# questions

title: cos-7-2-cos-7-4-cos-7-6
slug: cos-7-2-cos-7-4-cos-7-6-20562
question: <div class="question">$\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$</div>
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">is equal to zero</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">lies between 0 and 3</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is a negative number</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">lies between 3 and 6</span> </li> </ul>

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

answer:
<div class="solution">Let $\frac{2 \pi r}{7}=\theta$
<br/>$\Rightarrow \quad 2 \pi r=3 \theta+4 \theta$
<br/>$\Rightarrow \quad 4 \theta=2 \pi r-3 \theta$
<br/>$\Rightarrow \quad \sin 4 \theta=\sin (2 \pi r-3 \theta)$
<br/>$\Rightarrow \quad \sin 4 \theta=-\sin 3 \theta$
<br/>$\Rightarrow 2 \sin 2 \theta \cos 2 \theta=-\left[3 \sin \theta-4 \sin ^{3} \theta\right.$
<br/>$\Rightarrow 2 \times 2 \sin \theta \cos \theta\left(2 \cos ^{2} \theta-1\right)$
<br/>$=-3 \sin \theta+4 \sin ^{3} \theta$
<br/>$\Rightarrow \sin \theta\left[8 \cos ^{3} \theta-4 \cos \theta+3-4\left(1-\cos ^{2} \theta\right)\right]=0$
<br/>$\Rightarrow 8 \cos ^{3} \theta+4 \cos ^{2} \theta-4 \cos \theta-1=0$ (i)
<br/>Thus, $\cos \frac{2 \pi}{7}, \cos \frac{4 \pi}{7}$ and $\cos \frac{6 \pi}{7}$ are the roots of
<br/>the above equation. $\therefore \quad \cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}=-\frac{1}{2}$</div>


title: cos-lo-g-x-d-x-f-x-c-where-c-is-an-arbitrary-constant-here-f-x-is-equal-to
slug: cos-lo-g-x-d-x-f-x-c-where-c-is-an-arbitrary-constant-here-f-x-is-equal-to-63305
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">$\int \cos (\log x) d x=F(x)+C,$ where $C$ is an arbitrary constant. Here, $F(x)$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x}{2}[\cos (\log x)+\sin (\log x)]$</span> </div>

answer:
<div class="solution">Let $I=\int \cos (\log x) d x$
<br/>Put $\log x=t$
<br/>$\Rightarrow \quad x=e^{t}$
<br/>$\begin{aligned} \therefore \quad d x &amp;=e^{t} d t \\ \therefore \quad I &amp;=\int e^{t} \cos t d t \\ &amp;=\frac{e^{t}}{1^{2}+1^{2}}[\cos t+\sin t]+C \end{aligned}$
<br/>$\left[\because \int e^{a x} \cos b x d x=\frac{e^{a x}}{a^{2}+b^{2}}[a \cos b x+b \sin b x]+c\right]$
<br/>$\Rightarrow \quad I=\frac{e^{t}}{2}[\cos t+\sin t]+C$
<br/>$\quad \quad=\frac{x}{2}[\cos (\log x)+\sin (\log x]+C$
<br/>$\therefore f(x)=\frac{x}{2}[\cos (\log x)+\sin (\log x)]$</div>


title: cupric-compounds-are-more-stable-than-their-cuprous-counterparts-in-solid-state-this-is-because
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
slug:
cupric-compounds-are-more-stable-than-their-cuprous-counterparts-in-solid-state-this-is-because-61830

question:
<div class="question">Cupric compounds are more stable than their cuprous counterparts in solid state. This is because</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">the endothermic character of the 2nd IP of Cu is not so high, size of $\mathrm{Cu}^{2+}$  is less than $\mathrm{Cu}^{+}$, the lattice energy released for cupric compounds is much higher than $\mathrm{Cu}^{+}$</span> </div>

answer:
<div class="solution">Electronic configurations of cuprous $\left(\mathrm{Cu}^{*}\right)$ and cupric $\left(\mathrm{Cu}^{2}{ }^{+}\right)$ ions are as follows
<br/>$$
<br/>\begin{aligned}
<br/>\mathrm{Cu}^{+} &amp;=[\mathrm{Ar}] 3 d^{10} 4 s^{\circ} \\
<br/>\mathrm{Cu}^{2+} &amp;=[\mathrm{Ar}] 3 d^{9} 4 \mathrm{s}^{\circ}
<br/>\end{aligned}
<br/>$$
<br/>Thus, electronic configuration of $\mathrm{Cu}^{+}$ is more stable but it is less stable because in $\mathrm{Cu}^{2+}$ due to its small siag the nuclear charge is sufficient to hold 27 electrons but in $Cu^+$ such a condition is not true. Further, the IInd IP of Cu is not very high as compared to its Ist IP. Consequently a large amount of latsicy energy is released for cupric compounds as compared to $Cu^+$ compounds.</div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: db\MarksBatch1_P2.db
title:
current-through-mathrmabc-and-mathrmaprime-mathrmbprime-mathrmcprime-is-mathrmi-what-is-the-magnetic-field-at-mathrmp-mathrmbpmathrmpbprimemathrmrleft-1

slug:
current-through-mathrmabc-and-mathrmaprime-mathrmbprime-mathrmcprime-is-mathrmi-what-is-the-magnetic-field-at-mathrmp-mathrmbpmathrmpbprimemathrmrleft-1-94393

question:
<div class="question">Current through \(\mathrm{ABC}\) and \(\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}\) is \(\mathrm{I}\). What is the magnetic field at \(\mathrm{P}\) ? \(\mathrm{BP}=\mathrm{PB}^{\prime}=\mathrm{r}\left(\right.\) Here \(\mathrm{C}^{\prime} \mathrm{B}^{\prime} \mathrm{PBC}\) are collinear)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/eDzDjBUlXn78LbwfhdZ8gSiY8f3fLV_QiFkK1l6W4Mc.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{B}=\frac{1}{4 \pi} \frac{2 \mathrm{I}}{\mathrm{r}}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{B}=\frac{\mu_0}{4 \pi}\left(\frac{2 \mathrm{I}}{\mathrm{r}}\right)\)</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">\(\mathrm{B}=\frac{\mu_{\mathrm{o}}}{4 \pi}\left(\frac{\mathrm{I}}{\mathrm{r}}\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{B}=\frac{\mu_0}{4 \pi}\left(\frac{2 \mathrm{I}}{\mathrm{r}}\right)\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/H4mZs1rPT9AyY6mo4drscNbx2h1v7eMNIkLQKMT2QDs.original.fullsize.png"/><br/><br/>\(\mathrm{B}=2\left[\frac{\mu_0}{4 \pi} \frac{I}{\mathrm{r}}\right]\)</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P2.db
title:
current-through-mathrmabc-and-mathrmaprime-mathrmbprime-mathrmcprime-is-mathrmi-what-is-the-magnetic-field-at-mathrmp-mathrmbpmathrmpbprimemathrmrleft

slug:
current-through-mathrmabc-and-mathrmaprime-mathrmbprime-mathrmcprime-is-mathrmi-what-is-the-magnetic-field-at-mathrmp-mathrmbpmathrmpbprimemathrmrleft-58172

question:
<div class="question">Current through \(\mathrm{ABC}\) and \(\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}\) is \(\mathrm{I}\). What is the magnetic field at \(\mathrm{P}\) ? \(\mathrm{BP}=\mathrm{PB}^{\prime}=\mathrm{r}\left(\right.\) Here \(\mathrm{C}^{\prime} \mathrm{B}^{\prime} \mathrm{PBC}\) are collinear)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/eDzDjBUlXn78LbwfhdZ8gSiY8f3fLV_QiFkK1l6W4Mc.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{B}=\frac{1}{4 \pi} \frac{2 \mathrm{I}}{\mathrm{r}}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{B}=\frac{\mu_0}{4 \pi}\left(\frac{2 \mathrm{I}}{\mathrm{r}}\right)\)</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">\(\mathrm{B}=\frac{\mu_{\mathrm{o}}}{4 \pi}\left(\frac{\mathrm{I}}{\mathrm{r}}\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{B}=\frac{\mu_0}{4 \pi}\left(\frac{2 \mathrm{I}}{\mathrm{r}}\right)\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/H4mZs1rPT9AyY6mo4drscNbx2h1v7eMNIkLQKMT2QDs.original.fullsize.png"/><br/><br/>\(\mathrm{B}=2\left[\frac{\mu_0}{4 \pi} \frac{I}{\mathrm{r}}\right]\)</div>


tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: db\MarksBatch1_P2.db
title:
cyclo-18-carbon-is-an-allotrope-of-carbon-with-molecular-formula-c-18-it-is-a-ring-of-18-carbon-atoms-connected-by-single-and-triple-bonds-the-total-n-1

slug:
cyclo-18-carbon-is-an-allotrope-of-carbon-with-molecular-formula-c-18-it-is-a-ring-of-18-carbon-atoms-connected-by-single-and-triple-bonds-the-total-n-1-55596

question:
<div class="question">Cyclo [18] carbon is an allotrope of carbon with molecular formula $\mathrm{C}_{18}$. It is a ring of 18 carbon atoms, connected by single and triple bonds. The total number of triple bonds present in this cyclocarbon are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lyRGos3m_LB5twoUl3Dzub4JMVkO3xCfnVC8myEGUjQ.original.fullsize.png"/></div>


tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: db\MarksBatch1_P2.db
title:
cyclo-18-carbon-is-an-allotrope-of-carbon-with-molecular-formula-c-18-it-is-a-ring-of-18-carbon-atoms-connected-by-single-and-triple-bonds-the-total-n

slug:
cyclo-18-carbon-is-an-allotrope-of-carbon-with-molecular-formula-c-18-it-is-a-ring-of-18-carbon-atoms-connected-by-single-and-triple-bonds-the-total-n-88116

question:
<div class="question">Cyclo [18] carbon is an allotrope of carbon with molecular formula $\mathrm{C}_{18}$. It is a ring of 18 carbon atoms, connected by single and triple bonds. The total number of triple bonds present in this cyclocarbon are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lyRGos3m_LB5twoUl3Dzub4JMVkO3xCfnVC8myEGUjQ.original.fullsize.png"/></div>


title: cyclopentanol-on-reaction-with-nah-followed-by-cs-2-and-ch-3-i-produces-aan
slug: cyclopentanol-on-reaction-with-nah-followed-by-cs-2-and-ch-3-i-produces-aan-96848
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Cyclopentanol on reaction with NaH followed by $\mathrm{CS}_{2}$ and $\mathrm{CH}_{3} \mathrm{I}$ produces a/an</div>

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

answer:
<div class="solution">When cyclopentanol on reaction with NaH followed by $\mathrm{CS}_{2}$ and $\mathrm{CH}_{3} \mathrm{l}$ produces a xanthate.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a43.png"/> <br/>So, the option (d) is correct answer.</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P2.db
title:
dipole-moment-of-im-g-src-h-ttp-s-c-d-n-q-u-es-t-i-o-n-p-oo-l-g-e-t-ma-r-k-s-a-pp-p-y-q-w-bj-ee-q-o-bmk-p-c-n-qpssw-c-j-4-x-rb-c-ji-t-e-d-m-om-iw-e-k-

slug:
dipole-moment-of-im-g-src-h-ttp-s-c-d-n-q-u-es-t-i-o-n-p-oo-l-g-e-t-ma-r-k-s-a-pp-p-y-q-w-bj-ee-q-o-bmk-p-c-n-qpssw-c-j-4-x-rb-c-ji-t-e-d-m-om-iw-e-k-96474

question:
<div class="question">$$<br/>\text { Dipole moment of }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/QobmkpcnQPSSwcj4xRB-cjitedmOMiWekpRHK0eA3ag.original.fullsize.png"/><br/><br/>is $1.5 \mathrm{D}$. The dipole moment of<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/jSbVp-9Abpi3ZA-JHTeuNOd87OLTU_KqM7KmF1uM5ns.original.fullsize.png"/><br/><br/>is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1.5 \mathrm{D}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.25 \mathrm{D}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 \mathrm{D}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \mathrm{D}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.5 \mathrm{D}$</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints: Given for this molecule }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/QY-pqE9GV18xP2MbR_aukzXhse88zyg54wJWejYeAtc.original.fullsize.png"/><br/><br/>$\mu_1=1.5 \mathrm{D}$<br/>For<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9x3z6-2EiFuPIsCAr2lUtr8RYvWA9s26H_D3j79uY7k.original.fullsize.png"/><br/><br/>$\mu=0$ (as it is symmetrical)<br/>Hence for<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/wnPq3Yjbj7Yi0us6RmClSaiImknciUBS1mcg7xrLxWs.original.fullsize.png"/><br/><br/>$\mu$ will be $1.5 \mathrm{D}$</div>


title: dissolving-nacn-in-deionised-water-will-result-in-a-solution-having
slug: dissolving-nacn-in-deionised-water-will-result-in-a-solution-having-84582
question: <div class="question">Dissolving NaCN in de-ionised water will result in a solution having</div>
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{pH}    &lt; 7$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{pH}=7$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{pOH}=7$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{pH}&gt;7$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{pH}&gt;7$</span> </div>

answer:
<div class="solution">$\because$ On dissolving NaCN in de-ionised water, following reaction takes place.
<br/>$\mathrm{CN}^{-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{HCN}+\mathrm{OH}^{-}$
<br/>CN- ions come from the salt of strong base and weak acid.
<br/>$\therefore$ The solution becomes basic and $\mathrm{pH}&gt;7$</div>


title: domain-of-y-lo-g-10-2-3-x-x-2-is
slug: domain-of-y-lo-g-10-2-3-x-x-2-is-78340
question: <div class="question">Domain of $y=\sqrt{\log _{10} \frac{3 x-x^2}{2}}$ is</div>
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x &lt; 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 &lt; x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1 \leq x \leq 2$</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">$2 &lt; x &lt; 3$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\log _{10}\left(\frac{3 x-x^2}{2}\right) \geq 0<br/>$$<br/>or $\log _{10}\left(\frac{3 x-x^2}{2}\right) \geq \log _{10} 1$<br/>or $x^2-3 x+2 \leq 0$<br/>or $x \in[1,2]$ and $\frac{3 x-x^2}{2}&gt;0$<br/>$$<br/>\Rightarrow \mathrm{X} \in[1,2]<br/>$$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P2.db
title:
during-a-reversible-adiabatic-process-the-pressure-of-a-gas-is-found-to-be-proportional-to-the-cube-of-its-absolute-temperature-the-ratio-c-v-c-p-for-

slug:
during-a-reversible-adiabatic-process-the-pressure-of-a-gas-is-found-to-be-proportional-to-the-cube-of-its-absolute-temperature-the-ratio-c-v-c-p-for-20939

question:
<div class="question">During a reversible adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $\frac{C_{p}}{C_{V}}$ for the gas is</div>

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

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

answer:
<div class="solution">For a reversible adiabatic process.
<br/>$$
<br/>p T^{\frac{\gamma}{1-\gamma}}=\text { constant }
<br/>$$
<br/>According to question, during a reversible adiabatic process the pressure of a gas is found to be proportional to the cube of its absolute temperature.
<br/>$$
<br/>\begin{array}{c}
<br/>p \propto T^{3} \\
<br/>p T^{-3}=\text {constant }
<br/>\end{array}
<br/>$$
<br/>Equating equation (i) and (ii), we get
<br/>$$
<br/>\frac{\gamma}{1-\gamma}=-3
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>\gamma=-3(1-\gamma) \\
<br/>\gamma=-3+3 \gamma \\
<br/>-2 \gamma=-3 \\
<br/>\gamma=\frac{3}{2}
<br/>\end{array}
<br/>$$
<br/>As we know, the ratio of molar heat capacities at constant pressure and constant volume is represented by $y$. So, the ratio of $\frac{C}{C_{V}}$ for the gas is $\frac{3}{2}$</div>


title: during-electrolysis-of-molten-nacl-some-water-was-added-what-will-happen
slug: during-electrolysis-of-molten-nacl-some-water-was-added-what-will-happen-68425
tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">During electrolysis of molten $\mathrm{NaCl},$ some water was added. What will happen?</div>

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

answer:
<div class="solution">During electroysis of molten $\mathrm{NaCl}$, when water is added, the following
<br/>statements are true
<br/>(a) false
<br/>(b) $\because \mathrm{Na}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{NaOH}+\frac{1}{2} \mathrm{H}_{2}$
<br/>and discharge potential of hydrogen is less than of sodium. So, $\mathrm{H}_{2}$ will evolve.
<br/>(c) Some amount of caustic soda (NaOH) will be formed [As shown in (b)]
<br/>(d) Fire can likely take place because relations given given in (b) is highly exothermic.
<br/>$\therefore(b),(c)$ and $(d)$ are correct.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
title:
during-the-emission-of-a-positron-from-a-nucleus-the-mass-number-of-the-daughter-element-remains-the-same-but-the-atomic-number

slug:
during-the-emission-of-a-positron-from-a-nucleus-the-mass-number-of-the-daughter-element-remains-the-same-but-the-atomic-number-46680

question:
<div class="question">During the emission of a positron from a nucleus, the mass number of the daughter element remains the same but the atomic number</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">is decreased by 1 unit</span> </div>

answer:
<div class="solution">When a positron is emitted, a proton is converted into neutron and neutrino. Thus, the number of protons and hence the atomic number decreases by one unit. <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a9.png"/></div>


title: during-the-preparation-of-nh-3-in-habers-process-the-promoters-used-is-are
slug: during-the-preparation-of-nh-3-in-habers-process-the-promoters-used-is-are-47311
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">During the preparation of $\mathrm{NH}_3$ in Haber's process, the promoter(s) used is / are -</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Mo, Mix of $\mathrm{Al}_2 \mathrm{O}_3$ and $\mathrm{K}_2 \mathrm{O}$</span> </div>

answer:
<div class="solution">Mo was used as promoter earlier. Now $\mathrm{K}_2 \mathrm{O}$ and $\mathrm{Al}_2 \mathrm{O}_3$ are used as promoter.</div>


title: e-mi-ss-i-o-n-is-always-accompanied-by
slug: e-mi-ss-i-o-n-is-always-accompanied-by-32757
question: <div class="question">$\beta-emission$ is always accompanied by</div>
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">formation of antineutrino and $\alpha$ -particle</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">emission of $\alpha$ -particle and $\gamma$ -ray</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">formation of antineutrino and $y$ -ray</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">formation of antineutrino and positron</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">formation of antineutrino and $y$ -ray</span> </div>

answer:
<div class="solution">During the $\beta$ -emission, a neutron gets converted into a proton along with the formation of antineutrino and electron.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a10.png"/></div>


title: e-x-x-2-x-2-2-dx-is-equal-to
slug: e-x-x-2-x-2-2-dx-is-equal-to-48228
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">$\int \mathrm{e}^{\mathrm{x}}\left(\frac{2}{\mathrm{x}}-\frac{2}{\mathrm{x}^2}\right) \mathrm{dx}$ is equal to</div>

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

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

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


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
title:
each-of-a-and-b-can-take-values-1-or-2-with-equal-probability-the-probability-that-the-equation-a-x-2-b-x-1-0-has-real-roots-is-equal-to

slug:
each-of-a-and-b-can-take-values-1-or-2-with-equal-probability-the-probability-that-the-equation-a-x-2-b-x-1-0-has-real-roots-is-equal-to-52701

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

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

answer:
<div class="solution">The given equation
<br/>$$
<br/>a x^{2}+b x+1=0
<br/>$$
<br/>has real roots.
<br/>$\therefore$ Discriminant $(D) \geq 0$
<br/>$$
<br/>\Rightarrow \quad b^{2}-4 a \geq 0
<br/>$$
<br/>From Eq. (ii), we observe that
<br/>a has to be 1 and $b$ has to be 2 . 
<br/>So, the required probability $=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$</div>


tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P2.db
title:
each-of-the-two-inputs-a-and-b-can-assume-values-either-0-or-1-then-which-of-the-following-will-be-equal-to-a-b

slug:
each-of-the-two-inputs-a-and-b-can-assume-values-either-0-or-1-then-which-of-the-following-will-be-equal-to-a-b-19804

question:
<div class="question">Each of the two inputs $A$ and $B$ can assume values either 0 or 1. Then which of the following will be equal to $\bar{A} \cdot \bar{B} ?$</div>

