# questions

title: lim-n-n-n-1-2-n-1-is-equal-to
slug: lim-n-n-n-1-2-n-1-is-equal-to-35200
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\lim _{n \rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\ldots+\sqrt{n-1}}{n \sqrt{n}}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

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

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


title: lim-x-0-1-x-1-x-1
slug: lim-x-0-1-x-1-x-1-91153
question: <div class="question">$\lim _{x \rightarrow 0} \frac{\pi^{x}-1}{\sqrt{1+x}-1}$</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">does not exist</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">equals $\log _{e}\left(\pi^{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">equals 1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">lies between 10 and 11</span> </li> </ul>

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

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


title: lim-x-0-e-x-x-1-x
slug: lim-x-0-e-x-x-1-x-79554
question: <div class="question">$\lim _{x \rightarrow 0^{+}}\left(e^{x}+x\right)^{1 / x}$</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Does not exist finitely</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is 1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is $e^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">is 2</span> </li> </ul>

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

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text {  Let } L=\lim _{x \rightarrow 0^{+}}\left(e^{x}+x\right)^{1 / x}\\
<br/>&amp;\Rightarrow \log L=\lim _{x \rightarrow 0^{+}} \frac{\log \left(e^{x}+x\right)}{x}\\
<br/>&amp;\Rightarrow \log L=\lim _{x \rightarrow 0^{+}} \frac{\frac{1}{\left(e^{x}+x\right)} \cdot e^{x}+1}{1}\\
<br/>&amp;\text { [using L' Hospital rule] }\\
<br/>&amp;\Rightarrow \log L=\lim _{x \rightarrow 0^{+}} \frac{e^{x}+1}{e^{x}+x}\\
<br/>&amp;\Rightarrow \log L=2\\
<br/>&amp;\Rightarrow \quad L=e^{2}
<br/>\end{aligned}
<br/>$$</div>


title: lim-x-0-sin-x-2-t-a-n-x-is-equal-to
slug: lim-x-0-sin-x-2-t-a-n-x-is-equal-to-62476
question: <div class="question">$\lim _{x \rightarrow 0}(\sin x)^{2 \tan x}$ is equal to</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">does not exist</span> </li> </ul>

answer:
<div class="solution">Let $y=\lim _{x \rightarrow 0}(\sin x)^{2 \tan x}$
<br/>$\Rightarrow \log y=\lim _{x \rightarrow 0} \log (\sin x)^{2 \tan x}$
<br/>$=2 \lim _{x \rightarrow 0} \tan x \log \sin x$
<br/>$=2 \lim _{x \rightarrow 0} \frac{\log \sin x}{\cot x}$
<br/>$=2 \lim _{x \rightarrow 0} \frac{\frac{1}{\sin x} \cdot \cos}{-cosec^{2}x} x$
<br/>$=2 \lim _{x \rightarrow 0}(-\sin x \cos x)$
<br/>$=2 \times 0=0$
<br/>$\therefore \quad \log y=0 \Rightarrow y=e^{0}=1$</div>


title: lim-x-0-x-1-ln-1-x-1-x-is
slug: lim-x-0-x-1-ln-1-x-1-x-is-40052
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\lim _{x \rightarrow 0}\left(\frac{1}{x} \ln \sqrt{\frac{1+x}{1-x}}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 2$</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">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">does not exist</span> </li> </ul>

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


title: lim-x-0-x-n-ln-x-n-0
slug: lim-x-0-x-n-ln-x-n-0-78126
question: <div class="question">$\lim _{x \rightarrow 0^{+}}\left(x^{n} \ln x\right), n&gt;0$</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
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">exists and is zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">exists and is 1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">exists and is $e^{-1}$</span> </li> </ul>

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

answer:
<div class="solution">We have,
<br/>$\lim _{x \rightarrow 0^{+}} x^{n} \ln x$
<br/>$=\lim _{x \rightarrow 0^{+}} \frac{\ln x}{x^{-n}}$
<br/>$=\lim _{x \rightarrow 0^{+}} \frac{x}{-n x^{-n-1}} \quad$ [using L'Hospital's rule $]$
<br/>$=\lim _{x \rightarrow 0^{+}} \frac{-1}{n x^{-n}}=0 \quad\left[\because \frac{1}{x-n}=0,\right.$ when $\left.x=0\right]$</div>


title: lim-x-0-x-s-i-n-x-is-equal-to
slug: lim-x-0-x-s-i-n-x-is-equal-to-18966
question: <div class="question">$\lim _{x \rightarrow 0} \frac{\sin |x|}{x}$ is equal to</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
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=""> <span class="option-label">C</span> <span class="option-data">positive infinity</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">does not exist</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">does not exist</span> </div>

answer:
<div class="solution">Hints: $\operatorname{Lim}_{x \rightarrow 0} \frac{\sin |x|}{x}$<br/>LHL $=-1 \quad$ RHL $=1$<br/>Limit does not exist</div>


title: lim-x-1-2-x-1-x-1-x-1-x-is-equal-to
slug: lim-x-1-2-x-1-x-1-x-1-x-is-equal-to-34003
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{(1-\sqrt{x})}{(1-x)}}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">does not exist</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{\frac{2}{3}}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">ln 2</span> </li> </ul>

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

answer:
<div class="solution">We have, $\lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{1-\sqrt{x}}{1-x}}$
<br/>$=\lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{1-\sqrt{x}}{(1+\sqrt x)(1- \sqrt x)}}$
<br/>$=\lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{1}{1+\sqrt{x}}}$
<br/>$=\left(\frac{1+1}{2+1}\right)^{\frac{1}{1+1}}=\left(\frac{2}{3}\right)^{\frac{1}{2}}=\sqrt{\frac{2}{3}}$</div>


title: lim-x-1-l-n-x-1-x-1-1
slug: lim-x-1-l-n-x-1-x-1-1-45526
question: <div class="question">$\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{1}{(x-1)}\right)$</div>
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Does not exist</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

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

answer:
<div class="solution">Hint: 
<br/>$\operatorname{lt}_{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)=\operatorname{lt}_{x \rightarrow 1} \frac{(x-1)-\ln x}{(x-1) \ln x}=\frac{1}{2}$
<br/>Using L.H. rule twice</div>


title: lim-x-rightarrow-0-fracsin-leftpi-sin-2-xrightx2
slug: lim-x-rightarrow-0-fracsin-leftpi-sin-2-xrightx2-45906
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">\(\lim _{x \rightarrow 0} \frac{\sin \left(\pi \sin ^2 x\right)}{x^2}=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\pi^2\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(3 \pi\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2 \pi\)</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints : }=\operatorname{Lt}_{x \rightarrow 0} \frac{\sin \left(\pi \sin ^2 x\right)}{x^2}=\pi_{x \rightarrow 0} \frac{\sin ^2 x}{x^2}=\pi \\<br/>&amp; =\pi<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
lim-x-rightarrow-inftyleftxsqrtnleftxa1rightleftxa2right-ldotsleftxanrightright-where-a1-a2-ldots-an-are-positive-rational-numbers-the-limit

slug:
lim-x-rightarrow-inftyleftxsqrtnleftxa1rightleftxa2right-ldotsleftxanrightright-where-a1-a2-ldots-an-are-positive-rational-numbers-the-limit-60705

question:
<div class="question">\(\lim _{x \rightarrow \infty}\left\{x-\sqrt[n]{\left(x-a_1\right)\left(x-a_2\right) \ldots\left(x-a_n\right)}\right\}\) where \(a_1, a_2, \ldots, a_n\) are positive rational numbers. The limit</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 \(\frac{a_1+a_2+\ldots a_n}{n}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is \(\sqrt[n]{a_1 a_2 \ldots a_n}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">is \(\frac{n}{a_1+a_2+\ldots+a_n}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">is \(\frac{a_1+a_2+\ldots a_n}{n}\)</span> </div>

answer:
<div class="solution">Hint: \(\operatorname{lt}_{x \rightarrow \infty}\left\{x-\sqrt[x]{\left(x-a_1\right)\left(x-a_2\right) \ldots\left(x-a_n\right)}\right\}\)<br/>\(\begin{aligned} &amp; \operatorname{lt}_{x \rightarrow \infty}\left\{x-x^n \sqrt{\left(1-\frac{a_1}{x}\right)\left(1-\frac{a_2}{x}\right) \ldots\left(1-\frac{a_n}{x}\right)}\right\} \\ &amp; \Rightarrow \operatorname{lt}_{x \rightarrow \infty} x\left\{1-\left(1-\frac{a_1}{x}\right)^{1 / n}\left(1-\frac{a_2}{x}\right)^{1 / n} \ldots\left(1-\frac{a_n}{x}\right)^{1 / n}\right\} \\ &amp; \Rightarrow \operatorname{It}_{x \rightarrow \infty} x\left[1-\left(1-\frac{a_1}{n x}\right)\left(1-\frac{a_2}{n x}\right) \ldots\left(1-\frac{a_n}{x n}\right)\right] \\ &amp; \Rightarrow \operatorname{lt}_{x \rightarrow \infty} x\left[\frac{a_1+a_2+\ldots+a_n}{n x}\right]=\frac{a_1+a_2+\ldots+a_n}{n} \\ &amp; \end{aligned}\)</div>


title: lim-x-x-1-x-2-1-a-x-b-a-b-r-0-then
slug: lim-x-x-1-x-2-1-a-x-b-a-b-r-0-then-23562
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\lim _{x \rightarrow \infty}\left(\frac{x^2+1}{x+1}-a x-b\right),(a, b \in R)=0$. Then</div>

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

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

answer:
<div class="solution">$\lim _{x \rightarrow \infty} \frac{x^2+1-a x(x+1)-b(x+1)}{x+1}=0$<br/>$=\lim _{x \rightarrow \infty} \frac{(1-a) x^2-(a+b) x+1-b}{x+1}=0$<br/>for the limit to exist,<br/>$\begin{aligned}<br/>1-\mathrm{a} &amp;=0 \text { and }-(\mathrm{a}+\mathrm{b})=0 \\<br/>\therefore \mathrm{a}=1, \mathrm{~b} &amp;=-1<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
lines-x-y-1-and-3-y-x-3-intersect-the-ellipse-x-2-9-y-2-9-at-the-points-p-q-and-r-the-area-of-the-pqr-is

slug:
lines-x-y-1-and-3-y-x-3-intersect-the-ellipse-x-2-9-y-2-9-at-the-points-p-q-and-r-the-area-of-the-pqr-is-89269

question:
<div class="question">Lines $x+y=1$ and $3 y=x+3$ intersect the ellipse $x^{2}+9 y^{2}=9$ at the points $P, Q$ and $R$. The area of the $\triangle P Q R$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{36}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{18}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{9}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{18}{5}$</span> </div>

answer:
<div class="solution">Given equation of ellipse
<br/>$$
<br/>x^{2}+9 y^{2}=9
<br/>$$
<br/>$\Rightarrow \quad \frac{x^{2}}{9}+\frac{y^{2}}{1}=1$
<br/>and equation of lines
<br/>$$
<br/>x+y=1 \text { and } 3 y=x+3
<br/>$$
<br/>or $\quad \frac{x}{(-3)}+\frac{y}{1}=1$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a4.png"/> <br/> <br/>$\therefore$ Area of $\Delta P Q R=\left|\begin{array}{c|ccc}0 &amp; 1 &amp; 1 \\ -3 &amp; 0 &amp; 1 \\ 9 / 5 &amp; -4 / 5 &amp; 1\end{array}\right| \mid$
<br/>$=\frac{1}{2}\left[\frac{24}{5}+\frac{12}{5}\right]$
<br/>$=\frac{1}{2} \times \frac{36}{5}=\frac{18}{5}$</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$a+b$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
ln-a-a-bc-if-c-9-0-r-and-r-are-the-inradius-and-circumrodius-of-the-abc-respectively-then-2-r-r-is-equal-to

slug:
ln-a-a-bc-if-c-9-0-r-and-r-are-the-inradius-and-circumrodius-of-the-abc-respectively-then-2-r-r-is-equal-to-68409

question:
<div class="question">ln a$\Delta A B C,$ if $\angle C=90^{\circ}, r$ and $R$ are the inradius and circumrodius of the ABC respectively, then $2(r+R)$ is equal to</div>

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

answer:
<div class="solution">Tangents drawn from external points are of equal length.
<br/>$\Rightarrow \quad B D=B E=a-r$
<br/>$[\because C D=r]$
<br/>and $\quad A F=A E=b-r$
<br/>$[\because C F=r]$
<br/>$A E+B E=A B$
<br/>$\Rightarrow \quad b-r+a-r=2 R$
<br/>$1 \because A B$ is a diameter of circumcircle]
<br/>$\Leftrightarrow \quad b+a=2 R+2 r$
<br/>$\Rightarrow \quad 2(r+R)=a+b$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/5tderuuou8yjhmjmnk.png"/></div>


title: magnetic-field-intensity-h-at-the-centre-of-a-circular-loop-of-radius-r-carrying-current-i-e-mu-is
tags: ['Physics', 'Magnetic Properties of Matter', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
slug:
magnetic-field-intensity-h-at-the-centre-of-a-circular-loop-of-radius-r-carrying-current-i-e-mu-is-17557

question:
<div class="question">Magnetic field intensity $\mathrm{H}$ at the centre of a circular loop of radius $\mathrm{r}$ carrying current $\mathrm{I}$ e m.u is</div>

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

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

answer:
<div class="solution">$H=\frac{\mu_0 I}{2 r}=\frac{\mu_0}{4 \pi} \times \frac{2 \pi I}{r}$<br/>In e m.u system $\frac{\mu_0}{4 \pi}=1$. So $H=\frac{2 \pi I}{r}$</div>


tags: ['Chemistry', 'Practical Chemistry', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
match-the-flame-colours-of-the-alkaline-earth-metal-salts-in-the-bunsen-burner-beginarrayll-hline-texta-caicium-textp-brick-red-hline-textb-strontium-

slug:
match-the-flame-colours-of-the-alkaline-earth-metal-salts-in-the-bunsen-burner-beginarrayll-hline-texta-caicium-textp-brick-red-hline-textb-strontium-47099

question:
<div class="question">Match the flame colours of the alkaline earth metal salts in the Bunsen burner<br/>$\begin{array}{l|l}<br/>\hline \text{A. Caicium} &amp; \text{p. Brick red} \\<br/>\hline \text{B. Strontium} &amp; \text{q. Apple green} \\<br/>\hline \text{C. Barium} &amp; \text{r. Crimson} \\<br/>\hline<br/>\end{array}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">p, r, 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">B</span> <span class="option-data">r, p, q</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">q, r, p</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">p, q, r</span> </li> </ul>

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

answer:
<div class="solution">Flame colours are produced from the movement of the electrons in the metal ions present in the compounds.
<br/>Calcium gives brick red colour. Strontium gives crimson colour. Barium gives apple green colour.</div>


title: mathrmbrf3-self-ionises-as-following
slug: mathrmbrf3-self-ionises-as-following-62331
question: <div class="question">\(\mathrm{BrF}_3\) self ionises as following</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2 \mathrm{BrF}_3 \rightleftharpoons \mathrm{BrF}^{+}+\mathrm{BrF}_5^{-}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(2 \mathrm{BrF}_3 \rightleftharpoons \mathrm{BrF}_2^{+}+\mathrm{BrF}_4^{-}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{BrF}_3 \rightleftharpoons \mathrm{BrF}_4^{+}+\mathrm{BrF}_2^{-}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2 \mathrm{BrF}_3 \rightleftharpoons \mathrm{BrF}_3^{+}+\mathrm{BrF}_3^{-}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \mathrm{BrF}_3 \rightleftharpoons \mathrm{BrF}_2^{+}+\mathrm{BrF}_4^{-}\)</span> </div>

answer:
<div class="solution">Hint : \(2 \mathrm{BrF}_3 \rightleftharpoons \mathrm{BrF}_2^{+}+\mathrm{BrF}_4^{-}\)(Relatively more stable structures.)</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
mathrmn-objects-are-distributed-at-random-among-mathrmn-persons-the-number-of-ways-in-which-this-can-be-done-so-that-at-least-one-of-them-will-not-get

slug:
mathrmn-objects-are-distributed-at-random-among-mathrmn-persons-the-number-of-ways-in-which-this-can-be-done-so-that-at-least-one-of-them-will-not-get-90003

question:
<div class="question">\(\mathrm{n}\) objects are distributed at random among \(\mathrm{n}\) persons. The number of ways in which this can be done so that at least one of them will not get any object is</div>

