# questions

title: the-value-of-lt-x-0-1-3-x-2-1-5-x-2-x-2-1-is
slug: the-value-of-lt-x-0-1-3-x-2-1-5-x-2-x-2-1-is-84146
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\operatorname{Lt}_{x \rightarrow 0}\left(\frac{1+5 x^2}{1+3 x^2}\right)^{\frac{1}{x^2}}$ is</div>

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

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

answer:
<div class="solution">Hints: $\operatorname{Lim}_{x \rightarrow 0}\left(\frac{1+5 x^2}{1+3 x^2}\right)^{\frac{1}{x^2}}=\mathrm{e}^{\operatorname{Lim}_{x \rightarrow 0} \frac{1}{x^2}\left(\frac{1+5 x^2}{1+3 x^2}-1\right)}=\mathrm{e}^{\operatorname{Lim}_{x \rightarrow 0} \frac{2 x^2}{x^2\left(1+3 x^2\right)}}=\mathrm{e}^2$</div>


title: the-value-of-lt-x-0-x-2-s-i-n-2-x-c-o-s-x-1-is
slug: the-value-of-lt-x-0-x-2-s-i-n-2-x-c-o-s-x-1-is-98360
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\operatorname{Lt}_{x \rightarrow 0} \frac{\sin ^2 x+\cos x-1}{x^2}$ 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">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

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


title: the-value-of-n-1-10-2-n-1-2-sin-27-x-d-x-n-1-10-2-n-2-n-1-sin-27-x-d-x-is-equal-to
slug: the-value-of-n-1-10-2-n-1-2-sin-27-x-d-x-n-1-10-2-n-2-n-1-sin-27-x-d-x-is-equal-to-84576
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\sum_{\mathrm{n}=1}^{10} \int_{-2 n-1}^{-2 \pi} \sin ^{27} x d x+\sum_{n=1}^{10} \int_{2 n}^{2 n+1} \sin ^{27} x d x$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">27</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">54</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-54$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Hint:
<br/>$-\sum_{\mathrm{n}=1}^{10} \int_{2 n}^{2 n+1} \sin ^{27} x d x+\sum_{n=1}^{10} \int_{2 n}^{2 n+1} \sin ^{27} x d x$
<br/>$=0$</div>


title: the-value-of-n-1-13-i-n-i-n-1-i-1-is
slug: the-value-of-n-1-13-i-n-i-n-1-i-1-is-54079
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$i-1$</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right), i=\sqrt{-1}$ is</div>

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

answer:
<div class="solution">We have. 
<br/>$\begin{aligned} &amp; \sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right) \\=&amp; \sum_{n=1}^{13} i^{n}+\sum_{n=1}^{13} i^{n+1} \\=&amp; i \left(\frac{1-i^{13}}{1-i}\right)+i^{2}\left(\frac{1-i^{13}}{1-i}\right) \\=&amp; i \left(\frac{1-i}{1-i}\right)-\left(\frac{1-i}{1-i}\right) \\=&amp; i-1 \end{aligned}$</div>


title: the-value-of-n-for-which-x-n-y-n-x-n-1-y-n-1-is-the-geometric-mean-of-x-and-y-is
slug: the-value-of-n-for-which-x-n-y-n-x-n-1-y-n-1-is-the-geometric-mean-of-x-and-y-is-26465
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\mathrm{n}$ for which $\frac{\mathrm{x}^{\mathrm{n}+1}+\mathrm{y}^{\mathrm{n}+1}}{\mathrm{x}^{\mathrm{n}}+\mathrm{y}^{\mathrm{n}}}$ is the geometric mean of $\mathrm{x}$ and $\mathrm{y}$ is</div>

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

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

answer:
<div class="solution">Hints: $\frac{x^{n+1}+y^{n+1}}{x^n+y^n}=\sqrt{x y} \Rightarrow x^{n+1}+y^{n+1}=\sqrt{x y}\left(x^n+y^n\right)$<br/>$$<br/>\mathrm{x}^{\mathrm{n}+\frac{1}{2}}\left(\mathrm{x}^{\frac{1}{2}}-\mathrm{y}^{\frac{1}{2}}\right)=\mathrm{y}^{\mathrm{n}+\frac{1}{2}}\left(\mathrm{x}^{\frac{1}{2}}-\mathrm{y}^{\frac{1}{2}}\right),\left(\frac{\mathrm{x}}{\mathrm{y}}\right)^{\mathrm{n}+\frac{1}{2}}=1 \quad \mathrm{n}=-\frac{1}{2}<br/>$$</div>


title: the-value-of-r-2-r-1-2-r-1
slug: the-value-of-r-2-r-1-2-r-1-24544
question: <div class="question">The value of $\sum_{r=2}^{\infty} \frac{1+2+\quad+(r-1)}{r !}$</div>
tags: ['Mathematics', 'Binomial Theorem', '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">$e$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 e$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{e}{2}$</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">$\frac{3 e}{2}$</span> </li> </ul>

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

answer:
<div class="solution">$\sum_{r=2}^{\infty} \frac{(r-1) r}{2 r !}=\sum_{r=2}^{\infty} \frac{1}{2(r-2) !}$
<br/>$=\frac{1}{2}\left[\frac{1}{0 !}+\frac{1}{1 !}+\frac{1}{2 !}+\quad \infty\right]=\frac{1}{2} e$</div>


title: the-value-of-s-i-n-1-0-0-s-i-n-5-5-0-c-o-s-5-5-is
slug: the-value-of-s-i-n-1-0-0-s-i-n-5-5-0-c-o-s-5-5-is-10369
question: <div class="question">The value of $\frac{\sin 55^0-\cos 55^{\circ}}{\sin 10^0}$ is</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', '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">$\frac{1}{\sqrt{2}}$</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">1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\sqrt{2}$</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">$\sqrt{2}$</span> </div>

answer:
<div class="solution">Hints: $\frac{\sin 55-\sin 35}{\sin 10}=\frac{2 \cos 45 \cdot \sin 10}{\sin 10}=\sqrt{2}$</div>


title: the-value-of-such-that-the-system-of-equations-2-x-y-2-y-2-x-2-y-z-4-x-y-z-4-has-no-solution-is
tags: ['Mathematics', 'Determinants', '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
slug:
the-value-of-such-that-the-system-of-equations-2-x-y-2-y-2-x-2-y-z-4-x-y-z-4-has-no-solution-is-26512

question:
<div class="question">The value of $\lambda$ such that the system of equations $2x-y-2 y=2 ; x-2 y+z=-4$,$x+y+\lambda z=4,$ has no solution, 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">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</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">If the given system of equations has no solution. then $\left|\begin{array}{ccc}2 &amp; -1 &amp; -2 \\ 1 &amp; -2 &amp; 1 \\ 1 &amp; 1 &amp; \lambda\end{array}\right|=0$
<br/>$\Rightarrow 2(-2 \lambda-1)+1(\lambda-1)-2(1+2)=0$
<br/>$\Rightarrow \quad-4 \lambda-2+\lambda-1-6=0$
<br/>$\Rightarrow \quad-3 \lambda-9=0$
<br/>$\Rightarrow \quad \lambda=-3$</div>


title: the-value-of-t-a-n-3-6-c-o-t-5-4-c-o-t-7-0-t-a-n-2-0-is
slug: the-value-of-t-a-n-3-6-c-o-t-5-4-c-o-t-7-0-t-a-n-2-0-is-65940
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
question:
<div class="question">The value of $\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}$ 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">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Hints: $\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}=\frac{\tan 36^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\tan 20^{\circ}}=1+1=2$</div>


title: the-value-of-tan-5-2-tan-5-2-4-cot-5-4-is
slug: the-value-of-tan-5-2-tan-5-2-4-cot-5-4-is-31200
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$ is</div>

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

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

answer:
<div class="solution">$\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$
<br/>$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+4 \frac{\cos \frac{4 \pi}{5}}{\sin \frac{4 \pi}{5}}$
<br/>$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+\frac{4\left(\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}\right)}{2 \sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5}}$
<br/>$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+\frac{2\left(\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}\right)}{\sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5}}$
<br/>$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2\left[\frac{\sin ^{2} \frac{2 \pi}{5}+\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5} \sin \frac{2 \pi}{5}}\right]$
<br/>$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+\frac{2 \cos \frac{2 \pi}{5}}{\sin \frac{2 \pi}{5}}$
<br/>$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+\frac{2\left(\cos ^{2} \frac{\pi}{5}-\sin ^{2} \frac{\pi}{5}\right)}{2 \sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
<br/>$=\frac{\sin ^{2} \frac{\pi}{5}+\cos ^{2} \frac{\pi}{5}-\frac{\sin ^{2} \pi}{5}}{\sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
<br/>$=\frac{\cos ^{2} \frac{\pi}{5}}{\sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
<br/>$=\cot \frac{\pi}{5}$</div>


title: the-value-of-the-infinite-series-3-1-2-2-2-4-1-2-2-2-3-2-5-1-2-2-2-3-2-4-2
slug: the-value-of-the-infinite-series-3-1-2-2-2-4-1-2-2-2-3-2-5-1-2-2-2-3-2-4-2-86008
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the infinite series $\frac{1^{2}+2^{2}}{3 !}$ $+\frac{1^{2}+2^{2}+3^{2}}{4 !}+\frac{1^{2}+2^{2}+3^{2}+4^{2}}{5 !}+\ldots$</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">$5 e$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5 e}{6}-\frac{1}{2}$</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">$\frac{5 e}{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5 e}{6}-\frac{1}{2}$</span> </div>

answer:
<div class="solution">Given infinite series is $\frac{1^{2}+2^{2}}{3 !}+\frac{1^{2}+2^{2}+3^{2}}{4 !}$
<br/>$\begin{aligned} \frac{1^{2}+2^{2}}{3 !} &amp;+\frac{1^{2}+2^{2}+3^{2}}{4 !} \\ &amp;+\frac{1^{2}+2^{2}+3^{2}+4^{2}}{5 !}+\ldots \end{aligned}$
<br/>nth term, $t_{n}=\frac{1^{2}+2^{2}+3^{2}+4^{2}+\ldots+(r+1)^{2}}{(r+2) !}$
<br/>Now, $S_{n}=\Sigma t_{n}=\sum_{r=1}^{n} \frac{1^{2}+2^{2}+3^{2}+\ldots+(r+1)^{2}}{(r+2) !}$
<br/>$=\sum_{r=1}^{n} \frac{(r+1)(r+2)(2 n+3)}{6(r+2) !}$
<br/>$\left[\because \Sigma n^{2}=\frac{n(n+1)(2 n+1)}{6}\right]$
<br/>$=\sum_{r=1}^{n} \frac{(2 r+3)}{6 \cdot r !}=\frac{1}{6} \sum_{r=1}^{n}\left\{\frac{2}{(r-1) !}+\frac{3}{r !}\right\}$
<br/>$=\frac{1}{6}\left\{\left(\frac{2}{1}+\frac{3}{1 !}\right)+\left(\frac{2}{1 !}+\frac{3}{2 !}\right)+\left(\frac{2}{2 !}+\frac{3}{3 !}\right)+\ldots\right\}$
<br/>$=\frac{1}{6}\left\{\left(\frac{2}{1 !}+\frac{2}{1 !}+\frac{2}{2 !}+\ldots\right)+\left(\frac{3}{1 !}+\frac{3}{2 !}+\frac{3}{3 !}+\ldots\right)\right\}$
<br/>$=\frac{1}{6}\left\{2\left(1+\frac{1}{1 !}+\frac{1}{2 !}+\ldots\right)+3\left(\frac{1}{1 !}+\frac{1}{2 !}+\ldots\right)\right\}$
<br/>$=\frac{1}{6}\{2 e+3(e-1)\}$
<br/>$=\frac{1}{6}\{2 e+3 e-3\}$
<br/>$=\frac{1}{6}(5 e-3)=\frac{5 e}{6}-\frac{1}{2}$</div>


title: the-value-of-the-integral-0-1-e-x-2-d-x
slug: the-value-of-the-integral-0-1-e-x-2-d-x-40290
question: <div class="question">The value of the integral $\int_{0}^{1} e^{x^{2}} d x$</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">is less than 1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is greater than 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is less than or equal to 1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">lies in the closed interval $[1, e]$</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">lies in the closed interval $[1, e]$</span> </div>

answer:
<div class="solution">We know that, $0 \leq x \leq 1$ $\Rightarrow \quad x^{2} \leq 1$
<br/>$\begin{array}{ll}\Rightarrow &amp; e^{x^{2}} \leq e \\ \Rightarrow &amp; \int_{0}^{1} e^{x^{2}} d x \leq \int_{0}^{1} e d x \\ \Rightarrow &amp; \int_{0}^{1} e^{x^{2}} d x \leq e \\ \therefore \quad \int_{0}^{-} e^{x^{2}} \in[1, e]\end{array}$</div>


title: the-value-of-the-integral-0-2-1-t-a-n-x-101-1-d-x-is-equal-to
slug: the-value-of-the-integral-0-2-1-t-a-n-x-101-1-d-x-is-equal-to-35054
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{0}^{\pi / 2} \frac{1}{1+(\tan x)^{-101}} d x$ is equal to</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">$\frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{8}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Let $I=\int_{0}^{\pi / 2} \frac{1}{1+(\tan x)^{-101}} d x$
<br/>$=\int_{0}^{\pi / 2} \frac{d x}{1+\left\{\tan \left(\frac{\pi}{2}-x\right)\right\}^{101}}$
<br/>$=\int_{0}^{\pi / 2} \frac{d x}{1+(\cot x)^{101}}$
<br/>$=\int_{0}^{\pi / 2} \frac{\tan x^{10 t}}{\tan x^{101}+1} d x$
<br/>$=\int_{0}^{\pi / 2} \frac{1+\tan x^{101}-1}{1+\tan x^{101}}=[x]_{0}^{\pi / 2}-I$
<br/>$\Rightarrow \quad 1=\frac{\pi}{2}-I \Rightarrow 2 I=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$</div>


title: the-value-of-the-integral-0-2-sin-5-x-d-x-is
slug: the-value-of-the-integral-0-2-sin-5-x-d-x-is-48469
question: <div class="question">The value of the integral $\int_0^{\pi / 2} \sin ^5 x d x$ is</div>
tags: ['Mathematics', 'Definite Integration', '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">$\frac{4}{15}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{8}{5}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{8}{15}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{4}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{8}{15}$</span> </div>

answer:
<div class="solution">Hints : $I=\int_0^{\frac{\pi}{2}} \sin ^4 \mathrm{x} d \mathrm{dx} \quad \cos \mathrm{x}=\mathrm{f}, \sin \mathrm{d} \mathrm{x}=\mathrm{dt}$<br/>$=-\int_1^0\left(1-t^2\right) d t=\int_0^2\left(t^4-2 t^2+1\right) d t$<br/>$=\frac{1}{5}\left(\mathrm{t}^5\right)_0^1-\frac{2}{3}\left(\mathrm{t}^3\right)_0^1+(\mathrm{t})_0^1==\frac{1}{5}-\frac{2}{5}+1=\frac{3-10+15}{15}=\frac{8}{15}$</div>


title: the-value-of-the-integral-0-4-3-s-i-n-2-x-s-i-n-x-c-o-s-x-d-x-is-equal-to
slug: the-value-of-the-integral-0-4-3-s-i-n-2-x-s-i-n-x-c-o-s-x-d-x-is-equal-to-78120
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+\sin 2 x} d x$
<br/>is equal to</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">$\log _{e} 3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4} \log _{e} 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{4} \log _{e} 3$</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">$\frac{1}{4} \log _{e} 3$</span> </div>

answer:
<div class="solution">Let $I=\int_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+\sin 2 x} d x$
<br/>$=\int_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+2 \sin x \cos x} d x$
<br/>$=\int_{0}^{\pi / 4}-\frac{\sin x+\cos x}{(\sin x-\cos x)^{2}-4} d x$
<br/>Put $\sin x-\cos x=t$
<br/>$\Rightarrow \quad(\cos x+\sin x) d x=d t$
<br/>when $x=0 \Rightarrow t=-1$
<br/>and $x=\frac{\pi}{4} \Rightarrow t=0$
<br/>$\therefore$ Eq. (i) becomes, $I=-\int_{-1}^{0} \frac{d t}{t^{2}-4}=-\left.\frac{1}{4}\left[\log \mid \frac{t-2}{t+2}\right]\right|_{-1} ^{0}$
<br/>$=-\frac{1}{4}(\log 1-\log 3)$
<br/>$=-\frac{1}{4}(0-\log 3)=\frac{1}{4} \log 3$</div>


title: the-value-of-the-integral-1-1-e-x-x-2-c-o-s-x-x-2013-e-x-1-d-x-is-equal-to
slug: the-value-of-the-integral-1-1-e-x-x-2-c-o-s-x-x-2013-e-x-1-d-x-is-equal-to-68521
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{-1}^{1}\left\{\frac{x^{2013}}{e^{|x|}\left(x^{2}+\cos x\right)}+\frac{1}{e^{|x|}}\right\} d x$ is equal to</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">$1-e^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 e^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2\left(1-e^{-1}\right)$</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">$2\left(1-e^{-1}\right)$</span> </div>

answer:
<div class="solution">Let $\quad I=\int_{-1}^{1}\left\{\frac{x^{2013}}{e^{\mid x\left|(x^{2}+\cos x\right)}}+\frac{1}{e^{|x|}}\right\} d x$
<br/>$\Rightarrow \quad I=\int_{-1}^{1} \frac{x^{2013}}{e^{|x|+\left(x^{2}+\cos x\right)}} d x+\int_{-1}^{1} \frac{1}{e^{|x|}} d x$
<br/>Here, $\frac{x^{2013}}{e^{|x|}\left(x^{2}+\cos x\right)}$ is an odd function
<br/>and $\frac{1}{e^{\mid x|}}$ is an even function. $\left\{\because \int_{-a}^{a} f(x) d x=\left\{\begin{array}{cc}2 \int_{0}^{9}(x) d x ; &amp; f(x) \text { is even } \\ 0, &amp; f(x) \text { is odd }\end{array}\right\}\right.$
<br/>$\left.\therefore \quad 1=0+2 \int_{0}^{1} e^{-x} d x=-2 (e^{-x})\right\}_{0}^{1}=-2\left(e^{-1}\right)$
<br/>$=2\left(1-e^{-1}\right)$</div>


title: the-value-of-the-integral-1-1-e-x-x-2-c-o-s-x-x-2015-e-x-1-d-x-is-equal-to
slug: the-value-of-the-integral-1-1-e-x-x-2-c-o-s-x-x-2015-e-x-1-d-x-is-equal-to-42216
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{-1}^{1}\left\{\frac{x^{2015}}{e^{\mid x \mid}\left(x^{2}+\cos x\right)}+\frac{1}{e^{\mid{x} \mid}}\right\} d x$ is equal to</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">$1-e^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 e^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2\left(1-e^{-1}\right)$</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">$2\left(1-e^{-1}\right)$</span> </div>

answer:
<div class="solution">Let $I=\int_{-1}^{1}\left\{\frac{x^{2015}}{\left(e^{\mid x\mid} \left(x^{2}+\cos x\right)\right.}+\frac{1}{e^{|x|}}\right\} d x$
<br/>$\begin{aligned}=\int_{-1}^{1} \frac{x^{2015}}{e^{|x|}\left(x^{2}+\cos x\right)} &amp; d x+\int_{-1}^{1} \frac{1}{e^{|x|}} d x \\ \text { Let } f(x)=\frac{x^{2015}}{e^{|x|}\left(x^{2}+\cos x\right)} \text { and } g(x)=\frac{1}{e^{|x|}} \\ \text { Again, } f(-x)=\frac{(-x)^{2015}}{e^{|-x|}\left((-x)^{2}+\cos (-x)\right)} \\ &amp;=\frac{-x^{2015}}{e^{|x|}\left(x^{2}+\cos x\right)}=-f(x) \\ \text { and } \quad g(-x)=\frac{1}{e^{|-x|}}=\frac{1}{e^{|x|}}=g(x) \end{aligned}$
<br/>$\because f(x)$ is odd function and $g(x)$ is even function $\int_{-1}^{1} f(x) d x=0$ and $\int_{-1}^{1} g(x) d x=2 \int_{0}^{1} g(x) d x$
<br/>$\begin{aligned} I &amp;=2 \int_{0}^{1} \frac{1}{e^{\mid  x \mid}} d x \\ &amp;=2 \int_{0}^{1} \frac{1}{e^{x}} d x=2 \int_{0}^{1} e^{-x} d x \\ &amp;=-2\left[e^{-x}\right]_{0}^{1} \\ &amp;=-2\left[e^{-1}-e^{0}\right] \\ &amp;=-2\left(e^{-1}-1\right) \\ &amp;=2\left(1-e^{-1}\right) \end{aligned}$</div>


title: the-value-of-the-integral-1-2-e-x-lo-g-e-x-x-x-1-d-x-is
slug: the-value-of-the-integral-1-2-e-x-lo-g-e-x-x-x-1-d-x-is-30890
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{1}^{2} e^{x}\left(\log _{e} x+\frac{x+1}{x}\right) d x$ is</div>

