# questions

tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3: 4$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-frequency-of-the-first-overtone-of-a-closed-pipe-of-length-l-1-is-equal-to-that-of-the-first-overtone-of-an-open-pipe-of-length-l-2-the-ratio-of-t

slug:
the-frequency-of-the-first-overtone-of-a-closed-pipe-of-length-l-1-is-equal-to-that-of-the-first-overtone-of-an-open-pipe-of-length-l-2-the-ratio-of-t-55687

question:
<div class="question">The frequency of the first overtone of a closed pipe of length $l_{1}$ is equal to that of the first. overtone of an open pipe of length $l_{2}$. The ratio of their lengths $\left(l_{1}: l_{2}\right)$ is</div>

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

answer:
<div class="solution">Here, $\quad v_{1}=\frac{(2 n-1)}{4 l_{1}} v$
<br/>$$
<br/>v_{2}=\frac{n}{2 l_{2}} v
<br/>$$
<br/>Hence, $y_{1}=v_{2}$
<br/>$$
<br/>\frac{l_{1}}{l_{2}}=\frac{3}{4}
<br/>$$</div>


tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-frequency-v-of-the-radiation-emitted-by-an-atom-when-an-electron-jumps-from-one-orbit-to-another-is-given-by-k-e-where-k-is-a-constant-and-e-is-th

slug:
the-frequency-v-of-the-radiation-emitted-by-an-atom-when-an-electron-jumps-from-one-orbit-to-another-is-given-by-k-e-where-k-is-a-constant-and-e-is-th-82704

question:
<div class="question">The frequency $v$ of the radiation emitted by an atom when an electron jumps from one orbit to another is given by $=\mathrm{k} \delta \mathrm{E}$, where $\mathrm{k}$ is a constant and $\delta \mathrm{E}$ is the change in energy level due to the transition. Then dimension of $\mathrm{k}$ is</div>

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

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

answer:
<div class="solution">Hint: 
<br/>$v=k \delta E$
<br/>$[k]=\left[\frac{v}{\delta E}\right]=\frac{\left[T^{-1}\right]}{\left[M L^{2} T^{-2}\right]}$
<br/>$[k]=\left[M^{-} L^{-2} T^{1}\right]$</div>


title: the-function-f-x-2-x-2-t-a-n-x-2-where-x-denotes-the-greatest-integer-x-is
slug: the-function-f-x-2-x-2-t-a-n-x-2-where-x-denotes-the-greatest-integer-x-is-83551
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The function $f(x)=\frac{\tan \left\{\pi\left[x-\frac{\pi}{2}\right]\right\}}{2+[x]^{2}},$ where $[x]$ denotes the greatest integer $\leq x$, is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">continuous for all values of $x$</span> </div>

answer:
<div class="solution">Given, $f(x)=\frac{\tan \left\{\pi\left[x-\frac{\pi}{2}\right]\right\}}{2+[x]^{2}}$
<br/>Since. $\left[x-\frac{\pi}{2}\right]$ is an integer for all $x$, therefore $\pi\left[x-\frac{\pi}{2}\right]$ is an integral multiple of $\pi$ for all $x$. Hence, $\tan \left\{\pi\left[x-\frac{\pi}{2}\right]\right\}=0$ for all $x$
<br/>Also, $2+[x]^{2} \neq 0$ for all $x$
<br/>Hence, $f(x)=0$ for all $x$ Hence, $f(x)$ is continuous and derivable for all $x$.</div>


title: the-function-f-x-a-sin-x-b-e-x-is-differentiable-at-x-0-when
slug: the-function-f-x-a-sin-x-b-e-x-is-differentiable-at-x-0-when-23949
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The function $f(x)=a \sin |x|+b e^{| x \mid} \quad$ is differentiable at $x=0$ when</div>

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

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

answer:
<div class="solution">Given, $f(x)=a \sin |x|+b e^{|x|}$
<br/>We know that $\sin |x|$ and $e^{|x|}$ is not
<br/>differentiable at $x=0$.
<br/>Therefore, for $f(x)$ to differentiable at $x=0$, we
<br/>must have $a=b=0$.
<br/>$\therefore$
<br/>$$
<br/>a+b=0
<br/>$$</div>


title: the-function-f-x-sec-lo-g-x-1-x-2-is
slug: the-function-f-x-sec-lo-g-x-1-x-2-is-90564
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">even</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The function $\mathrm{f}(\mathrm{x})=\sec \left[\log \left(\mathrm{x}+\sqrt{1+\mathrm{x}^2}\right)\right]$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">odd</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">even</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">neither odd nor even</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">constant</span> </li> </ul>

answer:
<div class="solution">Hints : $\mathrm{f}(\mathrm{x})=\sec \left(\ell \ln \left(\mathrm{x}+\sqrt{1+\mathrm{x}^2}\right)\right)=\sec ($ odd function $)=$ even function<br/>$\because$ sec is an even function</div>


title: the-function-f-x-x-2-b-x-c-where-b-and-c-real-constants-describes
slug: the-function-f-x-x-2-b-x-c-where-b-and-c-real-constants-describes-82871
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The function $f(x)=x^{2}+b x+c,$ where $b$ and $c$ real constants, describes</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">one-to-one mapping</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">onto mapping</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">not one-to-one but onto mapping</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">neither one-to-one nor onto mapping</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">neither one-to-one nor onto mapping</span> </div>

answer:
<div class="solution">Given function is
<br/>$$
<br/>f(x)=x^{2}+b x+c
<br/>$$
<br/>It is a quadratic equation in $x$. So, we will get a parabola either downward or upward.
<br/> <br/>Hence, it is a many one mapping and not onto mapping. Hence, it is neither one-to-one nor onto mapping.</div>


title: the-function-fxa-xb-is-strictly-increasing-for-all-real-x-if
slug: the-function-fxa-xb-is-strictly-increasing-for-all-real-x-if-69907
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The function \(f(x)=a x+b\) is strictly increasing for all real \(x\) if</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a \( &gt; 0\)</span> </div>

answer:
<div class="solution">Hints : \(f^{\prime}(x)=a\)<br/>\(f^{\prime}(x) &gt; 0 \Rightarrow a &gt; 0\)</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-function-yek-x-satisfies-leftfracd2-yd-x2fracd-yd-xrightleftfracd-yd-xyrighty-fracd-yd-x-it-is-valid-for

slug:
the-function-yek-x-satisfies-leftfracd2-yd-x2fracd-yd-xrightleftfracd-yd-xyrighty-fracd-yd-x-it-is-valid-for-73599

question:
<div class="question">The function \(y=e^{k x}\) satisfies \(\left(\frac{d^2 y}{d x^2}+\frac{d y}{d x}\right)\left(\frac{d y}{d x}-y\right)=y \frac{d y}{d x}\). It is valid for</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">exactly one value of \(k\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">two distinct values of \(k\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">three distinct values of \(k\)</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">infinitely many values of \(k\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">three distinct values of \(k\)</span> </div>

answer:
<div class="solution">Hint : \(y=e^{k x}\)<br/>\(\Rightarrow \frac{d y}{d x}=k e^{k x}=k y \Rightarrow \frac{d^2 y}{d x^2}=k^2 e^{k x}=k^2 y\)<br/>Now LHS<br/>\(=\left(\frac{d^2 y}{d x^2}+\frac{d y}{d x}\right)\left(\frac{d y}{d x}-y\right)=\left(k^2 y+k y\right)(k y-y)=k(k+1)(k-1) y^2=k\left(k^2-1\right) y^2\)<br/>Now RHS \(y \frac{d y}{d x}=k y^2, A / q~ k\left(k^2-1\right) y^2=k y^2 \Rightarrow k\left[k^2-2\right]=0 \Rightarrow k=0, k= \pm \sqrt{2}\)</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 4$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-fundamental-frequency-of-a-closed-pipe-is-equal-to-the-frequency-of-the-second-harmonic-of-an-open-pipe-the-ratio-of-their-lengths-is

slug:
the-fundamental-frequency-of-a-closed-pipe-is-equal-to-the-frequency-of-the-second-harmonic-of-an-open-pipe-the-ratio-of-their-lengths-is-42838

question:
<div class="question">The fundamental frequency of a closed pipe is equal to the frequency of the second harmonic of an open pipe. The ratio of their lengths is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1: 2$</span> </li><li class="correct"> <span class="option-label">B</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><li class=""> <span class="option-label">C</span> <span class="option-data">$1: 8$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1: 16$</span> </li> </ul>

answer:
<div class="solution">Let the length of closed pipe is $L_{1}$ and that of pipe is $L_{2}$. Fundamental frequency of closed pipe, $v_{1}=\frac{v}{4 L_{1}}$
<br/>and frequency of second harmonic of open pipe.
<br/>$$
<br/>v_{2}=2 \times \frac{v}{2 L_{2}}
<br/>$$
<br/>Given,
<br/>$$
<br/>v_{1}=v_{2}
<br/>$$
<br/>$i, \theta$
<br/>$$
<br/>\frac{v}{4 L_{1}}=\frac{2 v}{2 L_{2}} \Rightarrow \frac{L_{1}}{L_{2}}=\frac{1}{4}
<br/>$$</div>


title: the-general-solution-of-the-different-equation-100-dx-2-d-2-y-20-dx-dy-y-0-is
slug: the-general-solution-of-the-different-equation-100-dx-2-d-2-y-20-dx-dy-y-0-is-29259
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The general solution of the different equation $100 \frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}-20 \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=\left(c_1+c_2 x\right) e^x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y=\left(c_1+c_2 x\right) e^{-x}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y=\left(c_1+c_2 x\right) e^{\frac{x}{10}}$</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">D</span> <span class="option-data">$y=c_1 e^x+c_2 e^{-x}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=\left(c_1+c_2 x\right) e^{\frac{x}{10}}$</span> </div>

answer:
<div class="solution">Hints: $100 p^2-20 p+1=$<br/>$$<br/>\begin{aligned}<br/>&amp; (10 P-1)^2=0, P=\frac{1}{10} \\<br/>&amp; y=\left(c_1+c_2 x\right) e^{\frac{x}{10}}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-general-solution-of-the-differential-equation-1-e-y-x-d-x-1-y-x-e-x-y-d-y-0-is-c-is-an-arbitrary-constant

slug:
the-general-solution-of-the-differential-equation-1-e-y-x-d-x-1-y-x-e-x-y-d-y-0-is-c-is-an-arbitrary-constant-87207

question:
<div class="question">The general solution of the differential equation $\left(1+e^{\frac{x}{y}}\right) d x+\left(1-\frac{x}{y}\right) e^{x / y} d y=0$ is$(C$ is an arbitrary constant $)$</div>

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

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

answer:
<div class="solution">$\left(1+e^{\frac{x}{y}}\right) d x+\left(1-\frac{x}{y}\right) e^{x / y} d y=0$
<br/>$\Rightarrow \quad\left(1+e^{x / y}\right) d x=-e^{x / y}\left(1-\frac{x}{y}\right) d y$
<br/>$\Rightarrow \quad \frac{d x}{d y}=\frac{-e^{x / y}(1-x / y)}{\left(1+e^{x / y}\right)}$ ...(i)
<br/>This is homogeneous differential equation.
<br/>So, put $\quad x=v y$
<br/>$\frac{d x}{d y}=v+y \frac{d v}{d y}$ ....(ii)
<br/>Therefore, $v+y \frac{d v}{d y}=\frac{-e^{v} (1-v)}{1+e^{v}}$
<br/>[from Eqs.(i) and (ii)]
<br/>$\Rightarrow \quad y \frac{d v}{d y}=\frac{-e^{v}+ve^{v}}{1+e^{v}}-v$
<br/>$=\frac{-e^{v}+ve^{v}-v\left(1+e^{v}\right)}{1+e^{v}}$
<br/>$=-e^{v}+v e^{v}-v-v e^{v}$
<br/>$\Rightarrow \quad y \frac{d v}{d y}=\frac{-\left(e^{v}+v\right)}{1+e^{v}}$
<br/>$\Rightarrow \quad \frac{1+e^{v}}{v+e^{r}} d v=-\frac{d y}{y} \quad$ [variable separation $]$
<br/>$\Rightarrow \quad \int \frac{1+e^{v}}{v+e^{v}} d v=-\int \frac{d y}{y} \quad$ [by integration $] \ldots$ (iii)
<br/>Let $\left.v+e^{v}=t \Rightarrow (1+e^{v}\right) d v=d t$
<br/>$\int \frac{1+e^{v}}{v+e^{v}} d v=\int \frac{d t}{t}=\log t=\log \left(v+e^{v}\right)$ ...(iv)
<br/>$\Rightarrow \log \left(v+e^{v}\right)=-\log y+\log C$
<br/>[from Eqs. (iii) and (iv)] $\Rightarrow \log \left(v+e^{v}\right)+\log y=\log C$
<br/>$\Rightarrow \quad y\left(v+e^{v}\right)=C$
<br/>$y\left(\frac{x}{y}+e^{x / y}\right)=C$
<br/>$y\left(\frac{x+ye^{x / y}}{y}\right)=C$
<br/>$\Rightarrow \quad x+y e^{x / y}=C$</div>


title: the-general-solution-of-the-differential-equation-d-x-d-y-2-x-2-y-1-x-y-1-is
slug: the-general-solution-of-the-differential-equation-d-x-d-y-2-x-2-y-1-x-y-1-is-63818
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The general solution of the differential equation $\frac{d y}{d x}=\frac{x+y+1}{2 x+2 y+1}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{e}|3 x+3 y+2|+3 x-6 y=C$</span> </div>

answer:
<div class="solution">Given differential equation, $\frac{d y}{d x}=\frac{x+y+1}{2 x+2 y+1}$
<br/>Put $x+y=v$ $\Rightarrow \quad 1+\frac{d y}{d x}=\frac{d v}{d x} \Rightarrow \frac{d y}{d x}=\frac{d v}{d x}-1$
<br/>$\therefore$ Eq.
<br/>(i) becomes, $\frac{d y}{d x}=\frac{v+1}{2 v+1}$
<br/>$\Rightarrow \quad \frac{d v}{d x}-1=\frac{v+1}{2 v+1} \Rightarrow \frac{d v}{d x}=\frac{v+1}{2 v+1}+1$
<br/>$\Rightarrow \quad \frac{d v}{d x}=\frac{v+1+2 v+1}{2 v+1} \Rightarrow \frac{d v}{d x}=\frac{3 v+2}{2 v+1}$
<br/>$\Rightarrow \frac{2 v+1}{3 v+2} d v=d x \Rightarrow \frac{\left[\frac{2}{3}(3 v+2)-\frac{1}{3}\right]}{3 v+2} d v=d x$
<br/>$\Rightarrow \quad\left[\frac{2}{3}-\frac{1}{3} \times \frac{1}{(3 v+2)}\right] d v=d x$
<br/>Integrating both sides, we get $\Rightarrow \quad \int\left(\frac{2}{3}-\frac{1}{3} \times \frac{1}{(3 v+2)}\right) d v=\int d x+C^{\prime}$
<br/>Where $c$ is a constant of integration.
<br/>$\Rightarrow \quad \frac{2}{3} v-\frac{1}{3} \frac{\log (3 v+2)}{3}=x+C^{\prime}$
<br/>$\Rightarrow \quad \frac{2}{3}(x+y)-\frac{1}{9} \log (3 x+3 y+2)=x+C$
<br/>$\Rightarrow \quad 6 x+6 y-\log (3 x+3 y+2)=9 x+C$
<br/>$\Rightarrow \quad 3 x-6 y+\log (3 x+3 y+2)=C$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-general-solution-of-the-differential-equation-d-x-d-y-e-y-x-e-y-x-is-where-c-is-an-arbitrary-constant

slug:
the-general-solution-of-the-differential-equation-d-x-d-y-e-y-x-e-y-x-is-where-c-is-an-arbitrary-constant-71215

question:
<div class="question">The general solution of the differential equation $\frac{d y}{d x}=e^{y+x}+e^{y-x}$ is<br/>where $c$ is an arbitrary constant</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}+\mathrm{c}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{e}^{-\mathrm{y}}=\mathrm{e}^{-\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{c}$</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">$e^{-y}=e^x+e^{-x}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$e^y=e^x+e^{-x}+c$</span> </li> </ul>

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

answer:
<div class="solution">Hints: $e^{-y} d y=\left(e^x+e^{-x}\right) d x$ Integrate<br/>$-\mathrm{e}^{-\mathrm{y}}=e^x-e^{-x}+c, \quad \mathrm{e}^{-\mathrm{y}}=e^{-x}-e^{+x}+c$</div>


title: the-general-solution-of-the-differential-equation-fracd2-yd-x28-fracd-yd-x16-y0-is
slug: the-general-solution-of-the-differential-equation-fracd2-yd-x28-fracd-yd-x16-y0-is-87928
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The general solution of the differential equation \(\frac{d^2 y}{d x^2}+8 \frac{d y}{d x}+16 y=0\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((\mathrm{A}+\mathrm{Bx}) \mathrm{e}^{-4 x}\)</span> </div>

answer:
<div class="solution">Hints: \(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}+8 \frac{\mathrm{dy}}{\mathrm{dx}}+16 \mathrm{y}=0\)<br/>auxilary equation \(\mathrm{m}^2+8 \mathrm{~m}+16=0 \Rightarrow \mathrm{m}=-4\)<br/>Solution \(y=(a x+b) e^{-4 x}\)</div>


title: the-general-solution-of-the-differential-equation-log-mathrmeleftfracd-yd-xrightxy-is
slug: the-general-solution-of-the-differential-equation-log-mathrmeleftfracd-yd-xrightxy-is-82125
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The general solution of the differential equation \(\log _{\mathrm{e}}\left(\frac{d y}{d x}\right)=x+y\) is</div>