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

answer:
<div class="solution">Hint : Total number of ways of distributing \(n\) objects randomly among \(n\) persons \(=\mathrm{n}^n\)<br/>Number of ways in which each person gets exactly one object \(={ }^n P_n=n\) !<br/>\(\therefore\) Required number of ways \(=n^n-n\) !</div>


tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
mathrmna2-mathrmco3-is-prepared-by-solvay-process-but-mathrmk2-mathrmco3-cannot-be-prepared-by-the-same-because

slug:
mathrmna2-mathrmco3-is-prepared-by-solvay-process-but-mathrmk2-mathrmco3-cannot-be-prepared-by-the-same-because-30133

question:
<div class="question">\(\mathrm{Na}_2 \mathrm{CO}_3\) is prepared by Solvay process but \(\mathrm{K}_2 \mathrm{CO}_3\) cannot be prepared by the same because</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{KHCO}_3\) is appreciably soluble</span> </div>

answer:
<div class="solution">Hint: \(\left(\mathrm{NH}_4\right) \mathrm{HCO}_3+\mathrm{KCl} \longrightarrow \mathrm{KHCO}_3(\mathrm{aq})+\mathrm{NH}_4 \mathrm{Cl}(\mathrm{aq})\)<br/>\(\mathrm{KHCO}_3\) being appreciably soluble cant be isolated from reaction medium easily.</div>


title: mathrmno2-is-not-obtained-on-heating
slug: mathrmno2-is-not-obtained-on-heating-41303
question: <div class="question">\(\mathrm{NO}_2\) is not obtained on heating</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', '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">\(\mathrm{AgNO}_3\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{KNO}_3\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\mathrm{Cu}\left(\mathrm{NO}_3\right)_2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{Pb}\left(\mathrm{NO}_3\right)_2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{KNO}_3\)</span> </div>

answer:
<div class="solution">Hints: \(\mathrm{KNO}_3 \stackrel{\Delta}{\longrightarrow} \mathrm{KNO}_2+\frac{1}{2} \mathrm{O}_2\)</div>


title: maximum-value-of-the-function-f-x-8-x-x-2-on-the-interval-16-is
slug: maximum-value-of-the-function-f-x-8-x-x-2-on-the-interval-16-is-12028
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Maximum value of the function $f(x)=\frac{x}{8}+\frac{2}{x}$ on the interval [1,6] is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{8}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{13}{12}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{17}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{17}{8}$</span> </div>

answer:
<div class="solution">$f(x)=\frac{x}{8}+\frac{2}{x}$
<br/>$\therefore \quad f^{\prime}(x)=\frac{1}{8}-\frac{2}{x^{2}}=\frac{x^{2}-16}{8 x^{2}}$
<br/>For maximum or minimum $f^{\prime}(x)$ must be vanish.
<br/>$$
<br/>\begin{array}{ll}
<br/>\therefore &amp; f^{\prime}(x)=0 \\
<br/>\Rightarrow &amp; \frac{x^{2}-16}{8 x^{2}}=0 \Rightarrow x=4,-4 \\
<br/>&amp; x \in[1,6] \\
<br/>\therefore &amp; x \neq-4
<br/>\end{array}
<br/>$$
<br/>Also, in $[1,4], f^{\prime}(x)     &lt; 0 \Rightarrow f(x)$ is decreasing. $\ln [4,6], f^{\prime}(x)&gt;0 \Rightarrow f(x)$ is increasing.
<br/>$f(1)=\frac{1}{8}+\frac{2}{1}=\frac{17}{8}$
<br/>$f(8)=\frac{6}{8}+\frac{2}{6}=\frac{3}{4}+\frac{1}{3}=\frac{13}{12}$
<br/>Hence, maximum value of $f(x)$ in [1,6] is $\frac{17}{8}$.</div>


tags: ['Mathematics', 'Statistics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
mean-of-n-observations-x-1-x-2-x-n-is-x-if-an-observation-x-q-is-replaced-by-x-q-then-the-new-mean-is

slug:
mean-of-n-observations-x-1-x-2-x-n-is-x-if-an-observation-x-q-is-replaced-by-x-q-then-the-new-mean-is-78948

question:
<div class="question">Mean of $n$ observations $x_{1}, x_{2}, \ldots, x_{n}$ is $\bar{x}$. If an observation $x_{q}$, is replaced by $x_{q}^{\prime}$ then the new mean is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\bar{x}-x_{q}+x_{q}^{\prime}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{(n-1) \bar{x}+x_{q}^{\prime}}{n}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{(n-1) \bar{x}-x_{q}^{\prime}}{n}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{n x-x_{q}+x_{q}^{\prime}}{n}$</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">$\frac{n x-x_{q}+x_{q}^{\prime}}{n}$</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>\bar{x}=\frac{x_{1}+x_{2}+\ldots+x_{q}+\ldots+x_{2}}{n}
<br/>$$
<br/>$\Rightarrow \quad \Sigma x=n \bar{x}$
<br/>If $x_{q}$, is replaced by $x_{q}^{\prime}$, then new total will be $\Sigma x'=\Sigma x-x_{4}-x_{4}^{\prime}$
<br/>New mean will be
<br/>$$
<br/>\begin{array}{l}
<br/>\bar{x}^{\prime}=\frac{\Sigma x^{\prime}}{n} \\
<br/>\bar{x}^{\prime}=\frac{\sum x-x_{q}+x_{i}^{\prime}}{n} \\
<br/>\bar{x}^{\prime}=\frac{n \bar{x}-x_{q}+x_{i}^{\prime}}{n}
<br/>\end{array}
<br/>$$</div>


title: metal-ion-responsible-for-the-minimata-disease-is
slug: metal-ion-responsible-for-the-minimata-disease-is-93793
question: <div class="question">Metal ion responsible for the Minimata disease is</div>
tags: ['Chemistry', 'Chemistry in Everyday Life', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Co}^{2+}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Hg}^{2+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Cu}^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$Z n^{2+}$</span> </li> </ul>

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

answer:
<div class="solution">Eating animal poisoned with mercury $\left(\mathrm{Hg}^{2+}\right)$ Causes deformity. known as minamata  (minimata) disease which is characterised by diarrhoea, hemolysis, impairment of various senses, meningitis and death.</div>


title: methoxybenzene-on-treatment-with-hi-produces-1
slug: methoxybenzene-on-treatment-with-hi-produces-1-62565
question: <div class="question">Methoxybenzene on treatment with $\mathrm{HI}$ produces</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">iodobenzene and methanol</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">phenol and methyl iodide</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">iodobenzene and methyl iodide</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">phenol and methanol</span> </li> </ul>

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

answer:
<div class="solution">Methoxybenzene on treatment with HI produces phenol and methyl iodide. This is due the more stable aryl oxygen bond. Here. methyl phenyl oxonium ion is formed by the protonation of ether. The bond between $\mathrm{O}-\mathrm{CH}_{3}$ is weaker than the bond between $\mathrm{O}-\mathrm{C}_{6} \mathrm{H}_{5}$ because the carbon of phenyl group is $s p^{2}$ -hybridised and there is a partial double bond character. Thus, the reaction yields phenol and alkyl halide.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a48.png"/></div>


title: mixing-of-two-different-ideal-gases-under-isothermal-reversible-condition-will-lead-to
slug: mixing-of-two-different-ideal-gases-under-isothermal-reversible-condition-will-lead-to-32928
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Mixing of two different ideal gases under isothermal reversible condition will lead to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">increase of entropy of the system</span> </div>

answer:
<div class="solution">On mixing of two different ideal gases under isothermal reversible conditions, $\Delta S_{\operatorname{mix}}$ is always positive $i . e .$, increasing.</div>


tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
molecular-velocities-of-two-gases-at-the-same-temperature-t-are-u-1-and-u-2-their-masses-are-m-1-and-m-2-respectively-which-of-the-following-expressio

slug:
molecular-velocities-of-two-gases-at-the-same-temperature-t-are-u-1-and-u-2-their-masses-are-m-1-and-m-2-respectively-which-of-the-following-expressio-59588

question:
<div class="question">Molecular velocities of two gases at the same temperature $(\mathrm{T})$ are $\mathrm{u}_{1}$ and $\mathrm{u}_{2} .$ Their masses are $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ respectively. Which of the following expressions is correct at temperature $\mathrm{T} ?$</div>

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

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

answer:
<div class="solution">We know $u \times \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}$ but as $\mathrm{T}$ is same for both gases
<br/>Hence u $\propto \frac{1}{\sqrt{\mathrm{m}}}$
<br/>$\therefore \frac{\mathrm{u}_{1}}{\mathrm{u}_{2}}=\sqrt{\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}}}$
<br/>or $\frac{\mathrm{u}_{1}^{2}}{\mathrm{u}_{2}^{2}}=\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}}$
<br/>or $\mathrm{m}_{1} \mathrm{u}_{1}^{2}=\mathrm{m}_{2} \mathrm{u}_{2}^{2}$</div>


title: n-2-h-4-and-h-2-o-2-show-similarity-in
slug: n-2-h-4-and-h-2-o-2-show-similarity-in-53312
tags: ['Chemistry', 'Hydrogen', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\mathrm{N}_{2} \mathrm{H}_{4}$ and $\mathrm{H}_{2} \mathrm{O}_{2}$ show similarity in</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Reducing nature, Oxidising nature, Hybridisation of central atoms</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/cKAcXnKaSSUyA_rugAzPnXSWsYoLnbx05LJ3r1iUTRY.original.fullsize.png"/></div>


title: n-c-4-h-10-o-h-2-so-4-k-2-cr-2-o-7-c-4-h-8-o-warm-i-2-naoh-chi-3-here-n-is
slug: n-c-4-h-10-o-h-2-so-4-k-2-cr-2-o-7-c-4-h-8-o-warm-i-2-naoh-chi-3-here-n-is-22349
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$$
<br/>\underset{\mathrm{N}}{\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{O}} \frac{\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}}{\mathrm{H}_{2} \mathrm{SO}_{4}} \mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O} \cdot \frac{\mathrm{I}_{2} / \mathrm{NaOH}}{\mathrm{Warm}}, \mathrm{CHI}_{3}
<br/>$$ Here, N is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q49_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/2018_che_q49_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/2018_che_q49_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_che_q49_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_che_q49_optb.png"/></span> </div>

answer:
<div class="solution">Given sequence of chemical equation is
<br/>$\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{O} \frac{\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}}{\mathrm{H}_{2} \mathrm{SO}_{4}}-\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O} \frac{\mathrm{I}_{2} / \mathrm{NaOH}}{\text { Warm }} \mathrm{CHI}_{3}$
<br/>$(\mathrm{N})$
<br/>Here N will be $=$ Butan- $2 \cdot 0$ l) Here,
<br/>Step I It involves the oxidation of secondary alcohol to corresponding alcohol in the presence of $\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}$ and $\mathrm{H}_{2} \mathrm{SO}_{4}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a49.png"/> <br/>Step II It involves the lodoform reaction. Ketone formed in step 1 have one methyl group linked to the carbonyl carbon atom are oxidised by sodium hypohalite to sodium salts of corresponding carboxylic acids having one carbon atom less than that of carbonyl compound.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a49a.png"/></div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
n-identical-droplets-are-charged-to-v-volt-each-if-they-coalesce-to-form-a-single-drop-then-its-potential-will-be

slug:
n-identical-droplets-are-charged-to-v-volt-each-if-they-coalesce-to-form-a-single-drop-then-its-potential-will-be-39468

question:
<div class="question">$n$ identical droplets are charged to $v$ volt each. If they coalesce to form a single drop, then its potential will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$n^{2 / 3} v$</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">$n^{1 / 3} v$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$n v$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$v / n$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } n \times \frac{4}{3} \pi r^3=\frac{4}{3} \pi \mathrm{R}^3 \\<br/>&amp; \Rightarrow \mathrm{R}=r n^{1 / 3} \\<br/>&amp; \mathrm{C}_0=4 \pi \varepsilon_0 r \\<br/>&amp; q_0=\mathrm{C}_0 \mathrm{~V}=\left(4 \pi \varepsilon_0 r\right) \mathrm{V}<br/>\end{aligned}<br/>$$<br/>Capacitance of Bigger drop,<br/>$$<br/>\mathrm{C}=4 \pi \varepsilon_0 \mathrm{R}<br/>$$<br/>So, $\mathrm{V}=\frac{n q_0}{\mathrm{C}}=\frac{n\left(4 \pi \varepsilon_0 r \mathrm{~V}\right)}{4 \pi \varepsilon_0 \mathrm{R}}=n\left(\frac{r}{\mathrm{R}}\right) \mathrm{V}=n\left(\frac{1}{n^{1 / 3}}\right) \mathrm{V}=n^{2 / 3} \mathrm{~V}$</div>


tags: ['Chemistry', 'Coordination Compounds', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
nickel-combines-with-a-uninegative-monodentate-ligand-leftmathrmxrightto-form-a-paramagnetic-complex-leftmathrmnix4right2-the-hybridisation-involved-a

slug:
nickel-combines-with-a-uninegative-monodentate-ligand-leftmathrmxrightto-form-a-paramagnetic-complex-leftmathrmnix4right2-the-hybridisation-involved-a-68969

question:
<div class="question">Nickel combines with a uninegative monodentate ligand \(\left(\mathrm{X}^{-}\right)\)to form a paramagnetic complex \(\left[\mathrm{NiX}_4\right]^{2-}\). The hybridisation involved and number of unpaired electrons present in the complex are respectively.</div>

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

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

answer:
<div class="solution">Hint : \(\mathrm{Ni}^{2+}\left(\mathrm{d}^8\right)\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Ti2bch-3sW6WGN8LwqMYl57ojgESfds6NlrLjzm8VzI.original.fullsize.png"/><br/><br/>it should be paramagnetic with 2 unpaired electrons</div>


title: nitric-acid-can-be-obtained-from-ammonia-via-the-formations-of-the-intermediate-compounds
slug: nitric-acid-can-be-obtained-from-ammonia-via-the-formations-of-the-intermediate-compounds-90277
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Nitric acid can be obtained from ammonia via the formations of the intermediate compounds</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">nitric oxide and nitrogen dioxide</span> </div>

answer:
<div class="solution">Ostwald process of for manufacture of nitric acid, $4 \mathrm{NH}_{3}+\mathrm{SO}_{2} \frac{\mathrm{Pt} \text { gauge }}{800-900^{\circ} \mathrm{C}} \underset{\text { nitric oxide }}{4 \mathrm{NO}}+6 \mathrm{H}_{2} \mathrm{O}$
<br/>$$
<br/>\Delta H=-21.5 \mathrm{kcal}
<br/>$$
<br/>$2 \mathrm{NO}+\mathrm{O}_{2} \stackrel{50^{\circ} \mathrm{C}}{\longrightarrow} \quad 2 \mathrm{NO}_{2}$
<br/>$4 \mathrm{NO}_{2}+2 \mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2} \longrightarrow \underset{\text { nitric acid }}{4 \mathrm{HNO}_{3}}$</div>


title: nitrogen-dioxide-is-not-produced-on-heating
slug: nitrogen-dioxide-is-not-produced-on-heating-35157
question: <div class="question">Nitrogen dioxide is not produced on heating</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{KNO}_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{AgNO}_{3}$</span> </li> </ul>

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

answer:
<div class="solution">Only KNO $_{3}$ on heating do not give $\mathrm{NO}_{2}$, but all other give
<br/>(a) $2 \mathrm{KNO}_{3} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{KNO}_{2}+\mathrm{O}_{2}$
<br/>(b) $2 \mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2} \stackrel{\mathrm{A}}{\longrightarrow} 2 \mathrm{P} \mathrm{bO}+4 \mathrm{NO}_{2}+\mathrm{O}_{2}$
<br/>(c) $2 \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2} \stackrel{\mathrm{A}}{\longrightarrow} 2 \mathrm{CuO}+4 \mathrm{NO}_{2}+\mathrm{O}_{2}$
<br/>(d) $2 \mathrm{AgNO}_{3} \frac{\Delta}{\text { Aed het }} 2 \mathrm{Ag}+\mathrm{O}_{2}+2 \mathrm{NO}_{2}$
<br/>Heavy metal nitrates liberate nitrogen dioxide on heating.</div>


title: number-of-common-tangents-of-y-x-2-and-y-x-2-4-x-4-is
slug: number-of-common-tangents-of-y-x-2-and-y-x-2-4-x-4-is-18553
question: <div class="question">Number of common tangents of $y=x^{2}$ and $y=-x^{2}+4 x-4$ is</div>
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">We have, equation of parabola $y=x^{2}$ Let $P\left(\alpha, \alpha^{2}\right)$ is a point on the parabola,
<br/>$\therefore$
<br/>$y-\alpha^{2}=2 \alpha(x-\alpha)$
<br/>$\because$
<br/>$\left[\because \frac{d y}{d x}=2 x \Rightarrow \frac{d y}{d x\left(\alpha, a^{2}\right)}=2 \alpha\right]$
<br/>$\Rightarrow \quad y=2 \alpha x-\alpha^{2}$
<br/>Also, given $y=-x^{2}+4 x-4$
<br/>$\therefore-x^{2}+4 x-4=2 a x-a^{2}$
<br/>$\Rightarrow \quad x^{2}+2 x(\alpha-2)+\left(4-a^{2}\right)=0$
<br/>Discriminant $=0$ 
<br/>$4(\alpha-2)^{2}-4\left(4-a^{2}\right)=0$
<br/>$\Rightarrow$
<br/>$(\alpha-2)^{2}-\left(4-\alpha^{2}\right)=0$
<br/>$\Rightarrow \quad \alpha^{2}-4 \alpha+4-4+\alpha^{2}=0$
<br/>$\alpha^{2}-2 \alpha=0$
<br/>$\alpha=0, \alpha=2$</div>


title: number-of-hydrogen-ions-present-in-10-milionth-part-of-133-cm-3-of-pure-water-at-2-5-c-is
slug: number-of-hydrogen-ions-present-in-10-milionth-part-of-133-cm-3-of-pure-water-at-2-5-c-is-32077
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Number of hydrogen ions present in 10 milionth part of $1.33 \mathrm{cm}^{3}$ of pure water at $25^{\circ} \mathrm{C}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6.023 million</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">60 million</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">8.01 million</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">80.23 million</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">8.01 million</span> </div>

answer:
<div class="solution">10 million $=10^{-7}$ 
<br/>$\therefore$ 10 $million^{th}$ part of $1.33 \mathrm{cm}^{3}=1.33 \times 10^{-7} \mathrm{cm}^{3}$ $=1.33 \times 10^{-7} \mathrm{mL}$
<br/>For pure water,
<br/>$$
<br/>\begin{aligned}
<br/>\left[\mathrm{H}^{+}\right] &amp;=10^{-7} \mathrm{mol} / \mathrm{L}
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow 1$ L water contains $\left[\mathrm{H}^{+}\right]=10^{-7} \mathrm{mol}$
<br/>or $1 \mathrm{mL}$ water contains $\left[\mathrm{H}^{+}\right]=\frac{10^{-7}}{1000}=10^{-10} \mathrm{mol}$
<br/>or 10 million $^{\text {th }}$ part of $1.33 \mathrm{cm}^{3}$ water contains $\begin{aligned}\left(\mathrm{H}^{+}\right] &amp;=1.33 \times 10^{-7} \times 10^{-10} \mathrm{mol} \\ &amp;=1.33 \times 10^{-17} \mathrm{mol} \end{aligned}$
<br/>$\therefore$ Number of $\mathrm{H}^{+}$ ions $=1.33 \times 10^{-17} \times N_{A}$
<br/>$=1.33 \times 10^{-17} \times 6.022 \times 10^{23}$
<br/>$=8.009 \times 10^{6}=8.01$ million</div>


title: number-of-intersecting-points-of-the-conics-4-x-2-9-y-2-1-and-4-x-2-y-2-4-is
slug: number-of-intersecting-points-of-the-conics-4-x-2-9-y-2-1-and-4-x-2-y-2-4-is-68576
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Number of intersecting points of the conics $4 x^{2}+9 y^{2}=1$ and $4 x^{2}+y^{2}=4$ is</div>

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

answer:
<div class="solution">For curve I. 
<br/>$4 x^{2}+9 y^{2}=1 \Rightarrow \frac{x^{2}}{\left(\frac{1}{2}\right)^{2}}+\frac{y^{2}}{\left(\frac{1}{3}\right)^{2}}=1$
<br/>which is an equation of ellipee with $a=\frac{1}{2}$ and $b=\frac{1}{3}$
<br/>For curve $I I, 4 x^{2}+y^{2}=4$ $\Rightarrow \quad \frac{x^{2}}{1}+\frac{y^{2}}{4}=1$
<br/>$\Rightarrow \quad \frac{x^{2}}{(1)^{2}}+\frac{y^{2}}{(2)^{2}}=1$
<br/>which is also an equation of ellipse with $a=1$ and $b=2$
<br/>Hence, there is no intersecting point.</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P2.db
title:
number-of-points-having-distance-5-fron-the-straight-line-x-2-y-1-0-and-a-distance-13-from-the-line-2-x-3-y-1-0-is

slug:
number-of-points-having-distance-5-fron-the-straight-line-x-2-y-1-0-and-a-distance-13-from-the-line-2-x-3-y-1-0-is-24662

question:
<div class="question">Number of points having distance $\sqrt{5}$ fron the straight line $x-2 y+1=0$ and a distance $\sqrt{13}$ from the line $2 x+3 y-1=0,$ is</div>

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

answer:
<div class="solution">Let the point be $(h, k)$
<br/>Then, $\left|\frac{h-2 k+1}{\sqrt{1^{2}+(-2)^{2}}}\right|=\sqrt{5}$
<br/>$h-2 k+1=\pm 5$
<br/>Also, $\left|\frac{2 h+3 k-1}{\sqrt{2^{2}+3^{2}}}\right|=$
<br/>$\Rightarrow \quad 2 h+3 k-1=\pm 13$
<br/>On solving Eqs
<br/>(i) and (ii), we get four points So. there are much 4 points.</div>


title: number-of-solutions-of-the-equation-tan-x-sec-x-2-cos-x-x-0-is
slug: number-of-solutions-of-the-equation-tan-x-sec-x-2-cos-x-x-0-is-41519
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Number of solutions of the equation $\tan x+\sec x=2 \cos x, x \in[0, \pi]$ is</div>

