# questions

title: suppose-m-0-2-x-2-c-o-s-x-d-x-n-0-4-x-1-2-s-i-n-x-c-o-s-x-d-x-then-the-value-of-m-n-equals
slug: suppose-m-0-2-x-2-c-o-s-x-d-x-n-0-4-x-1-2-s-i-n-x-c-o-s-x-d-x-then-the-value-of-m-n-equals-56333
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">Suppose $M=\int_{0}^{\pi / 2} \frac{\cos x}{x+2} d x$ $N=\int_{0}^{\pi / 4} \frac{\sin x \cos x}{(x+1)^{2}} d x$ Then, the value of $(M-N)$ equals</div>

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

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

answer:
<div class="solution">Given, $M=\int_{0}^{\pi / 2} \frac{\cos x}{(x+2)} d x$
<br/>and $\quad N=\int_{0}^{\sqrt{4}} \frac{\sin x \cos x}{(x+1)^{2}} d x$
<br/>$\Rightarrow \quad N=\int_{0}^{\pi / 4} \frac{1}{2} \frac{\sin 2 x}{(x+1)^{2}} d x$
<br/>Put $2 x=t \Rightarrow d x=\frac{d t}{2}$
<br/>$\begin{aligned} N &amp;=\int_{0}^{\pi / 2} \frac{\sin t}{4\left(\frac{t}{2}+1\right)^{2}} d t \\ &amp;=\int_{0}^{\pi / 2} \frac{\sin t}{(t+2)^{2}} d t \\ &amp;=\int_{0}^{\pi / 2} \frac{\sin x}{(x+2)^{2}} d x \end{aligned}$
<br/>$M-N=\int_{0}^{\pi / 2} \cos x \times \frac{1}{(x+2)} d x$
<br/>$=\left[\frac{\sin x}{x+2}\right]_{0}^{\pi / 2}-\int_{0}^{\pi / 2}-\frac{\sin x}{(x+2)^{2}} d x$
<br/>$-\int_{0}^{\pi / 2} \frac{\sin x}{(x+2)^{2}} d x$
<br/>$=\frac{\sin \frac{\pi}{2}}{\frac{\pi}{2}+2}=\frac{1}{\frac{\pi+4}{2}}=\frac{2}{\pi+4}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
suppose-that-f-x-is-a-differentiable-function-such-that-f-x-is-continuous-f-0-1-and-f-0-does-not-exist-let-g-x-x-f-x-then

slug:
suppose-that-f-x-is-a-differentiable-function-such-that-f-x-is-continuous-f-0-1-and-f-0-does-not-exist-let-g-x-x-f-x-then-22878

question:
<div class="question">Suppose that $f(x)$ is a differentiable function such that $f^{\prime}(x)$ is continuous, $f^{\prime}(0)=1$ and $f^{\prime \prime}(0)$ does not exist. Let $g(x)=x f^{\prime}(x) .$ Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$g^{\prime}(0)$ does not exist</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$g^{\prime}(0)=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$g^{\prime}(0)=1$</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">D</span> <span class="option-data">$g^{\prime}(0)=2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$g^{\prime}(0)=1$</span> </div>

answer:
<div class="solution">Given, $f^{\prime}(0)=1$ and $f^{\prime \prime}$ " (0) does not exist.
<br/>Also, given $g(x)=x f^{\prime}(x)$
<br/>$\therefore \quad g^{\prime}(x)=x f^{\prime \prime}(x)+f^{\prime}(x)$
<br/>Put $x=0,$ we get
<br/>$$
<br/>\begin{aligned}
<br/>g^{\prime}(0) &amp;=0 f^{\prime \prime}(0)+f^{\prime}(0) \\
<br/>&amp;=0+1=1
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
suppose-that-the-equation-f-x-x-2-b-x-c-0-has-two-distinct-real-roots-and-the-angle-between-the-tangent-to-the-curve-y-f-x-at-the-point-2-f-2-and-the-

slug:
suppose-that-the-equation-f-x-x-2-b-x-c-0-has-two-distinct-real-roots-and-the-angle-between-the-tangent-to-the-curve-y-f-x-at-the-point-2-f-2-and-the-48410

question:
<div class="question">Suppose that the equation $f(x)=x^{2}+b x+c=0$ has two distinct real roots $\alpha$ and $\beta$. The angle between the tangent to the curve $y=f(x)$ at the point $\left(\frac{\alpha+\beta}{2}, f\left(\frac{\alpha+\beta}{2}\right)\right)$ and the positive direction of the $x$ -axis is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$60^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$90^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0^{\circ}$</span> </div>