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

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

answer:
<div class="solution">Hints: \(\frac{d y}{d x}=e^x \cdot e^y \Rightarrow \int e^{-y} d y .=\int e^x d x \Rightarrow e^x+e^{-y}=c\)</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-general-value-of-the-real-angle-which-satisfies-the-equation-cos-i-sin-cos-2-i-sin-2-cos-n-t-sin-n-1-is-given-by-assuming-k-is-an-integer

slug:
the-general-value-of-the-real-angle-which-satisfies-the-equation-cos-i-sin-cos-2-i-sin-2-cos-n-t-sin-n-1-is-given-by-assuming-k-is-an-integer-50790

question:
<div class="question">The general value of the real angle $\theta$, which satisfies the equation. $(\cos \theta+i \sin \theta)(\cos 2 \theta+i \sin 2 \theta) \ldots$$(\cos n \theta+t \sin n \theta)=1$ is given by, (assuming $k$ is an integer)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4 k \pi}{n(n+1)}$</span> </div>

answer:
<div class="solution">We have,
<br/>$(\cos\theta+i\sin\theta)(cos2\theta+isin2\theta)...(cosn\theta+isinn\theta) = 1$
<br/>$\Rightarrow e^{i\theta}.e^{i(2\theta)}.e^{i(3\theta)}. ...e^{i(n\theta)} = 1$
<br/>$\Rightarrow e^{i\theta(1+2+3+...+n}=1$
<br/>$\Rightarrow e^{\frac{in(n+1)\theta}{2}}=1$
<br/>$\Rightarrow cos(\frac{n(n+1)}{2}\theta)+isin(\frac{n(n+1)}{2}\theta = 1 + 0i$
<br/>$\Rightarrow cos(\frac{n(n+1)}{2}\theta)=1$
<br/>$\Rightarrow \frac{n(n+1)}{2}\theta = 2k\pi$
<br/>$\Rightarrow \theta=\frac{4k}{n(n+1)}\pi$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.72</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-glass-prisms-p-1-and-p-2-are-to-be-combined-together-to-produce-dispersion-without-deviation-the-angles-of-the-prisms-p-1-and-p-2-are-selected-as-

slug:
the-glass-prisms-p-1-and-p-2-are-to-be-combined-together-to-produce-dispersion-without-deviation-the-angles-of-the-prisms-p-1-and-p-2-are-selected-as-15437

question:
<div class="question">The glass prisms $P_{1}$ and $P_{2}$ are to be combined together to produce dispersion without deviation. The angles of the prisms $P_{1}$ and $P_{2}$ are selected as $4^{\circ}$ and $3^{\circ}$ respectively. If the refractive index of prism $P_{1}$ is $1.54,$ then that of $P_{2}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.48</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.58</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.62</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1.72</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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 for prism $\mathrm{P}_{1}$
<br/>$$
<br/>\mu=1.54,A=4^{\circ}
<br/>$$
<br/>For prism $\mathrm{P}_{2}$
<br/>$$
<br/>\mu^{\prime}=?, A^{\prime}=3^{\circ}
<br/>$$
<br/>For no deviation $=\delta +\delta ^{\prime}=0$
<br/>$\Rightarrow \quad(\mu-1) A=\left(\mu^{\prime}-1\right) A^{\prime}$
<br/>$\Rightarrow \quad(1.54-1) 4=\left(\mu^{\prime}-1\right) 3$
<br/>On solving we get $\mu^{\prime}=1.72$</div>


title: the-graphs-of-the-polynomial-x-2-1-and-cos-x-intersect
slug: the-graphs-of-the-polynomial-x-2-1-and-cos-x-intersect-67752
question: <div class="question">The graphs of the polynomial $x^{2}-1$ and $\cos x$ intersect</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">at exactly two points</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">at exactly 3 points</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">at least 4 but at finitely many points.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">at infinitely many points.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">at exactly two points</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a64.png"/> <br/>It is clear from the graph that it intersects at exactly two points.</div>


title: the-greatest-and-least-values-of-f-x-tan-1-x-2-1-ln-x-on-3-1-3-are
slug: the-greatest-and-least-values-of-f-x-tan-1-x-2-1-ln-x-on-3-1-3-are-67530
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The greatest and least values of $f(x)=\tan ^{-1} x-\frac{1}{2} \ln x$ on $\left[\frac{1}{\sqrt{3}}, \sqrt{3}\right]$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{f}_{\min }=\sqrt{3}-1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}_{\max }=\pi / 6+\frac{1}{4} \ell \mathrm{n} 3$</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><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}_{\min }=\pi / 3-\frac{1}{4} \ell \mathrm{n} 3$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}_{\max }=\pi / 12+\ell \mathrm{n} 5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{f}_{\max }=\pi / 6+\frac{1}{4} \ell \mathrm{n} 3$, $\mathrm{f}_{\min }=\pi / 3-\frac{1}{4} \ell \mathrm{n} 3$</span> </div>

answer:
<div class="solution">$f(x)=\tan ^{-1} x-1 / 2 \ln x, x \in[1 / \sqrt{3}, \sqrt{3}]$
<br/>$f^{\prime}(x)=\frac{1}{1+x^{2}}-\frac{1}{2 x}=\frac{2 x-\left(1+x^{2}\right)}{2 x\left(1+x^{2}\right)}=\frac{-(x-1)^{2}}{2 x\left(1+x^{2}\right)}$
<br/>when $x&gt;0, f^{\prime}(x)     &lt; 0$
<br/>$f(x)$ is decreasing function.
<br/>$\begin{array}{l}
<br/>f_{\min }=f(\sqrt{3})=\tan ^{-1}(\sqrt{3})-1 / 2 \ln (\sqrt{3})=\pi / 3-\frac{1}{4} \ln 3 \\
<br/>f_{\max }=f(1 / \sqrt{3})=\pi / 6+1 / 4 \ln 3
<br/>\end{array}$</div>


title: the-greatest-integer-which-divides-p-1-p-2-p-3-p-q-for-all-p-n-and-fixed-q-n-is
slug: the-greatest-integer-which-divides-p-1-p-2-p-3-p-q-for-all-p-n-and-fixed-q-n-is-72836
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$q !$</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The greatest integer which divides $(p+1)(p+2)(p+3) \ldots(p+q)$ for all
<br/>$p \in N$ and fixed $q \in N$ is</div>

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

answer:
<div class="solution">$(p+1)(p+2)(p+3) \ldots(p+q)$ is the product of $q$ consecutive natural number ( $R$ q e $N$ ) The product of $q$ consecutive natural number is always divisible by $q!$</div>


title: the-ground-state-electronic-configuration-of-co-molecule-is
slug: the-ground-state-electronic-configuration-of-co-molecule-is-21420
question: <div class="question">The ground state electronic configuration of CO molecule is</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 \sigma^{2} 2 \sigma^{2} 1 \pi^{4} 3 \sigma^{2}$</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">B</span> <span class="option-data">$1 \sigma^{2} 2 \sigma^{2} 3 \sigma^{2} 1 \pi^{2} 2 \pi^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 \sigma^{2} 2 \sigma^{2} 1 \pi^{2} 3 \sigma^{2} 2 \pi^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 \sigma^{2} 1 \pi^{2} 2 \sigma^{2} 2 \sigma^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 \sigma^{2} 2 \sigma^{2} 1 \pi^{4} 3 \sigma^{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a25.png"/></div>


title: the-ground-state-magnetic-property-of-b-2-and-c-2-molecules-will-be
slug: the-ground-state-magnetic-property-of-b-2-and-c-2-molecules-will-be-52927
tags: ['Chemistry', 'Solid State', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The ground state magnetic property of $\mathrm{B}_{2}$ and
<br/>$\mathrm{C}_{2}$ molecules will be</div>

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

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

answer:
<div class="solution">According to molecular orbital theory, outer configuration for
<br/>$$
<br/>\left[\mathrm{C}_{2}\right]=\mathrm{C}_{12} \rightarrow \pi p_{x}^{2}=\pi p_{y}^{2}
<br/>$$
<br/>Thus, is diamagnetic in nature.
<br/>$$
<br/>\left[\mathrm{B}_{2}\right]=\mathrm{B}_{10}-&gt;\pi p_{x}^{1}=\pi p_{y}^{1}
<br/>$$
<br/>Thus, is paramagnetic in nature.</div>


title: the-h-3-o-ion-has-the-following-shape
slug: the-h-3-o-ion-has-the-following-shape-75984
question: <div class="question">The $\mathrm{H}_{3} \mathrm{O}^{+}$ion has the following shape</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Tetrahedral</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Pyramidal</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Triangular planar</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">"T" shaped</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{H}_{3} \mathrm{O}^{+}$
<br/>total valence $e^{-1}$ count
<br/>$\begin{array}{l}
<br/>\mathrm{O}+\mathrm{H}+\mathrm{H}+\mathrm{H}-\mathrm{e}^{-1}=8 \\
<br/>2,6 \quad 1 \quad 1 \quad 1
<br/>\end{array}$
<br/>using Lever rule
<br/>$2|8| 4 \leadsto$ sp $^{3}$ hybrid $(0)$
<br/>$\frac{8}{x}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/bWG4vBh0ldys9NWhirE-BZ1mUmvMnSchG4MaFGEyPPs.original.fullsize.png"/></div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Less</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-h-n-h-angle-in-ammonia-is-107-6-while-the-h-p-h-angle-in-phosphine-is-93-5-relative-to-phosphine-the-p-character-of-the-lonepair-on-ammonia-is-exp

slug:
the-h-n-h-angle-in-ammonia-is-107-6-while-the-h-p-h-angle-in-phosphine-is-93-5-relative-to-phosphine-the-p-character-of-the-lonepair-on-ammonia-is-exp-63460

question:
<div class="question">The H $-N-H$ angle in ammonia is $107.6^{\circ}$ while the $\mathrm{H}-\mathrm{P} -\mathrm{H}$ angle in phosphine is $93.5^{\circ} .$ Relative to phosphine, the $p$ -character of the lone-pair on ammonia is expected to be</div>

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

answer:
<div class="solution">Key point As the percentage of s character in a bond increases, the bond angle also increases. Therefore.
<br/>as the bond angle increases, the percentage of $p$ -character decreases.
<br/>In the given options, Ammonia ( $\mathrm{NH}_{3}$ ) has bond angle $=107.6^{\circ}$ and Phosphine $\left(\mathrm{PH}_{3}\right)$ has bond angle $=93.5^{\circ}$
<br/>$\because$ Bond angle $\propto \frac{1}{p-\text { character }}$ and $\propto s$ -character
<br/>Bond angle of $\mathrm{NH}_{3}&gt;\mathrm{PH}_{3}$ due to lone-pair lone-pair repulsion. Thus, the lone pair on $\mathrm{NH}_{3}$ has less $p$ -character. Hence, option (a) is the correct answer.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-halflife-for-decay-of-14-c-by-emission-is-5730-yr-the-fraction-of-14-c-decays-in-a-sample-that-is-22920-yr-old-would-be

slug:
the-halflife-for-decay-of-14-c-by-emission-is-5730-yr-the-fraction-of-14-c-decays-in-a-sample-that-is-22920-yr-old-would-be-56669

question:
<div class="question">The half-life for decay of ${ }^{14} \mathrm{~C, by~} \beta$ -emission is $5730 \mathrm{yr}$. The fraction of ${ }^{14} \mathrm{C}$ decays, in a sample that is 22920 yr old, would be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{8}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{16}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{8}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{15}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{15}{16}$</span> </div>

answer:
<div class="solution">$n=\frac{\text { total time }(t)}{\text { half -lite }(t / 2)}=\frac{22920}{5730}=4$
<br/>$$
<br/>\begin{aligned}
<br/>\text { Left amount, } N=&amp; N_{0}\left(\frac{1}{2}\right)^{n} \\
<br/>&amp;=N_{0}\left(\frac{1}{2}\right)^{4}=\frac{N_{0}}{16} \\
<br/>\therefore \text { Decayed fraction } &amp;=N_{0}-\frac{N_{0}}{16} \\
<br/>&amp;=\frac{16 N_{0}-N_{0}}{16}=\frac{15 N_{0}}{16}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-halflife-of-a-radioactive-element-is-10-hours-how-much-will-be-left-after-4-hours-in-1-g-atom-sample

slug:
the-halflife-of-a-radioactive-element-is-10-hours-how-much-will-be-left-after-4-hours-in-1-g-atom-sample-84140

question:
<div class="question">The half-life of a radioactive element is 10 hours. How much will be left after 4 hours in $1 \mathrm{~g}$ atom sample?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$45.6 \times 10^{23}$ atoms</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4.56 \times 10^{23}$ atoms</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.56 \times 10^{21}$ atoms</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4.56 \times 10^{20}$ atoms</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4.56 \times 10^{23}$ atoms</span> </div>

answer:
<div class="solution">$t_{1 / 2}=10 \mathrm{hr} . \quad \mathrm{K}=\frac{0.693}{10}$<br/>$4=\frac{2.303 \times 10}{0.693} \log \frac{1}{\mathrm{~N}}$<br/>$\log \frac{1}{\mathrm{~N}}=\frac{4 \times 0.693}{2.303 \times 10}=0.12036$<br/>$\log \mathrm{N}=-0.12036=\overline{1} .87964$<br/>$\mathrm{N}=7.575 \times 10^{-1} \mathrm{~g}$ atoms<br/>$\therefore$ No. of atoms $=7.575 \times 10^{-1} \times 6.023 \times 10^{23}$ atoms $=4.56 \times 10^{23}$ atoms</div>


title: the-halflife-of-c-14-is-5760-years-for-a-200-mg-sample-of-c-14-the-time-taken-to-change-to-25-mg-is
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
slug:
the-halflife-of-c-14-is-5760-years-for-a-200-mg-sample-of-c-14-the-time-taken-to-change-to-25-mg-is-85543

question:
<div class="question">The half-life of $C^{14}$ is 5760 years. For a $200 \mathrm{mg}$ sample of $\mathrm{C}^{14}$, the time taken to change to $25 \mathrm{mg}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">11520 years</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">23040 years</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">5760 years</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">17280 years</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">17280 years</span> </div>

answer:
<div class="solution">Given. Half-life of $\mathrm{C}^{4}, t_{1 / 2}=5760$ years Initial concentration of sample of $\mathrm{C}^{4}, \mathrm{N}_{0}=200 \mathrm{mg}$ Final concentration of sample of $\mathrm{C}^{4}, \mathrm{N}_{1}=2 \mathrm{S} \mathrm{mg}$ The given decay is radioactive and all radiouctive decay follows first order kinetics.
<br/>$$
<br/>t_{1 / 2}=\frac{0.693}{\lambda}
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>\lambda=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{5760} \mathrm{yr}^{-1}
<br/>$$
<br/>We know that, for first order reaction
<br/>$\begin{aligned} t &amp;=\frac{2.303}{\lambda} \log \frac{N_{0}}{N_{1}}=\frac{2303}{0.693} \log \frac{[200]}{125} \\ &amp;=\frac{2303 \times 5760}{0.693} \log 8 \\ &amp;=17,28678 \mathrm{yrs} \approx 17.280 \mathrm{yrs} \end{aligned}$
<br/>Therefore, the time taken to change 200 me to $25 \mathrm{mg}$ is $17,286.78 \mathrm{yr}$, which is very close to option (d) 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a55.png"/></div>


title: the-halflife-period-of-53-i-125-is-60-days-the-radioactivity-after-180-days-will-be
slug: the-halflife-period-of-53-i-125-is-60-days-the-radioactivity-after-180-days-will-be-72693
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The half-life period of ${ }_{53} I^{125}$ is 60 days. The radioactivity after 180 days will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$25 \%$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$12.5 \%$</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">$33.3 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.0 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$12.5 \%$</span> </div>

answer:
<div class="solution">Let, initial concentration $(a)=100$
<br/>Given, half-life $\left(t_{1 / 2}\right)=60$ days 
<br/>To find, radioactivity,
<br/>i.e. $(a-x)$ after time $T$ ( 180 days)
<br/>$$
<br/>\because \quad T=n \times t_{1 / 2}
<br/>$$
<br/>where, $n=$ no. of half-lives
<br/>$$
<br/>\begin{aligned}
<br/>180 &amp;=n \times 60 \\
<br/>n &amp;=\frac{180}{60}=3
<br/>\end{aligned}
<br/>$$
<br/>and $(a-x)=\frac{a}{2^{n}}=\frac{100}{2^{3}}=\frac{100}{8}$
<br/>$$
<br/>(a-x)=12.5 \%
<br/>$$
<br/>Hence, (b) is the correct answer.</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-heat-of-neutralisation-of-a-strong-base-acid-is-137-kcal-the-heat-released-when-06-mole-hcl-solution-is-added-to-a-025-mole-of-naoh-is

slug:
the-heat-of-neutralisation-of-a-strong-base-acid-is-137-kcal-the-heat-released-when-06-mole-hcl-solution-is-added-to-a-025-mole-of-naoh-is-29873

question:
<div class="question">The heat of neutralisation of a strong base acid is $13.7 \mathrm{kcal}$. The heat released when 0.6 mole $\mathrm{HCl}$ solution is added to a 0.25 mole of NaOH is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3.425 kcal</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8.22 kcal</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">11.645 kcal</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$13.7 \mathrm{kcal}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">3.425 kcal</span> </div>

answer:
<div class="solution">Given, the heat of neutralisation of a strong base and a strong acid is $13.7 \mathrm{kcal}$.
<br/>The reaction of neutralisation is as follows
<br/>$$
<br/>\begin{array}{l}
<br/>\mathrm{HCl}+\mathrm{NaOH} \longrightarrow \mathrm{NaCl}+\mathrm{H}_{2} \mathrm{O} = -137kcal \\
<br/>1 \mathrm{mol} \quad 1 \mathrm{mol} 
<br/>\end{array}
<br/>$$
<br/>According to question,
<br/>$$
<br/>\begin{array}{l}
<br/>\mathrm{HCl}+\mathrm{NaOH} \longrightarrow \mathrm{NaCl}+\mathrm{H}_{2} \mathrm{O} \\
<br/>0.6 \mathrm{mol} \quad 0.25 \mathrm{mol}
<br/>\end{array}
<br/>$$
<br/>In equation (i), NaOH acts as a limiting reagent. For 1 mole of NaOH and 1 mole of HCl, heat of neutralisation $=137 \mathrm{kcal}$
<br/>: For 0.25 mole of NaOH and 0.6 mole of $\mathrm{HCl}$, heat of neutralisation $=137 \times 0.25 \Rightarrow 3.425 \mathrm{kcal}$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-height-of-a-waterfall-is-50-m-if-g-98-ms-2-the-difference-between-the-temperature-at-the-top-and-the-bottom-of-the-waterfall-is

slug:
the-height-of-a-waterfall-is-50-m-if-g-98-ms-2-the-difference-between-the-temperature-at-the-top-and-the-bottom-of-the-waterfall-is-69870

question:
<div class="question">The height of a waterfall is $50 \mathrm{~m}$. If $g=9.8 \mathrm{~ms}^{-2}$ the difference between the temperature at the top and the bottom of the waterfall is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.17^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.17^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.117^{\circ} \mathrm{C}$</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">D</span> <span class="option-data">$1.43^{\circ} \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.117^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">$\frac{m g h}{J}=m s \Delta t \Rightarrow \Delta t=0.117^{\circ} \mathrm{C}$</div>


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-height-vertically-above-the-earths-surface-at-which-the-acceleration-due-to-gravity-becomes-1-of-its-value-at-the-surface-is-r-is-the-radius-of-th

slug:
the-height-vertically-above-the-earths-surface-at-which-the-acceleration-due-to-gravity-becomes-1-of-its-value-at-the-surface-is-r-is-the-radius-of-th-42991

question:
<div class="question">The height vertically above the earth's surface at which the acceleration due to gravity becomes $1 \%$ of its value at the surface is ( $\mathrm{R}$ is the radius of the Earth)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 \mathrm{R}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$9 \mathrm{R}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{R}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$20 \mathrm{R}$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } g^{\prime}=\frac{g}{\left(1+\frac{h}{\mathrm{R}}\right)^2} \Rightarrow \frac{g}{100}=\frac{g}{\left(1+\frac{h}{\mathrm{R}}\right)^2} \\<br/>&amp; 1+\frac{h}{\mathrm{R}}=10 \Rightarrow \frac{h}{\mathrm{R}}=9, h=9 \mathrm{R}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
the-human-eye-has-an-approximate-angular-resolution-of-58-1-0-4-rad-and-typical-photo-printer-prints-a-minimum-of-300-dip-dots-per-inch-1-inch-254-cm-

slug:
the-human-eye-has-an-approximate-angular-resolution-of-58-1-0-4-rad-and-typical-photo-printer-prints-a-minimum-of-300-dip-dots-per-inch-1-inch-254-cm-80369

question:
<div class="question">The human eye has an approximate angular resolution of $\theta=5.8 \times 10^{-4} \mathrm{rad}$ and typical photo printer prints a minimum of 300 dip (dots per inch, 1 inch $=2.54 \mathrm{~cm}$ ). At what minimal distance $d$ should a printed page be held so that one does not see the individual dots?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20.32 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$29.50 \mathrm{~cm}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$14.59 \mathrm{~cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6.85 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$14.59 \mathrm{~cm}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2MyM7L-E-5CCx_A5fTs5Mf9SCYDhO-ZhXu9uHk8o-uw.original.fullsize.png"/><br/><br/>$\theta=\frac{x}{d}$<br/>$5.8 \times 10^{-4}=\frac{2.54}{300} \times \frac{1}{d}$<br/>$d=14.59 \mathrm{~cm}$</div>


tags: ['Chemistry', 'Hydrogen', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-hydrides-of-the-first-elements-in-groups-15-17-namely-nh-3-h-2-o-and-hf-respectively-show-abnormally-high-values-for-melting-and-boiling-points-th

slug:
the-hydrides-of-the-first-elements-in-groups-15-17-namely-nh-3-h-2-o-and-hf-respectively-show-abnormally-high-values-for-melting-and-boiling-points-th-17710

question:
<div class="question">The hydrides of the first elements in groups $15-17,$ namely $\mathrm{NH}_{3}, \mathrm{H}_{2} \mathrm{O}$ and $\mathrm{HF}$ respectively show abnormally high values for melting and boiling points. This is due to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the ablility to form extensive intermolecula H-bonding</span> </div>

answer:
<div class="solution">Hydrides of $\mathrm{N}, \mathrm{O}$ and $\mathrm{F}$ because of the small size and high electronegativity of elements, have ability to form extensive intermolecular between two molecules) hydrogen bonding. Thus, a large amount of energy is required to break these bonds, le., the melting and boiling points of hydrides of these elements are abnormally high.</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
the-i-v-characteristics-of-a-metal-wire-at-two-different-temperatures-t-1-and-t-2-are-given-in-the-adjoining-figure-here-we-can-conclude-that-1

slug:
the-i-v-characteristics-of-a-metal-wire-at-two-different-temperatures-t-1-and-t-2-are-given-in-the-adjoining-figure-here-we-can-conclude-that-1-25403

question:
<div class="question">The $I$ - $V$ characteristics of a metal wire at two different temperatures $\left(T_{1}\right.$ and $T_{2}$ ) are given in the adjoining figure. Here, we can conclude that <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_q26.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$T_{1}&gt;T_{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$T_{1}     &lt; T_{2}$</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">$T_{1}=T_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$T_{1}=2 T_{2}$</span> </li> </ul>

