-mathrmca3leftmathrmpo4right2-in-water-is-y-moles-litre-its-solubility-product-is
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
slug:
the-solubility-of-mathrmca3leftmathrmpo4right2-in-water-is-y-moles-litre-its-solubility-product-is-17908

question:
<div class="question">The solubility of \(\mathrm{Ca}_3\left(\mathrm{PO}_4\right)_2\) in water is y moles / litre. Its solubility product is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(6 y^4\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(36 y^4\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(64 y^5\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(108 \mathrm{y}^5\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(108 \mathrm{y}^5\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints: } \mathrm{Ca}_3\left(\mathrm{PO}_4\right)_2(\mathrm{~s}) \rightleftharpoons 3 \underset{3 \mathrm{~s}}{\mathrm{Ca}^{2+}}(\mathrm{aq})+\underset{2 \mathrm{~s}}{2 \mathrm{PO}_4}{ }^{3-}(\mathrm{aq}) \\<br/>&amp; \mathrm{~K}_{\mathrm{Sp}}=\left[\mathrm{Ca}^{2+}\right]^3 \cdot\left[\mathrm{PO}_4^{3-}\right]^2 \\<br/>&amp; =(3 \mathrm{~s})^3 \cdot(2 \mathrm{~s})^2 \\<br/>&amp; =27 \mathrm{~s}^3 \times 4 \mathrm{~s}^2 \\<br/>&amp; =108 \mathrm{~s}^5 \\<br/>&amp;<br/>\end{aligned}\)</div>


title: the-solution-of-25-d-x-2-d-2-y-10-d-x-d-y-y-0-y-0-1-y-1-2-e-15-is
slug: the-solution-of-25-d-x-2-d-2-y-10-d-x-d-y-y-0-y-0-1-y-1-2-e-15-is-34946
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The solution of $25 \frac{d^{2} y}{d x^{2}}-10 \frac{d y}{d x}+y=0$, $y(0)=1, y(1)=2 e^{1 / 5}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=e^{5 x}+e^{-5 x}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y=(1+x) e^{5 x}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y=(1+x) e^{x / 5}$</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">D</span> <span class="option-data">$y=(1+x) e^{-x / 5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=(1+x) e^{x / 5}$</span> </div>

answer:
<div class="solution">Let $y=e^{\pi x}$ be the solution of given differential equation,
<br/>$\Rightarrow \quad \frac{d y}{d x}=m e^{n x} \Rightarrow \frac{d^{2} y}{d x^{2}}=m^{2} e^{m x}$
<br/>$\therefore \quad 25 \frac{d^{2} y}{d x^{2}}-10 \frac{d y}{d x}+y=0$
<br/>$\Rightarrow \quad 25 \mathrm{m}^{2} e^{m x}-10 m e^{m x}+e^{m x}=0$
<br/>$\Rightarrow \quad e^{\operatorname{mux}}\left(25 m^{2}-10 m+1\right)=0$
<br/>$\Rightarrow$ Auxiliary equation
<br/>$\Rightarrow \quad(5 m-1)^{2}=0$
<br/>$\Rightarrow \quad m=\frac{1}{5}, \frac{1}{5}$
<br/>since, roots are real and euqal.
<br/>$\therefore$ General solution is $y=\left(c_{1}+c_{2} x\right) e^{x / 5} \ldots .$
<br/>$$
<br/>\begin{array}{l}
<br/>y(0)=1 \Rightarrow c_{1}=1 \\
<br/>y(1)=2 e^{1 / 5} \Rightarrow 2 c^{1 / 5}=\left(c_{1}+c_{2}\right) e^{1 / 5}
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad c_{1}+c_{2}=2 \Rightarrow c_{1}=1$
<br/>Putting the value of $c_{1}$ and $c_{2}$ in Eq. (i), we get particular solution
<br/>$$
<br/>y=(1+x) e^{x / 5}
<br/>$$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
the-solution-of-cos-y-d-x-d-y-e-x-s-i-n-y-x-2-e-s-i-n-y-is-f-x-e-s-i-n-y-c-c-is-arbitrary-real-constant-where-f-x-is-equal-to

slug:
the-solution-of-cos-y-d-x-d-y-e-x-s-i-n-y-x-2-e-s-i-n-y-is-f-x-e-s-i-n-y-c-c-is-arbitrary-real-constant-where-f-x-is-equal-to-82738

question:
<div class="question">The solution of $\cos y \frac{d y}{d x}=e^{x+\sin y}+x^2 e^{\sin y}$ is $f(x)+e^{-\sin y}=C$ (C is arbitrary real constant) where $f(x)$ is equal to</div>

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

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

answer:
<div class="solution">$-e^{-\sin y} \cos y \frac{d y}{d x}=-\left[e^x+x^2\right] \Rightarrow d\left(e^{-\sin y}\right)+\left(e^x+x^2\right) d x=0 \Rightarrow e^{-\sin y}+e^x+\frac{x^3}{3}=C$</div>


title: the-solution-of-det-a-i-2-0-be-4-and-8-and-a-2-x-3-y-then
slug: the-solution-of-det-a-i-2-0-be-4-and-8-and-a-2-x-3-y-then-59639
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The solution of $\operatorname{det}\left(A-\lambda I_2\right)=0$ be 4 and 8 and $A=\left(\begin{array}{ll}2 &amp; 3 \\ x &amp; y\end{array}\right)$. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=4, y=10$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=5, y=8$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x=3, y=9$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x=-4, y=10$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x=-4, y=10$</span> </div>

answer:
<div class="solution">$\operatorname{det}\left(A-\lambda I_2\right)=0 \Rightarrow\left|\begin{array}{cc}2-\lambda &amp; 3 \\ x &amp; y-\lambda\end{array}\right|=0$<br/>$\begin{array}{ll}<br/>\Rightarrow &amp; (2-\lambda)(y-\lambda)-3 x=0 \\<br/>\Rightarrow &amp; 2 y-\lambda y-2 \lambda+\lambda^2-3 x=0 \\<br/>\Rightarrow &amp; \lambda^2-(y+2) \lambda+2 y-3 x=0 \\<br/>&amp; y+2=4+8 \Rightarrow y=10 \\<br/>&amp; \text { and } 2 y-3 x=4 \times 8 \Rightarrow x=-4<br/>\end{array}$</div>


title: the-solution-of-fracd-yd-xfracyxtan-fracyx-is
slug: the-solution-of-fracd-yd-xfracyxtan-fracyx-is-66405
question: <div class="question">The solution of \(\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}\) is</div>
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(x=c \sin (y / x)\)</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">\(x=c \sin (x y)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(y=c \sin (y / x)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(x y=c \sin (x / y)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x=c \sin (y / x)\)</span> </div>

answer:
<div class="solution">Hints: \(\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}\)<br/>\(\begin{aligned}<br/>&amp; \text { Put } \frac{y}{x}=\theta, y=\theta x \\<br/>&amp; \frac{d y}{d x}=\theta+\frac{x d \theta}{d x} \\<br/>&amp; \theta+x \cdot \frac{d \theta}{d x}=\theta+\tan \theta, \frac{d \theta}{\tan \theta}=\frac{d y}{x}<br/>\end{aligned}\)<br/>\(\begin{aligned}<br/>&amp; \int \cot \theta d \theta=\int \frac{d x}{x} \\<br/>&amp; \log \sin \theta=\log x+\log c \\<br/>&amp; \sin \theta=x . c ., \sin \frac{y}{x}=x . c \\<br/>&amp; x=c \cdot \sin \frac{y}{x}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-solution-of-the-differential-equation-d-x-d-y-x-l-o-g-e-x-y-x-1-under-the-condition-y-1-when-x-e-is

slug:
the-solution-of-the-differential-equation-d-x-d-y-x-l-o-g-e-x-y-x-1-under-the-condition-y-1-when-x-e-is-20720

question:
<div class="question">The solution of the differential equation $\frac{d y}{d x}+\frac{y}{x \log _{e} x}=\frac{1}{x}$ under the condition $y=1$ when $x=e$ is</div>

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

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

answer:
<div class="solution">Given, differential equation is $\frac{d y}{d x}+\frac{y}{x \log _{e} x}=\frac{1}{x}$
<br/>It is a linear equation of the form $\frac{d y}{d x}+P y=Q$
<br/>where $\quad P=\frac{1}{x \log _{e} x}$ and $Q=\frac{1}{x}$
<br/>$\therefore$ Integrating factor, $\mathrm{IF}=e$
<br/>$$
<br/>\begin{array}{l}
<br/>=e^{\int \frac{1}{x \log _{e} x} d x} \\
<br/>=e^{\left.\log \log _{e} x\right)} \\
<br/>=\log _{e} x
<br/>\end{array}
<br/>$$
<br/>$\therefore$ Solution of differential equation is $y \times \log _{e} x=\int \frac{1}{x} \log _{e} x d x$
<br/>$\Rightarrow y \times \log _{e} x=\frac{\left(\log _{e} x\right)^{2}}{2}+C$
<br/>When $y=1$ and $x=e,$ then
<br/>$$
<br/>\begin{aligned}
<br/>&amp; 1 \times \log _{e} e=\frac{\left(\log _{e} e\right)^{2}}{2}+C \\
<br/>\Rightarrow \quad 1 &amp;=\frac{1}{2}+C \\
<br/>\Rightarrow \quad C=\frac{1}{2}
<br/>\end{aligned}
<br/>$$
<br/>On putting $C=\frac{1}{2}$ in Eq. (i), we get
<br/>$$
<br/>y \times \log _{e} x=\frac{\left(\log _{e} x\right)^{2}}{2}+\frac{1}{2}
<br/>$$
<br/>$\Rightarrow \quad 2 y=\log _{e} x+\frac{1}{\log _{e} x}$</div>


title: the-solution-of-the-differential-equation-y-2-2-x-d-x-d-y-y-satisfies-x-1-y-1-then-the-solution-is
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
slug:
the-solution-of-the-differential-equation-y-2-2-x-d-x-d-y-y-satisfies-x-1-y-1-then-the-solution-is-60130

question:
<div class="question">The solution of the differential equation $\left(y^{2}+2 x\right) \frac{d y}{d x}=y$ satisfies $x=1, y=1$. Then the solution is</div>

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

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

answer:
<div class="solution">Given differential equation is 
<br/>$\left(y^{2}+2 x\right) \frac{d y}{d x}=y \Rightarrow \frac{d x}{d y}=\left(\frac{y^{2}+2 x}{y}\right)$
<br/>$\Rightarrow \quad \frac{d x}{d y}=y+\frac{2 x}{y}$
<br/>$\Rightarrow \quad \frac{d x}{d y}-\frac{2}{y} \cdot x=y$
<br/>$\mathrm{IF}=e^{\int \frac{2}{y} d y}=e^{-2 \mathrm{bg} \gamma}=\mathrm{y}^{-2}=\frac{1}{\mathrm{y}^{2}}$
<br/>$\therefore$ Complete solution is $x \cdot \frac{1}{y^{2}}=\int y \cdot \frac{1}{y^{2}} d y+C$
<br/>$\Rightarrow \quad \frac{x}{y^{2}}=\int \frac{d y}{y}+C=\log _{e} y+c$
<br/>$\Rightarrow \quad x=y^{2} \log _{e} y+C y^{2}$
<br/>$A t+x=1 y=1$ then from Eq $(i),$ we get
<br/>$$
<br/>1=0+C \Rightarrow C=1
<br/>$$
<br/>$\therefore$ From Eq. (i), we get
<br/>$$
<br/>x=y^{2}\left(\log _{e} y+1\right)
<br/>$$
<br/>Which is the required solution.</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-solution-of-the-differential-equation-y-d-x-d-y-x-x-2-y-2-x-2-y-2-x-2-y-2-is-where-c-is-a-constant

slug:
the-solution-of-the-differential-equation-y-d-x-d-y-x-x-2-y-2-x-2-y-2-x-2-y-2-is-where-c-is-a-constant-24842

question:
<div class="question">The solution of the differential equation $y \frac{d y}{d x}=x\left[\frac{y^{2}}{x^{2}}+\frac{\phi\left(\frac{y^{2}}{x^{2}}\right)}{\phi^{\prime}\left(\frac{y^{2}}{x^{2}}\right)}\right]$ is (where, $c$ is a constant)</div>

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

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

answer:
<div class="solution">Given differential equation can be rewritten as
<br/>$$
<br/>\frac{d y}{d x}=\frac{y}{x}+\frac{x \phi\left(\frac{y^{2}}{x^{2}}\right)}{y \phi^{\prime}\left(\frac{y^{2}}{x^{2}}\right)}
<br/>$$
<br/>Put $y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
<br/>$\therefore$ Given equation becomes.
<br/>$$
<br/>v+x \frac{d v}{d x}=\frac{v x}{x}+\frac{x \phi\left(\frac{v^{2} x^{2}}{x^{2}}\right)}{v x \phi^{\prime}\left(\frac{v^{2} x^{2}}{x^{2}}\right)}
<br/>$$
<br/>$\Rightarrow \quad x \frac{d v}{d x}=\frac{\phi\left(v^{2}\right)}{v \phi^{\prime}\left(v^{2}\right)}$
<br/>$\Rightarrow \quad \frac{v \psi^{\prime}\left(v^{2}\right)}{\phi\left(v^{2}\right)} d v=\frac{d x}{x}$
<br/>On integrating both sides, we get
<br/>$$
<br/>\frac{1}{2} \log \phi\left(v^{2}\right)=\log x+\log c_{1}
<br/>$$
<br/>$\Rightarrow \quad \log \phi\left(v^{2}\right)=2 \log x c_{1}$
<br/>$\Rightarrow \quad \phi\left(v^{2}\right)=\left(x c_{1}\right)^{2} \Rightarrow \phi\left(\frac{y^{2}}{x^{2}}\right)=x^{2} c_{1}^{2}$
<br/>$\Rightarrow \quad \phi\left(\frac{y^{2}}{x^{2}}\right)=x^{2} c \quad$ (put $\left.c_{1}^{2}=c\right]$</div>


title: the-solution-of-the-differential-equation-y-sin-y-x-d-x-x-sin-y-x-y-d-y-satisfying-y-4-1-is
slug: the-solution-of-the-differential-equation-y-sin-y-x-d-x-x-sin-y-x-y-d-y-satisfying-y-4-1-is-85304
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The solution of the differential equation $y \sin \left(\frac{x}{y}\right) d x=\left\{x \sin \left(\frac{x}{y}\right)-y\right\} d y \quad$ satisfying $y\left(\frac{\pi}{4}\right)=1 \mathrm{is}$</div>

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

answer:
<div class="solution">Given differential equation is
<br/>$$
<br/>\begin{array}{l}
<br/>y \sin \left(\frac{x}{y}\right) d x=\left\{x \sin \left(\frac{x}{y}\right)-y\right\} d y \\
<br/>\Rightarrow \quad \frac{d x}{d y}=\frac{x \sin \left(\frac{x}{y}\right)-y}{y \sin \left(\frac{x}{y}\right)}=\frac{x}{y}-\frac{1}{\sin \left(\frac{x}{y}\right)}
<br/>\end{array}
<br/>$$
<br/>On putting $v=\frac{x}{y} \Rightarrow x=y$
<br/>$\Rightarrow \quad \frac{d x}{d y}=v \cdot 1+y \frac{d v}{d y}$ in Eq. $0 .$ we get
<br/>$v+y \frac{d v}{d y}=v-\frac{1}{\sin v}$
<br/>$\Rightarrow \quad y \frac{d v}{d y}=-\frac{1}{\sin v}$
<br/>$\Rightarrow \quad-\int \sin v d v=\int \frac{d y}{y}$
<br/>(on integrating)
<br/>$\Rightarrow \quad \cos v=\log y+C$
<br/>$\Rightarrow \quad \cos \left(\frac{x}{y}\right)=\log y+C$
<br/>Given at $x=\frac{\pi}{4}, y=1$ then from $\mathrm{Eq}$. (i) $\Rightarrow \quad \cos \left(\frac{\pi}{4}\right)=\log (1)+C$
<br/>$\Rightarrow \quad\left(C=\frac{1}{\sqrt{2}}\right)$
<br/>On putting the value of $C$ in Eq. (0), we get $\cos \left(\frac{x}{y}\right)=\log _{e} y+\frac{1}{\sqrt{2}}$
<br/>Which is the required solution. So option D is correct.</div>


title: the-solution-of-the-equation-lo-g-101-lo-g-7-x-7-x-0-is
slug: the-solution-of-the-equation-lo-g-101-lo-g-7-x-7-x-0-is-11580
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The solution of the equation $\log _{101} \log _{7}(\sqrt{x+7}+\sqrt{x})=0$ is</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">7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">49</span> </li> </ul>

answer:
<div class="solution">Given, $\log _{101} \log _{7}(\sqrt{x+7}+\sqrt{x})=0$
<br/>$\therefore \quad \log _{7}(\sqrt{x+7}+\sqrt{x})=(101)^{0}$
<br/>$\Rightarrow \quad \log _{7}(\sqrt{x+7}+\sqrt{x})=1$
<br/>$\Rightarrow \quad(\sqrt{x+7}+\sqrt{x})=7^{1}$
<br/>$\Rightarrow \quad \sqrt{x+7}+\sqrt{x}=7$
<br/>On squaring both sides, we get
<br/>$(x+7)+x+2 \sqrt{x^{2}+7 x}=49$
<br/>$\Rightarrow \quad 2 x-42=-2 \sqrt{x^{2}+7 x}$
<br/>$\Rightarrow \quad x-21=-\sqrt{x^{2}+7 x}$
<br/>Again, squaring both sides, we get
<br/>$x^{2}+441-42 x=x^{2}+7 x$
<br/>$\Rightarrow \quad 49 x=441$
<br/>$\Rightarrow \quad x=\frac{441}{49} \Rightarrow x=9$</div>


title: the-solutions-set-of-inequation-cos-1-x-sin-1-x-is-1
slug: the-solutions-set-of-inequation-cos-1-x-sin-1-x-is-1-13488
question: <div class="question">The solutions set of inequation \(\cos ^{-1} x &lt; \sin ^{-1} x\) is</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'JEE Main']
source_db: db\MarksBatch2_P3.db
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">\(\left[\frac{1}{\sqrt{2}}, 1\right]\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\([0,1]\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\left(\frac{1}{\sqrt{2}}, 1\right)\)</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> </ul>