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

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

answer:
<div class="solution">Let $I=\int_{1}^{2} e^{x}\left(\log _{e} x+\frac{x+1}{x}\right) d x$
<br/>$\Rightarrow \quad I=\int_{1}^{2}\left(e^{x} \cdot \log _{e} x+e^{x}+\frac{e^{x}}{x}\right) d x$
<br/>$\Rightarrow \quad I=\int_{1}^{2} e^{x} \log _{e} x d x+\int_{1}^{2} e^{x} d x+\int_{1}^{2} \frac{e^{x}}{x} d x$
<br/>$-\int_{1}^{2} e^{x} \log _{e} x d x$
<br/>$\Rightarrow \quad I=\left(e^{2}-e^{1}\right)+\left(e^{2} \log _{e} 2-0\right)$
<br/>$=e^{2}\left(1+\log _{e} 2\right)-e$</div>


title: the-value-of-the-integral-1-5-x-3-1-x-d-x-is-equal-to
slug: the-value-of-the-integral-1-5-x-3-1-x-d-x-is-equal-to-94571
question: <div class="question">The value of the integral $\int_{1}^{5}[|x-3|+|1-x|] d x$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">$\int_{1}^{5}[|x-3|+|1-x|] d x$
<br/>$=\int_{1}^{5}|x-3| d x+\int_{1}^{5}|1-x| d x$
<br/>$=\int_{1}^{3}|x-3| d x+\int_{3}^{5}|x-3| d x+\int_{1}^{5}|1-x| d x$
<br/>$=\int_{1}^{3}-(x-3) d x+\int_{3}^{5}(x-3) d x$
<br/>$+\int_{1}^{5}-(1-x) d x$
<br/>$=\int_{1}^{3}(3-x) d x+\int_{3}^{5}(x-3) d x+\int_{1}^{5}(x-1) d x$
<br/>$=\left[3 x-\frac{x^{2}}{2}\right]^{3}+\left[\frac{x^{2}}{2}-3 x\right]_{3}^{5}+\left[\frac{x^{2}}{2}-x\right]_{0}^{5}$
<br/>$=\left(3 \times 3-\frac{9}{2}\right)-\left(3 \times 1-\frac{1}{2}\right)+\left(\frac{5 \times 5}{2}-3 \times 5\right)$
<br/>$\begin{aligned}-\left(\frac{3 \times 3}{2}-3 \times 3\right)+\left(\frac{5 \times 5}{2}-5\right)-\left(\frac{1}{2}-1\right) \\=\left(9-\frac{9}{2}\right)-\left(3-\frac{1}{2}\right)+\left(\frac{25}{2}-15\right)-\left(\frac{9}{2}-9\right) \\ &amp;+\left(\frac{25}{2}-5\right)-\left(-\frac{1}{2}\right) \\=&amp; \frac{9}{2}-\frac{5}{2}-\frac{5}{2}+\frac{9}{2}+\frac{15}{2}+\frac{1}{2} \\=&amp; \frac{9-5-5+9+15+1}{2}=\frac{24}{2}=12 \end{aligned}$</div>


title: the-value-of-the-integral-2-2-1-2-sin-x-e-x-d-x-is-equal-to
slug: the-value-of-the-integral-2-2-1-2-sin-x-e-x-d-x-is-equal-to-31718
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{-2}^{2}(1+2 \sin x) e^{|x|} d x$ is
<br/>equal to</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">$e^{2}-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2\left(e^{2}-1\right)$</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">D</span> <span class="option-data">1</span> </li> </ul>

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

answer:
<div class="solution">$\int_{-2}^{2}(1+2 \sin x) e^{|x|} d x$
<br/>$=\int_{-2}^{2} e^{|x|} d x+2 \int_{-2}^{2} \sin x e^{|x|} d x$
<br/>(Since, first function is even and second function is odd). $=2 \int_{0}^{2} e^{|x|} d x+2 \text{ instead of ,} 0=2\left[e^{|x|}\right]_{0}^{2}=2\left(e^{2}-1\right)$</div>


title: the-value-of-the-integral-6-2-s-i-n-x-c-o-s-x-1-s-i-n-2-x-c-o-s-2-x-d-x-is-equal-to
slug: the-value-of-the-integral-6-2-s-i-n-x-c-o-s-x-1-s-i-n-2-x-c-o-s-2-x-d-x-is-equal-to-47241
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2012']
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 the integral $\int_{\pi / 6}^{\pi / 2}\left(\frac{1+\sin 2 x+\cos 2 x}{\sin x+\cos x}\right) d x$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">16</span> </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">4</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Let $I=\int_{\pi / 6}^{\pi / 2}\left(\frac{1+\sin 2 x+\cos 2 x}{\sin x+\cos x}\right) d x$
<br/>$=\int_{\pi / 6}^{\pi / 2}\left(\frac{1+2 \sin x \cos x+2 \cos ^{2} x-1}{(\sin x+\cos x)}\right) d x$
<br/>$=\int_{\pi / 6}^{\pi / 2} \frac{2 \cos x(\sin x+\cos x)}{(\sin x+\cos x)} d x$
<br/>$=\int_{\pi / 6}^{\pi / 2} 2 \cos x d x=2[\sin x]_{\pi / 6}^{\pi / 2}$
<br/>$=2\left(\sin \frac{\pi}{2}-\sin \frac{\pi}{6}\right)=2\left(1-\frac{1}{2}\right)=2 \times \frac{1}{2}=1$</div>


title: the-value-of-the-integral-12-12-x-1-x-1-2-x-1-x-1-2-2-12-d-x-is-equal-to
slug: the-value-of-the-integral-12-12-x-1-x-1-2-x-1-x-1-2-2-12-d-x-is-equal-to-33140
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int_{-1 / 2}^{1 / 2}\left\{\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right\}^{1 / 2} d x$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\log _{e}\left(\frac{4}{3}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \log _{e}\left(\frac{3}{4}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4 \log _{e}\left(\frac{4}{3}\right)$</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">D</span> <span class="option-data">$\log _{e}\left(\frac{3}{4}\right)$</span> </li> </ul>

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

answer:
<div class="solution">$\int_{-1 / 2}^{1 / 2} \sqrt{\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2} d x=\int_{-1 / 2}^{y / 2}\left|\frac{x+1}{x-1}-\frac{x-1}{x+1}\right| d x$
<br/>$=\int_{-1 / 2}^{0} \frac{-4 x}{-(x-1)(x+1)} d x+\int_{0}^{1 / 2} \frac{4 x}{-(x-1)(x+1)} d x \int_{-\frac{1}{2}}^{0} \frac{4 x}{x^{2}-1}-\int_{0}^{\frac{1}{2}} \frac{4 x}{x^{2}-1} d x=4 \ln 4 / 3$</div>


title: the-value-of-the-integral-e-x-e-x-2-d-x-is
slug: the-value-of-the-integral-e-x-e-x-2-d-x-is-70307
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $\int \frac{d x}{\left(e^x+e^{-x}\right)^2}$ is</div>

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

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

answer:
<div class="solution">Hints: $\int \frac{e^{2 x} d x}{\left(e^{2 x}+1\right)^2} \quad e^x=t ; e^x d x=d t$<br/>$$<br/>=\frac{1}{2} \int \frac{2 \mathrm{tdt}}{\left(\mathrm{t}^2+1\right)^2}=\frac{1}{2}\left\{-\frac{1}{\left(t^2+1\right)}\right\}+c=-\frac{1}{2\left(\mathrm{e}^{2 x}+1\right)}+c<br/>$$</div>


title: the-value-of-the-integral-i-12014-2014-x-t-a-n-1-x-d-x-is
slug: the-value-of-the-integral-i-12014-2014-x-t-a-n-1-x-d-x-is-26148
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integral $I=\int_{1 / 2014}^{2014} \frac{\tan ^{-1} x}{x} d x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{4} \log 2014$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2} \log 2014$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\pi \log 2014$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2} \log 2014$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{2} \log 2014$</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>I=\int_{1 / 2014}^{2014} \frac{\tan ^{-1} x}{x} d x
<br/>$$
<br/>Let
<br/>$$
<br/>x=\frac{1}{t}
<br/>$$
<br/>$\Rightarrow$
<br/>$$
<br/>d x=\frac{-1}{t^{2}} d t
<br/>$$
<br/>Now,
<br/>$$
<br/>\begin{aligned}
<br/>I &amp;=\int_{2014}^{1 / 2014} \frac{\tan ^{-1}(1 / 7)}{1 / t}\left(\frac{-1}{t^{2}} d t\right) \\
<br/>&amp;=\int_{1 / 2014}^{2014} \frac{\cot ^{-1} t}{t} d t \\
<br/>&amp;=\int_{1 / 2014}^{2014} \frac{\cot ^{-1} x}{x} d x
<br/>\end{aligned}
<br/>$$
<br/>On adding Eqs. (i) and (ii), we get
<br/>$$
<br/>\begin{aligned}
<br/>2 I &amp;=\int_{1 / 2014}^{2014} \frac{\pi / 2}{x} d x=\frac{\pi}{2}(\log x)_{1 / 2014}^{2014} \\
<br/>&amp;=\frac{\pi}{2}(\log 2014-\log 1 / 2014)
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow$
<br/>$$
<br/>\begin{aligned}
<br/>I &amp;=\frac{\pi}{4}\left(\log 2014-\log \frac{1}{2014}\right) \\
<br/>&amp;=\frac{\pi}{4}(\log 2014+\log 2014) \\
<br/>&amp;=\frac{\pi}{4}(2 \log 2014)=\frac{\pi}{2} \log 2014
<br/>\end{aligned}
<br/>$$</div>


title: the-value-of-the-integration-4-4-sin-x-1-c-o-s-x-s-i-n-x-d-x
slug: the-value-of-the-integration-4-4-sin-x-1-c-o-s-x-s-i-n-x-d-x-38816
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the integration $\int_{-\pi / 4}^{\pi / 4}\left(\lambda|\sin x|+\frac{\mu \sin x}{1+\cos x}+\gamma\right) d x$</div>

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

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

answer:
<div class="solution">Let $I=\int_{-\frac{\pi}{4}}^{\pi / 4}\left(\lambda|\sin x|+\frac{\mu \sin x}{1+\cos x}+\gamma\right)$
<br/>$\begin{aligned} &amp;=\int_{-\pi / 4}^{\pi / 4}(\lambda|\sin x|+\gamma) d x+\mu \int_{-\frac{\pi}{4}}^{\pi / 4} \frac{\sin x}{1+\cos x} d x \\ \text { Let } f(x)=\frac{\sin x}{1+\cos x} &amp; \\ \Rightarrow f(-x)=\frac{\sin (-x)}{1+\cos (-x)} &amp;=\frac{-\sin x}{1+\cos x}=-f(x) \end{aligned}$
<br/>$f(x)$ is an odd function.
<br/>$\int_{-\pi}^{\pi / 4}\left(\frac{\sin x}{1+\cos x}\right) d x=0$
<br/>$\left.I=\int_{-\frac{\pi}{6}}^{\pi / 4} \lambda|\sin x|+\gamma\right) d x$
<br/>$\therefore I$ is independent of $\mu$</div>


title: the-value-of-the-limit-lim-x-1-l-o-g-x-s-i-n-e-x-1-1-is
slug: the-value-of-the-limit-lim-x-1-l-o-g-x-s-i-n-e-x-1-1-is-82975
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2009']
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 the limit $\lim _{x \rightarrow 1} \frac{\sin \left(e^{x-1}-1\right)}{\log x}$ 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">$e$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{e}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">$$<br/>\text { Hints } \begin{aligned}<br/>&amp;: \operatorname{Lt}_{h \rightarrow 0} \frac{\sin \left(e^h-1\right)}{\log (1+h)} \quad \text { Put } x=1+h \\<br/>&amp;=\operatorname{Lt}_{h \rightarrow 0} \frac{\sin \left(e^h-1\right)}{\left(e^h-1\right)} \cdot \frac{\left(e^h-1\right)}{\log (1+h)} \\<br/>&amp;=\operatorname{Lt}_{h \rightarrow 0} \frac{\sin \left(e^h-1\right)}{\left(e^h-1\right)} \cdot \frac{\left(e^h-1\right)}{h} \cdot \frac{h}{\log (1+h)} \\<br/>&amp;=1.1 .1 \\<br/>&amp;=1<br/>\end{aligned}<br/>$$</div>


title: the-value-of-the-sum-n-c-1-2-n-c-2-2-n-c-3-2-n-c-n-2-is
slug: the-value-of-the-sum-n-c-1-2-n-c-2-2-n-c-3-2-n-c-n-2-is-42437
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of the sum $\left({ }^{n} C_{1}\right)^{2}+\left({ }^{n} C_{2}\right)^{2}+\left({ }^{n} C_{3}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}$ is</div>

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

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

answer:
<div class="solution">We know that
<br/>$$
<br/>\begin{array}{r}
<br/>(1+x)^{n}={ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2} \\
<br/>\quad+\ldots+{ }^{n} C_{n} x^{n} \\
<br/>\text { and }(x+1)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} \\
<br/>\quad+{ }^{n} C_{2} x^{n-2}+\ldots+{ }^{n} C_{n}
<br/>\end{array}
<br/>$$
<br/>On multiplying Eqs. (i) and (ii), we get
<br/>$$
<br/>\begin{array}{r}
<br/>(1+x)^{2 n}=\left({ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots+{ }^{n} C_{n} x^{n}\right) \\
<br/>\quad \times\left({ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1}+{ }^{n} C_{2} x^{n-2}+\ldots+{ }^{n} C_{n}\right)
<br/>\end{array}
<br/>$$
<br/>Coefficient of $x^{n}$ in RHS
<br/>$$
<br/>=\left({ }^{n} C_{0}\right)^{2}+\left({ }^{n} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}
<br/>$$
<br/>and coefficient of $x^{n}$ in $\mathrm{LHS}={ }^{2 n} \mathrm{C}_{n}$
<br/>$$
<br/>\begin{array}{c}
<br/>\therefore\left({ }^{n} C_{0}\right)^{2}+\left({ }^{n} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}=\frac{2 n !}{n ! n !} \\
<br/>\Rightarrow\left({ }^{n} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}=\frac{(2 n) !}{n ! n !}-1 \\
<br/>\quad={ }^{2 n} C_{n}-1
<br/>\end{array}
<br/>$$</div>


title: the-value-of-x-2-2-x-3-7-13-x-2-d-x-is
slug: the-value-of-x-2-2-x-3-7-13-x-2-d-x-is-22848
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The value of $\int \frac{(x-2)}{\left\{(x-2)^{2}(x+3)^{7}\right\rangle^{1 / 3}} d x$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3}{20}\left(\frac{x-2}{x+3}\right)^{4 / 3}+C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{20}\left(\frac{x-2}{x+3}\right)^{3 / 4}+C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{5}{12}\left(\frac{x-2}{x+3}\right)^{4 / 3}+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{20}\left(\frac{x-2}{x+3}\right)^{5 / 3}+C$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{20}\left(\frac{x-2}{x+3}\right)^{4 / 3}+C$</span> </div>

answer:
<div class="solution">Let $I=\int \frac{(x-2)}{\left\{(x-2)^{2}(x+3)^{7}\right\}^{1 / 3}} d x$
<br/>$$
<br/>\begin{array}{l}
<br/>=\int \frac{d x}{(x-2)^{-1 / 3}(x+3)^{7 / 3}} \\
<br/>=\int \frac{d x}{(x-2)^{2}\left(\frac{x+3}{x-2}\right)^{7 / 3}} \\
<br/>\text { Put } \frac{x+3}{x-2}=t \Rightarrow \frac{-5 d x}{(x-2)^{2}}=d t
<br/>\end{array}
<br/>$$
<br/>$\therefore \quad I=-\frac{1}{5} \int \frac{d t}{t^{7 / 3}}=-\frac{1}{5} \int t^{-7 / 3} d t$
<br/>$=-\frac{1}{5}\left[\frac{t^{-7 / 3+1}}{-\frac{7}{3}+1}\right]+C=-\frac{1}{5}\left[\frac{t^{-4 / 3}}{-\frac{4}{3}}\right]+C$
<br/>$=\frac{3}{20 t^{4 / 3}}+C=\frac{3}{20}\left(\frac{x-2}{x+3}\right)^{4 / 3}+C$</div>


title: the-value-of-z-2-z-3-2-z-i-2-is-minimum-when-z-equals
slug: the-value-of-z-2-z-3-2-z-i-2-is-minimum-when-z-equals-57213
question: <div class="question">The value of $|z|^{2}+|z-3|^{2}+|z-i|^{2}$ is minimum when $z$ equals</div>
tags: ['Mathematics', 'Complex Number', '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">$2-\frac{2}{3} i$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45+3 i$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1+\frac{i}{3}$</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">$1-\frac{i}{3}$</span> </li> </ul>

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

answer:
<div class="solution">Let $z=x+i y$
<br/>$\therefore|z|^{2}+|z-3|^{2}+|z-i|^{2}$
<br/>$$
<br/>\begin{aligned}
<br/>=|x+i y|^{2}+\mid &amp;\mid x-3)+\left.i y\right|^{2} \\
<br/>&amp;+|x+i(y-1)|^{2}
<br/>\end{aligned}
<br/>$$
<br/>$=x^{2}+y^{2}+(x-3)^{2}+y^{2}+x^{2}+(y-1)^{2}$
<br/>$=x^{2}+y^{2}+x^{2}-6 x+9+y^{2}+x^{2}+y^{2}$
<br/>$=3 x^{2}+3 y^{2}-6 x-2 y+10$
<br/>$=3\left(x^{2}-2 x+1\right)+3\left(y^{2}-\frac{2}{3} y+\frac{1}{9}\right)$
<br/>$\quad=3(x-1)^{2}+3\left(y-\frac{1}{3}\right)^{2}+\frac{20}{3}$
<br/>It is minimum, when $x-1=0$ and $y-\frac{1}{3}=0$
<br/>$\therefore \quad x=1 \quad$ and $\quad y=\frac{1}{3}$
<br/>$\therefore \quad z=1+\frac{1}{3} i$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
the-values-of-a-b-c-for-which-the-function-f-x-x-s-i-n-a-1-x-s-i-n-x-x-0-c-x-0-b-x-12-x-b-x-2-12-x-x-0-is-continuous-at-x-0-are

slug:
the-values-of-a-b-c-for-which-the-function-f-x-x-s-i-n-a-1-x-s-i-n-x-x-0-c-x-0-b-x-12-x-b-x-2-12-x-x-0-is-continuous-at-x-0-are-18700

question:
<div class="question">The values of $a, b, c$ for which the function $f(x)=\left\{\begin{array}{l}\frac{\sin (a+1) x+\sin x}{x}, x &lt; 0 \\ c, x=0 \\ \frac{\left(x+b x^2\right)^{1 / 2}-\sqrt{x}}{b x^{1 / 2}}, x&gt;0\end{array}\right.$ is continuous at $x=0$, are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a=-2, b \in \mathbb{R}-\{0\}, c=0$</span> </div>

answer:
<div class="solution">$\quad f\left(0^{+}\right)=\underset{x \rightarrow 0^{+}}{L t} \frac{(1+b x)^{1 / 2}-1}{b}=0$<br/>$f(0)=c$<br/>$f\left(0^{-}\right)=a+2$<br/>$\Rightarrow c=0$ and $a=-2, b \in R-\{0\}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-values-of-h-and-s-of-a-certain-reaction-are-400-kj-mol-1-and-20-kj-mol-1-k-1-respectively-the-temperature-below-which-the-reaction-is-spontaneous-

slug:
the-values-of-h-and-s-of-a-certain-reaction-are-400-kj-mol-1-and-20-kj-mol-1-k-1-respectively-the-temperature-below-which-the-reaction-is-spontaneous-68888

question:
<div class="question">The values of $\Delta H$ and $\Delta S$ of a certain reaction are $-400 \mathrm{k} \mathrm{J} \mathrm{mol}^{-1}$ and $-20 \mathrm{kJ} \mathrm{mol}^{-1} \mathrm{K}^{-1}$ respectively. The temperature below which the reaction is spontaneous, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \mathrm{K}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$20^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$20 \mathrm{K}$</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">D</span> <span class="option-data">$120^{\circ} \mathrm{C}$</span> </li> </ul>

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

answer:
<div class="solution">Given, $\quad \Delta H=-400 \mathrm{kJ} \mathrm{mol}^{-1}$
<br/>$$
<br/>\Delta S=-20 \mathrm{kJ} \mathrm{mol}^{-1} \mathrm{K}^{-1}
<br/>$$
<br/>Gibbs Helmholtz equation is
<br/>$$
<br/>\Delta G=\Delta H-T \Delta S
<br/>$$
<br/>For a reaction to be spontaneous, $\Delta G$ must be $\therefore \quad 0 \geq \Delta H-T \Delta S$
<br/>$$
<br/>\begin{array}{r}
<br/>T \Delta S \leq \Delta H \\
<br/>T \leq \frac{\Delta H}{\Delta S} \leq \frac{-400}{-20} \leq 20 \mathrm{K}
<br/>\end{array}
<br/>$$</div>


title: the-variance-of-first-20-natural-numbers-is
slug: the-variance-of-first-20-natural-numbers-is-30989
question: <div class="question">The variance of first 20 natural numbers is</div>
tags: ['Mathematics', 'Statistics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$133 / 4$</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">$279 / 12$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">133/2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">399/4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$133 / 4$</span> </div>

answer:
<div class="solution">Variance of $n$ natural numbers $=\frac{n^{2}-1}{12}=\frac{(20)^{2}-1}{12} \quad$ $[\because n=20]$,
<br/>$=\frac{400-1}{12}=\frac{399}{12}=\frac{133}{4}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-variation-of-electric-field-along-the-z-axis-due-to-a-uniformly-charged-circular-ring-of-radius-a-in-x-y-plane-is-shown-in-the-figure-the-value-of

slug:
the-variation-of-electric-field-along-the-z-axis-due-to-a-uniformly-charged-circular-ring-of-radius-a-in-x-y-plane-is-shown-in-the-figure-the-value-of-11976