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

answer:
<div class="solution">$R$ increases with temperature and slope of $V \cdot I$ graph gives resistance so $T_{1}     &lt; T_{2}$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
the-i-v-characteristics-of-a-metal-wire-at-two-different-temperatures-t-1-and-t-2-are-given-in-the-adjoining-figure-here-we-can-conclude-that

slug:
the-i-v-characteristics-of-a-metal-wire-at-two-different-temperatures-t-1-and-t-2-are-given-in-the-adjoining-figure-here-we-can-conclude-that-64385

question:
<div class="question">The $I$ - $V$ characteristics of a metal wire at two different temperatures $\left(T_{1}\right.$ and $T_{2}$ ) are given in the adjoining figure. Here, we can conclude that <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_q26.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$T_{1}&gt;T_{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$T_{1}     &lt; T_{2}$</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">$T_{1}=T_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$T_{1}=2 T_{2}$</span> </li> </ul>

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

answer:
<div class="solution">$R$ increases with temperature and slope of $V \cdot I$ graph gives resistance so $T_{1}     &lt; T_{2}$</div>


title: the-important-advantages-of-lintz-and-donawitz-ld-process-for-the-manufacture-of-steel-is-are
slug: the-important-advantages-of-lintz-and-donawitz-ld-process-for-the-manufacture-of-steel-is-are-15681
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The important advantage(s) of Lintz and Donawitz (L.D.) process for the manufacture of steel is (are)</div>

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

answer:
<div class="solution">Lintz and Donawitz (L.D.) process is an important process for manufacturing of steel. In India, Rourkela steel plant is based upon this process.
<br/>This process is carried out in a converter similar to bessemer converter having lining (usually refractory made of $\mathrm{Mg} \mathrm{O}$ and $(\mathrm{CaO})$.
<br/>In this method scrap iron can be used along with flux in converter : On passing oxygen gas at $4-12$ atm pressure into converter, temperature of $2000-2500^{\circ} \mathrm{C}$ range produced which separates C. Si, Mn etc., impurities as slag from
<br/>iron quickly remaining pure iron is mixed with spiegeleisen (alloy Fe, $\mathrm{Mn}$, C) to produce better quality steel.</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
the-incentre-of-an-equilateral-triangle-is-11-and-the-equation-of-ene-side-is-3-x-4-y-3-0-then-the-equation-of-the-circumeircle-of-the-triangle-is

slug:
the-incentre-of-an-equilateral-triangle-is-11-and-the-equation-of-ene-side-is-3-x-4-y-3-0-then-the-equation-of-the-circumeircle-of-the-triangle-is-19839

question:
<div class="question">The incentre of an equilateral triangle is (1,1) and the equation of ene side is $3 x+4 y+3=0 .$ Then, the equation of the circumeircle of the triangle is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}-2 x-2 y-14=0$</span> </div>

answer:
<div class="solution">since, triangle is equilateral the refore incentre
<br/>(1,1) lies on the centroid of the $\Delta A B C$. $\therefore G D=$ Length of perpendicular from the point $G(1,1)$ to the line $3 x+4 y+3=0$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a68.png"/> <br/> <br/>$A G=2 G D=4$
<br/>$\therefore$ Equation of circumcircle with centre at (1,1)
<br/>$$
<br/>\begin{array}{l}
<br/>\text { and radius = } \begin{aligned}
<br/>4 \text { units } \\
<br/>(x-1)^{2}+(y-1)^{2} &amp;=4^{2} \\
<br/>\Rightarrow \quad x^{2}+y^{2}-2 x-2 y-14 &amp;=0
<br/>\end{aligned}
<br/>\end{array}
<br/>$$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
the-increase-in-rate-constant-of-a-chemical-reaction-with-increasing-temperature-is-are-due-to-the-facts-that

slug:
the-increase-in-rate-constant-of-a-chemical-reaction-with-increasing-temperature-is-are-due-to-the-facts-that-88214

question:
<div class="question">The increase in rate constant of a chemical reaction with increasing temperature is (are) due to the fact(s) that</div>

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

answer:
<div class="solution">The increase in rate constant of a chemical reaction with increasing temperature are due to the following facts. The number of collisions among the reactant molecules increases with increasing temperature.
<br/>The number of reactant molecules acquiring the activation energy increases with increasing temperature.</div>


title: the-increasing-order-of-o-n-o-bond-angle-in-the-species-no-2-no-2-and-no-2-is
slug: the-increasing-order-of-o-n-o-bond-angle-in-the-species-no-2-no-2-and-no-2-is-32482
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The increasing order of $\mathrm{O}-\mathrm{N}-\mathrm{O}$ bond angle in the species $\mathrm{NO}_{2}, \mathrm{NO}_{2}^{+}$ and $\mathrm{NO}_{2}{ }^{-}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{NO}_{2}{ }^{+}     &lt; \mathrm{NO}_{2} &lt; \mathrm{NO}_{2}^{-}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{NO}_{2}     &lt; \mathrm{NO}_{2}^{-} &lt; \mathrm{NO}_{2}^{+}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{NO}_{2}^{+}     &lt; \mathrm{NO}_{2}^{-} &lt; \mathrm{NO}_{2}$</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">Option D is correct. As the number of lone pair of electrons increases, bond angle decreases.
<br/>$\mathrm{NO}_{2}^{+}$ ion is isoelectronic with $\mathrm{CO}_{2}$, molecule. It is a linear ion and its central atom $\left(\mathrm{N}}\right)$ undergoes sp-hybridisation. Hence, its bond angle is 180 In $\mathrm{NO}_{2}^{-}$ ion, $\mathrm{N}$ -atom undergoes $\mathrm{sp}^{2}$ hybridisation The angle between hybrid orbital should be 120 but one lone pair of electrons is lying on N-atom hence bond angle decreases to $115^{\circ}$ In $\mathrm{NO}_{2}$ molecule, N-atom has one unpared electron in $s p^{2}$ -hybrid orbital. The bond angle should be $120^{\circ}$ but actually, it is $132^{\circ} .$ it may be due to one unpaired electron in $s p^{2}$ -hybrid orbtal
<br/>Therefore, the increasing order of bond angle is
<br/>$$
<br/>\begin{array}{rrr}
<br/>\mathrm{NO}_{2}^{-} &amp;      &lt; &amp; \mathrm{NO}_{2} &amp; &lt; &amp; \mathrm{NO}_{2}^{+} \\
<br/>115^{\circ} &amp; &amp; 132^{\circ} &amp; &amp; 180^{\circ}
<br/>\end{array}
<br/>$$</div>


title: the-indicated-atom-is-not-a-nucleophilic-site-in-1
slug: the-indicated-atom-is-not-a-nucleophilic-site-in-1-79143
question: <div class="question">The indicated atom is not a nucleophilic site in</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{BH}_{4}^{-}$  (atom = B)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{Mg}$  (atom = C)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{OH}$  (atom = O)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{NH}_{2}$  (atom = N)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{BH}_{4}^{-}$  (atom = B)</span> </div>

answer:
<div class="solution">In BH ${ }_{4}^{-}$ indicated atom is not a nucleophilic
<br/>site because there is no lone-pair on $\mathrm{B}^{\prime}$ atom.
<br/>Hence, the option (a) is correct answer.</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-initial-pressure-and-volume-of-a-given-mass-of-an-ideal-gas-with-c-v-c-p-taken-in-a-cylinder-fitted-with-a-piston-are-p-0-and-v-0-respectively-at-

slug:
the-initial-pressure-and-volume-of-a-given-mass-of-an-ideal-gas-with-c-v-c-p-taken-in-a-cylinder-fitted-with-a-piston-are-p-0-and-v-0-respectively-at-23071

question:
<div class="question">The initial pressure and volume of a given mass of an ideal gas $\left(\right.$ with $\frac{C_{p}}{C_{V}}=\gamma),$ taken in a cylinder fitted with a piston, are $p_{0}$ and $V_{0}$ respectively. At this stage the gas has the same temperature as that of the surrounding medium which is $T_{0}$. It is adiabatically compressed to a volume equal to $\frac{V_{0}}{2}$ Subsequently the gas is allowed to come to thermal equilibrium with the surroundings. What is the heat released to the surrounding?</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">$(2^{\gamma-1}-1) \frac{p_{0} V_{0}}{\gamma-1}$</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">$\gamma {p_{0}} {V_{0}} ln2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{{p_{0}} V_{0}}{2(\gamma-1)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$(2^{\gamma-1}-1) \frac{p_{0} V_{0}}{\gamma-1}$</span> </div>

answer:
<div class="solution">The equation of state for an ideal gas undergoing adiabatic process is given as,
<br/>$$
<br/>T V^{\gamma-1}=\text { constant }
<br/>$$
<br/>Let the temperature after adiabatic compression as given in the question be $T$ then,
<br/>$$
<br/>T_{0} V^{\gamma-1}=T\left(\frac{V_{0}}{2}\right)^{\gamma-1}
<br/>$$
<br/>$\Rightarrow \quad T=T_{0} 2^{\gamma-1}$
<br/>Now, heat released at volume $\frac{V_{0}}{2}$ to achieve
<br/>temperature $T_{0}$ The net heat released can be determined by the
<br/>equation.
<br/>$$
<br/>\begin{aligned}
<br/>\Delta Q &amp;=\mu C_{V} \Delta T \\
<br/>&amp;=\mu \times \frac{R}{\gamma-1} \quad\left(T_{0} 2^{\gamma-1}-T_{0}\right) \\
<br/>&amp;=\frac{\mu R T_{0}}{\gamma-1}\left(2^{\gamma-1}-1\right)\left(\because \mu T_{0} R=p_{0} V_{0}\right)
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Heat released $=\frac{p_{0} V_{0}}{\gamma-1}\left(2^{\gamma-1}-1\right)$</div>


title: the-inputs-to-the-digital-circuit-are-as-shown-below-the-output-y-is
slug: the-inputs-to-the-digital-circuit-are-as-shown-below-the-output-y-is-88277
tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The inputs to the digital circuit are as shown below. The output $Y$ is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q25.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A+B+\bar{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(A+B) \bar{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\bar{A}+\bar{B}+\bar{C}$</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">$\bar{A}+\bar{B}+C$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\bar{A}+\bar{B}+\bar{C}$</span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { The output of digital circuit can be given as, }\\
<br/>&amp;Y=\overline{A B}+\bar{C}\\
<br/>&amp;\text { It will be same as, } Y=\bar{A}+\bar{B}+\bar{C}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-insulated-plates-of-a-charged-parallel-plate-capacitor-with-small-separation-between-the-plates-are-approaching-each-other-due-to-electrostatic-at

slug:
the-insulated-plates-of-a-charged-parallel-plate-capacitor-with-small-separation-between-the-plates-are-approaching-each-other-due-to-electrostatic-at-62720

question:
<div class="question">The insulated plates of a charged parallel plate capacitor (with small separation between the plates) are approaching each other due to electrostatic attraction. Assuming no other force to be operative and no radiation taking place, which of the following graphs approximately shows the variation with time $(t)$ of the potential difference $(V)$ between the plates?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q34_opta.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q34_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/2018_phy_q34_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q34_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q34_opta.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a34.png"/> <br/> <br/>Force $=\frac{1}{2} \cdot \frac{\sigma}{\varepsilon_{0}} q$
<br/>$\therefore$ Relative acceleration of plates. $a_{\mathrm{rel}}=\frac{2 F}{M}$
<br/>Potential difference, $V=E d$ where, $d=d_{0}-\frac{1}{2} a_{\text {rel }} t^{2}$
<br/>(: plates are moving)
<br/>$\therefore$
<br/>$V=E\left[d_{0}-\frac{1}{2} a_{\mathrm{rel}} t^{2}\right]$
<br/>This is a equation of a parabola with downward concavity.</div>


title: the-integrating-factor-of-the-differential-equation-1-x-2-d-x-d-y-y-e-t-a-n-1-x-is
slug: the-integrating-factor-of-the-differential-equation-1-x-2-d-x-d-y-y-e-t-a-n-1-x-is-14584
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The integrating factor of the differential equation $\left(1+x^{2}\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}$ is</div>

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

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

answer:
<div class="solution">Given equation is
<br/>$$
<br/>\left(1+x^{2}\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}
<br/>$$
<br/>or it can be rewritten as
<br/>$$
<br/>\frac{d y}{d x}+\frac{1}{\left(1+x^{2}\right)} y=\frac{e^{\tan ^{-1} x}}{1+x^{2}}
<br/>$$
<br/>It is a linear differential equation of the form of
<br/>$$
<br/>\frac{d y}{d x}+P y=Q
<br/>$$
<br/>where, $P=\frac{1}{1+x^{2}}$ and $Q=\frac{e^{\tan ^{-1} x}}{1+x^{2}}$
<br/>$$
<br/>=e^{\int \frac{1}{1+x^{2}} d x}=e^{\tan ^{-1} x}
<br/>$$</div>


title: the-integrating-factor-of-the-differential-equation-3-x-lo-g-e-x-d-x-d-y-y-2-lo-g-e-x-is-given-by
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
slug:
the-integrating-factor-of-the-differential-equation-3-x-lo-g-e-x-d-x-d-y-y-2-lo-g-e-x-is-given-by-14459

question:
<div class="question">The integrating factor of the differential equation $3 x \log _{e} x \frac{d y}{d x}+y=2 \log _{e} x$ is given by</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\log _{e} x\right)^{1 / 3}$</span> </div>

answer:
<div class="solution">Given, $3 x \log _{e} x \frac{d y}{d x}+y=2 \log _{e} x$
<br/>Dividing both sides by $3 x \log _{e} x,$ we get $\frac{d y}{d x}+\frac{1}{3 x \log _{e} x} y=\frac{2 \log _{e} x}{3 x \log _{e} x}$
<br/>$\Rightarrow \frac{d y}{d x}+\frac{1}{3 x \log _{e} x} y=\frac{2}{3 x}$
<br/>which is linear form $\frac{d y}{d x}+P y=Q,$ where $P$ and $Q$ are function of $x$ and the integrating factor is given by the following formula $e^{\int P d x}$.
<br/>$\therefore$
<br/>$\mathrm{IF}=e^{\int \frac{1}{3 x \log _{t} x}} d x$
<br/>$\begin{aligned} \text { Put } \log _{e} x &amp;=t \Rightarrow \frac{1}{x} d x=d t \\ &amp;=e^{1 / 3} \int \frac{d t}{t}=e^{1 / 3} \log t \\ &amp;=e^{\log 1^{1 / 3}}=t^{1 / 3}=\left(\log _{e} x\right)^{-1 / 3} \end{aligned}$</div>


title: the-integrating-factor-of-the-differential-equation-d-x-d-y-3-x-2-tan-1-y-x-3-1-y-2-0-is
slug: the-integrating-factor-of-the-differential-equation-d-x-d-y-3-x-2-tan-1-y-x-3-1-y-2-0-is-29980
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The integrating factor of the differential equation $\frac{d y}{d x}+\left(3 x^{2} \tan ^{-1} y -x^{3}\right)\left(1+y^{2}\right)=0$ is</div>

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

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

answer:
<div class="solution">Given, $\frac{d y}{d x}=-\left(3 x^{2} \tan ^{-1} y-x^{3}\right)\left(1+y^{2}\right)$
<br/>$\Rightarrow \quad \frac{d y}{d x}=x^{3}\left(1+y^{2}\right)-3 x^{2}\left(\tan ^{-1} y\right)\left(1+y^{2}\right)$
<br/>$\Rightarrow \frac{1}{\left(1+y^{2}\right)} \cdot \frac{d y}{d x}=x^{3}-3 x^{2} \tan ^{-1} y$
<br/>$\Rightarrow \frac{1}{1+y^{2}} \cdot \frac{d y}{d x}+3 x^{2} \tan ^{-1} y=x^{3}$
<br/>Put $\tan ^{-1} y=t \Rightarrow \frac{1}{1+y^{2}} \cdot \frac{d y}{d x}=\frac{d t}{d x}$
<br/>$\therefore \quad \frac{d t}{d x}+3 t x^{2}=x^{3}$
<br/>which is linear differential equation in $t$.
<br/>Now.
<br/>$\mathrm{I} \quad \mathrm{F}=e^{\int 3 \mathrm{x}^{2} d x}=e^{x^{3}}$</div>


title: the-integrating-factor-of-the-differential-equation-x-lo-g-x-d-x-d-y-y-2-lo-g-x-is-given-by
slug: the-integrating-factor-of-the-differential-equation-x-lo-g-x-d-x-d-y-y-2-lo-g-x-is-given-by-79469
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The integrating factor of the differential equation $x \log x \frac{d y}{d x}+y=2 \log x$ is given by</div>

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

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

answer:
<div class="solution">Hints : $\frac{d y}{d x}+\frac{1}{x \log x} \cdot y=\frac{2}{x}$<br/>$$<br/>\begin{aligned}<br/>&amp; \text { If }=e^{\int \frac{1}{x \log x} d x}=e^{\int \frac{1 / x}{\log x} d x} \\<br/>&amp; =e^{\log (\log x)}=\log x<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
the-integrating-factor-of-the-first-order-differential-equation-x-2-x-2-1-d-x-d-y-x-x-2-1-y-x-2-1-is

slug:
the-integrating-factor-of-the-first-order-differential-equation-x-2-x-2-1-d-x-d-y-x-x-2-1-y-x-2-1-is-92029

question:
<div class="question">The integrating factor of the first order differential equation $x^{2}\left(x^{2}-1\right) \frac{d y}{d x}+x\left(x^{2}+1\right) y=x^{2}-1$ is</div>