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

answer:
<div class="solution">$\tan x+\sec x=2 \cos x$
<br/>$\Rightarrow \quad \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x$
<br/>$\Rightarrow \quad \sin x+1=2 \cos ^{2} x=2\left(1-\sin ^{2} x\right)$
<br/>$=2-2 \sin ^{2} x$
<br/>$\Rightarrow \quad 2 \sin ^{2} x+\sin x-1=0$
<br/>$\Rightarrow \quad 2 \sin ^{2} x+2 \sin x-\sin x-1=0$
<br/>$\Rightarrow \quad 2 \sin x(\sin x+2)-1(\sin x+1)=0$
<br/>$\Rightarrow \quad(\sin x+2)(2 \sin x-1)=0$
<br/>$\Rightarrow \quad \sin x=-2,$ which is not possible. or $\sin x=\frac{1}{2} \Rightarrow x=\frac{\pi}{6}, \frac{5 \pi}{6}$
<br/>$x \in[0, \pi]$
<br/>$\therefore$ number of solution $=2$.</div>


tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
of-the-following-acids-the-one-which-has-the-capability-to-form-complex-compound-and-also-possesses-oxidizing-and-reducing-properties-is

slug:
of-the-following-acids-the-one-which-has-the-capability-to-form-complex-compound-and-also-possesses-oxidizing-and-reducing-properties-is-78962

question:
<div class="question">Of the following acids the one which has the capability to form complex compound and also possesses oxidizing and reducing properties is :</div>

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

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

answer:
<div class="solution">$\mathrm{HNO}_2$<br/>Hints : Here oxidation state of $\mathrm{N}$ lies between -3 to +5</div>


title: of-the-following-compounds-the-most-acidic-is
slug: of-the-following-compounds-the-most-acidic-is-26211
question: <div class="question">Of the following compounds the most acidic is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{As}_2 \mathrm{O}_3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{P}_2 \mathrm{O}_5$</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">$\mathrm{Sb}_2 \mathrm{O}_3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Bi}_2 \mathrm{O}_3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}_2 \mathrm{O}_5$</span> </div>

answer:
<div class="solution">In a group as we go downwards, the oxide basic character increases hence maximum acidic oxide is $\mathrm{P}_2 \mathrm{O}_5$</div>


title: of-the-following-compounds-which-one-is-the-strongest-bronsted-acid-in-a-aqueous-solution
slug: of-the-following-compounds-which-one-is-the-strongest-bronsted-acid-in-a-aqueous-solution-73955
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Of the following compounds, which one is the strongest Bronsted acid in a aqueous solution?</div>

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

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

answer:
<div class="solution">$\because$ Higher be the oxidation number of central atom in oxo-acids, more strongly it behave as Bronsted-acid. Oxidation number for $\mathrm{Cl}$ in $\mathrm{HClO}_{3} \rightarrow+5$ 
<br/>Oxidation number for $\mathrm{Cl}$ in $\mathrm{HClO}_{2} \rightarrow+3$ 
<br/>Oxidation number for $\mathrm{Cl}$ in $\mathrm{HClO} \rightarrow+1$ 
<br/>Oxidation number for $\mathrm{Cl}$ in $\mathrm{HBrO} \rightarrow+1$ 
<br/>Hence, $\mathrm{HClO}_{3}$, is the strongest Bronsted acid in aqueous solution.</div>


title: of-the-following-molecules-which-have-shape-similar-to-co-2
slug: of-the-following-molecules-which-have-shape-similar-to-co-2-22552
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Of the following molecules, which have shape similar to $\mathrm{CO}_{2} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{HgCl}_{2}$, $\mathrm{C}_{2} \mathrm{H}_{2}$</span> </div>

answer:
<div class="solution">$\mathrm{HgCl}_{2}$ and $\mathrm{C}_{2} \mathrm{H}_{2}$ both show linear shape as of $\mathrm{CO}_{2}$.
<br/>($\because$ both have zero lone pair of electrons)
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_a38.png"/></div>


title: on-heating-chloric-acid-decomposes-to
slug: on-heating-chloric-acid-decomposes-to-72780
question: <div class="question">On heating, chloric acid decomposes to</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', '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">$\mathrm{HClO}_{4} . \mathrm{Cl}_{2}, \mathrm{O}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{HClO}_{2}$, $\mathrm{Cl}_{2}, \mathrm{O}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{HClO}, \mathrm{Cl}_{2} \mathrm{O}$ and $\mathrm{H}_{2} \mathrm{O}_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{HCl}, \mathrm{HClO}, \mathrm{Cl}_{2} \mathrm{O},$ and $\mathrm{H}_{2} \mathrm{O}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{HClO}_{4} . \mathrm{Cl}_{2}, \mathrm{O}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$</span> </div>

answer:
<div class="solution">$3 \mathrm{HClO}_{3} \stackrel{\mathrm{\Delta}}{\longrightarrow} \mathrm{HClO}_{4}+\mathrm{Cl}_{2}+2 \mathrm{O}_{2}+\mathrm{H}_{2} \mathrm{O}$</div>


tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
on-mixing-an-alkane-with-chlorine-and-irradiating-with-ultraviolet-light-it-forms-only-one-monochloroalkane-the-alkane-is

slug:
on-mixing-an-alkane-with-chlorine-and-irradiating-with-ultraviolet-light-it-forms-only-one-monochloroalkane-the-alkane-is-36456

question:
<div class="question">On mixing an alkane with chlorine and irradiating with ultra-violet light, it forms only one mono-chloro-alkane. The alkane is</div>

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

answer:
<div class="solution">Hints : Neopentane<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/EMny1mw3cbZO2aN9qx_hYdKcB_0OcqrusOzzwczQqhg.original.fullsize.png"/><br/><br/>contains all hydrogen atom equivalent</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
on-passing-c-ampere-of-current-for-time-t-sec-through-1-l-of-2-m-cuso-4-solution-atomic-weight-of-cu-635-the-amount-m-of-cu-in-gram-deposited-on-catho

slug:
on-passing-c-ampere-of-current-for-time-t-sec-through-1-l-of-2-m-cuso-4-solution-atomic-weight-of-cu-635-the-amount-m-of-cu-in-gram-deposited-on-catho-11718

question:
<div class="question">On passing 'C ampere of current for time 't' sec through 1 L of 2 (M) $\mathrm{CuSO}_{4}$ solution (atomic weight of $\mathrm{Cu}=63.5$ ), the amount ${ }^{\prime} \mathrm{m}^{\prime}$ of Cu (in gram) deposited on cathode will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$m=C t /(63.5 \times 96500)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$m=C t /(31.25 \times 96500)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$m=(C \times 96500) /(31.25 \times t)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$m=(31.25 \times C \times t) / 96500$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$m=(31.25 \times C \times t) / 96500$</span> </div>

answer:
<div class="solution">$m=\frac{E G}{F}$ $=\frac{\frac{63.5}{2} \times C \times t}{96500}=\frac{31.75 \times C \times t}{96500}$</div>


title: on-r-a-relation-is-defined-by-xpy-if-and-only-if-x-y-is-zero-or-irrational-then
slug: on-r-a-relation-is-defined-by-xpy-if-and-only-if-x-y-is-zero-or-irrational-then-17277
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">On $R$, a relation $\rho$ is defined by xpy if and only if $x-y$ is zero or irrational. Then.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\rho$ is equivalence relation</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\rho$ is reflexive but neither symmetric nor transitive</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\rho$ is reflexive and symmetric but not transitive</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\rho$ is symmetric and transitive but not reflexive</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\rho$ is reflexive and symmetric but not transitive</span> </div>

answer:
<div class="solution">On the set $R$ $x p y \Rightarrow x-y$ is zero or irrational number.<br/>Now, $x p x$<br/>$\Rightarrow x-x=0$<br/>$\Rightarrow \rho$ is reflexive. <br/>If $x \rho y \Rightarrow x-y$ is zero or irrational. <br/>$=-(y-x)$ is zero or irrational. <br/>$\Rightarrow$ yox is zero or irrational. <br/>$\Rightarrow \rho$ is symmetric.<br/>And if $x \rho y \Rightarrow x-y$ is 0 or irrational. <br/>$y \rho z \Rightarrow y-z$ is 0 or irrational. <br/>Then, $(x-y)+(y-z)=x-z$ may be 0 or rational. <br/>$\Rightarrow \rho$ is not transitive.</div>


title: on-r-the-relation-be-defined-by-xpy-holds-if-and-only-if-x-y-is-zero-or-irrational-then
slug: on-r-the-relation-be-defined-by-xpy-holds-if-and-only-if-x-y-is-zero-or-irrational-then-34908
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">On $R$, the relation $\rho$ be defined by 'xpy holds if and only if $x-y$ is zero or irrational'. Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\rho$ is reflexive and symmetric but not transitive</span> </div>

answer:
<div class="solution">We have, $x p y \Rightarrow x-y$ is zero or irrational. Now, $x-x=0$
<br/>$\Rightarrow(x, x) \in \rho$
<br/>$\therefore \rho$ is reflexive. 
<br/>Again, if $x-y$ is either zero or irrational, then $y-x$ will also be either zero or inational.
<br/>$(x, y) \in \rho$
<br/>$\Leftrightarrow \quad(y, x) \in \rho$
<br/>$\therefore \rho$ is symmetric. 
<br/>Again, $(2, \sqrt{3}) \in \rho$ and $(\sqrt{3}, 4) \in \rho$
<br/>$But (2,4) \notin \rho$
<br/>$\therefore p$ is not transitive.</div>


title: on-r-the-set-of-real-numbers-a-relation-is-defined-as-a-b-if-and-only-if-1-ab-0-then
slug: on-r-the-set-of-real-numbers-a-relation-is-defined-as-a-b-if-and-only-if-1-ab-0-then-55615
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">On $R$, the set of real numbers, a relation $\rho$ is defined as $a \rho b$ if and only if $1+a b&gt;0$. Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\rho$ is reflexive and symmetric but not transitive</span> </div>

answer:
<div class="solution">We know that, 
<br/>$1+a^{2}&gt;0, a \in R$
<br/>$\Rightarrow \quad(a, a) \in \rho$
<br/>$\therefore \rho$ is reflexive
<br/>Again, $\quad$ let $(a, b) \in \rho$
<br/>$\Rightarrow \quad 1+a b&gt;0$
<br/>$\Rightarrow \quad 1+b a&gt;0$
<br/>$\Rightarrow \quad(b, a) \in \rho$
<br/>$\therefore \rho$ is symmetric Now, $\quad(1,-0 \cdot 1) \in \rho$
<br/>and $(-0 \cdot 1,-9) \in \rho$
<br/>but $\quad(1,-9) \notin \rho$
<br/>$\therefore \rho$ is not transitive.</div>


title: on-set-a-1-2-3-relations-r-and-s-are-given-by-r-1-1-2-2-3-3-1-2-2-1-s-1-1-2-2-3-3-1-3-3-1-then
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
slug:
on-set-a-1-2-3-relations-r-and-s-are-given-by-r-1-1-2-2-3-3-1-2-2-1-s-1-1-2-2-3-3-1-3-3-1-then-67467

question:
<div class="question">On set $A=\{1,2,3\},$ relations $R$ and $S$ are given by $R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$
<br/>$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$
<br/>Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$R \cup S$ is reflexive and symmetric but not transitive</span> </div>

answer:
<div class="solution">We have, $R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$<br/>$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$<br/>$\therefore R \cup S=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)\}$<br/>Since, $(2,1) \in R \cup S,(1,3) \in R \cup S$<br/>but $(2,3) \in R \cup S$<br/>$\therefore \mathrm{R} \cup \mathrm{S}$ is reflexive and symmetric but not transitive.</div>


title: on-the-ellipse-4-x-2-9-y-2-1-the-points-at-which-the-tangents-are-parallel-to-the-line-8-x-9-y-are
tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
slug:
on-the-ellipse-4-x-2-9-y-2-1-the-points-at-which-the-tangents-are-parallel-to-the-line-8-x-9-y-are-15824

question:
<div class="question">On the ellipse $4 x^{2}+9 y^{2}=1$, the points at which the tangents are parallel to the line $8 x=9 y,$ are</div>

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

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

answer:
<div class="solution">Given equation of ellipse is
<br/>$$
<br/>4 x^{2}+9 y^{2}=1
<br/>$$
<br/>On differentiating. we get $\Rightarrow \quad 8 x+18 yy'=0$
<br/>$\Rightarrow \quad y'=\frac{-8 x}{18 y}=m$
<br/>Also, $8 x=9 y \quad$ [equation of line] 
<br/>On differentiating. we got $\Rightarrow \quad 8=9 y^{\prime}$
<br/>$\Rightarrow \quad y^{\prime}=\frac{8}{9}=m$
<br/>$[\because \text{tangents are parallel to this line}]$
<br/>So, by Eqs, we get $\frac{-8 x}{18 y}=\frac{8}{9}$
<br/>$\Rightarrow \quad-x=2 y$
<br/>$\Rightarrow \quad x=-2 y$
<br/>On substituting $x=-2 y$, we get
<br/>$$
<br/>\begin{array}{r}
<br/>4(-2 y)^{2}+9 y^{2}=1 \\
<br/>\Rightarrow \quad 16 y^{2}+9 y^{2}=1 \Rightarrow 25 y^{2}=1 \\
<br/>\Rightarrow \quad y=\pm \frac{1}{5} \quad \therefore \quad x=\mp \frac{2}{5}
<br/>\end{array}
<br/>$$
<br/>So, required points are $\left(\frac{2}{5},-\frac{1}{5}\right)$ and $\left(-\frac{2}{5}, \frac{1}{5}\right)$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2018']
answer: <div class="solution">Required number of ways $={}^{20} C_{2} \times 2!$ $= {}^{20}P_{2}$</div>
source_db: db\MarksBatch2_P2.db
title:
on-the-occasion-of-dipawali-festival-each-student-of-a-class-sends-greeting-cards-to-others-if-there-are-20-students-in-the-class-the-number-of-cards-

slug:
on-the-occasion-of-dipawali-festival-each-student-of-a-class-sends-greeting-cards-to-others-if-there-are-20-students-in-the-class-the-number-of-cards-96036

question:
<div class="question">On the occasion of Dipawali festival each student of a class sends greeting cards to others. If there are 20 students in the class. the number of cards sends by students is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">${ }^{20} \mathrm{P}_{2}$, $2 \times{ }^{20} \mathrm{C}_{2}$</span> </div>


title: on-the-set-r-of-real-numbers-the-relation-is-defined-by-x-y-x-y-r
slug: on-the-set-r-of-real-numbers-the-relation-is-defined-by-x-y-x-y-r-69383
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">On the set $R$ of real numbers, the relation $\rho$ is defined by $x \rho y,(x, y) \in R$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">If $|x-y|    &lt; 2$, then $\rho$ is reflexive but neither symmetric nor transitive.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II $x-y    &lt; 2$, then $\rho$ is reflexeve and symmetric bu not transitive.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$If(x \mid \geq y,$ then $\rho$ is reflexive and transitive but not symmetric.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">If $x&gt;|y|$, then $p$ is transitive but neither reflexve nor symmetric.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">If $x&gt;|y|$, then $p$ is transitive but neither reflexve nor symmetric.</span> </div>

answer:
<div class="solution">On the set $R$ of real numbers<br/>For reflexive,<br/>$x \rho x \Rightarrow(x, x) \in R$<br/>$\Rightarrow x &gt; |x|$ which is not true. $\Rightarrow \rho$ is not reflexive.<br/>For symmetric,<br/>$(x, y) \in R \Rightarrow x &gt; |y|$<br/>and $(y, x) \in R \Rightarrow y &gt; |x|$<br/>So, $x &gt; |y| \neq y &gt; |x|$ $\Rightarrow \rho$ is not symmetric. <br/>For transitive, $(x, y) \in R \Rightarrow x &gt; |y|(y, z) \in R \Rightarrow y &gt; |z|$<br/>$\Rightarrow x &gt; |z| \Rightarrow(x, z) \in R$<br/>$\Rightarrow \rho$ is transitive.</div>


title: on-the-set-r-of-real-numbers-we-define-x-p-y-if-and-only-if-x-y-0-then-the-relation-p-is
slug: on-the-set-r-of-real-numbers-we-define-x-p-y-if-and-only-if-x-y-0-then-the-relation-p-is-41962
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">On the set $R$ of real numbers we define $x P y$ if and only if $x y \geq 0$. Then, the relation $P$ is</div>