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/bOjNsQ3L13lWoAgKZZz_kMUh-bBYHmNugc3WqC9HSOE.original.fullsize.png"/><br/> <br/>The variation of electric field along the $Z$-axis due to a uniformly charged circular ring of radius ' $a$ ' in $X Y$ plane is shown in the figure. The value of coordinate $M$ will be</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">$\sqrt{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">D</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> </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">For $E_{z}$ to be maximum,
<br/>$$
<br/>\frac{d E_{z}}{d z}=0 \quad \therefore z=a / \sqrt{2} \quad \text { or } \frac{z}{a}=\frac{1}{\sqrt{2}}
<br/>$$</div>


title: the-vectors-a-and-b-are-such-that-a-b-a-b-the-angle-between-the-two-vectors-will-be
slug: the-vectors-a-and-b-are-such-that-a-b-a-b-the-angle-between-the-two-vectors-will-be-23694
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The vectors $\mathrm{A}$ and $\mathrm{B}$ are such that $|\mathbf{A}+\mathbf{B}|=|\mathbf{A}-\mathbf{B}| .$ The angle between the two vectors will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$60^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$90^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$45^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$90^{\circ}$</span> </div>

answer:
<div class="solution">Given, $|A+B|=|A-B|$<br/>$\therefore \quad(A+B)^{2}=(A-B)^{2}$<br/>$A^{2}+B^{2}+2 A B \cos \theta=A^{2}+B^{2}-2 A B \cos \theta$<br/>$2 A B \cos \theta+2$ AB $\cos \theta=0$<br/>$4 A B \cos \theta=0$<br/>Here, $\quad A \neq 0, B \neq 0$<br/>$\therefore \quad \cos \theta=0$<br/>$\theta=90^{\circ}$</div>


tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-vectors-are-given-by-a-i-2-j-2-k-and-b-3-i-6-j-2-k-another-vector-c-has-the-same-magnitude-as-b-but-has-the-same-direction-as-a-then-which-of-the-

slug:
the-vectors-are-given-by-a-i-2-j-2-k-and-b-3-i-6-j-2-k-another-vector-c-has-the-same-magnitude-as-b-but-has-the-same-direction-as-a-then-which-of-the-87338

question:
<div class="question">The vectors are given by $\dot{\mathbf{A}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{B}=3 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+2 \mathbf{k}$. Another vector $\mathbf{C}$ has the same magnitude as $\mathbf{B}$ but has the same direction as A. Then which of the following vectors represents C?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{7}{3}(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$</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">B</span> <span class="option-data">$\frac{3}{7}(i-2 \hat{j}+2 \hat{k})$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{9}(i-2 \hat{j}+2 \hat{k})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9}{7}(\vec{i}+2 \hat{j}+2 \hat{k})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{3}(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$</span> </div>

answer:
<div class="solution">(a) Given, $A=i+2 j+2 \hat{k}$ and $B=3 \hat{i}+6 j+2 \hat{k}$
<br/>So, $\quad C=\frac{\hat{i}+2 \hat{j}+2 \hat{k}}{\sqrt{1+4+4}} \times \sqrt{3^{2}+6^{2}+2^{2}}$
<br/>$=\frac{\hat{i}+2 \hat{j}+2 \hat{k}}{3} \times \sqrt{49}=\frac{7}{3}(\hat{i}+2 \hat{j}+2 \hat{\mathbf{k}})$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">95</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-velocity-of-a-car-travelling-on-a-straight-road-is-36-kmh-1-at-an-instant-of-time-now-travelling-with-uniform-acceleration-for-10-s-the-velocity-b

slug:
the-velocity-of-a-car-travelling-on-a-straight-road-is-36-kmh-1-at-an-instant-of-time-now-travelling-with-uniform-acceleration-for-10-s-the-velocity-b-65912

question:
<div class="question">The velocity of a car travelling on a straight road is $3.6 \mathrm{kmh}^{-1}$ at an instant of time. Now travelling with uniform acceleration for $10 \mathrm{s}$, the velocity becomes exactly double. If the wheel radius of the car is $25 \mathrm{cm}$, then which of the following is the closest to the number of revolutions that the wheel makes during this 10 s?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">84</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">95</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">126</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">135</span> </li> </ul>

answer:
<div class="solution">We have, $\theta=2 \pi n=\frac{\left(\frac{v_{f}^{2}}{r^{2}}-\frac{v_{1}^{2}}{r^{2}}\right)}{\left(2 \frac{a}{r}\right)}$
<br/>$$
<br/>n=\frac{v_{f}^{2}-v_{i}^{2}}{(2 a r)(2 \pi)}=95
<br/>$$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-velocity-of-a-particle-executing-a-simple-harmonic-motion-is-13-ms-1-when-its-distance-from-the-equilibrium-position-q-is-3-m-and-its-velocity-is-

slug:
the-velocity-of-a-particle-executing-a-simple-harmonic-motion-is-13-ms-1-when-its-distance-from-the-equilibrium-position-q-is-3-m-and-its-velocity-is-96591

question:
<div class="question">The velocity of a particle executing a simple harmonic motion is $13 \mathrm{ms}^{-1}$, when its distance from the equilibrium position $(Q)$ is $3 \mathrm{m}$ and its velocity is $12 \mathrm{ms}^{-1},$ when it is $5 \mathrm{m}$ away from $Q .$ The frequency of the simple harmonic motion is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{8 \pi}$</span> </div>

answer:
<div class="solution">The speed of a particle executing simple
<br/>harmonic motion is $v=\omega \sqrt{a^{2}-x^{2}}$
<br/>where, $a=$ Amplitude $\omega=$ Angular frequency $x=$ Displacement
<br/>or
<br/>$$
<br/>v^{2}=\omega^{2}\left(a^{2}-x^{2}\right)
<br/>$$
<br/>According to the question,
<br/>Here
<br/>$$
<br/>\begin{aligned}
<br/>v_{1}^{2} &amp;=\omega^{2}\left(a^{2}-x_{1}^{2}\right) \\
<br/>v_{2}^{2} &amp;=\omega^{2}\left(a^{2}-x_{2}^{2}\right) \\
<br/>v_{1}^{2}-v_{2}^{2} &amp;=\omega^{2}\left(x_{2}^{2}-x_{1}^{2}\right) \\
<br/>\omega &amp;=\sqrt{\frac{v_{1}^{2}-v_{2}^{2}}{x_{2}^{2}-x_{1}^{2}}} \\
<br/>v_{1}=&amp; 13 \mathrm{m} / \mathrm{s}
<br/>\end{aligned}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>v_{2} &amp;=12 \mathrm{m} / \mathrm{s} \\
<br/>x_{1} &amp;=3 \mathrm{m} \\
<br/>x_{2} &amp;=5 \mathrm{m} \\
<br/>\omega &amp;=\sqrt{\frac{(13)^{2}-(12)^{2}}{(5)^{2}-(3)^{2}}} \\
<br/>&amp;=\sqrt{\frac{169-144}{25-9}}=\sqrt{\frac{25}{16}}=\frac{5}{4} \\
<br/>\omega &amp;=2 \pi r \\
<br/>v &amp;=\frac{\omega}{2 \pi}=\frac{5 / 4}{2 \pi} \\
<br/>v &amp;=\frac{5}{8 \pi}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-velocity-of-sound-in-air-at-2-0-c-and-1-atm-pressure-is-3442-m-s-at-4-0-c-and-2-atm-pressure-the-velocity-of-sound-in-air-is-approximately

slug:
the-velocity-of-sound-in-air-at-2-0-c-and-1-atm-pressure-is-3442-m-s-at-4-0-c-and-2-atm-pressure-the-velocity-of-sound-in-air-is-approximately-25753

question:
<div class="question">The velocity of sound in air at $20^{\circ} \mathrm{C}$ and 1 atm pressure is $344.2 \mathrm{m} / \mathrm{s}$. At $40^{\circ} \mathrm{C}$ and $2 \mathrm{atm}$
<br/>pressure, the velocity of sound in air is approximately</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$350 \mathrm{m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$356 \mathrm{m} / \mathrm{s}$</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">$363 \mathrm{m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$370 \mathrm{m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$356 \mathrm{m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">$\quad T_{1}=273+20=293 \mathrm{K}$
<br/>$$
<br/>T_{2}=273+40=313 \mathrm{K}
<br/>$$
<br/>The velocity of sound wave in gases or air is given by
<br/>$$
<br/>v=\sqrt{\frac{\gamma R T_{1}}{M}}
<br/>$$
<br/>where, $\gamma=$ Ratio of $C_{p} / C_{V}$
<br/>$$
<br/>\begin{array}{l}R=\text { gas constant } \\ \qquad \begin{array}{l}v_{1}=\sqrt{\frac{\gamma R T_{1}}{M}} \\ v_{2}=\sqrt{\frac{\gamma R T_{2}}{M}}\end{array} \\ \frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}=\sqrt{\frac{293}{313}}\end{array}
<br/>$$
<br/>$\begin{aligned} \text { Given, } v_{1}=344.2 \mathrm{m} / \mathrm{s} &amp; \\ \begin{aligned} \therefore \quad v_{2} &amp;=v_{1} \sqrt{\frac{313}{293}} \\ &amp;=344.2 \times \sqrt{1068} \\ &amp;=344.2 \times 103 \mathrm{m} / \mathrm{s} \\ &amp;=35575 \mathrm{m} / \mathrm{s} \\ &amp;=356 \mathrm{m} / \mathrm{s} \end{aligned} \end{aligned}$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
the-velocity-v-of-a-particle-under-a-force-f-depends-on-its-distance-x-from-the-origin-with-x-0-v-x-1-find-how-the-magnitude-of-the-force-f-on-the-par

slug:
the-velocity-v-of-a-particle-under-a-force-f-depends-on-its-distance-x-from-the-origin-with-x-0-v-x-1-find-how-the-magnitude-of-the-force-f-on-the-par-47968

question:
<div class="question">The velocity $(v)$ of a particle (under a force $F$ ) depends on its distance $(x)$ from the origin (with $x&gt;0) v \propto\frac{1}{\sqrt{x}}$. Find how the magnitude
<br/>of the force $(F)$ on the particle depends on $x ?$</div>

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

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

answer:
<div class="solution">According to the question.
<br/>$$
<br/>v \propto \frac{1}{\sqrt{x}}
<br/>$$
<br/>or $v=\frac{k}{\sqrt{x}}$
<br/>$$
<br/>\frac{d v}{d t}=\frac{d}{d t} \cdot \frac{K}{\sqrt{x}}=k \cdot \frac{-1}{2} \cdot x^{-\frac{1}{2}-1} \cdot \frac{d x}{d t}
<br/>$$
<br/>$a=-\frac{k}{2} \cdot x^{-\frac{3}{2}} \cdot \frac{k}{\sqrt{x}}\left[\because \frac{d x}{d t}=v\right.$ and $v=\frac{k}{\sqrt{x}}$ from $\left.\mathrm{Eq}(i)\right]$
<br/>$$
<br/>=\frac{-k^{2}}{2} \cdot \frac{1}{x^{2}}
<br/>$$
<br/>$=m \frac{k^{2}}{2} \cdot \frac{1}{x^{2}}$
<br/>(magnitue
<br/>$F=\frac{k^{2}}{2} \cdot \frac{m}{x^{2}}$
<br/>$\therefore$
<br/>$F x \frac{1}{x^{2}}$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints:</div>
source_db: db\MarksBatch2_P3.db
title:
the-vertex-of-the-parabola-y26-x2-y130-is-beginaligned-y126-x12-y126x24leftfrac64rightx2-text-vertex-rightarrow21-endaligned

slug:
the-vertex-of-the-parabola-y26-x2-y130-is-beginaligned-y126-x12-y126x24leftfrac64rightx2-text-vertex-rightarrow21-endaligned-58927

question:
<div class="question">The vertex of the parabola \(y^2+6 x-2 y+13=0\) is<br/>\(\begin{aligned}<br/>&amp; (y-1)^2=-6 x-12 \\<br/>&amp; (y-1)^2=-6(x+2)=4\left(\frac{-6}{4}\right)(x+2) \\<br/>&amp; \text { Vertex } \rightarrow(-2,1)<br/>\end{aligned}\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((1,-1)\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\((-2,1)\)</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">C</span> <span class="option-data">\(\left(\frac{3}{2}, 1\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\left(-\frac{7}{2}, 1\right)\)</span> </li> </ul>

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


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-volume-of-ethyl-alcohol-density-115-g-cc-that-has-to-be-added-to-prepare-100-cc-of-05-m-ethyl-alcohol-solution-in-water-is

slug:
the-volume-of-ethyl-alcohol-density-115-g-cc-that-has-to-be-added-to-prepare-100-cc-of-05-m-ethyl-alcohol-solution-in-water-is-69448

question:
<div class="question">The volume of ethyl alcohol (density $1.15 \mathrm{g} / \mathrm{cc}$ ) that has to be added to prepare $100 \mathrm{cc}$ of $0.5 \mathrm{M}$ ethyl alcohol solution in water is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.15 \mathrm{cc}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{cc}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$2.15 \mathrm{cc}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.30 \mathrm{cc}$</span> </li> </ul>

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

answer:
<div class="solution">Molarity $=\frac{\text { mass } \times 1000}{\text { molecular weight } \times \text { volume of solution }}$
<br/>Molecular weight of ethyl alcohol,
<br/>$$
<br/>\begin{aligned}
<br/>\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} &amp;=12 \times 2+5+16+1 \\
<br/>&amp;=24+22=46 \mathrm{g} \mathrm{mol}^{-1}
<br/>\end{aligned}
<br/>$$
<br/>On substituting the values, we get
<br/>$$
<br/>\begin{array}{l}
<br/>0.5 \mathrm{M}=\frac{\text { mass } \times 1000}{46 \times 100} \\
<br/>\text { Mass }=\frac{0.5 \times 46}{10}=2.3 \mathrm{g}
<br/>\end{array}
<br/>$$
<br/>$\therefore$ Volume of ethyl alcohol required.
<br/>$$
<br/>V=\frac{\text { mass of ethyl alcohol }}{\text { density of ethyl alcohol }}=\frac{2.3 \mathrm{g}}{1.15 \mathrm{g} / \mathrm{cc}}=2 \mathrm{cc}
<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
the-water-equivalent-of-a-calorimeter-is-10-g-and-it-contains-50-g-of-water-at-1-5-c-some-amount-of-ice-initially-at-1-0-c-is-dropped-in-it-and-half-o

slug:
the-water-equivalent-of-a-calorimeter-is-10-g-and-it-contains-50-g-of-water-at-1-5-c-some-amount-of-ice-initially-at-1-0-c-is-dropped-in-it-and-half-o-57936

question:
<div class="question">The water equivalent of a calorimeter is $10 \mathrm{g}$ and it contains $50 \mathrm{g}$ of water at $15^{\circ} \mathrm{C}$. Some amount of ice, initially at $-10^{\circ} \mathrm{C}$ is dropped in it and half of the ice melts till equilibrium is reached. What was the initial amount of ice that was dropped (when specific heat of ice $=0.5 \mathrm{cal} \mathrm{gm}^{-1}{ }^{\circ} \mathrm{C}^{-1},$ specific heat of water $=1.0$
<br/>cal $\mathrm{gm}^{-1 \circ} \mathrm{C}^{-1}$ and latent heat of melting of ice $=80$ cal $\mathrm{gm}^{-1}$ )?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10 \mathrm{g}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$18 \mathrm{g}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$20 \mathrm{g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$30 \mathrm{g}$</span> </li> </ul>

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

answer:
<div class="solution">Let the mass of ice $=m$
<br/>Applying calorimetry principle, heat given $=$ heat taken $\left(m_{1}+m_{2}\right) s_{1}\left(t_{1}-t\right)=\frac{m L}{2}+m s_{2}\left(t-t_{2}\right)$
<br/>Putting values, we get $(10+50) \times 1 \times(15-0)=\frac{m}{2} \times 80+m \times 0.5[0-(-10)]$
<br/>$\Rightarrow 60 \times 15=40 m+\frac{m}{2} \times 10=45 m$
<br/>$\therefore \quad m=\frac{60 \times 15}{45}=20 \mathrm{g}$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-wavelength-of-second-balmer-line-in-hydrogen-spectrum-is-600-nm-the-wavelength-for-its-third-line-in-lyman-series-is

slug:
the-wavelength-of-second-balmer-line-in-hydrogen-spectrum-is-600-nm-the-wavelength-for-its-third-line-in-lyman-series-is-34639

question:
<div class="question">The wavelength of second Balmer line in hydrogen &amp;spectrum is 600 nm  The wavelength for its third line in Lyman series is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$800 \mathrm{nm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$600 \mathrm{nm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$400 \mathrm{nm}$</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">$\frac{1}{\lambda}=R\left[\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right]$<br/>For Balmer series,<br/>$\begin{array}{l}<br/>\therefore n_{1}=2, n_{2}=3,4,5 \ldots \\<br/>\frac{1}{\lambda_{B}}=R\left[\frac{1}{2^{2}}-\frac{1}{4^{2}}\right]<br/>\end{array}$<br/>$\begin{array}{l}<br/>=R\left[\frac{1}{4}-\frac{1}{16}\right] \\<br/>\frac{1}{\lambda_{B}}=R \cdot \frac{3}{16}<br/>\end{array}$<br/>Lyman series is obtained when $n_{1}=1, n_{2}=2,3,4, \ldots$ For third line in Lyman series,<br/>$\begin{aligned}<br/>\frac{1}{\lambda_{L}} &amp;=R\left[\frac{1}{1^{2}}-\frac{1}{4^{2}}\right]=R\left[1-\frac{1}{16}\right] \\<br/>\frac{1}{\lambda_{L}} &amp;=R \cdot \frac{15}{16} \\<br/>\frac{1 / \lambda_{B}}{\frac{1}{\lambda_{L}}} &amp;=\frac{R \cdot \frac{3}{16}}{R \cdot \frac{15}{16}} \\<br/>\frac{\lambda_{L}}{\lambda_{B}} &amp;=\frac{3}{15} \\<br/>\lambda_{L} &amp;=\frac{3}{15} \times \lambda_{\theta} \\<br/>\lambda_{B} &amp;=600 \mathrm{nm}=600 \times 10^{-9} \mathrm{m} \\<br/>\lambda_{L} &amp;=\frac{3}{15} \times 600 \times 10^{-9} \\<br/>&amp;=120 \times 10^{-9} \mathrm{m}=120 \mathrm{nm}<br/>\end{aligned}$<br/>Hence, none of the given options is correct.</div>


title: the-weight-of-oxalic-acid-that-will-be-required-to-prepare-a-1000-ml-n-20-solution-is
slug: the-weight-of-oxalic-acid-that-will-be-required-to-prepare-a-1000-ml-n-20-solution-is-86282
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The weight of oxalic acid that will be required to prepare a $1000 \mathrm{mL}(\mathrm{N} / 20)$ solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$126 / 100 \mathrm{g}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$63 / 40 \mathrm{g}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$63 / 20 \mathrm{g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$126 / 20 \mathrm{g}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$63 / 20 \mathrm{g}$</span> </div>

answer:
<div class="solution">$w=\frac{N E V}{1000}=\frac{1}{20} \times \frac{63 \times 1000}{1000}=\frac{63}{20} g$</div>


tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
the-well-known-compounds-lactic-acid-and-lactic-acid-have-the-same-molecular-formula-c-3-h-6-o-3-the-correct-relationship-between-them-is

slug:
the-well-known-compounds-lactic-acid-and-lactic-acid-have-the-same-molecular-formula-c-3-h-6-o-3-the-correct-relationship-between-them-is-93775

question:
<div class="question">The well known compounds, $(+)-$ lactic acid and $(-)-$ lactic acid, have the same molecular formula, $\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{3}$. The correct relationship between them is</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a6.png"/> <br/> <br/>Both are optical isomers because they rotate the plane of polarised light in opposite directions.</div>


tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-white-precipitate-y-obtained-on-passing-colourless-and-odourless-gas-x-through-an-ammoniacal-solution-of-nacl-loses-about-37-of-its-weight-on-heat

slug:
the-white-precipitate-y-obtained-on-passing-colourless-and-odourless-gas-x-through-an-ammoniacal-solution-of-nacl-loses-about-37-of-its-weight-on-heat-65299

question:
<div class="question">The white precipitate $(Y)$, obtained on passing colourless and odourless gas $(X)$ through an ammoniacal solution of $\mathrm{NaCl}$, loses about $37 \%$ of its weight on heating and a white residue $(\mathrm{Z})$ of basic nature is left. Identify $(\mathrm{X}),(\mathrm{Y})$ and $(\mathrm{Z})$ from following sets.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$$\mathrm{CO}_{2} \mathrm{NaHCO}_{3} \mathrm{Na}_{2} \mathrm{CO}_{3}
<br/>$$</span> </div>

answer:
<div class="solution">$\begin{array}{c}
<br/>\mathrm{NH}_{3}+\underset{(\mathrm{x})}{\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} \longrightarrow} \longrightarrow \mathrm{NH}_{4} \mathrm{HCO}_{3} \\
<br/>\mathrm{NH}_{4} \mathrm{HCO}_{3}+\mathrm{NaCl} \longrightarrow \mathrm{NaHCO}_{3} \downarrow+\mathrm{NH}_{4} \mathrm{Cl} \\
<br/>2 \mathrm{NaHCO}_{3} \stackrel{\Delta}{\longrightarrow} \underset{(\mathrm{y})}{\mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}} \\
<br/>{(\mathrm{z})}
<br/>\end{array}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0$</span> </div>
answer: <div class="solution">Work done by conservative force in a round trip is zero</div>
source_db: db\MarksBatch2_P3.db
title:
the-work-done-in-carrying-a-charge-q-once-around-a-circle-of-radius-r-about-a-charge-q-at-the-centre-is

slug:
the-work-done-in-carrying-a-charge-q-once-around-a-circle-of-radius-r-about-a-charge-q-at-the-centre-is-21373

question:
<div class="question">The work done in carrying a charge $\mathrm{Q}$ once around a circle of radius $\mathrm{r}$ about a charge $\mathrm{q}$ at the centre is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{q Q}{4 \pi \varepsilon_0 r}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{q Q}{4 \pi \varepsilon_0} \frac{1}{\pi r}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{q Q}{4 \pi \varepsilon_0}\left(\frac{1}{2 \pi r}\right)$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-work-function-of-cesium-is-227-ev-the-cutoff-voltage-which-stops-the-emission-of-electrons-from-a-cesium-cathode-irradiated-with-light-of-600-nm-w

slug:
the-work-function-of-cesium-is-227-ev-the-cutoff-voltage-which-stops-the-emission-of-electrons-from-a-cesium-cathode-irradiated-with-light-of-600-nm-w-39601

question:
<div class="question">The work function of Cesium is $2.27 \mathrm{eV}$. The cut-off voltage which stops the emission of electrons from a cesium cathode irradiated with light of $600 \mathrm{nm}$ wavelength is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.5 \mathrm{V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-0.2 \mathrm{V}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-0.5 \mathrm{V}$</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">Work function of cesium $=2.27 \mathrm{eV}$
<br/>The wavelength of incident light
<br/>$$
<br/>\lambda=600 \mathrm{nm}=600 \times 10^{-9} \mathrm{m}
<br/>$$
<br/>The energy of incident photon $=h v$
<br/>$$
<br/>\begin{aligned}
<br/>E &amp;=\frac{h c}{\lambda} \\
<br/>&amp;=\frac{6.6 \times 10^{-34} \times 3 \times 10^{8}}{600 \times 10^{-9} \times 1.6 \times 10^{-19}} e \mathrm{V} \\
<br/>&amp;=2.06 \mathrm{eV} \\
<br/>\frac{h c}{\lambda}    &lt; \phi
<br/>\end{aligned}
<br/>$$
<br/>$2.06 \mathrm{eV} &lt; 2.27 \mathrm{eV}$ 
<br/>So, no emission of electrons will take place. Hence, none of the given options is correct.</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
the-work-function-of-metals-is-in-the-range-of-2-ev-to-5-ev-find-which-of-the-following-wavelength-of-light-cannot-be-used-for-photoelectric-effect-co