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

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

answer:
<div class="solution">We have,
<br/>$x^{2}\left(x^{2}-1\right) \frac{d y}{d x}+x\left(x^{2}+1\right) y=x^{2}-1$
<br/>$\Rightarrow \quad \frac{d y}{d x}+\frac{x^{2}+1}{x\left(x^{2}-1\right)} y=\frac{1}{x^{2}}$
<br/>$\begin{aligned} I F=&amp; e^{\int \frac{x^{2}+1}{x\left(x^{2}-1\right)} d x} \\ &amp;=e^{\int \frac{x^{2}+1}{x(x-1)(x+1)}} d x \end{aligned}$
<br/>Let $\frac{x^{2}+1}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}$
<br/>$\Rightarrow x^{2}+1=A(x-1)(x+1)+B x(x+1)$ $\quad+\operatorname{Cx}(x-1)$
<br/>$\begin{array}{ll}\text { Put } x=0, &amp; \\ \therefore \quad &amp; 1=-A \\ \Rightarrow &amp; A=-1 \\ \Rightarrow &amp; 
<br/>\end{array}$
<br/>Put $x=1
<br/>\therefore \quad 2=2 B$
<br/>$\begin{array}{ll}\therefore &amp; B=1 \\ \Rightarrow &amp; x=-1\end{array}$
<br/>$\therefore \quad 2=2 C$
<br/>$\Rightarrow C=1$
<br/>$\therefore \frac{x^{2}+1}{x(x-1)(x+1)}=\frac{-1}{x}+\frac{1}{x-1}+\frac{1}{x+1}$
<br/>$\therefore \quad I F=e^{\int\left(\frac{-1}{x}+\frac{1}{x-1}+\frac{1}{x+1}\right) d x}$
<br/>$=e^{[-\log x+\ln (x-1)+\log (x+1)]}$
<br/>$=e^{\ln \left(\frac{x^{2}-1}{x}\right)}=\frac{x^{2}-1}{x}$
<br/>$=x-\frac{1}{x}$</div>


tags: ['Physics', 'Wave Optics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-intensity-of-a-sound-appears-to-an-observer-to-be-periodic-which-of-the-following-can-be-the-cause-of-it

slug:
the-intensity-of-a-sound-appears-to-an-observer-to-be-periodic-which-of-the-following-can-be-the-cause-of-it-57321

question:
<div class="question">The intensity of a sound appears to an observer to be periodic. Which of the following can be the cause of it?</div>

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

answer:
<div class="solution">The intensity of a sound source appears be periodic due to
<br/>(i) source intensity is periodic.
<br/>(ii) source is producing a sound composed of two nearby frequencies.</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">98</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-intensity-of-light-emerging-from-one-of-the-slits-in-a-youngs-double-slit-experiment-is-found-to-be-15-limes-the-intensity-of-light-emerging-from-

slug:
the-intensity-of-light-emerging-from-one-of-the-slits-in-a-youngs-double-slit-experiment-is-found-to-be-15-limes-the-intensity-of-light-emerging-from-90736

question:
<div class="question">The intensity of light emerging from one of the slits in a Young's double slit experiment is found to be $1.5$ limes the intensity of light emerging from the other slit. What will be the approximate ratio of intensity of an interference maximum to that of an interference minimum</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.25$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">98</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$9.9$</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>$I_{1}=1.51_{2}$
<br/>$\frac{\left.\right|_{1}}{I_{2}}=\frac{3}{2} \cdot \quad \frac{l_{\max }}{I_{\min }}=\left(\frac{\sqrt{L_{1}}+\sqrt{L_{2}}}{\sqrt{I_{1}}-\sqrt{L_{2}}}\right)^{2}=\left(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right)^{2}=98$</div>


tags: ['Physics', 'Magnetic Properties of Matter', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.6</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-intensity-of-magnetization-of-a-bar-magnet-is-50-1-0-4-am-1-the-magnetic-length-and-the-area-of-crosssection-of-the-magnet-are-12-cm-and-1-cm-2-re

slug:
the-intensity-of-magnetization-of-a-bar-magnet-is-50-1-0-4-am-1-the-magnetic-length-and-the-area-of-crosssection-of-the-magnet-are-12-cm-and-1-cm-2-re-14084

question:
<div class="question">The intensity of magnetization of a bar magnet is $5.0 \times 10^{4} \mathrm{Am}^{-1}$. The magnetic length and the area of cross-section of the magnet are $12 \mathrm{cm}$ and $1 \mathrm{cm}^{2}$ respectively. The magnitude of magnetic moment of this bar magnet is (in SI unit)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.24</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2.4</span> </li> </ul>

answer:
<div class="solution">We know that Intensity of magnetisation<br/>$I=\frac{M}{V}$<br/>[where $M=$ magnetic moment, $V=$ volume]<br/>So<br/>$\begin{aligned}<br/>M=I V &amp;=5.0 \times 10^{4} \times \frac{12}{100} \times \frac{1}{(100)^{2}} \\<br/>&amp;=60 \times 10^{4} \times 10^{-6}=0.6 \mathrm{Am}^{2}<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-intercept-on-the-line-yx-by-the-circle-x2y22-x0-is-a-b-equation-of-the-circle-with-a-b-as-diameter-is

slug:
the-intercept-on-the-line-yx-by-the-circle-x2y22-x0-is-a-b-equation-of-the-circle-with-a-b-as-diameter-is-32275

question:
<div class="question">The intercept on the line \(y=x\) by the circle \(x^2+y^2-2 x=0\) is \(A B\). Equation of the circle with \(A B\) as diameter is</div>

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

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

answer:
<div class="solution">Hints: \(2 x^2-2 x=0 \quad x(x+1)=0 \quad x=0,1 ; y=0,1\)<br/>\((0,0),(1,1)\) as diametric ends<br/>\(\begin{aligned}<br/>&amp; (x-0)(x-1)+(y+0)(y-1)=0 \\<br/>&amp; x^2+y^2-x-y=0<br/>\end{aligned}\)</div>


title: the-intermediate-image-formed-by-the-objective-of-a-compound-microscope-is
slug: the-intermediate-image-formed-by-the-objective-of-a-compound-microscope-is-45894
tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The intermediate image formed by the objective of a compound microscope is</div>

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

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

answer:
<div class="solution">The intermediate image formed by the objective of a compound microscope is real, inverted and magnified.</div>


title: the-intermediate-j-in-the-following-wittig-reaction-is
slug: the-intermediate-j-in-the-following-wittig-reaction-is-74717
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The intermediate $J$ in the following Wittig reaction is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q27.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q27_opta.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q27_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/2014_che_q27_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q27_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q27_opta.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a27.png"/></div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-ionization-energy-of-hydrogen-is-136-ev-the-energy-of-the-photon-released-when-an-electron-jumps-from-the-first-excited-state-n-2-to-the-ground-st

slug:
the-ionization-energy-of-hydrogen-is-136-ev-the-energy-of-the-photon-released-when-an-electron-jumps-from-the-first-excited-state-n-2-to-the-ground-st-28853

question:
<div class="question">The ionization energy of hydrogen is $13.6 \mathrm{eV}$. The energy of the photon released when an electron jumps from the first excited state $(n=2)$ to the ground state of a hydrogen atom is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3.4 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4.53 \mathrm{eV}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$10.2 \mathrm{eV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$13.6 \mathrm{eV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10.2 \mathrm{eV}$</span> </div>

answer:
<div class="solution">Energy required to removed electron in th $(n=2)$ stage The energy of the photon
<br/>$\begin{array}{l}
<br/>=R\left(\frac{1}{1^{2}}-\frac{1}{n^{2}}\right) \Rightarrow 13.6 \mathrm{eV}\left(\frac{1}{1}-\frac{1}{4}\right) \\
<br/>=13.6 \mathrm{eV}\left(\frac{4-1}{4}\right)=13.6 \mathrm{eV} \times \frac{3}{4}=\frac{40.8}{4} \mathrm{eV} \\
<br/>=10.2 \mathrm{eV}
<br/>\end{array}$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-ionization-energy-of-the-hydrogen-atom-is-136-ev-the-potential-energy-of-the-electron-in-n-2-state-of-hydrogen-atom-is

slug:
the-ionization-energy-of-the-hydrogen-atom-is-136-ev-the-potential-energy-of-the-electron-in-n-2-state-of-hydrogen-atom-is-56456

question:
<div class="question">The ionization energy of the hydrogen atom is 13.6 eV. The potential energy of the electron in $n=2$ state of hydrogen atom is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$+3.4 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-3.4 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$+6.8 \mathrm{eV}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-6.8 \mathrm{eV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-6.8 \mathrm{eV}$</span> </div>

answer:
<div class="solution">Given, $E_{n}=13.6 \mathrm{eV}$<br/>Energy of an electron in $n^{\text {th }}$ state<br/>$E_{n}=\frac{-13.6 z^{2} \mathrm{eV}}{n^{2}}$<br/>$\therefore$ Energy of an electron in $n=2$ state<br/>$E_{2}=\frac{18-6 z^{2}}{(2)^{2}}=-34 e V$<br/>So.<br/>$\begin{aligned}<br/>P E &amp;=2 E_{n=2} \\<br/>&amp;=2 \times(-3.4) \\<br/>&amp;=-6.8 \mathrm{eV}<br/>\end{aligned}$</div>


title: the-isomerisation-of-1-butyne-to-2-butyne-can-be-achieved-by-treatment-with
slug: the-isomerisation-of-1-butyne-to-2-butyne-can-be-achieved-by-treatment-with-60754
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The isomerisation of 1 -butyne to 2 -butyne can be achieved by treatment with</div>

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

answer:
<div class="solution">Isomerisation of 1 -butyne to 2 -butyne can be achived by treatment with ethanolic KOH
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a59.png"/></div>


title: the-iupac-name-of-is-8
slug: the-iupac-name-of-is-8-70483
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The IUPAC name of <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/B6ZsUNoihElF4umnhsi9S16ASLHPC8yue3_bhYLFN-0.original.fullsize.png"/><br/> is :</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2-Ethylbut-2-enal</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9DXWZwAsReGo-uW-y23xvCQXZgpw4H05liZhsvnFNa0.original.fullsize.png"/><br/></div>


title: the-iupac-name-of-the-compound-x-is
slug: the-iupac-name-of-the-compound-x-is-82213
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The IUPAC name of the compound $X$ is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q2.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2,2 -dimethyl-4-oxopentanenitrile</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a2.png"/> <br/>2,2-dimethyl-4-oxopentanenitrile</div>


title: the-iupac-name-of-the-following-molecule-is
slug: the-iupac-name-of-the-following-molecule-is-53609
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The IUPAC name of the following molecule is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q6.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">5,6 -dimethyihept-2-ene</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a6.png"/> <br/> <br/>The longest chain contains 7 C-atom, so the root word is hept and suffix -ene is added for double bond along with its position. Since, two methyl goups are also present at 5 and 6 positions, so the correct IUPAC name of the compound is 5.6 -dimethylhept-2-ene.</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
the-kinetic-energy-e-k-of-a-particle-moving-along-x-axis-varies-with-its-position-x-as-shown-in-the-figure-the-force-acting-on-the-particle-at-x-10-m-

slug:
the-kinetic-energy-e-k-of-a-particle-moving-along-x-axis-varies-with-its-position-x-as-shown-in-the-figure-the-force-acting-on-the-particle-at-x-10-m-43555

question:
<div class="question">The kinetic energy $\left(\mathrm{E}_{\mathrm{k}}\right)$ of a particle moving along $X$-axis varies with its position $(X)$ as shown in the figure. The force acting on the particle at $X=10 \mathrm{~m}$ is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8b2s3wWt6XYjxM3EvC7Mn128aVSy6ahtCztOaaAWwkQ.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \hat{\mathrm{i} N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$97.5 \hat{\mathrm{i} N}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-5 \hat{\mathrm{i} N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-5 \hat{\mathrm{i} N}$</span> </div>

answer:
<div class="solution">$$<br/>\frac{d k}{d x}=\frac{d\left(\frac{1}{2} m v^2\right)}{d x}=\frac{1}{2} m \not{2} v \frac{d v}{d x}=m a=F<br/>$$<br/>So, Force is slope at $x=10 m \quad F=-\frac{20}{4}=-5 \hat{i}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-kinetic-study-of-a-reaction-like-v-a-p-at-300-k-provides-the-following-curve-where-concentration-is-taken-in-mol-dm-3-and-time-in-min-identify-the

slug:
the-kinetic-study-of-a-reaction-like-v-a-p-at-300-k-provides-the-following-curve-where-concentration-is-taken-in-mol-dm-3-and-time-in-min-identify-the-76047

question:
<div class="question">The kinetic study of a reaction like $v A \rightarrow P$ at $300 \mathrm{K}$ provides the following curve, where concentration is taken in mol $\mathrm{dm}^{-3}$ and time in min.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q72.png"/> <br/>Identify the correct order $(n)$ and rate constant $(k)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$n=0, k=4.0 \mathrm{mol} \mathrm{dm}^{-3} \mathrm{min}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$n=1 / 2, k=2.0 \mathrm{mol}^{1 / 2} \mathrm{dm}^{-3 / 2} \mathrm{min}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$n=1, k=80 \mathrm{min}^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$n=2, k=16.0 \mathrm{dm}^{3} \mathrm{mol}^{-1} \mathrm{min}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$n=2, k=16.0 \mathrm{dm}^{3} \mathrm{mol}^{-1} \mathrm{min}^{-1}$</span> </div>

answer:
<div class="solution">From the given graph Slope $=\frac{(\text { rate })^{1 / 2}}{[A]}=4$
<br/>$\because$ rate $(r)=k[A]^{n}$
<br/>Where, $k=$ rate constant $[A]=$ concentration at time $t$ (in min).
<br/>$n=$ order.
<br/>According to the graph $(n=2)$ $\therefore \quad k=\frac{\text { rate }}{[A]^{n}}=[4]^{2}=16$
<br/>Hence, $n=2$ and
<br/>$$
<br/>k=16.0 \mathrm{dm}^{3} \mathrm{mol}^{-1} \min^{-1}
<br/>$$
<br/>and option (d) is the correct answer.</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-law-of-motion-of-a-body-moving-along-a-straight-line-is-x-2-1-vt-x-being-its-distance-from-a-fixed-point-on-the-line-at-time-t-and-v-is-its-veloci

slug:
the-law-of-motion-of-a-body-moving-along-a-straight-line-is-x-2-1-vt-x-being-its-distance-from-a-fixed-point-on-the-line-at-time-t-and-v-is-its-veloci-38488

question:
<div class="question">The law of motion of a body moving along a straight line is $x=\frac{1}{2}$ vt. $x$ being its distance from a fixed point on the line at time $t$ and $v$ is its velocity there, Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">acceleration $f$ is constant</span> </div>

answer:
<div class="solution">We have, $\quad x=\frac{1}{2} v t$ $\Rightarrow \quad x=\frac{1}{2} \frac{d x}{d t} t$
<br/>$\left[\because v=\frac{d x}{d t}\right]$
<br/>$\Rightarrow \quad \frac{2 d t}{t}=\frac{d x}{x}$
<br/>$\Rightarrow \quad 2 \cdot \int \frac{d t}{t}=\int \frac{d x}{x}$
<br/>$\Rightarrow \quad 2 \log |t|+\log |c|=\log |x|$
<br/>$\Rightarrow \quad \log \left(t^{2} \cdot c\right)=\log x$
<br/>$\Rightarrow \quad x=t^{2} c$
<br/>$\Rightarrow$
<br/>$\frac{d x}{d t}=2 t c$
<br/>$\Rightarrow$
<br/>$\frac{d^{2} x}{d t^{2}}=2$
<br/>$\Rightarrow$ acceleration $f$ is constant.</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-least-distance-of-vision-of-a-long-sighted-person-is-60-cm-by-using-a-spectacle-lens-this-distance-is-reduced-to-12-cm-the-power-of-the-lens-is

slug:
the-least-distance-of-vision-of-a-long-sighted-person-is-60-cm-by-using-a-spectacle-lens-this-distance-is-reduced-to-12-cm-the-power-of-the-lens-is-79532

question:
<div class="question">The least distance of vision of a long sighted person is $60 \mathrm{cm}$. By using a spectacle lens, this distance is reduced to $12 \mathrm{cm}$. The power of the lens is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$+5.0 D$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(20 / 3) \mathrm{D}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-(10 / 3)$ D</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$+2.0 \mathrm{D}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(20 / 3) \mathrm{D}$</span> </div>

answer:
<div class="solution">Here, $v=-60 \mathrm{cm}, u=-12 \mathrm{cm}$
<br/>$\therefore$ By using the relation, $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$
<br/>We have. $\frac{1}{f}=\frac{1}{-60}-\frac{1}{-12}$
<br/>$\Rightarrow \quad \frac{1}{f}=\frac{1}{15} \mathrm{cm}$ or $\frac{100}{15} \mathrm{m}$
<br/>So, the power of lens, $P=\frac{100}{15}=+\frac{20}{3} \mathrm{D}$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-least-positive-integer-n-such-that-cos-4-sin-4-sin-4-cos-4-n-is-an-identity-matrix-of-order-2-is

slug:
the-least-positive-integer-n-such-that-cos-4-sin-4-sin-4-cos-4-n-is-an-identity-matrix-of-order-2-is-61998

question:
<div class="question">The least positive integer $n$ such that $\left(\begin{array}{cc}\cos \frac{\pi}{4} &amp; \sin \frac{\pi}{4} \\ -\sin \frac{\pi}{4} &amp; \cos \frac{\pi}{4}\end{array}\right)^{n}$ is an identity matrix of order 2 is</div>

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

answer:
<div class="solution">We have, $\left(\begin{array}{cc}\cos \pi / 4 &amp; \sin \pi / 4 \\ -\sin \frac{\pi}{4} &amp; \cos \frac{\pi}{4}\end{array}\right)^{n}$
<br/>Let $A=\left(\begin{array}{cc}\frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}}\end{array}\right)$
<br/>$\Rightarrow \quad A=\left(\begin{array}{cc}k &amp; k \\ -k &amp; k\end{array}\right)$
<br/>$\Rightarrow A^{2}=\left(\begin{array}{cc}k &amp; k \\ -k &amp; k\end{array}\right)\left(\begin{array}{cc}k &amp; k \\ -k &amp; k\end{array}\right)=\left(\begin{array}{cc}0 &amp; 2 k^{2} \\ -2 k^{2} &amp; 0\end{array}\right)$
<br/>$A^{4}=\left(\begin{array}{cc}0 &amp; 2 k^{2} \\ -2 k^{2} &amp; 0\end{array}\right)\left(\begin{array}{cc}0 &amp; 2 k^{2} \\ -2 k^{2} &amp; 0\end{array}\right)$
<br/>$A^{4}=\left(\begin{array}{cc}-4 k^{4} &amp; 0 \\ 0 &amp; -4 k^{4}\end{array}\right)=\left(\begin{array}{cc}-4 \times \frac{1}{4} &amp; 0 \\ 0 &amp; -4 \times \frac{1}{4}\end{array}\right)$
<br/>$\Rightarrow A^{4}=\left(\begin{array}{cc}-1 &amp; 0 \\ 0 &amp; -1\end{array}\right)$
<br/>$\Rightarrow A^{B}=\left(\begin{array}{cc}-1 &amp; 0 \\ 0 &amp; -1\end{array}\right)\left(\begin{array}{cc}-1 &amp; 0 \\ 0 &amp; -1\end{array}\right)=\left(\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right)$</div>


title: the-least-positive-value-of-t-so-that-the-lines-x-t-y-16-0-and-y-x-are-concurrent-is
slug: the-least-positive-value-of-t-so-that-the-lines-x-t-y-16-0-and-y-x-are-concurrent-is-73048
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The least positive value of $t$, so that the lines $x=t+\alpha, y+16=0 \quad$ and $\quad y=\alpha x$ are concurrent, is</div>

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

answer:
<div class="solution">Consider the given equation of lines,
<br/>$$
<br/>\begin{array}{r}
<br/>x-(t+\alpha)=0 \\
<br/>y+16=0
<br/>\end{array}
<br/>$$
<br/>and
<br/>$$
<br/>-\alpha x+y=0
<br/>$$
<br/>Since, these lines are concurrent, therefore the system of equations is consistent.
<br/>Now,
<br/>$$
<br/>\left|\begin{array}{ccc}
<br/>1 &amp; 0 &amp; -(t+\alpha) \\
<br/>0 &amp; 1 &amp; 16 \\
<br/>-\alpha &amp; 1 &amp; 0
<br/>\end{array}\right|=0
<br/>$$
<br/>$$
<br/>\begin{array}{lr}
<br/>\Rightarrow &amp; 1(0-16)-(t+\alpha)(0+\alpha)=0 \\
<br/>\Rightarrow &amp; -16-\alpha(t+\alpha)=0 \\
<br/>\Rightarrow &amp; \alpha(t+\alpha)+16=0 \\
<br/>\Rightarrow &amp; \alpha^{2}+t \alpha+16=0
<br/>\end{array}
<br/>$$
<br/>Clearly, $\alpha$ should be real. $\therefore t^{2}-4 \times 16 \geq 0 \Rightarrow t^{2}-64 \geq 0$
<br/>Hence, least positive value of $t$ is 8.</div>


title: the-least-value-of-2-x-2-y-2-2-x-y-2-x-3-y-8-for-real-numbers-x-and-y-is
slug: the-least-value-of-2-x-2-y-2-2-x-y-2-x-3-y-8-for-real-numbers-x-and-y-is-51119
tags: ['Mathematics', 'Pair of Lines', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The least value of $2 x^{2}+y^{2}+2 x y+2 x-3 y+8$ for real
<br/>numbers $x$ and $y,$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-1 / 2$</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> </ul>