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

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

answer:
<div class="solution">By using the de-Morgan's law
<br/>$$
<br/>\overline{A+B}=\overline{A \cdot B}
<br/>$$
<br/>Hence, option (b) is the correct.</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P2.db
title:
electric-field-component-of-an-em-radiation-varies-with-time-as-e-a-cos-0-t-sin-t-cos-0-t-where-a-is-a-constant-and-1-0-15-sec-1-0-5-1-0-15-sec-1-this

slug:
electric-field-component-of-an-em-radiation-varies-with-time-as-e-a-cos-0-t-sin-t-cos-0-t-where-a-is-a-constant-and-1-0-15-sec-1-0-5-1-0-15-sec-1-this-91557

question:
<div class="question">Electric field component of an EM radiation varies with time as $\mathrm{E}=\mathrm{a}\left(\cos \omega_{0} \mathrm{t}+\sin \omega \mathrm{t} \cos \omega_{0} \mathrm{t}\right)$, where ' $\mathrm{a}$ ' is a constant and $\omega=10^{15} \mathrm{sec}^{-1}, \omega_{0}=5 \times 10^{15} \mathrm{sec}^{-1}$. This radiation falls on a metal whose stopping potential is $-2 \mathrm{ev}$. Then which of the following statement $(\mathrm{s})$ is/are true $?\left(\mathrm{~h}=6.62 \times 10^{-34} \mathrm{~J}-\mathrm{S}\right)$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">For light of frequency $\omega$, photoelectric effect is not possible</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">Stopping potential vs. frequency graph will be a straight line</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The work function of the metal is $-2 \mathrm{ev}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The maximum kinetic energy of the photo electrons is $1.95 \mathrm{ev}$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">For light of frequency $\omega$, photoelectric effect is not possible, Stopping potential vs. frequency graph will be a straight line</span> </div>

answer:
<div class="solution">Stopping potential $=-2 \mathrm{~V} \Rightarrow \mathrm{KE}_{\max }=2 \mathrm{ev}$ (Option D incorrect) $\mathrm{E}=\mathrm{a}\left(\cos \omega_{0} \mathrm{t}+\sin \omega \mathrm{t} \cos \omega_{0} \mathrm{t}\right)$ $=\mathrm{a} \cos \omega_{0} \mathrm{t}+\frac{\mathrm{a}}{2}\left\{\sin \left(\omega+\omega_{0}\right) \mathrm{t}+\sin \left(\omega-\omega_{0}\right) \mathrm{t}\right\}$
<br/>Corresponding frequencies are $\frac{\omega_{0}}{\alpha \pi}, \frac{\omega+\omega_{0}}{\alpha \pi} \&amp; \frac{\omega-\omega_{0}}{2 \pi}$
<br/>$\left(\frac{\omega+\omega_{0}}{2 \pi}\right)$ determines maximum KE \&amp; Stopping potential
<br/>Now Energy corresponding to $\frac{\omega+\omega_{0}}{2 \pi}$
<br/>Corresponding energy $=\mathrm{hf}$
<br/>Stopping potential $=V_{0}=2 V=3.95 \mathrm{ev}$
<br/>Now $\mathrm{KE}_{\max }=\mathrm{ev}_{0}=\mathrm{hf}-\phi \Rightarrow 2 \mathrm{ev}=3.95 \mathrm{ev}-\phi \Rightarrow \phi=1.95 \mathrm{ev}$
<br/>for $\mathrm{w}=10^{15} \mathrm{sec}^{-1}$
<br/>frequency $=\frac{10^{15}}{2 \pi}=\mathrm{f}_{1}$
<br/>When frequency is $\mathrm{w}, \mathrm{f}_{1}=\frac{\omega}{2 \pi}=\frac{10^{15}}{2 \pi} \quad \mathrm{E}=\frac{6.63 \times 10^{-34} \times 6 \times 10^{15}}{2 \pi \times 1.6 \times 10^{-19}} \mathrm{ev}=3.95 \mathrm{ev}$
<br/>Energy $=\mathrm{hf}_{1}=\frac{6.63 \times 10^{-34} \times 10^{15}}{2 \pi \times 1.6 \times 10^{-19}} \mathrm{ev}=0.66 \mathrm{ev}     &lt; \phi$ (work function)
<br/>Hence photoelectric effect not possible (option A correct)
<br/>Also ev $_{0}=h f-\phi$
<br/>(Straight line) (option B correct) $\quad \mathrm{KE}_{\max }=\mathrm{E}-\phi \Rightarrow \phi=\mathrm{E}-2 \mathrm{eV}=1.95 \mathrm{ev}$ (option $\mathrm{c}$ is incorrect)</div>


title: electron-is-an-alloy-of
slug: electron-is-an-alloy-of-77067
question: <div class="question">"Electron" is an alloy of</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{Mg}$ and $\mathrm{Zn}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{Fe}$ and $\mathrm{Mg}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Ni and $\mathrm{Zn}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Al}$ and $\mathrm{Zn}$</span> </li> </ul>

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

answer:
<div class="solution">Electron is an alloy of $\mathrm{Mg}(95 \%)+\mathrm{Zn}(4.5 \%)$ and $\mathrm{Cu}(0.5 \%)$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P2.db
title:
electrons-are-emitted-with-kinetic-energy-t-from-a-metal-plate-by-an-irradiation-of-light-of-intensity-j-and-frequency-v-then-which-of-the-following-w

slug:
electrons-are-emitted-with-kinetic-energy-t-from-a-metal-plate-by-an-irradiation-of-light-of-intensity-j-and-frequency-v-then-which-of-the-following-w-22683

question:
<div class="question">Electrons are emitted with kinetic energy $T$ from a metal plate by an irradiation of light of intensity $J$ and frequency v. Then, which of the following will be true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$T \propto J$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$T$ linearly increasing with $v$</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">$T \propto$ lime of irradiation</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Number of electrons emitted $\propto J$</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 answers are:
<span class="option-value">$T$ linearly increasing with $v$, Number of electrons emitted $\propto J$</span> </div>

answer:
<div class="solution">In photoelectric effect, if the incident light had a frequency less than a minimum frequency $v_{0}$, then no electrons are ejected regardless of the light's amplitude. This minimum frequency is also called the threshold frequency, and the value of $v_{0}$ depends on the metal. For frequency greater than $v_{0},$ electrons would be ejected from the metal. Furthermore, the kinetic energy of the photoelectrons is proportional to the light frequency 
<br/>The relationship between photoelectron kinetic energy $T$ and light frequency $v$ is shown in graph below
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a36.png"/> <br/>Graph shows that kinetic energy $T$ is linearly increasing with light frequency (v).
<br/>However, for a given photosensitive material and frequency of incident light, number of photoelectrons emitted per second is directly proportional to the intensity of incident light
<br/>i.e., Number of electrons emitted $\propto J$. Also, photoelectric emission is an instantaneous process.</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
title:
eleven-apples-are-distributed-among-a-girl-and-a-boy-then-which-one-of-the-following-statements-is-true

slug:
eleven-apples-are-distributed-among-a-girl-and-a-boy-then-which-one-of-the-following-statements-is-true-61140

question:
<div class="question">Eleven apples are distributed among a girl and a boy, Then, which one of the following statements is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">atleast one of them will receive 7 apples</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">the girl receives atleast 4 apples or the boy receives atleast 9 apples</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the girl receives atleast 5 apples or the boy receives atleast 8 apples</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">the girl receives atleast 4 apples or the boy receives atleast 8 apples</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">the girl receives atleast 4 apples or the boy receives atleast 8 apples</span> </div>

answer:
<div class="solution">When eleven apples are distributed among a gil and a boy, then the girl receives atieast
<br/>14 apples or the boys receives atieast 8 apples. (by hypothesis)</div>


tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P2.db
title:
eleven-equal-point-charges-all-of-them-having-a-charge-q-are-placed-at-all-the-hour-positions-of-a-circular-clock-of-radius-r-except-at-the-10-h-posit

slug:
eleven-equal-point-charges-all-of-them-having-a-charge-q-are-placed-at-all-the-hour-positions-of-a-circular-clock-of-radius-r-except-at-the-10-h-posit-59568

question:
<div class="question">Eleven equal point charges, all of them having a charge $+Q$, are placed at all the hour positions of a circular clock of radius $r$. except at the $10 \mathrm{h}$ position. What is the clectric field strength at the centre of the clock?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{Q}{4 \pi \varepsilon_{0} r^{2}}$ from the centre towards the mark 10</span> </div>

answer:
<div class="solution">The clock diagram is as shown below 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a19.png"/> <br/>In the above diagram, charge $+Q$ is not placed at
<br/>$10 \mathrm{h}$ position.
<br/>So, net electric ficld strength at centre $C$.
<br/>$$
<br/>\begin{array}{r}
<br/>\mathrm{E}_{\text {net }}=\mathrm{E}_{1}+\mathrm{E}_{2}+\mathrm{E}_{3}+\mathrm{E}_{4}+\mathrm{E}_{5}+\mathrm{E}_{6} \\
<br/>+\mathrm{E}_{7}+\mathrm{E}_{8}+\mathrm{E}_{9}+\mathrm{E}_{11}+\mathrm{E}_{12} \\
<br/>\Rightarrow \mathrm{E}_{\text {net }}=\mathrm{E}_{1}+\mathrm{E}_{2}+\mathrm{E}_{3}+\mathrm{E}_{4} \\
<br/>+\mathrm{E}_{5}+\mathrm{E}_{6}+\left(-\mathrm{E}_{1}-\mathrm{E}_{2}-\mathrm{E}_{3}-\mathrm{E}_{5}-\mathrm{E}_{6}\right)
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \mathrm{E}_{\mathrm{net}}=\mathrm{E}_{4}=\frac{Q}{4 \pi \varepsilon_{0} r^{2}},$ from centre towards the
<br/>mark $10 .$</div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P2.db
title:
equal-masses-of-ethane-and-hydrogen-are-mixed-in-a-empty-container-at-298-k-the-fraction-of-total-pressure-exerted-by-hydrogen-is

slug:
equal-masses-of-ethane-and-hydrogen-are-mixed-in-a-empty-container-at-298-k-the-fraction-of-total-pressure-exerted-by-hydrogen-is-47620

question:
<div class="question">Equal masses of ethane and hydrogen are mixed in a empty container at $298 \mathrm{~K}$. The fraction of total pressure exerted by hydrogen is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$15: 16$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1: 1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1: 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1: 6$</span> </li> </ul>

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

answer:
<div class="solution">Hint: 
<br/>Let mass of ethane and $\mathrm{H}_{2}$ are $\mathrm{xg}$.
<br/>$\begin{array}{c}
<br/>n_{\mathrm{H}_{2}}=\frac{\mathrm{x}}{2} \text { and } \mathrm{n}_{\mathrm{C}_{2} \mathrm{H}_{\mathrm{z}}}=\frac{\mathrm{x}}{30} \\
<br/>\frac{\mathrm{P}_{\mathrm{H}_{2}}}{\mathrm{P}_{\mathrm{T}}}=\frac{\mathrm{n}_{\mathrm{H}_{2}}}{\mathrm{n}_{\mathrm{H}_{2}}+\mathrm{n}_{\mathrm{C}_{2} \mathrm{H}}}=\frac{\frac{\mathrm{x}}{2}}{\frac{\mathrm{X}}{2}+\frac{\mathrm{x}}{30}}=\frac{15}{16}
<br/>\end{array}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P2.db
title:
equal-volumes-of-molar-hydrochloric-acid-and-sulphuric-acid-are-neutralised-by-dilute-naoh-solution-and-x-kcal-and-y-kcal-of-heat-are-liberated-respec

slug:
equal-volumes-of-molar-hydrochloric-acid-and-sulphuric-acid-are-neutralised-by-dilute-naoh-solution-and-x-kcal-and-y-kcal-of-heat-are-liberated-respec-59326

question:
<div class="question">Equal volumes of molar hydrochloric acid and sulphuric acid are neutralised by dilute $\mathrm{NaOH}$ solution and $\mathrm{x} \mathrm{kcal}$ and $\mathrm{y} \mathrm{kcal}$ of heat are liberated respectively. Which of the following is true?</div>

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

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

answer:
<div class="solution">Enthalpy of $1 \mathrm{~g}$ equivalent of strong acid and $1 \mathrm{~g}$ equivalent strong base $=13.7 \mathrm{kcal}$ Equal volume contains double eq. of $\mathrm{H}_2 \mathrm{SO}_4$ than $\mathrm{HCl}$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P2.db
title:
equal-weights-of-ch-4-and-h-2-are-mixed-in-an-empty-container-at-2-5-c-the-fraction-of-the-total-pressure-exerted-by-h-2-is

slug:
equal-weights-of-ch-4-and-h-2-are-mixed-in-an-empty-container-at-2-5-c-the-fraction-of-the-total-pressure-exerted-by-h-2-is-81730

question:
<div class="question">Equal weights of $\mathrm{CH}_{4}$ and $\mathrm{H}_{2}$ are mixed in an empty container at $25^{\circ} \mathrm{C}$. the fraction of the total pressure exerted by $\mathrm{H}_{2}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 9$</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">$8 / 9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$16 / 17$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 / 9$</span> </div>

answer:
<div class="solution">Ratio of number of moles of $\mathrm{CH}_{4}$ and $\mathrm{H}_{2}$ are
<br/>$$
<br/>\begin{array}{r}
<br/>\mathrm{CH}_{4}: \mathrm{H}_{2}=\frac{x}{16} ; \frac{x}{2} \\
<br/>=1: 8
<br/>\end{array}
<br/>$$
<br/>Hence, partial pressure exerted by
<br/>$$
<br/>\mathrm{H}_{2}=\frac{8}{1+8} \times p_{\text {total }}=\frac{8}{9} \times p_{\text {lotal }}
<br/>$$</div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15:16</span> </div>
source_db: db\MarksBatch1_P2.db
title:
equal-weights-of-ethane-and-hydrogen-are-mixed-in-an-empty-container-at-25-c-the-fraction-of-total-pressure-exerted-by-hydrogen-is

slug:
equal-weights-of-ethane-and-hydrogen-are-mixed-in-an-empty-container-at-25-c-the-fraction-of-total-pressure-exerted-by-hydrogen-is-24051

question:
<div class="question">Equal weights of ethane and hydrogen are mixed in an empty container at 25 C. The fraction of total pressure exerted by hydrogen 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: 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">01:16</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">15:16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, equal weights of ethane and hydrogen are mixed in an empty container at $25^{\circ} \mathrm{C}$
<br/>Initial gram weight $=\begin{array}{ll}\mathrm{C}_{2} \mathrm{H}_{6} &amp; \mathrm{H}_{2} \\ \mathrm{Wg} &amp; \mathrm{Wg}\end{array}$
<br/>Number of moles $=\frac{w}{30} \quad \frac{w}{2}$ According to Henry's law,
<br/>$$
<br/>\begin{array}{l}\frac{p_{\mathrm{H}_{2}}}{p_{\text {total }}}=\chi_{\mathrm{H}_{2}}=\frac{n_{\mathrm{H}_{2}}}{n_{\mathrm{H}_{2}}+n_{\mathrm{C}_{2} \mathrm{H}_{6}}}=\frac{\frac{w}{2}}{\frac{w}{2}+\frac{w}{30}} \\ \frac{p_{\mathrm{H}_{2}}}{p_{\text {total }}}=\frac{\frac{w}{2}}{\frac{15 w+w}{30}}=\frac{w}{2} \times \frac{30}{16 w}=\frac{15}{16}\end{array}
<br/>$$
<br/>So, the fraction of total pressure exerted by
<br/>hydrogen is 15: 16</div>


title: equation-of-a-tangent-to-the-hyperbola-5-x-2-y-2-5-and-which-passes-through-an-external-point-28-is
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P2.db
slug:
equation-of-a-tangent-to-the-hyperbola-5-x-2-y-2-5-and-which-passes-through-an-external-point-28-is-83450

question:
<div class="question">Equation of a tangent to the hyperbola $5 x^{2}-y^{2}=5$ and which passes through an external point (2,8) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 x-y+2=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 x+y-14=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$23 x-3 y-22=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x-23 y+178=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$3 x-y+2=0$, $23 x-3 y-22=0$</span> </div>

answer:
<div class="solution">Given equation of hyperbola can be write as
<br/>$\frac{x^{2}}{1}-\frac{y^{2}}{(\sqrt{5})^{2}}=1$
<br/>here, $a=1$ and $b=\sqrt{5}$
<br/>Since, $y=m x \pm \sqrt{a^{2} m^{2}-b^{2}}$
<br/>$y=m x \pm \sqrt{m^{2}-5}$
<br/>$\Rightarrow \quad(8-2 m)^{2}=\left(\sqrt{m^{2}-5}\right)^{2}$
<br/>$\Rightarrow \quad(8-2 m)^{2}=m^{2}-5$
<br/>$\Rightarrow 64+4 m^{2}-32 m=m^{2}-5$
<br/>$\Rightarrow 3 m^{2}-32 m+69=0$
<br/>$\Rightarrow \quad m=\frac{22}{3}, 3 \quad$ [by solving equation $]$
<br/>When $m=\frac{23}{3},$ then equation of tangent
<br/>$$
<br/>(y-8)=\frac{23}{3}(x-2)
<br/>$$
<br/>$\Rightarrow \quad 3 y-24=23 x-46$
<br/>$\Rightarrow \quad 23 x-3 y-46+24=0$
<br/>$\Rightarrow \quad 23 x-3 y-22=0$
<br/>When $m=3$, then equation of tangent
<br/>$$
<br/>\begin{aligned}
<br/>&amp; &amp;(y-8) &amp;=3(x-2) \\
<br/>\Rightarrow &amp; &amp; y-8 &amp;=3 x-6
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow \quad 3 x-y-6+8=0$
<br/>$\Rightarrow \quad 3 x-y+2=0$
<br/>Hence, option (a) and (c) are correct.</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: db\MarksBatch1_P2.db
title:
equilibrium-constants-for-the-following-reactions-at-1200-k-are-given-2-h-2-o-g-2-h-2-g-o-2-g-k-1-64-1-0-8-2-co-2-g-2-co-g-o-2-g-k-2-16-1-0-6-the-equi

slug:
equilibrium-constants-for-the-following-reactions-at-1200-k-are-given-2-h-2-o-g-2-h-2-g-o-2-g-k-1-64-1-0-8-2-co-2-g-2-co-g-o-2-g-k-2-16-1-0-6-the-equi-41013

question:
<div class="question">Equilibrium constants for the following reactions at $1200 \mathrm{K}$ are given
<br/>$2 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)$
<br/>$K_{1}=6.4 \times 10^{-8}$
<br/>$2 \mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g)$
<br/>$K_{2}=1.6 \times 10^{-6}$
<br/>The equilibrium constant for the reaction? $\mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g)$
<br/>at $1200 \mathrm{K}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.05</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">For the given reactions, $2 \mathrm{H}_{2} \mathrm{O} \rightleftharpoons 2 \mathrm{H}_{2}+\mathrm{O}_{2}, K_{1}=6.4 \times 10^{-8}$
<br/>or, $\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}, K_{1}^{\prime}=\sqrt{K_{1}}$
<br/>or $\mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}, \longrightarrow \mathrm{H}_{2} \mathrm{O}, K_{1}^{\prime \prime \prime}=1 / \sqrt{K_{1}} \ldots$
<br/>$2 \mathrm{CO}_{2} \rightleftharpoons 2 \mathrm{CO}+\mathrm{O}_{2}, K_{2}=1.6 \times 10^{-6}$
<br/>or $\mathrm{CO}_{2} \rightleftharpoons \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}, K_{2}^{\prime}=\sqrt{K_{2}}$
<br/>From (i) and (ii) [on adding] $K=\frac{\sqrt{K_{2}}}{\sqrt{K_{1}}}=\frac{\sqrt{1.6 \times 10^{-6}}}{\sqrt{6.4 \times 10^{-8}}}$
<br/>$K=\sqrt{\frac{10^{2}}{4}}=\sqrt{25}=5$</div>


title: equivalent-capacitance-between-a-and-b-in-the-figure-is-1
slug: equivalent-capacitance-between-a-and-b-in-the-figure-is-1-66242
tags: ['Physics', 'Capacitance', 'JEE Main']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Equivalent capacitance between $A$ and $B$ in the figure is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_q1.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \mu \mathrm{F}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$8 \mu \mathrm{F}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$12 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$16 \mu \mathrm{F}$</span> </li> </ul>