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

answer:
<div class="solution">Hints: \(\cos ^{-1} x  &lt; \sin ^{-1} x\)<br/>\(x \in\left(\frac{1}{\sqrt{2}}, 1\right), \quad \cos ^{-1} x &lt; \sin ^{-1} x\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/UROEt7XeVxBVIID3mCKR54q6c9W00HB9j3yb8ru1n20.original.fullsize.png"/><br/></div>


title: the-species-in-which-nitrogen-atom-is-in-a-state-of-sp-hybridisation-is
slug: the-species-in-which-nitrogen-atom-is-in-a-state-of-sp-hybridisation-is-91966
question: <div class="question">The species in which nitrogen atom is in a state of sp hybridisation is</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{NO}_3^{-}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{NO}_2\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\mathrm{NO}_2^{+}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{NO}_2^{-}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{NO}_2^{+}\)</span> </div>

answer:
<div class="solution">Hint : Steric Number in \(\mathrm{NO}_2^{+}=\frac{5+0-1}{2}=2\)<br/>\(\therefore\) sp hybridization</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-specific-heat-c-of-a-solid-at-low-temperature-shows-temperature-dependence-according-to-the-relation-c-d-t-3-where-d-is-a-constant-and-t-is-the-te

slug:
the-specific-heat-c-of-a-solid-at-low-temperature-shows-temperature-dependence-according-to-the-relation-c-d-t-3-where-d-is-a-constant-and-t-is-the-te-91199

question:
<div class="question">The specific heat $C$ of a solid at low temperature shows temperature dependence according to the relation $C=D T^{3}$, where $D$ is a constant and $T$ is the temperature is kelvin. A piece of this solid of mass $m \mathrm{kg}$ is taken and its temperature is raised from $20 \mathrm{K}$ to $30 \mathrm{K}$. The amount of heat required in the process in energy units is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \times 10^{4} \mathrm{Dm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(33 / 4) \times 10^{4} D m$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(65 / 4) \times 10^{4} D m$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(5 / 4) \times 10^{4} D m$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(65 / 4) \times 10^{4} D m$</span> </div>

answer:
<div class="solution">Amount of heat required, $Q=\int d Q$ $=\int_{T_{1}=20}^{T_{2}=30} \mathrm{mcdT}$
<br/>Given, $C=D T^{3}$
<br/>$\therefore \quad Q=\int_{20}^{30} m D T^{3} d T=m D \int_{20}^{30} T^{3} d T$
<br/>$$
<br/>=m D \frac{1}{4}\left[(30)^{2}-(20)^{2}\right]=\frac{65}{4} \times 10^{4} m D
<br/>$$</div>


title: the-stability-of-me-c-ch-2-is-more-than-that-of-mech-2-ch-ch-2-due-to-1
slug: the-stability-of-me-c-ch-2-is-more-than-that-of-mech-2-ch-ch-2-due-to-1-95369
tags: ['Chemistry', 'General Organic Chemistry', 'JEE Main']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The stability of Me $\mathcal{C}=\mathrm{CH}_{2}$ is more than that of $\mathrm{MeCH}_{2} \mathrm{CH}=\mathrm{CH}_{2}$ due to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">hyperconjugative effect of the Me groups</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a7.png"/> <br/> <br/>2-hyperconltjugate forms. Therefore, $\mathrm{Me} \underline{\mathcal{C}}=\mathrm{CH}_{2}$ is more stable than
<br/>$\mathrm{MeCH}_{2} \mathrm{CH}=\mathrm{CH}_{2}$</div>


title: the-stable-bivalency-of-pb-and-trivalency-of-bi-is
slug: the-stable-bivalency-of-pb-and-trivalency-of-bi-is-11777
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The stable bivalency of $\mathrm{Pb}$ and trivalency of $\mathrm{Bi}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">due to relativistic contraction of the $6 s$ orbitals of $\mathrm{Pb}$ and $\mathrm{Bi}$, leading to inert pair effect</span> </div>

answer:
<div class="solution">$_{82} \mathrm{Pb}=6 \mathrm{s}^{2} 6 p^{2}$
<br/>$_{83}{\mathrm{Bi}}=6 s^{2} 6 p^{3}$
<br/>Due to inertness of $6 s^{2}$ electrons in $\mathrm{Pb}$ and $\mathrm{Bi}$, they show bivalency and trivalency respectively instead of tetra and pentavalency.</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-standard-gibbs-free-energy-change-g-o-at-2-5-c-for-the-dissociation-of-n-2-o-4-g-to-no-2-g-is-given-equilibrium-constant-015-r-8314-jk-mol

slug:
the-standard-gibbs-free-energy-change-g-o-at-2-5-c-for-the-dissociation-of-n-2-o-4-g-to-no-2-g-is-given-equilibrium-constant-015-r-8314-jk-mol-45174

question:
<div class="question">The standard Gibbs free energy change $\left(\Delta G^{o}\right)$ at $25^{\circ} \mathrm{C}$ for the dissociation of $\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{g})$ to $\mathrm{NO}_{2}(g)$ is (given, equilibrium constant $=0.15, R=8.314 \mathrm{JK} / \mathrm{mol})$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.1 \mathrm{KJ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4.7 \mathrm{KJ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8.1 \mathrm{KJ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$38.2 \mathrm{KJ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4.7 \mathrm{KJ}$</span> </div>

answer:
<div class="solution">$$
<br/>\begin{array}{l}
<br/>=-2.303 \times 8.314 \times 298 \times \log 0.15 \\
<br/>=-2.303 \times 8.314 \times 298 \times(-0.82) \\
<br/>=4678.7 \mathrm{J}=4.67 \mathrm{kJ}
<br/>\end{array}
<br/>$$</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">+0.35</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-standard-reduction-potential-mathrmecirc-for-half-reations-are-beginarrayll-mathrmznmathrmzn2mathrmze-mathrmecirc076-mathrmv-mathrmfemathrmfe2math

slug:
the-standard-reduction-potential-mathrmecirc-for-half-reations-are-beginarrayll-mathrmznmathrmzn2mathrmze-mathrmecirc076-mathrmv-mathrmfemathrmfe2math-84324

question:
<div class="question">The standard reduction potential \(\mathrm{E}^{\circ}\) for half reations are<br/>\(\begin{array}{ll}<br/>\mathrm{Zn}=\mathrm{Zn}^{+2}+\mathrm{Ze} &amp; \mathrm{E}^{\circ}=+0.76 \mathrm{~V} \\<br/>\mathrm{Fe}=\mathrm{Fe}^{+2}+\mathrm{Ze} &amp; \mathrm{E}^{\circ}=+0.41 \mathrm{~V}<br/>\end{array}\)<br/>The EMF of hte cell reaction \(\mathrm{Fe}^{+2}+\mathrm{Zn}=\mathrm{Zn}^{+2}+\mathrm{Fe}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(-0.35 \mathrm{~V}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">+0.35</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\(+1.17 \mathrm{~V}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-1.17 \mathrm{~V}\)</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints: } \mathrm{E}_{\text {cell }}=\mathrm{E}_{\operatorname{Anu}([\mathrm{Qp})}^{\circ}-\mathrm{E}_{\text {cathode }(\mathrm{op})}^{\circ} \\<br/>&amp; =0.76-0.41 \\<br/>&amp; =+0.35 \mathrm{~V}<br/>\end{aligned}\)</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
the-state-of-hybridization-of-the-central-atom-and-the-number-of-lone-pairs-over-the-central-atom-in-pocl-3-are

slug:
the-state-of-hybridization-of-the-central-atom-and-the-number-of-lone-pairs-over-the-central-atom-in-pocl-3-are-62704

question:
<div class="question">The state of hybridization of the central atom and the number of lone pairs over the central atom in $\mathrm{POCl}_{3}$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$s p^{3}, 0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a23.png"/> <br/> <br/>$4 \sigma$ bonds $\Rightarrow s p^{3}$ hybridisation without lone pair of electrons.</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-stopping-potential-for-photoelectrons-from-a-metal-surface-is-v-1-when-monochromatic-light-of-frequency-v-1-is-incident-on-it-the-stopping-potenti

slug:
the-stopping-potential-for-photoelectrons-from-a-metal-surface-is-v-1-when-monochromatic-light-of-frequency-v-1-is-incident-on-it-the-stopping-potenti-28919

question:
<div class="question">The stopping potential for photoelectrons from a metal surface is $V_{1}$ when monochromatic light of frequency $v_1$ is incident on it. The stopping potential becomes $V_{2}$ when monochromatic light of another frequency is incident on the same metal surface. If $h$ be the Planck's constant and $e$ be the charge of an electron, then the frequency of light in the second case is</div>

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

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

answer:
<div class="solution">Here, $h v_{1}=\phi_{0}=e V_{1}$ ...(i)
<br/>and $h v_{2}=\phi_{0}+e V_{2}$...(ii)
<br/>From Eqs. (i) and (ii), we have
<br/>$\begin{aligned}
<br/>&amp; h\left(v_{2}-v_{1}\right)=e\left(V_{2}-v_{1}\right) \\
<br/>\Rightarrow \quad &amp; v_{2}=\frac{e}{h}\left(V_{2}-v_{1}\right)+v_{1}
<br/>\end{aligned}$</div>


title: the-straight-line-3-x-y-9-divides-the-line-segment-joining-the-points-1-3-and-2-7-in-the-ratio
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">Hints : Ratio $=-\frac{3+3-9}{6+7-9}=\frac{3}{4}$ internally</div>
source_db: db\MarksBatch2_P3.db
slug:
the-straight-line-3-x-y-9-divides-the-line-segment-joining-the-points-1-3-and-2-7-in-the-ratio-69850

question:
<div class="question">The straight line $3 x+y=9$ divides the line segment joining the points $(1,3)$ and $(2,7)$ in the ratio</div>

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

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


tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-straight-line-through-the-origin-which-divides-the-area-formed-by-the-curves-y-2-x-x-2-y-0-and-x-1-into-two-equal-halves-is

slug:
the-straight-line-through-the-origin-which-divides-the-area-formed-by-the-curves-y-2-x-x-2-y-0-and-x-1-into-two-equal-halves-is-26269

question:
<div class="question">The straight line through the origin which divides the area formed by the curves $y=2 x-x^{2}, y=0$ and $x=1$ into two equal halves is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/4uEs4--QOJdrfPRqL7lCdKIvKrBO-TAs5LlZ9A_BXlI.original.fullsize.png"/><br/> <br/>$y=2 x-x^{2}, y=0, x=1$
<br/>Area $=A=\int_{0}^{1}\left(2 x-x^{2}\right) d x$ or, $A=\frac{2}{3}$ sq units
<br/>Let $y=m x$ line divides area in two equal halves
<br/>$\therefore \frac{1}{2} \times 1 \times \mathrm{m}=\frac{1}{3}$ or, $\mathrm{m}=\frac{2}{3} \therefore$ equation of line is $\mathrm{y}=\frac{2}{3} \mathrm{x}$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
the-straight-line-x-y-1-0-meets-the-circle-x-2-y-2-6-x-8-y-0-at-a-and-b-then-the-equation-of-the-circle-of-which-a-b-is-a-diameter-is

slug:
the-straight-line-x-y-1-0-meets-the-circle-x-2-y-2-6-x-8-y-0-at-a-and-b-then-the-equation-of-the-circle-of-which-a-b-is-a-diameter-is-33885

question:
<div class="question">The straight line $x+y-1=0$ meets the circle $x^2+y^2-6 x-8 y=0$ at $A$ and $B$. Then the equation of the circle of which $A B$ is a diameter is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^2+y^2-2 y-6=0$</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">B</span> <span class="option-data">$x^2+y^2+2 y-6=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2\left(x^2+y^2\right)+2 y-6=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3\left(x^2+y^2\right)+2 y-6=0$</span> </li> </ul>

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

answer:
<div class="solution">Hints: $x^2+y^2-6 x-8 y+\lambda(x+y-1)=0$<br/>Centre $=\left(3-\frac{\lambda}{2} \cdot 4-\frac{\lambda}{2}\right)$ Lie on $x+y-1=0$<br/>$$<br/>\begin{aligned}<br/>&amp; 3-\frac{\lambda}{2}+4-\frac{\lambda}{2}-1=0, \lambda=6 \\<br/>&amp; x^2+y^2-6 x-8 y+6 x+6 y-6=0 ; \quad x^2+y^2-2 y-6=0<br/>\end{aligned}<br/>$$</div>


title: the-straight-lines-x-y-0-5-x-y-4-and-x-5-y-4-form
slug: the-straight-lines-x-y-0-5-x-y-4-and-x-5-y-4-form-34953
question: <div class="question">The straight lines $x+y=0,5 x+y=4$ and $x+5 y=4$ form</div>
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">an isosceles 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">an equilateral triangle</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a scalene triangle</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a right angled triangle</span> </li> </ul>

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

answer:
<div class="solution">Given lines are
<br/>$$
<br/>\begin{array}{r}
<br/>x+y=0 \\
<br/>5 x+y=4
<br/>\end{array}
<br/>$$
<br/>and
<br/>$$
<br/>x+5 y=4
<br/>$$
<br/>On solving above equations pairwise, we get
<br/>$$
<br/>\begin{array}{l}
<br/>A(1,-1), B\left(\frac{2}{3}, \frac{2}{3}\right), C(-1,1) \\
<br/>Now, A B=\sqrt{\left(\frac{2}{3}-1\right)^{2}+\left(\frac{2}{3}+1\right)^{2}}
<br/>\end{array}
<br/>$$
<br/> <br/>$$
<br/>\begin{array}{l}
<br/>=\sqrt{\frac{1}{9}+\frac{25}{9}}=\sqrt{\frac{26}{9}}=\frac{\sqrt{26}}{3} \\
<br/>\begin{aligned}
<br/>B C &amp;=\sqrt{\left(-1-\frac{2}{3}\right)^{2}+\left(1-\frac{2}{3}\right)^{2}} \\
<br/>&amp;=\sqrt{\frac{25}{9}+\frac{1}{9}}=\sqrt{\frac{26}{9}}=\frac{\sqrt{26}}{3} \\
<br/>C A=\sqrt{(1+1)^{2}+(-1-1)^{2}}
<br/>\end{aligned}
<br/>\end{array}
<br/>$$
<br/>$$
<br/>=\sqrt{4+4}=2 \sqrt{2}
<br/>$$
<br/>Here, we see that $A B=B C$
<br/>Hence, given lines form an isosceles triangle.</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-straight-lines-x2-y903-x5-y50-and-a-x-by-10-are-concurrent-if-the-straight-line-35-x22-y-10-passes-through-the-point

slug:
the-straight-lines-x2-y903-x5-y50-and-a-x-by-10-are-concurrent-if-the-straight-line-35-x22-y-10-passes-through-the-point-95691

question:
<div class="question">The straight lines \(x+2 y-9=0,3 x+5 y-5=0\) and \(a x+\) by \(-1=0\) are concurrent if the straight line \(35 x-22 y+\) \(1=0\) passes through the point</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((-a,-b)\)</span> </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">\((-a, b)\)</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