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

answer:
<div class="solution">$2 x^{2}+y^{2}+2 x y+2 x-3 y+8$
<br/>$=\frac{1}{2}\left|4 x^{2}+2 y^{2}+4 x y+4 x-6 y+16\right|$
<br/>$=\frac{1}{2}\left[\left(y^{2}-8 y\right)+\left(4 x^{2}+y^{2}+4 x y+4 x+2 y\right)+16\right].$
<br/>$=\frac{1}{2}\left|(y-4)^{2}+(2 x+y+1)^{2}-1\right|$
<br/>So, least value will be $-\frac{1}{2}$ at $y=4$ and $x=\frac{-5}{2}$</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
the-length-of-a-metal-wire-is-l-1-when-the-tension-is-t-1-and-l-2-when-the-tension-is-t-2-the-unstretched-length-of-wire-is

slug:
the-length-of-a-metal-wire-is-l-1-when-the-tension-is-t-1-and-l-2-when-the-tension-is-t-2-the-unstretched-length-of-wire-is-60547

question:
<div class="question">The length of a metal wire is $L_{1}$ when the tension is $T_{1}$ and $L_{2}$ when the tension is $T_{2}$ The unstretched length of wire is</div>

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

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

answer:
<div class="solution">Let the initial length of the metal wire is $L$.
<br/>The strain at tension $T_{1}$ is $\Delta L_{1}=L_{1}-L$
<br/>The strain at tension $T_{2}$ is $\Delta L_{2}=L_{2}-L$
<br/>Suppose, the Young's modulus of the wire is $Y$, then
<br/>$\frac{\frac{T_{1}}{A}}{\frac{\Delta L_{1}}{L}}=\rac{\frac{T_{2}}{A}}{\frac{\Delta L}{L}}$
<br/>where, $A$ is an cross-section of the wire assumed to be same at all the situations. $\begin{aligned} \Rightarrow \frac{T_{1}}{A} \times \frac{L}{\Delta L_{1}} &amp;=\frac{T_{2}}{A} \times \frac{L}{\Delta L_{2}} \\ \Rightarrow \quad \frac{T_{1}}{\left(L_{1}-L\right)} &amp;=\frac{T_{2}}{\left(L_{2}-L\right)} \\ &amp; T_{1}\left(L_{2}-L\right)=T_{2}\left(L_{1}-L\right) \cdot L=\frac{T_{2} L_{1}-T_{1} L_{2}}{T_{2}-T_{1}} \end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
the-length-of-an-open-organ-pipe-is-twice-the-length-of-another-closed-organ-pipe-the-fundamental-frequency-of-the-open-pipe-100-hz-the-frequency-of-t

slug:
the-length-of-an-open-organ-pipe-is-twice-the-length-of-another-closed-organ-pipe-the-fundamental-frequency-of-the-open-pipe-100-hz-the-frequency-of-t-27414

question:
<div class="question">The length of an open organ pipe is twice the length of another closed organ pipe. The fundamental frequency of the open pipe $100 \mathrm{Hz}$. The frequency of the third harmon of the closed pipe is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \mathrm{Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$200 \mathrm{Hz}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$300 \mathrm{Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$150 \mathrm{Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$300 \mathrm{Hz}$</span> </div>

answer:
<div class="solution">Let the length of closed organ pipe 1 be. Then,
<br/>$$
<br/>v_{n}=(2 n-1) \frac{v}{4 l}
<br/>$$
<br/>Its third harmonics (put $n=2$ ) in above question.
<br/>$$
<br/>v_{3}=\frac{3}{4} \frac{v}{1}
<br/>$$
<br/>Now, the length of open organ pipe is 21 , then
<br/>$$
<br/>v_{n}=n \frac{v}{2(2 l)}=\frac{n v}{4 l}
<br/>$$
<br/>Its fundamental frequency,
<br/>$$
<br/>v_{0}=\frac{v}{2 l}=100
<br/>$$
<br/>$$
<br/>v_{3}=\frac{3 v}{4 l}=3 \times 100=300 \mathrm{Hz}
<br/>$$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-length-of-conjugate-axis-of-a-hyperbola-is-greater-than-the-length-of-transverse-axis-then-the-eccentricity-e-is

slug:
the-length-of-conjugate-axis-of-a-hyperbola-is-greater-than-the-length-of-transverse-axis-then-the-eccentricity-e-is-32173

question:
<div class="question">The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity e is</div>

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

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

answer:
<div class="solution">Let the length of conjugate axis $=b$
<br/>and the length of transvers axis $=a$ Given that, $b&gt;a$
<br/>Now,$b^{2}&gt;a^{2}$
<br/>$\therefore$ Eccentricity $(e)=\sqrt{1+\frac{b^{2}}{a^{2}}}$
<br/>$e^{2}=\left(\sqrt{1+\frac{b^{2}}{a^{2}}}\right)^{2}&gt;2$
<br/>$e&gt;\sqrt{2}$</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
the-length-of-the-chord-of-the-parabola-y-2-4-a-x-a-0-which-passes-through-the-vertex-and-makes-an-acute-angle-with-the-axis-of-the-parabola-is-1

slug:
the-length-of-the-chord-of-the-parabola-y-2-4-a-x-a-0-which-passes-through-the-vertex-and-makes-an-acute-angle-with-the-axis-of-the-parabola-is-1-73921

question:
<div class="question">The length of the chord of the parabola $y^{2}=4 a x(a&gt;0)$ which passes through the vertex and makes an acute angle $\alpha$ with the axis of the parabola is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 a \cot \alpha \operatorname{cosec} \alpha$</span> </div>

answer:
<div class="solution">Hint: 
<br/>Equation of OP:-
<br/>$y=x \tan \alpha$
<br/>Solving with $y^{2}=4 a x$, we get :
<br/>$x^{2} \tan ^{2} \alpha=4 a x \quad \Rightarrow x=4 a \cot ^{2} \alpha$
<br/>Substituting, $y=4 a \cot \alpha$
<br/>$\therefore \mathrm{P} \equiv\left(4 \mathrm{a} \cot ^{2} \alpha, 4 \mathrm{a} \cot \alpha\right)$
<br/>So, $\overline{\mathrm{OP}}=\sqrt{16 \mathrm{a}^{2} \cot ^{4} \alpha+16 \mathrm{a}^{2} \cot ^{2} \alpha}=4 \mathrm{a} \cot \alpha \operatorname{cosec} \alpha\left(\right.$ as $0^{\circ}    &lt; \alpha &lt; 90^{\circ}$, so $\left.\cot \alpha&gt;0, \operatorname{cosec} \alpha&gt;0\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GaazMMB_eeEydRgN-EKaFP3fDDUp1Pjg3pF5aeUJZA8.original.fullsize.png"/></div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-length-of-the-chord-of-the-parabola-y-2-4-a-x-a-0-which-passes-through-the-vertex-and-makes-an-acute-angle-with-the-axis-of-the-parabola-is

slug:
the-length-of-the-chord-of-the-parabola-y-2-4-a-x-a-0-which-passes-through-the-vertex-and-makes-an-acute-angle-with-the-axis-of-the-parabola-is-62381

question:
<div class="question">The length of the chord of the parabola $y^{2}=4 a x(a&gt;0)$ which passes through the vertex and makes an acute angle $\alpha$ with the axis of the parabola is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 a \cot \alpha \operatorname{cosec} \alpha$</span> </div>

answer:
<div class="solution">Hint: 
<br/>Equation of OP:-
<br/>$y=x \tan \alpha$
<br/>Solving with $y^{2}=4 a x$, we get :
<br/>$x^{2} \tan ^{2} \alpha=4 a x \quad \Rightarrow x=4 a \cot ^{2} \alpha$
<br/>Substituting, $y=4 a \cot \alpha$
<br/>$\therefore \mathrm{P} \equiv\left(4 \mathrm{a} \cot ^{2} \alpha, 4 \mathrm{a} \cot \alpha\right)$
<br/>So, $\overline{\mathrm{OP}}=\sqrt{16 \mathrm{a}^{2} \cot ^{4} \alpha+16 \mathrm{a}^{2} \cot ^{2} \alpha}=4 \mathrm{a} \cot \alpha \operatorname{cosec} \alpha\left(\right.$ as $0^{\circ}    &lt; \alpha &lt; 90^{\circ}$, so $\left.\cot \alpha&gt;0, \operatorname{cosec} \alpha&gt;0\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GaazMMB_eeEydRgN-EKaFP3fDDUp1Pjg3pF5aeUJZA8.original.fullsize.png"/></div>


title: the-length-of-the-latus-rectum-of-the-ellipse-16-x225-y2400-is
slug: the-length-of-the-latus-rectum-of-the-ellipse-16-x225-y2400-is-76803
question: <div class="question">The length of the latus rectum of the ellipse \(16 x^2+25 y^2=400\) is</div>
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(5 / 16\) unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(32 / 5\) unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(16 / 5\) unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(5 / 32\) unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(32 / 5\) unit</span> </div>

answer:
<div class="solution">Hints : Length of latus rectum \(=2 \frac{b^2}{a}=\frac{2 \times 16}{5}=\frac{32}{5}\)<br/>\(\begin{aligned}<br/>&amp; 16 x^2+25 y^2=400 \\<br/>&amp; \frac{x^2}{25}+\frac{y^2}{16}=1 \\<br/>&amp; a^2=25 ; b^2=16<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">96</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-letters-of-the-word-cochin-are-permuted-and-all-permutations-are-arranged-in-an-alphabetical-order-as-in-an-english-dictionary-the-number-of-words

slug:
the-letters-of-the-word-cochin-are-permuted-and-all-permutations-are-arranged-in-an-alphabetical-order-as-in-an-english-dictionary-the-number-of-words-48083

question:
<div class="question">The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">96</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">48</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">183</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">267</span> </li> </ul>

answer:
<div class="solution">In a dictionary, the words at each stage are arranged in aiphabetical order.
<br/>In the given problem, we must consider the words beginning with $\mathrm{C}, \mathrm{C}, \mathrm{H}, I, \mathrm{N}, \mathrm{O}$ in order. 
<br/>So. Number of words starting with $C C=4 !$ 
<br/>Number of words starting with $\mathrm{CH}=4!$ 
<br/>Number of words starting with $\mathrm{Cl}=4!$ 
<br/>Number of words starting with $\mathrm{CN}=4!$ 
<br/>Next word staring with CO ie. COCHIN = 1
<br/>$\therefore$ Required number of words $=4 \times 41=96$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">96</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-letters-of-the-word-cochin-are-permuted-and-all-the-permutations-are-arranged-in-alphabetical-order-as-in-english-dictionary-the-number-of-words-t

slug:
the-letters-of-the-word-cochin-are-permuted-and-all-the-permutations-are-arranged-in-alphabetical-order-as-in-english-dictionary-the-number-of-words-t-72589

question:
<div class="question">The letters of the word 'COCHIN' are permuted and all the permutations are arranged in alphabetical order as in English dictionary. The number of words that appear before the word 'COCHIN', is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">360</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">192</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">96</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">48</span> </li> </ul>

answer:
<div class="solution">The number of words starting with $\mathrm{CC}=4!$
<br/>The number of words starting with $\mathrm{CH}=4!$
<br/>The number of words starting with $\mathrm{Cl}=4!$
<br/>The number of words starting with $\mathrm{CN}=4!$ COCHIN is the first word in the list of words beginning with CO.
<br/>$\therefore$ Number of words that appear before the word COCHIN = 96</div>


tags: ['Physics', 'Wave Optics', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$4: 1$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-light-beams-of-intensities-in-the-ratio-of-9-1-are-allowed-to-interfere-what-will-be-the-ratio-of-the-intensities-of-maxima-and-minima

slug:
the-light-beams-of-intensities-in-the-ratio-of-9-1-are-allowed-to-interfere-what-will-be-the-ratio-of-the-intensities-of-maxima-and-minima-54679

question:
<div class="question">The light beams of intensities in the ratio of $9: 1$ are allowed to interfere. What will be the ratio of the intensities of maxima and minima?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3: 1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4: 1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$25: 9$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$81: 1$</span> </li> </ul>

answer:
<div class="solution">Hints: $\frac{\mathrm{A}_1}{\mathrm{~A}_2}=\frac{3}{1}$<br/>$$<br/>\frac{I_{\max }}{I_{\min }}=\frac{16}{4}=\frac{4}{1}<br/>$$</div>


title: the-lim-x-3-x-1-3-x-1-4-x-equals
slug: the-lim-x-3-x-1-3-x-1-4-x-equals-44650
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The $\lim _{x \rightarrow \infty}\left(\frac{3 x-1}{3 x+1}\right)^{4 x}$ equals</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">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{e}^{-8 / 3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{e}^{-4 / 9}$</span> </li> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow \infty}\left(\frac{3 x-1}{3 x+1}\right)^{4 x}=1^{\infty}=e^{L}=e^{-\frac{8}{3}}$
<br/>$L=\lim _{x \rightarrow \infty} 4 x\left(\frac{3 x-1}{3 x+1}-1\right)=\lim _{x \rightarrow \infty} 4 x\left(\frac{-2}{3 x+1}\right)=\frac{-8}{3}$</div>


title: the-limit-of-n-1-1000-1-n-x-n-as-x
slug: the-limit-of-n-1-1000-1-n-x-n-as-x-88558
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The limit of $\sum_{n=1}^{1000}(-1)^{n} x^{n}$ as $x \rightarrow \infty$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">exists and approaches to $+\infty$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow-} \sum_{n=1}^{1000}(-1)^{n} x^{n}$
<br/>$=\lim _{x \rightarrow-}\left\{-x+x^{2}-x^{3}+x^{4}+\ldots+x^{1000}\right\}$
<br/>$=\lim _{x \rightarrow \infty}(-x) \cdot\left\{\frac{(-x)^{1000}-1}{(-x-1)}\right\}=\lim _{x \rightarrow-\infty} \frac{x^{1001}-x}{x+1}$
<br/>$=\lim _{x \rightarrow-} \frac{x^{1000}-1}{1+\left(\frac{1}{x}\right)}=+\infty$</div>


title: the-limit-of-the-interior-angle-of-a-regular-polygon-of-n-sides-as-n-is
slug: the-limit-of-the-interior-angle-of-a-regular-polygon-of-n-sides-as-n-is-14921
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The limit of the interior angle of a regular polygon of $n$ sides as $n \rightarrow \infty$ is</div>

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

answer:
<div class="solution">It is clear that, the limit of the interior angle of a regular polygon of $n$ sides as $n \rightarrow \infty$ is $\pi$ or $180^{\circ}$</div>


title: the-limit-of-x-1-1-x-1-x-2-1-as-x-0
slug: the-limit-of-x-1-1-x-1-x-2-1-as-x-0-68758
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The limit of $\left\{\frac{1}{x} \sqrt{1+x}-\sqrt{1+\frac{1}{x^{2}}}\right\}$ as $x \rightarrow 0$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">does not exist</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">is equal to $\frac{1}{2}$</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">is equal to 0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">is equal to 1</span> </li> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow 0}\left\{\frac{\sqrt{1+x}}{x}-\sqrt{1+\frac{1}{x^{2}}}\right\}$
<br/>$=\lim _{x \rightarrow 0}\left\{\frac{\sqrt{1+x}-\sqrt{1+x^{2}}}{x}\right\}\left(\frac{0}{0}\right.$ form $)$
<br/>$=\lim _{x \rightarrow 0} \frac{\frac{1}{2 \sqrt{1+x}}-\frac{x}{\sqrt{1+x^{2}}}}{1}$ (Use L-Hospital Rule)
<br/>$=\frac{1}{2 \sqrt{1+0}}-0=\frac{1}{2}$</div>


title: the-limit-of-x-2-1-e-x-1-2013-x-e-x-1-1-as-x-0
slug: the-limit-of-x-2-1-e-x-1-2013-x-e-x-1-1-as-x-0-75548
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The limit of $\left[\frac{1}{x^{2}}+\frac{(2013)^{x}}{e^{x}-1}-\frac{1}{e^{x}-1}\right]$ as $x \rightarrow 0$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">approaches $+\infty$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">approaches $-\infty$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is equal to $\log_{e}$, (2013)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">does not exist</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">approaches $+\infty$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow 0}\left\{\frac{1}{x^{2}}+\frac{(2013)^{x}}{e^{x}-1}-\frac{1}{e^{x}-1}\right\}$
<br/>$=\lim _{x \rightarrow 0}\left\{\frac{1}{x^{2}}+\frac{(2013)^{x}-1}{e^{x}-1}\right\}$
<br/>$=\lim _{x \rightarrow 0}\left\{\frac{1}{x^{2}}+\frac{(2013)^{x}-1}{x} \cdot \frac{x}{e^{x}-1}\right\}$
<br/>$=\lim _{x \rightarrow 0} \frac{1}{x^{2}}+\lim _{x \rightarrow 0} \frac{(2013)^{x}-1}{x} \cdot \lim _{x \rightarrow 0} \frac{x}{e^{x}-1}$
<br/>$=+\infty+\log (2013) \cdot 1$
<br/>$=+\infty$</div>


title: the-limit-of-x-sin-e-x-1-as-x-0
slug: the-limit-of-x-sin-e-x-1-as-x-0-60809
question: <div class="question">The limit of $x \sin \left(e^{\frac{1}{x}}\right)$ as $x \rightarrow 0$</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">is equal to 0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is equal to 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is equal to $\frac{e}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">does not exist</span> </li> </ul>

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

answer:
<div class="solution">$\lim _{x \rightarrow 0} x \sin e^{(1 / x)}$
<br/>$\begin{aligned} L H L &amp;=f(0-0)=\lim _{h \rightarrow 0}(-h) \sin e^{(-1 / h)} \\ &amp;=-0 \times \sin \left(e^{-\infty}\right)=-0 \times \sin (0)=0 \end{aligned}$
<br/>$=0 \times(\mathrm{a}$ finite number between $-1 \text{ to}+1)$
<br/>$$
<br/>=0 \quad(\because-1 \leq \sin x \leq 1)
<br/>$$
<br/>$\because \mathrm{LHL}=\mathrm{RHL}$
<br/>$\therefore \lim _{x \rightarrow 0} x \sin \left(e^{1 / x}\right)$ exist and equal to 0</div>


tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
the-line-a-a-is-on-charged-infinite-a-conducting-plane-which-is-perpendicular-to-the-plane-of-the-paper-the-plane-has-a-surface-density-of-charge-and--1

slug:
the-line-a-a-is-on-charged-infinite-a-conducting-plane-which-is-perpendicular-to-the-plane-of-the-paper-the-plane-has-a-surface-density-of-charge-and-1-68018

question:
<div class="question">The line $A A^{\prime}$ is on charged infinite $A$ conducting plane which is perpendicular to the plane of the paper. The plane has a surface density of charge $\sigma$ and $B$ is ball of mass $m$ with a like charge of magnitude $q . B$ is connected by string from a point on the line $A A^{\prime}$. The tangent of angle (0) formed between the line $A A^{\prime}$ and the string is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q7.png"/></div>