slug:
the-work-function-of-metals-is-in-the-range-of-2-ev-to-5-ev-find-which-of-the-following-wavelength-of-light-cannot-be-used-for-photoelectric-effect-co-62038

question:
<div class="question">The work function of metals is in the range of $2 \mathrm{eV}$ to $5 \mathrm{eV}$. Find which of the following wavelength of light cannot be used for photoelectric effect? (Consider, Plank constant $=4 \times 10^{-15} \mathrm{eVs},$ velocity of light $\left.=3 \times 10^{8} \mathrm{m} / \mathrm{s}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$510 \mathrm{nm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$650 \mathrm{nm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$400 \mathrm{nm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$570 \mathrm{nm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$650 \mathrm{nm}$</span> </div>

answer:
<div class="solution">Given that, $2 \mathrm{eV} \leq \phi \leq 5 \mathrm{eV}$
<br/>Wavelength corresponding to minimum and maximum values of work function are
<br/>$$
<br/>\begin{aligned}
<br/>\lambda_{\max } &amp;=\frac{h C}{E}=\frac{4 \times 10^{-15} \times 3 \times 10^{8}}{2} \\
<br/>&amp;=6 \times 10^{-7} \mathrm{m}=600 \mathrm{nm} \\
<br/>\text { and } \lambda_{\min } &amp;=\frac{h C}{E}=\frac{4 \times 10^{-15} \times 3 \times 10^{8}}{5} \\
<br/>&amp;=2.4 \times 10^{-7} \mathrm{m}=240 \mathrm{nm}
<br/>\end{aligned}
<br/>$$
<br/>Clearly, wavelength range would be
<br/>$$
<br/>240 \leq \lambda \leq 600
<br/>$$
<br/>Thus, wavelength $650 \mathrm{nm}$ is not suitable for photoelectric effect.</div>


tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-yield-of-a-acetanilide-in-the-reaction-100-conversion-of-2-moles-of-aniline-with-1-mole-of-acetic-anhydride-is

slug:
the-yield-of-a-acetanilide-in-the-reaction-100-conversion-of-2-moles-of-aniline-with-1-mole-of-acetic-anhydride-is-68972

question:
<div class="question">The yield of a acetanilide in the reaction (100\% conversion) of 2 moles of aniline with 1 mole of acetic anhydride is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2708</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$135 \mathrm{g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$67.5 \mathrm{g}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">177 g</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$135 \mathrm{g}$</span> </div>

answer:
<div class="solution">The reaction between aniline and acetic anhydride is as follows :
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a61.png"/></div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
there-are-7-greeting-cards-each-of-a-different-colour-and-7-envelopes-of-same-7-colours-as-that-of-the-cards-the-number-of-ways-in-which-the-cards-can

slug:
there-are-7-greeting-cards-each-of-a-different-colour-and-7-envelopes-of-same-7-colours-as-that-of-the-cards-the-number-of-ways-in-which-the-cards-can-12763

question:
<div class="question">There are 7 greeting cards, each of a different colour and 7 envelopes of same 7 colours as that of the cards. The number of ways in which the cards can be put in envelopes, so that exactly 4 of the cards go into envelopes of respective colour is,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{7} C_{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2^{7} C_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3!^{4} C_{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3!^{7} C_{3}^{4} C_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2^{7} C_{3}$</span> </div>

answer:
<div class="solution">Required number of ways
<br/>$$
<br/>\begin{array}{l}
<br/>={ }^{7} C_{4} \times D(3) \\
<br/>={ }^{7} C_{3} \times\left[3 \cdot\left(1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}\right)\right] \\
<br/>{\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r, a n d}\right.} \\
<br/>\left.D(n)=n\left[1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\ldots\right]\right] \\
<br/>={ }^{7} C_{3} \times 2=2\left({ }^{7} C_{3}\right)
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Sets and Relations', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">is 2</span> </div>
source_db: db\MarksBatch2_P3.db
title:
there-are-100-students-in-a-class-in-an-examination-50-of-them-failed-in-mathematics-45-failed-in-physics-40-failed-in-biology-and-32-failed-in-exactl-1

slug:
there-are-100-students-in-a-class-in-an-examination-50-of-them-failed-in-mathematics-45-failed-in-physics-40-failed-in-biology-and-32-failed-in-exactl-1-28258

question:
<div class="question">There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. Then, the number of students failing in all the three subjects</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">is 12</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is 4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is 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">cannot be determined from the given information</span> </li> </ul>

answer:
<div class="solution">According to question,<br/>$\begin{array}{c}<br/>a+b+c+d+e+f+g+h=100 \\<br/>h=1 \\<br/>a+b+d+e=50<br/>\end{array}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a34.png"/><br/>Venn diagram shows failed students $b+c+d+f=45$<br/>$\begin{array}{l}<br/>d+e+f+g=40 \\<br/>b+e+f=32<br/>\end{array}$<br/>On solving, the above equations, we get $d=2$</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">is 2</span> </div>
source_db: db\MarksBatch2_P3.db
title:
there-are-100-students-in-a-class-in-an-examination-50-of-them-failed-in-mathematics-45-failed-in-physics-40-failed-in-biology-and-32-failed-in-exactl

slug:
there-are-100-students-in-a-class-in-an-examination-50-of-them-failed-in-mathematics-45-failed-in-physics-40-failed-in-biology-and-32-failed-in-exactl-90641

question:
<div class="question">There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. Then, the number of students failing in all the three subjects</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">is 12</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is 4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is 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">cannot be determined from the given information</span> </li> </ul>

answer:
<div class="solution">According to question,<br/>$\begin{array}{c}<br/>a+b+c+d+e+f+g+h=100 \\<br/>h=1 \\<br/>a+b+d+e=50<br/>\end{array}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a34.png"/><br/>Venn diagram shows failed students $b+c+d+f=45$<br/>$\begin{array}{l}<br/>d+e+f+g=40 \\<br/>b+e+f=32<br/>\end{array}$<br/>On solving, the above equations, we get $d=2$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
there-are-n-elastic-balls-placed-on-a-smooth-horizontal-plane-the-masses-of-the-balls-are-m-fracm2-fracm22-ldots-fracm2n1-respectively-if-the-first-ba-1

slug:
there-are-n-elastic-balls-placed-on-a-smooth-horizontal-plane-the-masses-of-the-balls-are-m-fracm2-fracm22-ldots-fracm2n1-respectively-if-the-first-ba-1-60912

question:
<div class="question">There are \(n\) elastic balls placed on a smooth horizontal plane. The masses of the balls are \(m, \frac{m}{2}, \frac{m}{2^2}, \ldots . \frac{m}{2^{n-1}}\) respectively. If the first ball hits the second ball with velocity \(\mathrm{v}_0\), then the velocity of the \(\mathrm{n}^{\text {th }}\) ball will be,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{4}{3} v_0\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\left(\frac{4}{3}\right)^n v_0\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\left(\frac{4}{3}\right)^{n-1} v_0\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(v_0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left(\frac{4}{3}\right)^{n-1} v_0\)</span> </div>

answer:
<div class="solution">Hint : 1st Collision<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/iIk52JQ0L4PGQPuwPX_lr98sRIUuQbSS2IJ_XZOy45s.original.fullsize.png"/><br/><br/>\(\mathrm{V}_1=\frac{2 \times \mathrm{m}}{\frac{\mathrm{m}}{2}+\mathrm{m}} \mathrm{V}_0=\frac{4}{3} \mathrm{~V}_0\)<br/>2nd Collision<br/>\(V_2=\frac{2 \times \frac{m}{2}}{\frac{m}{2}+\frac{m}{2^2}} \times V_0=\left(\frac{4}{3}\right)^2 V_0\)<br/>3rd Collision<br/>\(\left(\frac{4}{3}\right)^3 \mathrm{~V}_0\)<br/>\(\ldots(n-1)\) collision, \(\therefore V_{n-1}=\left(\frac{4}{3}\right)^{n-1} V_0\)</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
there-are-n-elastic-balls-placed-on-a-smooth-horizontal-plane-the-masses-of-the-balls-are-m-fracm2-fracm22-ldots-fracm2n1-respectively-if-the-first-ba

slug:
there-are-n-elastic-balls-placed-on-a-smooth-horizontal-plane-the-masses-of-the-balls-are-m-fracm2-fracm22-ldots-fracm2n1-respectively-if-the-first-ba-96173

question:
<div class="question">There are \(n\) elastic balls placed on a smooth horizontal plane. The masses of the balls are \(m, \frac{m}{2}, \frac{m}{2^2}, \ldots . \frac{m}{2^{n-1}}\) respectively. If the first ball hits the second ball with velocity \(\mathrm{v}_0\), then the velocity of the \(\mathrm{n}^{\text {th }}\) ball will be,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{4}{3} v_0\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\left(\frac{4}{3}\right)^n v_0\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\left(\frac{4}{3}\right)^{n-1} v_0\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(v_0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left(\frac{4}{3}\right)^{n-1} v_0\)</span> </div>

answer:
<div class="solution">Hint : 1st Collision<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/iIk52JQ0L4PGQPuwPX_lr98sRIUuQbSS2IJ_XZOy45s.original.fullsize.png"/><br/><br/>\(\mathrm{V}_1=\frac{2 \times \mathrm{m}}{\frac{\mathrm{m}}{2}+\mathrm{m}} \mathrm{V}_0=\frac{4}{3} \mathrm{~V}_0\)<br/>2nd Collision<br/>\(V_2=\frac{2 \times \frac{m}{2}}{\frac{m}{2}+\frac{m}{2^2}} \times V_0=\left(\frac{4}{3}\right)^2 V_0\)<br/>3rd Collision<br/>\(\left(\frac{4}{3}\right)^3 \mathrm{~V}_0\)<br/>\(\ldots(n-1)\) collision, \(\therefore V_{n-1}=\left(\frac{4}{3}\right)^{n-1} V_0\)</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">BW BW ..... $=n ! \times n !$<br/>or<br/>WB WB ..... $=2(n !)^2$</div>
source_db: db\MarksBatch2_P3.db
title:
there-are-n-white-and-n-black-balls-marked-1-2-3-n-the-number-of-ways-in-which-we-can-arrange-these-balls-in-a-row-so-that-neighbouring-balls-are-of-d

slug:
there-are-n-white-and-n-black-balls-marked-1-2-3-n-the-number-of-ways-in-which-we-can-arrange-these-balls-in-a-row-so-that-neighbouring-balls-are-of-d-64809

question:
<div class="question">There are $n$ white and $n$ black balls marked $1,2,3, \ldots . n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours is</div>

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

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


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
there-are-two-coins-one-unbiased-with-probaility-2-1-or-getting-heads-and-the-other-one-is-biased-with-probability-4-3-of-getting-heads-a-coin-is-sele

slug:
there-are-two-coins-one-unbiased-with-probaility-2-1-or-getting-heads-and-the-other-one-is-biased-with-probability-4-3-of-getting-heads-a-coin-is-sele-74742

question:
<div class="question">There are two coins, one unbiased with probaility $\frac{1}{2}$ or getting heads and the other one is biased with probability $\frac{3}{4}$ of getting heads. A coin is selected at random and tossed. It shows heads up. Then, the probability that the unbiased coin wa selected is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{5}$</span> </li><li class=""> <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">$\frac{2}{5}$</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">$\frac{2}{5}$</span> </div>

answer:
<div class="solution">Let $E \rightarrow$ Event of head showing up
<br/>$E_{1} \rightarrow$ Event of biased coin chosen
<br/>$E_{2} \rightarrow$ Event of unbiased coin chosen Now, $P\left(E_{2}\right)=\frac{1}{2}$ and $P\left(E_{1}\right)=\frac{1}{2}$
<br/>Also, $P\left(\frac{E}{E_{2}}\right)=\frac{1}{2}$ and $P\left(\frac{E}{E_{1}}\right)=\frac{3}{4}$
<br/>(by conditional probability) By Baye's theorem
<br/>$$
<br/>\begin{aligned}
<br/>P\left(\frac{E_{2}}{E}\right)=&amp; \frac{P\left(E_{2}\right) \cdot P\left(\frac{E}{E_{2}}\right)}{P\left(E_{2}\right) \cdot P\left(\frac{E}{E_{2}}\right)+P\left(E_{1}\right) \cdot P\left(\frac{E}{E_{1}}\right)} \\
<br/>&amp;=\frac{\frac{1}{2} \times \frac{1}{2}}{\frac{1}{2} \times \frac{1}{2}+\frac{1}{2} \times \frac{3}{4}}=\frac{2}{5}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: db\MarksBatch2_P3.db
title:
there-is-a-group-of-265-persons-who-like-either-singing-or-dancing-or-painting-in-this-group-200-like-singing-110-like-dancing-and-55-like-painting-if

slug:
there-is-a-group-of-265-persons-who-like-either-singing-or-dancing-or-painting-in-this-group-200-like-singing-110-like-dancing-and-55-like-painting-if-36185

question:
<div class="question">There is a group of 265 persons who like either singing or dancing or painting. In this group 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 hike all three activities, then the number of persons who like only dancing and painting is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">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">B</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">30</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">40</span> </li> </ul>

answer:
<div class="solution">Let $D$ denotes dancing. $P$ denotes painting and $S$ denotes singing.
<br/>$\begin{array}{l}
<br/>\therefore \quad n(D \cup P \cup S)=265 \\
<br/>n(S)=200, n(D)=110, n(P)=55 \\
<br/>n(S \cap D)=60, n(S \cap P)=30 \\
<br/>\text { and } n(D \cap P \cap S)=10 \\
<br/>\because n(D \cup P \cup S)=n(D)+n(P)+n(S)-n(D \cap P) \\
<br/>-n(P \cap S)-n(S \cap D)+n(D \cap P \cap S) \\
<br/>\therefore \quad 265=110+55+200-n(D \cap P) \\
<br/>\Rightarrow \quad 265=285-n(D \cap P) \\
<br/>\Rightarrow \quad n(D \cap P)=20
<br/>\end{array}$
<br/>$\therefore$ Only person who like dancing and painting
<br/>$\begin{array}{l}
<br/>=n(D \cap P)-n(D \cap P \cap S) \\
<br/>=20-10=10
<br/>\end{array}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Zero</span> </div>
source_db: db\MarksBatch2_P3.db
title:
there-is-a-small-air-bubble-at-the-centre-of-a-solid-glass-sphere-of-radius-r-and-refractive-index-what-will-be-the-apparent-distance-of-the-bubble-fr

slug:
there-is-a-small-air-bubble-at-the-centre-of-a-solid-glass-sphere-of-radius-r-and-refractive-index-what-will-be-the-apparent-distance-of-the-bubble-fr-45434

question:
<div class="question">There is a small air bubble at the centre of a solid glass sphere of radius $r$ and refractive index $\mu .$ What will be the apparent distance of the bubble from the centre of the sphere, when viewed from outside?</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a5.png"/>'
<br/>As the object is at centre of the sphere.
<br/>$\therefore$ All rays will fall normally on surface, hence they do not deflect. Thus, a virtual image is formed at the
<br/>centre $O$.
<br/>$\therefore$ Apparent depth $=$ Real depth</div>


title: this-alcohol-is
slug: this-alcohol-is-47814
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OuIzkWd7tdEVUIxdwpFolXF9fHqmsqk2fwop2GSO72M.original.fullsize.png"/><br/> <br/>This alcohol 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/yqkfUJ6pWsTKHX48TzYb0-cxOeiw9BDFrs_il8BY4xI.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/bQApZWXE6yqOVLp1Twp6HUhJTo8ocxteHdZISCoNjec.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/6xOo3wjL690I6RMawi0hEbdBsQwtyR5FbWue52hgBUA.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/681yXnGw3BCaKVKJOnb7e6RPKJBhGEv7FihaAgF3CVw.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/6xOo3wjL690I6RMawi0hEbdBsQwtyR5FbWue52hgBUA.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Hint:
<br/>Hydride shift, here $\mathrm{D}^{-}$, shift decides the product
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sd59hYrkcvnp_HpqFGA-_UmDEQvn_PKv-8YH_njMJTk.original.fullsize.png"/></div>


title: this-product-in-the-above-reaction-is
slug: this-product-in-the-above-reaction-is-98324
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/z7nZD2oy9ro3VoAMCV7YMJsbT-PVIQ-MOJLQqLrnFME.original.fullsize.png"/><br/> <br/>This product in the above reaction 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/dp-8QNyItmxJqcCn0GDrV1qNGaJF5tePfyzJFtu0Y2A.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/5H4ff_Q6hdzXdim19ES7lnbnTDLpt5jkehleIKKVrCM.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/51pvrgNj7cbH-O9yauQyQwQFpPxYg7rukEnu9ZTK_Aw.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/gPRUexZ7LQGtpERYoz2g3CTjlkoZdLqs_TQN8qhBUNQ.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> </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/gPRUexZ7LQGtpERYoz2g3CTjlkoZdLqs_TQN8qhBUNQ.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/G-Yb1WegHReCz1Hw0lT0zCAIJPyzEWUBV_tATZzE9Gs.original.fullsize.png"/></div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
three-blocks-are-pushed-with-a-force-f-across-a-frictionless-table-as-shown-in-figure-let-n-1-be-the-contact-force-between-the-left-two-blocks-and-n-2-1

slug:
three-blocks-are-pushed-with-a-force-f-across-a-frictionless-table-as-shown-in-figure-let-n-1-be-the-contact-force-between-the-left-two-blocks-and-n-2-1-24354

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hivLux1gxm-68i8rx47G0ckDQm2RqVJiMEvcl5puoVo.original.fullsize.png"/><br/> <br/>Three blocks are pushed with a force $\mathrm{F}$ across a frictionless table as shown in figure. Let $\mathrm{N}_{1}$ be the contact force between the left two blocks and $\mathrm{N}_{2}$ be the contact force between the right two blocks. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$F&gt;N_{1}&gt;N_{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5rt-M6QIPQ290KVjI_hLaD-cd7Kogi_ADrT08xYP84w.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uokVwzVezMmEMbM4SfiojlwOy1gAftYBdeQUdkRD2Ow.original.fullsize.png"/><br/> <br/>$\mathrm{F}=6 \mathrm{a}, \mathrm{N}_{1}=5 \mathrm{a}, \mathrm{N}_{2}=3 \mathrm{a}$
<br/>$\mathrm{F}&gt;\mathrm{N}_{1}&gt;\mathrm{N}_{2}$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
three-blocks-are-pushed-with-a-force-f-across-a-frictionless-table-as-shown-in-figure-let-n-1-be-the-contact-force-between-the-left-two-blocks-and-n-2

slug:
three-blocks-are-pushed-with-a-force-f-across-a-frictionless-table-as-shown-in-figure-let-n-1-be-the-contact-force-between-the-left-two-blocks-and-n-2-88321

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/hivLux1gxm-68i8rx47G0ckDQm2RqVJiMEvcl5puoVo.original.fullsize.png"/><br/> <br/>Three blocks are pushed with a force $\mathrm{F}$ across a frictionless table as shown in figure. Let $\mathrm{N}_{1}$ be the contact force between the left two blocks and $\mathrm{N}_{2}$ be the contact force between the right two blocks. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$F&gt;N_{1}&gt;N_{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5rt-M6QIPQ290KVjI_hLaD-cd7Kogi_ADrT08xYP84w.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uokVwzVezMmEMbM4SfiojlwOy1gAftYBdeQUdkRD2Ow.original.fullsize.png"/><br/> <br/>$\mathrm{F}=6 \mathrm{a}, \mathrm{N}_{1}=5 \mathrm{a}, \mathrm{N}_{2}=3 \mathrm{a}$
<br/>$\mathrm{F}&gt;\mathrm{N}_{1}&gt;\mathrm{N}_{2}$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
three-blocks-of-masses-4-kg-2-kg-1-kg-respectively-are-in-contact-on-a-frictionless-table-as-shown-in-the-figure-if-a-force-of-14-n-is-applied-on-the-

slug:
three-blocks-of-masses-4-kg-2-kg-1-kg-respectively-are-in-contact-on-a-frictionless-table-as-shown-in-the-figure-if-a-force-of-14-n-is-applied-on-the-82563

question:
<div class="question">Three blocks of masses 4 kg. 2 kg. 1 kg respectively are in contact on a frictionless table as shown in the figure. If a force of $14 \mathrm{N}$ is applied on the 4 kg block, the contact force between the $4 \mathrm{kg}$ and the $2 \mathrm{kg}$ block will be <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_q35.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{N}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$6 \mathrm{N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 \mathrm{N}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$14 \mathrm{N}$</span> </li> </ul>