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

answer:
<div class="solution">Hint : \(\left|\begin{array}{lll}a &amp; b &amp; -1 \\ 1 &amp; 2 &amp; -9 \\ 3 &amp; 5 &amp; -5\end{array}\right|=0 \Rightarrow 35 a-22 b+1=0\)</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
the-stress-along-the-length-of-a-rod-with-rectangular-crosssection-is-1-of-the-youngs-modulus-of-its-material-what-is-the-approximate-percentage-of-ch-1

slug:
the-stress-along-the-length-of-a-rod-with-rectangular-crosssection-is-1-of-the-youngs-modulus-of-its-material-what-is-the-approximate-percentage-of-ch-1-53346

question:
<div class="question">The stress along the length of a rod (with rectangular cross-section) is $1 \%$ of the Young's modulus of its material. What is the approximate percentage of change of its volume? (Poisson's ratio of the material of the rod is $0.3 .$ )</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">$1 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.7 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.4 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.4 \%$</span> </div>

answer:
<div class="solution">$\because$ Stress, $\frac{F}{\Delta A}=1 \%$ of $Y=\frac{Y}{100}$
<br/>But Young's modulus, $Y=\frac{\text { stress }}{\text { strain }}=\frac{\frac{F}{\Delta A}}{\frac{\Delta l}{l}}$
<br/>$\therefore \quad Y=\frac{\frac{Y}{100}}{\frac{\Delta}{l}} \quad\left(\right.$ putting $\left.\frac{F}{\Delta A}=\frac{Y}{100}\right)$
<br/>$\therefore \quad \frac{\Delta l}{l}=\frac{1}{100}$
<br/>Poisson's ratio, $\sigma=\frac{-\frac{\Delta r}{r}}{\frac{\Delta l}{l}}$
<br/>$\therefore \quad \frac{\Delta r}{r}=-\sigma \cdot \frac{\Delta l}{l}=\frac{-0.3}{100}$
<br/>$$
<br/>\frac{\Delta r}{r}=\frac{-0.3}{100}
<br/>$$
<br/>$\therefore$ Change in volume, $\frac{\Delta V}{V}=\frac{2 \Delta r}{r}+\frac{\Delta l}{l}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{2 \times(-0.3)}{100}+\frac{1}{100} \\
<br/>=\frac{1-0.6}{100}=\frac{0.4}{100}
<br/>\end{array}
<br/>$$
<br/>$\Delta V \%=0.4 \%$</div>


title: the-structure-of-the-product-p-of-the-following-reaction-is
slug: the-structure-of-the-product-p-of-the-following-reaction-is-66935
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The structure of the product $P$ of the following reaction is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q62.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q62_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/2017_che_q62_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/2017_che_q62_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/2017_che_q62_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/2017_che_q62_optc.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a62.png"/> <br/> <br/>($\because$ Acetic anhydride is limiting reagent) Thus, we get one mole of acetanilide and molar mass of
<br/>(ii)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a62a.png"/> <br/> <br/>$\because O^{-}$ is more reactive than $-$ OMe and p-position is occupied by - OMe, the substitution occurs at ortho-position
<br/>w.r.t - O group.</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-structure-of-xef-6-is-experimentally-determined-to-be-distorted-octahedron-its-structure-according-to-vsepr-theory-is

slug:
the-structure-of-xef-6-is-experimentally-determined-to-be-distorted-octahedron-its-structure-according-to-vsepr-theory-is-53882

question:
<div class="question">The structure of $\mathrm{XeF}_{6}$ is experimentally determined to be distorted octahedron. Its structure according to VSEPR theory is</div>

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

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

answer:
<div class="solution">In $\mathrm{XeF}_{6}$. Xe, the central atom $\mathrm{F}$ contains, 8 valence electrons. Out of which 6 are utilised with fluorine in bonding (ie., it contains six bond pairs of electrons) while one pair remains as lone pair. Thus, the total pairs $=6+1=7$ Hence, its shape is pentagonal bipyramid according to VSEPR theory.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a55.png"/></div>


title: the-sum-1-1-2-2-50-50-equals
slug: the-sum-1-1-2-2-50-50-equals-29706
question: <div class="question">The sum $1 \times 1!+2 \times 2!+\ldots+50 \times 50$ ! equals</div>
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$51 !$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$51 !-1$</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">$51 !+1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \times 51 !$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$51 !-1$</span> </div>

answer:
<div class="solution">$1 \times 1!+2 \times 2!+\ldots+50 \times 50!$
<br/>$=(2-1) 1!+(3-1) 2!+(4-1) 3 !$
<br/>$+\ldots+(51-1) 50!$
<br/>$=(2 !-1!)+(3 !-2!)+(4 !-3 !)+(51!-50!)$
<br/>$+\ldots+(511-501)$
<br/>$=511-11=511-1$</div>


title: the-sum-of-all-real-roots-of-the-equation-x-2-2-x-2-2-0
slug: the-sum-of-all-real-roots-of-the-equation-x-2-2-x-2-2-0-16503
question: <div class="question">The sum of all real roots of the equation $|x-2|^2+|x-2|-2=0$</div>
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

answer:
<div class="solution">Hints: Put $1 x-21=y$<br/>$$<br/>\begin{aligned}<br/>&amp; y^2+y-2=0 \\<br/>&amp; (y-1)(y+2)=0 \\<br/>&amp; y=1 \\<br/>&amp; |x-2|=1 \\<br/>&amp; x-2= \pm 1 \\<br/>&amp; x=2 \pm 1 \\<br/>&amp; x=3,1<br/>\end{aligned}<br/>$$<br/>$$<br/>y=-2<br/>$$<br/>(Not possible)<br/>Sum $=4$</div>


title: the-sum-of-n-terms-of-the-following-series-1-3-3-3-5-3-7-3-is
slug: the-sum-of-n-terms-of-the-following-series-1-3-3-3-5-3-7-3-is-29108
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The sum of $n$ terms of the following series $1^{3}+3^{3}+5^{3}+7^{3}+\ldots$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$n^{2}\left(2 n^{2}-1\right)$</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">B</span> <span class="option-data">$n^{3}(n-1)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$n^{3}+8 n+4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 n^{4}+3 n^{2}$</span> </li> </ul>

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

answer:
<div class="solution">We have. $\mathrm{1}^{3}+3^{3}+5^{3}+7^{3}+\ldots$
<br/>$\begin{aligned} \text { Now, } T_{n} &amp;=(2 n-1)^{3} \\ &amp;=8 n^{3}-3(2 n)^{2}(1)+3(2 n)\left(0^{2}-(\pi)^{3}\right.\\ &amp;=8 n^{3}-12 n^{2}+6 n-1 \end{aligned}$
<br/>$\therefore \quad S_{n}=\Sigma T_{n}$
<br/>$=\frac{8 n^{2}(n+1)^{2}}{4}-\frac{12 n(n+0) 2 n+1}{6}$$+\frac{6 n(n+1)}{2}-n$
<br/>$=2 n^{2}(n+1)^{2}-2 n(n+1)(2 n+1)+3 n(n+1)-n$
<br/>$=n\left[2 n(n+1)^{2}-2(n+1)(2 n+1)+3(n+1)-1\right]$
<br/>$=n\left[2 n\left(n^{2}+1+2 n\right)-2\left(2 n^{2}+3 n+1\right)+3n+3-1\right].$
<br/>$=n\left[2 n^{3}+2 n+4 n^{2}-4 n^{2}-6 n-2+3 n+2\right]$
<br/>$=n\left[2 n^{3}-n\right]=n^{2}\left(2 n^{2}-1\right).$</div>


title: the-sum-of-the-infinite-series-1-2-1-4-13-6-135-is
slug: the-sum-of-the-infinite-series-1-2-1-4-13-6-135-is-68605
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The sum of the infinite series $1+\frac{1}{2 !}+\frac{1.3}{4 !}+\frac{1.3 .5}{6 !}+\ldots \ldots$. is</div>

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

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

answer:
<div class="solution">Hints : $\mathrm{T}_{\mathrm{n}}=\frac{1.3 .5 \ldots(2 n-1)}{\mid 2 n}$<br/><br/><br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{\mid 2 n}{\lfloor 2 n(2.4 \ldots 2 n)} \\<br/>&amp; =\frac{\lfloor 2 n}{2^{\mathrm{n}}\lfloor n \mid 2 n} \\<br/>&amp; =\frac{x^{\mathrm{n}}}{\lfloor n} \\<br/>&amp; \therefore \frac{x}{\lfloor 1}+\frac{x^2}{\lfloor 2}+\ldots=e^x-1 \quad \frac{1}{2}=x \\<br/>&amp; \exp =1+e^x-1=e^x=e^{1 / 2}<br/>\end{aligned}<br/>$$</div>


title: the-sum-of-the-infinite-series-1-3-1-3-6-1-3-3-6-9-1-3-5-3-6-9-12-1-3-5-7-is-equal-to
slug: the-sum-of-the-infinite-series-1-3-1-3-6-1-3-3-6-9-1-3-5-3-6-9-12-1-3-5-7-is-equal-to-38090
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The sum of the infinite series $1+\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 6}+\frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9}+\frac{1 \cdot 3 \cdot 5 \cdot 7}{3 \cdot 6 \cdot 9 \cdot 12}+\ldots$ is equal to</div>

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

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

answer:
<div class="solution">Given series, $1+\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 6}+\frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9}+\frac{1 \cdot 3 \cdot 5 \cdot 7}{3 \cdot 6 \cdot 9 \cdot 12}+\ldots \infty$
<br/>$=1+\frac{1}{3}+\frac{\frac{1}{2} \cdot \frac{3}{2}}{1 \cdot 2}\left(\frac{2}{3}\right)^{2}+\frac{1}{2} \cdot \frac{3}{1 \cdot 2 \cdot 3} \cdot \frac{5}{3}\left(\frac{2}{3}\right)^{3} \ldots \infty$
<br/>$=1+\frac{1}{2} \cdot \frac{2}{3}+\frac{\frac{1}{2}\left(\frac{1}{2}+1\right)}{2 !}\left(\frac{2}{3}\right)^{2}+\frac{\frac{1}{2}\left(\frac{1}{2}+1\right)\left(\frac{1}{2}+2\right)}{3 !}$
<br/>$=\left(1-\frac{2}{3}\right)^{-1 / 2}=3^{1 / 2}=\sqrt{3}$</div>


title: the-sum-of-the-series-1-2-1-25-c-0-2-3-1-25-c-1-3-4-1-25-c-2-26-27-1-25-c-25-is
slug: the-sum-of-the-series-1-2-1-25-c-0-2-3-1-25-c-1-3-4-1-25-c-2-26-27-1-25-c-25-is-19319
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The sum of the series
<br/>$\frac{1}{1 \times 2}^{25} C_{0}+\frac{1}{2 \times 3}{ }^{25} C_{1}+\frac{1}{3 \times 4}{ }^{25} C_{2}+\ldots \\
<br/>+\frac{1}{26 \times 27}{ }^{25} C_{25} \text { is }$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2^{27}-1}{26 \times 27}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{2^{27}-28}{26 \times 27}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}\left(\frac{2^{26}+1}{26 \times 27}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{2^{26}-1}{52}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2^{27}-28}{26 \times 27}$</span> </div>

answer:
<div class="solution">Given series is, $\frac{1}{1 \times 2} \stackrel{25}{C_{0}}+\frac{1}{2 \times 3}{ }^{25} C_{1}+\frac{1}{3 \times 4}{ }^{25} C_{2}$
<br/>$+\ldots+\frac{1}{26 \times 27}^{25} \mathrm{C}_{25}$
<br/>$\begin{aligned} \int_{0}^{x}(1+x)^{25} d x &amp;=\int_{0}^{x}\left[1^{25} C_{0}+{ }^{25} C_{1} x\right.\\ &amp;\left.+{ }^{25} C_{2} x^{2}+\ldots+{ }^{25} C_{25} x^{25}\right] d x \end{aligned}$
<br/>On integrating w. I.t. $x$, taking limits 0 to $x$, we get $\left[\frac{(1+x)^{26}}{26}\right]_{0}^{x}$
<br/>$=\left[2^{2} \mathrm{C}_{0} x+{ }^{25} \mathrm{C}_{1} \cdot \frac{x^{2}}{2}+{ }^{2} \mathrm{C}_{2} \frac{x^{3}}{3}+\ldots+{ }^{2} \mathrm{C}_{25} \cdot \frac{x^{26}}{26}\right]_{0}^{x}$
<br/>$\begin{aligned} \Rightarrow \quad &amp;\left\{\frac{1}{26}(1+x)^{26}-\frac{1}{26}\right\} \\ &amp;={ }^{25} \mathrm{C}_{0} x+{ }^{25} \mathrm{C}_{1} \cdot \frac{x^{2}}{2}+\ldots+{ }^{25} \mathrm{C}_{25} \cdot \frac{x^{26}}{26} \end{aligned}$
<br/>Again, integrating w.r.t. $x$, taking limits 0 to $1,$ we get
<br/>$\left.\frac{1}{26} \int_{0}^{1}[1+x)^{26}-1\right] d x$
<br/>$\begin{aligned} &amp;\left.\frac{1}{26} \int_{0}^{1}[1+x)^{26}-1\right] d x \\=&amp; \int_{0}^{1}\left[2^{25} C_{0} x+{ }^{25} C_{1} \cdot \frac{x^{2}}{2}+\ldots+{ }^{25} C_{25} \frac{x^{26}}{26}\right] d x \\ \Rightarrow \quad &amp; \frac{1}{26}\left[\frac{(1+x)^{27}}{27}-x\right]_{0}^{1} \end{aligned}$
<br/>$=\left[{ }^{25} \mathrm{C}_{0} \frac{x^{2}}{2}+{ }^{25} \mathrm{C}_{1} \cdot \frac{x^{3}}{2 \times 3}+\ldots+{ }^{25} \mathrm{C}_{25} \frac{x^{27}}{26 \times 27}\right]_{0}^{1}$
<br/>$\begin{aligned} \Rightarrow \quad \frac{1}{26}\left\{\frac{2^{27}}{27}-1-\frac{1}{27}\right\}=\frac{1}{2} &amp; C_{0}+\frac{1}{2 \times 3} \cdot{ }^{25} C_{1} \\ &amp;+\ldots+\frac{1}{26 \times 27} \cdot{ }^{25} C_{25} \end{aligned}$
<br/>$\begin{aligned} \therefore \quad \frac{1}{1 \times 2} \cdot^{25} \mathrm{C}_{0} &amp;+\frac{1}{2 \times 3}{ }^{25} \mathrm{C}_{1}+\frac{1}{3 \times 4}{ }^{25} \mathrm{C}_{2} \\ \quad &amp;+\ldots+\frac{1}{26 \times 27} \cdot{ }^{25} \mathrm{C}_{25}=\frac{2^{27}-28}{26 \times 27} \end{aligned}$</div>


title: the-sum-of-the-series-frac112frac123frac134ldots-infty-is
slug: the-sum-of-the-series-frac112frac123frac134ldots-infty-is-55904
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The sum of the series \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\ldots . \infty\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2 \log _{\mathrm{e}} 2+1\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(2 \log _{\mathrm{e}} 2\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 \log _{\mathrm{e}} 2-1\)</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">\(\log _e 2-1\)</span> </li> </ul>

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

answer:
<div class="solution">\(\begin{aligned}\text {Hints: } <br/>&amp; s=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4} \ldots \ldots \infty \\<br/>= &amp; \left(\frac{1}{1}-\frac{1}{2}\right)-\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right) \ldots . \\<br/>= &amp; \frac{1}{1}-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}+\frac{1}{5} \ldots \\<br/>= &amp; \frac{1}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-\ldots \\<br/>= &amp; 2\left[\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} \ldots .\right]-1=2 \log 2-1<br/>\end{aligned}\)</div>


title: the-sum-of-the-series-n-1-sin-720-n-is
slug: the-sum-of-the-series-n-1-sin-720-n-is-36828
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The sum of the series $\sum_{n=1}^{\infty} \sin \left(\frac{n ! \pi}{720}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin \left(\frac{\pi}{180}\right)+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{540}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)+\sin \left(\frac{\pi}{360}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)$ $+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{720}\right)$</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">$\sin \left(\frac{\pi}{180}\right)+\sin \left(\frac{\pi}{360}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sin \left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{120}\right)$ $+\sin \left(\frac{\pi}{360}\right)+\sin \left(\frac{\pi}{720}\right)$</span> </div>

answer:
<div class="solution">We know that
<br/>$$
<br/>\begin{array}{c}
<br/>\sin n \pi=0, \forall n \in N \\
<br/>\therefore E=\sum_{n=1}^{\infty} \sin \left(\frac{n ! \pi}{720}\right)=\sin \left(\frac{\pi}{720}\right)+\sin \left(\frac{2 ! \pi}{720}\right) \\
<br/>+\sin \left(\frac{3 ! \pi}{720}\right)+\sin \left(\frac{4 ! \pi}{720}\right)+\sin \left(\frac{5 ! \pi}{720}\right) \\
<br/>+\sin \left(\frac{6 ! \pi}{720}\right)+\ldots+\sin \left(\frac{720 ! \pi}{720}\right) \\
<br/>+\sin \left(\frac{\pi}{720}\right)+\sin \left(\frac{\pi}{320}\right)+\sin \left(\frac{\pi}{120}\right)+\sin \left(\frac{\pi}{30}\right) \\
<br/>\end{array}
<br/>$$
<br/>Here, we see that after five terms of the above series all angles are multiple of $\pi,$ i.e., $\sin n \pi$, so all further values are zero.
<br/>$$
<br/>\begin{aligned}
<br/>\therefore \quad E=\sin \left(\frac{\pi}{720}\right)+\sin \left(\frac{\pi}{320}\right)+\sin \left(\frac{\pi}{120}\right) \\
<br/>&amp;+\sin \left(\frac{\pi}{30}\right)+\sin \left(\frac{\pi}{6}\right)
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
the-system-of-equations-x-y-3-z-0-2-x-y-z-0-5-x-7-y-z-0-ha-s-in-f-ini-t-e-l-y-man-yso-l-u-t-i-o-n-s-in-r-then

slug:
the-system-of-equations-x-y-3-z-0-2-x-y-z-0-5-x-7-y-z-0-ha-s-in-f-ini-t-e-l-y-man-yso-l-u-t-i-o-n-s-in-r-then-85816

question:
<div class="question">The system of equations $$ \begin{array}{l} \lambda x+y+3 z=0 \\ 2 x+\mu y-z=0 \\ 5 x+7 y+z=0 \end{array} $$ has infinitely many solutions in $R$. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\lambda=2, \mu=3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\lambda=1, \mu=2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\lambda=1, \mu=3$</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">$\lambda=3, \mu=1$</span> </li> </ul>