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

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

answer:
<div class="solution">The diagram is as follows
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a7.png"/> <br/> <br/>The electric field due to charged infinite conducting sheet is $E=\frac{\sigma}{\varepsilon_{0}}$
<br/> <br/>Now, force (electric force) on the charged ball is $F=q E=\frac{q \sigma}{\varepsilon_{0}}$
<br/>The resultant of electric force and $m g$ balance the tension produced in the string. So, tan $\theta=\frac{F_{e}}{m g}=\frac{q \sigma}{\frac{\varepsilon_{0}}{m g}}=\frac{q \sigma}{\varepsilon_{0} m g}$</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
the-line-a-a-is-on-charged-infinite-a-conducting-plane-which-is-perpendicular-to-the-plane-of-the-paper-the-plane-has-a-surface-density-of-charge-and-

slug:
the-line-a-a-is-on-charged-infinite-a-conducting-plane-which-is-perpendicular-to-the-plane-of-the-paper-the-plane-has-a-surface-density-of-charge-and-59673

question:
<div class="question">The line $A A^{\prime}$ is on charged infinite $A$ conducting plane which is perpendicular to the plane of the paper. The plane has a surface density of charge $\sigma$ and $B$ is ball of mass $m$ with a like charge of magnitude $q . B$ is connected by string from a point on the line $A A^{\prime}$. The tangent of angle (0) formed between the line $A A^{\prime}$ and the string is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q7.png"/></div>

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

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

answer:
<div class="solution">The diagram is as follows
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_a7.png"/> <br/> <br/>The electric field due to charged infinite conducting sheet is $E=\frac{\sigma}{\varepsilon_{0}}$
<br/> <br/>Now, force (electric force) on the charged ball is $F=q E=\frac{q \sigma}{\varepsilon_{0}}$
<br/>The resultant of electric force and $m g$ balance the tension produced in the string. So, tan $\theta=\frac{F_{e}}{m g}=\frac{q \sigma}{\frac{\varepsilon_{0}}{m g}}=\frac{q \sigma}{\varepsilon_{0} m g}$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
the-line-a-b-cuts-off-equal-intercepts-2-a-from-the-axes-from-any-point-p-on-the-line-a-b-perpendiculars-pr-and-ps-are-drawn-on-the-axes-locus-of-midp

slug:
the-line-a-b-cuts-off-equal-intercepts-2-a-from-the-axes-from-any-point-p-on-the-line-a-b-perpendiculars-pr-and-ps-are-drawn-on-the-axes-locus-of-midp-30776

question:
<div class="question">The line $A B$ cuts off equal intercepts $2 a$ from the axes. From any point $P$ on the line $A B$ perpendiculars $P R$ and $P S$ are drawn on the axes. Locus of mid-point of $R S$ is</div>

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

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

answer:
<div class="solution">$\Rightarrow$
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{x}{2 a}+\frac{y}{2 a}=1 \\
<br/>x+y=2 a
<br/>\end{array}
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a32.png"/> <br/> <br/>Let the coordinates of the mid-point of $\mathrm{RS}$ be $(\mathrm{h}, \mathrm{k})$
<br/>So. $R$ and $S$ are $(2 h, 0)$ and $(0,2 k)$ Hence, the mid-point of $R S$ is $(h, k)$
<br/>P lies on line AB
<br/>Then. from Eq  we have
<br/>$$
<br/>2 h+2 k=2 a
<br/>$$
<br/>$\Rightarrow \quad h+k=a$
<br/>So, the locus of $(h, k)$ is $x+y=a$.</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-line-joining-a-b-cos-b-sin-and-b-a-cos-a-sin-where-a-b-is-produced-to-the-point-m-x-y-so-that-a-m-mb-b-a-then-x-cos-2-y-sin-2-is-equal-to

slug:
the-line-joining-a-b-cos-b-sin-and-b-a-cos-a-sin-where-a-b-is-produced-to-the-point-m-x-y-so-that-a-m-mb-b-a-then-x-cos-2-y-sin-2-is-equal-to-49788

question:
<div class="question">The line joining $A(b \cos \alpha, b \sin \alpha)$
<br/>and
<br/>$B(a \cos \beta, a \sin \beta),$ where $a \neq b,$ is produced to the
<br/>point $M(x, y)$ so that $A M: M B=b: a$. Then, $x \cos \frac{\alpha+\beta}{2}+y \sin \frac{\alpha+\beta}{2}$ is equal to</div>

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

answer:
<div class="solution">Since, $A M: B M=b: a$ $\therefore M$ divides $A B$ extemally in the ratio $b: a$...(i)
<br/>$\therefore \quad x=\frac{b \cdot a \cos \beta-a b \cos \alpha}{b-a}$
<br/>$y=\frac{b a \sin \beta-a b \sin \alpha}{b-a}$
<br/>(ii)
<br/>Divide Eq. (i) by Eq, (ii), we get
<br/>$\frac{x}{y}=\frac{\cos \beta-\cos \alpha}{\sin \beta-\sin \alpha}=\frac{2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}}{-2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}}$
<br/>$\therefore \quad x \cos \frac{\alpha+\beta}{2}+y \sin \frac{\alpha+\beta}{2}=0$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-line-segment-joining-the-foci-of-the-hyperbola-x-2-y-2-1-0-is-one-of-the-diameters-of-a-circle-the-equation-of-the-circle-is

slug:
the-line-segment-joining-the-foci-of-the-hyperbola-x-2-y-2-1-0-is-one-of-the-diameters-of-a-circle-the-equation-of-the-circle-is-66602

question:
<div class="question">The line segment joining the foci of the hyperbola $x^{2}-y^{2}+1=0$ is one of the diameters of a circle. The equation of the circle is</div>

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

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

answer:
<div class="solution">Given, equation of hyperbola is
<br/>$$
<br/>x^{2}-y^{2}+1=0
<br/>$$
<br/>$\Rightarrow \quad y^{2}-x^{2}=1$
<br/>On comparing it with $\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1,$ we get
<br/>$$
<br/>a=b=1
<br/>$$
<br/>Now, $e=\sqrt{1+\frac{a^{2}}{b^{2}}}=\sqrt{1+\frac{1}{1}}=\sqrt{2}$
<br/>$\therefore$ Foci $=(0, \pm b e)=(0, \pm \sqrt{2})$
<br/>since, line joining foci of hyperbola is diameter of circle.
<br/>$\therefore$ Centre of circle $=\left(\frac{0+0}{2}, \frac{\sqrt{2}-\sqrt{2}}{2}\right)=(0,0)$
<br/>$\begin{aligned} \text { and radius } &amp;=\frac{1}{2} \sqrt{(0-0)^{2}+(\sqrt{2}-(-\sqrt{2}))^{2}} \\ &amp;=\frac{1}{2} \sqrt{(2 \sqrt{2})^{2}}=\sqrt{2} \end{aligned}$
<br/>$\therefore$ Equation of circle will be
<br/>$$
<br/>(x-0)^{2}+(y-0)^{2}=(\sqrt{2})^{2}
<br/>$$
<br/>$\Rightarrow \quad x^{2}+y^{2}=2$</div>


title: the-line-through-the-points-a-b-and-a-b-passes-through-the-point
slug: the-line-through-the-points-a-b-and-a-b-passes-through-the-point-17492
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The line through the points $(a, b)$ and $(-a,-b)$
<br/>passes through the point</div>

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

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

answer:
<div class="solution">Let the given points be $A(a, b)$ and $B(-a,-b)$. The equation of line passing through the points A and $B$ is
<br/>$$
<br/>\begin{array}{ll}
<br/>y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right) \\
<br/>\Rightarrow \quad y-b=\frac{-b-b}{-a-a}(x-a) \\
<br/>\Rightarrow \quad y-b=\frac{2 b}{2 a}(x-a)
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad y-b=\frac{b}{a}(x-a)$
<br/>$\Rightarrow \quad a y-a b=b x-a b$
<br/>$$
<br/>\Rightarrow \quad b x=a y ...(i)
<br/>$$
<br/>Since, from the given points $\left(a^{2}, a b\right)$ and $(a, b)$ are satisty the Eq. (i), but ( $a^{2}$, ab) is the required answer because $(a, b)$ is already given in question.</div>


title: the-line-x-2-y-4-z-4-0-x-y-z-8-0-intersect-the-plane-x-y-2-z-1-0-at-the-point
slug: the-line-x-2-y-4-z-4-0-x-y-z-8-0-intersect-the-plane-x-y-2-z-1-0-at-the-point-85348
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The line $x-2 y+4 z+4=0, x+y+z-8=0$ intersect the plane $x-y+2 z+1=0$ at the point</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(-2,5,1)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(2,-5,1)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(2,5,-1)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(2,5,1)$</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">$(2,5,1)$</span> </div>

answer:
<div class="solution">Solving the \(3 \mathrm{eq}^n\) s will give as the required point of intersection
<br/>\(\begin{aligned} &amp; x-2 y+4 z=-4 \\ &amp; x+y+z=8 \\ &amp; x-y+2 z=-1 \\ &amp; D=\left|\begin{array}{ccc} 1 &amp; -2 &amp; 4 \\ 1 &amp; 1 &amp; 1 \\ 1 &amp; -1 &amp; 2 \end{array}\right|=1(3)+2(1)+4(-2)=-3 \\ &amp; \begin{aligned} D_1=\left|\begin{array}{rrr} -4 &amp; -2 &amp; 4 \\ 8 &amp; 1 &amp; 1 \\ -1 &amp; -1 &amp; 2 \end{array}\right|= &amp; -4(3)+2(17)+4(-7) \\ &amp; =-6 \end{aligned} \\ &amp; D_2=\left|\begin{array}{ccc} 1 &amp; -4 &amp; 4 \\ 1 &amp; 8 &amp; 1 \\ 1 &amp; -1 &amp; 2 \end{array}\right|=-15 \\ &amp; D_3=\left|\begin{array}{ccc} 1 &amp; -2 &amp; -4 \\ 1 &amp; 1 &amp; 8 \\ 1 &amp; -1 &amp; -1 \end{array}\right|=-3 \\ &amp; x=\frac{D_1}{D}=2, \quad y=\frac{D_2}{D}=5, \quad z=\frac{D_3}{D}=1 \\ &amp; \end{aligned}\)
<br/>\((2, S, 1)\) is the required point.</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
the-line-x-2-y-intersects-the-ellipse-4-x-2-y-2-1-at-the-points-p-and-q-the-equation-of-the-circle-with-pq-as-diameter-is

slug:
the-line-x-2-y-intersects-the-ellipse-4-x-2-y-2-1-at-the-points-p-and-q-the-equation-of-the-circle-with-pq-as-diameter-is-60818

question:
<div class="question">The line $x=2 y$ intersects the ellipse $\frac{x^{2}}{4}+y^{2}=1$ at the points $P$ and $Q .$ The equation of the circle with $P Q$ as diameter is</div>

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

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

answer:
<div class="solution">Solving $x=2 y$
<br/>and $\frac{x^{2}}{4}+y^{2}=1$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a55.png"/> <br/> <br/>Put $x=2 y$ in Eq. (ii), we get $\frac{(2 y)^{2}}{4}+y^{2}=1 \Rightarrow \frac{4 y^{2}}{4}+y^{2}=1$
<br/>$=$
<br/>$2 y^{2}=1 \Rightarrow y=\pm \frac{1}{\sqrt{2}}$
<br/>$\therefore$ From $\mathrm{E} q$ (i), $x=\pm \sqrt{2}$
<br/>$\therefore \quad P\left(\sqrt{2} \frac{1}{\sqrt{2}}\right)$ and $Q\left(-\sqrt{2},-\frac{1}{\sqrt{2}}\right)$
<br/>Equation of circle with $P Q$ as diameter is
<br/>$$
<br/>\begin{array}{l}
<br/>(x-\sqrt{2})(x+\sqrt{2})+\left(y-\frac{1}{\sqrt{2}}\right)\left(y+\frac{1}{\sqrt{2}}\right) \\
<br/>\Rightarrow \quad x^{2}-2+y^{2}-\frac{1}{2}=0 \Rightarrow x^{2}+y^{2}=\frac{5}{2}
<br/>\end{array}
<br/>$$</div>


title: the-line-y-2-t-2-intersects-the-ellipse-9-x-2-4-y-2-1-in-real-points-if
slug: the-line-y-2-t-2-intersects-the-ellipse-9-x-2-4-y-2-1-in-real-points-if-20105
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The line $\mathrm{y}=2 \mathrm{t}^2$ intersects the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ in real points if</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$|t| \leq 1$</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">$|t| &lt; 1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$|t|&gt;1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|t| \geq 1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|t| \leq 1$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \frac{x^2}{9}+\frac{y^2}{4}=1 ; \mathrm{y}=2 \mathrm{t}^2 \\<br/>&amp; \frac{x^2}{9}+\frac{4 t^4}{4}=1 \Rightarrow \frac{x^2}{9}+t^4=1 \Rightarrow x^2=9\left(1-t^4\right) \\<br/>&amp; x^2 \geq 0 \Rightarrow 9\left(1-t^4\right) \geq 0 \Rightarrow t^4-1 \leq 0 \\<br/>&amp; \Rightarrow\left(t^2-1\right)\left(t^2+1\right) \leq 0 \\<br/>&amp; \Rightarrow t^2-1 \leq 0 \quad\left(\because t^2+1&gt;0\right) \\<br/>&amp; \Rightarrow|t| \leq 1<br/>\end{aligned}<br/>$$</div>


title: the-line-y-x-5-touches
slug: the-line-y-x-5-touches-47201
question: <div class="question">The line $y=x+5$ touches</div>
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">the parabola $y^2=20 x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">the ellipse $9 x^2+16 y^2=144$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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 hyperbola $\frac{x^2}{29}-\frac{y^2}{4}=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the circle $x^2+y^2=25$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">the parabola $y^2=20 x$, the ellipse $9 x^2+16 y^2=144$, the hyperbola $\frac{x^2}{29}-\frac{y^2}{4}=1$</span> </div>

answer:
<div class="solution">(A) $y^2=20 x=4(5) x \quad \therefore$ Tangent $: y=m x+\frac{a}{m}$ for $m=1, a=5$<br/>$\therefore \mathrm{y}=\mathrm{x}+5$ is a tangent to $\mathrm{y}^2=20 \mathrm{x}$<br/>(B) $9 x^2+16 y^2=144$<br/>$c^2=a^2 m^2+b^2 \quad y=x+5 \Rightarrow m=1 ; c=5$<br/>$\Rightarrow \frac{x^2}{16}+\frac{y^2}{9}=1$<br/>$\therefore 5^2=16(1)^2+9 \Rightarrow$ True<br/>$\therefore y=x+5$ is a tangent to $9 x^2+16 y^2=144$<br/>(C) $c^2=a^2 m^2-b^2$<br/>$\frac{x^2}{29}-\frac{y^2}{4}=1 \rightarrow a^2=29 ; b^2=4$<br/>$5^2=29(1)^2-4 \rightarrow$ True<br/>$\therefore \mathrm{y}=\mathrm{x}+5$ is a tangent<br/>(D) <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/bWeVGesyy95n-iNmzq3VBibBr6kxnSEHUmjzF0isjfU.original.fullsize.png"/><br/><br/>$\therefore \mathrm{y}=\mathrm{x}+5$ is NOT tangent to $\mathrm{x}^2+\mathrm{y}^2=25$<br/>(A), (B), (C) are correct.</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-line-y-x-intersects-the-hyperbola-9-x-2-25-y-2-1-at-the-points-p-and-q-the-eccentricity-of-ellipse-with-pq-as-major-axis-and-minor-axis-of-length-

slug:
the-line-y-x-intersects-the-hyperbola-9-x-2-25-y-2-1-at-the-points-p-and-q-the-eccentricity-of-ellipse-with-pq-as-major-axis-and-minor-axis-of-length-29107

question:
<div class="question">The line $y=x$ intersects the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ at the points $P$ and $Q .$ The eccentricity of ellipse with $P Q$ as major axis and minor axis of length $\frac{5}{\sqrt{2}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{5}}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5}{\sqrt{3}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{5}{9}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{2 \sqrt{2}}{3}$</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">$\frac{2 \sqrt{2}}{3}$</span> </div>

answer:
<div class="solution">Given equation of hyperbola and line are $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ and $y=x$ respectively.
<br/>For intersection point of both curve put $y=x$, we get
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{x^{2}}{9}-\frac{x^{2}}{25}=1 \\
<br/>\Rightarrow \quad x^{2}=\frac{9 \times 25}{16}=\left(\frac{15}{4}\right)^{2} \\
<br/>\Rightarrow \quad x=\pm \frac{15}{4} \text { and } y=\pm \frac{15}{4}
<br/>\end{array}
<br/>$$
<br/>$\therefore$ Intersetion points $P\left(\frac{15}{4}, \frac{15}{4}\right)$
<br/>and $\quad Q\left(\frac{-15}{4}, \frac{-15}{4}\right)$
<br/>Since, $P Q$ is major axis, then its length
<br/>$$
<br/>=2 \sqrt{2} \cdot \frac{15}{4}=\frac{15}{\sqrt{2}}
<br/>$$
<br/>and length of minor axis is $\frac{5}{\sqrt{2}}$ (given)
<br/>i.e., Major axis, $2 a=\frac{15}{\sqrt{2}} \Rightarrow a=\frac{15}{2 \sqrt{2}}$
<br/>and minor axis, $2 b=\frac{5}{\sqrt{2}} \Rightarrow b=\frac{5}{2 \sqrt{2}}$
<br/>$\therefore$ Eccentricity of an ellipse
<br/>$$
<br/>=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{1-\left(\frac{b}{a}\right)^{2}}
<br/>$$
<br/>$=\sqrt{1-\left(\frac{1}{3}\right)^{2}}=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{3}$</div>


title: the-line-y-x-is-tangent-to-the-ellipse-2-x-2-3-y-2-1-then-is
slug: the-line-y-x-is-tangent-to-the-ellipse-2-x-2-3-y-2-1-then-is-85840
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The line $y=x+\lambda$ is tangent to the ellipse $2 x^{2}+3 y^{2}=1 .$ Then $, \lambda$ is</div>

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

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

answer:
<div class="solution">The equation of the line is $y=x+\lambda$
<br/>On comparing it with $y=m x+c,$ we get
<br/>$$
<br/>m=1 \text { and } c=\lambda
<br/>$$
<br/>The equation of the ellipse is
<br/>$$
<br/>2 x^{2}+3 y^{2}=1
<br/>$$
<br/>or
<br/>$$
<br/>\frac{x^{2}}{\left(\frac{1}{\sqrt{2}}\right)^{2}}+\frac{y^{2}}{\left(\frac{1}{\sqrt{3}}\right)^{2}}=1
<br/>$$
<br/>On comparing it with $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ we get
<br/>$$
<br/>a^{2}=\frac{1}{2} \text { and } b^{2}=\frac{1}{3}
<br/>$$
<br/>If the line touches the ellipse, then
<br/>$$
<br/>c^{2}=a^{2} m^{2}+b^{2} \Rightarrow \lambda^{2}=\frac{1}{2} \cdot 1+\frac{1}{3}=\frac{5}{6}
<br/>$$
<br/>$\Rightarrow \quad \lambda^{2}=\frac{5}{6} \Rightarrow \lambda=\sqrt{\frac{5}{6}}$</div>


title: the-linear-system-of-equations-8-x-3-y-5-z-0-5-x-8-y-3-z-0-3-x-5-y-8-z-0-has
slug: the-linear-system-of-equations-8-x-3-y-5-z-0-5-x-8-y-3-z-0-3-x-5-y-8-z-0-has-40187
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The linear system of equations
<br/>$\left.\begin{array}{l}8 x-3 y-5 z=0 \\ 5 x-8 y+3 z=0 \\ 3 x+5 y-8 z=0\end{array}\right\}$ has</div>