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

answer:
<div class="solution">For every real number $x, x^{2} \geq 0$
<br/>$\therefore$ $(x, x) \in P$
<br/>Hence, $P$ is reflexive.
<br/>Now let $(x,y) \in P \\
<br/>\Rightarrow xy \geq 0 \\
<br/>\Rightarrow yx \geq 0$
<br/>$(y, x) \in P$
<br/>Hence, $P$ is symmetric. 
<br/>Again (-1,0)$\in P$ and (0,2)$\in P$
<br/>But (-1,2)$\notin P$
<br/>$as(-1)(2)=-2    &lt; 0$
<br/>Pis not transitive.</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
one-kg-of-copper-is-drawn-into-a-wire-of-1-mm-diameter-and-a-wire-of-2-mm-diameter-the-resistance-of-the-two-wires-will-be-in-the-ratio

slug:
one-kg-of-copper-is-drawn-into-a-wire-of-1-mm-diameter-and-a-wire-of-2-mm-diameter-the-resistance-of-the-two-wires-will-be-in-the-ratio-67074

question:
<div class="question">One $\mathrm{Kg}$ of copper is drawn into a wire of $1 \mathrm{~mm}$ diameter and a wire of $2 \mathrm{~mm}$ diameter. The resistance of the two wires will be in the ratio</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2: 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1: 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$16: 1$</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">$4: 1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$16: 1$</span> </div>

answer:
<div class="solution">Mass $=\left(\pi \mathrm{r}_1^2 \ell_1\right) \sigma$ (Ist wire)<br/>$\begin{aligned} &amp; \text { Mass }=\left(\pi \mathrm{r}_1^2 \ell_2\right) \sigma(2 \text { nd wire) } \\ &amp; \left(\pi \mathrm{r}_1^2 \ell_1\right) \sigma=\left(\pi \mathrm{r}_2^2 \ell_2\right) \sigma\end{aligned}$<br/>$\begin{aligned} &amp; \left(\pi \mathrm{r}_1^2 \ell_1\right) \sigma=\left(\pi \mathrm{r}_2^2 \ell_2\right) \sigma \\ &amp; \frac{\ell_1}{\ell_2}=\left(\frac{r_2}{r_1}\right)^2\end{aligned}$<br/>$\frac{R_1}{R_2}=\frac{\rho \frac{\ell_1}{A_1}}{\rho \frac{\ell_2}{A_2}}=\frac{\ell_1}{\ell_2} \times \frac{A_2}{A_1}=\frac{\ell_1}{\ell_2} \times\left(\frac{r_2}{r_1}\right)^2$<br/>$\begin{aligned} &amp; =\left(\frac{r_2}{r_1}\right)^4 \\ &amp; \Rightarrow 16: 1\end{aligned}$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
one-mole-of-a-diatomic-ideal-gas-undergoes-a-process-shown-in-p-v-diagram-the-total-heat-given-to-the-gas-ln-2-07-is

slug:
one-mole-of-a-diatomic-ideal-gas-undergoes-a-process-shown-in-p-v-diagram-the-total-heat-given-to-the-gas-ln-2-07-is-77993

question:
<div class="question">One mole of a diatomic ideal gas undergoes a process shown in $\mathrm{P}-\mathrm{V}$ diagram. The total heat given to the gas $(\ln 2=0.7)$ is<br/> <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/f8vT5SmqtSBdmcs4gLsGcPacM5As4wpO5eF766EesdU.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{P}_0 \mathrm{~V}_0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3.9 \mathrm{P}_0 \mathrm{~V}_0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.1 \mathrm{P}_0 \mathrm{~V}_0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.4 \mathrm{P}_0 \mathrm{~V}_0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.9 \mathrm{P}_0 \mathrm{~V}_0$</span> </div>

answer:
<div class="solution">For a Diatomic gas the total heat given is $\Delta \mathrm{Q}_{\mathrm{AB}}+\Delta \mathrm{Q}_{\mathrm{BC}}$<br/>$$<br/>\begin{aligned}<br/>&amp;\Delta \mathrm{Q}_{\mathrm{AB}}(\text { Isochoric Process })=\mathrm{nC}_{\mathrm{V}} \Delta \mathrm{T} \\<br/>&amp;=\frac{\mathrm{P}_{\mathrm{f}} \mathrm{V}_{\mathrm{f}}-\mathrm{P}_{\mathrm{i}} \mathrm{V}_{\mathrm{i}}}{\gamma-1}=\frac{2 \mathrm{P}_0 \mathrm{~V}_0-\mathrm{P}_0 \mathrm{~V}_0}{\frac{7}{5}-1}=\frac{\mathrm{P}_0 \mathrm{~V}_0}{\left(\frac{2}{5}\right)}=\frac{5}{2} \mathrm{P}_0 \mathrm{~V}_0=2.5 \mathrm{P}_0 \mathrm{~V}_0<br/>\end{aligned}<br/>$$<br/>$\Delta \mathrm{Q}_{\mathrm{BC}}$ (Isothermal Process)<br/>$$<br/>Q_{B C}=n R T \ell n \frac{V_f}{V_i} \quad=P V \ell n \frac{V_f}{V_i}<br/>$$<br/>$$<br/>=2 \mathrm{P}_0 \times \mathrm{V}_0 \ln \left(\frac{2 \mathrm{~V}_0}{\mathrm{~V}_0}\right)=2 \mathrm{P}_0 \mathrm{~V}_0 \ell n 2=1.4 \mathrm{P}_0 \mathrm{~V}_0<br/>$$<br/>$$<br/>\Rightarrow \Delta \mathrm{Q}_{\text {total }}=3.9 \mathrm{P}_0 \mathrm{~V}_0<br/>$$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 R$</span> </div>
source_db: db\MarksBatch2_P2.db
title:
one-mole-of-a-monoatomic-ideal-gas-undergoes-a-quasistatic-process-which-is-depicted-by-a-straight-line-joining-points-v-0-t-0-and-2-v-0-3-t-0-in-a-v-

slug:
one-mole-of-a-monoatomic-ideal-gas-undergoes-a-quasistatic-process-which-is-depicted-by-a-straight-line-joining-points-v-0-t-0-and-2-v-0-3-t-0-in-a-v-41010

question:
<div class="question">One mole of a monoatomic ideal gas undergoes a quasistatic process, which is depicted by a straight line joining points $\left(V_{0}, T_{0}\right)$ and $\left(2 V_{0}, 3 T_{0}\right)$ in a ${V} \cdot T$ diagram. What is the value of the heat capacity of the gas at the point $\left(V_{0}, T_{0}\right) ?$</div>

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

answer:
<div class="solution">Heat capacity of an ideal gas in a thermodynamic process,
<br/>$\because$ Ideal gas is monoatomic,
<br/>$$
<br/>\begin{aligned}
<br/>C_{V} &amp;=\frac{f R}{2}=\frac{3 R}{2} \\
<br/>C_{\text {process }} &amp;=C_{V}+\frac{p}{n} \cdot \frac{d V}{d T} \\
<br/>&amp;=\frac{3}{2} R+\frac{p}{n} \cdot \frac{V_{0}}{2 T} \\
<br/>\therefore C_{\text {process }}=\frac{3}{2} R+\frac{n R T}{n 2 T} &amp;[\because p V=n R T] \\
<br/>=&amp; \frac{3}{2} R+\frac{R}{2}=2 R
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
one-mole-of-a-van-der-waals-gas-obeying-the-equation-p-v-2-a-v-b-rt-undergoes-the-quasistatic-cyclic-process-which-is-shown-in-the-p-v-diagram-the-net-1

slug:
one-mole-of-a-van-der-waals-gas-obeying-the-equation-p-v-2-a-v-b-rt-undergoes-the-quasistatic-cyclic-process-which-is-shown-in-the-p-v-diagram-the-net-1-83427

question:
<div class="question">One mole of a van der Waals' gas obeying the equation $\left(p+\frac{a}{V^{2}}\right)(V-b)=R T$ undergoes the quasi-static cyclic process which is shown in the $p \cdot V$ diagram. The net heat absorbed by the gas in this process is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q14.png"/></div>

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

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

answer:
<div class="solution">For the cyclic process
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a14.png"/> <br/>Heat absorbed = Work done
<br/>$$
<br/>\begin{array}{l}
<br/>=\text { Area }=\frac{1}{2}(\Delta p) \times \Delta V \\
<br/>=\frac{1}{2}\left(p_{1}-p_{2}\right) \times\left(V_{1}-V_{2}\right) \\
<br/>=\frac{1}{2}\left(p_{1}-p_{2}\right)\left(V_{1}-V_{2}\right)
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
one-mole-of-a-van-der-waals-gas-obeying-the-equation-p-v-2-a-v-b-rt-undergoes-the-quasistatic-cyclic-process-which-is-shown-in-the-p-v-diagram-the-net

slug:
one-mole-of-a-van-der-waals-gas-obeying-the-equation-p-v-2-a-v-b-rt-undergoes-the-quasistatic-cyclic-process-which-is-shown-in-the-p-v-diagram-the-net-76303

question:
<div class="question">One mole of a van der Waals' gas obeying the equation $\left(p+\frac{a}{V^{2}}\right)(V-b)=R T$ undergoes the quasi-static cyclic process which is shown in the $p \cdot V$ diagram. The net heat absorbed by the gas in this process is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q14.png"/></div>

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

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

answer:
<div class="solution">For the cyclic process
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a14.png"/> <br/>Heat absorbed = Work done
<br/>$$
<br/>\begin{array}{l}
<br/>=\text { Area }=\frac{1}{2}(\Delta p) \times \Delta V \\
<br/>=\frac{1}{2}\left(p_{1}-p_{2}\right) \times\left(V_{1}-V_{2}\right) \\
<br/>=\frac{1}{2}\left(p_{1}-p_{2}\right)\left(V_{1}-V_{2}\right)
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
one-mole-of-an-ideal-monoatomic-gas-expands-along-the-polytrope-p-v-3-constant-from-v-1-to-v-2-at-a-constant-pressure-p-1-the-temperature-during-the-p

slug:
one-mole-of-an-ideal-monoatomic-gas-expands-along-the-polytrope-p-v-3-constant-from-v-1-to-v-2-at-a-constant-pressure-p-1-the-temperature-during-the-p-78326

question:
<div class="question">One mole of an ideal monoatomic gas expands along the polytrope $PV^3=$ constant from $V_1$ to $V_2$ at a constant pressure $P_1$. The temperature during the process is such that molar specific heat $C_V=\frac{3 R}{2}$. The total heat absorbed during the process can be expressed as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad P_1 V_1\left(\frac{V_1^2}{V_2^2}+1\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{P}_1 \mathrm{~V}_1\left(\frac{\mathrm{V}_1^2}{\mathrm{~V}_2^2}-1\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{P}_1 \mathrm{~V}_1\left(\frac{\mathrm{V}_1^3}{\mathrm{~V}_2^2}-1\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad P_1 V_1\left(\frac{V_1}{V_2^2}-1\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}_1 \mathrm{~V}_1\left(\frac{\mathrm{V}_1^2}{\mathrm{~V}_2^2}-1\right)$</span> </div>

answer:
<div class="solution">Assuming $P_1$ as initial pressure.<br/>$\mathrm{PV}^3=$ constant $n=3$<br/>$$<br/>\mathrm{C}=\mathrm{C}_{\mathrm{V}}+\frac{\mathrm{R}}{1-\mathrm{n}}=\frac{3 \mathrm{R}}{2}+\frac{\mathrm{R}}{1-3}=\mathrm{R}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp;T_i=\frac{P_1 V_1}{R} \\<br/>&amp;P_2 V_2{ }^3=P_1 V_1^3 \\<br/>&amp;P_2=P_1 \times\left(\frac{V_1}{V_2}\right)^3<br/>\end{aligned}<br/>$$<br/>$$<br/>T_2=\frac{P_2 V_2}{R}=\frac{P_1}{R} \times\left(\frac{V_1}{V_2}\right)^3 \times V_2=\frac{P_1 V_1^3}{R V_2^2}<br/>$$<br/>$$<br/>\mathrm{Q}=\mathrm{nC} \Delta \mathrm{T}<br/>$$<br/>$$<br/>=1 \times R \times\left(\frac{P_1 V_1^3}{R V_2^2}-\frac{P_1 V_1}{R}\right)<br/>$$<br/>$$<br/>\mathrm{Q}=\mathrm{P}_1 \mathrm{~V}_1\left(\frac{\mathrm{V}_1^2}{\mathrm{~V}_2^2}-1\right)<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
one-mole-of-an-ideal-monoatomic-gas-is-heated-at-a-constant-pressure-from-0-c-to-10-0-c-then-the-change-in-the-internal-enery-of-the-gas-is-given-r-83

slug:
one-mole-of-an-ideal-monoatomic-gas-is-heated-at-a-constant-pressure-from-0-c-to-10-0-c-then-the-change-in-the-internal-enery-of-the-gas-is-given-r-83-89803

question:
<div class="question">One mole of an ideal monoatomic gas is heated at a constant pressure from $0^{\circ} \mathrm{C}$ to $100^\circ C$. Then the change in the internal enery of the gas is (Given, $R=8.32 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.83 \times 10^{3} \mathrm{J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$46 \times 10^{3} \mathrm{J}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.08 \times 10^{3} \mathrm{J}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1.25 \times 10^{3} \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.25 \times 10^{3} \mathrm{J}$</span> </div>

answer:
<div class="solution">We know that $\Delta U=n C_{v} \Delta T$
<br/>$$
<br/>\begin{array}{l}
<br/>\text { where } \begin{aligned}
<br/>\Delta T &amp;=\left(T_{2}-T_{1}\right) \\
<br/>T_{1} &amp;=0^{\circ} \mathrm{C}=273 \mathrm{K} \\
<br/>T_{2} &amp;=100^{\circ} \mathrm{C}=373 \mathrm{K} \\
<br/>n &amp;=1 \text { (monoatomic gas) }
<br/>\end{aligned} \\
<br/>\begin{aligned}
<br/>R=8.32 \mathrm{J} \mathrm{mol}^{-} \mathrm{k}^{-1}=1 \times \frac{3}{2} R \times(373-273) \\
<br/>=1 \times \frac{3}{2} \times 8.32 \times 100=3 \times 8.32 \times 50
<br/>\end{aligned}
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow 1248 \mathrm{kJ}$
<br/>of
<br/>$$
<br/>=1.25 \times 10^{3} \mathrm{J}
<br/>$$</div>


title: one-of-the-products-of-the-following-reaction-is-p-structure-of-p-is
slug: one-of-the-products-of-the-following-reaction-is-p-structure-of-p-is-20578
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">One of the products of the following reaction is $P$.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q41.png"/> <br/>Structure of $P$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q41_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q41_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q41_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q41_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/2019_che_q41_optc.png"/></span> </div>

answer:
<div class="solution">2,2,2 -trichloroacetophenone reacts with aqueous KOH to form potassium Salt of benzoic acid, which undergoes hydrolysis to form benzo acid as the final product.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a41.png"/> <br/>The product of the above reaction $P$ is benzoic acid. Thus, option (c) is correct answer.</div>


title: orthoand-parahydrogens-have
slug: orthoand-parahydrogens-have-34950
question: <div class="question">Ortho-and para-hydrogens have</div>
tags: ['Chemistry', 'Hydrogen', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">No Solution Available</div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Identical chemical properties but different physical properties</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Identical physical and chemical properties</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Identical physical properties but different chemical properties</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Different physical and chemical properties</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Identical chemical properties but different physical properties</span> </div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">25200</span> </div>
source_db: db\MarksBatch2_P2.db
title:
out-of-7-consonants-and-4-vowels-the-number-of-words-not-necessarily-meaningful-that-can-be-made-each-consisting-of-3-consonants-and-2-vowels-is

slug:
out-of-7-consonants-and-4-vowels-the-number-of-words-not-necessarily-meaningful-that-can-be-made-each-consisting-of-3-consonants-and-2-vowels-is-93203

question:
<div class="question">Out of 7 consonants and 4 vowels, the number of words (not necessarily meaningful) that can be made, each consisting of 3 consonants and 2 vowels, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">24800</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">25100</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">25200</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">25400</span> </li> </ul>

answer:
<div class="solution">3 consonants can be selected from 7 consonante
<br/>$$
<br/>={ }^{7} C_{3} \text { ways }
<br/>$$
<br/>2 vowels can be selected from 4 vowels
<br/>$$
<br/>={ }^{4} C_{2} \text { ways }
<br/>$$
<br/>$\therefore$ Required number of words
<br/>$$
<br/>={ }^{7} C_{3} \times{ }^{4} C_{2} \times 5 !
<br/>$$
<br/>[selected 5 letters can be arrange in $51,$ so got, a different words]
<br/>$$
<br/>=35 \times 6 \times 120=25200
<br/>$$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">25200</span> </div>
source_db: db\MarksBatch2_P2.db
title:
out-of-7-consonants-and-4-vowels-words-are-formed-each-having-3-consonants-and-2-vowels-the-number-of-such-words-that-can-be-formed-is

slug:
out-of-7-consonants-and-4-vowels-words-are-formed-each-having-3-consonants-and-2-vowels-the-number-of-such-words-that-can-be-formed-is-44259

question:
<div class="question">Out of 7 consonants and 4 vowels, words are formed each having 3 consonants and 2 vowels. The number of such words that can be formed is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">210</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">25200</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">2520</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">302400</span> </li> </ul>

answer:
<div class="solution">Out of 7 consonants, the number of ways of selecting 3 consonants $={ }^{7} C_{3}$ 
<br/>Similarly, number of ways of selecting
<br/>2 vowels out of 4 vowels $={ }^{4} C_{2}$
<br/>$\therefore$ Total number of words formed
<br/>$$
<br/>\begin{array}{l}
<br/>={ }^{7} C_{3} \times{ }^{4} C_{2} \times{ }^{5} P_{5} \\
<br/>=\frac{7 \times 6 \times 5}{3 \times 2 \times 1} \times \frac{4 \times 3}{2 \times 1} \times 51 \\
<br/>=7 \times 5 \times 2 \times 3 \times 120=25200
<br/>\end{array}
<br/>$$</div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
out-of-the-following-outer-electronic-configurations-of-atoms-the-highest-oxidation-state-is-achieved-by-which-one-1

slug:
out-of-the-following-outer-electronic-configurations-of-atoms-the-highest-oxidation-state-is-achieved-by-which-one-1-64107

question:
<div class="question">Out of the following outer electronic configurations of atoms, the highest oxidation state is achieved by which one?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(n-1) d^{8} n s^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(n-1) d^{5} n s^{2}$</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">$(n-1) d^{3} n s^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(n-1) d^{5} n s^{1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(n-1) d^{5} n s^{2}$</span> </div>

answer:
<div class="solution">Out of the given outer electronic configuration of atoms, the highest oxidation state is achieved by $(n-1) d^{5} n s^{2}$ i.e. 7
<br/>A large number of oxidation state is due to the fact the $(n-1) d$ -electrons may get involved along with $n s$ electrons in bonding as electrons in $(n-1) d-$ orbitals are in an energy state comparable to $n s$ -electrons. Oxidation state of other options are as follows:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a68.png"/></div>


tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
out-of-the-following-outer-electronic-configurations-of-atoms-the-highest-oxidation-state-is-achieved-by-which-one

slug:
out-of-the-following-outer-electronic-configurations-of-atoms-the-highest-oxidation-state-is-achieved-by-which-one-46495

question:
<div class="question">Out of the following outer electronic configurations of atoms, the highest oxidation state is achieved by which one?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(n-1) d^{8} n s^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(n-1) d^{5} n s^{2}$</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">$(n-1) d^{3} n s^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(n-1) d^{5} n s^{1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(n-1) d^{5} n s^{2}$</span> </div>

answer:
<div class="solution">Out of the given outer electronic configuration of atoms, the highest oxidation state is achieved by $(n-1) d^{5} n s^{2}$ i.e. 7
<br/>A large number of oxidation state is due to the fact the $(n-1) d$ -electrons may get involved along with $n s$ electrons in bonding as electrons in $(n-1) d-$ orbitals are in an energy state comparable to $n s$ -electrons. Oxidation state of other options are as follows:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a68.png"/></div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
oxidation-of-allyl-alcohol-with-a-peracid-gives-a-compound-of-molecular-formula-c-3-h-6-o-2-which-contains-an-asymmetric-carbon-atom-the-structure-of-

slug:
oxidation-of-allyl-alcohol-with-a-peracid-gives-a-compound-of-molecular-formula-c-3-h-6-o-2-which-contains-an-asymmetric-carbon-atom-the-structure-of-64176

question:
<div class="question">Oxidation of allyl alcohol with a peracid gives a compound of molecular formula $\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{2}$, which contains an asymmetric carbon atom. The structure of the compound is</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/2019_che_q75_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/2019_che_q75_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/2019_che_q75_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/2019_che_q75_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/2019_che_q75_opta.png"/></span> </div>

answer:
<div class="solution">Oxidation of allyl alcohol with a peracid gives a compound of molecular formula $\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}_{2},$ which contains an asymmetric carbon atom. The structure of the compound is
<br/>So, the option (a) is correct answer.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a75.png"/></div>


title: oxidation-states-of-cr-in-k-2-cr-2-o-7-and-cro-5-are-respectively
slug: oxidation-states-of-cr-in-k-2-cr-2-o-7-and-cro-5-are-respectively-97211
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Oxidation states of $\mathrm{Cr}$ in $\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7$ and $\mathrm{CrO}_5$ are, respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$+6,+5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$+6,+10$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$+6,+6$</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">None of these</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/iFisOrO98W8bba1IpXjvhe2dLfB3AWw6Ai1YHh8bies.original.fullsize.png"/><br/><br/>$\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7: 2(+1)+2 \mathrm{x}+7(-2)=0 \Rightarrow \mathrm{x}=+6$</div>