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

answer:
<div class="solution">Given, system of equations
<br/>$$
<br/>\begin{array}{l}
<br/>\lambda x+y+3 z=0 \\
<br/>2 x+\mu y-z=0 \\
<br/>5 x+7 y+z=0
<br/>\end{array}
<br/>$$
<br/>System has infinitely many solutions in $R$, if
<br/>$$
<br/>\left|\begin{array}{ccc}
<br/>\lambda &amp; 1 &amp; 3 \\
<br/>2 &amp; \mu &amp; -1 \\
<br/>5 &amp; 7 &amp; 1
<br/>\end{array}\right|=0
<br/>$$
<br/>$\Rightarrow \quad \lambda(\mu+7)-1(2+5)+3(14-5 \mu)=0$
<br/>$\Rightarrow \quad \lambda(\mu+7)-7+3(14-5 \mu)=0$
<br/>$\Rightarrow \quad \lambda \mu+7 \lambda-7+42-15 \mu=0$
<br/>$\Rightarrow \quad \lambda \mu+7 \lambda-15 \mu+35=0$
<br/>By checking the options, we get option (c) $(\lambda=1, \mu=3)$ satisfies the given equation.</div>


title: the-system-of-linear-equations-x-y-z-3-x-y-2-z-6-x-y-z-has
slug: the-system-of-linear-equations-x-y-z-3-x-y-2-z-6-x-y-z-has-91266
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The system of linear equations $\lambda x+y+z=3$
<br/>$x-y-2 z=6$
<br/>$-x+y+z=\mu$ has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">infinite number of solutions for $\lambda \neq-1$ and all $\mu$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">infinite number of solutions for $\lambda=-1$ and $\mu=3$</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">no solution for $\lambda, x-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">unique solution for $\lambda=-1$ and $\mu=3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">infinite number of solutions for $\lambda=-1$ and $\mu=3$</span> </div>

answer:
<div class="solution">Augmented matrix [A:B] = $\left[\begin{array}{rrrr}1 &amp; -1 &amp; -2 &amp; 6 \\ -1 &amp; 1 &amp; 1 &amp; \mu \\ \lambda &amp; 1 &amp; 1 &amp; 3\end{array}\right]$
<br/>$\sim\left[\begin{array}{cccc}1 &amp; -1 &amp; -2 &amp; 6 \\ 0 &amp; 0 &amp; -1 &amp; \mu+6 \\ \lambda+1 &amp; 0 &amp; -1 &amp; 9\end{array}\right]$
<br/>Applying
<br/>$$
<br/>-\left[\begin{array}{cccc}1 &amp; -1 &amp; -2 &amp; R_{2}+R_{1}, R_{3} \rightarrow R_{3}+R_{1} \\ 0 &amp; 0 &amp; -1 &amp; \mu+6 \\ \lambda+1 &amp; 0 &amp; 0 &amp; 3-\mu\end{array}\right]
<br/>$$
<br/>Infinite number of solutions for $\lambda=-1$ and $\mu=3$</div>


title: the-system-that-contains-the-maximum-number-of-atoms-is
slug: the-system-that-contains-the-maximum-number-of-atoms-is-81467
question: <div class="question">The system that contains the maximum number of atoms is</div>
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.25 \mathrm{g}$ of $\mathrm{NH}_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$8 \mathrm{g}$ of $\mathrm{O}_{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{g}$ of $\mathrm{H}_{2}$</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">$4 \mathrm{g}$ of He</span> </li> </ul>

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

answer:
<div class="solution">Number of atoms $=\frac{\text { mass }}{\text { molar mass }} \times N_{A}$ x Number of atoms in 1 mole
<br/>$\therefore$ Number of atoms in $4.25 \mathrm{g} \mathrm{NH}_{3}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{4.25}{17} \times N_{A} \times 4 \\
<br/>=N_{A}
<br/>\end{array}
<br/>$$
<br/>Number of atoms in $8 \mathrm{g}$
<br/>$$
<br/>\mathrm{O}_{2}=\frac{8}{32} \times \mathrm{N}_{A} \times 2=\frac{N_{\mathrm{A}}}{2}
<br/>$$
<br/>Number of atoms in $2 \mathrm{g}$
<br/>$$
<br/>\mathrm{H}_{2}=\frac{2}{2} \times \mathrm{N}_{A} \times 2=2 \mathrm{N}_{\mathrm{A}}
<br/>$$
<br/>Number of atoms in $4 \mathrm{g}$
<br/>$$
<br/>\mathrm{He}=\frac{4}{4} \times N_{A} \times 1=N_{A}
<br/>$$
<br/>Thus, $2 \mathrm{g} \mathrm{H}_{2}$ contains the maximum number of atoms among the given.</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-tangent-at-point-a-cos-theta-b-sin-theta-0-theta-fracpi2-to-the-ellipse-fracx2a2fracy2b21-meets-the-xaxis-at-t-and-yaxis-at-t1-then-the-value-of-m

slug:
the-tangent-at-point-a-cos-theta-b-sin-theta-0-theta-fracpi2-to-the-ellipse-fracx2a2fracy2b21-meets-the-xaxis-at-t-and-yaxis-at-t1-then-the-value-of-m-63048

question:
<div class="question">The tangent at point \((a \cos \theta, b \sin \theta), 0  &lt; \theta &lt; \frac{\pi}{2}\), to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) meets the \(x\)-axis at \(T\) and \(y\)-axis at \(T_1\). Then the value of \(\min _{0 &lt; \theta &lt; \frac{\pi}{2}}(O T)\left(O T_1\right)\) is</div>

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

answer:
<div class="solution">Hint: Tangent : \(x(b \cos \theta)+y(a \sin \theta)=a b\)<br/>\(\therefore \mathrm{OT} \cdot \mathrm{OT}_1=\frac{\mathrm{ab}}{\cos \theta \sin \theta}=\frac{2 \mathrm{ab}}{\sin 2 \theta}\)</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-temperature-of-a-blackbody-radiation-enclosed-in-a-container-of-volume-v-is-increased-from-10-0-c-to-100-0-c-the-heat-required-in-the-process-is

slug:
the-temperature-of-a-blackbody-radiation-enclosed-in-a-container-of-volume-v-is-increased-from-10-0-c-to-100-0-c-the-heat-required-in-the-process-is-80845

question:
<div class="question">The temperature of a blackbody radiation enclosed in a container of volume $V$ is increased from $100^{\circ} \mathrm{C}$ to $1000^{\circ} \mathrm{C}$. The heat required in the process is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.79 \times 10^{-4}$ cal</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$9.21 \times 10^{-5}$ cal</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.17 \times 10^{-4} \mathrm{cal}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Information not sufficient</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Information not sufficient</span> </div>

answer:
<div class="solution">Information in the statement of the questions is insufficient to calculate heat required to raise the temperature from $100^{\circ} \mathrm{C}$ to $1000^{-\mathrm{C}}$.</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 v$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-temperature-of-anideal-gas-is-increased-from-120-k-to-480-k-if-at-120-k-the-root-mean-square-speed-of-gas-molecules-is-v-then-at-480-k-it-will-be

slug:
the-temperature-of-anideal-gas-is-increased-from-120-k-to-480-k-if-at-120-k-the-root-mean-square-speed-of-gas-molecules-is-v-then-at-480-k-it-will-be-76097

question:
<div class="question">The temperature of anideal gas is increased from $120 \mathrm{~K}$ to $480 \mathrm{~K}$. If at $120 \mathrm{~K}$, the root mean square speed of gas molecules is $v$, then at $480 \mathrm{~K}$ it will be</div>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \frac{\mathrm{V}_1}{\mathrm{~V}_2}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}} \\<br/>&amp; \frac{\mathrm{V}_1}{\mathrm{~V}_2}=\sqrt{\frac{120}{480}}-=\sqrt{\frac{1}{4}}=\frac{1}{2} \\<br/>&amp; \mathrm{~V}_2=2 v<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-temperature-of-the-water-of-a-pond-is-0-c-while-that-of-the-surrounding-atmosphere-is-2-0-c-if-the-density-of-ice-is-coefficient-of-thermal-conduc

slug:
the-temperature-of-the-water-of-a-pond-is-0-c-while-that-of-the-surrounding-atmosphere-is-2-0-c-if-the-density-of-ice-is-coefficient-of-thermal-conduc-39417

question:
<div class="question">The temperature of the water of a pond is $0^{\circ} \mathrm{C}$ while that of the surrounding atmosphere is $-20^{\circ} \mathrm{C}$. If the density of ice is $\rho$ coefficient of thermal conductivity is $k$ and latent heat of melting is $L,$ then the thickness $Z$ of ice layer formed increases as a function of time $t$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$Z^{2}=\frac{60 k}{\rho L} t$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$Z=\sqrt{\frac{40 k}{\rho L} t}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$Z^{2}=\frac{40 k}{\rho l} \sqrt{t}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$Z^{2}=\frac{40 k}{\rho L} t$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$Z^{2}=\frac{40 k}{\rho L} t$</span> </div>

answer:
<div class="solution">Temperature of water $=0^{\circ} \mathrm{C}$ 
<br/>Temperature of surrounding atmosphere $=-20^{\circ} \mathrm{C}$
<br/>Density of ice $=\rho$
<br/>Coefficient of thermal conductivity $=k$
<br/>latent heat of melting $=L$ 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a22.png"/> <br/> <br/>The rate of heat flow $H=\frac{d Q}{d t}=\frac{k A\left(T_{1}-T_{2}\right)}{x}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{k f(0-(-20)]}{x} \\
<br/>=\frac{k A \times 20}{x} \\
<br/>\frac{d Q}{d t}=\frac{d m}{d t} \cdot L \quad(\because m L=Q) \\
<br/>d m=\rho A \cdot d x \\
<br/>\frac{d Q}{d t}=\rho A \cdot \frac{d x}{d t} L=\frac{k A \cdot 20}{x} \\
<br/>\int_{0}^{Z} x \cdot d x=\frac{20 k}{\rho L} \int_{0}^{t} d t \Rightarrow Z^{2}=\frac{40 k}{\rho L} t
<br/>\end{array}
<br/>$$
<br/>So, thickness $Z$ increases as a function of time $t$ according to the above equation.</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
the-three-sides-of-a-right-angled-triangle-are-in-gp-geometric-progression-if-the-two-acute-angles-be-and-then-tan-and-tan-are-1

slug:
the-three-sides-of-a-right-angled-triangle-are-in-gp-geometric-progression-if-the-two-acute-angles-be-and-then-tan-and-tan-are-1-85278

question:
<div class="question">The three sides of a right angled triangle are in GP (geometric progression). If the two acute angles be $\alpha$ and $\beta$, then tan $\alpha$ and tan $\beta$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$</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">$\sqrt{5}$ and $\frac{1}{\sqrt{5}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{5}}{2}$ and $\frac{2}{\sqrt{5}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$</span> </div>

answer:
<div class="solution">Let $\Delta A B C$ be a right angled triangle at $B$. Let $\angle A$ and $\angle C$ be $\alpha$ and $\beta$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a13.png"/> <br/>Since, sides are in GP so sides are $a,$ ar, ar $^{2}$
<br/>$\begin{array}{lr}\text { Now, } &amp; \begin{array}{r}A C^{2}=A B^{2}+B C^{2} \\ \left(a r^{2}\right)^{2}=a^{2}+(a \cdot r)^{2}\end{array} \\ \Rightarrow \quad a^{2} r^{4}=a^{2}+a^{2} r^{2} \\ \Rightarrow \quad r^{4}-r^{2}-1=0 \\ \Rightarrow \quad r^{2}=\frac{1+\sqrt{5}}{2} \\ \Rightarrow \quad r=\sqrt{\frac{\sqrt{5}+1}{2}}\end{array}$
<br/>$\therefore \quad \tan \alpha=\frac{B C}{A B}=\frac{a r}{a}$
<br/>$\Rightarrow \quad r=\sqrt{\frac{\sqrt{5}+1}{2}}$
<br/>$\begin{aligned} \text { and } \tan \beta &amp;=\frac{A B}{B C}=\frac{a}{a r}=\frac{1}{r} \\ &amp;=\frac{1}{\sqrt{\frac{\sqrt{5}+1}{2}}}=\sqrt{\frac{\sqrt{5}-1}{2}} \end{aligned}$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
the-three-sides-of-a-right-angled-triangle-are-in-gp-geometric-progression-if-the-two-acute-angles-be-and-then-tan-and-tan-are

slug:
the-three-sides-of-a-right-angled-triangle-are-in-gp-geometric-progression-if-the-two-acute-angles-be-and-then-tan-and-tan-are-54584

question:
<div class="question">The three sides of a right angled triangle are in GP (geometric progression). If the two acute angles be $\alpha$ and $\beta$, then tan $\alpha$ and tan $\beta$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$</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">$\sqrt{5}$ and $\frac{1}{\sqrt{5}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{5}}{2}$ and $\frac{2}{\sqrt{5}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$</span> </div>

answer:
<div class="solution">Let $\Delta A B C$ be a right angled triangle at $B$. Let $\angle A$ and $\angle C$ be $\alpha$ and $\beta$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a13.png"/> <br/>Since, sides are in GP so sides are $a,$ ar, ar $^{2}$
<br/>$\begin{array}{lr}\text { Now, } &amp; \begin{array}{r}A C^{2}=A B^{2}+B C^{2} \\ \left(a r^{2}\right)^{2}=a^{2}+(a \cdot r)^{2}\end{array} \\ \Rightarrow \quad a^{2} r^{4}=a^{2}+a^{2} r^{2} \\ \Rightarrow \quad r^{4}-r^{2}-1=0 \\ \Rightarrow \quad r^{2}=\frac{1+\sqrt{5}}{2} \\ \Rightarrow \quad r=\sqrt{\frac{\sqrt{5}+1}{2}}\end{array}$
<br/>$\therefore \quad \tan \alpha=\frac{B C}{A B}=\frac{a r}{a}$
<br/>$\Rightarrow \quad r=\sqrt{\frac{\sqrt{5}+1}{2}}$
<br/>$\begin{aligned} \text { and } \tan \beta &amp;=\frac{A B}{B C}=\frac{a}{a r}=\frac{1}{r} \\ &amp;=\frac{1}{\sqrt{\frac{\sqrt{5}+1}{2}}}=\sqrt{\frac{\sqrt{5}-1}{2}} \end{aligned}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
the-time-taken-for-10-completion-of-a-first-order-reactin-is-20-mins-then-for-19-completion-the-reaction-will-take-1

slug:
the-time-taken-for-10-completion-of-a-first-order-reactin-is-20-mins-then-for-19-completion-the-reaction-will-take-1-19366

question:
<div class="question">The time taken for $10 \%$ completion of a first order reactin is 20 mins. Then, for $19 \%$ completion, the reaction will take</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$40 \mathrm{mins}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">60 mins</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30 \mathrm{mins}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">50 mins</span> </li> </ul>

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

answer:
<div class="solution">$t=\frac{2.303}{\lambda} \log \frac{\mathrm{N}_0}{\mathrm{~N}}$<br/>$\begin{aligned} &amp; 20=\frac{2.303}{\lambda} \log \frac{100}{90} \\ &amp; t=\frac{2.303}{\lambda} \log \frac{100}{81}\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OpaORw-7o5Sl_iRh5ghrN-7I31VTtm3b-JUPSxJ-aN8.original.fullsize.png"/><br/><br/>equation (i) / (ii)<br/>$\therefore t=40 \mathrm{~min}$.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
title:
the-time-taken-for-10-completion-of-a-first-order-reactin-is-20-mins-then-for-19-completion-the-reaction-will-take

slug:
the-time-taken-for-10-completion-of-a-first-order-reactin-is-20-mins-then-for-19-completion-the-reaction-will-take-36018

question:
<div class="question">The time taken for $10 \%$ completion of a first order reactin is 20 mins. Then, for $19 \%$ completion, the reaction will take</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$40 \mathrm{mins}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">60 mins</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30 \mathrm{mins}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">50 mins</span> </li> </ul>