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

answer:
<div class="solution">We have, 
<br/>$$
<br/>\begin{aligned}
<br/>&amp; 8x-3 y-5 z=0 \\
<br/>&amp; 5 x-8 y+3 z=0 \\
<br/>&amp; 3 x+5 y-8 z=0 \\
<br/>&amp;D=\left|\begin{array}{ccc}
<br/>8 &amp; -3 &amp; -5 \\
<br/>5 &amp; -8 &amp; 3 \\
<br/>3 &amp; 5 &amp; -8
<br/>\end{array}\right| \\
<br/>&amp;=8(64-15)+3(-40-9)-5(25+24) \\
<br/>=&amp; 8 \times 49+3 \times(-49)-5 \times 49 \\
<br/>=&amp; 0
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ The system has infinitely many non-zero solutions.</div>


title: the-locus-of-points-x-y-in-the-plane-satisfying-sin-2-xsin-2-y1-consists-of
slug: the-locus-of-points-x-y-in-the-plane-satisfying-sin-2-xsin-2-y1-consists-of-42765
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The locus of points \((x, y)\) in the plane satisfying \(\sin ^2 x+\sin ^2 y=1\) consists of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">infinitely many lines with slope \(\pm 1\)</span> </div>

answer:
<div class="solution">\(\begin{array}{r}\text {Hint: } <br/>\sin ^2 y=\cos ^2 x \\<br/>\sin y= \pm \cos x<br/>\end{array}\)<br/>\(\begin{aligned}<br/>&amp; \text { If } \sin y=\cos x=\sin \left(\frac{\pi}{2}-x\right) \\<br/>&amp; \Rightarrow y=n \pi+(-1)^n\left(\frac{\pi}{2}-x\right)<br/>\end{aligned}\)<br/>\(\text { If } \sin y=-\cos x=\sin \left(x-\frac{\pi}{2}\right)\)<br/>\(\Rightarrow \mathrm{y}=\mathrm{n} \pi+(-1)^{\mathrm{n}}\left(\mathrm{x}-\frac{\pi}{2}\right)\)</div>


title: the-locus-of-the-center-of-a-variable-circle-which-always-touches-two-given-circles-externally-is-1
tags: ['Mathematics', 'Hyperbola', 'JEE Main']
source_db: db\MarksBatch2_P3.db
slug:
the-locus-of-the-center-of-a-variable-circle-which-always-touches-two-given-circles-externally-is-1-23136

question:
<div class="question">The locus of the center of a variable circle which always touches two given circles externally is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uGZ9juDApa3ce3MFoEUVBdILakRT6kg5WcijawMj1tg.original.fullsize.png"/><br/> <br/>$\begin{array}{l}
<br/>\mathrm{oo}_{1}=\mathrm{r}+\mathrm{r}_{1} \\
<br/>\mathrm{oo}_{2}=\mathrm{r}+\mathrm{r}_{2} \\
<br/>\therefore \mathrm{oo}_{2}-\mathrm{oo}_{1}=\mathrm{r}_{2}-\mathrm{r}_{1}
<br/>\end{array}$
<br/>hence, locus is hyperbola</div>


title: the-locus-of-the-center-of-a-variable-circle-which-always-touches-two-given-circles-externally-is
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
slug:
the-locus-of-the-center-of-a-variable-circle-which-always-touches-two-given-circles-externally-is-97791

question:
<div class="question">The locus of the center of a variable circle which always touches two given circles externally is</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uGZ9juDApa3ce3MFoEUVBdILakRT6kg5WcijawMj1tg.original.fullsize.png"/><br/> <br/>$\begin{array}{l}
<br/>\mathrm{oo}_{1}=\mathrm{r}+\mathrm{r}_{1} \\
<br/>\mathrm{oo}_{2}=\mathrm{r}+\mathrm{r}_{2} \\
<br/>\therefore \mathrm{oo}_{2}-\mathrm{oo}_{1}=\mathrm{r}_{2}-\mathrm{r}_{1}
<br/>\end{array}$
<br/>hence, locus is hyperbola</div>


title: the-locus-of-the-centre-of-a-circle-which-passes-through-two-variable-points-a-0a-0-is
slug: the-locus-of-the-centre-of-a-circle-which-passes-through-two-variable-points-a-0a-0-is-50923
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The locus of the centre of a circle which passes through two variable points \((a, 0),(-a, 0)\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x=0\)</span> </div>

answer:
<div class="solution">Hint: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/LFG9qOSNZ8KzlFiRjrXqF_1gm4MYEDEpcBye_aqYA8E.original.fullsize.png"/><br/><br/>Centre lies on \(y\)-axis locus \(x=0\)</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
title:
the-locus-of-the-centre-of-the-circles-which-touch-both-the-circles-x-2-y-2-a-2-and-x-2-y-2-4-ax-externally-is

slug:
the-locus-of-the-centre-of-the-circles-which-touch-both-the-circles-x-2-y-2-a-2-and-x-2-y-2-4-ax-externally-is-25034

question:
<div class="question">The locus of the centre of the circles which touch both the circles $x^{2}+y^{2}=a^{2}$ and $x^{2}+y^{2}=4$ax externally is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a circle</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a parabola</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">an ellipse</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a hyperbola</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>Let, centre $\equiv(\mathrm{h}, \mathrm{k})$ and radius $=\mathrm{r}$ for the variable circle So, using $\mathrm{C}_{1} \mathrm{C}_{2}=\mathrm{r}_{1}+\mathrm{r}_{2}$ for both cases we have:
<br/>$\begin{array}{l}
<br/>\mathrm{h}^{2}+\mathrm{k}^{2}=(\mathrm{r}+\mathrm{a})^{2} \rightarrow(1) \text { and }(\mathrm{h}-2 \mathrm{a})^{2}+\mathrm{k}^{2}=(\mathrm{r}+2 \mathrm{a})^{2} \rightarrow(2) \\
<br/>\text { Eq. }(2)-\text { Eq. }(1), \text { gives }: \mathrm{r}=\frac{\mathrm{a}-4 \mathrm{~h}}{2} \rightarrow(3)
<br/>\end{array}$
<br/>Substitute $(3)$ in $(1)$ to get:
<br/>$12 h^{2}-4 k^{2}-24 a h+9 a^{2}=0$
<br/>$\therefore$ locus $: 12 x^{2}-4 y^{2}-24 a x+9 a^{2}=0$ i.e. a hyperbola</div>


title: the-locus-of-the-middle-points-of-all-chords-of-the-parabola-y24-ax-passing-through-the-vertex-is
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
slug:
the-locus-of-the-middle-points-of-all-chords-of-the-parabola-y24-ax-passing-through-the-vertex-is-48292

question:
<div class="question">The locus of the middle points of all chords of the parabola \(y^2=4\) ax passing through the vertex is</div>

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

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

answer:
<div class="solution">\(\text {Hints: } 2 \mathrm{~h}=\mathrm{x}, 2 \mathrm{k}=\mathrm{y}\)<br/>\(\begin{aligned}<br/>&amp; \mathrm{y}^2=4 \mathrm{ax} \\<br/>&amp; \mathrm{k}^2=2 \mathrm{ah} \\<br/>&amp; \mathrm{y}^2=2 \mathrm{ax}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=0$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-locus-of-the-midpoints-of-all-chords-of-the-parabola-y-2-4-a-x-through-its-vertex-is-another-parabola-with-directrix

slug:
the-locus-of-the-midpoints-of-all-chords-of-the-parabola-y-2-4-a-x-through-its-vertex-is-another-parabola-with-directrix-55157

question:
<div class="question">The locus of the mid-points of all chords of the parabola $y^{2}=4 a x$ through its vertex is another parabola with directrix</div>

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

answer:
<div class="solution">Let $P\left(a t^{2}, 2 a t\right)$ be a moving point on the parabola $y^{2}=4 a x$ and $S(a, 0)$ be its focus. Let $Q(h, k)$ be the mid-point of SP. 
<br/>Then, $\quad h=\frac{a t^{2}+a}{2}$
<br/>and $\quad k=\frac{2 a t+0}{2}$
<br/>$\Rightarrow \quad \frac{2 h}{a}=t^{2}+1$
<br/>and $\quad t=\frac{k}{a}$
<br/>$\Rightarrow \quad \frac{2 h}{a}=\frac{k^{2}}{a^{2}}+1$
<br/>$\Rightarrow \quad k^{2}=2 a h-a^{2} \quad[$ on eliminating t $\mid$
<br/>Thus, the locus of $(h, k)$ is $y^{2}=2 a x-a^{2}$ 
<br/>Now. $\quad y^{2}=2 a x-a^{2}$
<br/>$\Rightarrow \quad(y-0)^{2}=2 a\left(x-\frac{a}{2}\right)$
<br/>The equation of the directrix of this parabola is
<br/>$$
<br/>x-\frac{a}{2}=-\frac{a}{2}, i \cdot e, x=0
<br/>$$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-locus-of-the-midpoints-of-chords-of-the-circle-x-2-y-2-1-which-subtends-a-right-angle-at-the-origin-is

slug:
the-locus-of-the-midpoints-of-chords-of-the-circle-x-2-y-2-1-which-subtends-a-right-angle-at-the-origin-is-60479

question:
<div class="question">The locus of the mid-points of chords of the circle $x^{2}+y^{2}=1$, which subtends a right angle at the origin, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}=\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}=\frac{1}{2}$</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">$x y=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}-y^{2}=0$</span> </li> </ul>

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

answer:
<div class="solution">Let $(h, k)$ be the coordinates of the mid-point of a chord which subtends a right angle at the origin. Then, equation of the chord is
<br/>$$
<br/>\begin{array}{l}
<br/>\quad h x+h y-1=h^{2}+k^{2}-1 \quad[u \operatorname{sing} T=S] \\
<br/>\Rightarrow \quad h x+k y=h^{2}+k^{2}
<br/>\end{array}
<br/>$$
<br/>The combined equation of the pair of lines joining the origin to the points of intersection of $x^{2}+y^{2}=1$ and $h x+k y=h^{2}+k^{2}$ is
<br/>$$
<br/>x^{2}+y^{2}-1\left(\frac{h x+h y}{h^{2}+k^{2}}\right)^{2}=0
<br/>$$
<br/>Lines given by the above equation are at right angle. Therefore, coefficient of $x^{2}+$ coefficient of $y^{2}=0$ 
<br/>ie. $\quad h^{2}+k^{2}=\frac{1}{2}$
<br/>$\therefore x^{2}+y^{2}=\frac{1}{2}$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-locus-of-the-midpoints-of-the-chords-of-an-ellipse-x-2-4-y-2-4-that-are-drawn-from-the-positive-end-of-the-minor-axis-is

slug:
the-locus-of-the-midpoints-of-the-chords-of-an-ellipse-x-2-4-y-2-4-that-are-drawn-from-the-positive-end-of-the-minor-axis-is-46266

question:
<div class="question">The locus of the mid-points of the chords of an ellipse $x^{2}+4 y^{2}=4$ that are drawn from the positive end of the minor axis, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a circle with centre $\left(\frac{1}{2}, 0\right)$ and radius 1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a parabola with focus $\left(\frac{1}{2}, 0\right)$ and directrix $x=-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">an ellipse with centre $\left(0, \frac{1}{2}\right),$ major axis 1 and minor axis $\frac{1}{2}$</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">a hyperbola with centre $\left(0, \frac{1}{2}\right),$ transverse axis 1 and conjugate axis $\frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">an ellipse with centre $\left(0, \frac{1}{2}\right),$ major axis 1 and minor axis $\frac{1}{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a6.png"/> <br/> <br/>Given equation of an ellipse is
<br/>$$
<br/>x^{2}+4 y^{2}=4
<br/>$$
<br/>$\Rightarrow \quad \frac{x^{2}}{4}+\frac{y^{2}}{1}=1$
<br/>$\therefore$ Coordinate of positive end of minor axis is $B(0,1)$ Let mid-point of the chord $B P$ is $M(h, k)$ 
<br/>Then, $\quad(h, k)=\left(\frac{0+x}{2}, \frac{1+y}{2}\right)$
<br/>$\Rightarrow \quad h=\frac{x}{2} \quad \Rightarrow x=2 h$
<br/>and $\quad k=\frac{1+y}{2} \Rightarrow y=2 k-1$
<br/>$\therefore$
<br/>$P(x, y)=\{2 h,(2 k-1)\}$
<br/>Since, the point '$P$' lies ellipses so form Eq. (i). we get $(2 h)_{1}^{2}+4(2 k-1)^{2}=4$
<br/>$\Rightarrow \quad 4 h^{2}+4(2 k-1)^{2}=4$
<br/>$\Rightarrow \quad h^{2}+4 k^{2}+1-4 k=1$
<br/>$\Rightarrow \quad h^{2}+4 k^{2}-4 k=0$
<br/>Thus, required locus is an ellipse whose equation is $x^{2}+4 y^{2}-4 y=0$
<br/>$\Rightarrow \quad \frac{(x-0)^{2}}{1}+\frac{\left(y-\frac{1}{2}\right)^{2}}{\left(\frac{1}{4}\right)}=1$
<br/>whose centre $\left(0, \frac{1}{2}\right)$ and major and minor axis $\frac{1}{2}$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-locus-of-the-midpoints-of-the-chords-of-the-circle-x-2-y-2-2-x-2-y-2-0-which-make-an-angle-of-9-0-at-the-centre-is

slug:
the-locus-of-the-midpoints-of-the-chords-of-the-circle-x-2-y-2-2-x-2-y-2-0-which-make-an-angle-of-9-0-at-the-centre-is-73156

question:
<div class="question">The locus of the mid-points of the chords of the circle $x^{2}+y^{2}+2 x-2 y-2=0,$ which make an angle of $90^{\circ}$ at the centre is</div>

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

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

answer:
<div class="solution">Given, equation of circle is $x^{2}+y^{2}+2 x-2 y-2=0$
<br/>$\Rightarrow \quad(x+1)^{2}+(y-1)^{2}=4$
<br/>$\therefore$ Centre (-1,1) and radius $=2$ 
<br/>Let $(h, k)$ be the mid-point of chord.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a20.png"/> <br/> <br/>From figure,
<br/>$$
<br/>O P=\sqrt{(h+1)^{2}+(k-1)^{2}}
<br/>$$
<br/>$\ln \Delta O A P$
<br/>$$
<br/>\begin{aligned}
<br/>\sin 45^{\circ}=&amp; \frac{O P}{O A} \\
<br/>\Rightarrow \quad \frac{1}{\sqrt{2}}=\frac{\sqrt{(h+1)^{2}+(k-1)^{2}}}{2}
<br/>\end{aligned}
<br/>$$
<br/>On squaring both sides, we get
<br/>$$
<br/>(h+1)^{2}+(k-1)^{2}=2
<br/>$$
<br/>$\Rightarrow \quad h^{2}+k^{2}+2 h-2 k=0$
<br/>$\therefore$ Locus of $P$ will be
<br/>$$
<br/>x^{2}+y^{2}+2 x-2 y=0
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-locus-of-the-point-of-intersection-of-the-straight-lines-a-x-b-y-k-and-a-x-b-y-k-1-where-k-is-a-nonzero-real-variable-is-given-by

slug:
the-locus-of-the-point-of-intersection-of-the-straight-lines-a-x-b-y-k-and-a-x-b-y-k-1-where-k-is-a-nonzero-real-variable-is-given-by-84107

question:
<div class="question">The locus of the point of intersection of the straight lines $\frac{x}{a}+\frac{y}{b}=K$ and $\frac{x}{a}-\frac{y}{b}=\frac{1}{K}$,
<br/>where $K$ is a non-zero real variable, is given by</div>

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


title: the-locus-of-the-vertices-of-the-family-of-parabolas-6-y-2-a-3-x-2-3-a-2-x-12-a-is
slug: the-locus-of-the-vertices-of-the-family-of-parabolas-6-y-2-a-3-x-2-3-a-2-x-12-a-is-38764
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The locus of the vertices of the family of parabolas $6 y=2 a^{3} x^{2}+3 a^{2} x-12 a$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x y=\frac{105}{64}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x y=\frac{64}{105}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x y=\frac{35}{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x y=\frac{16}{35}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x y=\frac{105}{64}$</span> </div>

answer:
<div class="solution">$\begin{aligned}
<br/>6 y+12 a &amp;=2 a\left(a^{2} x^{2}+\frac{3}{2} a x\right) \\
<br/>&amp;=2 a\left(a^{2} x^{2}+2 \cdot \frac{3}{4} a x+\frac{9}{16}\right)-\frac{9}{8} a \\
<br/>\Rightarrow 6 y+\frac{105}{8} a &amp;=2 a\left(a x+\frac{3}{4}\right)^{2}
<br/>\end{aligned}$
<br/>Let Vertices be ( $h, k), \quad h=-\frac{3}{4 a}, k=-\frac{35 a}{16}$
<br/>$\therefore \mathrm{hk}=\frac{105}{64}$
<br/>Hence, locus is $x y=\frac{105}{64}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
the-magnetic-field-at-the-point-of-intersection-of-diagonals-of-a-square-wire-loop-of-side-mathrml-carrying-a-current-mathrmi-is-1

slug:
the-magnetic-field-at-the-point-of-intersection-of-diagonals-of-a-square-wire-loop-of-side-mathrml-carrying-a-current-mathrmi-is-1-21428

question:
<div class="question">The magnetic field at the point of intersection of diagonals of a square wire loop of side \(\mathrm{L}\) carrying a current \(\mathrm{I}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\mu_0 \mathrm{I}}{\pi \mathrm{L}}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{2 \mu_0 \mathrm{I}}{\pi \mathrm{L}}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\sqrt{2} \mu_0 I}{\pi \mathrm{L}}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{2 \sqrt{2} \mu_{\mathrm{o}} \mathrm{I}}{\pi \mathrm{L}}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">\(\frac{2 \sqrt{2} \mu_{\mathrm{o}} \mathrm{I}}{\pi \mathrm{L}}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned} &amp; \text {Hints: } B=4\left[\frac{\mu_0}{4 \pi} \frac{I}{\left(\frac{\mathrm{L}}{2}\right)}\left(\sin 45^{\circ}+\sin 45^{\circ}\right)\right] \\ &amp; =\frac{\mu_{\mathrm{o}}}{\pi} \frac{2 \mathrm{I}}{\mathrm{L}} \cdot \frac{2}{\sqrt{2}} ; \quad B=\frac{\mu_{\mathrm{o}}}{\pi} \frac{2 \sqrt{2} \mathrm{I}}{\mathrm{L}}\end{aligned}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/HtIBfzUGg9OrkDtPgUEhvZTXE0Acd6XdqFYwzP-vm5s.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
the-magnetic-field-at-the-point-of-intersection-of-diagonals-of-a-square-wire-loop-of-side-mathrml-carrying-a-current-mathrmi-is

slug:
the-magnetic-field-at-the-point-of-intersection-of-diagonals-of-a-square-wire-loop-of-side-mathrml-carrying-a-current-mathrmi-is-51037

question:
<div class="question">The magnetic field at the point of intersection of diagonals of a square wire loop of side \(\mathrm{L}\) carrying a current \(\mathrm{I}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\mu_0 \mathrm{I}}{\pi \mathrm{L}}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{2 \mu_0 \mathrm{I}}{\pi \mathrm{L}}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\sqrt{2} \mu_0 I}{\pi \mathrm{L}}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{2 \sqrt{2} \mu_{\mathrm{o}} \mathrm{I}}{\pi \mathrm{L}}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">\(\frac{2 \sqrt{2} \mu_{\mathrm{o}} \mathrm{I}}{\pi \mathrm{L}}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned} &amp; \text {Hints: } B=4\left[\frac{\mu_0}{4 \pi} \frac{I}{\left(\frac{\mathrm{L}}{2}\right)}\left(\sin 45^{\circ}+\sin 45^{\circ}\right)\right] \\ &amp; =\frac{\mu_{\mathrm{o}}}{\pi} \frac{2 \mathrm{I}}{\mathrm{L}} \cdot \frac{2}{\sqrt{2}} ; \quad B=\frac{\mu_{\mathrm{o}}}{\pi} \frac{2 \sqrt{2} \mathrm{I}}{\mathrm{L}}\end{aligned}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/HtIBfzUGg9OrkDtPgUEhvZTXE0Acd6XdqFYwzP-vm5s.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-magnetic-field-due-to-a-current-in-a-straight-wire-segment-of-length-l-at-a-point-on-its-perpendicular-bisector-at-a-distano-r-r-l

slug:
the-magnetic-field-due-to-a-current-in-a-straight-wire-segment-of-length-l-at-a-point-on-its-perpendicular-bisector-at-a-distano-r-r-l-49476

question:
<div class="question">The magnetic field due to a current in a straight wire segment of length $L$ at a point on
<br/>its perpendicular bisector at a distano $r(r&gt;&gt;L)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">decreases as $\frac{1}{r}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">decreases as $\frac{1}{r^{2}}$</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">C</span> <span class="option-data">decreases as $\frac{1}{r^{3}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">approaches a finite limit as $r \rightarrow \infty$</span> </li> </ul>