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

answer:
<div class="solution">$\frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}=\frac{1}{4}+\frac{1}{4}=\frac{2}{4}$
<br/>$C=2 \mu F$
<br/>$2 \mu \mathrm{F}$ will be parallel to $4 \mu \mathrm{F}$. so the resultant capacitance
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a1.png"/> <br/> <br/>$\begin{aligned} \frac{1}{C^{\prime \prime}} &amp;=\frac{1}{4 \mu F}+\frac{1}{4 \mu F}=\frac{2}{4} \mu F \\ C^{-} &amp;=2 \mu F \\ C_{A Q} &amp;=2 \mu F+6 \mu F \\ &amp;=8 \mu F \end{aligned}$</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
title:
equivalent-conductivity-at-infinite-dilution-for-sodiumpotassium-oxalate-coo-2-na-k-will-be-given-molar-conductivities-of-oxalate-k-and-na-ions-at-inf

slug:
equivalent-conductivity-at-infinite-dilution-for-sodiumpotassium-oxalate-coo-2-na-k-will-be-given-molar-conductivities-of-oxalate-k-and-na-ions-at-inf-62310

question:
<div class="question">Equivalent conductivity at infinite dilution for $\quad$ sodium-potassium oxalate $\left[\left(\mathrm{COO}^{-}\right)_{2} \mathrm{Na}^{+} \mathrm{K}^{+}\right]$ will be [given molar
<br/>conductivities of oxalate, $\mathrm{K}^{+}$ and $\mathrm{Na}^{+}$ ions at infinite dilution are $148.2, \quad 50.1,$ $73.5 \mathrm{S} \mathrm{cm}^{2} \mathrm{mol}^{-1}$ respectively $]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$271.8 \mathrm{S} \mathrm{cm}^{2} \mathrm{eq}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$67.95 \mathrm{Scm}^{2} \mathrm{eq}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$543.6 \mathrm{Scm}^{2} \mathrm{eq}^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$135.9 \mathrm{Scm}^{2} \mathrm{eq}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$135.9 \mathrm{Scm}^{2} \mathrm{eq}^{-1}$</span> </div>

answer:
<div class="solution">$\lambda_{m}^{-}=\lambda_{m}^{-}$ (oxalate) $+\lambda_{m}^{-}\left(\mathrm{Na}^{+}\right)+\lambda_{m}^{-}\left(\mathrm{K}^{+}\right)$
<br/>$$
<br/>\begin{array}{l}
<br/>=(148 \cdot 2+73 \cdot 5+50 \cdot 1) \\
<br/>=271.85 \mathrm{cm}^{2} \mathrm{mol}^{-1}
<br/>\end{array}
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>\begin{aligned}
<br/>\lambda_{e q}^{-} &amp;=\frac{\lambda_{m}^{-}}{n \cdot \text { factor }}=\frac{271.8}{2} \\
<br/>&amp;=135.9 \mathrm{Scm}^{2} \mathrm{eq}^{-1}
<br/>\end{aligned}
<br/>$$</div>


title: ethelene-can-be-separated-from-acetylene-by-passing-the-mixture-through
slug: ethelene-can-be-separated-from-acetylene-by-passing-the-mixture-through-24717
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Ethelene can be separated from acetylene by passing the mixture through :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">fuming $\mathrm{H}_2 \mathrm{SO}_4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">pyrogallol</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">ammoniacal $\mathrm{Cu}_2 \mathrm{Cl}_2$</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">Charcoal powder</span> </li> </ul>

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

answer:
<div class="solution">Hints : $\mathrm{H}-\mathrm{C} \equiv \mathrm{C}-\mathrm{H}+\mathrm{Cu}_2 \mathrm{Cl}_2 \rightarrow \mathrm{Cu}^{+} \mathrm{C}^{-} \equiv \mathrm{C}^{-} \mathrm{Cu}^{+} \downarrow$<br/>Red ppt.<br/><br/>$$<br/>\mathrm{H}_2 \mathrm{C}=\mathrm{CH}_2+\mathrm{Cu}_2 \mathrm{Cl}_2 \rightarrow \mathrm{No} . \mathrm{ppt}<br/>$$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P2.db
title:
experimental-investigations-show-that-the-intensity-of-solar-radiation-is-maximum-for-a-wavelength-480-nm-in-the-visible-region-estimate-the-surface-t

slug:
experimental-investigations-show-that-the-intensity-of-solar-radiation-is-maximum-for-a-wavelength-480-nm-in-the-visible-region-estimate-the-surface-t-97417

question:
<div class="question">Experimental investigations show that the intensity of solar radiation is maximum for a wavelength $480 \mathrm{~nm}$ in the visible region. Estimate the surface temperature of sun. Given Wein's constant $b=2.88 \times 10^{-3} \mathrm{mK}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4000 \mathrm{~K}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$6000 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8000 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10^6 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6000 \mathrm{~K}$</span> </div>

answer:
<div class="solution">Hints: $\lambda_{\mathrm{m}} \times \mathrm{T}=\mathrm{b}$<br/>$\lambda_{\mathrm{m}}=480 \mathrm{~nm}$<br/>$\mathrm{T}=\frac{b}{\lambda_m}=\frac{2.88 \times 10^{-3}}{480 \times 10^{-9}}=6000 \mathrm{~K}$</div>


tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P2.db
title:
extraction-of-a-metal-m-from-its-sulfide-ore-m-2-s-involves-the-following-chemical-reactions-2-m-2-s-3-o-2-heat-2-m-2-o-2-so-2-m-2-s-2-m-2-o-heat-6-m-

slug:
extraction-of-a-metal-m-from-its-sulfide-ore-m-2-s-involves-the-following-chemical-reactions-2-m-2-s-3-o-2-heat-2-m-2-o-2-so-2-m-2-s-2-m-2-o-heat-6-m-34087

question:
<div class="question">Extraction of a metal ( $M$ ) from its sulfide ore $\left(M_{2} S\right)$ involves the following chemical reactions:
<br/>$\begin{array}{l}
<br/>2 \mathrm{M}_{2} \mathrm{~S}+3 \mathrm{O}_{2} \stackrel{\text { heat }}{\longrightarrow} 2 \mathrm{M}_{2} \mathrm{O}+2 \mathrm{SO}_{2} \uparrow \\
<br/>\mathrm{M}_{2} \mathrm{~S}+2 \mathrm{M}_{2} \mathrm{O} \stackrel{\text { heat }}{\longrightarrow} 6 \mathrm{M}+\mathrm{SO}_{2} \uparrow
<br/>\end{array}$
<br/>The metal (M) may be</div>

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

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

answer:
<div class="solution">$\begin{array}{l}
<br/>2 \mathrm{Cu}_{2} \mathrm{~S}+3 \mathrm{O}_{2} \stackrel{\Delta}{\longrightarrow} \mathrm{Cu}_{2} \mathrm{O}+\mathrm{SO}_{2} \\
<br/>2 \mathrm{Cu}_{2} \mathrm{O}+\mathrm{Cu}_{2} \mathrm{~S} \longrightarrow 6 \mathrm{Cu}+\mathrm{SO}_{2}
<br/>\end{array}$</div>


title: extraction-of-gold-au-involves-the-formation-of-complex-ions-x-and-y-x-and-y-respectively-are
slug: extraction-of-gold-au-involves-the-formation-of-complex-ions-x-and-y-x-and-y-respectively-are-59118
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">"Extraction of gold $(\mathrm{Au})$ involves the formation of complex ions $X$ and $Y$
<br/><img src="https://panel-qbank.s3.ap-south-1.amazonaws.com/previous-yr/images/wbjee/2015_che_q42.png"/> <br/>$X$ and $Y$ respectively are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$Au(CN) ^{-}_{2}$ and $\mathrm{Zn} (\mathrm{CN)}_{4}^{2-}$</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">$Au(CN) ^{3-}_{4}$ and $\mathrm{Zn} (\mathrm{CN)}_{4}^{2-}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$Au(CN) ^{-}_{3}$ and $\mathrm{Zn} (\mathrm{CN)}_{6}^{4-}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$Au(CN) ^{-}_{4}$ and $\mathrm{Zn} (\mathrm{CN)}_{3}^{-}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$Au(CN) ^{-}_{2}$ and $\mathrm{Zn} (\mathrm{CN)}_{4}^{2-}$</span> </div>

answer:
<div class="solution">Extraction of gold (Au) involves the following sequence of reaction,
<br/>$\begin{array}{l}
<br/>\text { Gold ore } \frac{\text { Roasting }}{\mathrm{CN}^{-}, \mathrm{H}_{2} \mathrm{O}, \mathrm{O}_{2}} \quad \mathrm{HO}^{-}+\mathrm{Au}(\mathrm{CN})_{2}^{-} 
<br/>\quad \stackrel{\mathrm{Zn}}{\longrightarrow} \mathrm{Zn}(\mathrm{CN})_{4}^{2-}+\mathrm{Au} \\
<br/>Y
<br/>\end{array}$
<br/>Hence, $X=\mathrm{Au}(\mathrm{CN})_{2}^{-}, Y=\mathrm{Zn}(\mathrm{CN})_{4}^{2-}$</div>


title: f-x-is-real-valued-function-such-that-2-f-x-3-f-x-15-4-x-for-all-x-r-then-f-2
slug: f-x-is-real-valued-function-such-that-2-f-x-3-f-x-15-4-x-for-all-x-r-then-f-2-40815
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">$f(x)$ is real valued function such that $2 f(x)+3 f(-x)=15-4 x$ for all $x \in R$. Then $f(2)=$</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">22</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">$2 f(x)+3 f(-x)=15-4 x$...(1)
<br/>Replace $x$ by $-x$
<br/>$2 f(-x)+3 f(x)=15+4 x$ ...(2)
<br/>After solving (1) and (2), we get
<br/>$\begin{array}{l}
<br/>f(x)=4 x+3 \\
<br/>\therefore f(2)=8+3=11
<br/>\end{array}$</div>


title: f-x-r-x-x-0-x-1-is-defined-as-f-x-1-2-x-1-2-x-1-then
slug: f-x-r-x-x-0-x-1-is-defined-as-f-x-1-2-x-1-2-x-1-then-19570
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">$f: X \rightarrow \mathbb{R}, X=\{x \mid 0  &lt; x &lt; 1\}$ is defined as $f(x)=\frac{2 x-1}{1-|2 x-1|}$. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{f}$ is only injective</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f$ is only surjective</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}$ is bijective</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">f is neither injective nor surjective</span> </li> </ul>

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

answer:
<div class="solution">Put $2 \mathrm{x}-1=\mathrm{t}$, then the function becomes<br/>$\begin{aligned}<br/>&amp;f(t)=\frac{t}{1-|t|},-1  &lt; t &lt; 1 \\<br/>&amp;\therefore f(t)=\left\{\begin{array}{l}<br/>\frac{t}{1+t},-1 &lt; t \leq 0 \\<br/>\frac{t}{1-t}, 0 &lt; t &lt; 1<br/>\end{array}\right.<br/>\end{aligned}$<br/>$\because$ It is continuous and $f\left(-1^{+}\right)=-\infty, f\left(1^{-}\right)=+\infty$<br/>$\therefore$ Range $=(-\infty,-\infty)=\mathbb{R}$<br/>It is differentiable also,<br/>$f^{\prime}(t)=\left\{\begin{array}{l}<br/>\frac{1}{(1+t)^2},-1 &lt; t &lt; 0 \\<br/>\frac{1}{(1-t)^2}, 0 &lt; t &lt; 1<br/>\end{array}\right.$<br/>$\because f^{\prime}(t) &gt; 0 \forall-1 &lt; t &lt; 1$<br/>$\therefore$ f is injective</div>


title: f-x-x-1-f-x-x-1-f-x-x-x-f-x-dx
slug: f-x-x-1-f-x-x-1-f-x-x-x-f-x-dx-39059
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">$\int \frac{\mathrm{f}(\mathrm{x}) \varphi^{\prime}(\mathrm{x})+\varphi(\mathrm{x}) \mathrm{f}^{\prime}(\mathrm{x})}{(\mathrm{f}(\mathrm{x}) \varphi(\mathrm{x})+1) \sqrt{\mathrm{f}(\mathrm{x}) \varphi(\mathrm{x})-1}} \mathrm{dx}=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin ^{-1} \sqrt{\frac{\mathrm{f}(\mathrm{x})}{\varphi(\mathrm{x})}}+\mathrm{c}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos ^{-1} \sqrt{(f(x))^{2}-(\varphi(x))^{2}}+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{2} \tan ^{-1} \sqrt{\frac{f(x) \varphi(x)-1}{2}}+c$</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">$\sqrt{2} \tan ^{-1} \sqrt{\frac{f(x) \varphi(x)+1}{2}}+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{2} \tan ^{-1} \sqrt{\frac{f(x) \varphi(x)-1}{2}}+c$</span> </div>

answer:
<div class="solution">Hint:
<br/>Let $f(x) \phi(x)=t$
<br/>$\int \frac{d t}{(t+1) \sqrt{t-1}}$
<br/>Let $t-1=p^{2}, d t=2 p d p$
<br/>$\Rightarrow \int \frac{2 d p}{p^{2}+2}=\sqrt{2} \tan ^{-1}\left|\sqrt{\frac{f(x) \phi(x)-1}{2}}\right|+c$</div>


title: f-x-x-x-is-continuous-for
slug: f-x-x-x-is-continuous-for-44108
question: <div class="question">$f(x)=x+|x|$ is continuous for</div>
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad x \in(-\infty, \infty)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x \in(-\infty, \infty)-\{0\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">only $x&gt;0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">no value of $x$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad x \in(-\infty, \infty)$</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints: } f(x)=\left\{\begin{array}{lc}<br/>2 x ; &amp; x \geq 0 \\<br/>0 ; &amp; x &lt; 0<br/>\end{array}\right.<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2y8bu7Cchp2ogxl0C6QgIiMH8D5VPenR11VjbsBxcrY.original.fullsize.png"/><br/></div>


title: ferric-ion-forms-a-prussian-blue-precipitate-due-to-the-formation-of
slug: ferric-ion-forms-a-prussian-blue-precipitate-due-to-the-formation-of-89681
question: <div class="question">Ferric ion forms a prussian blue precipitate due to the formation of</div>
tags: ['Chemistry', 'Coordination Compounds', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{K}_{4}\left[\mathrm{Fe}\left(\mathrm{CN})_{6}\right]\right.$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{K}_{3}\left[\mathrm{Fe}\left(\mathrm{CN})_{6}\right)\right.$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$Fe(CNS)_{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$Fe_{4}[Fe(CN)_{6}]_{3}$</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">$Fe_{4}[Fe(CN)_{6}]_{3}$</span> </div>

answer:
<div class="solution">Ferric ion forms a prussian blue precipitate due to the formation of $\mathrm{Fe}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{3}$. This complex is formed during the determine action of presence of nitrogen in the given sample. In this method, to portion of sodium fusion extract. freshly prepared ferrous sulphate, FeSO $_{4}$ solution is added and warmed. Then about 2 to 3 drops of $\mathrm{FeCl}_{3}$ solution are added and acidified with conc. HCl. The appearance of a prussian blue colour indicate the presence of nitrogen.
<br/>$$
<br/>\mathrm{FeSO}_{4}+2 \mathrm{NaOH} \longrightarrow \mathrm{Fe}(\mathrm{OH})_{2}+\mathrm{Na}_{2} \mathrm{SO}_{4}
<br/>$$
<br/>$6 \mathrm{NaCN}+\mathrm{Fe}(\mathrm{OH})_{2} \longrightarrow \mathrm{Na}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]+2 \mathrm{NaOH}$
<br/>$$
<br/>\begin{array}{r}
<br/>3 \mathrm{Na}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]+4 \mathrm{FeCl}_{3} \longrightarrow \\
<br/>\mathrm{Fe}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{3}+12 \mathrm{NaCl}
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P2.db
title:
figure-below-shows-the-distancetime-graph-of-the-motion-of-a-car-if-follows-from-the-graph-that-the-car-is

slug:
figure-below-shows-the-distancetime-graph-of-the-motion-of-a-car-if-follows-from-the-graph-that-the-car-is-54552

question:
<div class="question">Figure below shows the distance-time graph of the motion of a car. If follows from the graph that the car is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uIKZ4Z2IvrO1pyRMHpjR4PFqmaRHQBmrXwZUE-e9qfo.original.fullsize.png"/><br/></div>

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

answer:
<div class="solution">Slope is increasing with constant rate. i.e motion is uniformaly accelerated<br/>$\mathrm{x}=1.2 \mathrm{t}^2 \Rightarrow \mathrm{v}=2.4 \mathrm{t} \Rightarrow \mathrm{a}=2.4 \mathrm{~m} / \mathrm{s}^2$</div>


title: find-the-correct-statements-about-photoelectric-effect
slug: find-the-correct-statements-about-photoelectric-effect-28281
tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Find the correct statement(s) about photoelectric effect. <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q58.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">There is no significant Ime delay between the absorption of a suitable tadiation and the emission of electrons</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Einstein analysis gives a threshold frequency above which no electron can be emitted</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The maximum kinetic energy of the emitted photoelectrons is proportional to the frequency of incident radiation</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">The maximum kinetic energy of electrons does not depend on the intensity of radiation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">There is no significant Ime delay between the absorption of a suitable tadiation and the emission of electrons, The maximum kinetic energy of electrons does not depend on the intensity of radiation</span> </div>

answer:
<div class="solution">There is no significant time delay between the absorption of a suitable radiation and the emission of electrons and the maximum kinetic energy of electrons does not depend on the intensity of radiation.
<br/>So, (a) and (d) are correct options.</div>


title: find-the-equivalent-capacitance-between-a-and-b-of-the-following-arrangement
slug: find-the-equivalent-capacitance-between-a-and-b-of-the-following-arrangement-86812
tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3C</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Find the equivalent capacitance between $A$ and $B$ of the following arrangement :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/L5Ymn4Wpk-YWuDz7oVuCGZuOv4IkBSbk3TSdsP4m4hE.original.fullsize.png"/></div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8k-Mii7cAWBo9hCxUPI3ce_QQZJ0SO9x77xA0kQxHds.original.fullsize.png"/><br/><br/>Cnet. $=C+C+C=3 C$</div>


title: find-the-maximum-value-of-z-when-z-z-3-2-where-z-being-a-complex-number
slug: find-the-maximum-value-of-z-when-z-z-3-2-where-z-being-a-complex-number-72581
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Find the maximum value of $|z|$ when $\left|z-\frac{3}{z}\right|=2,$ where $z$ being a complex number.</div>

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

answer:
<div class="solution">We have, $|z|=\left|z-\frac{3}{z}+\frac{3}{z}\right|$
<br/>$\Rightarrow \quad|z| \leq\left|z-\frac{3}{z}\right|+\left|\frac{3}{z}\right|$
<br/>[using triangle inequality] $\Rightarrow \quad|z| \leq 2+\left|\frac{3}{z}\right|$
<br/>$\left[\because\left|z-\frac{3}{z}\right|=2\right]$
<br/>$\Rightarrow \quad|z| \leq 2+\frac{3}{|z|}$
<br/>$\begin{array}{lr}\Rightarrow &amp; |z|^{2}-2|z|-3 \leq 0 \\ \Rightarrow &amp; (|z|-3)(|z|+1) \leq 0\end{array}$
<br/>$\Rightarrow \quad-1 \leq|z| \leq 3$
<br/>Thus, the maximum value of $|z|$ is 3 .</div>


title: find-the-right-conditions-for-fraunhoffer-difiraction-due-to-a-single-slit
slug: find-the-right-conditions-for-fraunhoffer-difiraction-due-to-a-single-slit-83021
tags: ['Physics', 'Wave Optics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">Find the right condition(s) for Fraunhoffer difiraction due to a single slit.</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Source is near to the slit and the incidont beam is parallel, Source is at infinity and the incident beam is parallel</span> </div>

answer:
<div class="solution">For Fraunhofer diffraction, incident beam should
<br/>be parallel. This can be achieved when
<br/>(a) Source is placed at focus of converging lens.
<br/>(b) Source is at infinite distance.</div>


tags: ['Physics', 'Current Electricity', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch1_P2.db
title:
five-equal-resistance-each-of-resistance-mathrmr-are-connected-as-shown-in-figure-below-a-battery-of-mathrmv-volt-is-connected-between-a-and-b-the-cur

slug:
five-equal-resistance-each-of-resistance-mathrmr-are-connected-as-shown-in-figure-below-a-battery-of-mathrmv-volt-is-connected-between-a-and-b-the-cur-62214

question:
<div class="question">Five equal resistance, each of resistance \(\mathrm{R}\), are connected as shown in figure below. A battery of \(\mathrm{V}\) volt is connected between \(A\) and \(B\). The current flowing in \(\mathrm{FC}\) will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ViAk-Cd8FsqZOTkDgS8N85KeTUgmx9ZndNrvt8dygkg.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{3 \mathrm{~V}}{\mathrm{R}}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{V}{R}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{\mathrm{V}}{2 \mathrm{R}}\)</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">D</span> <span class="option-data">\(\frac{2 \mathrm{~V}}{\mathrm{R}}\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8pvtfU37u6QbYyMLDKHYuJCZK2aOf3mYVKxP-KNSVh0.original.fullsize.png"/><br/><br/>\(\mathrm{I}=\frac{\mathrm{V}}{\mathrm{R}} \therefore \text { Current in } \mathrm{FC}=\frac{\mathrm{I}}{2}=\frac{\mathrm{V}}{2 \mathrm{R}}\)</div>