title: ozonolysis-of-an-alkene-produces-only-one-dicarbonyl-compound-the-structure-of-the-alkene-is
slug: ozonolysis-of-an-alkene-produces-only-one-dicarbonyl-compound-the-structure-of-the-alkene-is-91561
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Ozonolysis of an alkene produces only one dicarbonyl compound. The structure of the alkene is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}_{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q29_optb.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q29_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_{3}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}=\mathrm{CH}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q29_optb.png"/></span> </div>

answer:
<div class="solution">Only gives one compound on ozonolysis.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_a29.png"/> <br/> <br/>All other options give more than one aldehyde/ketone on ozonolysis.</div>


title: p-4-o-10-is-the-anhydride-of
slug: p-4-o-10-is-the-anhydride-of-61801
question: <div class="question">$\mathrm{P}_4 \mathrm{O}_{10}$ is the anhydride of</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_3 \mathrm{PO}_2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{H}_3 \mathrm{PO}_3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_3 \mathrm{PO}_4$</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">D</span> <span class="option-data">$\mathrm{H}_4 \mathrm{P}_2 \mathrm{O}_7$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{H}_3 \mathrm{PO}_4$</span> </div>

answer:
<div class="solution">Hints: $4 \mathrm{H}_3 \mathrm{PO}_4 \longrightarrow \mathrm{P}_4 \mathrm{O}_{10}+6 \mathrm{H}_2 \mathrm{O}$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
p-br-2-c-2-h-4-br-2-nh-3-nanh-2-q-q-hg-2-20-h-2-so-4-r-zn-hg-hc-s-the-species-p-q-r-and-s-respectively-are-1

slug:
p-br-2-c-2-h-4-br-2-nh-3-nanh-2-q-q-hg-2-20-h-2-so-4-r-zn-hg-hc-s-the-species-p-q-r-and-s-respectively-are-1-70672

question:
<div class="question">$[P] \stackrel{\text { Br }_{2}}{\longrightarrow} C_{2} \mathrm{H}_{4}$ Br $_{2} \frac{\text { NaNH }_{2}}{\mathrm{NH}_{3}}$ ? $[Q]$
<br/>$[Q] \frac{20 \% \mathrm{H}_{2} \mathrm{SO}_{4}}{\mathrm{Hg}^{2+} \cdot \Delta}[R] \stackrel{\mathrm{Zn}-\mathrm{Hg} / \mathrm{HC}}{\longrightarrow}[S]$
<br/>The species $P, Q, R$ and $S$ respectively are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">ethene, ethyne, ethanal, ethane</span> </div>

answer:
<div class="solution">Given reactions are
<br/>$[P] \stackrel{\mathrm{Br}_{2}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Br}_{2} \frac{\mathrm{NaNH}_{2}}{\mathrm{NH}_{3}},[Q]$
<br/>$[Q] \frac{20 \% / \mathrm{H}_{2} \mathrm{SO}_{4}}{\mathrm{Hg}^{2+}, \Delta},[\mathrm{R}] \stackrel{\mathrm{Zn} \cdot \mathrm{Hg} / \mathrm{HG}}{\longrightarrow}[S]$
<br/>$[P]$ is an alkene with two carbon atoms i.e. ethene. When it reacts with Br $y$, addition reaction occurs and $\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Br}_{2}$ is formed.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a71.png"/> <br/>$\mathrm{C}_{\mathrm{H}}, \mathrm{Br}_{2}$(1, 2-dibromoethane) on reaction with $\operatorname{NaNH}_{2} / \mathrm{NH},$ gives ethyne $(\mathrm{C}, \mathrm{H}, \mathrm{y})$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a71a.png"/> <br/>Ethyne IQlin presence of $20 \% \mathrm{H}_{2} \mathrm{SO}_{4}, \mathrm{Hg}^{2+} \mathrm{at} 333 \mathrm{K}$
<br/>gives ethanal.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a71b.png"/> <br/> <br/>Ethanal $[R]$ undergoes reduction in presence of $$
<br/>\text { Zn-Hg /HCl to give ethane }[S]
<br/>$$
<br/>$\mathrm{CH}_{3}\mathrm{CHO} \xrightarrow[\mathrm{HCl}]{\mathrm{Zn-Hg}} \mathrm{CH}_{3}-\mathrm{CH}_{3}$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
p-br-2-c-2-h-4-br-2-nh-3-nanh-2-q-q-hg-2-20-h-2-so-4-r-zn-hg-hc-s-the-species-p-q-r-and-s-respectively-are

slug:
p-br-2-c-2-h-4-br-2-nh-3-nanh-2-q-q-hg-2-20-h-2-so-4-r-zn-hg-hc-s-the-species-p-q-r-and-s-respectively-are-63762

question:
<div class="question">$[P] \stackrel{\text { Br }_{2}}{\longrightarrow} C_{2} \mathrm{H}_{4}$ Br $_{2} \frac{\text { NaNH }_{2}}{\mathrm{NH}_{3}}$ ? $[Q]$
<br/>$[Q] \frac{20 \% \mathrm{H}_{2} \mathrm{SO}_{4}}{\mathrm{Hg}^{2+} \cdot \Delta}[R] \stackrel{\mathrm{Zn}-\mathrm{Hg} / \mathrm{HC}}{\longrightarrow}[S]$
<br/>The species $P, Q, R$ and $S$ respectively are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">ethene, ethyne, ethanal, ethane</span> </div>

answer:
<div class="solution">Given reactions are
<br/>$[P] \stackrel{\mathrm{Br}_{2}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Br}_{2} \frac{\mathrm{NaNH}_{2}}{\mathrm{NH}_{3}},[Q]$
<br/>$[Q] \frac{20 \% / \mathrm{H}_{2} \mathrm{SO}_{4}}{\mathrm{Hg}^{2+}, \Delta},[\mathrm{R}] \stackrel{\mathrm{Zn} \cdot \mathrm{Hg} / \mathrm{HG}}{\longrightarrow}[S]$
<br/>$[P]$ is an alkene with two carbon atoms i.e. ethene. When it reacts with Br $y$, addition reaction occurs and $\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Br}_{2}$ is formed.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a71.png"/> <br/>$\mathrm{C}_{\mathrm{H}}, \mathrm{Br}_{2}$(1, 2-dibromoethane) on reaction with $\operatorname{NaNH}_{2} / \mathrm{NH},$ gives ethyne $(\mathrm{C}, \mathrm{H}, \mathrm{y})$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a71a.png"/> <br/>Ethyne IQlin presence of $20 \% \mathrm{H}_{2} \mathrm{SO}_{4}, \mathrm{Hg}^{2+} \mathrm{at} 333 \mathrm{K}$
<br/>gives ethanal.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a71b.png"/> <br/> <br/>Ethanal $[R]$ undergoes reduction in presence of $$
<br/>\text { Zn-Hg /HCl to give ethane }[S]
<br/>$$
<br/>$\mathrm{CH}_{3}\mathrm{CHO} \xrightarrow[\mathrm{HCl}]{\mathrm{Zn-Hg}} \mathrm{CH}_{3}-\mathrm{CH}_{3}$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
p-is-the-extremity-of-the-latusrectum-of-ellipse-3-x-2-4-y-2-48-in-the-first-quadrant-the-eccentric-angle-of-p-is

slug:
p-is-the-extremity-of-the-latusrectum-of-ellipse-3-x-2-4-y-2-48-in-the-first-quadrant-the-eccentric-angle-of-p-is-14987

question:
<div class="question">$P$ is the extremity of the latusrectum of ellipse $3 x^{2}+4 y^{2}=48$ in the first quadrant. The eccentric angle of $P$ is</div>

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

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

answer:
<div class="solution">Given equation of ellipse is
<br/>$$
<br/>\begin{array}{l}
<br/>\quad 3 x^{2}+4 y^{2}=48 \\
<br/>\Rightarrow \quad \frac{3 x^{2}}{48}+\frac{4 y^{2}}{48}=1 \quad \Rightarrow \quad \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 \\
<br/>\text { Here, } \quad a=4 \text { and } b=2 \sqrt{3}
<br/>\end{array}
<br/>$$
<br/>$\therefore$
<br/>$e=\sqrt{1-\frac{b^{2}}{a^{2}}}$
<br/>$=\sqrt{1-\frac{12}{16}}=\sqrt{\frac{4}{16}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$
<br/>$\therefore$ Coordinates of $P$ are $=(2,3)$ $\left.(4 \cos \theta, 2 \sqrt{3} \sin \theta)=(2,3) \quad [\because p=\left(a e, \frac{b^{2}}{a}\right)\right]$
<br/>By comparing, we get
<br/>$$
<br/>4 \cos \theta=2 \text { and } 2 \sqrt{3} \sin \theta=3
<br/>$$
<br/>$\Rightarrow \quad \cos \theta=\frac{1}{2}$ and $\sin \theta=\frac{\sqrt{3}}{2}$
<br/>$\Rightarrow \quad \cos \theta=\cos \frac{\pi}{3}$ and $\sin \theta=\sin \frac{\pi}{3}$
<br/>$\theta=\frac{\pi}{3}$ and $\theta=\frac{\pi}{3}$
<br/>$\therefore$ eccentric angle of $P$ is $\frac{\pi}{3}$</div>


title: paracetamol-is
slug: paracetamol-is-44094
question: <div class="question">Paracetamol is</div>
tags: ['Chemistry', 'Chemistry in Everyday Life', '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">Methyl salicylate</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Phenyl salicylate</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">N-acetyl p-amino phenol</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Acetyl salicylic acid</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">N-acetyl p-amino phenol</span> </div>

answer:
<div class="solution">Hints: Fact<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/VJdC8zQSacZzbwCo-LlvWiVzUIoCWCUZQIFq-O5ZAxI.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Motion In Two Dimensions', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
particle-a-moves-along-x-axis-with-a-uniform-velocity-of-magnitude-10-m-s-particle-b-moves-with-uniform-velocity-20-m-s-along-a-direction-making-an-an-1

slug:
particle-a-moves-along-x-axis-with-a-uniform-velocity-of-magnitude-10-m-s-particle-b-moves-with-uniform-velocity-20-m-s-along-a-direction-making-an-an-1-82192

question:
<div class="question">Particle $A$ moves along $X$ -axis with a uniform velocity of magnitude $10 \mathrm{m} / \mathrm{s}$. Particle $B$ moves with uniform velocity $20 \mathrm{m} / \mathrm{s}$ along a direction making an angle of $60^{\circ}$ with the positive direction of $X$ -axis as shown in the figure. The relative velocity of $B$ with respect to that of $A$ is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q21.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10 \mathrm{m} / \mathrm{s}$ along $X$ -axis</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$10 \sqrt{3} \mathrm{m} / \mathrm{s}$ along $Y$ -axis (perpendicular to $X$ -axis)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \sqrt{5} \mathrm{m} / \mathrm{s}$ along the bisection of the velocities of A and $B$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$30 \mathrm{m} / \mathrm{s}$ along negative $X$ -axis</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10 \sqrt{3} \mathrm{m} / \mathrm{s}$ along $Y$ -axis (perpendicular to $X$ -axis)</span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { The component of velocity of } B \text { along } x \text { -direction }\\
<br/>&amp;\begin{aligned}
<br/>\mathbf{v}_{B x} &amp;=20 \cos 60^{\circ}=20 \times \frac{1}{2}=10 \mathrm{m} / \mathrm{s} \\
<br/>\mathbf{v}_{A} &amp;=10 \hat{\mathbf{i}} \\
<br/>\mathbf{v}_{B} &amp;=10 \hat{\mathbf{i}}+10 \sqrt{3} \hat{\mathbf{j}} \\
<br/>\mathbf{v}_{B A} &amp;=\mathbf{v}_{B}-\mathbf{v}_{A}=10 \hat{\mathbf{i}}+10 \sqrt{3} \hat{\mathbf{j}}-10 \hat{\mathbf{i}}=10 \sqrt{3} \hat{\mathbf{j}}
<br/>\end{aligned}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
particle-a-moves-along-x-axis-with-a-uniform-velocity-of-magnitude-10-m-s-particle-b-moves-with-uniform-velocity-20-m-s-along-a-direction-making-an-an

slug:
particle-a-moves-along-x-axis-with-a-uniform-velocity-of-magnitude-10-m-s-particle-b-moves-with-uniform-velocity-20-m-s-along-a-direction-making-an-an-29975

question:
<div class="question">Particle $A$ moves along $X$ -axis with a uniform velocity of magnitude $10 \mathrm{m} / \mathrm{s}$. Particle $B$ moves with uniform velocity $20 \mathrm{m} / \mathrm{s}$ along a direction making an angle of $60^{\circ}$ with the positive direction of $X$ -axis as shown in the figure. The relative velocity of $B$ with respect to that of $A$ is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_phy_q21.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10 \mathrm{m} / \mathrm{s}$ along $X$ -axis</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$10 \sqrt{3} \mathrm{m} / \mathrm{s}$ along $Y$ -axis (perpendicular to $X$ -axis)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \sqrt{5} \mathrm{m} / \mathrm{s}$ along the bisection of the velocities of A and $B$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$30 \mathrm{m} / \mathrm{s}$ along negative $X$ -axis</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10 \sqrt{3} \mathrm{m} / \mathrm{s}$ along $Y$ -axis (perpendicular to $X$ -axis)</span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { The component of velocity of } B \text { along } x \text { -direction }\\
<br/>&amp;\begin{aligned}
<br/>\mathbf{v}_{B x} &amp;=20 \cos 60^{\circ}=20 \times \frac{1}{2}=10 \mathrm{m} / \mathrm{s} \\
<br/>\mathbf{v}_{A} &amp;=10 \hat{\mathbf{i}} \\
<br/>\mathbf{v}_{B} &amp;=10 \hat{\mathbf{i}}+10 \sqrt{3} \hat{\mathbf{j}} \\
<br/>\mathbf{v}_{B A} &amp;=\mathbf{v}_{B}-\mathbf{v}_{A}=10 \hat{\mathbf{i}}+10 \sqrt{3} \hat{\mathbf{j}}-10 \hat{\mathbf{i}}=10 \sqrt{3} \hat{\mathbf{j}}
<br/>\end{aligned}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Rotational Motion', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
particle-is-moving-in-an-elliptical-orbit-as-shown-in-figure-if-p-l-and-r-denote-the-linear-momentum-angular-momentum-and-position-vector-of-the-parti-1

slug:
particle-is-moving-in-an-elliptical-orbit-as-shown-in-figure-if-p-l-and-r-denote-the-linear-momentum-angular-momentum-and-position-vector-of-the-parti-1-31046

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0SoE3gGgWu2Kjbs1Usr91_vNWyRAjHLTis7APWnVbYw.original.fullsize.png"/><br/><br/> particle is moving in an elliptical orbit as shown in figure. If $\vec{p}, \vec{L}$ and $\vec{r}$ denote the linear momentum, angular momentum and position vector of the particle (from focus $\mathrm{O}$ ) respectively at a point $\mathrm{A}$, then the direction of $\vec{\alpha}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{L}}$ is along</div>

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

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

answer:
<div class="solution">$\vec{\alpha}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{L}}$<br/>$\vec{L}$ is along Z-axis<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/3mrJY9qxcwWbqQ3gIpP1dBZGGGRknRa3wNEo8c1DjhA.original.fullsize.png"/></div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
particle-is-moving-in-an-elliptical-orbit-as-shown-in-figure-if-p-l-and-r-denote-the-linear-momentum-angular-momentum-and-position-vector-of-the-parti

slug:
particle-is-moving-in-an-elliptical-orbit-as-shown-in-figure-if-p-l-and-r-denote-the-linear-momentum-angular-momentum-and-position-vector-of-the-parti-72506

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0SoE3gGgWu2Kjbs1Usr91_vNWyRAjHLTis7APWnVbYw.original.fullsize.png"/><br/><br/> particle is moving in an elliptical orbit as shown in figure. If $\vec{p}, \vec{L}$ and $\vec{r}$ denote the linear momentum, angular momentum and position vector of the particle (from focus $\mathrm{O}$ ) respectively at a point $\mathrm{A}$, then the direction of $\vec{\alpha}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{L}}$ is along</div>