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

answer:
<div class="solution">We know that, $F=m a$
<br/>$$
<br/>a=\frac{F}{m}=\frac{14}{7}=2 \mathrm{ms}^{-2}
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_a35.png"/> <br/> <br/>Hence, from the figure
<br/>$$
<br/>14-N=4 a
<br/>$$
<br/>$$
<br/>\begin{array}{r}
<br/>14-N=8 \\
<br/>N=6 \mathrm{N}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
three-bodies-of-the-same-material-and-having-masses-m-m-and-3-m-are-at-temperatures-4-0-c-5-0-c-and-6-0-c-respectively-if-the-bodies-are-brought-in-th

slug:
three-bodies-of-the-same-material-and-having-masses-m-m-and-3-m-are-at-temperatures-4-0-c-5-0-c-and-6-0-c-respectively-if-the-bodies-are-brought-in-th-93123

question:
<div class="question">Three bodies of the same material and having masses $m, m$ and $3 m$ are at temperatures $40^{\circ} \mathrm{C}, \quad 50^{\circ} \mathrm{C} \quad$ and $\quad 60^{\circ} \mathrm{C}$ respectively. If the bodies are brought in thermal contact, the final temperature will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$45^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$54^{\circ} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$52^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$48^{\circ} \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$54^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">Let the final temperature after the masses in thermal contact is $\theta$, then from the principle of calonimetry.
<br/>Heat lost $=$ Heat gained
<br/>$\begin{aligned} \Rightarrow 3 m s(60-\theta) &amp;=m s(\theta-50)+m s(\theta-40) \\ \Rightarrow \quad 3(60-\theta) &amp;=\theta-50+\theta-40 \\ \Rightarrow \quad 180-3 \theta &amp;=2 \theta-90 \\ \theta &amp;=\frac{270}{5}=54^{\circ} \mathrm{C} \end{aligned}$</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">60</span> </div>
source_db: db\MarksBatch2_P3.db
title:
three-capacitors-3-f-6-f-and-6-f-are-connected-in-aeries-to-a-source-of-120-v-the-potential-difference-in-volt-across-the-3-f-capacitor-will-be

slug:
three-capacitors-3-f-6-f-and-6-f-are-connected-in-aeries-to-a-source-of-120-v-the-potential-difference-in-volt-across-the-3-f-capacitor-will-be-61589

question:
<div class="question">Three capacitors $3 \mu \mathrm{F}, 6 \mu \mathrm{F}$ and $6 \mu \mathrm{F}$ are connected in aeries to a source of $120 \mathrm{V}$. The potential difference, in volt, across the $3 \mu \mathrm{F}$ capacitor will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">30</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">40</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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> </ul>

answer:
<div class="solution">The combination of three charges in series
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}} \\
<br/>\frac{1}{C}=\frac{1}{3}+\frac{1}{6}+\frac{1}{6} \Rightarrow C=\frac{6}{4}=1.5 \mu \mathrm{F}
<br/>\end{array}
<br/>$$
<br/>The charge of this circuit
<br/>$$
<br/>\begin{array}{l}
<br/>q=C V=1.5 \times 120 \\
<br/>q=180 \mu \mathrm{C}
<br/>\end{array}
<br/>$$
<br/>The potential difference across the $3 \mu \mathrm{F}$
<br/>$$
<br/>q=C V
<br/>$$
<br/>$$
<br/>V=\frac{q}{C}=\frac{180}{3}=60 \mathrm{V}
<br/>$$</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
three-capacitors-of-capacitance-1020-and-50-f-are-connected-in-series-to-a-10-v-source-the-potential-difference-across-the-20-f-capacitor-is

slug:
three-capacitors-of-capacitance-1020-and-50-f-are-connected-in-series-to-a-10-v-source-the-potential-difference-across-the-20-f-capacitor-is-19761

question:
<div class="question">Three capacitors of capacitance 1.0,2.0 and $5.0 \mu \mathrm{F}$ are connected in series to a $10 \mathrm{V}$ source. The potential difference across the $2.0 \mu \mathrm{F}$ capacitor is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{100}{17} \mathrm{V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{20}{17} \mathrm{v}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{50}{17} \mathrm{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10 \mathrm{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{50}{17} \mathrm{V}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a9.png"/> <br/> <br/>When the capacitors are connected in series, the resultant capacitance of combination
<br/>$$
<br/>\frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{1}{1}+\frac{1}{2}+\frac{1}{5}=\frac{17}{10} \\
<br/>C=\frac{10}{17} \mu \mathrm{F}
<br/>\end{array}
<br/>$$
<br/>The charge will be same on all the capacitors in series
<br/>$$
<br/>Q=C V=\frac{10}{17} \times 10=\frac{100}{17}
<br/>$$
<br/>The potential difference across $2.0 \mu \mathrm{F}$ capacitor
<br/>$$
<br/>V^{*}=\frac{Q}{C}=\frac{100 / 17}{2}=\frac{50}{17} \text { volt}
<br/>$$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
three-concentric-metallic-shells-a-b-and-c-of-radii-a-b-and-c-a-b-c-have-surface-charge-densities-and-respectively-the-potential-of-shell-b-is

slug:
three-concentric-metallic-shells-a-b-and-c-of-radii-a-b-and-c-a-b-c-have-surface-charge-densities-and-respectively-the-potential-of-shell-b-is-40451

question:
<div class="question">Three concentric metallic shells A, B and C of radii $a, b$ and $c(a &lt; b &lt; c)$ have surface charge densities $+\sigma,-\sigma$ and $+\sigma$ respectively. The potential of shell B is<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/OYuD0-ig4j2tGp871QhyUNd5PLDJuwWDM1qJANtet7o.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad(a+b+c) \frac{\sigma}{\varepsilon_0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sigma \mathrm{c}}{\varepsilon_0}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{a^2}{c}-\frac{b^2}{c}+c\right) \frac{\sigma}{e_0}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left(\frac{a^2}{b}-b+c\right) \frac{\sigma}{e_0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{a^2}{b}-b+c\right) \frac{\sigma}{e_0}$</span> </div>

answer:
<div class="solution">$V_b=V_{A, b}+V_{B, b}+V_{C, b}$<br/>$=\frac{K \sigma 4 \pi a^2}{b}-\frac{K \sigma 4 \pi b^2}{b}+\frac{K \sigma 4 \pi c^2}{c}$<br/>$=\frac{\sigma}{\varepsilon_0}\left(\frac{a^2}{b}-b+c\right)$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
three-identical-convex-lenses-each-of-focal-length-f-are-placed-in-a-straight-line-separated-by-a-distance-f-from-each-other-an-object-is-located-at-m

slug:
three-identical-convex-lenses-each-of-focal-length-f-are-placed-in-a-straight-line-separated-by-a-distance-f-from-each-other-an-object-is-located-at-m-84549

question:
<div class="question">Three identical convex lenses each of focal length \(f\) are placed in a straight line separated by a distance \(f\) from each other. An object is located at \(\mathrm{f} / 2\) in front of the leftmost lens. Then,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ZFT3Clo6gRk89WOI3nH1D-BkebjvTreZR2zUpTFqDlU.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Final image will be at \(f / 2\) behind the rightmost lens and its magnification will be -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">Final image will be at \(f / 2\) behind the rightmost lens and its magnification will be +1 .</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Final image will be at \(f\) behind the rightmost lens and its magnification will be -1 .</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Final image will be at \(f\) behind the rightmost lens and its magnification will be +1 .</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Final image will be at \(f / 2\) behind the rightmost lens and its magnification will be -1 .</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/U76JsHF5grWW2aZUHd4-kxLH1LGGfp7jCm0esAibRUA.original.fullsize.png"/><br/><br/>For first lens<br/>\(\begin{aligned}<br/>&amp; u=-\frac{f}{2} \\<br/>&amp; \frac{1}{f}=\frac{1}{v}-\frac{1}{u} \Rightarrow \frac{1}{f}=\frac{1}{v}-\frac{1}{\frac{-f}{2}} \Rightarrow \frac{1}{v}=\frac{1}{f}-\frac{2}{f} \Rightarrow \frac{1}{v}=\frac{-1}{f} \\<br/>&amp; v=-f \\<br/>&amp; m_1=\frac{v}{u}=\frac{-f}{\frac{-f}{2}}=2<br/>\end{aligned}\)<br/>For second lens<br/>\(u=-(f+f)=-2 f\)<br/>\(\begin{aligned}<br/>&amp; \frac{1}{v}-\frac{1}{u}=\frac{1}{f} \Rightarrow \frac{1}{v}-\frac{1}{-2 f}=\frac{1}{f} \Rightarrow \frac{1}{v}=\frac{1}{f}-\frac{1}{2 f} \Rightarrow \frac{1}{v}=\frac{1}{2 f} \\<br/>&amp; v=2 f \\<br/>&amp; m_1=\frac{v}{u}=\frac{2 f}{-2 f}=-1<br/>\end{aligned}\)<br/>For third lens<br/>\(\begin{aligned}<br/>&amp; u=f \\<br/>&amp; f=f \\<br/>&amp; \frac{1}{v}-\frac{1}{u}=\frac{1}{f} \Rightarrow \frac{1}{v}=\frac{1}{f}+\frac{1}{u} \Rightarrow \frac{1}{v}=\frac{1}{f}+\frac{1}{f} \Rightarrow \frac{1}{v}=\frac{2}{f} \Rightarrow v=\frac{f}{2}<br/>\end{aligned}\)<br/>\(m_3=\frac{v}{u}=\frac{\frac{f}{2}}{f}=\frac{1}{2}\)<br/>Total magnification \(=m_1 m_2 m_3=2 \times(-1) \times \frac{1}{2}=-1\)</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
three-identical-square-plates-rotate-about-the-axes-shown-in-the-figure-in-such-a-way-that-their-kinetic-energies-are-equal-each-of-the-rotation-axes--1

slug:
three-identical-square-plates-rotate-about-the-axes-shown-in-the-figure-in-such-a-way-that-their-kinetic-energies-are-equal-each-of-the-rotation-axes-1-48027

question:
<div class="question">Three identical square plates rotate about the axes shown in the figure in such a way that their kinetic energies are equal. Each of the rotation axes passes through the centre of the square. Then the ratio of angular speeds $\omega_{1}: \omega_{2}: \omega_{3}$ is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q51.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">01:01:01</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{2}: \sqrt{2}: 1$</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">C</span> <span class="option-data">$1: \sqrt{2}: 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1: 2: \sqrt{2}$</span> </li> </ul>

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

answer:
<div class="solution">We know that
<br/>$$
<br/>K=\frac{1}{2} \mathrm{l} \omega^{2}
<br/>$$
<br/>where $K=$ kinetic energy
<br/>$$
<br/>\begin{array}{l}
<br/>I=\text { moment of inertia } \\
<br/>\omega=\text { angular speed }
<br/>\end{array}
<br/>$$
<br/>So, $\quad \omega \propto \frac{1}{\sqrt{I}}$
<br/>$$
<br/>\begin{aligned}
<br/>\omega_{1}: \omega_{2}: \omega_{3} &amp;=\frac{1}{\sqrt{I_{1}}}: \frac{1}{\sqrt{I_{2}}}: \frac{1}{\sqrt{I_{3}}} \\
<br/>&amp;=\frac{1}{\sqrt{1}}: \frac{1}{\sqrt{1}}: \frac{1}{\sqrt{2}}=\sqrt{2}: \sqrt{2}: 1
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
three-identical-square-plates-rotate-about-the-axes-shown-in-the-figure-in-such-a-way-that-their-kinetic-energies-are-equal-each-of-the-rotation-axes-

slug:
three-identical-square-plates-rotate-about-the-axes-shown-in-the-figure-in-such-a-way-that-their-kinetic-energies-are-equal-each-of-the-rotation-axes-97094

question:
<div class="question">Three identical square plates rotate about the axes shown in the figure in such a way that their kinetic energies are equal. Each of the rotation axes passes through the centre of the square. Then the ratio of angular speeds $\omega_{1}: \omega_{2}: \omega_{3}$ is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q51.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">01:01:01</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{2}: \sqrt{2}: 1$</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">C</span> <span class="option-data">$1: \sqrt{2}: 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1: 2: \sqrt{2}$</span> </li> </ul>

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

answer:
<div class="solution">We know that
<br/>$$
<br/>K=\frac{1}{2} \mathrm{l} \omega^{2}
<br/>$$
<br/>where $K=$ kinetic energy
<br/>$$
<br/>\begin{array}{l}
<br/>I=\text { moment of inertia } \\
<br/>\omega=\text { angular speed }
<br/>\end{array}
<br/>$$
<br/>So, $\quad \omega \propto \frac{1}{\sqrt{I}}$
<br/>$$
<br/>\begin{aligned}
<br/>\omega_{1}: \omega_{2}: \omega_{3} &amp;=\frac{1}{\sqrt{I_{1}}}: \frac{1}{\sqrt{I_{2}}}: \frac{1}{\sqrt{I_{3}}} \\
<br/>&amp;=\frac{1}{\sqrt{1}}: \frac{1}{\sqrt{1}}: \frac{1}{\sqrt{2}}=\sqrt{2}: \sqrt{2}: 1
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
three-infinite-plane-sheets-carrying-uniform-charge-densities-2-4-are-placed-parallel-to-xz-plane-at-y-a-3-a-4a-respectively-the-electric-field-at-the

slug:
three-infinite-plane-sheets-carrying-uniform-charge-densities-2-4-are-placed-parallel-to-xz-plane-at-y-a-3-a-4a-respectively-the-electric-field-at-the-28504

question:
<div class="question">Three infinite plane sheets carrying uniform charge densities $-\sigma, 2 \sigma, 4 \sigma$ are placed parallel to $X Z$ plane at $Y=a, 3 a$, 4a respectively. The electric field at the point $(0,2 a, 0)$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{7 \sigma}{2 \varepsilon_{0}} \hat{j}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/I-3gxMkSPk3XfRMJPTzROU218KdXkLg1ZTpxuCoVhrA.original.fullsize.png"/><br/> <br/>$\begin{aligned} \mathrm{E}_{\text {net }} &amp;=7 \mathrm{E} \\ &amp;=7 \frac{\sigma}{2 \varepsilon_{0}} \text { alone }-\text { ve } \mathrm{Y} \text {-axis } \\ &amp; \therefore \overrightarrow{\mathrm{E}}=-\frac{7 \sigma}{2 \varepsilon_{0}} \hat{\mathrm{j}} \end{aligned}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
three-lines-are-drawn-from-the-origin-o-with-direction-cosines-proportional-to-111-230-and-1-0-3-the-three-lines-are

slug:
three-lines-are-drawn-from-the-origin-o-with-direction-cosines-proportional-to-111-230-and-1-0-3-the-three-lines-are-26897

question:
<div class="question">Three lines are drawn from the origin $O$ with direction cosines proportional to (1,-1,1) (2,-3,0) and $(1,0,3) .$ The three lines are</div>

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

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

answer:
<div class="solution">$\quad$ $\Delta=\left|\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; -3 &amp; 0 \\ 1 &amp; 0 &amp; 3\end{array}\right|$
<br/>$=1(-9-0)-(-1) \cdot(6-0)+1(0-(-3))$
<br/>$=-9+6+3=0$
<br/>since, $\Delta=0$
<br/>$\therefore$ Lines are coplanar.</div>


title: three-numbers-are-chosen-at-random-from-1-to-20-the-probability-that-they-are-consecutive-is
slug: three-numbers-are-chosen-at-random-from-1-to-20-the-probability-that-they-are-consecutive-is-72051
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">Three numbers are chosen at random from 1 to 20 . The probability that they are consecutive is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{190}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{120}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3}{190}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5}{190}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{190}$</span> </div>

answer:
<div class="solution">Hints: Total number of cases ; ${ }^{20} \mathrm{C}_3=\frac{20 \times 19 \times 18}{2 \times 3}=20 \times 19 \times 3=1140$<br/>Total number of favourable cases $=18$<br/>$\therefore$ Required probability $=\frac{18}{1140}=\frac{3}{190}$</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
three-particles-each-of-mass-m-grams-situated-at-the-vertices-of-an-equilateral-a-bc-of-side-a-cm-as-shown-in-the-figure-the-moment-of-inertia-of-the--1

slug:
three-particles-each-of-mass-m-grams-situated-at-the-vertices-of-an-equilateral-a-bc-of-side-a-cm-as-shown-in-the-figure-the-moment-of-inertia-of-the-1-73558

question:
<div class="question">Three particles, each of mass ' $m$ ' grams situated at the vertices of an equilateral $\triangle A B C$ of side ' $a$ ' $\mathrm{cm}$ (as shown in the figure). The moment of inertia of the system about a line $\mathrm{AX}$ perpendicular to $\mathrm{AB}$ and in the plane of $\mathrm{ABC}$ in $\mathrm{g}-\mathrm{cm}^2$ units will be<br/> <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/S9zLAXTSN2hHR9l1fM2m1M1YMAwZ4W-vXsH7YOpxsmo.original.fullsize.png"/></div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GwLG36TkLPUkUWkYSuhZnrmH2VnSgOGUFPGGvgCpPZk.original.fullsize.png"/><br/><br/>$\mathrm{I}=\mathrm{~ma}^2+\mathrm{m}\left(\mathrm{a} \cos 60^{\circ}\right)^2=m \mathrm{a}^2+\frac{\mathrm{ma}^2}{4}=\frac{5 \mathrm{ma}^2}{4}$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
three-particles-each-of-mass-m-grams-situated-at-the-vertices-of-an-equilateral-a-bc-of-side-a-cm-as-shown-in-the-figure-the-moment-of-inertia-of-the-

slug:
three-particles-each-of-mass-m-grams-situated-at-the-vertices-of-an-equilateral-a-bc-of-side-a-cm-as-shown-in-the-figure-the-moment-of-inertia-of-the-73298

question:
<div class="question">Three particles, each of mass ' $m$ ' grams situated at the vertices of an equilateral $\triangle A B C$ of side ' $a$ ' $\mathrm{cm}$ (as shown in the figure). The moment of inertia of the system about a line $\mathrm{AX}$ perpendicular to $\mathrm{AB}$ and in the plane of $\mathrm{ABC}$ in $\mathrm{g}-\mathrm{cm}^2$ units will be<br/> <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/S9zLAXTSN2hHR9l1fM2m1M1YMAwZ4W-vXsH7YOpxsmo.original.fullsize.png"/></div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GwLG36TkLPUkUWkYSuhZnrmH2VnSgOGUFPGGvgCpPZk.original.fullsize.png"/><br/><br/>$\mathrm{I}=\mathrm{~ma}^2+\mathrm{m}\left(\mathrm{a} \cos 60^{\circ}\right)^2=m \mathrm{a}^2+\frac{\mathrm{ma}^2}{4}=\frac{5 \mathrm{ma}^2}{4}$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
three-unequal-positive-numbers-a-b-c-are-such-that-a-b-c-are-in-gp-while-lo-g-2-a-5-c-lo-g-5-c-7-b-lo-g-7-b-2-a-are-in-ap-then-a-b-c-are-the-lengths-o

slug:
three-unequal-positive-numbers-a-b-c-are-such-that-a-b-c-are-in-gp-while-lo-g-2-a-5-c-lo-g-5-c-7-b-lo-g-7-b-2-a-are-in-ap-then-a-b-c-are-the-lengths-o-98289

question:
<div class="question">Three unequal positive numbers a, b, c are such that a, b, c are in G.P. while $\log \left(\frac{5 c}{2 a}\right), \log \left(\frac{7 b}{5 c}\right), \log \left(\frac{2 a}{7 b}\right)$ are in A.P. Then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the lengths of the sides of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">an isosceles triangle</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">an equilateral triangle</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">a scalene 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">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">a scalene triangle</span> </div>

answer:
<div class="solution">$\log \frac{5 c}{2 a}+\log \frac{2 a}{7 b}=2 \log \frac{7 b}{5 c} \Rightarrow \log \frac{5 c}{7 b}=\log \frac{49 b^{2}}{25 c^{2}}$
<br/>$\begin{array}{l}
<br/>\Rightarrow 5^{3} \mathrm{c}^{3}=7 \mathrm{~b}^{3} \Rightarrow 5 \mathrm{c}=7 \mathrm{~b} \Rightarrow \mathrm{c}=\frac{7}{5} \mathrm{~b} \\
<br/>\because \mathrm{b}^{2}=\mathrm{ac}=\mathrm{a} \cdot \frac{7}{5} \mathrm{~b} \Rightarrow \mathrm{a}=\frac{5 \mathrm{~b}}{7}
<br/>\end{array}$
<br/>Sides are $\frac{5 b}{7}, b, \frac{7}{5} b$</div>


tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
three-vectors-a-a-i-j-k-b-i-b-j-k-and-c-i-j-c-k-are-mutually-perpendicular-i-j-and-k-are-unit-vectors-along-x-y-and-z-axes-respectively-the-respective

slug:
three-vectors-a-a-i-j-k-b-i-b-j-k-and-c-i-j-c-k-are-mutually-perpendicular-i-j-and-k-are-unit-vectors-along-x-y-and-z-axes-respectively-the-respective-37533

question:
<div class="question">Three vectors $\mathbf{A}=a \mathbf{i}+\mathbf{j}+\mathbf{k} ; \mathbf{B}=\mathbf{i}+b \mathbf{j}+\mathbf{k}$
<br/>and $\mathbf{C}=\mathbf{i}+\mathbf{j}+c \mathbf{k}$ are mutually perpendicular (i, $\mathbf{j}$ and $\mathbf{k}$ are unit vectors along $X, Y$ and $Z$ axes respectively). The respective values of $a$, $b$ and $c$ are</div>

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

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

answer:
<div class="solution">Three vectors are mutually perpendicular, so the scalar or dot product of two vectors will zero. So,<br/>$\mathbf{A} \cdot \mathbf{B}=0, \mathbf{B} \mathbf{C}=0,$<br/>So, $\mathbf{A} \cdot \mathbf{B}=(a \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \cdot(\hat{\mathbf{i}}+b \hat{\mathbf{j}}+\hat{\mathbf{k}})=0$<br/>$\Rightarrow a+b+1=0$<br/>$\mathbf{B} \cdot \mathbf{C}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}} \cdot \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}=$<br/>$\Rightarrow 1+b+c=0$<br/>$\mathbf{A} \cdot \mathbf{C}=(a \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+c \hat{\mathbf{k}})=0$<br/>$\mathbf{A} \cdot \mathbf{C}=0$<br/>$\Rightarrow \quad a+1+c=0$<br/>(iii Addition of Eqs. (i), (ii) and (iii),<br/>$\begin{array}{l}<br/>2(a+b+c)+3=0 \\<br/>a+b+c=-\frac{3}{2}<br/>\end{array}$<br/>$\Rightarrow \quad-1+c=-\frac{3}{2}$<br/>$\begin{aligned}<br/>c &amp;=-\frac{3}{2}+1 \\<br/>&amp;=-\frac{1}{2}=0.5 \\<br/>a &amp;=-\frac{1}{2} \\<br/>b &amp;=-\frac{1}{2} \\<br/>c &amp;=-\frac{1}{2}<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1 \%$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
time-period-t-of-a-simple-pendulum-of-length-l-is-given-by-t-2-g-l-if-the-length-is-increased-by-2-then-an-approximate-change-in-the-time-period-is

slug:
time-period-t-of-a-simple-pendulum-of-length-l-is-given-by-t-2-g-l-if-the-length-is-increased-by-2-then-an-approximate-change-in-the-time-period-is-50530

question:
<div class="question">Time period $T$ of a simple pendulum of length $l$ is given by $T=2 \pi \sqrt{\frac{l}{g}}$. If the length is increased by $2 \%$, then an approximate change in the time period is</div>