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

answer:
<div class="solution">$t=\frac{2.303}{\lambda} \log \frac{\mathrm{N}_0}{\mathrm{~N}}$<br/>$\begin{aligned} &amp; 20=\frac{2.303}{\lambda} \log \frac{100}{90} \\ &amp; t=\frac{2.303}{\lambda} \log \frac{100}{81}\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OpaORw-7o5Sl_iRh5ghrN-7I31VTtm3b-JUPSxJ-aN8.original.fullsize.png"/><br/><br/>equation (i) / (ii)<br/>$\therefore t=40 \mathrm{~min}$.</div>


title: the-time-taken-for-an-electron-to-complete-one-revolution-in-bohr-orbit-of-hydrogen-atom-is
slug: the-time-taken-for-an-electron-to-complete-one-revolution-in-bohr-orbit-of-hydrogen-atom-is-71985
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The time taken for an electron to complete one revolution in Bohr orbit of hydrogen atom is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4 \pi^{2} m r^{2}}{n h}$</span> </div>

answer:
<div class="solution">Let the distance travelled in $T$ time $=2 \pi r$ (Circumference of orbit) 
<br/>$\therefore \quad$ Velocity $(v)=\frac{2 \pi r}{T}$
<br/>Also, $v=\frac{n \cdot h}{2 \pi m r}$
<br/>$\therefore \quad \frac{1}{T \text { (Time period) }}=$ frequency $v=\text{v} \times \frac{1}{2 \pi r}$
<br/>From Eq.
<br/>$$
<br/>\begin{array}{l}
<br/>\mathrm{v}=\frac{1}{T}=\frac{n h}{2 \pi \times 2 \pi m r} \\
<br/>v=\frac{n h}{4 \pi^{2} r^{2} m}
<br/>\end{array}
<br/>$$
<br/>$T$ (time taken in one revolution)
<br/>$$
<br/>=\frac{4 \pi^{2} r^{2} m}{n h}
<br/>$$</div>


tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-total-number-of-alkyl-bromides-including-stereoismers-formed-in-the-reaction-me-3-c-ch-ch-2-hbr-will-be

slug:
the-total-number-of-alkyl-bromides-including-stereoismers-formed-in-the-reaction-me-3-c-ch-ch-2-hbr-will-be-89775

question:
<div class="question">The total number of alkyl bromides (including stereoismers) formed in the reaction $\mathrm{Me}_{3} \mathrm{C}-\mathrm{CH}=\mathrm{CH}_{2}+\mathrm{HBr} \rightarrow$ will be</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">No bromide forms</span> </li> </ul>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/cH8bl0NFHWHidYT59vSNOL9B84IADlTSqUaszVQ4mx0.original.fullsize.png"/></div>


title: the-total-number-of-aromatic-species-generated-in-the-following-reactions-is
slug: the-total-number-of-aromatic-species-generated-in-the-following-reactions-is-52279
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The total number of aromatic species generated in the following reactions is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q74.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">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">Criteria for aromaticity (Huckel rule) The cyclic $\pi$ molecular orbital (electron cloud) formed by overlap of $p$ -orbitals must contalu $(4 n+2) \pi$ -electrons, where $n=$ integer 0,1,2,3 Pte. This is known as Huckel rule.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a74.png"/></div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">840</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-total-number-of-injections-oneone-into-mappings-from-a-1-a-2-a-3-a-4-b-1-b-2-b-3-b-4-b-5-b-6-b-7-is

slug:
the-total-number-of-injections-oneone-into-mappings-from-a-1-a-2-a-3-a-4-b-1-b-2-b-3-b-4-b-5-b-6-b-7-is-38042

question:
<div class="question">The total number of injections (one-one into mappings) from $\quad\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$
<br/>$\left(b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">400</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">420</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">800</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">840</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$
<br/>$\begin{aligned} B &amp;=\left\{b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}\right\} \\ n(A) &amp;=4, n(B)=7 \end{aligned}$
<br/>The total number of infections $={ }^{7} P_{4}$ $=\frac{71}{3 !}=7 \cdot 6 \cdot 5 \cdot 4=840$</div>


title: the-total-number-of-isomeric-linear-dipeptides-which-can-be-synthesised-from-racemic-alanine-is
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P3.db
slug:
the-total-number-of-isomeric-linear-dipeptides-which-can-be-synthesised-from-racemic-alanine-is-34123

question:
<div class="question">The total number of isomeric linear dipeptides which can be synthesised from racemic alanine 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">Total number of isomeric linear dipeptides which can be synthesised from racemic mixture of alanine are four $(4),$ by $(R)$ type and
<br/>(S) type.
<br/>($\because$ Has two chiral centres)
<br/>i.e.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a71.png"/> <br/>The possible isomers are
<br/>$$
<br/>(R R),(S R),(R S) \text { and }(S S)
<br/>$$
<br/>Hence, total (4) isomeric linear dipeptides for alanine can be synthesised and option
<br/>(d) is the correct answer.</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-total-number-of-tangents-through-the-point-3-5-that-can-be-drawn-to-the-ellipses-3-x-2-5-y-2-32-and-25-x-2-9-y-2-450-is

slug:
the-total-number-of-tangents-through-the-point-3-5-that-can-be-drawn-to-the-ellipses-3-x-2-5-y-2-32-and-25-x-2-9-y-2-450-is-67548

question:
<div class="question">The total number of tangents through the point $(3,5)$ that can be drawn to the ellipses $3 x^2+5 y^2=32$ and $25 x^2+9 y^2=450$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</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">Hints : $(3,5)$ lies outside the ellipse $3 x^2+5 y^2=32$ and on the ellipse $25 x^2+9 y^2=450$. Therefore there will be 2 tangents for the first ellipse and one tangent for the second ellipse.</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
the-transverse-axis-of-a-hyperbola-is-along-the-x-axis-and-its-length-is-2-a-the-vertex-of-the-hyperbola-bisects-the-line-segment-joining-the-centre-a

slug:
the-transverse-axis-of-a-hyperbola-is-along-the-x-axis-and-its-length-is-2-a-the-vertex-of-the-hyperbola-bisects-the-line-segment-joining-the-centre-a-77791

question:
<div class="question">The transverse axis of a hyperbola is along the $x$ -axis and its length is $2 a$. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is</div>

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

answer:
<div class="solution">Let e be the eccentricity of hyperbola and length of conjugate axis be $2 h$. since, vertex $(a, 0)$ bisects the join of centre (0,0) and focus $(a e, 0)$
<br/>$$
<br/>\begin{array}{l}
<br/>a=\frac{a e+0}{2} \Rightarrow e=2 \\
<br/>b^{2}=a^{2}\left(c^{2}-1\right)=3 a^{2}
<br/>\end{array}
<br/>$$
<br/>$\therefore$ Required hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
<br/>$\Rightarrow \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{3 a^{2}}=1 \Rightarrow 3 x^{2}-y^{2}=3 a^{2}$</div>


title: the-trigonometric-equation-sin-1-x-2-sin-1-2-a-has-a-real-solution-if
slug: the-trigonometric-equation-sin-1-x-2-sin-1-2-a-has-a-real-solution-if-16761
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The trigonometric equation $\sin ^{-1} x=2 \sin ^{-1} 2 a$ has a real solution, if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$|a|&gt;\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2 \sqrt{2}}    &lt; |a| &lt; \frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$|a|&gt;\frac{1}{2 \sqrt{2}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$|a| \leq \frac{1}{2 \sqrt{2}}$</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> </ul>

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

answer:
<div class="solution">We know that, $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$
<br/>$\Rightarrow \frac{-\pi}{2} \leq 2 \sin ^{-1} 2 a \leq \frac{\pi}{2}$
<br/>$\Rightarrow \frac{-\pi}{4} \leq \sin ^{-1} 2 a \leq \frac{\pi}{4}$
<br/>$\Rightarrow \sin \left(\frac{-\pi}{4}\right) \leq 2 a \leq \sin \left(\frac{\pi}{4}\right)$
<br/>$\Rightarrow \frac{-1}{\sqrt{2}} \leq 2 a \leq \frac{1}{\sqrt{2}}$
<br/>$\Rightarrow \frac{-1}{2 \sqrt{2}} \leq a \leq \frac{1}{2 \sqrt{2}} \Rightarrow|a| \leq \frac{1}{2 \sqrt{2}}$</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-two-halfcell-reactions-of-an-electrochemical-cell-is-given-as-ag-e-ag-e-ax-ag-03995-v-fe-3-fe-3-e-e-fe-3-fv-2-07120-v-the-value-of-cell-emf-will-b

slug:
the-two-halfcell-reactions-of-an-electrochemical-cell-is-given-as-ag-e-ag-e-ax-ag-03995-v-fe-3-fe-3-e-e-fe-3-fv-2-07120-v-the-value-of-cell-emf-will-b-12630

question:
<div class="question">$\begin{array}{lllll}\text {The two } &amp; \text { half-cell } &amp; \text { reactions } &amp; \text { of } &amp; \text { an }\end{array}$ electrochemical cell is given as $\mathrm{Ag}^{+}+e^{-} \longrightarrow \mathrm{Ag} ; \quad E_{\mathrm{Ax}^{+} / \mathrm{Ag}}^{\circ}=-0.3995 \mathrm{V}$ $\mathrm{Fe}^{3+} \longrightarrow \mathrm{Fe}^{3 *}+e^{-} ; E_{\mathrm{Fe}^{3+} \mathrm{Fv}^{2+}}=-0.7120 \mathrm{V}$. The value of cell EMF will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-0.3125 \mathrm{V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.3125 \mathrm{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.114 \mathrm{V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-1.114 \mathrm{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.3125 \mathrm{V}$</span> </div>

answer:
<div class="solution">Species with more negative $E^{\circ}$ (standard reduction potential) generally acts as reducing agent while with less negative $E^{\circ}$ acts as oxidising agent. Thus, the overall reaction is
<br/>$\begin{aligned} \mathrm{Ag}^{+}+\mathrm{Fe}^{2+} &amp; \longrightarrow \mathrm{Fe}^{3+}+\mathrm{Ag} \\ \Delta E^{\circ} &amp;=E_{\mathrm{OA}}^{\circ}-E_{\mathrm{RA}}^{\circ} \\ &amp;=-0.3995-(-0.7120) \mathrm{V} \\ &amp;=-0.3995+0.7120 \mathrm{V} \\ &amp;=+0.3125 \mathrm{V} \end{aligned}$</div>


title: the-two-structures-written-below-represent
slug: the-two-structures-written-below-represent-45231
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">$$<br/>\text { The two structures written below represent }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hrAFEQAINridkjxcLEpskq9XXG93z_ksmmWmHaqtmMY.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OmeC5SPDVw1rs2HXTMmsCEMo5L7zbcVU5xnJx9rOxOE.original.fullsize.png"/><br/><br/>I \&amp; II are same Fischer projection because $180^{\circ}$ rotation doesn't change configuration</div>


title: the-unit-vector-in-zox-plane-making-angles-4-5-and-6-0-respectively-with-2-i-2-j-k-and-j-k-is
slug: the-unit-vector-in-zox-plane-making-angles-4-5-and-6-0-respectively-with-2-i-2-j-k-and-j-k-is-95322
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The unit vector in ZOX plane, making angles $45^{\circ}$ and $60^{\circ}$ respectively with $\vec{\alpha}=2 \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{\beta}=j-\hat{k}$ is</div>

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

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

answer:
<div class="solution">Hint: 
<br/>Let the vector be $\vec{r}=x \hat{i}+z \hat{k} \Rightarrow|r|=1$
<br/>$\vec{r} \cdot \vec{a}=|r| |\vec{\alpha}| \cos 45^{\circ}$
<br/>$\therefore 2 x-z=\frac{3}{\sqrt{2}}$
<br/>$\vec{r} \cdot \vec{\beta}=|\vec{r}| |\vec{\beta}| \cos 60^{\circ}$
<br/>$z=-\frac{1}{\sqrt{2}}$
<br/>$\therefore \mathrm{x}=\frac{1}{\sqrt{2}}$
<br/>$\therefore \overrightarrow{\mathrm{r}}=\frac{1}{\sqrt{2}} \hat{i}-\frac{1}{\sqrt{2}} \hat{k}$</div>


title: the-units-of-surface-tension-and-viscosity-of-liquids-respectively-are
slug: the-units-of-surface-tension-and-viscosity-of-liquids-respectively-are-48958
question: <div class="question">The units of surface tension and viscosity of liquids respectively are</div>
tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{kg} \mathrm{m}^{-1} \mathrm{s}^{-1}, Nm^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{kg} \mathrm{s}^{-2}, \mathrm{kg} \mathrm{m}^{-1} \mathrm{s}^{-1}$</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">$\mathrm{Nm}^{-1}, \mathrm{kg} \mathrm{m}^{-1} \mathrm{s}^{-2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{kg} \mathrm{s}^{-1}, \mathrm{kg} \mathrm{m}^{-2} \mathrm{s}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{kg} \mathrm{s}^{-2}, \mathrm{kg} \mathrm{m}^{-1} \mathrm{s}^{-1}$</span> </div>

answer:
<div class="solution">From the formula, Surface tension, $(\gamma)$
<br/>$$
<br/>=\frac{\Delta W}{\Delta A}=\frac{\mathrm{J}}{\mathrm{m}^{2}}=\frac{\mathrm{kg} \mathrm{m}^{2} \mathrm{s}^{-2}}{\mathrm{m}^{2}}=\mathrm{kg} \mathrm{s}^{-2}
<br/>$$
<br/>Form the formula, $F=\eta \cdot A \cdot \frac{d V}{d x}$ or, $\eta$ (coefficient of viscosity)
<br/>$$
<br/>=\frac{F}{A \cdot \frac{d V}{d x}}=\frac{\mathrm{N}}{\mathrm{m}^{2} \cdot \frac{\mathrm{ms}^{-1}}{\mathrm{m}}}=\mathrm{Nm}^{-2} \cdot \mathrm{s}
<br/>$$
<br/>or $\mathrm{kg} \mathrm{ms}^{-2} \cdot \mathrm{m}^{-2} \cdot \mathrm{s}=\mathrm{kg} \mathrm{m}^{-1} \mathrm{s}^{-1}$</div>


title: the-value-of-0-2-c-o-s-x-s-i-n-x-s-i-n-x-c-o-s-x-c-o-s-x-s-i-n-x-d-x-is
slug: the-value-of-0-2-c-o-s-x-s-i-n-x-s-i-n-x-c-o-s-x-c-o-s-x-s-i-n-x-d-x-is-25405
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x}+(\sin x)^{\cos x}} d x$ is</div>

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

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

answer:
<div class="solution">$I=\int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x}+(\sin x)^{\cos x} d x}$<br/>$I=\int_0^{\pi / 2} \frac{(\sin x)^{\cos x} d x}{(\sin x)^{\cos x}+(\cos x)^{\sin x}}$<br/>$\Rightarrow 2 \mathrm{I}=\pi / 2$<br/>$\Rightarrow I=\pi / 4$</div>


title: the-value-of-0-5-max-x-2-6-x-8-d-x-is
slug: the-value-of-0-5-max-x-2-6-x-8-d-x-is-98328
question: <div class="question">The value of $\int_{0}^{5} \max \left\{x^{2}, 6 x-8\right\} d x$ is</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">43</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">72</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">125</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">43</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">69</span> </li> </ul>

answer:
<div class="solution">$\int_{0}^{5} \max \left\{x^{2}, 6 x-8\right\} d x$
<br/>Let $f(x)=x^{2}-6 x+8$
<br/>$\quad=(x-4)(x-2)$
<br/>$\therefore$ For $x \in[0,2] \cup[4,5], f(x)&gt;0$ and in $x \in[2,4], f(x)     &lt; 0$
<br/>$\int_{0}^{5} \max \left(x^{2}, 6 x-8\right)=\int_{0}^{2} x^{2} d x+\int_{2}^{4}(6 x-8) d x+\int_{4}^{5} x^{2} d x=\frac{8}{3}+20+\frac{61}{3}=43$</div>


title: the-value-of-0-x-2-4-x-2-9-d-x-is
slug: the-value-of-0-x-2-4-x-2-9-d-x-is-39486
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\int_0^{\infty} \frac{d x}{\left(x^2+4\right)\left(x^2+9\right)}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{60}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{\pi}{20}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{40}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{80}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{60}$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } \left.\int_0^{\infty} \frac{\mathrm{dx}}{\left(\mathrm{x}^2+4\right)\left(\mathrm{x}^2+9\right)}=\int_0^{\pi / 2} \frac{\sec ^2 \theta}{\left(\tan ^2 \theta+4\right)\left(\tan ^2 \theta+9\right)} \mathrm{d} \theta \text { (putting } \mathrm{x}=\tan \theta\right) \\<br/>&amp; =\frac{1}{5} \int_0^{\pi / 2} \frac{\left\{\left(9+\tan ^2 \theta\right)-\left(4+\tan ^2 \theta\right)\right\} \sec ^2 \theta}{\left(\tan ^2 \theta+4\right)\left(\tan ^2 \theta+9\right)} \mathrm{d} \theta<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>=\frac{1}{5}\left[\int_0^{\pi / 2} \frac{\sec ^2 \theta}{4+\tan ^2 \theta} \mathrm{d} \theta-\int_0^{\pi / 2} \frac{\sec ^2 \theta}{9+\tan ^2 \theta} \mathrm{d} \theta\right]<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{1}{5}\left[\left.\frac{1}{2} \tan ^{-1}\left(\frac{\tan \theta}{2}\right)\right|_0 ^{\pi / 2}-\left.\frac{1}{3} \tan ^{-1}\left(\frac{\tan \theta}{3}\right)\right|_0 ^{\pi / 2}\right] \\<br/>&amp; =\frac{1}{5}\left[\frac{1}{2} \cdot \frac{\pi}{2}-\frac{1}{3} \cdot \frac{\pi}{2}\right]=\left(\frac{\pi}{2}\right)\left(\frac{1}{5}\right)\left(\frac{1}{2}-\frac{1}{3}\right)=\frac{\pi}{2} \cdot \frac{1}{5} \cdot \frac{1}{6}=\frac{\pi}{60}<br/>\end{aligned}<br/>$$</div>