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch1_P2.db
title:
five-identical-capacitors-of-capacitance-20-f-each-are-connected-to-a-battery-of-150-v-in-a-combination-as-shown-in-the-diagram-what-is-the-total-amou

slug:
five-identical-capacitors-of-capacitance-20-f-each-are-connected-to-a-battery-of-150-v-in-a-combination-as-shown-in-the-diagram-what-is-the-total-amou-39379

question:
<div class="question">Five identical capacitors, of capacitance $20 \mu \mathrm{F}$ each, are connected to a battery of $150 \mathrm{V}$, in a combination as shown in the diagram. What is the total amount of charge stored?  <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_q18.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$15 \times 10^{-3} \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$12 \times 10^{-3} \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \times 10^{-3} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3 \times 10^{-3} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">The given circuit shows a balanced Wheatstone bridge. Now, the circuit becomes
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_phy_a18.png"/> <br/>In the branch $A B,$ both capacitor are arranged in the series combination. Hence, its cquivalent capacitance is given by
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{1}{C_{A B}}=\frac{1}{20}+\frac{1}{20}=\frac{2}{20}=\frac{1}{10} \\
<br/>C_{A B}=10 \mu \mathrm{F}
<br/>\end{array}
<br/>$$
<br/>Similarly, in branch $\mathrm{CD}$,
<br/>$$
<br/>C_{C D}=10 \mu \mathrm{F}
<br/>$$
<br/>Now, $C_{A B}$ and $C_{C D}$ are connected in parallel combination. Hence, the equivalent capacitance of the circuit,
<br/>$$
<br/>C_{\mathrm{eq}}=10+10=20 \mu \mathrm{F}
<br/>$$
<br/>We know that, charge $Q=C_{\mathrm{eq}} V$
<br/>$$
<br/>Q=20 \times 10^{-6} \times 150 \mathrm{V}
<br/>$$
<br/>Amount of charge stored $Q=3 \times 10^{-3} C$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1080</span> </div>
source_db: db\MarksBatch1_P2.db
title:
five-letter-words-having-distinct-letters-are-to-be-constructed-using-the-letters-of-the-word-equation-so-that-each-word-contains-exactly-three-vowels

slug:
five-letter-words-having-distinct-letters-are-to-be-constructed-using-the-letters-of-the-word-equation-so-that-each-word-contains-exactly-three-vowels-71858

question:
<div class="question">Five letter words, having distinct letters, are to be constructed using the letters of the word 'EQUATION' so that each word contains exactly three vowels and two consonants. How many of them have all the vowels together?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3600</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1800</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1080</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">900</span> </li> </ul>

answer:
<div class="solution">Vowels = E, U, I, O, A
<br/>Consonants = Q, T, N
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/zkoEFvuBggY1CB1Dd6xoMbLyU22KPgIynyWl0VmiI9A.original.fullsize.png"/><br/> <br/>$=10 \times 3 \times 6 \times 6=1080$</div>


title: five-numbers-are-in-ap-with-common-difference-0-if-the-1-st-3-rd-and-4-th-terms-are-in-gp-then
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
slug:
five-numbers-are-in-ap-with-common-difference-0-if-the-1-st-3-rd-and-4-th-terms-are-in-gp-then-61963

question:
<div class="question">Five numbers are in AP with common difference $\neq 0 .$ If the 1 st, 3 rd and 4 th terms are in GP, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">the 5th term is always $0 .$</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">the 1 st term is always $0 .$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the middle term is always $0 .$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the middle term is always -2.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the 5th term is always $0 .$</span> </div>

answer:
<div class="solution">Let the tine number is in AP are
<br/>$$
<br/>(a-2 d),(a-d), a,(a+d),(a+2 d)
<br/>$$
<br/>where, $d \neq 0$
<br/>Given, 1 st, $3 r d$ and 4 th terms are in GP.
<br/>$$
<br/>\begin{array}{lr}
<br/>\Rightarrow \quad a^{2}=(a-2 d)(a+d) \\
<br/>\Rightarrow \quad a^{2}=a^{2}-2 a d+a d-2 d^{2} \\
<br/>\Rightarrow \quad 2 d^{2}+a d=0 \Rightarrow d(2 d+a)=0 \\
<br/>\because \quad d \neq 0
<br/>\end{array}
<br/>$$
<br/>$\therefore \quad a+2 d=0 \Rightarrow a=-2 d$
<br/>Hence, terms are
<br/>$$
<br/>-4 d,-3 d,-2 d,-d, 0
<br/>$$
<br/>$\therefore$ The fifth term is always $0 .$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
title:
five-numbers-are-in-hp-the-middle-term-is-1-and-the-ratio-of-the-second-and-the-fourth-terms-is-2-1-then-the-sum-of-the-first-three-terms-is

slug:
five-numbers-are-in-hp-the-middle-term-is-1-and-the-ratio-of-the-second-and-the-fourth-terms-is-2-1-then-the-sum-of-the-first-three-terms-is-32250

question:
<div class="question">Five numbers are in HP. The middle term is 1 and the ratio of the second and the fourth terms is 2: 1 . Then, the sum of the first three terms is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{11}{2}$</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">5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{14}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{11}{2}$</span> </div>

answer:
<div class="solution">Let $a-2 d, a-d, a, a+d, a+2 d$ are in $A P$. then $\frac{1}{a-2 d} \cdot \frac{1}{a-d} \cdot \frac{1}{a} \cdot \frac{1}{a+d} \cdot \frac{1}{a+2 d}$ are in $\mathrm{HP}$.
<br/>Given, middle term $=1$ $\Rightarrow \quad \frac{1}{a}=1 \Rightarrow a=1$
<br/>and $\quad \frac{\text { Second term }}{\text { Fourth term }}=\frac{2}{1}$
<br/>$\Rightarrow \quad \frac{1}{a-d} \times a+d=\frac{2}{1}$
<br/>$\Rightarrow \quad a=3 d \Rightarrow d=\frac{1}{3} \quad(\because a=1)$
<br/>$\therefore$ Sum of first three terms $=\frac{1}{1-\frac{2}{3}}+\frac{1}{1-\frac{1}{3}}+1=3+\frac{3}{2}+1=\frac{11}{2}$</div>


title: for-0-p-1-and-for-any-positive-a-b-let-i-p-a-b-p-j-p-a-p-b-p-then
slug: for-0-p-1-and-for-any-positive-a-b-let-i-p-a-b-p-j-p-a-p-b-p-then-46227
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For $0 \leq p \leq 1$ and for any positive $a, b$, let $I(p)=(a+b)^{p}, J(p)=a^{P}+b^{P},$ then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$l(p) \leq J(p)$</span> </div>

answer:
<div class="solution">Here, let $p=\frac{1}{m}$
<br/>then $\left(a^{p}+b^{p}\right)^{1 / p}=\left(a^{1 / m}+b^{1 / m}\right)^{m}$
<br/>$=a+b+k, k \geq 0$
<br/>$\therefore \quad a^{p}+b^{p} \geq(a+b)^{p} \geq(a+b)$
<br/>$\Rightarrow \quad J(p) \geq I(p)$</div>


title: for-0-p-q-2-if-sin-p-cos-q-2-then-the-value-of-tan-2-p-q-is-equal-to
slug: for-0-p-q-2-if-sin-p-cos-q-2-then-the-value-of-tan-2-p-q-is-equal-to-12016
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For $0 \leq P, Q \leq \frac{\pi}{2},$ if $\sin P+\cos Q=2$, then the
<br/>value of $\tan \left(\frac{\vec{P}+Q}{2}\right)$ is equal to</div>

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

answer:
<div class="solution">Given, $0 \leq P, Q \leq \frac{\pi}{2}$
<br/>and $\sin P+\cos Q=2$
<br/>This equation hold only when, $\quad P=\frac{\pi}{2}$
<br/>and $\quad Q=0$
<br/>$\mathrm{LHS}=\sin P+\cos Q=\sin \frac{\pi}{2}+\cos 0$
<br/>$$
<br/>=1+1=2=\mathrm{RHS}
<br/>$$
<br/>$\therefore \quad \tan \left(\frac{P+Q}{2}\right)=\tan \left(\frac{\frac{\pi}{2}+0}{2}\right)=\tan \frac{\pi}{4}=1$</div>


title: for-2-x-2-3-the-value-of-d-x-d-tan-1-1-s-i-n-x-c-o-s-x-is-equal-to
slug: for-2-x-2-3-the-value-of-d-x-d-tan-1-1-s-i-n-x-c-o-s-x-is-equal-to-87867
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For $-\frac{\pi}{2}     &lt; x &lt; \frac{3 \pi}{2}, \quad$ the $\quad$ value $\quad$ of $\frac{d}{d x}\left\{\tan ^{-1} \frac{\cos x}{1+\sin x}\right\}$ 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="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sin x}{(1+\sin x)^{2}}$</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">$\frac{d}{d x}\left\{\tan ^{-1} \frac{\cos x}{1+\sin x}\right\}$
<br/>$=\frac{1}{1+\left(\frac{\cos x}{1+\sin x}\right)^{2}} \frac{d}{d x} \frac{\cos x}{1+\sin x}$
<br/>$=\frac{(1+\sin x)^{2}}{1+\sin ^{2} x+2 \sin x+\cos ^{2} x}$
<br/>$=\frac{(1+\sin x)(-\sin x)-\cos (\cos x)}{(1+\sin x)^{2}}$
<br/>$=\frac{1}{2+2 \sin x} \times\left[-\left(\sin x+\sin ^{2} x+\cos ^{2} x\right)\right]$
<br/>$=\frac{1}{2(1+\sin x)} \times-(1+\sin x)=-\frac{1}{2}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$e$</span> </div>
source_db: db\MarksBatch1_P2.db
title:
for-a-chemical-reaction-at-2-7-c-the-activation-energy-is-600-r-the-ratio-of-the-rate-constants-at-32-7-c-to-that-of-at-2-7-c-will-be

slug:
for-a-chemical-reaction-at-2-7-c-the-activation-energy-is-600-r-the-ratio-of-the-rate-constants-at-32-7-c-to-that-of-at-2-7-c-will-be-58019

question:
<div class="question">For a chemical reaction at $27^{\circ} \mathrm{C},$ the activation energy is $600 \mathrm{R}$. The ratio of the rate constants at $327^{\circ} \mathrm{C}$ to that of at $27^{\circ} \mathrm{C}$
<br/>will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">40</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$e$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$e^{2}$</span> </li> </ul>

answer:
<div class="solution">From Arrhenius equation
<br/>$$
<br/>\begin{array}{l}
<br/>\operatorname{In} \frac{k_{2}}{k_{1}}=\frac{E_{a}}{R}\left[\frac{1}{T_{1}}-\frac{1}{T_{2}}\right] \\
<br/>\operatorname{In} \frac{k_{2}}{k_{1}}=\frac{600 \mathrm{R}}{R}\left(\frac{1}{300}-\frac{1}{600}\right)
<br/>\end{array}
<br/>$$
<br/>or $\ln \frac{k_{2}}{k_{1}}=600\left(\frac{2-1}{600}\right)$
<br/>of $\ln \frac{k_{2}}{k_{1}}=1$ or $\frac{k_{2}}{k_{1}}=e$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch1_P2.db
title:
for-a-matrix-a-1-2-3-0-1-2-0-0-1-if-u-1-u-2-and-u-3-are-3-1-column-matrices-satisfying-a-u-1-1-0-0-a-u-2-2-3-0-a-u-3-2-3-1-and-u-is-3-3-matrix-whose-c

slug:
for-a-matrix-a-1-2-3-0-1-2-0-0-1-if-u-1-u-2-and-u-3-are-3-1-column-matrices-satisfying-a-u-1-1-0-0-a-u-2-2-3-0-a-u-3-2-3-1-and-u-is-3-3-matrix-whose-c-95116

question:
<div class="question">For a matrix $A=\left(\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 2 &amp; 1 &amp; 0 \\ 3 &amp; 2 &amp; 1\end{array}\right)$ if $U_{1},U_{2}$ and $U_{3}$ are $3 \times 1$ column matrices satisfying $A U_{1}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), A U_{2}=\left(\begin{array}{l}2 \\ 3 \\ 0\end{array}\right), A U_{3}=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)$ and $U$ is $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$ and $U_{3}$. Then, sum of he elements of $U^{-1}$ is</div>

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

answer:
<div class="solution">Let $U_{i}=\left(\begin{array}{l}a_{i} \\ b_{1} \\ c_{i}\end{array}\right),$ where $i=1,2,3$
<br/>$\therefore \quad A U_{1}=\left(\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 2 &amp; 1 &amp; 0 \\ 3 &amp; 2 &amp; 1\end{array}\right)\left(\begin{array}{l}a_{1} \\ b_{1} \\ c_{1}\end{array}\right)$
<br/>$\Rightarrow\left(\begin{array}{c}a_{1} \\ 2 a_{1}+b_{1} \\ 3 a_{1}+2 b_{1}+c_{1}\end{array}\right)=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right) \quad\left[\because A U_{1}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)\right]$
<br/>$\therefore \quad a_{1}=1,2 a_{1}+b_{1}=0$
<br/>$\Rightarrow \quad 2+b_{1}=0$
<br/>$\Rightarrow \quad b_{1}=-2$
<br/>and $3 a_{1}+2 b_{1}+c_{1}=0$
<br/>$\Rightarrow \quad 3+2(-2)+c_{1}=0$
<br/>$\Rightarrow \quad 3-4+c_{1}=0$
<br/>$\Rightarrow \quad c_{1}=1$
<br/>Similarly, $A U_{2}=\left(\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 2 &amp; 1 &amp; 0 \\ 3 &amp; 2 &amp; 1\end{array}\right)\left(\begin{array}{l}a_{2} \\ b_{2} \\ c_{2}\end{array}\right)$
<br/>$\Rightarrow\left(\begin{array}{c}a_{2} \\ 2 a_{2}+b_{2} \\ 3 a_{2}+2 b_{2}+c_{2}\end{array}\right)=\left(\begin{array}{l}2 \\ 3 \\ 0\end{array}\right) \quad\left[\because A U_{2}=\left(\begin{array}{l}2 \\ 3 \\ 0\end{array}\right)\right]$
<br/>$\therefore a_{2}=2$ and $2 a_{2}+b_{2}=3$
<br/>$\Rightarrow \quad 2 \times 2+b_{2}=3$
<br/>$\Rightarrow \quad 4+b_{2}=3$
<br/>$\Rightarrow \quad b_{2}=-1$
<br/>and $\quad 3 a_{2}+2 b_{2}+c_{2}=0$
<br/>$\Rightarrow \quad 3 \times 2+2(-1)+c_{2}=0$
<br/>$\Rightarrow \quad 6-2+c_{2}=0$
<br/>$\Rightarrow \quad c_{2}=-4$
<br/>Now, $A U_{3}=\left(\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 2 &amp; 1 &amp; 0 \\ 3 &amp; 2 &amp; 1\end{array}\right)\left(\begin{array}{l}a_{3} \\ b_{3} \\ c_{3}\end{array}\right)$
<br/>$\Rightarrow\left(\begin{array}{c}a_{3} \\ 2 a_{3}+b_{3} \\ 3 a_{3}+2 b_{3}+c_{3}\end{array}\right)=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right) \quad\left[\because A U_{3}=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)\right]$
<br/>$\therefore \quad a_{3}=2$ and $2 a_{3}+b_{3}=3$
<br/>$\Rightarrow \quad 2(2)+b_{3}=3$
<br/>$\Rightarrow \quad 4+b_{3}=3$
<br/>$\Rightarrow \quad b_{3}=-1$
<br/>and $3 a_{3}+2 b_{3}+c_{3}=1$
<br/>$\Rightarrow \quad 3 \times 2+2(-1)+c_{3}=1$
<br/>$\Rightarrow \quad 6-2+c_{3}=1$
<br/>$\Rightarrow$
<br/>$c_{3}=-3$
<br/>$\therefore \quad U=\left(\begin{array}{ccc}1 &amp; 2 &amp; 2 \\ -2 &amp; -1 &amp; -1 \\ 1 &amp; -4 &amp; -3\end{array}\right)$
<br/>$\Rightarrow|U|=1(3-4)-2(6+1)+2(8+1)$
<br/>$=-1-14+18=3$
<br/>Also, adj $(U)=\left[\begin{array}{ccc}-1 &amp; -7 &amp; 9 \\ -2 &amp; -5 &amp; 6 \\ 0 &amp; -3 &amp; 3\end{array}\right]^{T}=\left[\begin{array}{ccc}-1 &amp; -2 &amp; 0 \\ -7 &amp; -5 &amp; -1 \\ 9 &amp; 6 &amp; 3\end{array}\right]$
<br/>$\therefore$
<br/>$U^{-1}=\frac{1}{|U|}$ adj $(U)$
<br/>$\Rightarrow$
<br/>$$
<br/>U^{-1}=\left[\begin{array}{rrr}
<br/>-\frac{1}{3} &amp; -\frac{2}{3} &amp; 0 \\
<br/>-\frac{7}{3} &amp; -\frac{5}{3} &amp; -1 \\
<br/>3 &amp; 2 &amp; 1
<br/>\end{array}\right]
<br/>$$
<br/>Hence, the sum of all elements of $U^{-1}$ is $0 .$</div>


tags: ['Physics', 'Electromagnetic Waves', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P2.db
title:
for-a-plane-eletromagnetic-wave-the-electric-field-is-given-by-e-90-sin-05-1-0-3-x-15-1-0-11-t-k-v-m-the-corresponding-magnetic-field-b-will-be

slug:
for-a-plane-eletromagnetic-wave-the-electric-field-is-given-by-e-90-sin-05-1-0-3-x-15-1-0-11-t-k-v-m-the-corresponding-magnetic-field-b-will-be-16788

question:
<div class="question">For a plane eletromagnetic wave, the electric field is given by $\overrightarrow{\mathrm{E}}=90 \sin \left(0.5 \times 10^{3} \mathrm{x}+1.5 \times 10^{11} \mathrm{t}\right) \mathrm{k} . \mathrm{v} / \mathrm{m} .$ The corresponding magnetic field $\vec{B}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\vec{B}=3 \times 10^{-7} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} \mathrm{t}\right) \hat{i} \mathrm{~T}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\vec{B}=3 \times 10^{-7} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} \mathrm{t}\right) \hat{j} \mathrm{~T}$</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">C</span> <span class="option-data">$\vec{B}=27 \times 10^{9} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} \mathrm{t}\right) \hat{j} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{\mathrm{B}}=3 \times 10^{-7} \sin \left(0.5 \times 10^{3} \mathrm{x}+1.5 \times 10^{11} \mathrm{t}\right) \hat{\mathrm{k}} \mathrm{T}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{B}=3 \times 10^{-7} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} \mathrm{t}\right) \hat{j} \mathrm{~T}$</span> </div>

answer:
<div class="solution">$E=90 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{k} v / m$<br/>Direction of propagation is $-\hat{i}$<br/>$\&amp; E_{0}=90$<br/>$\frac{\mathrm{E}_{0}}{\mathrm{~B}_{0}}=\mathrm{C} \Rightarrow \mathrm{B}_{0}=\frac{\mathrm{E}_{0}}{\mathrm{C}}=\frac{90}{3 \times 10^{8}} \mathrm{~T}=3 \times 10^{-7} \mathrm{~T}$<br/>$\overrightarrow{\mathrm{E}}$ is along $\hat{\mathrm{k}} \&amp; \overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{B}}$ should be along $-\hat{\mathrm{i}} \Rightarrow \overrightarrow{\mathrm{B}}$ is should be along $\hat{\mathrm{j}}$ Hence equation of magnetic field is<br/>$\mathrm{B}=3 \times 10^{-7} \sin \left(0.5 \times 10^{3} \mathrm{x}+1.5 \times 10^{11} \mathrm{t}\right) \hat{\mathrm{j}} \mathrm{T}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch1_P2.db
title:
for-a-reaction-2-a-b-p-when-concentration-of-b-alone-is-doubled-rate-does-not-change-and-when-concentrations-of-both-a-and-b-is-doubled-rate-increases

slug:
for-a-reaction-2-a-b-p-when-concentration-of-b-alone-is-doubled-rate-does-not-change-and-when-concentrations-of-both-a-and-b-is-doubled-rate-increases-61799

question:
<div class="question">For a reaction $2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{P}$, when concentration of $\mathrm{B}$ alone is doubled, rate does not change and when concentrations of both $\mathrm{A}$ and $\mathrm{B}$ is doubled, rate increases by a factor of 4 . The unit of rate constant is,</div>