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

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

answer:
<div class="solution">$\vec{\alpha}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{L}}$<br/>$\vec{L}$ is along Z-axis<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/3mrJY9qxcwWbqQ3gIpP1dBZGGGRknRa3wNEo8c1DjhA.original.fullsize.png"/></div>


title: pbcl-2-is-insoluble-in-cold-water-addition-of-hcl-increases-its-solubility-due-to
slug: pbcl-2-is-insoluble-in-cold-water-addition-of-hcl-increases-its-solubility-due-to-29975
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\mathrm{PbCl}_{2}$ is insoluble in cold water. Addition of $\mathrm{HCl}$ increases its solubility due to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">formation of soluble complex anions like $\left[\mathrm{PbCl}_{3}\right]^{-}$</span> </div>

answer:
<div class="solution">$\mathrm{PbCl}_{2}$ react with HCl as follows $\mathrm{PbCl}_{2}(s)+\mathrm{Cl}^{-} \stackrel{\text { Cold }}{\longrightarrow}\left[\mathrm{PbCl}_{3}\right]^{-}(a q)$
<br/>$\mathrm{PbCl}_{2}(s)+2 \mathrm{Cl}^{-} \frac{\text { Excess }}{\text { of HCl }}\left[\mathrm{PbCl}_{4}\right]^{2-}(a q)$
<br/>Thus, addition of excess of $\mathrm{Cl}^{-}$ ions change the $\mathrm{PbCl}_{2}$ as soluble complex of $\left[\mathrm{PbCl}_{4}\right]^{-2}$. Hence, becomes soluble.</div>


title: pick-the-correct-statement
slug: pick-the-correct-statement-12998
question: <div class="question">Pick the correct statement.</div>
tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Relative lowering of vapour pressure is independent of $\mathrm{T}$.</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><li class=""> <span class="option-label">B</span> <span class="option-data">Osmotic pressure always depends on the nature of solute.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Elevation of boiling point is independent of nature of the solvent.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Lowering of freezing point is proportional to the molar concentration of solute.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Relative lowering of vapour pressure is independent of $\mathrm{T}$.</span> </div>

answer:
<div class="solution">Relative lowering of vapour pressure is $\frac{\Delta \mathrm{P}}{\mathrm{P}^{\circ}}$, which is equal to mole fraction of solute which is independent of temperature.</div>


title: platinum-palladium-and-iridium-are-called-noble-metals-because
slug: platinum-palladium-and-iridium-are-called-noble-metals-because-13484
question: <div class="question">Platinum, Palladium and Iridium are called noble metals because</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints: Fact</div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Alfred Nobel discovered them</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">They are shining lustrous and pleasing to look at</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">They are found in native state</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">They are inert towards many common reagents.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">They are inert towards many common reagents.</span> </div>


title: pnitron-n-dimethylaniline-cannot-be-represented-by-the-resonating-structures
slug: pnitron-n-dimethylaniline-cannot-be-represented-by-the-resonating-structures-81179
tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">p-nitro-N, N - dimethylaniline cannot be represented by the resonating structures
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/-QrorlbhT45L9COSejzXXiacqECgfQQos4y41RqqyYI.original.fullsize.png"/></div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ScsoZqNz1NFIyZdZhTYjCBE9PLJfALqAaOe-AQCDJAk.original.fullsize.png"/><br/> <br/>Both the structures are impossible as $\mathrm{N}^{\circ}$ is pentavalent and we known that $\mathrm{N}$ can not expand its valency beyond 4 due to absence of vacant d-orbitals in its valence shell.</div>


title: point-out-the-false-statement
slug: point-out-the-false-statement-64320
question: <div class="question">Point out the false statement.</div>
tags: ['Chemistry', 'Surface Chemistry', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Colloidal sols are homogeneous</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Colloids carry + ve or - ve charges</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Colloids show Tyndall effect</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The size range of colloidal particles is $10-1000 \mathrm{A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Colloidal sols are homogeneous</span> </div>

answer:
<div class="solution">Option (a) is false as colloidal sols are not homogeneous.
<br/>Colloidal sols are heterogeneous mixture of dispersed phase and dispersion medium.</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
pq-is-a-double-ordinate-of-the-hyperbola-a-2-x-2-b-2-y-2-1-such-that-opq-is-an-equilateral-triangle-o-being-the-centre-of-the-hyperbola-then-the-eccen

slug:
pq-is-a-double-ordinate-of-the-hyperbola-a-2-x-2-b-2-y-2-1-such-that-opq-is-an-equilateral-triangle-o-being-the-centre-of-the-hyperbola-then-the-eccen-40395

question:
<div class="question">$P Q$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\triangle O P Q$ is an equilateral triangle, $O$ being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{e} &gt; 2 / \sqrt{3}$</span> </div>

answer:
<div class="solution">$\tan 30^{\circ}=\frac{b \tan \theta}{a \sec \theta}$<br/>$\Rightarrow \frac{b}{a}=\frac{1}{\sin \theta \sqrt{3}}$<br/>$e=\sqrt{1+\frac{1}{3 \sin ^2 \theta}} &gt; \sqrt{1+\frac{1}{3}}$<br/>$\Rightarrow e &gt; \frac{2}{\sqrt{3}}\left(0  &lt; \sin ^2 \theta &lt; 1\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uqNRXOXXZnnhO2dcM-MwasaIb8EhGQUw5TVkovm_VUo.original.fullsize.png"/></div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">zero</span> </div>
source_db: db\MarksBatch2_P2.db
title:
pressurevolume-p-v-work-done-by-an-ideal-gaseous-system-at-constant-volume-is-where-e-is-internal-energy-of-the-system

slug:
pressurevolume-p-v-work-done-by-an-ideal-gaseous-system-at-constant-volume-is-where-e-is-internal-energy-of-the-system-24807

question:
<div class="question">Pressure-volume $(p V)$ work done by an ideal gaseous system at constant volume is (where $E$ is internal energy of the system)</div>

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

answer:
<div class="solution">From first law of thermodynamics
<br/>$$
<br/>\Delta E=q+W
<br/>$$
<br/>where, work do net $(W)=p \Delta V$
<br/>$$
<br/>\begin{array}{l}
<br/>=p \times \Delta V \\
<br/>=0
<br/>\end{array} \quad(\because \Delta V=0)
<br/>$$</div>


title: product-of-any-r-consecutive-natural-numbers-is-always-divisible-by
slug: product-of-any-r-consecutive-natural-numbers-is-always-divisible-by-68481
question: <div class="question">Product of any $r$ consecutive natural numbers is always divisible by</div>
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{r}$ !</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">B</span> <span class="option-data">$(r+4) !$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(r+1) !$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(r+2) !$</span> </li> </ul>

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

answer:
<div class="solution">Hints : $(n+1)(n+2) \ldots \ldots .(n+r)$<br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{(n+r) !}{n !} \\<br/>&amp; =\frac{(n+r) !}{n ! r !} r !=r !^{n+r} C_n<br/>\end{aligned}<br/>$$</div>


title: radioactivity-of-a-sample-z-22-decreases-90-after-10-years-what-will-be-the-half-life-of-the-sample
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
slug:
radioactivity-of-a-sample-z-22-decreases-90-after-10-years-what-will-be-the-half-life-of-the-sample-75982

question:
<div class="question">Radioactivity of a sample ( $\mathrm{z}=22$ ) decreases $90 \%$ after 10 years. What will be the half life of the sample?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5 years</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 years</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3 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><li class=""> <span class="option-label">D</span> <span class="option-data">10 years</span> </li> </ul>

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

answer:
<div class="solution">Hints: $t=10$ yrs $\quad t_{\frac{1}{2}}=?$<br/>$$<br/>\lambda=\frac{2.303}{\mathrm{t}} \log \frac{\mathrm{N}_0}{\mathrm{~N}_{\mathrm{t}}}<br/>$$<br/>Since radioactivity decreases $90 \%$ in 10 yrs. $\Rightarrow N_0=100 \&amp; N_t=10$<br/>Thus $\lambda=\frac{2.303}{10} \log \frac{100}{10} \Rightarrow \lambda=\frac{2.303}{10}$<br/>since $\mathrm{t}_{\frac{1}{2}}=\frac{0.693}{\lambda}=\frac{2.303 \times \log 2}{\lambda} \Rightarrow \mathrm{t}_{\frac{1}{2}}=\frac{2.303 \times \log 2}{2.303 / 10}$<br/>$\Rightarrow \mathrm{t}_{\frac{1}{2}}=(\log 2) \times 10 \simeq 3$ years</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
radon-222-has-a-halflife-of-38-days-if-one-starts-with-0064-kg-of-radon222-the-quantity-of-radon-222-left-after-19-days-will-be

slug:
radon-222-has-a-halflife-of-38-days-if-one-starts-with-0064-kg-of-radon222-the-quantity-of-radon-222-left-after-19-days-will-be-69432

question:
<div class="question">Radon- 222 has a half-life of 3.8 days. If one starts with $0.064 \mathrm{kg}$ of radon-222, the quantity of radon- 222 left after 19 days will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.002 \mathrm{kg}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.062 \mathrm{kg}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.032 \mathrm{kg}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.024 \mathrm{kg}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.002 \mathrm{kg}$</span> </div>

answer:
<div class="solution">Half life of radon $=3 \cdot 8$ days <br/>Total half-lives in 19 days<br/>$\begin{array}{l}<br/>=\frac{19}{3 \cdot 8} \\<br/>=\frac{190}{38}<br/>\end{array}$<br/>$=5$ half-lives<br/>Initially mass of radon $=0.064 \mathrm{kg}$<br/>$0 \cdot 064 \frac{T_{1}}{T_{3}} 0 \cdot 032 \stackrel{T_{2}}{\longrightarrow}$<br/>$0 \cdot 016 \stackrel{T_{4}}{\longrightarrow} 0 \cdot 004 \stackrel{T_{5}}{\longrightarrow} 0 \cdot 002$<br/>So in five half-life periods, the Radon sample will reduce to $0.002 \mathrm{kg}$<br/>$\text {Or } \frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{5}$<br/>$\Rightarrow \quad N=0 \cdot 064 \times \frac{1}{32}$<br/>$=0 \cdot 002 \mathrm{kg}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
ram-is-visiting-a-friend-ram-knows-that-his-friend-has-2-children-and-1-of-them-is-a-boy-assuming-that-a-child-is-equally-likely-to-be-a-boy-or-a-girl

slug:
ram-is-visiting-a-friend-ram-knows-that-his-friend-has-2-children-and-1-of-them-is-a-boy-assuming-that-a-child-is-equally-likely-to-be-a-boy-or-a-girl-77563

question:
<div class="question">Ram is visiting a friend. Ram knows that his friend has 2 children and 1 of them is a boy. Assuming that a child is equally likely to be a boy or a girl, then the probability that the other child is a girl, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 2$.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 / 3$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$7 / 10$</span> </li> </ul>

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

answer:
<div class="solution">Total sample space $=\left\{B_{1} B_{2}, B_{1} G_{1}, G_{1} B_{2}\right\}$
<br/>Favourable ways $=\left\{B_{1} G_{1}, G_{1} B_{2}\right\}$
<br/>$\therefore$ Required probability $=\frac{2}{3}$</div>


title: reaction-of-benzene-with-me-3-ccocl-in-the-presence-of-anhydrous-alcl-3-gives-1
slug: reaction-of-benzene-with-me-3-ccocl-in-the-presence-of-anhydrous-alcl-3-gives-1-41717
tags: ['Chemistry', 'Hydrocarbons', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Reaction of benzene with $\mathrm{Me}_{3} \mathrm{CCOCl}$ in the presence of anhydrous AlCl ${ }_{3}$ gives</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q47_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/2013_che_q47_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/2013_che_q47_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/2013_che_q47_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q47_optb.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a47.png"/></div>


title: reaction-of-formaldehyde-and-ammonia-gives
slug: reaction-of-formaldehyde-and-ammonia-gives-73291
question: <div class="question">Reaction of formaldehyde and ammonia gives</div>
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Hexamethylene tetramine</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Bakelite</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Urea</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Triethylene Tetramine</span> </li> </ul>

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

answer:
<div class="solution">Hints: \(6 \mathrm{HCHO}+4 \mathrm{NH}_3 \rightarrow\left(\mathrm{CH}_2\right)_6 \mathrm{~N}_4+6 \mathrm{H}_2 \mathrm{O}\)</div>


title: reaction-of-r-oh-with-r-mgx-produces-1
slug: reaction-of-r-oh-with-r-mgx-produces-1-46089
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Reaction of $\mathrm{R} \mathrm{OH}$ with $\mathrm{R}^{\prime} \mathrm{MgX}$ produces :</div>

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

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/jkrMl2UCfzsjZtuDGZL2pK5UBMpJLBmyPo5yZuXELuU.original.fullsize.png"/><br/></div>


title: reaction-of-silver-nitrate-solution-with-phosphorous-acid-produces
slug: reaction-of-silver-nitrate-solution-with-phosphorous-acid-produces-15213
question: <div class="question">Reaction of silver nitrate solution with phosphorous acid produces:</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', '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">Silver phosphite</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Phosphoric acid</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Metallic silver</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Silver phosphate</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Phosphoric acid, Metallic silver</span> </div>

answer:
<div class="solution">$2 \mathrm{AgNO}_{3}+\mathrm{H}_{3} \mathrm{PO}_{3}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{H}_{3} \mathrm{PO}_{4}+2 \mathrm{Ag}+2 \mathrm{HNO}_{3}$</div>


title: reduction-of-the-lactol-s-oh-with-sodium-borohydride-gives
slug: reduction-of-the-lactol-s-oh-with-sodium-borohydride-gives-50712
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Reduction of the lactol S
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q74.png"/> <br/>OH with sodium borohydride gives</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q74_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q74_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q74_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_74_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q74_optc.png"/></span> </div>

answer:
<div class="solution">The reaction occurs as follows:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a74.png"/></div>


title: reflection-of-the-line-bara-za-barz0-in-the-real-axis-is-given-by
slug: reflection-of-the-line-bara-za-barz0-in-the-real-axis-is-given-by-66273
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Reflection of the line \(\bar{a} z+a \bar{z}=0\) in the real axis is given by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\mathrm{az}+\overline{\mathrm{az}}=0\)</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">\(\bar{a} z-a \bar{z}=0\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(a z-\overline{a z}=0\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{\mathrm{a}}{\mathrm{z}}+\frac{\overline{\mathrm{a}}}{\overline{\mathrm{z}}}=0\)</span> </li> </ul>

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

answer:
<div class="solution">Hint: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_kZxZE9gu1R420nh-WstEue9iqy4qLNpQsxuGJH09Oc.original.fullsize.png"/><br/><br/>Let, \(a=\alpha+i \beta\) and \(z=x+i y\)<br/>\(\begin{aligned}<br/>&amp; \therefore \bar{a} z+a \bar{z}=0 \\<br/>&amp; \Rightarrow(\alpha-i \beta)(x+i y)+(\alpha+i \beta)(x-i y)=0 \\<br/>&amp; \Rightarrow 2(\alpha x+\beta y)=0 \Rightarrow \alpha x+\beta y=0 \text { which passes through origin } \therefore \text { slope }=-\frac{\alpha}{\beta}<br/>\end{aligned}\)<br/>\(\therefore\) Reflected line's slope \(=\frac{\alpha}{\beta}\)<br/>\(\therefore\) Reflected line's equation : \(\alpha x -\beta y=0\)<br/>\(\begin{aligned}<br/>&amp; \Rightarrow\left(\frac{a+\bar{a}}{2}\right) x-\left(\frac{a-\bar{a}}{2 i}\right) y=0 \\<br/>&amp; \Rightarrow\left(\frac{a+\bar{a}}{2}\right)\left(\frac{z+\bar{z}}{2}\right)-\left(\frac{a-\bar{a}}{2 i}\right)\left(\frac{z-\bar{z}}{2 i}\right)=0 \\<br/>&amp; \Rightarrow a z+\bar{a} \bar{z}=0 \\<br/>&amp; \Rightarrow a z+\bar{a} z=0<br/>\end{aligned}\)</div>


title: ribose-and-2-deoxyribose-can-be-differentiated-by
slug: ribose-and-2-deoxyribose-can-be-differentiated-by-55391
question: <div class="question">Ribose and 2 -deoxyribose can be differentiated by</div>
tags: ['Chemistry', 'Biomolecules', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Fehling's reagent</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Tollen's reagent</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Barfoed's reagent</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Osazone formation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Osazone formation</span> </div>

answer:
<div class="solution">Ribose forms osazone with $\mathrm{Ph}-\mathrm{NH}-\mathrm{NH}_{2}$ whereas in deoxyribose, one - OH group is missing, due to which, it cannot form osazone.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a32.png"/></div>


title: roasted-copper-pyrite-on-smelting-with-sand-produces
slug: roasted-copper-pyrite-on-smelting-with-sand-produces-16646
question: <div class="question">Roasted copper pyrite on smelting with sand produces</div>
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{FeSiO}_{3}$ as fusible slag and $\mathrm{Cu}_{2} \mathrm{S}$ as matte</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{CaSiO}_{3}$ as infusible slag and $\mathrm{Cu}_{2} \mathrm{O}$ as matte</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Ca}_{3}\left(\mathrm{PO}_{4}\right)_{2}$ as fusible slag and $\mathrm{Cu}_{2} \mathrm{S}$ as matte</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Fe}_{3}\left(\mathrm{PO}_{4}\right)_{2}$ as infusible slag and $\mathrm{Cu}_{2} \mathrm{S}$ as matte</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{FeSiO}_{3}$ as fusible slag and $\mathrm{Cu}_{2} \mathrm{S}$ as matte</span> </div>

answer:
<div class="solution">Roasted copper pyrite on smelting with sand produces $\mathrm{FeSiO}_{3}$ as fusible slag and $\mathrm{Cu}_{2}$ S as matte
<br/>For removing the gangue, FeS, silica present in the lining of the Bessemer converter, acts as a flux and forms slag (iron silicate) on reaction with FeO.
<br/>$\mathrm{FeS}+3 \mathrm{O}_{2} \longrightarrow 2 \mathrm{FeO}+2 \mathrm{SO}_{2}$
<br/>$\mathrm{FeO}+\mathrm{SiO}_{2} \longrightarrow \mathrm{FeSiO}_{3}$
<br/>Copper matte mainly contains $\mathrm{Cu}_{2} \mathrm{S}$ and FeS.</div>


title: rolles-theorem-is-applicable-in-the-interval-22-for-the-function
slug: rolles-theorem-is-applicable-in-the-interval-22-for-the-function-87093
question: <div class="question">Rolle's theorem is applicable in the interval [-2,2] for the function</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)=x^{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f(x)=4 x^{4}$</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">$f(x)=2 x^{3}+3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(x)=\pi|x|$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)=4 x^{4}$</span> </div>

answer:
<div class="solution">If we take $f(x)=4 x^{4}$, then
<br/>(i) $f(x)$ is continuous in (-2,2)
<br/>(ii) $f(x)$ is differentiable in (-2,2)
<br/>(iii) $f(-2)=f(2)$ So, $f(x)=4 x^{4}$ satisfies all the conditions of Rolle's theorem therefore $\exists$ a point $c$ such that
<br/>$f^{\prime}(c)=0$
<br/>$\Rightarrow \quad 16 c^{3}=0 \Rightarrow c=0 \in(-2,2)$</div>


tags: ['Physics', 'Wave Optics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
s-1-and-s-2-are-the-coherent-point-sources-of-light-located-in-the-x-y-plane-at-points-00-and-0-3-respectively-here-is-the-wavelength-of-light-at-whic

slug:
s-1-and-s-2-are-the-coherent-point-sources-of-light-located-in-the-x-y-plane-at-points-00-and-0-3-respectively-here-is-the-wavelength-of-light-at-whic-54275

question:
<div class="question">$S_{1}$ and $S_{2}$ are the coherent point sources of light located in the $x y$ -plane at points (0,0) and $(0,3 \lambda)$ respectively. Here $\lambda$ is the wavelength of light. At which one of the following points (given as coordinates), the intensity of interference will be maximum?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(3 \lambda, 0)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(4 \lambda, 0)$</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">$(5 \lambda / 4,0)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(2 \lambda / 3,0)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(4 \lambda, 0)$</span> </div>

answer:
<div class="solution">The given situation can be show as
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_phy_a10.png"/> <br/> <br/>The intensity will be maximum at those given points where the path difference between the two interfering waves is an integral multiple of wavelength $i . e ., \Delta x=n \lambda$
<br/>For the the given points, the intensily will be maximum for $(4 \lambda, 0)$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
s-and-t-are-the-foci-of-an-ellipse-and-b-is-end-point-of-the-minor-axis-if-stb-is-an-equilateral-triangle-the-eccentricity-of-the-ellipse-is

slug:
s-and-t-are-the-foci-of-an-ellipse-and-b-is-end-point-of-the-minor-axis-if-stb-is-an-equilateral-triangle-the-eccentricity-of-the-ellipse-is-18583

question:
<div class="question">S and $T$ are the foci of an ellipse and B is end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is</div>