title: the-value-of-1-3-i-1-3-i-64-1-3-i-1-3-i-64-is
slug: the-value-of-1-3-i-1-3-i-64-1-3-i-1-3-i-64-is-77598
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\left(\frac{1+\sqrt{3} i}{1-\sqrt{3 i}}\right)^{64}+\left(\frac{1-\sqrt{3} i}{1+\sqrt{3} i}\right)^{64}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">-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">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$i$</span> </li> </ul>

answer:
<div class="solution">We know that, $\omega=\frac{-1+\sqrt{3} i}{2} 1-\sqrt{3} i=-2 \omega$
<br/>and $\omega^{2}=\frac{-1-\sqrt{3} i}{2} 1+\sqrt{3} i=-2 \omega^{2}$
<br/>Now, $\begin{aligned}\left(\frac{1+\sqrt{3} i}{1-\sqrt{3 i}}\right)^{64}+\left(\frac{1-\sqrt{3} i}{1+\sqrt{3} i}\right)^{64} \\ &amp;=\left(\frac{-2 \omega^{2}}{-2 \omega}\right)^{64}+\left(\frac{-2 \omega}{-2 \omega^{2}}\right)^{64} \end{aligned}$
<br/>$=\omega^{64}+\frac{1}{\omega^{64}}=\omega+\omega^{2}$
<br/>$$
<br/>\left[\because \cdot \omega^{3}=1\right]
<br/>$$
<br/>$$
<br/>\left.=-1 \quad \mid \because 1+\omega+\omega^{2}=0\right]
<br/>$$</div>


title: the-value-of-1-cos-6-1-cos-3-1-cos-2-3-1-cos-7-6-is
slug: the-value-of-1-cos-6-1-cos-3-1-cos-2-3-1-cos-7-6-is-19972
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $(1+\cos \pi / 6)(1+\cos \pi / 3)(1+\cos 2 \pi / 3)(1+\cos 7 \pi / 6)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{8}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{16}$</span> </div>

answer:
<div class="solution">Hints : $\left(1+\cos \frac{\pi}{6}\right)\left(1+\cos \frac{\pi}{3}\right)\left(1+\cos \frac{2 \pi}{3}\right)\left(1+\cos \frac{7 \pi}{6}\right)$<br/>$$<br/>=\left(1+\frac{\sqrt{3}}{2}\right)\left(1+\frac{1}{2}\right)\left(1-\frac{1}{2}\right)\left(1-\frac{\sqrt{3}}{2}\right)=\left(1-\frac{3}{4}\right)\left(1-\frac{1}{4}\right)=\frac{1}{4} \times \frac{3}{4}=\frac{3}{16}<br/>$$</div>


title: the-value-of-2-cot-1-2-1-cot-1-3-4-is
slug: the-value-of-2-cot-1-2-1-cot-1-3-4-is-27716
question: <div class="question">The value of $2 \cot ^{-1} \frac{1}{2}-\cot ^{-1} \frac{4}{3}$ is</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{\pi}{8}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3 \pi}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2}$</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>

answer:
<div class="solution">$2 \cot ^{-1} \frac{1}{2}-\cot ^{-1} \frac{4}{3}$<br/>$\begin{array}{l}<br/>=2 \tan ^{-1} 2-\tan ^{-1} \frac{3}{4} \\<br/>=\pi+\tan ^{-1} \frac{4}{-3}-\tan ^{-1} \frac{3}{4} \\<br/>=\pi-\tan ^{-1} \frac{4}{3}-\tan ^{-1} \frac{3}{4}<br/>\end{array}$<br/>$=\pi-\left\{\tan ^{-1} \frac{\frac{4}{3}+\frac{3}{4}}{1-\frac{4}{3} \cdot \frac{3}{4}}\right\}$<br/>$=\pi-\tan ^{-1} \infty=\pi-\frac{\pi}{2}=\frac{\pi}{2}$</div>


title: the-value-of-3-2-5-4-7-6-is
slug: the-value-of-3-2-5-4-7-6-is-69201
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\frac{2}{3 !}+\frac{4}{5 !}+\frac{6}{7 !}+\ldots \ldots$. is</div>

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

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

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>\text { Hints: } t_n=\frac{2 n}{(2 n+1) !}=\frac{2 n+1}{(2 n+1) !}-\frac{1}{(2 n+1) !}=\frac{1}{(2 n) !}-\frac{1}{(2 n+1) !} \\<br/>\sum_{n=1}^{\infty} t_n=\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}+\ldots \ldots \infty=\mathrm{e}^{-1}<br/>\end{gathered}<br/>$$</div>


title: the-value-of-1000-1-2-1-2-3-1-3-4-1-999-1000-1-is
slug: the-value-of-1000-1-2-1-2-3-1-3-4-1-999-1000-1-is-28608
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">999</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $1000\left[\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{999 \times 1000}\right]$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1000</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">999</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1001</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{999}$</span> </li> </ul>

answer:
<div class="solution">$1000\left[\frac{1}{1 \times 2} \times \frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{999 \times 1000}\right]$
<br/>$=1000\left\{\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)\right.$
<br/>$\left.+\ldots+\left(\frac{1}{999}-\frac{1}{1000}\right)\right]$
<br/>$=1000\left(1-\frac{1}{1000}\right)$
<br/>$=1000 \times \frac{999}{1000}$
<br/>$=999$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-value-of-a-for-which-sum-of-the-squares-of-the-roots-of-the-equation-x-2-a-2-x-a-1-0-assumes-the-least-value-is

slug:
the-value-of-a-for-which-sum-of-the-squares-of-the-roots-of-the-equation-x-2-a-2-x-a-1-0-assumes-the-least-value-is-39652

question:
<div class="question">The value of a for which sum of the squares of the roots of the equation $x^2-(a-2) x-a-1=0$ assumes the least value is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Let $\alpha, \beta$ be the roots, then<br/>$\begin{aligned}<br/>&amp;\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta \\<br/>&amp;=(a-2)^2+2(a+1) \\<br/>&amp;=a^2-2 a+6 \\<br/>&amp;=(a-1)^2+5 \geq 5 \\<br/>&amp;\therefore \alpha^2+\beta^2 \text { is minimum if } a=1 .<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-value-of-a-for-which-the-scalar-triple-product-formed-by-the-vectors-vecahatia-hatjhatk-vecbetahatja-hatk-and-vecgammaa-hatihatk-is-maximum-is

slug:
the-value-of-a-for-which-the-scalar-triple-product-formed-by-the-vectors-vecahatia-hatjhatk-vecbetahatja-hatk-and-vecgammaa-hatihatk-is-maximum-is-70543

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

answer:
<div class="solution">Hint : \(\Delta=1+\mathrm{a}^3-\mathrm{a}\) For \(\max ; \frac{\mathrm{d} \Delta}{\mathrm{da}}=0\)<br/>\(\frac{\mathrm{d} \Delta}{\mathrm{da}}=3 \mathrm{a}^2-1 \quad \mathrm{a}= \pm \frac{1}{\sqrt{3}}\)<br/>\(\frac{\mathrm{d}^2 \Delta}{\mathrm{da}^2}=6 \mathrm{a}  &lt; 0\) for \(\mathrm{a}=-\frac{1}{\sqrt{3}}\)</div>


title: the-value-of-c-o-t-2-x-c-o-t-x-t-a-n-x-is
slug: the-value-of-c-o-t-2-x-c-o-t-x-t-a-n-x-is-36561
question: <div class="question">The value of $\frac{\cot x-\tan x}{\cot 2 x}$ is</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Hints: $\frac{\cos ^2 x-\sin ^2 x}{\sin x \cos x} \times \frac{\sin 2 x}{\cos 2 x}=\frac{2 \cos 2 x}{\sin 2 x} \times \frac{\sin 2 x}{\cos 2 x}=2$</div>


title: the-value-of-cos-1-5-cos-7-2-1-0-sin-7-2-1-0-is
slug: the-value-of-cos-1-5-cos-7-2-1-0-sin-7-2-1-0-is-25190
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\cos 15^{\circ} \cos 7 \frac{1^0}{2} \sin 7 \frac{1^0}{2}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{8}$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } \cos 15^{\circ} \cos 7 \frac{1}{2}^0 \sin 7 \frac{1}{2}^0=\frac{1}{2}\left(2 \sin 7 \frac{1}{2}^0 \cos 7 \frac{1}{2}^0\right) \cdot\left(\cos 15^{\circ}\right) \\<br/>&amp; \frac{1}{2}\left(\sin 15^{\circ}\right)\left(\cos 15^{\circ}\right)=\frac{1}{4}\left(2 \sin 15^{\circ} \cos 15^{\circ}\right)=\frac{1}{4} \times \sin 30^0=\frac{1}{8}<br/>\end{aligned}<br/>$$</div>


title: the-value-of-cos-1-5-cos-7-2-1-sin-7-2-1-is
slug: the-value-of-cos-1-5-cos-7-2-1-sin-7-2-1-is-85243
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\cos 15^{\circ} \cos 7 \frac{1^{\circ}}{2} \sin 7 \frac{1^{\circ}}{2}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{8}$</span> </div>

answer:
<div class="solution">We have, $\cos 15^{\circ} \cdot \cos 7 \frac{1}{2} \cdot \sin 7 \frac{1}{2}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{1}{2} \cos 15\left(2 \cos 7 \frac{1}{2} \cdot \sin 7 \frac{1}{2}\right) \\
<br/>=\frac{1}{2} \cos 15\left[\sin 2\left(7 \frac{1}{2}\right)\right] \quad\left[\begin{array}{c}
<br/>\because 2 \sin \theta \cdot \cos \theta \\
<br/>=\sin 20
<br/>\end{array}\right] \\
<br/>=\frac{1}{2} \cos 15^{\circ} \sin 15 \\
<br/>=\frac{1}{2} \times \frac{1}{2}\left(2 \sin 15^{\circ} \cos 15^{\circ}\right) \\
<br/>=\frac{1}{4} \sin 30^{\circ}=\frac{1}{4} \times \frac{1}{2}=\frac{1}{8}
<br/>\end{array}
<br/>$$</div>


title: the-value-of-cos-2-7-5-cos-2-4-5-cos-2-1-5-cos-2-3-0-cos-2-6-0-is
slug: the-value-of-cos-2-7-5-cos-2-4-5-cos-2-1-5-cos-2-3-0-cos-2-6-0-is-49982
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\cos ^{2} 75^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 15^{\circ}$ $-\cos ^{2} 30^{\circ}-\cos ^{2} 60^{\circ}$ is</div>

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

answer:
<div class="solution">Given $\cos ^{2} 75^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 15^{\circ}-\cos ^{2} 30^{\circ}\cos^260^\circ$
<br/>$=\left(\frac{\sqrt{3}}{2 \sqrt{2}}-\frac{1}{2 \sqrt{2}}\right)^{2}+\left(\frac{1}{\sqrt{2}}\right)^{2}+\left(\frac{\sqrt{3}}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}\right)^{2}$ $-\left(\frac{\sqrt{3}}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}$
<br/>$=\left(\frac{3+1-2 \sqrt{3}}{8}\right)+\frac{1}{2}+\left(\frac{3+1+2 \sqrt{3}}{8}\right)-\frac{3}{4}-\frac{1}{4}$
<br/>$=\frac{1}{2}-\frac{\sqrt{3}}{4}+\frac{1}{2}+\frac{1}{2}+\frac{\sqrt{3}}{4}-\frac{3}{4}-\frac{1}{4}=\frac{3}{2}-1=\frac{1}{2}$</div>


title: the-value-of-cos-15circsin-15circ-is
slug: the-value-of-cos-15circsin-15circ-is-69580
question: <div class="question">The value of \(\cos 15^{\circ}-\sin 15^{\circ}\) is</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{\sqrt{2}}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(-\frac{1}{\sqrt{2}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{2 \sqrt{2}}\)</span> </li> </ul>

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

answer:
<div class="solution">Hints : \(\cos 15^{\circ}-\sin 15^{\circ}=\sqrt{2} \cos 60^{\circ}=\frac{1}{\sqrt{2}}\)</div>


title: the-value-of-det-a-where-a-1-cos-1-cos-1-cos-0-cos-1-lies
slug: the-value-of-det-a-where-a-1-cos-1-cos-1-cos-0-cos-1-lies-40094
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of det $A$, where $A=\left(\begin{array}{ccc}1 &amp; \cos \theta &amp; 0 \\ -\cos \theta &amp; 1 &amp; \cos \theta \\ -1 &amp; -\cos \theta &amp; 1\end{array}\right)$, lies</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">in the closed interval [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">B</span> <span class="option-data">in the closed interval [0,1]</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">in the open interval (0,1)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">in the open interval (1, 2)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">in the closed interval [1,2]</span> </div>

answer:
<div class="solution">We have, $|A|=\left|\begin{array}{ccc}1 &amp; \cos \theta &amp; 0 \\ -\cos \theta &amp; 1 &amp; \cos \theta \\ -1 &amp; -\cos \theta &amp; 1\end{array}\right|$
<br/>$=1[1-(-\cos \theta)(\cos \theta)]$
<br/>$-\cos \theta[-\cos \theta+\cos \theta]+0\left(\cos ^{2} \theta+1\right)$
<br/>$=1+\cos ^{2} \theta$
<br/>Now, we know that $-1 \leq \cos \theta \leq 1$
<br/>$\Rightarrow \quad 0 \leq \cos ^{2} \theta \leq 1$
<br/>$\Rightarrow \quad 1 \leq 1+\cos ^{2} \theta \leq 2$
<br/>$\Rightarrow \quad 1 \leq|A| \leq 2$
<br/>$\therefore \quad|A| \in[1,2]$</div>


title: the-value-of-determinant-1-a-2-b-2-2-ab-2-b-2-ab-1-a-2-b-2-2-a-2-b-2-a-1-a-2-b-2-is
slug: the-value-of-determinant-1-a-2-b-2-2-ab-2-b-2-ab-1-a-2-b-2-2-a-2-b-2-a-1-a-2-b-2-is-10755
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of determinant $\left|\begin{array}{ccc}1+a^{2}-b^{2} &amp; 2 a b &amp; -2 b \\ 2 a b &amp; 1-a^{2}+b^{2} &amp; 2 a \\ 2 b &amp; -2 a &amp; 1-a^{2}-b^{2}\end{array}\right|$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(1+a^{2}+b^{2}\right)^{3}$</span> </div>

answer:
<div class="solution">Let
<br/>$\Delta=\left|\begin{array}{ccc}1+a^{2}-b^{2} &amp; 2 a b &amp; -2 b \\ 2 a b &amp; 1-a^{2}+b^{2} &amp; 2 a \\ 2 b &amp; -2 a &amp; 1-a^{2}-b^{2}\end{array}\right|$
<br/>Apply $\mathrm{C}_{\mathrm{1}} \rightarrow \mathrm{C}_{1}-\mathrm{bC}_{3}$ and $\mathrm{C}_{2} \rightarrow \mathrm{aC}_{3}+\mathrm{C}_{2}$
<br/>$=\left|\begin{array}{ccc}1+a^{2}-b^{2}+2 b^{2} &amp; 2 a b-2 a b &amp; -2 b \\ 2 a b-2 a b &amp; 1-a^{2}+b^{2}+2 a^{2} &amp; 2 a \\ 2 b-b+a^{2} b+b^{3} &amp; -2 a+a-a^{3}-a b^{2} &amp; 1-a^{2}-b^{2}\end{array}\right|$
<br/>$=\left|\begin{array}{ccc}\left(1+a^{2}+b^{2}\right) &amp; 0 &amp; -2 b \\ 0 &amp; \left(1+a^{2}+b^{2}\right) &amp; 2 a \\ b\left(1+a^{2}+b^{2}\right) &amp; -a\left(1+a^{2}+b^{2}\right) &amp; \left(1-a^{2}-b^{2}\right)\end{array}\right|$
<br/>$=\left(1+a^{2}+b^{2}\right)^{2}\left|\begin{array}{ccc}1 &amp; 0 &amp; -2 b \\ 0 &amp; 1 &amp; 2 a \\ b &amp; -a &amp; \left(1-a^{2}-b^{2}\right)\end{array}\right|$
<br/>$=\left(1+a^{2}+b^{2}\right)^{2}\left\{\left(1-a^{2}-b^{2}+2 a^{2}\right)+2 b^{2}\right\}$
<br/>$=\left(1+a^{2}+b^{2}\right)^{2}\left(1+a^{2}+b^{2}\right)=\left(1+a^{2}+b^{2}\right)^{3}$</div>


title: the-value-of-f-0-so-that-the-function-f-x-x-4-1-c-o-s-1-c-o-s-x-is-continuous-everywhere-is
slug: the-value-of-f-0-so-that-the-function-f-x-x-4-1-c-o-s-1-c-o-s-x-is-continuous-everywhere-is-60694
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $f(0)$ so that the function $f(x)=\frac{1-\cos (1-\cos \mathrm{x})}{\mathrm{x}^4}$ is continuous everywhere is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{6}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{8}$</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints: } \operatorname{Lim}_{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^4}<br/>$$<br/><br/>$$<br/>\operatorname{Lim}_{x \rightarrow 0} \frac{2 \sin ^2\left(\frac{2 \sin ^2\left(\frac{x}{2}\right)}{2}\right)}{x^4}=2 \operatorname{Lim}_{x \rightarrow 0} \frac{\sin ^2\left(\sin ^2\left(\frac{x}{2}\right)\right)\left(\sin ^2\left(\frac{x}{2}\right)\right)^2}{x^4\left(\sin ^2\left(\frac{x}{2}\right)\right)^2}=2 \operatorname{Lim}_{x \rightarrow 0} \frac{\sin ^4\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)^4 2^4}=\frac{1}{2^3}=\frac{1}{8}<br/>$$</div>


title: the-value-of-for-which-the-curve-7-x-5-2-7-y-3-2-2-4-x-3-y-24-2-representa-a-parabola-is
slug: the-value-of-for-which-the-curve-7-x-5-2-7-y-3-2-2-4-x-3-y-24-2-representa-a-parabola-is-10226
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lambda$ for which the curve $(7 x+5)^{2}+(7 y+3)^{2}=\lambda^{2}(4 x+3 y-24)^{2}$ representa a parabola is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pm \frac{6}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\pm \frac{7}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pm \frac{1}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pm \frac{2}{5}$</span> </li> </ul>