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

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

answer:
<div class="solution">Hint :
<br/>Rate law: $\mathrm{r}=\mathrm{k}[\mathrm{A}]^{\alpha}[\mathrm{B}]^{\beta} \quad \mathrm{B}$ follows first order kinetics. $\therefore \beta=1$
<br/>$\frac{r_{2}}{r_{1}}=\frac{[2 A]^{\alpha}[2 B]^{p}}{[A]^{\alpha}[B]^{\beta}}$
<br/>or, $4=2^{a+\beta}$
<br/>or, $\alpha+\beta=2$
<br/>or, $\alpha=1$
<br/>$\therefore$ overall order of reaction $=2$
<br/>Hence unit of rate constant $=\mathrm{L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{S}^{-1}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P2.db
title:
for-a-reversible-chemical-reaction-where-the-forward-process-is-exothermic-which-of-the-following-statements-is-correct

slug:
for-a-reversible-chemical-reaction-where-the-forward-process-is-exothermic-which-of-the-following-statements-is-correct-98166

question:
<div class="question">For a reversible chemical reaction where the forward process is exothermic, which of the following statements is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The backward reaction has higher activation energy than the forward reaction</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6t3iDuN_0S5HM0ioNYijX-eUJB9vQZ8zrbYHbGosfgs.original.fullsize.png"/><br/></div>


title: for-a-spontaneous-process-the-correct-statements-is-are
slug: for-a-spontaneous-process-the-correct-statements-is-are-25517
question: <div class="question">For a spontaneous process, the correct statement(s) is (are)</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\Delta G_{\text {system }}\right)_{T, p}&gt;0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\Delta S_{\text {system }}\right)+\left(\Delta S_{\text {surroundings }}\right)&gt;0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\Delta G_{\text {system }}\right)_{T, p}    &lt; 0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\Delta U_{\text {system }}\right)_{T, v}&gt;0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\left(\Delta S_{\text {system }}\right)+\left(\Delta S_{\text {surroundings }}\right)&gt;0$, $\left(\Delta G_{\text {system }}\right)_{T, p}    &lt; 0$</span> </div>

answer:
<div class="solution">For a process to be spontaneous.
<br/>$\begin{aligned} \Delta S_{\text {system }}+\Delta S_{\text {surrounding }} &amp;&gt;0 \\ &amp;\left(\Delta G_{\text {system }}\right)_{T, p}    &lt; 0 \\ &amp;\left(\Delta U_{\text {system }}\right)_{T, V} &lt; 0 \end{aligned}$.</div>


title: for-a-spontaneous-reaction-at-all-temperatures-which-of-the-following-is-correct
slug: for-a-spontaneous-reaction-at-all-temperatures-which-of-the-following-is-correct-21550
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For a spontaneous reaction at all temperatures which of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\Delta \mathrm{H}$ is negative and $\Delta \mathrm{S}$ is positive</span> </div>

answer:
<div class="solution">For spontancity $\Delta G$ must be negative $\because \Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \Delta \mathrm{S}$
<br/>To be negative at all Temperature, $\Delta \mathrm{H}     &lt; 0, \Delta \mathrm{S}&gt;0$</div>


title: for-a-van-der-waals-gas-the-term-v-2-ab-represents-some
slug: for-a-van-der-waals-gas-the-term-v-2-ab-represents-some-57362
tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">energy</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For a van der Waals' gas, the term $\left(\frac{a b}{V^{2}}\right)$ represents some</div>

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

answer:
<div class="solution">Dimensions of $\left(\frac{a b}{V^{2}}\right)$ are shown below.
<br/>$$
<br/>\begin{aligned}
<br/>\frac{a}{V^{2}} \times b &amp;=\text { pressure } \times \text { volume } \\
<br/>&amp;=\frac{\text { force }}{\text { area }} \times \text { volume } \\
<br/>&amp;=\frac{\mathrm{MLT}^{-2}}{\mathrm{L}^{2}} \times \mathrm{L}^{3}=\mathrm{ML}^{2} \mathrm{T}^{-2}
<br/>\end{aligned}
<br/>$$
<br/>The units of energy (joule) is,
<br/>$$
<br/>1 \mathrm{J}=1 \mathrm{kg} \mathrm{m}^{2} \mathrm{s}^{-2}=\mathrm{ML}^{2} \mathrm{T}^{-2}
<br/>$$
<br/>Therefore, the dimensions of $\left(\frac{a b}{V^{2}}\right)$ is same as that of energy. So, the option (b) is correct.</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.4</span> </div>
source_db: db\MarksBatch1_P2.db
title:
for-air-at-room-temperature-the-atmospheric-pressure-is-10-1-0-5-nm-2-and-density-of-ait-is-12-kgm-3-for-a-tube-of-length-10-m-closed-at-one-end-the-l-1

slug:
for-air-at-room-temperature-the-atmospheric-pressure-is-10-1-0-5-nm-2-and-density-of-ait-is-12-kgm-3-for-a-tube-of-length-10-m-closed-at-one-end-the-l-1-30913

question:
<div class="question">For air at room temperature, the atmospheric pressure is $1.0 \times 10^{5} \mathrm{Nm}^{-2}$ and density of ait is $1.2 \mathrm{kgm}^{-3}$. For a tube of length $1.0 \mathrm{m}$, closed at one end, the lowest frequency $\begin{array}{lllllll}\text { generated is } &amp; 84 &amp; \text { Hz. } &amp; \text { The } &amp; \text { value } &amp; \text { of } \boldsymbol{\gamma}\end{array}$ (ratio of two specific heats) for air is</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">1.5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1.4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The tube is closed at one end. So, it will act like a closed organ pipe. The lowest frequency produced in the tube
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a20.png"/> <br/> <br/>$$
<br/>\begin{array}{l}
<br/>l=\frac{\lambda}{4}, \lambda=4 \\
<br/>l=1 \mathrm{m} \\
<br/>f=84 \mathrm{Hz} \\
<br/>V=n \lambda=84 \times 4
<br/>\end{array}
<br/>$$
<br/>When wave propagate in air, the velocity
<br/>Here,
<br/>$$
<br/>\begin{array}{l}
<br/>v=\sqrt{\frac{\gamma p}{\rho}} \\
<br/>p=1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>\rho &amp;=1.2 \mathrm{kg} / \mathrm{m}^{3} \\
<br/>V^{2} &amp;=\frac{\gamma P}{\rho} \\
<br/>\gamma &amp;=\frac{\mathrm{v}^{2} \cdot \rho}{\mathrm{P}} \\
<br/>&amp;=\frac{(84 \times 4)^{2} \times 1.2}{1.0 \times 10^{5}} \\
<br/>&amp;=\frac{135472.2}{10^{5}}=1.354 \\
<br/>&amp;=1.4
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.4</span> </div>
source_db: db\MarksBatch1_P2.db
title:
for-air-at-room-temperature-the-atmospheric-pressure-is-10-1-0-5-nm-2-and-density-of-ait-is-12-kgm-3-for-a-tube-of-length-10-m-closed-at-one-end-the-l

slug:
for-air-at-room-temperature-the-atmospheric-pressure-is-10-1-0-5-nm-2-and-density-of-ait-is-12-kgm-3-for-a-tube-of-length-10-m-closed-at-one-end-the-l-37447

question:
<div class="question">For air at room temperature, the atmospheric pressure is $1.0 \times 10^{5} \mathrm{Nm}^{-2}$ and density of ait is $1.2 \mathrm{kgm}^{-3}$. For a tube of length $1.0 \mathrm{m}$, closed at one end, the lowest frequency $\begin{array}{lllllll}\text { generated is } &amp; 84 &amp; \text { Hz. } &amp; \text { The } &amp; \text { value } &amp; \text { of } \boldsymbol{\gamma}\end{array}$ (ratio of two specific heats) for air is</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">1.5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1.4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The tube is closed at one end. So, it will act like a closed organ pipe. The lowest frequency produced in the tube
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_phy_a20.png"/> <br/> <br/>$$
<br/>\begin{array}{l}
<br/>l=\frac{\lambda}{4}, \lambda=4 \\
<br/>l=1 \mathrm{m} \\
<br/>f=84 \mathrm{Hz} \\
<br/>V=n \lambda=84 \times 4
<br/>\end{array}
<br/>$$
<br/>When wave propagate in air, the velocity
<br/>Here,
<br/>$$
<br/>\begin{array}{l}
<br/>v=\sqrt{\frac{\gamma p}{\rho}} \\
<br/>p=1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\begin{aligned}
<br/>\rho &amp;=1.2 \mathrm{kg} / \mathrm{m}^{3} \\
<br/>V^{2} &amp;=\frac{\gamma P}{\rho} \\
<br/>\gamma &amp;=\frac{\mathrm{v}^{2} \cdot \rho}{\mathrm{P}} \\
<br/>&amp;=\frac{(84 \times 4)^{2} \times 1.2}{1.0 \times 10^{5}} \\
<br/>&amp;=\frac{135472.2}{10^{5}}=1.354 \\
<br/>&amp;=1.4
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">[0,1]</span> </div>
source_db: db\MarksBatch1_P2.db
title:
for-all-real-values-of-a-0-a-1-a-2-a-3-satisfying-a-0-2-a-1-3-a-z-4-a-3-0-the-equation-a-0-a-1-x-a-2-x-2-a-3-x-3-0-has-a-real-root-in-the-interval

slug:
for-all-real-values-of-a-0-a-1-a-2-a-3-satisfying-a-0-2-a-1-3-a-z-4-a-3-0-the-equation-a-0-a-1-x-a-2-x-2-a-3-x-3-0-has-a-real-root-in-the-interval-49393

question:
<div class="question">For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{z}}{3}+\frac{a_{3}}{4}=0,$ the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval</div>

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

answer:
<div class="solution">Let $f(x)=\frac{a_{3} x^{4}}{4}+\frac{a_{2} x^{3}}{3}+\frac{a_{1} x^{2}}{2}+a_{0} x$
<br/>$f(0)=0, f(1)=\frac{a_{3}}{4}+\frac{a_{2}}{3}+\frac{a_{1}}{2}+a_{0}=0$
<br/>$\Rightarrow f(0)=f(1)$
<br/>$\Rightarrow f^{\prime}(x)=0$ has atleast one real root in |0,1| $\therefore f^{\prime}(x)=a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0}$
<br/>Hence, $a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0}$ must has a real root in the interval $[0,1 \mid$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch1_P2.db
title:
for-an-ideal-gas-with-initial-pressure-and-volume-p-i-and-v-i-respectively-a-reversible-isothermal-expansion-happens-when-its-volume-becomes-v-0-then-

slug:
for-an-ideal-gas-with-initial-pressure-and-volume-p-i-and-v-i-respectively-a-reversible-isothermal-expansion-happens-when-its-volume-becomes-v-0-then-46004

question:
<div class="question">For an ideal gas with initial pressure and volume $p_{i}$ and $V_{i}$ respectively, a reversible isothermal expansion happens, when its volume becomes $V_{0}$. Then, it is compressed to its original volume $V_{i}$ by a reversible adiabatic process. If the final pressure is $p_{f}$ then which of the following statement(s) is/are true?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p_{f}&gt;p_{i}$</span> </div>

answer:
<div class="solution">In $p-V$ diagram, the slope of an adiabatic curve at any point in steeper than that of isothermal curve at that point.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a22.png"/> <br/>Thus, $p_{f}&gt;p_{i}$</div>


title: for-any-complex-number-z-the-minimum-value-of-z-z-1-is
slug: for-any-complex-number-z-the-minimum-value-of-z-z-1-is-96461
question: <div class="question">For any complex number $z$, the minimum value of $|z|+|z-1|$ is</div>
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1</span> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>\text { Hints }: 1=|z-(z-1)| \\<br/>\qquad 1 \leq|z|+|z-1|<br/>\end{gathered}<br/>$$</div>


title: for-any-nonzero-complex-number-z-the-minimum-value-of-z-z-1-is
slug: for-any-nonzero-complex-number-z-the-minimum-value-of-z-z-1-is-92797
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For any non-zero complex number $z$, the minimum value of $|z|+|z-1|$ is</div>

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

answer:
<div class="solution">$|z|+|z-1| a$ is complex number
<br/>$$
<br/>\begin{array}{c}
<br/>|z|+|-(1-z)| \quad|a|=|-a| \\
<br/>|z|+|1-z| \quad\{\because|a|=|-a|\} \\
<br/>|z|+|1-z| \geq|z+1-z| \\
<br/>|z|+|1-z| \geq|1| \\
<br/>|z|+|1-z| \geq 1 \\
<br/>|z|+|-(z-1)| \geq 1 \\
<br/>|z|+|z-1| \geq 1
<br/>\end{array}
<br/>$$
<br/>Minimum value of $|z|+|z-1|=1$</div>


title: for-any-two-real-numbers-a-and-b-we-define-a-r-b-if-and-only-if-sin-2-a-cos-2-b-1-the-relation-r-is
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
slug:
for-any-two-real-numbers-a-and-b-we-define-a-r-b-if-and-only-if-sin-2-a-cos-2-b-1-the-relation-r-is-22419

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

answer:
<div class="solution">Let the given relation defined as<br/>$R=\left\{(a, b) \mid \sin ^{2} a+\cos ^{2} b=1\right\}$<br/>For reflexive, $\sin ^{2} a+\cos ^{2} a=1$<br/>$\left(\because \sin ^{2} \theta+\cos ^{2} \theta=1, \forall \theta \in R\right)$<br/>$\Rightarrow { }_{a} R_{a} \Rightarrow(a, a) \in R$<br/>$\Rightarrow R$ is reflexive.<br/>For symmetric, $\sin ^{2} a+\cos ^{2} b=1$<br/>$\Rightarrow 1-\cos ^{2} a+1-\sin ^{2} b=1$<br/>$\Rightarrow \sin ^{2} b+\cos ^{2} a=1$<br/>$\Rightarrow$ bRa<br/>Hence, $R$ is symmetric.<br/>For transitive<br/>Let $a R b, b R c$ $\Rightarrow \sin ^{2} a+\cos ^{2} b=1$<br/>and $\sin ^{2} b+\cos ^{2} c=1$<br/>On adding Eqs. (0) and (ii), we get $\sin ^{2} a+\left(\sin ^{2} b+\cos ^{2} b\right)+\cos ^{2} c=2$<br/>$\Rightarrow \sin ^{2} a+\cos ^{2} c+1=2$<br/>$\Rightarrow \sin ^{2} a+\cos ^{2} c=1$<br/>Hence, $A$ is transitive also. Therefore, relation $A$ is an equivalence relation.</div>


title: for-any-two-real-numbers-and-we-define-r-if-and-only-if-sec-2-tan-2-1-the-relation-r-is
slug: for-any-two-real-numbers-and-we-define-r-if-and-only-if-sec-2-tan-2-1-the-relation-r-is-23052
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For any two real numbers $\theta$ and $\phi$ we define $\theta$. R $\phi$ if and only if $\sec ^{2} \theta-\tan ^{2} \phi=1$. The relation $R$ is</div>

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

answer:
<div class="solution">Given relation is defined as $\theta R \phi$ such that $\sec ^{2} \theta-\tan ^{2} \phi=1$<br/>For Reflexive<br/>When $\theta R \theta$<br/>$\sec ^{2} \theta-\tan ^{2} \theta=1$<br/>$\Rightarrow 1=1,$ which is true.<br/>Thus, it is reflexive.<br/>For Symmetric When $\theta R \phi$<br/>$\begin{array}{l}<br/>\sec ^{2} \theta-\tan ^{2} \phi=1 \\<br/>\Rightarrow\left(1+\tan ^{2} \theta\right)-\left(\sec ^{2} \phi-1\right)=1 \\<br/>\Rightarrow 2+\tan ^{2} \theta-\sec ^{2} \phi=1 \\<br/>\Rightarrow \sec ^{2} \phi-\tan ^{2} \theta=1<br/>\end{array}$<br/>$\Rightarrow \phi F \theta$<br/>Thus, it is symmetric.<br/>For Transitive<br/>When $\theta R \phi$ and $\phi$ Fiy, then<br/>$\sec ^{2} \theta-\tan ^{2} \phi=1$<br/>and $\sec ^{2} \phi-\tan ^{2} \psi=1$<br/>Now, $\theta$ R $\psi$<br/>Then. $\sec ^{2} \theta-\tan ^{2} \psi=1$<br/>$\Rightarrow \sec ^{2} \theta-\tan ^{2} \psi+1=1+1$<br/>$\Rightarrow \sec ^{2} \theta-\tan ^{2} \psi+\sec ^{2} \phi-\tan ^{2} \phi=1+1$<br/>$\Rightarrow \theta R \phi$ and $\phi R \psi$<br/>Thus, it is transitive. Hence, it is an equivalence relation.</div>


title: for-any-two-sets-a-and-b-a-a-b-equals
slug: for-any-two-sets-a-and-b-a-a-b-equals-98387
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For any two sets $\mathrm{A}$ and $\mathrm{B}, \mathrm{A}-(\mathrm{A}-\mathrm{B})$ equals</div>

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

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

answer:
<div class="solution">Hints: $A-(A-B)=A-\left(A \cap B^c\right)=A \cap\left(A \cap B^c\right)^c=A \cap\left(A^c \cup B\right)=\left(A \cap A^c\right) \cup(A \cap B)=A \cap B$</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch1_P2.db
title:
for-any-vector-x-where-i-j-k-have-their-usual-meanings-the-value-of-x-i-2-x-j-2-x-k-2-where-i-j-k-have-their-usual-meanings-is-equal-to

slug:
for-any-vector-x-where-i-j-k-have-their-usual-meanings-the-value-of-x-i-2-x-j-2-x-k-2-where-i-j-k-have-their-usual-meanings-is-equal-to-14983

question:
<div class="question">For any vector $\mathbf{x},$ where $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$ have their usual meanings the value of $(\mathbf{x} \times \hat{\mathbf{i}})^{2}+(\mathbf{x} \times \mathbf{j})^{2}+(\mathbf{x} \times \mathbf{k})^{2}$ where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ have their usual meanings, is equal to</div>

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

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

answer:
<div class="solution">Let $\mathbf{x}=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}$
<br/>Then, $\mathbf{x} \times \mathbf{i}=-\beta \hat{\mathbf{k}}+\gamma \hat{\mathbf{j}}$
<br/>$\begin{aligned} \mathbf{x} \times \hat{\mathbf{j}} &amp;=\hat{\mathbf{k}}-\gamma \hat{\mathbf{i}} \\ \mathbf{x} \times \mathbf{k} &amp;=-\alpha \hat{\mathbf{j}}+\beta \hat{\mathbf{i}} \end{aligned}$
<br/>$\begin{aligned} \text {Now, }(\mathbf{x} \times \mathbf{i})^{2} &amp;=(\mathbf{x} \times \mathbf{i}) \cdot(\mathbf{x} \times \hat{\mathbf{i}}) \\ &amp;=(-\beta \hat{\mathbf{k}}+\gamma \hat{\mathbf{j}}) \cdot(-\beta \hat{\mathbf{k}}+\gamma \hat{\mathbf{j}}) \\ &amp;=\beta^{2}+\gamma^{2} \end{aligned}$
<br/>Similarly, $(\mathbf{x} \times \mathbf{j})^{2}=\alpha^{2}+\gamma^{2}$
<br/>and $(\mathbf{x} \times \mathbf{k})^{2}=\alpha^{2}+\beta^{2}$
<br/>$\therefore(\mathbf{x} \times \mathbf{i})^{2}+(\mathbf{x} \times \mathbf{j})^{2}+(\mathbf{x} \times \mathbf{\mathbf { k }})^{2}$
<br/>$=\beta^{2}+\gamma^{2}+\alpha^{2}+\gamma^{2}+\alpha^{2}+\beta^{2}$
<br/>$=2\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)=2|\mathbf{x}|^{2}$</div>


title: for-bcl-3-alcl-3-and-gacl-3-the-increasing-order-of-ionic-character-is
slug: for-bcl-3-alcl-3-and-gacl-3-the-increasing-order-of-ionic-character-is-28432
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For $\mathrm{BCl}_{3}, \mathrm{AlCl}_{3}$ and $\mathrm{GaCl}_{3}$ the increasing order of ionic character is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{BCl}_{3}     &lt; \mathrm{GaCl}_{3} &lt; \mathrm{AlCl}_{3}$</span> </div>

answer:
<div class="solution">Since, ionic character is inversely proportional to polarising power of cation therefore correct order is
<br/>$$
<br/>\mathrm{AICl}_{3}&gt;\mathrm{GaCl}_{3}&gt;\mathrm{BCl}_{3}
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch1_P2.db
title:
for-different-values-of-the-locus-of-the-point-of-intersection-of-the-two-straight-lines-3-x-y-4-3-0-and-3-x-y-4-3-0-is

slug:
for-different-values-of-the-locus-of-the-point-of-intersection-of-the-two-straight-lines-3-x-y-4-3-0-and-3-x-y-4-3-0-is-69232

question:
<div class="question">For different values of $\alpha$, the locus of the point of intersection of the two straight lines $\sqrt{3} x-y-4 \sqrt{3} \alpha=0$ and $\sqrt{3} \alpha x+\alpha y-4 \sqrt{3}=0$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">a hyperbola with eccentricity 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">an ellipse with eccentricity $\sqrt{\frac{2}{3}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a hyperbola with eccentricity $\sqrt{\frac{19}{16}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">an ellipse with eccentricity $\frac{3}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a hyperbola with eccentricity 2</span> </div>

answer:
<div class="solution">Hints : $\sqrt{3} x-y=4 \sqrt{3} \alpha$<br/>(1) $\sqrt{3} x+y=\frac{4 \sqrt{3}}{\alpha}$<br/>(1) $x(2) \Rightarrow 3 x^2-y^2=48 \Rightarrow \frac{x^2}{16}-\frac{y^2}{48}=1$<br/>$$<br/>\mathrm{e}=\sqrt{\frac{48+16}{16}}=2<br/>$$</div>


title: for-each-n-n-2-3-n-1-is-divisible-by-where-n-is-a-set-of-natural-numbers
slug: for-each-n-n-2-3-n-1-is-divisible-by-where-n-is-a-set-of-natural-numbers-34035
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: db\MarksBatch1_P2.db
question:
<div class="question">For each $n \in N, 2^{3 n}-1$ is divisible by<br/>where $\mathrm{N}$ is a set of natural numbers</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : } 2^{3 \mathrm{n}}=(8)^{\mathrm{n}}=(1+7)^{\mathrm{n}}=1+{ }^{\mathrm{n}} \mathrm{C}_1 7+{ }^{\mathrm{n}} \mathrm{C}_2 7^2 \ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} 7^{\mathrm{n}} \\<br/>&amp; 2^{3 \mathrm{n}}-1=7\left[{ }^{\mathrm{n}} \mathrm{C}_1+{ }^{\mathrm{n}} \mathrm{C}_2 7+\ldots\right]<br/>\end{aligned}<br/>$$</div>


title: if-1-log-9left31x2right-log-3left43mathrmx1right-are-in-ap-then-x-equals
slug: if-1-log-9left31x2right-log-3left43mathrmx1right-are-in-ap-then-x-equals-10635
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(1, \log _9\left(3^{1-x}+2\right), \log _3\left(4.3^{\mathrm{x}}-1\right)\) are in A.P, then \(x\) equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\log _3 4\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(1-\log _3 4\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1-\log _4 3\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\log _4 3\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(1-\log _3 4\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint: } 2 \log _9\left(3^{1-x}+2\right)=\log _3\left(4.3^x-1\right)+1 \\<br/>&amp; \Rightarrow \log _3\left(3^{1-x}+2\right)=\log _3 3\left(4.3^x-1\right) \\<br/>&amp; \Rightarrow 3^{1-x}+2=3\left(4.3^x-1\right) \\<br/>&amp; \Rightarrow \frac{3}{3^x}+2=4.3^{x+1}-3 \\<br/>&amp; \Rightarrow \frac{3}{3^x}+2=12 \cdot 3^x-3 \\<br/>&amp; \Rightarrow 12 \cdot\left(3^x\right)^2-5\left(3^x\right)-3=0 \\<br/>&amp; \Rightarrow\left(4\left(3^x\right)-3\right)\left(3\left(3^x\right)+1\right)=0 \\<br/>&amp; \because 3^x &gt; 0 \quad \therefore 4\left(3^x\right)-3=0 \\<br/>&amp; \therefore 3\left(3^x\right)+1 \neq 0 \quad \Rightarrow 3^x=\frac{3}{4}<br/>\end{aligned}\)<br/>\(\begin{aligned} &amp; \Rightarrow x=\log _3 \frac{3}{4}=\log _3 3-\log _3 4 \\ &amp; \Rightarrow \mathrm{x}=1-\log _3 4\end{aligned}\)</div>


title: if-1-x-x-2-n-a-0-a-1-x-a-2-n-x-2-n-then-the-value-of-a-0-a-2-a-4-a-2-n-is
slug: if-1-x-x-2-n-a-0-a-1-x-a-2-n-x-2-n-then-the-value-of-a-0-a-2-a-4-a-2-n-is-95965
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\left(1-x+x^2\right)^n=a_0+a_1 x+\ldots . .+a_{2 n} x^{2 n}$, then the value of $a_0+a_2+a_4+\ldots \ldots .+a_{2 n}$ is</div>