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

answer:
<div class="solution">Given, $T=2 \pi \sqrt{\frac{1}{g}}$
<br/>On differentiating w.r.t. $l,$ we get
<br/>$\begin{aligned} \frac{d T}{d l} &amp;=\frac{2 \pi}{\sqrt{g}} \cdot \frac{1}{2 \sqrt{I}} \\ \Delta T &amp;=\frac{d T}{d l} \Delta l=\frac{\pi}{\sqrt{g} l} \cdot\left(\frac{2 I}{100}\right) \\ &amp;=2 \pi \sqrt{\frac{1}{g}} \cdot \frac{1}{100}=\frac{T}{100} \\ \frac{\Delta T}{T} &amp;=\frac{1}{100}=1 \% \end{aligned}$
<br/>Hence, approximate change in the time period is $1 \%$.</div>


tags: ['Chemistry', 'Practical Chemistry', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
title:
to-a-solution-of-a-colourless-efflorescent-sodium-salt-when-dilute-acid-is-added-a-colourless-gas-is-evolved-along-with-formation-of-a-white-precipita

slug:
to-a-solution-of-a-colourless-efflorescent-sodium-salt-when-dilute-acid-is-added-a-colourless-gas-is-evolved-along-with-formation-of-a-white-precipita-60085

question:
<div class="question">To a solution of a colourless efflorescent sodium salt, when dilute acid is added, a colourless gas is evolved along with formation of a white precipitate. Acidified dichromate solution turns green when the colourless gas is passed through it. The sodium salt is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\underset{\text { (colouriess, efflorescent) }}{\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}(\mathrm{~s})}+2 \mathrm{H}^{+}(\mathrm{dil} .) \rightarrow \mathrm{S}(\mathrm{s})+2 \mathrm{Na}^{+}(\mathrm{aq})+\mathrm{SO}_{2} \uparrow+\mathrm{H}_{2} \mathrm{O}(\ell)$
<br/>$\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}($ aq. $)+3 \mathrm{SO}_{2}(\mathrm{~g})+2 \mathrm{H}^{+}(\mathrm{aq} .) \longrightarrow 2 \mathrm{~K}^{+}($ aq. $)+\underset{(\text { green })}{2 \mathrm{Cr}^{+3}}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)+3 \mathrm{SO}_{4}^{-2}(\mathrm{aq})$</div>


tags: ['Chemistry', 'Practical Chemistry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
to-a-solution-of-colourless-sodium-salt-a-solution-of-lead-nitrate-was-added-to-have-a-white-precipitate-which-dissolves-in-warm-water-and-reprecipita

slug:
to-a-solution-of-colourless-sodium-salt-a-solution-of-lead-nitrate-was-added-to-have-a-white-precipitate-which-dissolves-in-warm-water-and-reprecipita-37771

question:
<div class="question">To a solution of colourless sodium salt, a solution of lead nitrate was added to have a white precipitate which dissolves in warm water and reprecipitates on cooling. Which of the following acid radical is present in the salt?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{Cl}^{-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{SO}_4{ }^{2-}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{S}^{2-}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{NO}_3^{-}$</span> </li> </ul>

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

answer:
<div class="solution">$2 \mathrm{NaCl}+\mathrm{Pb}\left(\mathrm{NO}_3\right)_2 \rightarrow \mathrm{PbCl}_2(\mathrm{ppt})+2 \mathrm{NaNO}_3$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
to-determine-the-coefficient-of-friction-between-a-rough-surface-and-a-block-the-surface-is-kept-inclined-at-4-5-and-the-block-is-released-from-rest-t

slug:
to-determine-the-coefficient-of-friction-between-a-rough-surface-and-a-block-the-surface-is-kept-inclined-at-4-5-and-the-block-is-released-from-rest-t-42358

question:
<div class="question">To determine the coefficient of friction between a rough surface and a block, the surface is kept inclined at $45^{\circ}$ and the block is released from rest. The block takes a time $t$ in moving a distance $d$. The rough surface is then replaced by a smooth surface and the same experiment is repeated. The block now takes a time $t / 2$ in moving down the same distance d. The coefficient of friction is</div>

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

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

answer:
<div class="solution">If the same wedge is made rough then time taken by it to come down becomes $n$ times more. The coefficient of friction <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a17.png"/> <br/>$$
<br/>\begin{array}{l}
<br/>\mu=\left[1-\frac{1}{n^{2}}\right] \tan \theta \Rightarrow \mu=\left[1-\frac{1}{2^{2}}\right] \tan 45^{\circ} \\
<br/>\mu=\frac{4-1}{4} \Rightarrow \mu=\frac{3}{4}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
to-determine-the-composition-of-a-bimetallic-alloy-a-sample-is-first-weighed-in-air-and-then-in-water-these-weights-are-found-to-be-w-1-and-w-2-respec

slug:
to-determine-the-composition-of-a-bimetallic-alloy-a-sample-is-first-weighed-in-air-and-then-in-water-these-weights-are-found-to-be-w-1-and-w-2-respec-98453

question:
<div class="question">To determine the composition of a bimetallic alloy, a sample is first weighed in air and then in water. These weights are found to be $w_{1}$ and $w_{2}$ respectively. If the densities of the two constituents metals are $\rho_{1}$ and $\rho_{2}$ respectively. then the weight of the first metal in the sample is (where $\rho_{w}$ is the density of water)</div>

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

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

answer:
<div class="solution">By Archimedes' Principle
<br/>$$
<br/>F=v \rho_{w} g \Rightarrow\left(w_{1}-w_{2}\right) g=v \rho_{w} g
<br/>$$
<br/>Let, the total volume be $v$ and first metal weight be $x$
<br/>$$
<br/>\begin{array}{l}
<br/>w_{1}-w_{2}=\left(v_{1}+v_{2}\right) \rho_{w} \\
<br/>w_{1}-w_{2}=v_{1} \rho_{w}+v_{2} g_{w} \\
<br/>w_{1}-w_{2}=\left(\frac{x}{\rho_{1}} \rho_{w}+\frac{w_{1}-x}{\rho_{2}} \rho_{w}\right)
<br/>w_{1}-w_{2}=\frac{x \rho_{\mathcal{P}} \rho_{w}+\left(w_{1}-x\right) \rho_{w} \rho_{1}}{\rho_{\jmath} \rho_{2}}
<br/>\end{array} \quad\left(\because v=\frac{m}{\rho}\right)
<br/>$$
<br/>$$
<br/>\begin{aligned} w_{1} \rho_{1} \rho_{2}-w_{2} \rho_{1} \rho_{2} &amp;=x \rho_{2} \rho_{w}+w_{1} \rho_{w} \rho_{1}-x \rho_{w} \rho_{1} \\ x\left(\rho_{2}-\rho_{1}\right) \rho_{w} &amp;=\rho_{1}\left[w_{1}\left(\rho_{2}-\rho_{w}\right)-w_{2} \rho_{2}\right] \end{aligned}\\
<br/>x=\frac{\rho_{1}}{\rho_{w}\left(\rho_{2}-\rho_{1}\right)}\left[w_{1}\left(\rho_{2}-\rho_{w}\right)-w_{z} \rho_{2}\right]
<br/>$$</div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
to-observe-an-elevation-of-boiling-point-of-00-5-c-the-amount-of-a-solute-mol-wt-100-to-be-added-to-100-g-of-water-k-b-05-is

slug:
to-observe-an-elevation-of-boiling-point-of-00-5-c-the-amount-of-a-solute-mol-wt-100-to-be-added-to-100-g-of-water-k-b-05-is-62275

question:
<div class="question">To observe an elevation of boiling point of $0.05^{\circ} \mathrm{C},$ the amount of a solute (mol. wt. $=100$ ) to be added to $100 \mathrm{g}$ of water $\left(K_{b}=0.5\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{g}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.5 \mathrm{g}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1 \mathrm{g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.75 \mathrm{g}$</span> </li> </ul>

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

answer:
<div class="solution">Elevation of boiling point,
<br/>$$
<br/>\Delta T_{b}=\frac{w \times K_{b} \times 1000}{M \times W}
<br/>$$
<br/>(Here, w and $W=$ weights of solute and solvent respectively.
<br/>$M=$ molecular weight of solute and $K_{b}=$ constant
<br/>On substituting values, we get
<br/>of
<br/>$$
<br/>\begin{aligned}
<br/>0.05 &amp;=\frac{W \times 0.5 \times 1000}{100 \times 100} \\
<br/>w &amp;=\frac{0.05 \times 100 \times 100}{0.5 \times 1000}=1 \mathrm{g}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
to-which-of-the-following-quantities-the-radius-of-the-circular-path-of-a-charged-particle-moving-at-right-angles-to-a-uniform-magnetic-field-is-direc

slug:
to-which-of-the-following-quantities-the-radius-of-the-circular-path-of-a-charged-particle-moving-at-right-angles-to-a-uniform-magnetic-field-is-direc-37651

question:
<div class="question">To which of the following quantities, the radius of the circular path of a charged particle moving at right angles to a uniform magnetic field is directly proportional?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">momentum of the particle</span> </div>

answer:
<div class="solution">If velocity of particle, $\mathrm{v}$ is perpendicular to magnetic field, $\mathbf{B}$ i.e. $\theta=90^{\circ},$ then particle will experience maximum magnetic force, i.e. $F_{\max }=q v B .$ This force acts in a direction perpendicular to the motion of charged particle. Therefore the trajectory of the particle is a circle. In this case path of charged particle is circular and magnetic force provides the necessary centripetal force,
<br/>i.e.
<br/>$$
<br/>q v B=\frac{m v^{2}}{r}
<br/>$$
<br/>$\Rightarrow$ Radius of path, $r=\frac{m v}{q B}$
<br/>$$
<br/>r=\frac{p}{q B} \quad[\because p=m v]
<br/>$$
<br/>where, $p=$ momentum of the particle
<br/>$\therefore r$ momentum
<br/>Hence, option (d) is correct</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
to-which-of-the-following-the-angular-velocity-of-the-electron-in-the-n-th-bohr-orbit-is-proportional

slug:
to-which-of-the-following-the-angular-velocity-of-the-electron-in-the-n-th-bohr-orbit-is-proportional-98112

question:
<div class="question">To which of the following the angular velocity of the electron in the $n$ -th Bohr orbit is proportional?</div>

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

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

answer:
<div class="solution">According to the Bohr's atomic model.<br/>$\text {Angular momentum, } L=m v=\frac{n h}{2 \pi}....(i)$<br/>since, angular velocity $\omega=\frac{v}{r}$<br/>$\Rightarrow v=r \omega$<br/>From Eq. (i), we get<br/>$\begin{aligned}<br/>m(r \omega) r &amp;=\frac{n h}{2 \pi} \\<br/>m \omega r^{2} &amp;=\frac{n h}{2 \pi} \\<br/>\omega &amp;=\frac{n h}{2 \pi m r^{2}}....(ii)<br/>\end{aligned}$<br/>since, the radius of the electron in $n^{t h}$ orbit of<br/>Bohr's atomic model is given as, $r=\frac{n^{2} h^{2} \varepsilon_{0}}{\pi m z e^{2}} \ldots$ (iii)<br/>Squaring the Eq. (iii) and substituting its value in Eq. (ii), we get<br/>$\omega=\frac{n h(\pi n z e)^{2}}{2 \pi m\left(n^{4} h^{4} \varepsilon_{0}^{2}\right)} \Rightarrow \omega \propto \frac{1}{n^{3}}$</div>


title: to-write-the-decimal-number-37-in-binary-how-many-binary-digits-are-required
slug: to-write-the-decimal-number-37-in-binary-how-many-binary-digits-are-required-88414
tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$6$</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">To write the decimal number 37 in binary, how many binary digits are required?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$6$</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">C</span> <span class="option-data">$7$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4$</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6webiQL9ra45EgmWbDIKJrNVnE0Ol6anpfjR9BSl9WI.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
transforming-to-parallel-axes-through-a-point-p-q-the-equation-2-x-2-3-x-y-4-y-2-x-18-y-25-0-becomes-2-x-2-3-x-y-4-y-2-1-then

slug:
transforming-to-parallel-axes-through-a-point-p-q-the-equation-2-x-2-3-x-y-4-y-2-x-18-y-25-0-becomes-2-x-2-3-x-y-4-y-2-1-then-11664

question:
<div class="question">Transforming to parallel axes through a point $(p, q), \quad$ the $\quad$ equation $2 x^{2}+3 x y+4 y^{2}+x+18 y+25=0$ becomes
<br/>$2 x^{2}+3 x y+4 y^{2}=1 .$ Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p=-2, q=3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$p=2, q=-3$</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">$p=3, q=-4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$p=-4, q=3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p=2, q=-3$</span> </div>

answer:
<div class="solution">Given equations are
<br/>$$
<br/>\begin{array}{r}
<br/>2 x^{2}+3 x y+4 y^{2}+x+18 y+25=0 ...(i)\\
<br/>2 x^{2}+3 x y+4 y^{2}+1=0 ....(ii)
<br/>\end{array}
<br/>$$
<br/>Let the origin be transferred to $(p, q)$ axes being parallel to the previous axes; then the equation
<br/>(i) becomes.
<br/>$$
<br/>\begin{array}{l}
<br/>2\left(x^{\prime}+p\right)^{2}+3\left(x^{\prime}+p\right)\left(y^{\prime}+q\right) \\
<br/>+4\left(y^{\prime}+q\right)^{2}+\left(x^{\prime}+p\right)+18\left(y^{\prime}+q\right)+25=0 \\
<br/>\Rightarrow 2 x^{2}+2 p^{2}+4 x^{\prime} p+3 x^{\prime} y^{\prime}+3 x^{\prime} q+3 p y^{\prime} \\
<br/>+3 p q+4 y^{2}+4 q^{2}+8 y^{\prime} q+x^{\prime}+p \\
<br/>+18 y^{\prime}+18 q+25=0 \\
<br/>\Rightarrow 2 x^{2}+4 y^{a}+3 x^{\prime} y^{\prime}+(4 p+3 q+1) x^{\prime} \\
<br/>1+(3 p+8 q+18) y^{\prime}+2 p^{2}+3 p q \\
<br/>+4 q^{2}+p+25=0
<br/>\end{array}
<br/>$$
<br/>From Eq. (ii) coefficient of $x^{\prime}$ and $y^{\prime}$ must be
<br/>zero.
<br/>$\therefore$
<br/>$$
<br/>4 p+3 q+1=0 ...(iii)
<br/>$$
<br/>$$
<br/>3 p+8 q+18=0 ...(iv)
<br/>$$
<br/>By solving Eqs. (iii) and (iv), we get
<br/>$$
<br/>p=2, q=-3
<br/>$$</div>


title: treatment-of-with-nanh-2-liq-nh-3-gives-1
slug: treatment-of-with-nanh-2-liq-nh-3-gives-1-58300
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">Treatment of <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q54.png"/> $\text { with } \mathrm{NaNH}_{2} \text { /liq. } \mathrm{NH}_{3}$ gives</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/2013_che_q54_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q54_optb.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q54_optc.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q54_optd.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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q54_optd.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a54.png"/></div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
twenty-metres-of-wire-is-available-to-fence-off-a-flower-bed-in-the-form-of-a-circular-sector-what-must-the-radius-of-the-circle-be-if-the-area-of-the

slug:
twenty-metres-of-wire-is-available-to-fence-off-a-flower-bed-in-the-form-of-a-circular-sector-what-must-the-radius-of-the-circle-be-if-the-area-of-the-79119

question:
<div class="question">Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed be greatest?</div>

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

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

answer:
<div class="solution">Let, radius $=r$, arc length $=\ell$<br/>$2 \mathrm{r}+\ell=20 \Rightarrow \ell=20-2 \mathrm{r}$<br/>$\Rightarrow \mathrm{A}=\frac{1}{2}\mathrm{r}^{2} \theta = \frac{1}{2} \ell \mathrm{r} \Rightarrow \frac{\mathrm{dA}}{\mathrm{dr}}=\frac{1}{2}(20-4 \mathrm{r})=0 \Rightarrow \mathrm{r}=5 \quad \therefore \mathrm{r}=5 \mathrm{~m}\right.$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/JRbJrKnFUlvZADQFyXNXNZKKvSVzhvIuDZAe_uq25ls.original.fullsize.png"/></div>


tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
two-aromatic-compounds-having-formula-mathrmc7-mathrmh8-mathrmo-which-are-easily-identifiable-by-mathrmfecl3-solution-test-violet-colouration-are

slug:
two-aromatic-compounds-having-formula-mathrmc7-mathrmh8-mathrmo-which-are-easily-identifiable-by-mathrmfecl3-solution-test-violet-colouration-are-19580

question:
<div class="question">Two aromatic compounds having formula \(\mathrm{C}_7 \mathrm{H}_8 \mathrm{O}\) which are easily identifiable by \(\mathrm{FeCl}_3\) solution test (violet colouration) are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">o- cresol and benzyl alcohol</span> </div>

answer:
<div class="solution">Hints: \(\mathrm{O}\) - cresol contains phenolic group, thus it gives violet coloration with \(\mathrm{FeCl}_3\) where as benzylalchol donot contains phenolic group, hence no coloration with \(\mathrm{FeCl}_3\). Hence Identifiable</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
two-base-balls-masses-m1100-mathrmg-and-m250-mathrmg-are-thrown-both-of-them-move-with-uniform-velocity-but-the-velocity-of-m2-is-15-times-that-of-m1-

slug:
two-base-balls-masses-m1100-mathrmg-and-m250-mathrmg-are-thrown-both-of-them-move-with-uniform-velocity-but-the-velocity-of-m2-is-15-times-that-of-m1-47144

question:
<div class="question">Two base balls (masses: \(m_1=100 \mathrm{~g}\), and \(m_2=50 \mathrm{~g}\) ) are thrown. Both of them move with uniform velocity, but the velocity of \(m_2\) is 1.5 times that of \(m_1\). The ratio of de Broglie wavelengths \(\lambda\left(m_1\right): \lambda\left(m_2\right)\) is given by</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(3: 4\)</span> </div>

answer:
<div class="solution">Hint : \(\frac{\lambda_1}{\lambda_2}=\frac{m_2 \mathrm{~V}_2}{m_1 \mathrm{~V}_1}=\frac{50 \times 1.5 \mathrm{~V}_1}{100 \times \mathrm{V}_1}=\frac{1.5}{2}=\frac{3}{4}\)</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
two-black-bodies-a-and-b-have-equal-surface-areas-and-are-maintained-at-temperatures-2-7-c-and-17-7-c-respectively-what-will-be-the-ratio-of-the-therm

slug:
two-black-bodies-a-and-b-have-equal-surface-areas-and-are-maintained-at-temperatures-2-7-c-and-17-7-c-respectively-what-will-be-the-ratio-of-the-therm-27897

question:
<div class="question">Two black bodies $A$ and $B$ have equal surface areas and are maintained at temperatures $27^{\circ} \mathrm{C}$ and $177^{\circ} \mathrm{C}$ respectively. What will be the ratio of the thermal energy radiated per second by $A$ to that by $B ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4: 9$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2: 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$16: 81$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$27:177$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$16: 81$</span> </div>

answer:
<div class="solution">According to the question, Area of both bodies $A$ and $B=A$
<br/>Temperature of body $A=27^{\circ} \mathrm{C}=27+273 \mathrm{K}$
<br/>Temperature of body $B=177^{\circ} \mathrm{C}=177+273 \mathrm{K}$
<br/>Now, by Stefan-Boltzmann law, thermal energy radiated per second by a body
<br/>$$
<br/>Q=\sigma A T^{4}
<br/>$$
<br/>where, $A=$ Area
<br/>$T=$ temperature and $\quad \sigma=$ Stefan-Boltzmam's constant
<br/>So, the ratio of thermal energy radiated per second by $A$ to that by $B$ is
<br/>$$
<br/>\frac{Q_{1}}{Q_{2}}=\frac{\sigma A}{\sigma A}\left(\frac{T_{1}}{T_{2}}\right)^{4}
<br/>$$
<br/>Now, $\frac{Q_{1}}{Q_{2}}=\left(\frac{T_{1}}{T_{2}}\right)^{4}$
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{Q_{1}}{Q_{2}}=\left(\frac{273+27}{273+177}\right)^{4} \\
<br/>\frac{Q_{1}}{Q_{2}}=\left(\frac{300}{450}\right)^{4}
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\frac{Q_{1}}{Q_{2}}=\left(\frac{2}{3}\right)^{4}=\left(\frac{16}{81}\right)
<br/>$$
<br/>Ratio of thermal energy radiated per second
<br/>$$
<br/>Q_{1}: Q_{2}=16: 81
<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
two-black-bodies-at-temperatures-32-7-c-and-42-7-c-are-kept-in-an-evacuated-chamber-at-2-7-c-the-ratio-of-their-rates-of-loss-of-heat-are

slug:
two-black-bodies-at-temperatures-32-7-c-and-42-7-c-are-kept-in-an-evacuated-chamber-at-2-7-c-the-ratio-of-their-rates-of-loss-of-heat-are-17989

question:
<div class="question">Two black bodies at temperatures $327^{\circ} \mathrm{C}$ and $427^{\circ} \mathrm{C}$ are kept in an evacuated chamber at $27^{\circ} \mathrm{C}$. The ratio of their rates of loss of heat are :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{6}{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{6}{7}\right)^2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{6}{7}\right)^3$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{243}{464}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{243}{464}$</span> </div>

answer:
<div class="solution">Hints: Rate ofloss of heat $\propto\left(\mathrm{T}^4-\mathrm{T}_0^4\right)$<br/>$$<br/>\frac{\mathrm{E}_1}{\mathrm{E}_2}=\frac{\mathrm{T}_1^4-\mathrm{T}_0^4}{\mathrm{~T}_2^4-\mathrm{T}_0^4}=\frac{(600)^4-(300)^4}{(700)^4-(300)^4}=\frac{6^4-3^4}{7^4-3^4}<br/>$$<br/><br/>$$<br/>\frac{E_1}{E_2}=\frac{243}{464}<br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
two-blocks-of-2-kg-and-1-kg-are-in-contact-on-a-frictionless-table-if-a-force-of-3-n-is-applied-on-2-kg-block-then-the-force-of-contact-between-the-tw-1

slug:
two-blocks-of-2-kg-and-1-kg-are-in-contact-on-a-frictionless-table-if-a-force-of-3-n-is-applied-on-2-kg-block-then-the-force-of-contact-between-the-tw-1-40309

question:
<div class="question">Two blocks of $2 \mathrm{~kg}$ and $1 \mathrm{~kg}$ are in contact on a frictionless table. If a force of $3 \mathrm{~N}$ is applied on $2 \mathrm{~kg}$ block, then the force of contact between the two blocks will be :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Ua9vbh_hS921PTPp-RPqXh_u3jBcYsg9YCzqfVyij14.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">Hints: Common acceleration $=\frac{3}{3}=1 \mathrm{~m} / \mathrm{sec}^2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/VNczpMjsBPK9g6tl-vRGyHIFEEZWSF6wWjizp_uYUls.original.fullsize.png"/><br/><br/>$$<br/>\mathrm{N}_1=1 \mathrm{~N}<br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
two-blocks-of-2-kg-and-1-kg-are-in-contact-on-a-frictionless-table-if-a-force-of-3-n-is-applied-on-2-kg-block-then-the-force-of-contact-between-the-tw

slug:
two-blocks-of-2-kg-and-1-kg-are-in-contact-on-a-frictionless-table-if-a-force-of-3-n-is-applied-on-2-kg-block-then-the-force-of-contact-between-the-tw-88516

question:
<div class="question">Two blocks of $2 \mathrm{~kg}$ and $1 \mathrm{~kg}$ are in contact on a frictionless table. If a force of $3 \mathrm{~N}$ is applied on $2 \mathrm{~kg}$ block, then the force of contact between the two blocks will be :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Ua9vbh_hS921PTPp-RPqXh_u3jBcYsg9YCzqfVyij14.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">Hints: Common acceleration $=\frac{3}{3}=1 \mathrm{~m} / \mathrm{sec}^2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/VNczpMjsBPK9g6tl-vRGyHIFEEZWSF6wWjizp_uYUls.original.fullsize.png"/><br/><br/>$$<br/>\mathrm{N}_1=1 \mathrm{~N}<br/>$$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
two-bodies-of-masses-m-1-and-m-2-are-separated-by-a-distance-r-the-distance-of-the-centre-of-mass-of-the-bodies-from-the-mass-m-1-is-1

slug:
two-bodies-of-masses-m-1-and-m-2-are-separated-by-a-distance-r-the-distance-of-the-centre-of-mass-of-the-bodies-from-the-mass-m-1-is-1-92861

question:
<div class="question">Two bodies of masses $m_{1}$ and $m_{2}$ are separated by a distance $R$. The distance of the centre of mass of the bodies from the mass $m_{1}$ is</div>