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

answer:
<div class="solution">Given curve is
<br/>$$
<br/>\begin{array}{l}
<br/>(7 x+5)^{2}+(7 y+3)^{2}=\lambda^{2}(4 x+3 y-24)^{2} \\
<br/>\Rightarrow 49 x^{2}+25+70 x+49 y^{2}+9+42 y \\
<br/>=\lambda^{2}\left(16 x^{2}+9 y^{2}+576+24 x y-144 y-192 x\right) \\
<br/>\Rightarrow \left(49-16 \lambda^{2}\right) x^{2}+\left(49-9 \lambda^{2}\right) y^{2}+\left(70+192 \lambda^{2}\right) x \\
<br/>+\left(42+144 \lambda^{2}\right) y-24 \lambda^{2} x y+\left(25-576 \lambda^{2}\right)=0
<br/>\end{array}
<br/>$$
<br/>On comparing with $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0,$ we get
<br/>$a=49-16 \lambda^{2}, b=\left(49-9 \lambda^{2}\right), h=-12 \lambda^{2}$
<br/>Condition for parabola is ab $=h^{2}$.
<br/>$$
<br/>\begin{array}{l}
<br/>\therefore \quad\left(49-16 \lambda^{2}\right)\left(49-9 \lambda^{2}\right)=\left(-12 \lambda^{2}\right)^{2} \\
<br/>\Rightarrow \quad 2401-441 \lambda^{2}-784 \lambda^{2}+144 \lambda^{4}=144 \lambda^{4} \\
<br/>\Rightarrow \quad 2401-1225 \lambda^{2}=0 \\
<br/>\Rightarrow \quad \lambda^{2}=\frac{2401}{1225}
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad \lambda^{2}=\frac{(49)^{2}}{(35)^{2}}$
<br/>$\Rightarrow \quad \lambda=\pm \frac{49}{35}$
<br/>$\Rightarrow \quad \lambda=\pm \frac{7}{5}$</div>


title: the-value-of-for-which-the-straight-line-3-x-2-y-1-1-z-3-may-lie-on-the-plane-x-2-y-0-is
slug: the-value-of-for-which-the-straight-line-3-x-2-y-1-1-z-3-may-lie-on-the-plane-x-2-y-0-is-76264
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lambda$ for which the straight line $\frac{x-\lambda}{3}=\frac{y-1}{2+\lambda}=\frac{z-3}{-1}$ may lie on the plane $x-2 y=0,$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">there is no such $\lambda$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there is no such $\lambda$</span> </div>

answer:
<div class="solution">If a line $\frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}}$ lies
<br/>on a plane $a_{2} x+b_{2} y+c_{2} z=d,$ then
<br/>(i) $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$
<br/>(ii) $a_{2} x_{1}+b_{2} y_{1}+c_{2} z_{1}=d$
<br/>According to the question. $3(1)+(2+\lambda)(-2)+(-1)(0)=0$
<br/>$\Rightarrow \quad 3-4-2 \lambda=0$
<br/>$\Rightarrow \quad \lambda=-1 / 2$
<br/>Also, $\quad \lambda-2=0$
<br/>$\Rightarrow \quad \lambda=2$
<br/>So. there is no such value of $\lambda$</div>


title: the-value-of-for-which-the-two-vectors-a-5-i-j-k-and-b-i-2-j-k-are-perpendicular-to-each-other-is
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
slug:
the-value-of-for-which-the-two-vectors-a-5-i-j-k-and-b-i-2-j-k-are-perpendicular-to-each-other-is-56654

question:
<div class="question">The value of ' $\lambda$ ' for which the two vectors $\vec{a}=5 \hat{i}+\lambda \hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$ are perpendicular to each other is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</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">-3</span> </li> </ul>

answer:
<div class="solution">Hints : For $\vec{a} \perp \vec{b}$<br/>$\begin{aligned}<br/>&amp; \vec{a} \cdot \vec{b}=0 \\<br/>&amp; \text { i.e., } 5-2 \lambda+1=0 \\<br/>&amp; \lambda=3<br/>\end{aligned}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-value-of-h-for-cooling-2-mole-of-an-ideal-monoatomic-gas-from-22-5-c-to-12-5-c-at-constant-pressure-will-be-given-c-p-2-5-r

slug:
the-value-of-h-for-cooling-2-mole-of-an-ideal-monoatomic-gas-from-22-5-c-to-12-5-c-at-constant-pressure-will-be-given-c-p-2-5-r-45132

question:
<div class="question">The value of $\Delta H$ for cooling 2 mole of an ideal monoatomic gas from $225^{\circ} \mathrm{C}$ to $125^{\circ} \mathrm{C}$ at constant pressure will be [given $\left.C_{p}=\frac{5}{2} R\right]$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-500 R$</span> </div>

answer:
<div class="solution">$\Delta H=-n C_{p} \Delta T=-2 \times \frac{5}{2} R(225-125)$
<br/>$$
<br/>=-5 R(100)=-500 R
<br/>$$</div>


title: the-value-of-i-0-4-tan-n-1-x-d-x-2-1-0-2-tan-n-1-2-x-d-x-is
slug: the-value-of-i-0-4-tan-n-1-x-d-x-2-1-0-2-tan-n-1-2-x-d-x-is-49023
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $I=\int_{0}^{\frac{\pi}{4}}\left(\tan ^{n+1} x\right) d x$
<br/>$+\frac{1}{2} \int_{0}^{\frac{\pi}{2}} \tan ^{n+1}\left(\frac{x}{2}\right) d x$ is</div>

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

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

answer:
<div class="solution">.Given. $I=\int_{0}^{\pi / 4}\left(\tan ^{n+1} x\right) d x+\frac{1}{2} \int_{0}^{\pi / 2} \tan ^{n-1}\left(\frac{x}{2}\right) d x$
<br/>In second integral, putt $=\frac{x}{2} \Rightarrow d x=2 d t$
<br/>$\Rightarrow \quad$ Also, when $x=0$ then $t=0$
<br/>When $x=\pi / 2,$ then $t=\pi / 4$
<br/>Then, $\begin{aligned} I &amp;=\int_{0}^{\pi / 4}\left(\tan ^{n+1} x\right) d x+\int_{0}^{\pi / 4} \tan ^{n-1} t d t \\ &amp; I=\int_{0}^{\pi / 4} \tan ^{n+1} x \cdot d x+\int_{0}^{\pi / 4} \tan ^{n-1} x d x \\ &amp; \quad\left\{\because \int_{a}^{b} f(x) d x=\int_{a}^{b} f(y) d y\right\} \end{aligned}$
<br/>$\Rightarrow \quad I=\int_{0}^{\pi / 4}\left(\tan ^{n+1} x+\tan ^{n-1} x\right) d x$
<br/>$\Rightarrow \quad I=\int_{0}^{\pi / 4} \tan ^{n-1} x \cdot\left(\tan ^{2} x+1\right) d x$
<br/>$\Rightarrow \quad I=\int_{0}^{\pi / 4} \tan ^{n-1} x\left(\sec ^{2} x\right) d x$
<br/>Put $\quad t=\tan x$
<br/>$\Rightarrow \quad d t=\sec ^{2} x d x$
<br/>Also, when $x=0,$ then $t=0$
<br/>when $x=\pi / 4,$ then $t=1$ $I=\int_{0}^{1} t^{n-1} d t=\left[\frac{t^{n}}{n}\right]_{0}^{1}=\frac{1}{n}$</div>


title: the-value-of-i-2-2-sin-x-d-x-is
slug: the-value-of-i-2-2-sin-x-d-x-is-57128
question: <div class="question">The value of $I=\int_{-\pi / 2}^{\pi / 2}|\sin x| d x$ is</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
answer: <div class="solution">Hints: $I=2 \int_0^{\frac{\pi}{2}} \sin x d x=2(1)=2$</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-2 &lt; $ I $ &lt; 2$</span> </li> </ul>


title: the-value-of-i-2-5-2-e-t-a-n-1-s-i-n-x-e-t-a-n-1-c-o-s-x-e-t-a-n-1-s-i-n-x-d-x-is
slug: the-value-of-i-2-5-2-e-t-a-n-1-s-i-n-x-e-t-a-n-1-c-o-s-x-e-t-a-n-1-s-i-n-x-d-x-is-75180
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of
<br/>$I=\int_{\pi / 2}^{5 \pi / 2} \frac{e^{\tan ^{-1}}(\sin x)}{e^{\tan ^{-1}(\sin x)}+e^{\tan ^{-1}(\cos x)}} d x,$ is</div>

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

answer:
<div class="solution">Let
<br/>$I=\int_{\pi / 2}^{5 \pi / 2} \frac{e^{\tan ^{-1}}(\sin x)}{e^{\tan ^{-1}(\sin x)}+e^{\tan ^{-1}(\cos x)}} d x$
<br/>$=\int_{0}^{5 \pi / 2} \frac{e^{\tan ^{-1}(\sin x)}}{e^{\tan ^{-1}(\sin x)}+e^{\tan ^{-1}(\cos x)}} d x$
<br/>$-\int_{0}^{\pi / 2} \frac{e^{\tan ^{-1}}(\sin x)}{e^{\tan ^{-1}(\sin x)}+e^{\tan ^{-1}(\cos x)}} d x \ldots(\mathrm{i}$
<br/>$I=\int_{0}^{5 \pi / 2} \frac{e^{\tan ^{-1}(\cos x)}}{e^{\tan ^{-1}(\cos x)}+e^{\tan ^{-1}(\sin x)}} d x$
<br/>$\quad-\int_{0}^{\pi / 2} \frac{e^{\tan ^{-1}}(\cos x)}{e^{\tan ^{-1}(\cos x)}+e^{\tan ^{-1}(\sin x)}} d x \ldots$ (ii)
<br/>$\quad\left[\right.$ using, $\left.\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right]$
<br/>On adding Eqs. (i) and (ii), we get
<br/>$$
<br/>\begin{array}{l}
<br/>2 I=\int_{0}^{5 \pi / 2} d x-\int_{0}^{\pi / 2} d x \\
<br/>\Rightarrow \quad 2 I=(x)_{0}^{5 \pi / 2}-(x)_{0}^{\mathrm{v} / 2} \\
<br/>\Rightarrow \quad 2 I=\frac{5 \pi}{2}-\frac{\pi}{2}=2 \pi \quad \Rightarrow \quad I=\pi
<br/>\end{array}
<br/>$$</div>


title: the-value-of-int0pi-sin-50-x-cos-49-x-d-x-is
slug: the-value-of-int0pi-sin-50-x-cos-49-x-d-x-is-54484
question: <div class="question">The value of \(\int_0^\pi \sin ^{50} x \cos ^{49} x d x\) is</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\pi / 4\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\pi / 2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Hints: \(I=\int_0^\pi \sin ^{50} x \cdot \cos ^{49} x d x \int_0^4 f(x)=\int_0^4 f(a-x)\)<br/>\(\begin{aligned}<br/>&amp; I=\int_0^\pi \sin ^{50} x\left(-\cos ^{49}(x)\right)=-\int_0^\pi \sin ^{50} x \cdot \cos ^{49} x \\<br/>&amp; =I=-I \\<br/>&amp; I=0<br/>\end{aligned}\)</div>


title: the-value-of-int01-2-fracd-xsqrt1x2-n-is-n-in-mathbbn
slug: the-value-of-int01-2-fracd-xsqrt1x2-n-is-n-in-mathbbn-89063
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of \(\int_0^{1 / 2} \frac{d x}{\sqrt{1-x^{2 n}}}\) is \((n \in \mathbb{N})\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">less than or equal to \(\frac{\pi}{6}\)</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">greater than or equal to 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">less than \(\frac{1}{2}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">greater than \(\frac{\pi}{6}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">less than or equal to \(\frac{\pi}{6}\)</span> </div>

answer:
<div class="solution">Hint : For 0 to \(\frac{1}{2}\)<br/>\(\begin{aligned} &amp; x^2 \geq x^{2 n} \\ &amp; \Rightarrow-x^2 \leq-x^{2 n} \\ &amp; \Rightarrow 1-x^2 \leq 1-x^{2 n} \\ &amp; \Rightarrow \frac{1}{\sqrt{1-x^2}} \geq \frac{1}{\sqrt{1-x^{2 n}}} \\ &amp; \Rightarrow \int_0^{1 / 2} \frac{1}{\sqrt{1-x^2}} d x \geq \int_0^{1 / 2} \frac{1}{\sqrt{1-x^{2 n}}} d x \\ &amp; \Rightarrow\left[\sin ^{-1} x\right]_0^{1 / 2} \geq \mathrm{I} \\ &amp; \Rightarrow \frac{\pi}{6} \geq I \\ &amp; \end{aligned}\)</div>


title: the-value-of-int22x-cos-xsin-x1-d-x-is
slug: the-value-of-int22x-cos-xsin-x1-d-x-is-31582
question: <div class="question">The value of \(\int_{-2}^2(x \cos x+\sin x+1) d x\) is</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
answer: <div class="solution">Hints: \(\int_{-2}^2(x \cos x+\sin x+1) d x=\int_{-2}^2 d x=4\)</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-2</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>


title: the-value-of-integral-1-1-x-2-x-2-d-x-is
slug: the-value-of-integral-1-1-x-2-x-2-d-x-is-89807
question: <div class="question">The value of integral $\int_{-1}^1 \frac{|x+2|}{x+2} d x$ is</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1</span> </li> </ul>

answer:
<div class="solution">Hints : $I=\int_{-1}^1 \frac{|x+2|}{x+2} d x \quad, \quad x+2=v \Rightarrow d x=d v$<br/>$$<br/>\therefore \mathrm{I}=\int_1^3 \frac{|v|}{V} d v=\int_1^3 \frac{v}{V} d v=\int_1^3 d v=2<br/>$$</div>


title: the-value-of-integral-6-3-x-x-s-in-x-s-in-x-x-cos-x-d-x-is
slug: the-value-of-integral-6-3-x-x-s-in-x-s-in-x-x-cos-x-d-x-is-94510
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{(sinx-xcosx)}{x(x+sinx)}dx$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\log _{e}\left\{\frac{2(\pi+3)}{(2 \pi+3 \sqrt{3})}\right\}$</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">$\log _{e}\left\{\frac{\pi+3}{2(2 \pi+3 \sqrt{3})}\right\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\log _{e}\left\{\frac{2 \pi+3 \sqrt{3}}{2(\pi+3)}\right\}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log _{e}\left\{\frac{2(2 \pi+3 \sqrt{3})}{\pi+3}\right\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{e}\left\{\frac{2(\pi+3)}{(2 \pi+3 \sqrt{3})}\right\}$</span> </div>

answer:
<div class="solution">Let $l=\int_{\pi / 6}^{\pi / 3} \frac{(\sin x-x \cos x)}{x(x+\sin x)} d x$
<br/>$\Rightarrow \quad I=\int_{\pi / 6}^{\pi / 3} \frac{(x+\sin x)-x(1+\cos x)}{x(x+\sin x)} d x$
<br/>$\Rightarrow \quad I=\int_{\pi / 6}^{\pi / 3}\left(\frac{1}{x}-\frac{1+\cos x}{x+\sin x}\right) d x$
<br/>$\Rightarrow \quad 1=[\log \times]_{\pi / 6}^{\pi / 3}-\int_{\pi / 6}^{\pi / 3} \frac{1+\cos x}{x+\sin x} \cdot d x$
<br/>$\left\{\begin{array}{l}\text { put } t=x+\sin x \\ \text { ot }=(1+\cos x) d x\end{array}\right.$ in lind term
<br/>$\Rightarrow \quad I=\left(\log \frac{\pi}{3}-\log \frac{\pi}{6}\right)-\int\left(\frac{\pi}{3}+\frac{\sqrt{3}}{2}\right) \frac{d t}{t}$
<br/>$\Rightarrow \quad I=\log 2-\left[\log t \mid\left(\frac{\pi}{3}+\frac{\sqrt{3}}{2}\right)\right.$
<br/>$\Rightarrow \quad 1=\log 2-\left[\log \left(\frac{\pi}{3}+\frac{\sqrt{3}}{2}\right)-\log \left(\frac{\pi}{6}+\frac{1}{2}\right)\right]$
<br/>$\Rightarrow \quad I=\log 2-\log \left(\frac{2 \pi+3 \sqrt{3}}{\pi+3}\right)$
<br/>$\left(\because \log m-\log n=\log \frac{m}{n}\right)$
<br/>$I=\log \left(\frac{2(\pi+3)}{2 \pi+3 \sqrt{3}}\right)$
<br/>$=\log \left(\frac{2 \pi+6}{2 \pi+3 \sqrt{3}}\right)$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-value-of-k-in-order-that-f-x-sin-x-cos-x-k-x-5-decreases-for-all-positive-real-values-of-x-is-given-by

slug:
the-value-of-k-in-order-that-f-x-sin-x-cos-x-k-x-5-decreases-for-all-positive-real-values-of-x-is-given-by-36882

question:
<div class="question">The value of $K$ in order that $f(x)=\sin x-\cos x-K x+5$  decreases for all positive real values of $x$ is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$K    &lt; 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$K \geq 1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$K&gt;\sqrt{2}$</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">$K    &lt; \sqrt{2}$</span> </li> </ul>