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

answer:
<div class="solution">By Biot-Savart's law, the magnetic field due to a current carrying wire is
<br/>$$
<br/>B=\frac{\mu_{0}}{4 \pi} \cdot \frac{I d l \sin \theta}{r^{2}}
<br/>$$
<br/>So,
<br/>$$
<br/>B=\frac{1}{r^{2}}
<br/>$$
<br/>So, the magnetic field due to current decreases as inverse square of distance of the point of observation.</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
title:
the-magnetic-flux-linked-with-a-coil-satisfies-the-relation-4-t-2-6-t-9-wb-where-t-is-the-time-in-second-the-emf-induced-in-the-coil-at-t-2-s-is

slug:
the-magnetic-flux-linked-with-a-coil-satisfies-the-relation-4-t-2-6-t-9-wb-where-t-is-the-time-in-second-the-emf-induced-in-the-coil-at-t-2-s-is-35146

question:
<div class="question">The magnetic flux linked with a coil satisfies the relation $\phi=4 t^{2}+6 t+9$ Wb, where $t$ is the
<br/>time in second. The emf induced in the coil at $t=2 \mathrm{s}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$22 \mathrm{V}$</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">B</span> <span class="option-data">$18 \mathrm{V}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$16 \mathrm{V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$40 \mathrm{V}$</span> </li> </ul>

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

answer:
<div class="solution">Given $\phi=4 t^{2}+6 x+9$ and $t=2 \mathrm{s}$ We know that $\varepsilon=\frac{d \phi}{d t}$
<br/>Here, $\frac{d \phi}{d t}=8 t+6$
<br/>$$
<br/>\begin{aligned}
<br/>\left(\frac{d \phi}{d t}\right)_{t=2} &amp;=8 \times 2+6 \\
<br/>\varepsilon &amp;=22 \mathrm{V}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
the-magnetic-flux-through-a-loop-of-resistance-10-is-given-by-5-t-2-4-t-1-weber-how-much-current-is-induced-in-the-loop-after-02-sec

slug:
the-magnetic-flux-through-a-loop-of-resistance-10-is-given-by-5-t-2-4-t-1-weber-how-much-current-is-induced-in-the-loop-after-02-sec-90161

question:
<div class="question">The magnetic flux through a loop of resistance $10 \Omega$ is given by $\phi=5 t^2-4 t+1$ Weber. How much current is induced in the loop after $0.2 \mathrm{sec}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.4 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.2 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.04 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.02 \mathrm{~A}$</span> </li> </ul>

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

answer:
<div class="solution">Hints : $\phi=5 t^2-4 t+1$<br/>$$<br/>\frac{d \phi}{d t}=10 t-4<br/>$$<br/>$$<br/>\mathrm{I}=\frac{e}{\mathrm{R}}=\frac{-d \phi / d t}{\mathrm{R}}=-\frac{10 t-4}{10}<br/>$$<br/>At $t=0.2 \mathrm{sec}$<br/>$$<br/>\mathrm{I}=\frac{-(10 \times 0.2-4)}{10}=-\frac{(2-4)}{10}=+\frac{2}{10}=+0.2 \mathrm{~A}=0.2 \mathrm{~A}<br/>$$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1: 1</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-magnets-of-two-suspended-coll-galvanometers-are-of-the-same-strength-so-that-they-produce-identical-uniform-magnetic-fields-in-the-region-of-the-c

slug:
the-magnets-of-two-suspended-coll-galvanometers-are-of-the-same-strength-so-that-they-produce-identical-uniform-magnetic-fields-in-the-region-of-the-c-56149

question:
<div class="question">The magnets of two suspended coll galvanometers are of the same strength so that they produce identical uniform magnetic fields in the region of the coils. The coil of the first one is in the shape of a square of side a and that of the second one is circular of radius $\frac{a}{\sqrt{\pi}},$ When the same current is passed through the coils, the ratio of the torque experienced by the first coil to that experienced by the second one is</div>

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

answer:
<div class="solution">In galvanometer $I \sim \theta=G \theta$ Where, $\quad G=\frac{k}{N A B}$
<br/>When coil is in square of shape,
<br/>$$
<br/>G=\frac{k}{N A B}=\frac{k}{N a^{2} B}
<br/>$$
<br/>When coil is in circular shape of radius $a / \sqrt{\pi}$
<br/>$$
<br/>\begin{aligned}
<br/>G &amp;=\frac{k}{N A B}=\frac{k}{N \pi r^{2} \cdot B} \\
<br/>&amp;=\frac{k}{N \pi\left(\frac{a}{\sqrt{\pi}}\right)^{2} \cdot B}=\frac{k}{N a^{2} B}
<br/>\end{aligned}
<br/>$$
<br/>So, value of $G$ is same for both type of shape of coil. So, the torque $e=N B L A \sin \theta$ will be same in both the case
<br/>$$
<br/>e: e=1: 1
<br/>$$</div>


title: the-main-reason-that-sicl-4-is-easily-hydrolysed-as-compared-to-ccl-4-is-that
slug: the-main-reason-that-sicl-4-is-easily-hydrolysed-as-compared-to-ccl-4-is-that-40595
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The main reason that $\mathrm{SiCl}_{4}$ is easily hydrolysed as compared to $\mathrm{CCl}_{4}$ is that</div>

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

answer:
<div class="solution">It is less reactive than other oxides of chlorine and does not react with $P, S,$ coal or paper at room temperature.
<br/> <br/>The main reason that $\mathrm{SiCl}_{4}$ is easily hydrolysed as compared to $\mathrm{CCl}_{4}$, is that Si can extend its coordination number beyond four because it has vacant $d$ -orbital. On the other hand, carbon has no $d$ -orbital thus, it cannot extend its coordination number beyond four, so its halides are not attacked (hydrolysed) by water. The vacant $d$ -orbitals of Si can coordinate with water molecules and hence their halides are hydrolysed by water.
<br/>The hydrolysis of $\mathrm{SiCl}_{4}$ occurs due to coordination of $-\mathrm{OH}$ with empty $3 d$ -orbitals of Si-atom of $\mathrm{SiCl}_{4}$ molecule.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a42.png"/></div>


title: the-major-product-of-the-above-reaction-is-1
slug: the-major-product-of-the-above-reaction-is-1-89934
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The major product of the above reaction is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q44.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q44_opta.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q44_optb.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q44_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q44_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q44_optb.png"/></span> </div>

answer:
<div class="solution">$\mathrm{HBr} \rightleftharpoons \mathrm{H}^{+}+\mathrm{Br}^{-}$
<br/>For 1,3 -butadiene at room temperature, there is formation of 1,4 -addition product.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a44.png"/></div>


title: the-major-product-of-the-following-reaction-is-f-3-c-ch-ch-2-hbr
slug: the-major-product-of-the-following-reaction-is-f-3-c-ch-ch-2-hbr-26406
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The major product of the following reaction is $\mathrm{F}_3 \mathrm{C}-\mathrm{CH}=\mathrm{CH}_2+\mathrm{HBr} \rightarrow$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{F}_3 \mathrm{C}-\mathrm{CH}_2-\mathrm{CH}_2 \mathrm{Br}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{F}_3 \mathrm{C}-\mathrm{CH}(\mathrm{Br})-\mathrm{CH}_3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sVmIrn11i5UVWPv0wrcvJ3bZPCIQtaw1_SEnZib0rLg.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/-P8kz9CI-78bG2eodl4KEo9b7J6zX3s03gCGRwgdYCY.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{F}_3 \mathrm{C}-\mathrm{CH}_2-\mathrm{CH}_2 \mathrm{Br}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/QALydchvpP2sxrzlL7x8sFkQ11upiNe487U9BEAgj14.original.fullsize.png"/></div>


tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-major-products-obtained-during-ozonolysis-of-23-dimethyl-1-butene-and-subsequent-reductions-with-z-n-and-h-2-o-te

slug:
the-major-products-obtained-during-ozonolysis-of-23-dimethyl-1-butene-and-subsequent-reductions-with-z-n-and-h-2-o-te-91908

question:
<div class="question">The major products obtained during ozonolysis of 2,3 -dimethyl- 1 -butene and subsequent reductions with $Z \mathrm{n}$ and $\mathrm{H}_{2} \mathrm{O}$ Ã te</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">methanal and 3 -methy -2 -butanone</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_a35.png"/></div>


title: the-major-products-obtained-form-the-following-reaction-of-1-mole-of-hexadeuteriobenzene-isare
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
slug:
the-major-products-obtained-form-the-following-reaction-of-1-mole-of-hexadeuteriobenzene-isare-57903

question:
<div class="question">The major product(s) obtained form the following reaction of 1 mole of hexadeuteriobenzene is/are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q79.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q79_opta.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q79_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/2017_che_q79_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_79_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q79_opta.png"/></span> </div>

answer:
<div class="solution">The reaction for 1 mole of hexadeuterisobenzene occurs as follows :
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a79.png"/></div>


title: the-major-products-obtained-in-the-following-reaction-isare-1
slug: the-major-products-obtained-in-the-following-reaction-isare-1-50074
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The major products obtained in the following reaction is/are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q77.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q77_opta.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q77_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/2018_che_q77_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/2018_che_q77_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 answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q77_opta.png"/>, <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q77_optd.png"/></span> </div>

answer:
<div class="solution">The major product $($ s) obtained in the given reaction are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a77.png"/> <br/>The mechanism for the above given reaction is as follows. When the $\pi$ -electrons of the alkene approach a molecule of $\mathrm{Br}_{2}$, one of the bromine atom accepts them and releases the shared electrons to the other
<br/>bromine atom. A cyclic bromonium ion is formed.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a77a.png"/> <br/>In next step $\mathrm{Br}^{-}$ attacks a carbon atom of the bromonium ion. This release the strain in the three  membered ring and form a vicinal dibromide.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a77b.png"/></div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
the-mathrmph-of-an-aqueous-solution-of-mathrmch3-mathrmcoona-of-concentrated-mathrmcmathrmm-is-given-by

slug:
the-mathrmph-of-an-aqueous-solution-of-mathrmch3-mathrmcoona-of-concentrated-mathrmcmathrmm-is-given-by-41427

question:
<div class="question">The \(\mathrm{pH}\) of an aqueous solution of \(\mathrm{CH}_3 \mathrm{COONa}\) of concentrated \(\mathrm{C}(\mathrm{M})\) is given by</div>

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

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

answer:
<div class="solution">Hints : In case of Hydrolysis of salt of weak acid and strong base, the \(\mathrm{pH}\) is given by \(\frac{1}{2} \mathrm{pK}_{\mathrm{w}}+\frac{1}{2} \mathrm{pK}_{\mathrm{a}}+\frac{1}{2} \log \mathrm{c}\)</div>


title: the-mathrmsp3-mathrmd2-hybridization-of-central-atom-of-a-molecule-would-lead-to
slug: the-mathrmsp3-mathrmd2-hybridization-of-central-atom-of-a-molecule-would-lead-to-18108
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints : Fact</div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The \(\mathrm{sp}^3 \mathrm{~d}^2\) hybridization of central atom of a molecule would lead to</div>

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


title: the-maximum-and-minimum-values-of-cos-6-sin-6-are-respectively
slug: the-maximum-and-minimum-values-of-cos-6-sin-6-are-respectively-67953
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The maximum and minimum values of $\cos ^{6} \theta+\sin ^{6} \theta$ are respectively</div>

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

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

answer:
<div class="solution">Let $f(\theta)=\sin ^{6} \theta+\cos ^{6} \theta$
<br/>$\Rightarrow \quad f(\theta)=\left(\sin ^{2} \theta\right)^{3}+\left(\cos ^{2} \theta\right)^{3}$
<br/>$=\left(\sin ^{2} \theta+\cos ^{2} \theta\right)$
<br/>$\left(\sin ^{4} \theta+\cos ^{4} \theta-\sin ^{2} \theta-\cos ^{2} \theta\right)$
<br/>$\left[\because a^{3}+b^{3}=(a+b)\left(a^{2}+b^{2}-a b\right]\right.$
<br/>$=1 \cdot\left\{\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-3 \sin ^{2} \theta \cdot \cos ^{2} \theta\right\}$
<br/>$=1 \cdot\left(1-\frac{3}{4} \cdot 4 \sin ^{2} \theta \cdot \cos ^{2} \theta\right)$
<br/>$=1-\frac{3}{4}(\sin 2 \theta)^{2}(\because \sin 2 A=2 \sin A \cos A)$
<br/>$=1-\frac{3}{8}(1-\cos 4 \theta)$
<br/>$=1-\frac{3}{8}+\frac{3}{8} \cos 4 \theta$
<br/>$f(\theta)=\frac{5}{8}+\frac{3}{8} \cdot \cos 4 \theta$
<br/>$-1 \leq \cos 4 \theta \leq 1$
<br/>$\Rightarrow \quad \frac{-3}{8} \leq \frac{3}{8} \cos 4 \theta \leq \frac{3}{8}$
<br/>$\Rightarrow \quad \frac{5}{8}-\frac{3}{8} \leq \frac{5}{8}+\frac{3}{8} \cos 48 \leq \frac{5}{8}+\frac{3}{8}$
<br/>$\Rightarrow \quad \frac{1}{4} \leq f(\theta) \leq 1$
<br/>So, the maximum value is 1 and minimum value is $\frac{1}{4}$</div>


title: the-maximum-number-of-atoms-that-can-be-in-one-plane-in-the-molecule-pnitrobenzonitrile-are
slug: the-maximum-number-of-atoms-that-can-be-in-one-plane-in-the-molecule-pnitrobenzonitrile-are-90110
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The maximum number of atoms that can be in one plane in the molecule p-nitrobenzonitrile are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">13</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/TWrLJMT7y7vsMWqHbxsjoHs3ik2aGWy7PRBLPD8YwsY.original.fullsize.png"/><br/> <br/>All atoms are on the same plane as all central atoms are either $\mathrm{sp}^{2}$ or sp hybridised $\therefore$ total number of atoms on same plane $\Rightarrow \mathrm{C}_{6} \mathrm{H}_{4}+\mathrm{NO}_{2}+\mathrm{CN}=15$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">30</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-maximum-number-of-electrons-in-an-atom-in-which-the-last-electron-filled-has-the-quantum-numbers-n-3-2-and-m-1-is

slug:
the-maximum-number-of-electrons-in-an-atom-in-which-the-last-electron-filled-has-the-quantum-numbers-n-3-2-and-m-1-is-50988

question:
<div class="question">The maximum number of electrons in an atom in which the last electron filled has the quantum numbers $n=3, \ell=2$ and $m=-1$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">17</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">27</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">28</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">30</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>$\mathrm{n}=3$ and $\mathrm{I}=2$ means $3 \mathrm{~d}$ -orbital. Since five orientations of $\mathrm{d}$ -orbitals are degenerate, $\mathrm{m}=-1$ can be assigned to any one. For maximum number of electrons as asked, answer should be atomic number 30.</div>


title: the-maximum-value-of-f-x-e-s-i-n-x-e-c-o-s-x-x-r-is
slug: the-maximum-value-of-f-x-e-s-i-n-x-e-c-o-s-x-x-r-is-37905
question: <div class="question">The maximum value of $f(x)=e^{\sin x}+e^{\cos x} ; x \in R$ is</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{e}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{\mathrm{e}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{e}^{1 / \sqrt{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 e^{-1 / \sqrt{2}}$</span> </li> </ul>

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

answer:
<div class="solution">$f(x)=e^{\sin x}+e^{\cos x}$<br/>$\begin{aligned}<br/>&amp;f^{\prime}(x)=e^{\sin x} \cdot \cos x-e^{\cos x} \cdot \sin x \\<br/>&amp;\begin{aligned}<br/>f^{\prime \prime}(x) &amp;=e^{\sin x} \cos ^2 x+e^{\cos x} \cdot \sin ^2 x-\sin x \cdot e^{\sin x}-\cos x \cdot e^{\cos x} \\<br/>&amp;=e^{\sin x}\left(1-\sin x-\sin ^2 x\right)+e^{\cos x}\left(1-\cos ^2 x-\cos x\right) \\<br/>f^{\prime}(\pi / 4) &amp;=0 \text { and } f^{\prime \prime}(\pi / 4)  &lt; 0 \\<br/>f^{\prime}(x) &amp;=0 \text { at } x=\pi / 4+2 n \pi \text { or } 5 \pi / 4+2 n \pi(n \in z)<br/>\end{aligned}<br/>\end{aligned}$<br/>$f^{\prime}(\pi / 4+2 n \pi)=0$ and $f^{\prime \prime}(\pi / 4+2 n \pi) &lt; 0$<br/>$\therefore f_{\max }=f(\pi / 4+2 n \pi), n \in Z=2 e^{1 / \sqrt{2}}$</div>


title: the-maximum-value-of-x-when-the-complex-number-z-satisfies-the-condition-z-z-2-2-is
slug: the-maximum-value-of-x-when-the-complex-number-z-satisfies-the-condition-z-z-2-2-is-51800
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The maximum value of $|x|$, when the complex number $z$ satisfies the condition $\left|z+\frac{2}{z}\right|=2$ is</div>

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

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

answer:
<div class="solution">We have the identity,
<br/>$$
<br/>\begin{aligned}|z|=\left|z+\frac{2}{z}-\frac{2}{z}\right| \leq\left|z+\frac{2}{z}\right|+\left|\frac{2}{z}\right| \\ \Rightarrow \quad|z| \leq 2+\frac{2}{|z|} \end{aligned}
<br/>$$
<br/>$\Rightarrow \quad|z|^{2} \leq 2|z|+2$
<br/>$\Rightarrow \quad|z|^{2}-2|z|+1 \leq 3$
<br/>\begin{array}{ll}
<br/>\Rightarrow &amp; (|z|-1)^{2} \leq 3 \\
<br/>\Rightarrow &amp; -\sqrt{3} \leq|z|-1 \leq \sqrt{3} \\
<br/>\hline
<br/>\end{array}
<br/>$\Rightarrow \quad-\sqrt{3}+1 \leq|z| \leq \sqrt{3}+1$
<br/>Hence, the maximum value $=\sqrt{3}+1$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-mean-and-variance-of-a-binomial-distribution-are-4-and-2-respectively-then-the-probability-of-exactly-two-succsses-is

slug:
the-mean-and-variance-of-a-binomial-distribution-are-4-and-2-respectively-then-the-probability-of-exactly-two-succsses-is-91119

question:
<div class="question">The mean and variance of a binomial distribution are 4 and 2 respectively. Then the probability of exactly two succsses is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{7}{64}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{21}{128}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{32}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9}{32}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{64}$</span> </div>

answer:
<div class="solution">$\mathrm{np}=4, \mathrm{npq}=2$
<br/>$\therefore \mathrm{q}=\frac{2}{4}=\frac{1}{2} \Rightarrow \mathrm{p}=\frac{1}{2} \therefore \mathrm{n}=8$
<br/>$\mathrm{p}(\mathrm{x}=2)={ }^{8} \mathrm{C}_{2}\left(\frac{1}{2}\right)^{8}=\frac{28}{256}=\frac{7}{64}$</div>