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

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

answer:
<div class="solution">The bodies are separated by a distance $R$. so the coordinates of $m_{1}$ and $m_{2}$ will be (0,0) and $(R, 0)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a13.png"/> <br/> <br/>From the formula of centre of mass,
<br/>$$
<br/>\begin{aligned}
<br/>X_{\mathrm{cm}} &amp;=\frac{m_{1} x_{1}+m_{2} x_{2}}{m_{1}+m_{2}} \\
<br/>&amp;=\frac{m_{1} \times 0+m_{2} \times R}{m_{1}+m_{2}} \\
<br/>X_{\mathrm{cm}} &amp;=\frac{m_{2} \cdot R}{m_{1}+m_{2}}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
two-bodies-of-masses-m-1-and-m-2-are-separated-by-a-distance-r-the-distance-of-the-centre-of-mass-of-the-bodies-from-the-mass-m-1-is

slug:
two-bodies-of-masses-m-1-and-m-2-are-separated-by-a-distance-r-the-distance-of-the-centre-of-mass-of-the-bodies-from-the-mass-m-1-is-30320

question:
<div class="question">Two bodies of masses $m_{1}$ and $m_{2}$ are separated by a distance $R$. The distance of the centre of mass of the bodies from the mass $m_{1}$ is</div>

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

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

answer:
<div class="solution">The bodies are separated by a distance $R$. so the coordinates of $m_{1}$ and $m_{2}$ will be (0,0) and $(R, 0)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a13.png"/> <br/> <br/>From the formula of centre of mass,
<br/>$$
<br/>\begin{aligned}
<br/>X_{\mathrm{cm}} &amp;=\frac{m_{1} x_{1}+m_{2} x_{2}}{m_{1}+m_{2}} \\
<br/>&amp;=\frac{m_{1} \times 0+m_{2} \times R}{m_{1}+m_{2}} \\
<br/>X_{\mathrm{cm}} &amp;=\frac{m_{2} \cdot R}{m_{1}+m_{2}}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
two-cells-a-and-b-of-emf-2-v-and-15-v-respectively-are-connected-as-shown-in-figure-through-an-external-resistance-10-the-internal-resistance-of-each-

slug:
two-cells-a-and-b-of-emf-2-v-and-15-v-respectively-are-connected-as-shown-in-figure-through-an-external-resistance-10-the-internal-resistance-of-each-49561

question:
<div class="question">Two cells $A$ and $B$ of emf $2 \mathrm{V}$ and $1.5 \mathrm{V}$ respectively, are connected as shown in figure through an external resistance $10 \Omega$ The internal resistance of each cell is $5 \Omega$. The potential difference $E_{A}$ and $E_{B}$ across the terminals of the cells $A$ and $B$ respectively are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q35.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$E_{A}=20 \mathrm{V}, E_{B}=1.5 \mathrm{V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$E_{A}=2.125 \mathrm{V}, E_{B}=1.375 \mathrm{V}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$E_{A}=1.875 \mathrm{V}, E_{B}=1.625 \mathrm{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$E_{A}=1.875 \mathrm{V}, E_{B}=1.375 \mathrm{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$E_{A}=1.875 \mathrm{V}, E_{B}=1.625 \mathrm{V}$</span> </div>

answer:
<div class="solution">The figure can be redrawn as, 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a35.png"/> <br/> <br/>The current through the circuit
<br/>$$
<br/>\begin{aligned}
<br/>i &amp;=\frac{\text { net emf }}{\text { effective resistance }} \\
<br/>&amp;=\frac{2-1.5}{5+5+10}=\frac{0.5}{20} \\
<br/>&amp;=\frac{1}{40}=0.025 \mathrm{A}
<br/>\end{aligned}
<br/>$$
<br/>The terminal potential difference of the batteries
<br/>and
<br/>$$
<br/>\begin{aligned}
<br/>V_{A} &amp;=\varepsilon_{A}-i I_{A}=2-0.025 \times 5 \\
<br/>&amp;=2-0.0125=1.875 \mathrm{V} \\
<br/>V_{B} &amp;=\varepsilon_{B}+i I_{B}
<br/>\end{aligned}
<br/>$$
<br/>$$
<br/>=1.5+0.025 \times 5
<br/>$$
<br/>$$
<br/>=1,5+0.0125=1.625 \mathrm{V}
<br/>$$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
two-cells-with-the-same-emf-e-and-different-internal-resistances-r1-and-r2-are-connected-in-series-to-an-external-resistance-r-the-value-of-r-so-that-

slug:
two-cells-with-the-same-emf-e-and-different-internal-resistances-r1-and-r2-are-connected-in-series-to-an-external-resistance-r-the-value-of-r-so-that-43442

question:
<div class="question">Two cells with the same e.m.f. \(E\) and different internal resistances \(r_1\) and \(r_2\) are connected in series to an external resistance \(R\). The value of \(R\) so that the potential difference across the first cell be zero is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\sqrt{r_1 r_2}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(r_1+r_2\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\mathrm{r}_1-\mathrm{r}_2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{r_1+r_2}{2}\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/TodE7DDcwNUogLWejDavnergZeS6AhGJo9h8BFrGFz0.original.fullsize.png"/><br/><br/>\(I=\frac{2 E}{R+r_1+r_2}\)<br/>Potential difference across first cell \(\mathrm{V}=\mathrm{E}-\mathrm{Ir}_1=0\)<br/>\(\begin{aligned}<br/>&amp; E-\frac{2 E r_1}{R+r_1+r_2}=0 \\<br/>&amp; {\left[\frac{R+r_1+r_2-2 r_1}{R+r_1+r_2}\right]=0} \\<br/>&amp; \Rightarrow R+r_2-r_1=0 \\<br/>&amp; R=r_1-r_2<br/>\end{aligned}\)</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-y</span> </div>
source_db: db\MarksBatch2_P3.db
title:
two-charges-each-equal-to-q-are-kept-at-a-0-and-a-0-a-charge-q-is-placed-at-the-origin-if-q-is-given-a-small-displacement-along-y-direction-the-force-

slug:
two-charges-each-equal-to-q-are-kept-at-a-0-and-a-0-a-charge-q-is-placed-at-the-origin-if-q-is-given-a-small-displacement-along-y-direction-the-force-18716

question:
<div class="question">Two charges, each equal to $-q$ are kept at $(-a, 0)$ and $(a, 0)$. A charge $q$ is placed at the origin. If $q$ is given a small displacement along y direction, the force acting on q is proportional to</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/t7TQ3LqJc-JSEbuvMknpZS5icu7vTFH6H9NU7ErQnIo.original.fullsize.png"/><br/><br/>$\vec{F}=+q$ (Electric field due to the two negative charges $=\vec{E}$ )<br/>$$<br/>\vec{F}=+q\left[\frac{k(-2 q) \vec{y}}{a^3}\right] \Rightarrow \vec{F}=-\frac{2 k q^2 y}{a^3} \Rightarrow \vec{F} \propto-\vec{y}<br/>$$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
two-charges-q-and-q-are-placed-at-a-distance-a-in-a-uniform-electric-fleld-the-dipole-moment-of-the-combination-is-2-q-a-cos-i-sin-where-is-the-angle-

slug:
two-charges-q-and-q-are-placed-at-a-distance-a-in-a-uniform-electric-fleld-the-dipole-moment-of-the-combination-is-2-q-a-cos-i-sin-where-is-the-angle-60235

question:
<div class="question">Two charges $+q$ and $-q$ are placed at a distance $a$ in a uniform electric fleld. The dipole moment of the combination is $2 q a(\cos \theta i+\sin \theta)$. where, $\theta$ is the angle between the direction of the field and the line joining the two charges. Which of the following statement(s) is/ase correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">The net force on the dipole vanishes, The torque is independent of the choice of coordinates, The net force is independent of a</span> </div>

answer:
<div class="solution">The situation can be diagrammatically as,
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a36.png"/> <br/> <br/>The dipole moment,
<br/>$$
<br/>\mathbf{P}=2 q a(\cos \theta \hat{\mathbf{i}}+\sin \theta \hat{\mathbf{j}})
<br/>$$
<br/>The net force on the dipole is always zero while net torque on the dipole is not zero.</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
two-circle-s-1-p-x-2-p-y-2-2-g-x-2-f-y-d-0-and-s-2-x-2-y-2-2-gx-2-f-y-d-0-have-a-common-chord-pq-the-equation-of-pq-is

slug:
two-circle-s-1-p-x-2-p-y-2-2-g-x-2-f-y-d-0-and-s-2-x-2-y-2-2-gx-2-f-y-d-0-have-a-common-chord-pq-the-equation-of-pq-is-60905

question:
<div class="question">Two circle $S_1=p x^2+p y^2+2 g^{\prime} x+2 f^{\prime} y+d=0$ and $S_2=x^2+y^2+2 g x+2 f y+d^{\prime}=0$ have a common chord $P Q$. The equation of $\mathrm{PQ}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$S_1-S_2=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$S_1+S_2=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{S}_1-\mathrm{pS}_2=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{S}_1+\mathrm{pS}_2=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{S}_1-\mathrm{pS}_2=0$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{S}_1}{\mathrm{p}}-\mathrm{S}_2=0 \Rightarrow \mathrm{S}_1-\mathrm{pS}_2=0$</div>


tags: ['Physics', 'Wave Optics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
two-coherent-monochromatic-beams-of-intensities-i-and-4-i-respectively-are-superposed-the-maximum-and-minimum-intensitien-in-the-reaulting-pattern-are

slug:
two-coherent-monochromatic-beams-of-intensities-i-and-4-i-respectively-are-superposed-the-maximum-and-minimum-intensitien-in-the-reaulting-pattern-are-29727

question:
<div class="question">Two coherent monochromatic beams of intensities $I$ and $4 I$ respectively are superposed. The maximum and minimum intensitien in the reaulting pattern are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">9I and I</span> </div>

answer:
<div class="solution">We know that The maximum intensities
<br/>$$
<br/>I_{\max }=\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}
<br/>$$
<br/>The minimum intensities
<br/>$$
<br/>I_{\min }=\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}
<br/>$$
<br/>So, the ratio of the maximum and minimum of intensities is
<br/>$$
<br/>\begin{aligned}
<br/>\frac{I_{\max }}{I_{\min }} &amp;=\frac{\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}}{\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}} \\
<br/>\frac{I_{\max }}{I_{\min }} &amp;=\frac{(\sqrt{4 I}+\sqrt{I})^{2}}{(\sqrt{4 I}-\sqrt{I})^{2}} \\
<br/>&amp;=\left(\frac{3 \sqrt{I}}{\sqrt{I}}\right)^{2} \\
<br/>=&amp; \frac{9}{1}=9: 1
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
two-coils-of-selfinductances-6-mh-and-8-mh-are-connected-in-series-and-are-adjusted-for-highest-coefficient-of-coupling-equivalent-selfinductance-l-fo

slug:
two-coils-of-selfinductances-6-mh-and-8-mh-are-connected-in-series-and-are-adjusted-for-highest-coefficient-of-coupling-equivalent-selfinductance-l-fo-84851

question:
<div class="question">Two coils of self-inductances $6 \mathrm{mH}$ and $8 \mathrm{mH}$ are connected in series and are adjusted for highest coefficient of coupling. Equivalent self-inductance $L$ for the assembly is approximately</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$50 \mathrm{mH}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$36 \mathrm{mH}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$28 \mathrm{mH}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$18 \mathrm{mH}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$28 \mathrm{mH}$</span> </div>

answer:
<div class="solution">Given, $L_{1}=6 \mathrm{mH}=6 \times 10^{-3} \mathrm{H}$
<br/>$$
<br/>L_{2}=8 \mathrm{mH}=8 \times 10^{-3} \mathrm{H}
<br/>$$
<br/>If two coills $L_{1}$ and $L_{2}$ are connected in series, the equivalent self-inductance $L$ for assembly
<br/>$$
<br/>L=L_{1}+L_{2}+2 \sqrt{L_{1} L_{2}}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>=6 \times 10^{-3}+8 \times 10^{-3}+2 \sqrt{6 \times 8 \times 10^{-6}} \\
<br/>=14 \times 10^{-3}+2 \sqrt{48} \times 10^{-3} \\
<br/>=14 \times 10^{-3}+2 \times 7 \times 10^{-3} \\
<br/>=28 \times 10^{-3} \mathrm{H}=28 \mathrm{mH}
<br/>\end{array} \quad(\because \sqrt{48}=6.92=7)
<br/>$$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
two-coins-are-available-one-fair-and-the-other-twoheaded-choose-a-coin-and-toss-it-once-assume-that-the-unbiased-coin-is-chosen-with-probability-4-3-g

slug:
two-coins-are-available-one-fair-and-the-other-twoheaded-choose-a-coin-and-toss-it-once-assume-that-the-unbiased-coin-is-chosen-with-probability-4-3-g-50266

question:
<div class="question">Two coins are available, one fair and the other two-headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probability $\frac{3}{4}$, Given that the outcome is head, the probability that the two-headed coin was chosen, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{2}{5}$</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">$\frac{1}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{7}$</span> </li> </ul>

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

answer:
<div class="solution">Let $F$ denotes fair coin $T$ denotes two headed $H$ denotes head occurs
<br/>$$
<br/>\begin{array}{lc}
<br/>\therefore \quad P(F)=\frac{3}{4}, P(T)=1-\frac{3}{4}=\frac{1}{4} \\
<br/>\therefore \quad P\left(\frac{T}{H}\right)=\frac{P\left(\frac{H}{T}\right) \cdot P(T)}{P\left(\frac{H}{T}\right) \cdot P(T)+P\left(\frac{H}{F}\right) \cdot P(F)}
<br/>\end{array}
<br/>$$
<br/>By Baye's theorem
<br/>$$
<br/>=\frac{1 \cdot \frac{1}{4}}{1 \cdot \frac{1}{4}+\frac{1}{2} \cdot \frac{3}{4}}=\frac{2}{5}
<br/>$$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
two-decks-of-playing-cards-are-well-shuffled-and-26-cards-are-randomly-distributed-to-a-player-then-the-probability-that-the-player-gets-all-distinct-

slug:
two-decks-of-playing-cards-are-well-shuffled-and-26-cards-are-randomly-distributed-to-a-player-then-the-probability-that-the-player-gets-all-distinct-65784

question:
<div class="question">Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then, the probability that the player gets all distinct cards is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{52} C_{26} /{ }^{104} C_{26}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \times{ }^{52} \mathrm{C}_{26} /{ }^{104} \mathrm{C}_{26}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{3} \times{ }^{52} C_{26} /{ }^{104} C_{26}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2^{26} \times{ }^{52} \mathrm{C}_{26} /{ }^{104} \mathrm{C}_{26}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2^{26} \times{ }^{52} \mathrm{C}_{26} /{ }^{104} \mathrm{C}_{26}$</span> </div>

answer:
<div class="solution">Since, these are 52 distinct cards in decks and each distinct card is 2 in number. Therefore, 2 decks will also contain only 52 distinet cards two each.
<br/>$\therefore$ Probability that the player gets all distinct cards
<br/>$$
<br/>=\frac{{ }^{52} C_{26} \times 2^{26}}{104_{C_{26}}}
<br/>$$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
two-dice-are-tossed-once-the-probability-of-getting-an-even-number-at-the-first-die-or-a-total-of-8-is

slug:
two-dice-are-tossed-once-the-probability-of-getting-an-even-number-at-the-first-die-or-a-total-of-8-is-90813

question:
<div class="question">Two dice are tossed once. The probability of getting an even number at the first die or a total of 8 is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{36}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{36}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{11}{36}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Hints: $A=$ getting even no on 1st dice<br/>$$<br/>\mathrm{B}=\text { getting sum } 8<br/>$$<br/>So $|\mathrm{A}|=18 \quad|\mathrm{~B}|=5 \quad|\mathrm{~A} \cap \mathrm{B}|=3$<br/><br/>$$<br/>\text { So } P(A \cup B)=\frac{18+5-3}{36}=\frac{20}{36} \text {}<br/>$$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 4$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
two-elements-a-and-b-with-atomic-numbers-z-a-and-z-b-are-used-to-produce-characteristic-xrays-with-frequencies-v-a-and-v-b-respectively-if-z-a-z-b-1-2

slug:
two-elements-a-and-b-with-atomic-numbers-z-a-and-z-b-are-used-to-produce-characteristic-xrays-with-frequencies-v-a-and-v-b-respectively-if-z-a-z-b-1-2-87806

question:
<div class="question">Two elements $A$ and $B$ with atomic numbers $Z_{A}$ and $Z_{B}$ are used to produce characteristic X-rays with frequencies $v_{A}$ and $v_{B}$ respectively. If $Z_{A}: Z_{B}=1: 2,$ then $v_{A}: v_{B}$ will be</div>

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

answer:
<div class="solution">Given $Z_{A}: Z_{B}=1: 2$ We know<br/>$v \propto Z^{2}$<br/>Hence,<br/>$\begin{array}{l}<br/>\frac{v_{A}}{v_{B}}=\frac{(1)^{2}}{(2)^{2}} \\<br/>v_{A}: v_{B}=1: 4<br/>\end{array}$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
two-equal-resistances-400-each-are-connected-in-series-with-a-8-v-battery-if-the-resistance-of-first-one-increases-by-05-the-change-required-in-the-re

slug:
two-equal-resistances-400-each-are-connected-in-series-with-a-8-v-battery-if-the-resistance-of-first-one-increases-by-05-the-change-required-in-the-re-94449

question:
<div class="question">Two equal resistances, $400 \Omega$ each, are connected in series with a $8 \mathrm{V}$ battery. If the resistance of first one increases by $0.5 \%$, the change required in the resistance of the second one in order to keep the potential difference across it unaltered is to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">increase it by $2 \Omega$</span> </div>

answer:
<div class="solution">In series combination,
<br/>$$
<br/>R=R_{1}+R_{2}
<br/>$$
<br/>Same current will pass in $A_{1}$ and $R_{2}$. According to question, $R_{1}$ is increased by $0.5 \%$
<br/>$\therefore$ Increment in the resistance
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{0.5}{100} \times 400 \\
<br/>=2.0 \Omega
<br/>\end{array}
<br/>$$
<br/>So to keep the potential unchanged in second resistance the change required will be $2.0 ~ \Omega$ increment.</div>


tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
two-gases-x-molecular-weight-m-x-and-y-molecular-weight-m-y-m-y-m-x-are-at-the-same-temperature-t-in-two-different-containers-their-root-mean-square-v

slug:
two-gases-x-molecular-weight-m-x-and-y-molecular-weight-m-y-m-y-m-x-are-at-the-same-temperature-t-in-two-different-containers-their-root-mean-square-v-55467

question:
<div class="question">Two gases $X$ (molecular weight $M_{X}$ ) and Y (molecular weight $M_{Y} ; M_{Y}&gt;M_{X}$ ) are at the same temperature $T$ in two different containers. Their root mean square velocities are $C_{X}$ and $C_{Y}$ respectively. If the average kinetic energies per molecule of two gases $X$ and $Y$ are $E_{x}$ and $E_{Y}$ respectively then which of the following relation(s) is(are) true?</div>