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

answer:
<div class="solution">Hints : $\frac{\mathrm{b}}{\mathrm{ae}}=\sqrt{3} ; \quad \mathrm{b}=\sqrt{3} \mathrm{ae}$<br/>$$<br/>\mathrm{e}^2=\frac{\mathrm{a}^2-3 \mathrm{a}^2 \mathrm{e}^2}{\mathrm{a}^2}=1-3 \mathrm{e}^2 ; \quad 4 \mathrm{e}^2=1 \Rightarrow \mathrm{e}=\frac{1}{2}<br/>$$</div>


title: s-i-n-x-3-c-o-s-x-dx-equals-where-c-is-an-arbitrary-constant
slug: s-i-n-x-3-c-o-s-x-dx-equals-where-c-is-an-arbitrary-constant-55793
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\int \frac{\mathrm{dx}}{\sin x+\sqrt{3} \cos x}$ equals<br/><br/>where $\mathrm{c}$ is an arbitrary constant</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2} \ln \left|\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right|+c$</span> </div>

answer:
<div class="solution">Hints : $\int \frac{d x}{\sin x+\sqrt{3} \cos x}=\int \frac{d x}{2\left(\frac{1}{2} \sin x+\frac{\sqrt{3}}{2} \cos x\right)}=\frac{1}{2} \int \frac{d x}{\sin \left(x+\frac{\pi}{3}\right)}$<br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{1}{2} \int \operatorname{cosec}\left(\mathrm{x}+\frac{\pi}{3}\right) \mathrm{dx}=\frac{1}{2} \log \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{c} \\<br/>&amp; =\frac{1}{2} \ln \left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{6}\right)\right|+\mathrm{c}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
same-quantity-of-ice-is-filled-in-each-of-the-two-metal-containers-p-and-q-having-the-same-size-shape-and-wall-thickness-but-made-of-different-materia

slug:
same-quantity-of-ice-is-filled-in-each-of-the-two-metal-containers-p-and-q-having-the-same-size-shape-and-wall-thickness-but-made-of-different-materia-79164

question:
<div class="question">Same quantity of ice is filled in each of the two metal containers $P$ and $Q$ having the same size, shape and wall thickness but made of different materials. The containers are kept in identical surroundings. The ice in $P$ melts completely in time $t_{1}$ whereas in $Q$ takes a time $t_{2}$. The ratio of thermal conductivities of the materials of $P$ and $Q$ is</div>

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

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

answer:
<div class="solution">We know that Relation between temperature gradient (TG) and thermal conductivity $(K)$
<br/>So,
<br/>$$
<br/>\frac{d \theta}{d t}=-K A \frac{d \theta}{d x}=-K A \times(\mathrm{TG})
<br/>$$
<br/>Le. $\mathrm{TG} \propto \frac{1}{K}\left(\frac{d \theta}{d t}=\right.$ constant $)$
<br/>or $K \times \frac{1}{T}$
<br/>$$
<br/>\begin{array}{l}
<br/>K_{1} \times \frac{1}{t_{1}} \\
<br/>K_{2} \times \frac{1}{t_{2}}
<br/>\end{array}
<br/>$$
<br/>From the Eqs. (i) and (ii), we get
<br/>$$
<br/>\frac{K_{1}}{K_{2}}=\frac{1 / t_{1}}{1 / t_{2}}
<br/>$$
<br/>$$
<br/>K_{1}: K_{2}=t_{2}: t_{1}
<br/>$$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
sand-t-are-the-foci-of-an-ellipse-and-b-is-the-end-point-of-the-minor-axis-if-stb-is-equilateral-triangle-the-eccentricity-of-the-ellipse-is

slug:
sand-t-are-the-foci-of-an-ellipse-and-b-is-the-end-point-of-the-minor-axis-if-stb-is-equilateral-triangle-the-eccentricity-of-the-ellipse-is-36228

question:
<div class="question">Sand $T$ are the foci of an ellipse and $B$ is the end point of the minor axis. If $S T B$ is equilateral triangle, the eccentricity of the ellipse is</div>

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

answer:
<div class="solution">Equation of the ellipse is
<br/>$$
<br/>\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a40.png"/> <br/>Foci are $S(a e, 0), T(-a e, 0)$
<br/>$B(0, b)$ is the end of the minor axis. STB is an equilateral triangle.
<br/>$$
<br/>\begin{aligned}
<br/>S B &amp;=S T \\
<br/>S B^{2} &amp;=S T^{2}
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow \quad a^{2} e^{2}+b^{2}=4 a^{2} e^{2}$
<br/>$\Rightarrow \quad b^{2}=3 a^{2} e^{2}$
<br/>$\Rightarrow \quad a^{2}\left(1-e^{2}\right)=3 a^{2} e^{2}$
<br/>$\Rightarrow \quad 1-e^{2}=3 e^{2}$
<br/>$\Rightarrow \quad 4 e^{2}=1$
<br/>$\Rightarrow \quad e^{2}=\frac{1}{4}$
<br/>$\Rightarrow \quad e=\frac{1}{2}$
<br/>Eccentricity of the ellipse, $e=\frac{1}{2}$</div>


title: select-the-molecule-in-which-all-the-atoms-may-lie-on-a-single-plane-is
slug: select-the-molecule-in-which-all-the-atoms-may-lie-on-a-single-plane-is-31327
question: <div class="question">Select the molecule in which all the atoms may lie on a single plane is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">4-Nitrobenzaldehyde</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4-Methoxybenzaldehyde</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4-Methylnitrobenzene</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4-Nitroacetophenone</span> </li> </ul>

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/V-me_0HyISRHWO7l5anZj-bRrv9SgpAOKeqkWN6VRBg.original.fullsize.png"/><br/><br/>All atoms other than Hydrogen are \(\mathrm{sp}^2\) hybridised.</div>


title: silicon-oil-is-obtained-from-the-hydrolysis-and-polymerisation-of
slug: silicon-oil-is-obtained-from-the-hydrolysis-and-polymerisation-of-27112
question: <div class="question">Silicon oil is obtained from the hydrolysis and polymerisation of</div>
tags: ['Chemistry', 'Polymers', '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">trimetylchlorosilane and dimethyldichlorosilane</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">trimethylchlorosilane and methyldichlorosilane</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">methyltrichlorosilane and dimethyldichlorosilane</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">triethylchlorosilane and dimethyidichlorosilane</span> </li> </ul>

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

answer:
<div class="solution">Silicon oil in a polymer of trimethylchloro silane and dimethyldichloro silane. These are useful as broad spectrum antifoaming agents. 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a53.png"/></div>


tags: ['Chemistry', 'Coordination Compounds', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
silver-chloride-dissolves-in-excess-of-ammonium-hydroxide-solution-the-cation-present-in-the-resulting-solution-is-1

slug:
silver-chloride-dissolves-in-excess-of-ammonium-hydroxide-solution-the-cation-present-in-the-resulting-solution-is-1-22439

question:
<div class="question">Silver chloride dissolves in excess of ammonium hydroxide solution. The cation present in the resulting solution is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}$</span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\begin{array}{l}
<br/>\text { Silver chloride dissolves in excess of } \\
<br/>\text { ammonium hydroxide solution. The cation } \\
<br/>\text { present in the resulting solution is }\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+} \\
<br/>\text {In aqueous solution, silver chloride exist as } \mathrm{Ag}^{+} \\
<br/>\text {=and } \mathrm{Cl}^{-} \text {. } \mathrm{Ag}^{+} \text {present in solution reacts with } \mathrm{NH}_{3}
<br/>\end{array}\\
<br/>&amp;\begin{array}{l}
<br/>\text { form }\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+} \\
<br/>\mathrm{Ag}^{+} \mathrm{Cl}^{-}+2 \mathrm{NH}_{3} \longrightarrow\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+} \mathrm{Cl}^{-}
<br/>\end{array}
<br/>\end{aligned}
<br/>$$</div>


title: simplest-form-of-2-2-2-2-c-o-s-4-x-2-is
slug: simplest-form-of-2-2-2-2-c-o-s-4-x-2-is-64537
question: <div class="question">Simplest form of $\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 4 x}}}}$ is</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sec \frac{x}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sec x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\operatorname{cosec} x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\text { Hints : } \begin{aligned}<br/>&amp; \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2.2 \cos ^2 2 x}}}}=\frac{2}{\sqrt{2+\sqrt{2+2 \cos 2 x}}}=\frac{2}{\sqrt{2+\sqrt{2.2 \cos ^2 x}}} \\<br/>&amp; =\frac{2}{\sqrt{2+2 \cos x}}=\frac{2}{2 \cos \frac{x}{2}}=\sec \frac{x}{2}<br/>\end{aligned}<br/>$$</div>


title: sio-2-is-attacked-by-which-one-ones-of-the-following
slug: sio-2-is-attacked-by-which-one-ones-of-the-following-45163
question: <div class="question">$\mathrm{SiO}_{2}$ is attacked by which one / ones of the following?</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">HF</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">conc. HCl</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">hot $\mathrm{NaOH}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">Fluorine</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">HF, hot $\mathrm{NaOH}$, Fluorine</span> </div>

answer:
<div class="solution">Hint:
<br/>$\mathrm{SiO}_{2}(\mathrm{~s})+\mathrm{F}_{2}(\mathrm{~g}) \rightarrow \mathrm{SiF}_{4}(\mathrm{aq} .)+\mathrm{O}_{2}(\mathrm{~g})$
<br/>$\mathrm{SiO}_{2}(\mathrm{~s})+6 \mathrm{HF}$ (aq.) $\rightarrow \mathrm{H}_{2} \mathrm{SiF}_{6}$ (aq.) $+2 \mathrm{H}_{2} \mathrm{O}(\ell)$
<br/>$\mathrm{SiO}_{2}(\mathrm{~s})+2 \mathrm{NaOH}($ aq. $) \rightarrow \mathrm{Na}_{2} \mathrm{SiO}_{3}(\mathrm{~s})+\mathrm{H}_{2} \mathrm{O}(\ell)$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
six-molecules-of-an-ideal-gas-have-velocities-13556-and-5-mathrmm-mathrms-respectively-at-any-given-temperature-if-overlinev-and-vmathrmrms-represent-

slug:
six-molecules-of-an-ideal-gas-have-velocities-13556-and-5-mathrmm-mathrms-respectively-at-any-given-temperature-if-overlinev-and-vmathrmrms-represent-93984

question:
<div class="question">Six molecules of an ideal gas have velocities \(1,3,5,5,6\) and \(5 \mathrm{~m} / \mathrm{s}\) respectively. At any given temperature, if \(\overline{V}\) and \(V_{\mathrm{rms}}\) represent average and rms speed of the molecules, then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(V_{r m s} &gt; \overline{V}\)</span> </div>

answer:
<div class="solution">Hint : \(\overline{V}=\frac{1+3+5+5+6+5}{6}=\frac{25}{6}=4.16\)<br/>\(\begin{aligned}<br/>&amp; \overline{V}_{\mathrm{ms}}=\sqrt{\frac{1+9+25 \times 3+36}{6}}=\sqrt{\frac{121}{6}}=\frac{11}{\sqrt{6}}=4.48 \\<br/>&amp; \mathrm{~V}_{\mathrm{rms}} &gt; \overline{V}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: db\MarksBatch2_P2.db
title:
six-numbers-are-in-ap-such-that-their-sum-is-3-the-first-term-is-4-times-the-third-term-then-the-fifth-term-is

slug:
six-numbers-are-in-ap-such-that-their-sum-is-3-the-first-term-is-4-times-the-third-term-then-the-fifth-term-is-49813

question:
<div class="question">Six numbers are in AP such that their sum is 3 . The first term is 4 times the third term. Then, the fifth term is</div>

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

answer:
<div class="solution">Let the numbers be $a-5 d t, a-3 d, a-d, a+d, a+3 d, a+5 d$
<br/>Also, given $T_{1}=4 T_{3},$ where $T_{1}, T_{3}$ are respectively, first and third terms of AP. $\Rightarrow \quad a-5 d=4(a-d)$
<br/>$\Rightarrow \quad d=-3 a=-\frac{3}{2}$
<br/>$\therefore$ The fifth term $a+3 d=\frac{1}{2}+3\left(-\frac{3}{2}\right)=\frac{1}{2}-\frac{9}{2}=-4$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
six-positive-numbers-are-in-gp-such-that-their-product-is-1000-if-the-fourth-term-is-1-then-the-last-term-is

slug:
six-positive-numbers-are-in-gp-such-that-their-product-is-1000-if-the-fourth-term-is-1-then-the-last-term-is-65709

question:
<div class="question">Six positive numbers are in GP, such that their product is 1000 . If the fourth term is 1 , then the last term is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1000</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">100</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{100}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{1000}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{100}$</span> </div>

answer:
<div class="solution">Let the six terms of GP are
<br/>$$
<br/>\frac{a}{r^{5}} \cdot \frac{a}{r^{3}}, \frac{a}{r}, a r, a r^{3}, a r^{5}
<br/>$$
<br/>Now, according to the question $\frac{a}{r^{5}} \cdot \frac{a}{r^{3}} \cdot \frac{a}{r} \cdot a r \cdot a r^{3} \cdot a r^{5}=1000$
<br/>$\Rightarrow \quad a^{6}=(10)^{3} \Rightarrow a^{2}=10$
<br/>Also, given fourth term $=a r=1$
<br/>$\Rightarrow \quad a^{2} r^{2}=1$
<br/>$\Rightarrow \quad r^{2}=\frac{1}{10} \quad \ldots$ (ii)
<br/>$\therefore$ Last term $=a r^{5}=\left[a^{2}\left(r^{2}\right)^{5}\right]^{1 / 2}$
<br/>$=\left[10 \times \frac{1}{10^{5}}\right]^{1 / 2}[$ from Eqs. (1) and $(i i)]$
<br/>$=\left(\frac{1}{10^{4}}\right)^{1 / 2}=\frac{1}{100}$</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
six-wires-each-of-resistance-r-are-connected-so-as-to-form-a-tetrahedron-the-equivalent-resistance-of-the-combination-when-current-enters-through-one-

slug:
six-wires-each-of-resistance-r-are-connected-so-as-to-form-a-tetrahedron-the-equivalent-resistance-of-the-combination-when-current-enters-through-one-16458

question:
<div class="question">Six wires, each of resistance $r$, are connected so as to form a tetrahedron. The equivalent resistance of the combination when current enters through one corner and leaves through some other corner is</div>

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

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

answer:
<div class="solution">Six wires each of resistance $r$ form a tetrahedron as shown in the following figure
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a15.png"/> <br/> <br/>It is circuit of Wheatstone the equivalent resistance of upper circuit $\frac{1}{R}=\frac{1}{2 r}+\frac{1}{2 r}=\frac{2}{2 r}=\frac{1}{r}$
<br/>$$
<br/>R=r
<br/>$$
<br/>It will in parallel with outer resistance $\frac{1}{R_{e q}}=\frac{1}{r}+\frac{1}{r}=\frac{2}{r}$
<br/>$R_{e q}=\frac{r}{2}$</div>


title: sodium-nitroprusside-is
slug: sodium-nitroprusside-is-88909
question: <div class="question">Sodium nitroprusside is</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">Fact</div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Na}_4\left[\mathrm{Fe}(\mathrm{CN})_5 \mathrm{NO}_2\right]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Na}_2\left[\mathrm{Fe}(\mathrm{CN})_5 \mathrm{NO}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Na}_3\left[\mathrm{Fe}(\mathrm{CN})_5 \mathrm{NO}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Na}_4\left[\mathrm{Fe}(\mathrm{CN})_5 \mathrm{NO}_3\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Na}_2\left[\mathrm{Fe}(\mathrm{CN})_5 \mathrm{NO}\right]$</span> </div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
solubility-products-k-sp-of-the-salts-of-types-mx-mx-2-and-m-3-x-at-temperature-t-are-40-1-0-8-32-1-0-14-and-27-1-0-15-respectively-solubilities-in-mo

slug:
solubility-products-k-sp-of-the-salts-of-types-mx-mx-2-and-m-3-x-at-temperature-t-are-40-1-0-8-32-1-0-14-and-27-1-0-15-respectively-solubilities-in-mo-33000

question:
<div class="question">Solubility products $\left(\mathrm{K}_{\mathrm{sp}}\right)$ of the salts of types $\mathrm{MX}, \mathrm{MX}_{2}$ and $\mathrm{M}_{3} \mathrm{X}$ at temperature $\mathrm{T}$ are $4.0 \times 10^{-8}, 3.2 \times 10^{-14}$ and $2.7 \times$ $10^{-15}$ respectively. Solubilities (in $\mathrm{mol} \mathrm{dm}^{-3}$ ) of the salts at temperature $\mathrm{T}$ are in the order</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad M X&gt;M X_{2}&gt;M_{3} X$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$M_{3} X&gt;M X_{2}&gt;M X$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{MX}_{2}&gt;\mathrm{M}_{3} \mathrm{X}&gt;\mathrm{MX}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{MX}&gt;\mathrm{M}_{3} \mathrm{X}&gt;\mathrm{MX}_{2}$</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">$\mathrm{MX}&gt;\mathrm{M}_{3} \mathrm{X}&gt;\mathrm{MX}_{2}$</span> </div>

answer:
<div class="solution">$\begin{array}{ll}
<br/>\mathrm{MX} &amp; \mathrm{K}_{\mathrm{sp}}=4 \times 10^{-8}=\mathrm{S}^{2} \therefore \mathrm{S}=2 \times 10^{-4} \\
<br/>\mathrm{MX}_{2} &amp; \mathrm{~K}_{\mathrm{sp}}=3.2 \times 10^{-14}=4 \mathrm{~S}^{3}
<br/>\end{array}$
<br/>$\therefore \mathrm{S}^{3}=\frac{3.2 \times 10^{-14}}{4}$
<br/>$\mathrm{~S}^{3}=8 \times 10^{-15} \quad \mathrm{~S}=2 \times 10^{-5}$
<br/>$\mathrm{M}_{3} \mathrm{X} \quad \mathrm{K}_{\mathrm{sp}}=2.7 \times 10^{-15}=27 \mathrm{~S}^{4} \quad \therefore \mathrm{S}=10^{-4}$
<br/>$\therefore \mathrm{MX}&gt;\mathrm{M}_{3} \mathrm{X}&gt;\mathrm{MX}_{2}$</div>


title: solution-of-the-differential-equation-x-d-y-y-d-x-0-represents-a
slug: solution-of-the-differential-equation-x-d-y-y-d-x-0-represents-a-36847
question: <div class="question">Solution of the differential equation $x d y-y d x=0$ represents a</div>
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">parabola</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">circle</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">hyperbola</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">straight line</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Hints: $x . d y-y . d x=0 \Rightarrow x d y=y d x$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{d y}{y}=\frac{d x}{x} \Rightarrow \log y=\log x+\log c \\<br/>&amp; y=x c<br/>\end{aligned}<br/>$$</div>


title: solution-of-x-y-2-d-x-d-y-a-2-a-being-a-constant-is
slug: solution-of-x-y-2-d-x-d-y-a-2-a-being-a-constant-is-80976
question: <div class="question">Solution of $(x+y)^{2} \frac{d y}{d x}=a^{2}$ ('a' being a constant) is</div>
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{(x+y)}{a}=\tan \frac{y+C}{a}, C$ is an arbitrary 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">B</span> <span class="option-data">$x y=a \tan C x, C$ is an arbitrary constant</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x}{a}=\tan \frac{y}{C}, C$ is an arbitrary constant</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x y=\tan (x+C), C$ is an arbitrary constant</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{(x+y)}{a}=\tan \frac{y+C}{a}, C$ is an arbitrary constant</span> </div>

answer:
<div class="solution">We have, $(x+y)^{2} \frac{d y}{d x}=a^{2}$
<br/>Let $x+y=v$
<br/>$\therefore$
<br/>$1+\frac{d y}{d x}=\frac{d v}{d x}$
<br/>$\frac{d y}{d x}=\frac{d y}{d x}-1$
<br/>$v^{2}\left(\frac{d v}{d x}-1\right)=a^{2}$
<br/>$\Rightarrow$
<br/>$v^{2} \frac{d v}{d x}=v^{2}+a^{2}$
<br/>$\quad \frac{d v}{d x}=\frac{v^{2}+a^{2}}{v^{2}}$
<br/>$\Rightarrow \quad \frac{v^{2}}{v^{2}+d^{2}} d v=d x$
<br/>On integrating both sides, we get $\int \frac{v^{2}}{v^{2}+a^{2}} d v=\int d x$
<br/>$\int\left(1-\frac{a^{2}}{v^{2}+a^{2}}\right) \mathrm{d} v=x+C^{\prime}$
<br/>$\Rightarrow \quad v-\frac{a^{2}}{a} \tan ^{-1} \frac{v}{a}=x+C^{\prime}$
<br/>$\Rightarrow \quad x+y-a \tan ^{-1} \frac{x+y}{a}=x+C^{\prime}$
<br/>$\Rightarrow \quad y=a \tan ^{-1} \frac{x+y}{a}+C^{\prime}$
<br/>$=\frac{y-C^{\prime}}{a}=\tan ^{-1} \frac{x+y}{a}$
<br/>$\Rightarrow \quad\left(\frac{x+y}{a}\right)=\tan \left(\frac{y+C}{a}\right)$
<br/>where $-C^{\prime}=C$.</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
sound-waves-are-passing-through-two-routesone-in-atraight-path-and-the-other-along-a-semicireular-path-of-radius-r-and-are-again-combined-into-one-pip

slug:
sound-waves-are-passing-through-two-routesone-in-atraight-path-and-the-other-along-a-semicireular-path-of-radius-r-and-are-again-combined-into-one-pip-23951

question:
<div class="question">Sound waves are passing through two routes-one in atraight path and the other along a semicireular path of radius $r$ and are again combined into one pipe and superposed as shown in the figure. If the velocity of sound waves in the pipe is $v$, then frequencies of resultant waves of maximum amplitude will be integral multiples of <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q51.png"/></div>