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

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

answer:
<div class="solution">Hints: $x=1$<br/>$$<br/>\begin{gathered}<br/>1=a_0+a_1+a_2+a_3+\ldots \ldots \ldots+a_{2 n} \\<br/>x=-1,3^{\mathrm{n}}=a_0-a_1+a_2-a_3+\ldots \ldots \ldots \ldots+a_{2 n} \\<br/>1+3^{\mathrm{n}}=2\left[a_0+a_2+a_4+\ldots \ldots \ldots \ldots+a_{2 \mathrm{n}}\right] \\<br/>\Rightarrow a_0+a_2+a_4+\ldots \ldots \ldots \ldots+a_{2 \mathrm{n}}=\frac{1+3^{\mathrm{n}}}{2}<br/>\end{gathered}<br/>$$</div>


title: if-2-2-3-then-the-value-of-4-cos-4-sin-2-2-4-cot-cos-2-4-2-is
slug: if-2-2-3-then-the-value-of-4-cos-4-sin-2-2-4-cot-cos-2-4-2-is-14593
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\theta \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$ then the value of $\sqrt{4 \cos ^{4} \theta+\sin ^{2} 2 \theta}+4 \cot \theta \cos ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)$ is</div>

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

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

answer:
<div class="solution">$\sqrt{4 \cos ^{4} \theta+\sin ^{2} 2 \theta}+4 \cot \theta \cos ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)$
<br/>$=\sqrt{4 \cos ^{4} \theta+(2 \sin \theta \cos \theta)^{2}} +2 \cot \theta\left[2 \cos ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)\right]$
<br/>$=\sqrt{4 \cos ^{4} \theta+4 \sin ^{2} \theta \cos ^{2} \theta} +2 \cot \theta\left[1+\cos \left(\frac{\pi}{2}-\theta\right)\right]$
<br/>$=\sqrt{4 \cos ^{2} \theta\left(\cos ^{2} \theta+\sin ^{2} \theta\right)}+2 \cot \theta(1+\sin \theta)$
<br/>$=|2 \cos \theta|+2 \cot \theta+2 \cos \theta$
<br/>$=-2 \cos \theta+2 \cot \theta+2 \cos \theta$
<br/>$\quad\left[\right.$ for $\left.\theta \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right),|\cos \theta|=-\cos \theta\right]$
<br/>$=2 \cot \theta$</div>


title: if-2-2-x-2-x-d-x-a-2-2-x-c-then-a-is-equal-to
slug: if-2-2-x-2-x-d-x-a-2-2-x-c-then-a-is-equal-to-33768
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\int 2^{2^{x}} \cdot 2^{x} d x=A \cdot 2^{2^{x}}+C,$ then $A$ is equal to</div>

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

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

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { (d) Let } I=\int 2^{2^{x}} \cdot 2^{x} d x\\
<br/>&amp;\begin{array}{l}
<br/>\text { Put } \quad t=2^{x} \\
<br/>\Rightarrow \quad d t=2^{x} \log 2 d x \\
<br/>\therefore \quad I=\int \frac{2^{t}}{\log 2} d t \\
<br/>=\frac{2^{t}}{(\log 2)^{2}}+C=\frac{2^{2^{x}}}{(\log 2)^{2}}+C \\
<br/>\therefore \quad A=\frac{1}{(\log 2)^{2}}
<br/>\end{array}
<br/>\end{aligned}
<br/>$$</div>


title: if-2-3-i-2-3-50-3-25-x-i-y-where-x-and-y-are-real-then-the-ordered-pair-x-y-is
slug: if-2-3-i-2-3-50-3-25-x-i-y-where-x-and-y-are-real-then-the-ordered-pair-x-y-is-42613
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)$ where $x$ and $y$
<br/>are real, then the ordered pair $(x, y)$ is</div>

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

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

answer:
<div class="solution">Let $z=\frac{3}{2}+i \frac{\sqrt{3}}{2}$
<br/>$\therefore$
<br/>$r=\sqrt{\frac{9}{4}+\frac{3}{4}}=\sqrt{\frac{12}{4}}=\sqrt{3}$
<br/>$\left.\begin{array}{rl}\theta=\tan ^{-1}\left(\frac{\sqrt{3}}{2}\right. \\ \frac{3}{2}\end{array}\right)=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}$
<br/>$\frac{3}{2}+\frac{i \sqrt{3}}{2}=\sqrt{3} e^{\frac{i \pi}{6}}$
<br/>$\therefore$
<br/>$\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=\left(\sqrt{3} e^{\frac{i \pi}{6}}\right)^{50}$
<br/>$=(\sqrt{3})^{50}\left(e^{\frac{i \pi}{6}}\right)^{50}=3^{25} e^{\frac{50 \pi}{6}}$
<br/>$\begin{aligned} \Rightarrow\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)^{50} &amp;=3^{25} e^{\frac{i 25 \pi}{3}} \\ &amp;=3^{25}\left(\cos \frac{25 \pi}{3}+i \sin \frac{25 \pi}{3}\right) \\ &amp;=3^{25}(\cos 1500+i \sin 1500) \end{aligned}$
<br/>$=3^{25}[\cos (360 \times 4+60)+i \sin (360$
<br/>$\times 4+60)]$
<br/>$=3^{25}(\cos 60+i \sin 60)$
<br/>$\Rightarrow\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}\left(\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)$
<br/>According to question, $\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)$
<br/>$3^{25}\left(\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)=3^{25}(x+6 y)$
<br/>which is true only when $x=\frac{1}{2}, y=\frac{\sqrt{3}}{2}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">45</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-2-i-and-5-2-i-are-the-roots-of-the-equation-x-2-a-x-b-x-2-c-x-d-0-where-a-b-c-and-d-are-real-constants-then-product-of-all-the-roots-of-the-equatio

slug:
if-2-i-and-5-2-i-are-the-roots-of-the-equation-x-2-a-x-b-x-2-c-x-d-0-where-a-b-c-and-d-are-real-constants-then-product-of-all-the-roots-of-the-equatio-57394

question:
<div class="question">If $(2+i)$ and $(\sqrt{5}-2 i)$ are the roots of the equation $\left(x^{2}+a x+b\right)\left(x^{2}+c x+d\right)=0$ where $a, b, c$ and $d$ are real constants, then product of all the roots of the equation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">40</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$9 \sqrt{5}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">45</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">35</span> </li> </ul>

answer:
<div class="solution">If one root of a quadratic equation is of the form
<br/>$a+i b,$ then other root will be $a-i b$ So, all the roots are $2 \pm 1, \sqrt{5} \pm 2 i$
<br/>$\therefore$ Product of all the roots $=(2+i)(2-i)(\sqrt{5}+2 i)(\sqrt{5}-2 i)$
<br/>$=(4+1)(5+4)=5 \times 9=45$</div>


title: if-2-lo-g-x-1-lo-g-x-2-1-lo-g-2-then-x
slug: if-2-lo-g-x-1-lo-g-x-2-1-lo-g-2-then-x-50149
question: <div class="question">If $2 \log (x+1)-\log \left(x^{2}-1\right)=\log 2$, then $x=$</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">only 3</span> </div>
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">only 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">$-1$ and 3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">only -1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 3</span> </li> </ul>

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


title: if-2-r-n-then-n-c-r-2-n-c-r-1-n-c-r-2-is-equal-to
slug: if-2-r-n-then-n-c-r-2-n-c-r-1-n-c-r-2-is-equal-to-62511
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">$$
<br/>\text { If }(2 \leq r \leq n), \text { then }{ }^{n} C_{r}+2 \cdot{ }^{n} C_{r+1}+{ }^{n} C_{r+2} \text { is }
<br/>$$
<br/>equal to</div>

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

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

answer:
<div class="solution">${ }^{n} C_{r}+2 \cdot{ }^{n} C_{r+1}+{ }^{n} C_{r+2}$
<br/>$={ }^{n} C_{r}+{ }^{n} C_{r+1}+{ }^{n} C_{r+1}+{ }^{n} C_{r+2}$
<br/>$={ }^{n+1} C_{r+1}+{ }^{n+1} C_{r+2}$
<br/>$\quad\left(\because{ }^{n} C_{r}+{ }^{n} C_{r+1}={ }^{n+1} C_{r+1}\right)$
<br/>$={ }^{n+2} C_{r+2}$</div>


title: if-3-c-o-s-a-4-c-o-s-b-5-1-2-a-0-2-b-0-then-value-of-2-sin-a-4-sin-b-is
slug: if-3-c-o-s-a-4-c-o-s-b-5-1-2-a-0-2-b-0-then-value-of-2-sin-a-4-sin-b-is-63305
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\frac{\cos A}{3}=\frac{\cos B}{4}=\frac{1}{5},-\frac{\pi}{2} &lt; A &lt; 0,-\frac{\pi}{2} &lt; B &lt; 0$ then value of $2 \sin A+4 \sin B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-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">0</span> </li> </ul>

answer:
<div class="solution">Hints : $\cos A=\frac{3}{5} \quad \sin A=-\frac{4}{5}$<br/>$$<br/>\begin{aligned}<br/>&amp; \cos B=\frac{4}{5} \quad \sin B=-\frac{3}{5} \\<br/>&amp; =2\left(-\frac{4}{5}\right)+4\left(-\frac{3}{5}\right)=-\frac{20}{5}=-4<br/>\end{aligned}<br/>$$</div>


title: if-4-a29-b2c212-a-b0-then-the-family-of-straight-lines-a-xb-yc0-is-concurrent-at
slug: if-4-a29-b2c212-a-b0-then-the-family-of-straight-lines-a-xb-yc0-is-concurrent-at-42687
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(4 a^2+9 b^2-c^2+12 a b=0\), then the family of straight lines \(a x+b y+c=0\) is concurrent at</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((2,3)\) or \((-2,-3)\)</span> </div>

answer:
<div class="solution">Hint : \(\because 2 a+3 b-c=0\) or \(2 a+3 b+c=0\)<br/>\(\begin{aligned}<br/>&amp; \Rightarrow c= \pm(2 a+3 b) \\<br/>&amp; \because \quad a x+b y+c=0 \\<br/>&amp; \Rightarrow \quad a x+b y \pm(2 a+3 b)=0 \\<br/>&amp; \Rightarrow \quad a(x \pm 2)+b(y \pm 3)=0 \text { (family of lines) } \\<br/>&amp; \Rightarrow \quad(-2,-3) \quad \text { or }(2,3)<br/>\end{aligned}\)</div>


title: if-5-c-r-1-6-c-r-1-4-c-r-1-then-the-value-of-r-is
slug: if-5-c-r-1-6-c-r-1-4-c-r-1-then-the-value-of-r-is-15160
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\frac{1}{{ }^{5} C_{r}}+\frac{1}{{ }^{6} C_{r}}=\frac{1}{{ }^{4} C_{r}},$ then the value of $r$ is</div>

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

answer:
<div class="solution">Given, $\frac{1}{5 C_{r}} +\frac{1}{6 C_{r}} =\frac{1}{4 C_{r}}$
<br/>$\Rightarrow \quad \frac{r !(5-r) !}{5 !}+\frac{r !(6-r) !}{6 !}=\frac{r !(4-r) !}{4 !}$
<br/>$\Rightarrow \quad \frac{(5-r) !}{5}+\frac{(6-r) !}{30}=(4-r) !$
<br/>$\Rightarrow \quad \frac{(5-r)}{5}+\frac{(6-r)(5-r)}{30}=1$
<br/>$\Rightarrow \quad 30-6 r+30-11r+r^{2}=30$
<br/>$\Rightarrow \quad r^{2}-17 r+30=0$
<br/>$\Rightarrow \quad r^{2}-15 r-2 r+30=0$
<br/>$\Rightarrow \quad r(r-15)-2(r-15)=0$
<br/>$\Rightarrow \quad(r-2)(r-15)=0$
<br/>$\therefore \quad r=15,2$
<br/>Hence, the value of $r$ is 2 .</div>


title: if-5-cos-2-2-cos-2-2-1-0-when-0-then-the-values-of-are
slug: if-5-cos-2-2-cos-2-2-1-0-when-0-then-the-values-of-are-84718
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $5 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1=0$, when $(0 &lt; \theta &lt; \pi)$, then the values of $\theta$ are :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{3} \pm \pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{3}, \cos ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\cos ^{-1}\left(\frac{3}{5}\right) \pm \pi$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{3}, \pi-\cos ^{-1}\left(\frac{3}{5}\right)$</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 answer is:
<span class="option-value">$\frac{\pi}{3}, \pi-\cos ^{-1}\left(\frac{3}{5}\right)$</span> </div>

answer:
<div class="solution">Hints : $5 \cos 2 \theta+1+\cos \theta+1=0$<br/>$$<br/>\begin{aligned}<br/>&amp; 5\left(2 \cos ^2 \theta-1\right)+\cos \theta+2=0 \\<br/>&amp; 10 \cos ^2 \theta+\cos \theta-3=0 \\<br/>&amp; (5 \cos \theta+3)(2 \cos \theta-1)=0 \\<br/>&amp; \cos \theta=\frac{1}{2} \\<br/>&amp; \theta=\frac{\pi}{3}<br/>\end{aligned}<br/>$$<br/>$$<br/>\mid \begin{aligned}<br/>&amp; \cos \theta=-\frac{3}{5} \\<br/>&amp; \theta=\cos ^{-1}\left(-\frac{3}{5}\right) \\<br/>&amp; =\pi-\cos ^{-1}\left(\frac{3}{5}\right)<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
if-5-distinct-balls-are-placed-at-random-into-5-cells-then-the-probability-that-exactly-one-cell-remains-empty-is

slug:
if-5-distinct-balls-are-placed-at-random-into-5-cells-then-the-probability-that-exactly-one-cell-remains-empty-is-53670

question:
<div class="question">If 5 distinct balls are placed at random into 5 cells, then the probability that exactly one cell remains empty, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$48 / 125$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$12 / 125$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 / 125$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 125$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$48 / 125$</span> </div>

answer:
<div class="solution">Clearly. 5 distinct balls can be placed into 5 olls in $5^{5}$ ways. Now, the number of ways of selecting one cell ${ }^{5} C_{1}$
<br/>Let the selected cell be empty, Now, we are left with 5 balls and 4 cells. Now, number of ways of placing 5 distinct balls into 4 cells such that each cell have atleast one $=4^{5}-{ }^{4} C_{1}(4-1)^{5}+{ }^{4} C_{2}(4-2)^{5}-{ }^{4} C_{3}(4-3)^{6}$
<br/>$=1024-4 \times 3^{5}+6 \times 2^{5}-4$
<br/>$=1024-972+192-4=1216-976$
<br/>$=240$
<br/>Thus, the number of ways of placing 5 distinct balls such that exactly one cell remains empty $={ }^{5} C_{1} \times 240$
<br/>$\therefore$ Required probability $=\frac{{ }^{5} C_{1} \times 240}{5^{5}}=\frac{240}{625}=\frac{48}{125}$</div>


title: if-642736-are-the-pth-q-th-and-r-th-terms-of-a-gp-then-p-2-q-is-equal-to
slug: if-642736-are-the-pth-q-th-and-r-th-terms-of-a-gp-then-p-2-q-is-equal-to-90418
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3 R$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If 64,27,36 are the Pth $Q$ th and $R$ th terms of a GP, then $P+2 Q$ is equal to</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="correct"> <span class="option-label">C</span> <span class="option-data">$3 R$</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">$4 R$</span> </li> </ul>

answer:
<div class="solution">Let a be the first term and \(r\) be the common ration of a GP. \(\therefore P^{\text {th }}, Q^{\text {th }}\) and \(R^{\text {th }}\) terms of a GP are respectively ar \({ }^{\mathrm{P}-1}\), ar \(\mathrm{Q}^{-1}\) and \({a r}^{\mathrm{R}-1}\).
<br/>According to question, \(a r^{P-1}=64 \ldots . .\) (i) 
<br/>\(a r^{Q-1}=27 \ldots . .\) (ii) 
<br/>\(a r^{R-1}=36 \ldots . .\) (iii)
<br/>Dividing Eq. (i) by Eq. (ii), we get 
<br/>\(\mathrm{r}^{\mathrm{P}-\mathrm{Q}}=\left(\frac{4}{3}\right)^{3} \ldots \ldots\) (iv)
<br/>Dividing Eq. (ii) by Eq. (iii), we get 
<br/>\(\begin{array}{l}
<br/>\mathrm{r}^{\mathrm{Q}-\mathrm{R}}=\frac{3}{4} \\
<br/>\Rightarrow \mathrm{r}^{3 \mathrm{Q}-3 \mathrm{R}}=\left(\frac{3}{4}\right)^{3}
<br/>\end{array}\)
<br/>Multiplying Eq. (iv) and Eq. (v), we get
<br/>\(\begin{array}{l}
<br/>\mathrm{r}^{\mathrm{P}-\mathrm{Q}} \times \mathrm{r}^{3 \mathrm{Q}-3 \mathrm{R}}=1 \\
<br/>\Rightarrow \mathrm{r}^{\mathrm{P}-\mathrm{Q}+3 \mathrm{Q}-3 \mathrm{R}}=1 \\
<br/>\Rightarrow \mathrm{r}^{\mathrm{P}+2 \mathrm{Q}-3 \mathrm{R}}=\mathrm{r}^{0} \\
<br/>\Rightarrow \mathrm{P}+2 \mathrm{Q}-3 \mathrm{R}=0 \\
<br/>\Rightarrow \mathrm{P}+2 \mathrm{Q}=3 \mathrm{R}
<br/>\end{array}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
if-a-0-b-0-then-the-maximum-area-of-the-parallelogram-whose-three-vertices-are-o-0-0-a-a-cos-b-sin-and-b-a-cos-b-sin-is

slug:
if-a-0-b-0-then-the-maximum-area-of-the-parallelogram-whose-three-vertices-are-o-0-0-a-a-cos-b-sin-and-b-a-cos-b-sin-is-86225

question:
<div class="question">If $a&gt;0, b&gt;0$ then the maximum area of the parallelogram whose three vertices are $O(0,0), A(a \cos \theta, b \sin \theta)$ and $\mathrm{B}(\mathrm{a} \cos \theta,-\mathrm{b} \sin \theta)$ is</div>