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

answer:
<div class="solution">We have, 
<br/>$\begin{aligned} f(x) &amp;=\sin x-\cos x-K x+5 \\ \Rightarrow f^{\prime}(x) &amp;=\cos x+\sin x-K \end{aligned}$
<br/>For decreasing, $f^{\prime}(x)    &lt; 0$ $\Rightarrow \cos x+\sin x-K &lt; 0$
<br/>$\begin{array}{ll}\Rightarrow &amp; K&gt;\cos x+\sin x \\ \Rightarrow &amp; \quad K&gt;\sqrt{2}\left[\frac{1}{\sqrt{2}} \cos x+\frac{1}{\sqrt{2}} \sin x\right] \\ \Rightarrow &amp; K&gt;\sqrt{2} \sin \left(\frac{\pi}{4}+x\right) \\ \therefore \quad &amp; K&gt;\sqrt{2}\end{array}$</div>


title: the-value-of-l-o-g-3-12-1-l-o-g-4-12-1-is
slug: the-value-of-l-o-g-3-12-1-l-o-g-4-12-1-is-93418
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
answer: <div class="solution">Hints : $\log _{12} 3+\log _{12} 4=\log _{12} 12=1$</div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\left(\frac{1}{\log _3 12}+\frac{1}{\log _4 12}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$</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">2</span> </li> </ul>


title: the-value-of-l-o-g-81-3-l-o-g-3-5-l-o-g-25-27-l-o-g-49-7-is
slug: the-value-of-l-o-g-81-3-l-o-g-3-5-l-o-g-25-27-l-o-g-49-7-is-10621
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\frac{\log _3 5 \times \log _{25} 27 \times \log _{49} 7}{\log _{81} 3}$ 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">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Hints: $\frac{\left(\frac{\log 5}{\log 3} \times \frac{3 \log 3}{2 \log 5} \times \frac{\log 7}{2 \log 7}\right)}{\left(\frac{\log 3}{4 \log 3}\right)}=3$</div>


title: the-value-of-lim-n-n-1-n-n-is
slug: the-value-of-lim-n-n-1-n-n-is-12078
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{n \rightarrow \infty} \frac{(n !)^{n}}{\frac{1}{n}}$ 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">$\frac{1}{e^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2 e} \quad$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{e}$</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> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow \infty} \frac{(n !)^{1 / n}}{n}=\lim _{n \rightarrow-}\left(\frac{n !}{n^{n}}\right)^{1 / n}$
<br/>We have, $\frac{n !}{n^{n}}=\frac{1 \cdot 2 \cdot 3}{n \cdot n \cdot n} \frac{n}{n}$
<br/>$\therefore \quad\left\{\frac{n !}{n^{n}}\right\}^{1 / n}=\left\{\frac{1}{n} \cdot \frac{2}{n} \cdot \frac{3}{n} \cdots \frac{r}{n} \cdots \frac{n}{n}\right\}^{1 / n}$
<br/>$\Rightarrow \lim _{n \rightarrow \infty}\left\{\frac{n !}{n^{n}}\right\}^{1 / n}=\lim _{n \rightarrow \infty}\left\{\frac{1}{n} \cdot \frac{2}{n} \cdot \frac{3}{n} \cdots \frac{r}{n} \cdots \frac{n}{n}\right\}^{1 / n}$
<br/>Let $\quad A=\lim _{n \rightarrow \infty}\left\{\frac{n !}{n^{n}}\right\}^{1 / n}$
<br/>Then, $A=\lim _{n \rightarrow-}\left\{\frac{1}{n} \cdot \frac{2}{n} \cdot \frac{3}{n} \cdots \frac{r}{n} \cdots \frac{n}{n}\right\}^{1 / n}$
<br/>$\begin{aligned} \Rightarrow \quad \log A &amp;=\lim _{n \rightarrow \infty} \frac{1}{n} \Sigma \log \left(\frac{r}{n}\right)=\int_{0}^{1} \log x d x \\ &amp;=\left[x \log x-\int \frac{1}{x} \cdot x d x\right]_{0}^{1} \end{aligned}$
<br/>Integrating by parts $=[x \log x-x]_{0}^{1}=-1$
<br/>$\Rightarrow \quad A=e^{-1}=\frac{1}{e}$</div>


title: the-value-of-lim-n-n-1-sec-2-4-n-sec-2-4-n-2-sec-2-4-n-n-i-s
slug: the-value-of-lim-n-n-1-sec-2-4-n-sec-2-4-n-2-sec-2-4-n-n-i-s-82916
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of
<br/>$\lim _{n \rightarrow \infty} \frac{1}{n}\left\{\sec ^{2} \frac{\pi}{4 n}+\sec ^{2} \frac{2 \pi}{4 n}+\ldots+\sec ^{2} \frac{n \pi}{4 n}\right] is$</div>

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

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

answer:
<div class="solution">We have, $\lim _{n \rightarrow \infty} \frac{1}{n}\left\{\sec ^{2} \frac{\pi}{4 n}+\sec ^{2} \frac{2 \pi}{4 n}+\ldots+\sec ^{2} \frac{n \pi}{4 n}\right\}$
<br/>$\quad=\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n} \sec ^{2}\left(\frac{r \pi}{4 n}\right)=\int_{0}^{1} \sec ^{2}\left(\frac{\pi x}{4}\right)$
<br/>$=\frac{4}{\pi}\left[\tan \left(\frac{\pi x}{4}\right)\right]_{0}^{1}$
<br/>$=\frac{4}{\pi} \times 1=\frac{4}{\pi}$</div>


title: the-value-of-lim-n-n-2-1-2-n-n-2-2-2-n-2-n-1-is
slug: the-value-of-lim-n-n-2-1-2-n-n-2-2-2-n-2-n-1-is-44548
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{n \rightarrow\infty}\left[\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{1}{2 n}\right]$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{n \pi}{4}$</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">$\frac{\pi}{4 n}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2 n}$</span> </li> </ul>

answer:
<div class="solution">$\lim _{n \rightarrow-}\left[\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{1}{2 n}\right]$
<br/>$=\lim _{n \rightarrow-}\left[\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+n^{2}}\right]$
<br/>$=\lim _{n \rightarrow-} \sum_{r=1}^{-\infty} \frac{n}{n^{2}+r^{2}}$
<br/>$=\lim _{n \rightarrow-\infty} \frac{1}{n} \sum_{r=1}^{-\infty} \frac{1}{1+\left(\frac{r}{n}\right)^{2}}$
<br/>$\quad=\int_{0}^{1} \frac{1}{1+x^{2}} d x$
<br/>$\quad=\left[\tan ^{-1} x\right]_{0}^{1}$
<br/>$\quad=\tan ^{-1} 1-\tan ^{-1} 0$
<br/>$\quad=\frac{\pi}{4}-0=\frac{\pi}{4}$</div>


title: the-value-of-lim-n-n-2-1-2-n-n-2-2-2-n-n-2-n-2-n-is
slug: the-value-of-lim-n-n-2-1-2-n-n-2-2-2-n-n-2-n-2-n-is-75987
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{n \rightarrow \infty}\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\ldots \ldots . . \frac{n}{n^2+n^2}\right]$ is</div>

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

answer:
<div class="solution">$$<br/>\text { Hints : } \lim _{n \rightarrow \infty}\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\ldots+\frac{n}{n^2+n^2}\right]<br/>$$<br/><br/>$$<br/>=\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{n}{n^2+r^2}=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^n \frac{1}{1+\left(\frac{r}{n}\right)^2}=\int_0^1 \frac{d x}{1+x^2}=\left[\tan ^{-1} x\right]_0^1=\frac{\pi}{4}<br/>$$</div>


title: the-value-of-lim-n-n-32-n-1-n-2-2-n-1-is
slug: the-value-of-lim-n-n-32-n-1-n-2-2-n-1-is-53738
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{n \rightarrow \infty}\left\{\frac{\sqrt{n+1}+\sqrt{n+2}+\ldots+\sqrt{2 n-1}}{n^{3 / 2}}\right\}$ is</div>

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

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

answer:
<div class="solution">$\because \lim _{n \rightarrow \infty \left[\frac{\sqrt{n+1}+\sqrt{n+2}+\ldots+\sqrt{2 n-1}}{n^{\frac{3}{2}}}\right]}$
<br/>$=\lim _{n \rightarrow \infty 1}\left[\sqrt{1+\frac{1}{n}}+\sqrt{1+\frac{2}{n}}+\ldots+\sqrt{1+\frac{n-1}{n}}\right] \frac{1}{n}$
<br/>$=\lim _{n \rightarrow \infty 1} \sum_{r=1}^{n-1} \frac{1}{n} \sqrt{1+\frac{r}{n}}$
<br/>$=\int_{0}^{1} \sqrt{1+x} d x=\left[\frac{2}{3}(1+x)^{3 / 2}\right]_{0}^{1}=\frac{2}{3}(2 \sqrt{2}-1)$</div>


title: the-value-of-lim-n-rightarrow-infty-sumr1n-fracr3r4n4-text-is
slug: the-value-of-lim-n-rightarrow-infty-sumr1n-fracr3r4n4-text-is-53003
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of<br/>\(\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r^3}{r^4+n^4} \text { is }\)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{4} \log _{\mathrm{e}} 2\)</span> </div>

answer:
<div class="solution">Hints: \(\operatorname{Lt}_{n \rightarrow \infty} \cdot \sum \frac{n^3\left(\frac{r}{n}\right)^3}{n^4\left[\left(\frac{r}{n}\right)^4+1\right]}\)<br/>\(\begin{aligned}<br/>&amp; =\frac{1}{4} \cdot \int_0^1 \frac{x^3}{1+x^4} d x=\frac{1}{4}\left[\log \left(1+x^4\right)\right]_0^1 \\<br/>&amp; =\frac{1}{4}(\log 2-\log 1)=\frac{1}{4} \log 2<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-value-of-lim-n-rightarrow-inftyleftleftfrac12-cdot-3frac122-cdot-3rightleftfrac122-cdot-32frac123-cdot-32rightldotsleftfrac12n-cdot-3nfrac12n1-cdo

slug:
the-value-of-lim-n-rightarrow-inftyleftleftfrac12-cdot-3frac122-cdot-3rightleftfrac122-cdot-32frac123-cdot-32rightldotsleftfrac12n-cdot-3nfrac12n1-cdo-79068

question:
<div class="question">The value of \(\lim _{n \rightarrow \infty}\left[\left(\frac{1}{2 \cdot 3}+\frac{1}{2^2 \cdot 3}\right)+\left(\frac{1}{2^2 \cdot 3^2}+\frac{1}{2^3 \cdot 3^2}\right)+\ldots+\left(\frac{1}{2^n \cdot 3^n}+\frac{1}{2^{n+1} \cdot 3^n}\right)\right]\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{3}{8}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{3}{10}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{3}{14}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{3}{16}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{3}{10}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>\text{Hint : } &amp; \lim _{n \rightarrow \infty} \sum_{x=1}^n\left(\frac{1}{2^x \times 3^x}+\frac{1}{2^{x+1} \times 3^x}\right) \\<br/>&amp; =\lim _{n \rightarrow \infty} \sum_{x=1}^n \frac{1}{6^x}\left(1+\frac{1}{2}\right) \\<br/>&amp; =\frac{3}{2} \times \underbrace{\lim _{n \rightarrow \infty}\left(\frac{1}{6}+\frac{1}{6^2}+\ldots \cdots \cdot \frac{1}{6^n}\right)}_{\text {infinite GP }}=\frac{3}{2} \times \frac{\frac{1}{6}}{1-\frac{1}{6}}=\frac{3}{10}<br/>\end{aligned}\)</div>


title: the-value-of-lim-x-0-p-x-x-q-is
slug: the-value-of-lim-x-0-p-x-x-q-is-86342
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{x \rightarrow 0^{+}} \frac{x}{p}\left[\frac{q}{x}\right]$ is</div>

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

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

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text {  Given, } \lim _{x \rightarrow 0^{+}} \frac{x}{p}\left[\frac{q}{x}\right]\\
<br/>&amp;\begin{array}{l}
<br/>=\lim _{x \rightarrow 0^{+}} \frac{x}{p}\left(\frac{q}{x}-\{\frac{q}{x}\right\}\right) \\
<br/>=\frac{[q]}{p}-\frac{x}{p}\left\{\frac{q}{x}\right\} \\
<br/>=\frac{[q]}{p}-0=\frac{[q]}{p}
<br/>\end{array}
<br/>\end{aligned}
<br/>$$</div>


title: the-value-of-lim-x-0-x-1-y-a-e-s-i-n-2-t-d-t-x-y-a-e-s-i-n-2-t-d-t-is-equal-to
slug: the-value-of-lim-x-0-x-1-y-a-e-s-i-n-2-t-d-t-x-y-a-e-s-i-n-2-t-d-t-is-equal-to-24330
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]$ is equal to</div>

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

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

answer:
<div class="solution">We have
<br/>$\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]$
<br/>$=\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t+\int_{a}^{x+y} e^{\sin ^{2} t} d t\right]$
<br/>$=\lim _{x \rightarrow 0} \frac{1}{x} \int_{y}^{x+y} e^{\sin ^{2} t} d t$
<br/>$=\lim _{x \rightarrow 0} \frac{\int_y^{x+y}e^{sin^{2t}}dt}{x}$
<br/>$=\lim _{x \rightarrow 0} \frac{e^{\sin ^{2}(x+y)}(1+0)-0}{1}=e^{\sin ^{2} y}$</div>


title: the-value-of-lim-x-0-x-s-i-n-x-0-x-2-c-o-s-t-2-d-t-is
slug: the-value-of-lim-x-0-x-s-i-n-x-0-x-2-c-o-s-t-2-d-t-is-84620
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \cos \left(t^{2}\right) d t}{x \sin x}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log _{e} 2$</span> </li> </ul>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \cos \left(t^{2}\right) d t}{x \sin x} \quad\left[\frac{0}{0}\text{ form}\right]$
<br/>$=\lim _{x \rightarrow 0} \frac{\cos \left(x^{4}\right) \times 2 x}{\sin x+x \cos x} \quad$ [L' Hospital's rule]
<br/>$=\lim _{x \rightarrow 0} \frac{2\left[\cos x^{4}-x \sin \left(x^{4}\right) \times 4 x^{3}\right]}{\cos x+\cos x-x \sin x}$
<br/>[L' Hospital's rule]
<br/>$=\frac{2[\cos 0-0]}{\cos 0+\cos 0-0}=\frac{2}{1+1}=1$</div>


title: the-value-of-lim-x-2-2-x-x-2-3-t-2-d-t-is
slug: the-value-of-lim-x-2-2-x-x-2-3-t-2-d-t-is-67533
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{x \rightarrow 2} \int_{2}^{x} \frac{3 t^{2}}{(x-2)} d t$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">$\lim _{x \rightarrow 2} \int_{2}^{x} \frac{3 t^{2}}{(x-2)}$ dt $=\frac{\lim _{x \rightarrow 2} \int_{2}^{x} 3 t^{2} d t}{\lim _{x \rightarrow 2}(x-2)}$
<br/>$=\frac{\lim _{x \rightarrow 2} 3 x^{2}}{1} \quad$ [using L' Hospital's nule]
<br/>$=3 \times(2)^{2}=12$</div>


title: the-value-of-lim-x-rightarrow-1-fracxx2ldotsxnnx1-is
slug: the-value-of-lim-x-rightarrow-1-fracxx2ldotsxnnx1-is-33567
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of \(\lim _{x \rightarrow 1} \frac{x+x^2+\ldots+x^n-n}{x-1}\) is</div>

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

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

answer:
<div class="solution">Hints: \(\operatorname{Lt}_{x \rightarrow 1} \frac{(x-1)+\left(x^2-1\right)+\left(x^3-1\right) \ldots \ldots \ldots \ldots .\left(x^4-1\right)}{x-1}\)<br/>\(=1+2+3 \ldots \ldots \ldots \ldots \ldots \ldots+n=\frac{\mathrm{n}(\mathrm{n}+1)}{2}\)</div>