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

answer:
<div class="solution">Given, molecular weight of $X=M_{X}$ 
<br/>Mol. wt. of $Y=M_{Y}$ and $M_{Y}&gt;M_{x}$
<br/>Root mean square velocitios $=C_{X}$ and $C_{Y}$ respectively 
<br/>Average $\mathrm{KE}$/molecule $=E_{X}$ and $E_{Y}$ respectively We know that. Root mean square velocity.
<br/>$$
<br/>C=\sqrt{\frac{3 R T}{M}} \text { or } C \propto \sqrt{\frac{1}{M}}
<br/>$$
<br/>$\therefore \quad \frac{C_{X}}{C_{Y}}=\frac{M_{Y}}{M_{X}}$
<br/>since $\quad M_{Y}&gt;M_{X}$
<br/>$\therefore$
<br/>$$
<br/>C_{X}&gt;C_{Y}
<br/>$$
<br/>KE/molecules $=\frac{3}{2} n R T$
<br/>$\Rightarrow \quad E_{X} \not \subset E_{Y} \not = \frac{3}{2} R T$
<br/>Further. $\quad E_{X}=E_{Y}=\frac{3}{2} \boldsymbol{k}_{\boldsymbol{B}} \boldsymbol{T}$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
two-identical-blocks-of-ice-move-in-opposite-directions-with-equal-speed-and-collide-with-cach-other-what-will-be-the-minimum-speed-required-to-make-b

slug:
two-identical-blocks-of-ice-move-in-opposite-directions-with-equal-speed-and-collide-with-cach-other-what-will-be-the-minimum-speed-required-to-make-b-90985

question:
<div class="question">Two identical blocks of ice move in opposite directions with equal speed and collide with cach other. What will be the minimum speed required to make both the blocks melt completely, if the initial temperatures of the blocks were $-8^{\circ} \mathrm{C}$ each? (Specific heat of ice is $2100 \mathrm{Jkg}^{-1} \mathrm{K}^{-1}$ and latent heat of fusion of ice is $3.36 \times 10^{5} \mathrm{Jkg}^{-1}$ )</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$840 \mathrm{ms}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$420 \mathrm{ms}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$84 m s^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$84 \mathrm{ms}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$840 \mathrm{ms}^{-1}$</span> </div>

answer:
<div class="solution">Maximum loss in $K . E=$ Total $K . E$ energy before collision of two ice blocks
<br/>$$
<br/>K . E_{\max } \text { loss }=\frac{1}{2} m v^{2}+\frac{1}{2} m v^{2}=m v^{2}
<br/>$$
<br/>This maximum loss in kinetic energy will be equal to loss in heat to melt two ice block.
<br/>$$
<br/>\Rightarrow \quad m v^{2}=(m s \Delta \theta+m L) 2
<br/>$$
<br/>According to question,
<br/>$$
<br/>\begin{aligned}
<br/>L &amp;=336 \times 10^{5} \mathrm{J} \mathrm{kg}^{-1} \\
<br/>S=&amp; 2100 \mathrm{J} \mathrm{kg}^{-1} \mathrm{K}^{-1} \\
<br/>\Delta \theta=8^{\circ} \mathrm{C}
<br/>\end{aligned}
<br/>$$
<br/>From Eq.
<br/>(i) $v=\sqrt{2(s-\Delta \theta+L)}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\sqrt{2\left(2100 \times 8+336 \times 10^{5}\right)} \\
<br/>v=840 \mathrm{ms}^{-1}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
title:
two-identical-cells-each-of-emf-e-and-internal-resistance-r-are-connected-in-parallel-with-an-external-resistance-r-to-get-maximum-power-developed-acr

slug:
two-identical-cells-each-of-emf-e-and-internal-resistance-r-are-connected-in-parallel-with-an-external-resistance-r-to-get-maximum-power-developed-acr-27240

question:
<div class="question">Two identical cells each of emf $\mathrm{E}$ and internal resistance $\mathrm{r}$ are connected in parallel with an external resistance $\mathrm{R}$. To get maximum power developed across $R$, the value of $R$ is</div>

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

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

answer:
<div class="solution">$R_{e q}=\frac{r}{2}$<br/>$I=\frac{2 E}{r+2 R}$<br/>For max. power consumption. I should be max. So denominator should be min. for that<br/>$r+2 R=(\sqrt{r}, \quad)$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
two-identical-equiconvex-lenses-each-of-focal-length-f-are-placed-side-by-side-in-contact-with-each-other-with-a-layer-of-water-in-between-them-as-sho

slug:
two-identical-equiconvex-lenses-each-of-focal-length-f-are-placed-side-by-side-in-contact-with-each-other-with-a-layer-of-water-in-between-them-as-sho-43444

question:
<div class="question">Two identical equiconvex lenses, each of focal length $f$ are placed side by side in contact with each other with a layer of water in between them as shown in the figure. If refractive index of the material of the lenses is greater than that of water, how the combined focal length $F$ is related to $f ?$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q4.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{f}{2}    &lt; F &lt; f$</span> </div>

answer:
<div class="solution">The given combination of lenses
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a4.png"/> <br/> <br/>Here, $f_{1}=$ focal length of equiconvex lenses of glass. $f_{2}=$ focal length of lens formed by water (concave). The focal length of the combination.
<br/>$$
<br/>\therefore \quad \frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}}+\frac{1}{f_{3}}=\frac{1}{f_{1}}-\frac{1}{f_{2}}+\frac{1}{f_{1}}=\frac{2}{f_{1}}-\frac{1}{f_{2}}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>\frac{1}{F}=\frac{2 f_{2}-f_{1}}{f_{1} f_{2}} \Rightarrow F &amp;=\frac{f_{1} f_{2}}{2 f_{2}-f_{1}} \\
<br/>F &amp;=\frac{f_{1}}{2-\frac{f_{1}}{f_{2}}}
<br/>\end{aligned}
<br/>$$
<br/>$\left[\because f_{3}=f_{1}\right]$
<br/>Here, $\frac{1}{f_{1}}=\left(\mu_{g}-1\right) \frac{2}{R},$ for $L_{1}$ and $L_{3}$
<br/>and $\frac{1}{f_{2}}=\left(\mu_{w}-1\right)\left(-\frac{2}{R}\right),$ for $L_{2}$
<br/>$$
<br/>\text { Given, } \mu_{w} &lt; \mu_{g}
<br/>$$
<br/>Thus, $\frac{f_{1}}{f_{2}} &lt; 1$
<br/>So.
<br/>$$
<br/>F&gt;\frac{f_{1}}{2}
<br/>$$
<br/>From Eqs.
<br/>(i) and (ii), we get
<br/>$$
<br/>\frac{f_{1}}{2} &lt; F &lt; f_{1} \quad \text { or } \frac{f}{2} &lt; F &lt; f
<br/>$$</div>


tags: ['Physics', 'Wave Optics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
title:
two-identical-light-waves-propagating-in-the-same-direction-have-a-phase-difference-after-they-superpose-the-intensity-of-the-resulting-wave-will-be-p

slug:
two-identical-light-waves-propagating-in-the-same-direction-have-a-phase-difference-after-they-superpose-the-intensity-of-the-resulting-wave-will-be-p-14298

question:
<div class="question">Two identical light waves, propagating in the same direction, have a phase difference $\delta$. After they superpose the intensity of the resulting wave will be proportional to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos \delta$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos (\delta / 2)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\cos ^2(\delta / 2)$</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">$\cos ^2 \delta$</span> </li> </ul>

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

answer:
<div class="solution">$I=4 I_0 \cos ^2\left(\frac{\delta}{2}\right) \Rightarrow I \propto \cos ^2\left(\frac{\delta}{2}\right)$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
two-identical-springs-are-connected-to-mass-mathrmm-as-shown-mathrmk-spring-constant-if-the-period-of-the-configuration-in-a-is-2-mathrms-the-period-o

slug:
two-identical-springs-are-connected-to-mass-mathrmm-as-shown-mathrmk-spring-constant-if-the-period-of-the-configuration-in-a-is-2-mathrms-the-period-o-58802

question:
<div class="question">Two identical springs are connected to mass \(\mathrm{m}\) as shown \((\mathrm{k}=\) spring constant). If the period of the configuration in (a) is \(2 \mathrm{~S}\), the period of the configuration (b) is</div>

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

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

answer:
<div class="solution">\(\begin{aligned} &amp; \text {Hints: } \frac{\mathrm{T}_1}{\mathrm{~T}_2}=\sqrt{\frac{\mathrm{k}_2}{\mathrm{k}_1}} \Rightarrow \frac{2}{\mathrm{~T}}=\sqrt{\frac{2 \mathrm{k}}{\frac{\mathrm{k}}{2}}}=2 \\ &amp; \therefore \mathrm{T}=1 \mathrm{~S}\end{aligned}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0l5EkFSmtartzOmFLKUuJIPB5LoZJyRurXlHA5pNWOA.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$v=c$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
two-infinite-linecharges-parallel-to-each-other-are-moving-with-a-constant-velocity-v-in-the-same-direction-as-shown-in-the-figure-the-separation-betw-1

slug:
two-infinite-linecharges-parallel-to-each-other-are-moving-with-a-constant-velocity-v-in-the-same-direction-as-shown-in-the-figure-the-separation-betw-1-57616

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8qF_D8cxY0oJEvgIMravf0Jl66wI-SInUSu27DoJ6M4.original.fullsize.png"/><br/><br/>Two infinite line-charges parallel to each other are moving with a constant velocity $v$ in the same direction as shown in the figure. The separation between two line-charges is $d$. The magnetic attraction balances the electric repulsion when, [c = speed of light in free space]</div>

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

answer:
<div class="solution">$\mathrm{F}_{\mathrm{E}}=\mathrm{q}_2 \mathrm{E}_1=\lambda_2 \ell \frac{2 \mathrm{k} \lambda_1}{\mathrm{~d}}$<br/>$\mathrm{~F}_{\mathrm{B}}=\frac{\mu_0 \mathrm{i}_1 \mathrm{i}_2 \ell}{2 \pi \mathrm{d}}=\frac{\mu_0 \lambda_1 \mathrm{~V} \lambda_2 \mathrm{~V}}{2 \pi \mathrm{d}} \times \ell$<br/>$\mathrm{F}_{\mathrm{E}}=\mathrm{F}_{\mathrm{B}}$<br/>$\Rightarrow \frac{2 \mathrm{k} \lambda_1 \lambda_2 \ell}{\mathrm{d}}=\frac{\mu_0 \lambda_1 \lambda_2 \mathrm{~V}^2 \ell}{2 \pi \mathrm{d}}$<br/>$\Rightarrow \frac{2 \lambda_1 \lambda_2 \ell}{4 \pi \varepsilon_0}=\frac{\mu_0 \lambda_1 \lambda_2 \mathrm{~V}^2 \ell}{2 \pi}$<br/>$\mathrm{V}^2=\frac{1}{\mu_0 \varepsilon_0}=\mathrm{c}^2 \Rightarrow \mathrm{V}=\mathrm{c}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$v=c$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
two-infinite-linecharges-parallel-to-each-other-are-moving-with-a-constant-velocity-v-in-the-same-direction-as-shown-in-the-figure-the-separation-betw

slug:
two-infinite-linecharges-parallel-to-each-other-are-moving-with-a-constant-velocity-v-in-the-same-direction-as-shown-in-the-figure-the-separation-betw-61070

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8qF_D8cxY0oJEvgIMravf0Jl66wI-SInUSu27DoJ6M4.original.fullsize.png"/><br/><br/>Two infinite line-charges parallel to each other are moving with a constant velocity $v$ in the same direction as shown in the figure. The separation between two line-charges is $d$. The magnetic attraction balances the electric repulsion when, [c = speed of light in free space]</div>

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

answer:
<div class="solution">$\mathrm{F}_{\mathrm{E}}=\mathrm{q}_2 \mathrm{E}_1=\lambda_2 \ell \frac{2 \mathrm{k} \lambda_1}{\mathrm{~d}}$<br/>$\mathrm{~F}_{\mathrm{B}}=\frac{\mu_0 \mathrm{i}_1 \mathrm{i}_2 \ell}{2 \pi \mathrm{d}}=\frac{\mu_0 \lambda_1 \mathrm{~V} \lambda_2 \mathrm{~V}}{2 \pi \mathrm{d}} \times \ell$<br/>$\mathrm{F}_{\mathrm{E}}=\mathrm{F}_{\mathrm{B}}$<br/>$\Rightarrow \frac{2 \mathrm{k} \lambda_1 \lambda_2 \ell}{\mathrm{d}}=\frac{\mu_0 \lambda_1 \lambda_2 \mathrm{~V}^2 \ell}{2 \pi \mathrm{d}}$<br/>$\Rightarrow \frac{2 \lambda_1 \lambda_2 \ell}{4 \pi \varepsilon_0}=\frac{\mu_0 \lambda_1 \lambda_2 \mathrm{~V}^2 \ell}{2 \pi}$<br/>$\mathrm{V}^2=\frac{1}{\mu_0 \varepsilon_0}=\mathrm{c}^2 \Rightarrow \mathrm{V}=\mathrm{c}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
two-infinite-parallel-metal-planes-contain-electric-charges-with-charge-densities-and-respectively-and-they-are-separated-by-a-small-distance-in-air-i

slug:
two-infinite-parallel-metal-planes-contain-electric-charges-with-charge-densities-and-respectively-and-they-are-separated-by-a-small-distance-in-air-i-14849

question:
<div class="question">Two infinite parallel metal planes, contain electric charges with charge densities $+\sigma$ and $-\sigma$ respectively and they are separated by a small distance in air. If the permittivity of air is $\varepsilon_{0}$, then the magnitude of the field between the two planes with its direction will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sigma / \varepsilon_{0}$ towards the negatively charged plane</span> </div>

answer:
<div class="solution">$E=\frac{\sigma}{2 \varepsilon_{0}}+\frac{\sigma}{2 \varepsilon_{0}}$
<br/>$E=\frac{\sigma}{\varepsilon_{0}}$ towards negative charged plate.</div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
two-long-parallel-wires-separated-by-01-m-carry-currents-of-1-a-and-2-a-respectively-in-opposite-directions-a-third-currentcarrying-wire-parallel-to-b-1

slug:
two-long-parallel-wires-separated-by-01-m-carry-currents-of-1-a-and-2-a-respectively-in-opposite-directions-a-third-currentcarrying-wire-parallel-to-b-1-28241

question:
<div class="question">Two long parallel wires separated by $0.1 \mathrm{m}$ carry currents of 1 A and 2 A, respectively in opposite directions. A third current-carrying wire parallel to both of them is placed in the same plane such that it feels no net magnetic force. It is placed at a distance of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.5 \mathrm{m}$ from the 1 st wire, towards the 2nd wire</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.2 \mathrm{m}$ from the 1 st wire, towards the 2nd wire</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.1 \mathrm{m}$ from the 1 st wire, away from the 2nd wire</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.2 \mathrm{m}$ from the 1 st wire, away from the 2nd wire</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$0.1 \mathrm{m}$ from the 1 st wire, away from the 2nd wire</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a37.png"/> <br/> <br/>According to question, the third wire should not feel magnetic force due to wire (1) and
<br/>(2). It can be only possible in the condition
<br/>that fields due to wire (1) and (2) must act opposite to each other and must be equal. The magnetic field due to long straight wire,
<br/>$$
<br/>B=\frac{\mu_{0} \cdot I}{2 \pi x}
<br/>$$
<br/>$\therefore \quad B_{1}=B_{2}$
<br/>$$
<br/>\frac{\mu_{j_{1}}}{2 \pi x}=\frac{\mu_{d_{2}}}{2 \pi(0 \cdot 1+x)}
<br/>$$
<br/>Here
<br/>$$
<br/>\begin{aligned}
<br/>i_{1} &amp;=1 \mathrm{A} \\
<br/>i_{2} &amp;=2 \mathrm{A} \\
<br/>\frac{\mu_{0} \times 1}{2 \pi x} &amp;=\frac{\mu_{0} \times 2}{2 \pi(0 \cdot 1+x)} \\
<br/>2 x &amp;=(0 \cdot 1+x) \\
<br/>x &amp;=0 \cdot 1 \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
two-long-parallel-wires-separated-by-01-m-carry-currents-of-1-a-and-2-a-respectively-in-opposite-directions-a-third-currentcarrying-wire-parallel-to-b

slug:
two-long-parallel-wires-separated-by-01-m-carry-currents-of-1-a-and-2-a-respectively-in-opposite-directions-a-third-currentcarrying-wire-parallel-to-b-43734

question:
<div class="question">Two long parallel wires separated by $0.1 \mathrm{m}$ carry currents of 1 A and 2 A, respectively in opposite directions. A third current-carrying wire parallel to both of them is placed in the same plane such that it feels no net magnetic force. It is placed at a distance of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.5 \mathrm{m}$ from the 1 st wire, towards the 2nd wire</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.2 \mathrm{m}$ from the 1 st wire, towards the 2nd wire</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.1 \mathrm{m}$ from the 1 st wire, away from the 2nd wire</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.2 \mathrm{m}$ from the 1 st wire, away from the 2nd wire</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$0.1 \mathrm{m}$ from the 1 st wire, away from the 2nd wire</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a37.png"/> <br/> <br/>According to question, the third wire should not feel magnetic force due to wire (1) and
<br/>(2). It can be only possible in the condition
<br/>that fields due to wire (1) and (2) must act opposite to each other and must be equal. The magnetic field due to long straight wire,
<br/>$$
<br/>B=\frac{\mu_{0} \cdot I}{2 \pi x}
<br/>$$
<br/>$\therefore \quad B_{1}=B_{2}$
<br/>$$
<br/>\frac{\mu_{j_{1}}}{2 \pi x}=\frac{\mu_{d_{2}}}{2 \pi(0 \cdot 1+x)}
<br/>$$
<br/>Here
<br/>$$
<br/>\begin{aligned}
<br/>i_{1} &amp;=1 \mathrm{A} \\
<br/>i_{2} &amp;=2 \mathrm{A} \\
<br/>\frac{\mu_{0} \times 1}{2 \pi x} &amp;=\frac{\mu_{0} \times 2}{2 \pi(0 \cdot 1+x)} \\
<br/>2 x &amp;=(0 \cdot 1+x) \\
<br/>x &amp;=0 \cdot 1 \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">In series $\mathrm{K}_{\text {eff }}=\frac{K_1 K_2}{K_1+K_2}$</div>
source_db: db\MarksBatch2_P3.db
title:
two-massless-springs-of-force-constants-k-1-and-k-2-are-joined-end-to-end-the-resultant-force-constant-k-of-the-system-is

slug:
two-massless-springs-of-force-constants-k-1-and-k-2-are-joined-end-to-end-the-resultant-force-constant-k-of-the-system-is-86721

question:
<div class="question">Two massless springs of force constants $\mathrm{K}_1$ and $\mathrm{K}_2$ are joined end to end. The resultant force constant $\mathrm{K}$ of the system is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{K}=\frac{\mathrm{K}_1+\mathrm{K}_2}{\mathrm{~K}_1 \mathrm{~K}_2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{K}=\frac{\mathrm{K}_1-\mathrm{K}_2}{\mathrm{~K}_1 \mathrm{~K}_2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{K}=\frac{\mathrm{K}_1 \mathrm{~K}_2}{\mathrm{~K}_1+\mathrm{K}_2}$</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">$\mathrm{K}=\frac{\mathrm{K}_1 \mathrm{~K}_2}{\mathrm{~K}_1-\mathrm{K}_2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{K}=\frac{\mathrm{K}_1 \mathrm{~K}_2}{\mathrm{~K}_1+\mathrm{K}_2}$</span> </div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
two-metal-wires-of-identical-dimensions-are-connected-in-series-if-1-and-2-are-the-electrical-conductivities-of-the-metal-wires-respectively-the-effec

slug:
two-metal-wires-of-identical-dimensions-are-connected-in-series-if-1-and-2-are-the-electrical-conductivities-of-the-metal-wires-respectively-the-effec-81405

question:
<div class="question">Two metal wires of identical dimensions are connected in series. If $\sigma_{1}$ and $\sigma_{2}$ are the electrical conductivities of the metal wires respectively, the effective conductivity of the combination is</div>

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

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

answer:
<div class="solution">$R_{e q}=R_{1}+R_{2}$
<br/>$\frac{1}{\sigma_{e q}} \frac{2 l}{A}=\frac{1}{\sigma_{1}} \frac{1}{A}+\frac{1}{\sigma_{2}} \frac{1}{A}\left[\sigma_{e q}=\frac{2 \sigma_{1} \sigma_{2}}{\sigma_{1}+\sigma_{2}}\right]$</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
title:
two-metallic-spheres-of-equal-outer-radii-are-found-to-have-same-moment-of-inertia-about-their-respective-diameters-then-which-of-the-following-statem

slug:
two-metallic-spheres-of-equal-outer-radii-are-found-to-have-same-moment-of-inertia-about-their-respective-diameters-then-which-of-the-following-statem-52344

question:
<div class="question">Two metallic spheres of equal outer radii are found to have same moment of inertia about their respective diameters. Then which of the following statement(s) is/are true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The two spheres have equal masses</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The ratio of their masses is nearly $1.67: 1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The spheres are made of different materials</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Their rotational kinetic energies will be equal when rotated with equal uniform angular speed about their respective diameters</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Their rotational kinetic energies will be equal when rotated with equal uniform angular speed about their respective diameters</span> </div>

answer:
<div class="solution">Hint: 
<br/>Inner radius is not given
<br/>$\therefore$ only option (D) is correct</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: db\MarksBatch2_P3.db
title:
two-mirrors-at-an-angle-produce-5-images-of-a-point-the-number-of-images-produced-when-is-decreased-to-3-0-is

slug:
two-mirrors-at-an-angle-produce-5-images-of-a-point-the-number-of-images-produced-when-is-decreased-to-3-0-is-56608

question:
<div class="question">Two mirrors at an angle $\theta^{\circ}$ produce 5 images of a point. The number of images produced when $\theta$ is decreased to $\theta^{\circ}-30^{\circ}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">12</span> </li> </ul>

answer:
<div class="solution">Hints: No. of images $=5$<br/>$\therefore \theta=60^{\circ}$<br/>New angle $=\theta-30^{\circ}=30^{\circ}$. No of images $=\frac{360^{\circ}}{30^{\circ}}-1=11$</div>