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

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

answer:
<div class="solution">Area $\mathrm{OABC}=2 \times$ area $\triangle \mathrm{OAB}$
<br/>$\begin{array}{l}
<br/>=2 \times\left(\frac{1}{2} \times \mathrm{OM} \times \mathrm{AB}\right) \\
<br/>=2 \times \frac{1}{2}|\mathrm{a} \cos \theta \times 2 \mathrm{~b} \sin \theta|=2 \times \mathrm{ab}|\sin \theta \cos \theta| \\
<br/>=\mathrm{ab}|\sin 2 \theta|
<br/>\end{array}$
<br/>$\therefore$ Maximum is ab when $\theta=\pi / 4$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/73FBTtrTPxi888I5kAlL5eo6NlfoBoWIg1O2jJuh4gg.original.fullsize.png"/></div>


title: if-a-1-0-1-i-and-a-2018-a-c-b-d-then-a-d-equals
slug: if-a-1-0-1-i-and-a-2018-a-c-b-d-then-a-d-equals-29141
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $A=\left(\begin{array}{ll}1 &amp; 1 \\ 0 &amp; i\end{array}\right)$ and $A^{2018}=\left(\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right)$, then $(a+d)$ equals</div>

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

answer:
<div class="solution">$A^2=\left(\begin{array}{cc}1 &amp; 1+i \\ 0 &amp; -1\end{array}\right), A^3=\left(\begin{array}{cc}1 &amp; i \\ 0 &amp; -i\end{array}\right), A^4=I$<br/>$\begin{aligned}<br/>&amp;\therefore A^{2018}=A^{2016} \times A^2=A^2=\left(\begin{array}{ll}<br/>1 &amp; 1+i \\<br/>0 &amp; -1<br/>\end{array}\right) \\<br/>&amp;\therefore a+d=1+(-1)=0<br/>\end{aligned}$</div>


title: if-a-1-4-2-1-then-a-1-is
slug: if-a-1-4-2-1-then-a-1-is-17559
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{rr}1 &amp; 2 \\ -4 &amp; -1\end{array}\right]$ then $\mathrm{A}^{-1}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{7}\left[\begin{array}{rr}-1 &amp; -2 \\ 4 &amp; 1\end{array}\right]$</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">$\frac{1}{7}\left[\begin{array}{rr}1 &amp; 2 \\ -4 &amp; -1\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{7}\left[\begin{array}{rr}-1 &amp; -2 \\ 4 &amp; -1\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Does not exist</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{7}\left[\begin{array}{rr}-1 &amp; -2 \\ 4 &amp; 1\end{array}\right]$</span> </div>

answer:
<div class="solution">Hints: $|\mathrm{A}|=-1+8=7$<br/>$$<br/>\begin{aligned}<br/>&amp; \operatorname{adj}(\mathrm{A})=\left[\begin{array}{rr}<br/>+(-1) &amp; -(2) \\<br/>-(-4) &amp; +(1)<br/>\end{array}\right]=\left[\begin{array}{rr}<br/>-1 &amp; -2 \\<br/>4 &amp; 1<br/>\end{array}\right] \\<br/>&amp; \mathrm{A}^{-1}=\frac{1}{7}\left[\begin{array}{rr}<br/>-1 &amp; -2 \\<br/>4 &amp; 1<br/>\end{array}\right] \quad \operatorname{Both}(\mathrm{A} \text { and } C)<br/>\end{aligned}<br/>$$</div>


title: if-a-2-2-b-6-a-4-5-then
slug: if-a-2-2-b-6-a-4-5-then-56428
question: <div class="question">If $\mathrm{a}=2 \sqrt{2}, \mathrm{~b}=6, \mathrm{~A}=45^{\circ}$, then</div>
tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">no triangle is possible</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">one triangle is possible</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">two triangle are possible</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">either no triangle or two triangles are possible</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">no triangle is possible</span> </div>

answer:
<div class="solution">Hints: $\mathrm{a}=2 \sqrt{2} ; \mathrm{b}=6 ; \mathrm{A}=45^0$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}} \Rightarrow \sin \mathrm{B}=\frac{\mathrm{b}}{\mathrm{a}} \sin \mathrm{A} \\<br/>&amp; \Rightarrow \sin \mathrm{B}=\frac{6}{2 \sqrt{2}} \sin 45^{\circ}=\frac{3}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}=\frac{3}{2} \Rightarrow \text { No triangle is possible since } \sin \mathrm{B}&gt;1<br/>\end{aligned}<br/>$$</div>


title: if-a-2-a-2-ab-ab-b-c-b-2-b-2-b-c-c-2-a-c-c-a-c-2-k-a-2-b-2-c-2-then-k
slug: if-a-2-a-2-ab-ab-b-c-b-2-b-2-b-c-c-2-a-c-c-a-c-2-k-a-2-b-2-c-2-then-k-60587
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{ccc}a^{2} &amp; b c &amp; c^{2}+a c \\ a^{2}+a b &amp; b^{2} &amp; c a \\ a b &amp; b^{2}+b c &amp; c^{2}\end{array}\right|=k a^{2} b^{2} c^{2}$, then $k=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-4$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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">Hint: 
<br/>$\left|\begin{array}{ccc}a^{2} &amp; b c &amp; c^{2}+a c \\ a^{2}+a b &amp; b^{2} &amp; c a \\ a b &amp; b^{2}+b c &amp; c^{2}\end{array}\right|=(a b c)\left|\begin{array}{ccc}a &amp; c &amp; a+c \\ a+b &amp; b &amp; a \\ b &amp; b+c &amp; c\end{array}\right|$
<br/>opening through $\mathrm{R}-1=4 \mathrm{a}^{2} \mathrm{~b}^{2} \mathrm{C}^{2}$</div>


title: if-a-2-a-i-0-then-the-inverse-of-the-matrix-a-is
slug: if-a-2-a-i-0-then-the-inverse-of-the-matrix-a-is-22128
question: <div class="question">If $A^2-A+I=0$, then the inverse of the matrix $A$ is</div>
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{A}-\mathrm{I}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{I}-\mathrm{A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{A}+\mathrm{I}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{A}$</span> </li> </ul>

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

answer:
<div class="solution">Hints : $\mathrm{A}^2-\mathrm{A}+\mathrm{I}=0 \Rightarrow \mathrm{A}^2=\mathrm{A}-\mathrm{I} \Rightarrow \mathrm{A}^2 \cdot \mathrm{A}^{-1}=\mathrm{A} \cdot \mathrm{A}^{-1}-\mathrm{A}^{-1} \Rightarrow \mathrm{A}=\mathrm{I}-\mathrm{A}^{-1} \Rightarrow \mathrm{A}^{-1}=\mathrm{I}-\mathrm{A}$</div>


title: if-a-5-0-0-5-x-x-0-x-5-x-5-and-a-2-25-then-x-is-equal-to
slug: if-a-5-0-0-5-x-x-0-x-5-x-5-and-a-2-25-then-x-is-equal-to-16084
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $A=\left(\begin{array}{ccc}5 &amp; 5 x &amp; x \\ 0 &amp; x &amp; 5 x \\ 0 &amp; 0 &amp; 5\end{array}\right)$ and $\left|A^{2}\right|=25,$ then $|x|$ is equal to</div>

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

answer:
<div class="solution">Given
<br/>$$
<br/>\begin{aligned}
<br/>A &amp;=\left[\begin{array}{ccc}
<br/>5 &amp; 5 x &amp; x \\
<br/>0 &amp; x &amp; 5 x \\
<br/>0 &amp; 0 &amp; 5
<br/>\end{array}\right] \\
<br/>\therefore \quad A^{2}=A \cdot A
<br/>&amp;=\left[\begin{array}{ccc}
<br/>5 &amp; 5 x &amp; x \\
<br/>0 &amp; x &amp; 5 x \\
<br/>0 &amp; 0 &amp; 5
<br/>\end{array}\right]\left[\begin{array}{ccc}
<br/>5 &amp; 5 x &amp; x \\
<br/>0 &amp; x &amp; 5 x \\
<br/>0 &amp; 0 &amp; 5
<br/>\end{array}\right] \\
<br/>&amp;\left[\begin{array}{ccc}
<br/>25 &amp; 25 x+5 x^{2} &amp; 5 x+25 x^{2}+5 x \\
<br/>0 &amp; x^{2} &amp; 5 x^{2}+25 \\
<br/>0 &amp; 0 &amp; 25
<br/>\end{array}\right] \\
<br/>&amp;=25\left(25 x^{2}-0\right) \\
<br/>&amp;=25\left(25 x^{2}\right)
<br/>\end{aligned}
<br/>$$
<br/>since, given that $|A|^{2}=25$
<br/>Now, $25\left(25 x^{2}\right)=25$
<br/>$\begin{array}{lr}\Rightarrow &amp; 25 x^{2}=1 \\ \Rightarrow &amp; x^{2}=\frac{1}{25} \\ \Rightarrow &amp; x=\pm \frac{1}{5}\end{array}$
<br/>$\Rightarrow \quad |x|=\frac{1}{5}$</div>


title: if-a-5-n-4-n-1-n-n-and-b-16-n-1-n-n-then
slug: if-a-5-n-4-n-1-n-n-and-b-16-n-1-n-n-then-53645
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $A=\left\{5^{n}-4 n-1: n \in N\right\}$ and
<br/>$B=\{16(n-1): n \in N],$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A=B$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$A \cap B=\phi$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$A \subseteq B$</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">$B \subseteq A$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A \subseteq B$</span> </div>

answer:
<div class="solution">We have,
<br/>$\begin{aligned} A &amp;=5^{n}-4 n-1=(1+4)^{n}-4 n-1 \\ &amp;=\left({ }^{n} C_{0}+{ }^{n} C_{1} \times 4+{ }^{n} C_{2} \times 4^{2}+\ldots+{ }^{n} C_{n} \times 4^{n}\right) \\ &amp;=4^{2}\left({ }^{n} C_{2}+{ }^{n} C_{3} \times 4+\ldots+{ }^{n} C_{n} \times 4^{n-2}\right) \end{aligned}$
<br/>$\therefore$ A contains some multiples of 16 
<br/>Clearly, $B$ contains all multiples of 16 including $0 .$
<br/>$A \subseteq B$</div>


title: if-a-and-b-are-arbitrary-positive-real-numbers-then-the-least-possible-value-of-5-b-6-a-3-a-10-b-is
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P1.db
slug:
if-a-and-b-are-arbitrary-positive-real-numbers-then-the-least-possible-value-of-5-b-6-a-3-a-10-b-is-92166

question:
<div class="question">If $a$ and $b$ are arbitrary positive real numbers, then the least possible value of $\frac{6 a}{5 b}+\frac{10 b}{3 a}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">$\frac{6}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{10}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{68}{15}$</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>$\frac{6 a}{5 b}+\frac{10 b}{3 a} \geq 2 \sqrt{\frac{6 a}{5 b} \times \frac{10 b}{3 a}}, \quad \frac{6 a}{5 b}+\frac{10 b}{3 a} \geq 2 \times 2 \geq 4$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-a-and-b-are-coefficients-of-xmathrmn-in-the-expansions-of-1x2-mathrmn-and-1x2-mathrmn1-respectively-then-mathrma-mathrmb-is-equal-to

slug:
if-a-and-b-are-coefficients-of-xmathrmn-in-the-expansions-of-1x2-mathrmn-and-1x2-mathrmn1-respectively-then-mathrma-mathrmb-is-equal-to-35835

question:
<div class="question">If \(A\) and \(B\) are coefficients of \(x^{\mathrm{n}}\) in the expansions of \((1+x)^{2 \mathrm{n}}\) and \((1+x)^{2 \mathrm{n}-1}\) respectively, then \(\mathrm{A} / \mathrm{B}\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">9</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">\(\begin{array}{r}\text {Hints: } <br/>A={ }^{2 n} C_n \\<br/>B={ }^{2 n-1} C_n<br/>\end{array}\)<br/>\(\frac{\mathrm{A}}{\mathrm{B}}=\frac{{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}}{{ }^{2 \mathrm{n}-1} \mathrm{C}_{\mathrm{n}}}=\frac{2 \mathrm{n}}{\mathrm{n}}=2\)</div>


title: if-a-and-b-are-two-matrices-such-that-a-b-b-and-b-a-a-then-a-2-b-2-equals
slug: if-a-and-b-are-two-matrices-such-that-a-b-b-and-b-a-a-then-a-2-b-2-equals-28361
tags: ['Mathematics', 'Matrices', '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_P1.db
question:
<div class="question">If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A,$ then $A^{2}+B^{2}$ equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">248</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{BA}$</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">$A B$</span> </li> </ul>

answer:
<div class="solution">Given, $A B=B$ and $B A=A$ $\begin{aligned} \text { Now, } A^{2}+B^{2} &amp;=A \cdot A+B \cdot B \\ &amp;=A(B A)+B(A B)=(A B) A+(\ln ) \\ &amp;=B A+A B=A+B \end{aligned}$</div>


title: if-a-and-b-square-matrices-of-the-same-order-and-a-b-3-i-then-a-1-is-equal-to
slug: if-a-and-b-square-matrices-of-the-same-order-and-a-b-3-i-then-a-1-is-equal-to-81755
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ square matrices of the same order and $A B=3 I$, then $A^{-1}$ is equal to</div>

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

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

answer:
<div class="solution">Hints: $\mathrm{AB}=3 \mathrm{I}, \mathrm{A}^{-1} \cdot \mathrm{AB}=3 \cdot \mathrm{A}^{-1} \mathrm{I} \Rightarrow \mathrm{B}=3 \mathrm{~A}^{-1} \Rightarrow A^{-1}=\frac{1}{3} \mathrm{~B}$</div>


title: if-a-b-1-2-3-and-the-equation-a-x-2-b-x-1-0-has-real-roots-then
slug: if-a-b-1-2-3-and-the-equation-a-x-2-b-x-1-0-has-real-roots-then-51779
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If  $a, b \in\{1,2,3\}$  and the equation $a x^{2}+b x+1=0$ has real roots, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">number of possible ordered pairs $(a, b)$ is 3, $a    &lt; b$</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>a x^{2}+b x+1=0
<br/>$$
<br/>For real roots, $D \geq 0$
<br/>$$
<br/>\therefore \quad b^{2}-4 a \geq 0
<br/>$$
<br/>$\Rightarrow \quad b^{2} \geq 4 a$
<br/>$\therefore(a, b)=(1,2),(1,3),(2,3)$
<br/>$\therefore$ Number of ordered pairs $(a, b)=3$ and $a$ is always less than $b$.</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(1,-1)</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-a-b-and-c-are-in-ap-then-the-straight-line-a-x-2-b-y-c-0-will-always-pass-through-a-fixed-point-whose-coordinates-are

slug:
if-a-b-and-c-are-in-ap-then-the-straight-line-a-x-2-b-y-c-0-will-always-pass-through-a-fixed-point-whose-coordinates-are-68129

question:
<div class="question">If $a, b$ and $c$ are in $\mathrm{AP}$, then the straight line $a x+2 b y+c=0$ will always pass through a fixed point whose coordinates are</div>

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

answer:
<div class="solution">Given that, $a$, $b$ and $c$ are in $\mathrm{AP}$
<br/>$\therefore \quad 2 b=a+c$
<br/>$\Rightarrow \quad c=2 b-a$
<br/>and equation of straight line is
<br/>$a x+2 b y+c=0$
<br/>$\Rightarrow \quad a x+2 b y+(2 b-a)=0$
<br/>$\Rightarrow \quad a(x-1)+b(2 y+2)=a \cdot 0+b \cdot 0$
<br/>On comparing, we get
<br/>$x-1=0 \Rightarrow x=1$
<br/>and $\quad 2 y+2=0$
<br/>$\Rightarrow \quad y=-1$
<br/>Hence, the required fixed point is (1,-1)
<br/>$$</div>


title: if-a-b-and-c-are-in-arithmetic-progression-then-the-roots-of-the-equation-a-x-2-2-h-x-c-0-are
slug: if-a-b-and-c-are-in-arithmetic-progression-then-the-roots-of-the-equation-a-x-2-2-h-x-c-0-are-97127
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $a, b$ and $c$ are in arithmetic progression, then the roots of the equation $a x^{2}-2 h x+c=0$ are</div>

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

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

answer:
<div class="solution">Since, $a, b$ and $c$ are in $\mathrm{AP}$.
<br/>$\therefore$
<br/>$2 b=a+c$
<br/>Given, quadratic equation, $a x^{2}-2 b x+c=0$
<br/>$\Rightarrow \quad a x^{2}-(a+c) x+c=0$
<br/>$2 b=a+c)$
<br/>$\Rightarrow \quad a x^{2}-a x-c x+c=0$
<br/>$\begin{array}{lr}\Rightarrow &amp; a x(x-1)-c(x-1)=0 \\ \Rightarrow &amp; (x-1)(a x-c)=0 \\ \Rightarrow &amp; x=1, \frac{c}{a}\end{array}$</div>


tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-a-b-and-c-are-positive-numbers-in-a-gp-then-the-roots-of-the-quadratic-equation-lo-g-e-a-x-2-2-lo-g-e-b-x-lo-g-e-c-0-are

slug:
if-a-b-and-c-are-positive-numbers-in-a-gp-then-the-roots-of-the-quadratic-equation-lo-g-e-a-x-2-2-lo-g-e-b-x-lo-g-e-c-0-are-36672

question:
<div class="question">If $a, b$ and $c$ are positive numbers in a GP, then the roots of the quadratic equation $\left(\log _{e} a\right) x^{2}-\left(2 \log _{e} b\right) x+\left(\log _{e} c\right)=0$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and $\log _{a} c$</span> </div>

answer:
<div class="solution">Since, $a, b$ and $c$ are in $\mathrm{GP}$.<br/>$\therefore b^{2}=a c$<br/>Given equation is<br/>$(\log _{e} \text { a) } x^{2}-\left(2 \log _{e} \text { b) } x+\log _{e} c\right)=0$<br/>Put $x=1,$ we get<br/>$\begin{aligned}<br/>&amp; \log _{e} a-2 \log _{e} b+\log _{e} c=0 \\<br/>\Rightarrow &amp; 2 \log _{e} b=\log _{e} a+\log _{0} c \\<br/>\Rightarrow \quad &amp; \log _{e} b^{2}=\log _{e} a c<br/>\end{aligned}$<br/>$\Rightarrow \quad b^{2}=a c,$ which is true.<br/>Hence, one of the root of given equation is $1 .$<br/>Let another root be $\alpha$.<br/>$\therefore$ Sum of roots, $1+\alpha=\frac{2 \log _{e} b}{\log _{e} a}=\frac{\log _{e} b^{2}}{\log _{e} a}$<br/>$\begin{aligned}<br/>\Rightarrow \quad a &amp;=\frac{\log _{e} a c}{\log _{e} a}-1 \\<br/>&amp;=\frac{\left(\log _{e} a+\log _{e} c\right)}{\log _{e} a}-1 \\<br/>&amp;=\frac{\log _{e} c}{\log _{c} a}=\log _{a} c<br/>\end{aligned}$<br/>Hence, roots are 1 and $\log _{a} c .$</div>


title: if-a-b-are-odd-integers-then-the-roots-of-the-equation-2-a-x-2-2-a-b-x-b-0-a-0-are
slug: if-a-b-are-odd-integers-then-the-roots-of-the-equation-2-a-x-2-2-a-b-x-b-0-a-0-are-15027
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">Discriminant is a perfect square $\Rightarrow$ Rational roots</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $a, b$ are odd integers, then the roots of the equation $2 a x^2+(2 a+b) x+b=0, a \neq 0$ are</div>

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

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


title: if-a-b-are-two-events-such-that-p-a-b-4-3-and-8-1-p-a-b-8-3-then
slug: if-a-b-are-two-events-such-that-p-a-b-4-3-and-8-1-p-a-b-8-3-then-38893
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $A, B$ are two events such that $P(A \cup B) \geq \frac{3}{4}$ and $\frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8},$ then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$P(A)+P(B) \leq \frac{11}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$P(A) \cdot P(B) \leq \frac{3}{8}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$P(A)+P(B) \geq \frac{7}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$P(A)+P(B) \leq \frac{11}{8}$, $P(A)+P(B) \geq \frac{7}{8}$</span> </div>

answer:
<div class="solution">We know,
<br/>$$
<br/>P(A \cup B)=P(A)+P(B)-P(A \cap B)
<br/>$$
<br/>$\Rightarrow P(A)+P(B)=P(A \cup B)+P(A \cap B)$
<br/>Given, $\quad \frac{3}{4} \leq P(A \cup B) \leq 1$
<br/>and $\quad \frac{1}{8} \leq P(A \cap B) \leq \frac{3}{8}$
<br/>So. $\quad \frac{7}{8} \leq P(A \cup B)+P(A \cap B) \leq \frac{11}{8}$
<br/>$\therefore P(A)+P(B) \geq \frac{7}{8}$ and $P(A)+P(B) \leq \frac{11}{8}$</div>


title: if-a-b-c-0-where-a-b-c-are-nonzero-scalars-then-the-vectors-are
slug: if-a-b-c-0-where-a-b-c-are-nonzero-scalars-then-the-vectors-are-13302
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{a}(\vec{\alpha} \times \vec{\beta})+\mathrm{b}(\vec{\beta} \times \vec{\gamma})+\mathrm{c}(\vec{\gamma} \times \vec{\alpha})=\overrightarrow{0}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are non-zero scalars, then the vectors $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are</div>