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

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

answer:
<div class="solution">Path difference $=(\pi r-2 r)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a51.png"/> <br/> <br/>$=(\pi-2) r=n \lambda$ (wheref $=$ frequency)
<br/>$$
<br/>\begin{aligned}
<br/>v &amp;=f \times \lambda \\
<br/>\Rightarrow \quad \frac{v}{\lambda} &amp;=f \Rightarrow f=n \left[\frac{v}{(\pi-2) r}\right]^{n}
<br/>\end{aligned}
<br/>$$
<br/>So, multiples integral $=\left[\frac{v}{(\pi-2) r}\right]$</div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
stackrelmathrmlxrightarrow-text-ii-mathrmh3-mathrmooplustext-i-mathrmphmgbrstackrelmathrmmxrightarrow-mathrmcro3-mathrmhoplusstackrelmathrmnxrightarro

slug:
stackrelmathrmlxrightarrow-text-ii-mathrmh3-mathrmooplustext-i-mathrmphmgbrstackrelmathrmmxrightarrow-mathrmcro3-mathrmhoplusstackrelmathrmnxrightarro-55032

question:
<div class="question">\(\stackrel{\mathrm{L}}{=}~\xrightarrow [\text { (ii) } \mathrm{H}_3 \mathrm{O}^{\oplus}]{\text { (i) } \mathrm{PhMgBr}}\stackrel{\mathrm{M}}{=}~\xrightarrow []{\mathrm{CrO}_3 / \mathrm{H}^{\oplus}}\stackrel{\mathrm{N}}{=}~\xrightarrow []{\mathrm{Ph}_3 \mathrm{P}=\mathrm{CH}_2}\mathrm{Ph}_2 \mathrm{C}= \mathrm{CH}_2\)<br/>"L" in the above sequence of reaction is/are (where \(L \neq M \neq N\))</div>

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

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/luD0UpSnX2HZbV0w-2RpW9elUnLqNcmPJJD8M_aBVWo.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Statistics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
standard-deviation-of-n-observations-a-1-a-2-a-3-a-n-is-then-the-standard-deviation-of-the-observations-a-1-a-2-a-n-is

slug:
standard-deviation-of-n-observations-a-1-a-2-a-3-a-n-is-then-the-standard-deviation-of-the-observations-a-1-a-2-a-n-is-29055

question:
<div class="question">Standard deviation of $n$ observations $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ is ${\sigma} .$ Then, the standard deviation of the observations $\lambda a_{1}, \lambda a_{2}, \ldots, \lambda a_{n}$ is</div>

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

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

answer:
<div class="solution">Let $x_{i}=a_{1}, a_{2}, a_{3}, \ldots, a_{n}$
<br/>and $y_{i}=\lambda a_{1}, \lambda a_{2}, \lambda a_{3}, \ldots, \lambda a_{n-}$
<br/>$$
<br/>y_{i}=\lambda x_{i}
<br/>$$
<br/>$\because$
<br/>$\begin{aligned} \bar{y} &amp;=\frac{\Sigma y_{i}}{N} \\ &amp;=\frac{\Sigma \lambda x_{i}}{n} \end{aligned}$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=\lambda \frac{\Sigma x_{i}}{n} \\
<br/>&amp;=\lambda \bar{x} \\
<br/>\sigma_{y} &amp;=\sqrt{\frac{\Sigma\left(y_{i}-\bar{y}\right)^{2}}{N}} \\
<br/>&amp;=\sqrt{\frac{\Sigma(\lambda x_{i}-\lambda \bar{x})^{2}}{n}} \\
<br/>&amp;=\sqrt{\frac{\lambda^{2} \Sigma\left(x_{i}-\bar{x}\right)^{2}}{n}} \\
<br/>&amp;=\sqrt{\lambda^{2} \sigma_{x}^{2}}=|\lambda| \sigma_{x}=|\lambda| \sigma
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
straight-lines-x-y-7-and-x-4-y-2-intersect-at-b-points-a-and-c-are-so-chosen-on-these-two-lines-such-that-a-b-a-c-the-equation-of-line-a-c-passing-thr

slug:
straight-lines-x-y-7-and-x-4-y-2-intersect-at-b-points-a-and-c-are-so-chosen-on-these-two-lines-such-that-a-b-a-c-the-equation-of-line-a-c-passing-thr-29195

question:
<div class="question">Straight lines $x-y=7$ and $x+4 y=2$ intersect at $B$. Points $A$ and $C$ are so chosen on these two lines such that $A B=A C$. The equation of line $A C$ passing through (2,-7) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x-y-9=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$23 x+7 y+3=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x-y-11=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7 x-6 y-56=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$x-y-9=0$, $23 x+7 y+3=0$</span> </div>

answer:
<div class="solution">Given equations of lines are
<br/>$x-y=7$ ....(i)
<br/>and $x+4y=2$ ...(ii)
<br/>By solving Eqs.(i) and (ii), we get the point $B(6,-1)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a72.png"/> <br/>Let the slope of $A C$ be $m$ then, $A B=A C$
<br/>$\therefore \quad\left|\frac{m+\frac{1}{4}}{1-\frac{m}{4}}\right|=\left|\frac{-\frac{1}{4}-1}{1-\frac{1}{4}}\right|$
<br/>$\Rightarrow \quad m=\frac{-23}{7}, 1$
<br/>When $m=-\frac{23}{7},$ then equation of line $y+7=\frac{-23}{7}(x-2)$
<br/>$\Rightarrow \quad 7 y+49=-23 x+46$
<br/>$\Rightarrow \quad 23 x+7 y+3=0$
<br/>When $m=1$, then equation of line
<br/>$y+7=(x-2)$
<br/>$\Rightarrow \quad x-y-9=0$</div>


title: structure-of-m-is
slug: structure-of-m-is-18268
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/PyD-mhqjalE7MuU1DiyeFlRFeUb5CQizW_stRvtoms0.original.fullsize.png"/><br/><br/>Structure of \(M\) is,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{Ph}-\mathrm{C} \equiv \mathrm{CH}\)</span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/xpDM4-OJ2mW6VpMnXYdeYd5TzqFdF-ghbN5hGIHRZnI.original.fullsize.png"/><br/><br/>Benzaldehyde \((\mathrm{O})\) gives Cinnamic acid \((\mathrm{Ph}-\mathrm{CH}=\mathrm{CH}-\mathrm{COOH})\) on reaction with \(\left(\mathrm{CH}_3 \mathrm{CO}\right)_2 \mathrm{O}\) and \(\mathrm{CH}_3 \mathrm{COONa}\) known as Perkin's condensation</div>


title: sulphan-is
slug: sulphan-is-19116
question: <div class="question">Sulphan' is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a mixture of $\mathrm{SO}_{3}$ and $\mathrm{H}_{2} \mathrm{SO}_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$100 \%$ conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a mixture of gypsum and conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$100 \%$ oleum (a mixture of $100 \% \mathrm{SO}_{3}$ in $100 \%$ $\left.\mathrm{H}_{2} \mathrm{SO}_{4}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$100 \%$ oleum (a mixture of $100 \% \mathrm{SO}_{3}$ in $100 \%$ $\left.\mathrm{H}_{2} \mathrm{SO}_{4}\right)$</span> </div>

answer:
<div class="solution">$\mathrm{H}_{2} \mathrm{SO}_{4}$ saturated with $\mathrm{SO}_{3}$ is called oleum of sulphan.
<br/>$$
<br/>\mathrm{H}_{2} \mathrm{SO}_{4}+\mathrm{SO}_{3} \longrightarrow \mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{7}
<br/>$$</div>


title: sulphuryl-chloride-so-2-cl-2-reacts-with-white-phosphorus-p-4-to-give
slug: sulphuryl-chloride-so-2-cl-2-reacts-with-white-phosphorus-p-4-to-give-37382
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Sulphuryl chloride $\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right.$ ) reacts with white phosphorus $\left(\mathrm{P}_{4}\right)$ to give</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$P C l_{5}, S O_{2}$</span> </div>

answer:
<div class="solution">When sulphuryl chloride $\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)$, reacts with white phosphorus $\left(\mathrm{P}_{4}\right),$ it give phosphorus pentachloride $\left(\mathrm{PCl}_{5}\right)$ and sulphur dioxide $\left(\mathrm{SO}_{2}\right)$. The reaction is as follows:
<br/>$$
<br/>\mathrm{P}_{4}+10 \mathrm{SO}_{2} \mathrm{Cl}_{2} \longrightarrow 4 \mathrm{PCl}_{5}+10 \mathrm{SO}_{2}
<br/>$$</div>


title: sum-of-n-terms-of-the-following-series-1-3-3-3-5-3-7-3-is
slug: sum-of-n-terms-of-the-following-series-1-3-3-3-5-3-7-3-is-54920
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Sum of $n$ terms of the following series $1^3+3^3+5^3+7^3+\ldots \ldots \ldots$ is</div>

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

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

answer:
<div class="solution">Hints : $\sum(2 n-1)^3$<br/>$$<br/>\begin{aligned}<br/>&amp; \sum\left\{\left(8 n^3-3.4 n^2+3.2 n-1\right)\right\} \\<br/>= &amp; 2 n^2(n+1)^2-2 n(n+1)(2 n+1)+3 n(n+1)-n \\<br/>= &amp; 2 n^4+4 n^3+2 n^2-2 n\left[2 n^2+3 n+1\right]+3 n^2+3 n-n \\<br/>= &amp; 2 n^4+4 n^3+2 n^2-4 n^3-6 n^2-2 n+3 n^2+3 n-n \\<br/>= &amp; 2 n^4-n^2 \\<br/>= &amp; n^2\left(2 n^2-1\right)<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
sum-of-the-last-30-coeffivients-in-the-expansion-of-1-x-59-when-expanded-in-ascending-powers-of-x-is

slug:
sum-of-the-last-30-coeffivients-in-the-expansion-of-1-x-59-when-expanded-in-ascending-powers-of-x-is-72290

question:
<div class="question">Sum of the last 30 coeffivients in the expansion of $(1+x)^{59}$, when expanded in ascending powers of $x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{59}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2^{58}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{30}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2^{29}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2^{58}$</span> </div>

answer:
<div class="solution">Hints: Total terms $=60$<br/>Sum of first 30 terms $=\frac{\text { Sum of all the terms }}{2}=\frac{2^{59}}{2}=2^{58}$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
title:
supose-in-a-hypothetical-world-the-angular-momentum-is-quantized-to-be-even-integral-multiples-of-2-h-the-largest-possible-wavelength-emitted-by-hydro

slug:
supose-in-a-hypothetical-world-the-angular-momentum-is-quantized-to-be-even-integral-multiples-of-2-h-the-largest-possible-wavelength-emitted-by-hydro-87400

question:
<div class="question">Supose in a hypothetical world the angular momentum is quantized to be even integral multiples of $\frac{\mathrm{h}}{2 \pi}$. The largest possible wavelength emitted by hydrogen atoms in visible range in a world according to Bohr's model will be, (Consider hc $=1242 \mathrm{Mev}-\mathrm{fm}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$153 \mathrm{~nm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$409 \mathrm{~nm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$121 \mathrm{~nm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$487 \mathrm{~nm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$487 \mathrm{~nm}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/vOqTq-hUptjtxTgonrgNN6PAxQ_0b2LTIZQ47L4YADE.original.fullsize.png"/><br/><br/>$\frac{1}{\lambda}=R\left[\frac{1}{(2)^2}-\frac{1}{(4)^2}\right]$<br/>$\frac{1}{\lambda}=\mathrm{R}\left[\frac{3}{16}\right], \quad \lambda=\frac{912 \times 16}{3}=4864 \mathrm{~A}^0$<br/>$\lambda \approx 487 \mathrm{~nm}$</div>


tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
suppose-a-gaseous-mixture-of-mathrmhe-mathrmne-mathrmar-and-mathrmkr-is-treated-with-photons-of-the-frequency-appropriate-to-ionize-ar-what-ions-will-

slug:
suppose-a-gaseous-mixture-of-mathrmhe-mathrmne-mathrmar-and-mathrmkr-is-treated-with-photons-of-the-frequency-appropriate-to-ionize-ar-what-ions-will-96566

question:
<div class="question">Suppose a gaseous mixture of \(\mathrm{He}, \mathrm{Ne}, \mathrm{Ar}\) and \(\mathrm{Kr}\) is treated with photons of the frequency appropriate to ionize Ar. What ion(s) will be present in the mixture?</div>

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

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

answer:
<div class="solution">Hint: \(\mathrm{He} &gt; \mathrm{Ne} &gt; \mathrm{Ar} &gt; \mathrm{Kr} &gt; \mathrm{Xe} &gt; \mathrm{Rn}\) (Order of lonization energy)<br/>Energy of photon is sufficient to ionize \(\mathrm{Ar}\), hence \(\mathrm{Kr}\) will also ionize.<br/>Therefore mixture contains \(\mathrm{Ar}^{+}\)and \(\mathrm{Kr}^{+}\)</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
suppose-a-machine-produces-metal-parts-that-contain-some-defective-parts-witl-probability-005-how-many-parts-should-be-produced-in-order-that-the-prob

slug:
suppose-a-machine-produces-metal-parts-that-contain-some-defective-parts-witl-probability-005-how-many-parts-should-be-produced-in-order-that-the-prob-87877

question:
<div class="question">Suppose a machine produces metal parts that contain some defective parts witl probability $0.05 .$ How many parts should be produced in order that the probability 0 atleast one part being defective is $1 / 20$ more? (Given that, $\log _{10} 95=1.977$ an $\log _{10} 2=0.3$ )</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">15, 14</span> </div>

answer:
<div class="solution">Given probability of delective pat $=0.05=\frac{1}{20}$
<br/>Probability of non-defective part $=1-0.05=0.95=\frac{19}{20}$
<br/>We know that, $P(X=r)={ }^{n} Cr p^{r} q^{n-r}$
<br/>where, $\quad p=\frac{1}{20}, q=\frac{19}{20}$
<br/>$r \geq 1$ and $n=?$
<br/>Also, $P(X \geq 1) \geq \frac{1}{2}$
<br/>$\Rightarrow 1-P(X=0) \geq \frac{1}{2}$
<br/>$\Rightarrow 1-{ }^{n} C_{0}\left(\frac{1}{20}\right)^{0}\left(\frac{19}{20}\right)^{n-0} \geq \frac{1}{2}$
<br/>$\Rightarrow \quad 1-\frac{1}{2} \geq\left(\frac{19}{20}\right)^{n}$
<br/>$\Rightarrow$
<br/>$\frac{1}{2} \geq\left(\frac{19}{20}\right)^{n}$
<br/>$\Rightarrow$
<br/>$\frac{1}{2} \geq\left(\frac{95}{100}\right)^{n}$
<br/>$\Rightarrow \quad \log 2^{-1} \geq n[\log 95-\log 100 \mid$
<br/>$\Rightarrow \quad-\log 2 \geq n[\log 95-2]$
<br/>$\Rightarrow \quad-(0.3) \geq n[1977-2]$
<br/>$\Rightarrow \quad n \geq \frac{03}{0.023}$
<br/>$\Rightarrow \quad n \geq \frac{300}{23}$
<br/>$\Rightarrow \quad n \geq 1304$
<br/>$\therefore \quad n=14,15$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
suppose-f-mathbbr-rightarrow-mathbbr-be-given-by-fxleftbeginarraycc1-quad-text-if-x1-eleftx101rightx12-sin-frac1x1endarrayright-text-if-x-neq-1-text-t

slug:
suppose-f-mathbbr-rightarrow-mathbbr-be-given-by-fxleftbeginarraycc1-quad-text-if-x1-eleftx101rightx12-sin-frac1x1endarrayright-text-if-x-neq-1-text-t-98958

question:
<div class="question">Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) be given by \(f(x)=\left\{\begin{array}{cc}1, \quad \text { if } x=1 \\ e^{\left(x^{10}-1\right)}+(x-1)^2 \sin \frac{1}{x-1}\end{array}\right., \text { if } x \neq 1 \text { then }\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(f^{\prime}(1)\) does not exist</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(f^{\prime}(1)\) exists and is zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(f^{\prime}(1)\) exist and is 9</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(f^{\prime}(1)\) exists and is 10</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(f^{\prime}(1)\) does not exist</span> </div>

answer:
<div class="solution">Hint: \(f: R \rightarrow R f(x)=\left\{\begin{array}{cc} 1 &amp; \text { if } x=1 &amp; \\  e^{\left(x^{10}-1\right)} &amp;+(x-1)^2  &amp;\cdot \sin \left(\frac{1}{x-1}\right) \text { if } x \neq 1\end{array}\right.\)<br/>\(\begin{aligned}<br/>&amp; f\left(1^{-}\right)=1+0=1 \\<br/>&amp; f\left(1^{+}\right)=1+0=1 \\<br/>&amp; f(1)=1 \\<br/>&amp; \therefore \quad f(x) \text { is continuous at } x=1 \\<br/>&amp; f^{\prime}(x)=\left\{e^{\left(x^{10}-1\right)} \cdot\left(10 x^5\right)+2(x-1) \cdot \sin \left(\frac{1}{x-1}\right)-(x-1)^2 \cdot \cos \left(\frac{1}{x-1}\right) \cdot\left(\frac{1}{x-1}\right)^2\right\}<br/>\end{aligned}\)<br/>which does not exist.</div>