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

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

answer:
<div class="solution">$\therefore \vec{\alpha} \times \vec{\beta}, \vec{\beta} \times \vec{\lambda}, \vec{\gamma} \times \vec{\alpha}$ are coplanar $\Rightarrow \vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are coplanar</div>


title: if-a-b-c-and-a-b-c-then-the-angle-included-between-a-and-b-is
slug: if-a-b-c-and-a-b-c-then-the-angle-included-between-a-and-b-is-67876
tags: ['Physics', 'Mathematics in Physics', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Zero</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}+\vec{b}=\vec{c}$ and $a+b=c$, then the angle included between $\vec{a}$ and $\vec{b}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$90^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$180^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$120^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Hints: Here $\vec{a}+\vec{b}=\vec{c} \&amp; c=a+b$<br/>Now, $c=\sqrt{a^2+b^2+2 a b \cos \theta}$<br/>$\begin{aligned}<br/>&amp; (a+b)=\sqrt{a^2+b^2+2 a b \cos \theta} \\<br/>&amp; a^2+b^2+2 a b=a^2+b^2+2 a b \cos \theta \\<br/>&amp; \cos \theta=1, \theta=0^{\circ}<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-a-b-c-are-in-g-p-and-lo-g-a-lo-g-2-b-lo-g-2-b-lo-g-3-c-lo-g-3-c-lo-g-a-are-in-a-p-then-a-b-c-are-the-lengths-of-the-sides-of-a-triangle-which-is

slug:
if-a-b-c-are-in-g-p-and-lo-g-a-lo-g-2-b-lo-g-2-b-lo-g-3-c-lo-g-3-c-lo-g-a-are-in-a-p-then-a-b-c-are-the-lengths-of-the-sides-of-a-triangle-which-is-49568

question:
<div class="question">If $a, b, c$ are in G. P. and $\log a .-\log 2 b, \log 2 b-\log 3 c, \log 3 c-\log a$ are in A. P., then $a, b, c$ are the lengths of the sides of a triangle which is</div>

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

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

answer:
<div class="solution">$$<br/>\because 2 \log \frac{2 b}{3 c}=\log \frac{a}{2 b}+\log \frac{3 c}{a} \Rightarrow \frac{4 b^2}{9 c^2}=\frac{3 c}{2 b} \Rightarrow 2 b=3 c \Rightarrow \frac{c}{b}=\frac{2}{3} \text { (common ratio) }<br/>$$<br/>$\therefore$ Sides are a, $\frac{2 \mathrm{a}}{3}, \frac{4 \mathrm{a}}{9}$<br/>Using Cosine rule<br/>$\Rightarrow \cos \mathrm{A} &lt; 0 \Rightarrow$ obteuse angled tiangle</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">H.P.</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-a-b-c-are-in-gp-a-1-b-1-c-1-then-for-any-real-number-x-with-x-0-x-1-lo-g-a-x-lo-g-b-x-lo-g-c-x-are-in

slug:
if-a-b-c-are-in-gp-a-1-b-1-c-1-then-for-any-real-number-x-with-x-0-x-1-lo-g-a-x-lo-g-b-x-lo-g-c-x-are-in-31736

question:
<div class="question">If $a, b, c$ are in G.P. $(a&gt;1, b&gt;1, c&gt;1)$, then for any real number $x($ with $x&gt;0, x \neq 1), \log _{\mathrm{a}} x, \log _{\mathrm{b}} x, \log _{\mathrm{c}} x$ are in</div>

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

answer:
<div class="solution">Hints : $a, b, c$ are in G.P.<br/>$\Rightarrow \log _x a, \log _x b, \log _x c$ are in A.P.<br/>$\Rightarrow \frac{1}{\log _x a}, \frac{1}{\log _x b}, \frac{1}{\log _x c}$ arein H.P.<br/>$\Rightarrow \log _a X, \log _b X, \log _c X$ are in H.P.</div>


title: if-a-b-c-are-real-then-both-the-roots-of-the-equation-x-b-x-c-x-c-x-a-x-a-x-b-0-are-always
slug: if-a-b-c-are-real-then-both-the-roots-of-the-equation-x-b-x-c-x-c-x-a-x-a-x-b-0-are-always-22368
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">real</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $a, b, c$ are real, then both the roots of the equation $(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0$ are always</div>

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

answer:
<div class="solution">Hints : $3 x^2-2 x(a+b+c)+a b+b c+c a=0$<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{D}=4(a+b+c)^2-4.3(a b+b c+c a) \\<br/>&amp; =4\left(a^2+b^2+c^2-a b-b c-c a\right) \\<br/>&amp; =2\left[(a-b)^2+(b-c)^2+(c-a)^2\right] \\<br/>&amp; =\left[(a-b)^2+(b-c)^2+(c-a)^2\right] \\<br/>&amp; \geq 0<br/>\end{aligned}<br/>$$</div>


title: if-a-b-c-have-scalar-magnitudes-of-543-units-respectively-then-the-angle-between-a-and-c-is-1
slug: if-a-b-c-have-scalar-magnitudes-of-543-units-respectively-then-the-angle-between-a-and-c-is-1-21657
tags: ['Physics', 'Mathematics in Physics', 'JEE Main']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathbf{A}=\mathbf{B}+\mathbf{C}$ have scalar magnitudes of 5,4,3 units respectively, then the angle between $A$ and $\mathbf{C}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\cos ^{-1}(3 / 5)$</span> </div>

answer:
<div class="solution">Here, triangle is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_phy_a17.png"/> <br/> <br/>$\theta=\cos ^{-1}\left(\frac{|\mathbf{C}|}{|\mathbf{A}|}\right)$
<br/>$\theta=\cos ^{-1}\left(\frac{3}{5}\right)$</div>


title: if-a-b-c-o-s-x-2-s-i-n-2-x-d-x-lo-g-e-a-b-cos-x-a-b-c-o-s-x-a-c-then
slug: if-a-b-c-o-s-x-2-s-i-n-2-x-d-x-lo-g-e-a-b-cos-x-a-b-c-o-s-x-a-c-then-89826
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\int \frac{\sin 2 x}{(a+b \cos x)^{2}} d x=\alpha\left[\log _{e}|a+b \cos x|+\frac{a}{a+b \cos x}\right]+c$, then $\alpha=$</div>

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

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

answer:
<div class="solution">$\int \frac{\sin 2 x}{(a+b \cos x)^{2}} d x$
<br/>Put $a+b \cos x=t$
<br/>$-b \sin x d x=d t$
<br/>$=\int \frac{2 \sin x \cos x}{t^{2}} \cdot \frac{d t}{-b \sin x}=\int \frac{-2}{b} \cdot\left(\frac{t-a}{b}\right) \cdot \frac{1}{t^{2}} d t=\frac{-2}{b^{2}} \int\left(\frac{1}{t}-\frac{a}{t^{2}}\right) d t \quad \therefore \alpha=\frac{-2}{b^{2}}$
<br/>$=-\frac{2}{b^{2}}\left[\ln |t|+\frac{a}{t}\right]+c$
<br/>$=\frac{-2}{b^{2}}\left[\ln |a+b \cos x|+\frac{a}{a+b \cos x}\right]+c$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
if-a-denote-the-greatest-integer-which-is-less-than-or-equal-to-a-then-the-value-of-the-integral-2-2-sin-x-cos-x-d-x-is

slug:
if-a-denote-the-greatest-integer-which-is-less-than-or-equal-to-a-then-the-value-of-the-integral-2-2-sin-x-cos-x-d-x-is-18315

question:
<div class="question">If [a] denote the greatest integer which is less than or equal to a. Then, the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}[\sin x \cos x] d x$ is</div>

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

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

answer:
<div class="solution">Let $/=\int_{-\pi / 2}^{\pi / 2}[\sin x \cdot \cos x] d x=\int_{-\pi / 2}^{\pi / 2}\left[\frac{1}{2} \sin 2 x\right] d x$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a67.png"/> <br/> <br/>Put $\theta=2 x \Rightarrow d \theta=2 d x$
<br/>Also, when $\quad x=-\pi / 2,$ then $\theta=-\pi$ when $\quad x=\frac{\pi}{2},$ then $\theta=\pi$
<br/>$\begin{aligned} \text { Then, } &amp; I=\frac{1}{2} \int_{\pi}^{\pi}\left[\frac{1}{2} \sin \theta\right] d \theta \\ &amp;=\frac{1}{2}\left[\int_{\pi}^{0}(-1) d x+\int_{0}^{\pi}(0) d x\right] \\ &amp;=\frac{1}{2}[-x]_{\pi}^{2}+0=-\frac{\pi}{2} \end{aligned}$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-a-hyperbola-passes-through-the-point-mathrmpsqrt2-sqrt3-and-foci-at-pm-20-then-the-tangent-to-this-hyperbola-at-mathrmp-is

slug:
if-a-hyperbola-passes-through-the-point-mathrmpsqrt2-sqrt3-and-foci-at-pm-20-then-the-tangent-to-this-hyperbola-at-mathrmp-is-90953

question:
<div class="question">If a hyperbola passes through the point \(\mathrm{P}(\sqrt{2}, \sqrt{3})\) and foci at \(( \pm 2,0)\) then the tangent to this hyperbola at \(\mathrm{P}\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{y}=\mathrm{x} \sqrt{6}-\sqrt{3}\)</span> </div>

answer:
<div class="solution">Hint: \(a e=2, a^2+b^2=4\)<br/>\(\Rightarrow \frac{2}{a^2}-\frac{3}{b^2}=1 \Rightarrow b^2=3 ; a^2=1\)<br/>\(\frac{x^2}{1}-\frac{y^2}{3}=1\). Tangent at \(P(\sqrt{2}, \sqrt{3})\) is<br/>\(\sqrt{6 x}-y=\sqrt{3}\)</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-a-i-j-k-b-i-j-k-and-c-is-unit-vector-perpendicular-to-a-and-coplanar-with-a-and-b-then-unit-vector-d-perpendicular-to-both-a-and-c-is

slug:
if-a-i-j-k-b-i-j-k-and-c-is-unit-vector-perpendicular-to-a-and-coplanar-with-a-and-b-then-unit-vector-d-perpendicular-to-both-a-and-c-is-38701

question:
<div class="question">If $\vec{a}=\hat{i}+\hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ is unit vector perpendicular to $\vec{a}$ and coplanar with $\vec{a}$ and $\vec{b}$, then unit vector $\vec{d}$ perpendicular to both $\vec{a}$ and $\vec{c}$ is</div>

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

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

answer:
<div class="solution">$\because \overrightarrow{\mathrm{d}}$ is normal to the plane of $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$<br/>$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ or $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}=\pm 2(\hat{\mathrm{j}}+\hat{\mathrm{k}}) \therefore$ unit vector $\overrightarrow{\mathrm{d}}=\pm \frac{1}{\sqrt{2}}(\hat{\mathrm{j}}+\hat{\mathrm{k}})$</div>


title: if-a-is-a-square-matrix-then
slug: if-a-is-a-square-matrix-then-79078
question: <div class="question">If $A$ is a square matrix then,</div>
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{A}+\mathrm{A}^{\mathrm{T}}$ is 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><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{AA}^{\mathrm{T}}$ is skew - symmetric</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{A}^{\mathrm{T}}+\mathrm{A}$ is skew-symmetric</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{A}^{\mathrm{T}} \mathrm{A}$ is skew symmetric</span> </li> </ul>

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

answer:
<div class="solution">Hints: $\left(\mathrm{A}+\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\mathrm{A}=\mathrm{A}+\mathrm{A}^{\mathrm{T}}$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-a-n-0-be-the-n-th-term-of-a-gp-then-lo-g-a-n-lo-g-a-n-3-lo-g-a-n-6-lo-g-a-n-1-lo-g-a-n-4-lo-g-a-n-7-lo-g-a-n-2-lo-g-a-n-5-lo-g-a-n-8-is-equal-to

slug:
if-a-n-0-be-the-n-th-term-of-a-gp-then-lo-g-a-n-lo-g-a-n-3-lo-g-a-n-6-lo-g-a-n-1-lo-g-a-n-4-lo-g-a-n-7-lo-g-a-n-2-lo-g-a-n-5-lo-g-a-n-8-is-equal-to-68564

question:
<div class="question">If $a_{n}(&gt;0)$ be the $n^{\text {th }}$ term of a G.P. then $\left|\begin{array}{lll}\log a_{n} &amp; \log a_{n+1} &amp; \log a_{n+2} \\ \log a_{n+3} &amp; \log a_{n+4} &amp; \log a_{n+5} \\ \log a_{n+6} &amp; \log a_{n+7} &amp; \log a_{n+8}\end{array}\right|$ 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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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">$\log a_{n}=\log \left(a r^{n-1}\right)=\log a+(n-1) \log r$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/e_sD8wRo7ExMXGkIYU3XW6sKkbTzhgBIMwglIIaSv8M.original.fullsize.png"/><br/> <br/>$=\left|\begin{array}{ccc}\log \mathrm{a}+(\mathrm{n}-1) \log \mathrm{r} &amp; \log \mathrm{a}+\mathrm{nlogr} &amp; \log \mathrm{a}+(\mathrm{n}+1) \log \mathrm{r} \\ 3 \log \mathrm{r} &amp; 3 \log \mathrm{r} &amp; 3 \log \mathrm{r} \\ 6 \log r &amp; 6 \log r &amp; 6 \log r\end{array}\right|$
<br/>= 0</div>


tags: ['Physics', 'Motion In Two Dimensions', 'JEE Main']
source_db: db\MarksBatch2_P1.db
title:
if-a-person-can-throw-a-stone-to-maximum-height-of-mathrmh-metre-vertically-then-the-maximum-distance-through-which-it-can-be-thrown-horizontally-by-t-1

slug:
if-a-person-can-throw-a-stone-to-maximum-height-of-mathrmh-metre-vertically-then-the-maximum-distance-through-which-it-can-be-thrown-horizontally-by-t-1-14196

question:
<div class="question">If a person can throw a stone to maximum height of \(\mathrm{h}\) metre vertically, then the maximum distance through which it can be thrown horizontally by the same person is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\mathrm{h}}{2}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{h}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{~h}\)</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">D</span> <span class="option-data">\(3 \mathrm{~h}\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fvdCdVLIReqA1PCts394j7JgkKDzrEpuvA9vtzpw7e0.original.fullsize.png"/><br/><br/>\(\mathrm{h}=\frac{\mathrm{u}^2}{2 \mathrm{~g}} \Rightarrow \mathrm{u}^2=2 \mathrm{gh}\)<br/>Maximum horizontal distance<br/>\(\begin{aligned}<br/>&amp; \mathrm{R}_{\max }=\frac{\mathrm{u}^2}{\mathrm{~g}}\left(\text { when } \theta=45^{\circ}\right) \\<br/>&amp; \mathrm{R}_{\max }=2 \mathrm{~h}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Motion In Two Dimensions', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-a-person-can-throw-a-stone-to-maximum-height-of-mathrmh-metre-vertically-then-the-maximum-distance-through-which-it-can-be-thrown-horizontally-by-t

slug:
if-a-person-can-throw-a-stone-to-maximum-height-of-mathrmh-metre-vertically-then-the-maximum-distance-through-which-it-can-be-thrown-horizontally-by-t-75460

question:
<div class="question">If a person can throw a stone to maximum height of \(\mathrm{h}\) metre vertically, then the maximum distance through which it can be thrown horizontally by the same person is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\mathrm{h}}{2}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{h}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{~h}\)</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">D</span> <span class="option-data">\(3 \mathrm{~h}\)</span> </li> </ul>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/fvdCdVLIReqA1PCts394j7JgkKDzrEpuvA9vtzpw7e0.original.fullsize.png"/><br/><br/>\(\mathrm{h}=\frac{\mathrm{u}^2}{2 \mathrm{~g}} \Rightarrow \mathrm{u}^2=2 \mathrm{gh}\)<br/>Maximum horizontal distance<br/>\(\begin{aligned}<br/>&amp; \mathrm{R}_{\max }=\frac{\mathrm{u}^2}{\mathrm{~g}}\left(\text { when } \theta=45^{\circ}\right) \\<br/>&amp; \mathrm{R}_{\max }=2 \mathrm{~h}<br/>\end{aligned}\)</div>


title: if-a-r-cos-2-r-i-sin-2-r-19-then-the-value-of-a-1-a-4-a-7-a-2-a-5-a-8-a-3-a-6-a-9-is-equal-to
slug: if-a-r-cos-2-r-i-sin-2-r-19-then-the-value-of-a-1-a-4-a-7-a-2-a-5-a-8-a-3-a-6-a-9-is-equal-to-91951
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $a_{r}=(\cos 2 r \pi+i \sin 2 r \pi)^{1 / 9}$, then the value of
<br/>$\left|\begin{array}{lll}a_{1} &amp; a_{2} &amp; a_{3} \\ a_{4} &amp; a_{5} &amp; a_{6} \\ a_{7} &amp; a_{8} &amp; a_{9}\end{array}\right|$ 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">-1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">We have,
<br/>$a_{r}=(\cos 2 r \pi+i \sin 2 r \pi)^{1 / 9}=e^{\frac{2 r \pi i}{9}}$
<br/>Now, $\left|\begin{array}{lll}a_{1} &amp; a_{2} &amp; a_{3} \\ a_{4} &amp; a_{5} &amp; a_{6} \\ a_{7} &amp; a_{8} &amp; a_{9}\end{array}\right|=\left|\begin{array}{ccc}\ e^{\frac{2 \pi i}{9}} &amp; e^{\frac{4 \pi i}{9}} &amp; e^{\frac{6 \pi i}{9}} \\ e^{\frac{8 \pi i}{9}} &amp; e^{\frac{10 \pi i}{9}} &amp; e^{\frac{12 \pi i}{9}} \\ e^{\frac{14 \pi i}{9}} &amp; e^{\frac{16 \pi i}{9}} &amp; e^{\frac{18 \pi i}{9}}\end{array}\right|$
<br/>$=e^{\frac{2 \pi i}{9}} \times e^{\frac{8 \pi i}{9}}\left|\begin{array}{ccc}1 &amp; e^{\frac{2 \pi i}{9}} &amp; e^{\frac{4 \pi i}{9}} \\ 1 &amp; e^{\frac{2 \pi i}{9}} &amp; e^{\frac{4 \pi i}{9}} \\ e^{\frac{14 \pi i}{9}} &amp; e^{\frac{16 \pi i}{9}} &amp; e^{\frac{18 \pi i}{9}}\end{array}\right|$
<br/>$=e^{\frac{2 \pi i}{9}} \times e^{\frac{8 \pi}{9}} \times 0 \quad\left(\because R_{1}\right.$ and $R_{2}$ are identical $)$
<br/>$=0$</div>


title: if-a-species-has-16-protons-18-electrons-and-16-neutrons-find-the-species-and-its-charge
slug: if-a-species-has-16-protons-18-electrons-and-16-neutrons-find-the-species-and-its-charge-59280
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If a species has 16 protons, 18 electrons and 16 neutrons, find the species and its charge</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{S}^{1-}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Si}^{2-}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{P}^{3-}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{S}^{2-}$</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> </ul>

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

answer:
<div class="solution">Hints : $16 \mathrm{p}$ means $\mathrm{z}=16$<br/>$18 \mathrm{e}^{-}$means, 2 unit negative charge is present.<br/>Hence species is $\mathrm{S}^{-2}$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-a-straight-line-passes-through-the-point-and-the-portion-of-the-line-intercepted-between-the-axes-is-divided-equally-at-that-point-then-a-x-y-is

slug:
if-a-straight-line-passes-through-the-point-and-the-portion-of-the-line-intercepted-between-the-axes-is-divided-equally-at-that-point-then-a-x-y-is-55808

question:
<div class="question">If a straight line passes through the point $(\alpha, \beta)$ and the portion of the line intercepted between the axes is divided equally at that point, then $\frac{x}{a}+\frac{y}{\beta}$ 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">4</span> </li> </ul>

answer:
<div class="solution">Let the line be $A B=\frac{x}{a}+\frac{y}{b}=1$
<br/>$\ldots(0$
<br/>$\therefore$ The coordinates of $A$ and $B$ are respectively
<br/>$(a, 0)$ and $(0, b)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a60.png"/> <br/> <br/>Since, $(\alpha, \beta)$ is the mid-point of $A B$
<br/>$\begin{array}{ll}\therefore \quad &amp; \alpha=\frac{a}{2}, \beta=\frac{b}{2} \\ \therefore &amp; a=2 x \text { and } b=2 \beta\end{array}$
<br/>Crom Eq. (i), $\frac{x}{2 x}+\frac{y}{2 \beta}=1 \Rightarrow \frac{x}{\alpha}+\frac{y}{\beta}=2$</div>