answer:
<div class="solution">Since, $\alpha$ and $\beta$ are the roots of
<br/>$$
<br/>\begin{aligned}
<br/>f(x) &amp;=x^{2}+b x+c \\
<br/>\therefore \quad \alpha+\beta &amp;=-b \text { and } \alpha \beta=c \\
<br/>\therefore \quad f\left(\frac{\alpha+\beta}{2}\right) &amp;=\left(\frac{\alpha+\beta}{2}\right)^{2}+b\left(\frac{\alpha+\beta}{2}\right)+c \\
<br/>&amp;=\left(-\frac{b}{2}\right)^{2}+b\left(-\frac{b}{2}\right)+c \\
<br/>&amp;=\frac{b^{2}}{4}-\frac{b^{2}}{2}+c=-\frac{b^{2}}{4}+c
<br/>\end{aligned}
<br/>$$
<br/>Now, $\quad \frac{d y}{d x}=f^{\prime}(x)=2 x+b$
<br/>At point $\left(\frac{\alpha+\beta}{2}, f\left(\frac{\alpha+\beta}{2}\right)\right)$
<br/>i.e., $\begin{aligned}\left(-\frac{b}{2},-\frac{b^{2}}{4}+c\right) \\ \frac{d y}{d x} &amp;=2\left(-\frac{b}{2}\right)+b=0 \end{aligned}$
<br/>Hence, the slope of the tangent to the curve and the positive direction of $x$ -axis is $0^{\circ}$.</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
suppose-that-z-1-z-2-z-3-are-three-vertices-of-an-equilateral-triangle-in-the-angand-plane-let-2-1-3-i-and-be-a-nonzero-complex-number-the-points-a-z-

slug:
suppose-that-z-1-z-2-z-3-are-three-vertices-of-an-equilateral-triangle-in-the-angand-plane-let-2-1-3-i-and-be-a-nonzero-complex-number-the-points-a-z-17458

question:
<div class="question">Suppose that $z_{1}, z_{2}, z_{3}$ are three vertices of an equilateral triangle in the angand plane. Let $\alpha=\frac{1}{2}(\sqrt{3}+i)$ and $\beta$ be a non-zero complex number. The points $a z_{1}+\beta, \alpha z_{2}+\beta, \alpha z_{3}+\beta$ will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the vertices of an equilateral triangle</span> </div>

answer:
<div class="solution">Since, $z_{1}, z_{2}$ and $z_{3}$ are the vertices of an
<br/>equilateral triangle, therefore
<br/>$$
<br/>\begin{aligned}
<br/>\left|z_{1}-z_{2}\right| &amp;=\left|z_{2}-z_{3}\right| \\
<br/>&amp;=\left|z_{3}-z_{1}\right|=k \\
<br/>\alpha &amp;=\frac{1}{2}(\sqrt{3}+i)
<br/>\end{aligned}
<br/>$$
<br/>Also,
<br/>$\Rightarrow \quad|\alpha|=\frac{1}{2} \sqrt{3+1}=\frac{1}{2} \times 2=1$
<br/>Let $A=\alpha z_{1}+\beta, B=\alpha z_{2}+\beta$
<br/>and
<br/>$$
<br/>C=\alpha z_{3}+\beta
<br/>$$
<br/>Now. $|A B|=\left|\alpha z_{2}+\beta-\left(\alpha z_{1}+\beta\right)\right|$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=\left|\alpha\left(z_{2}-z_{1}\right)\right| \\
<br/>&amp;=|\alpha|\left|z_{2}-z_{1}\right| \\
<br/>&amp;=|1|\left|z_{2}-z_{1}\right| \\
<br/>&amp;=1\left|z_{2}-z_{1}\right| \\
<br/>&amp;=\left|z_{2}-z_{1}\right|=k \\
<br/>\text { Similarly, } &amp; B C=C A=k
<br/>\end{aligned}
<br/>$$
<br/>Hence, the points $\alpha z_{1}+\beta, \alpha z_{2}+\beta$ and $\alpha z_{3}+\beta$
<br/>are the vertices of an equilateral triangle.</div>


tags: ['Chemistry', 'Solid State', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
suppose-the-mass-of-a-single-agatom-is-m-ag-metal-crystallises-in-fcc-lattice-with-unit-cell-of-length-a-the-density-of-ag-metal-in-terms-of-a-and-m-i

slug:
suppose-the-mass-of-a-single-agatom-is-m-ag-metal-crystallises-in-fcc-lattice-with-unit-cell-of-length-a-the-density-of-ag-metal-in-terms-of-a-and-m-i-34801

question:
<div class="question">Suppose the mass of a single Ag-atom is $m$. Ag metal crystallises in fcc lattice with unit cell of length $a$. The density of Ag metal in terms of $a$ and $m$ is</div>

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

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

answer:
<div class="solution">Given that,
<br/>Mass of single Ag-atom $=m$
<br/>Unit cell length $=a$
<br/>Total atoms present per unit cell of fcc lattice $=4$ 
<br/>Therefore, mass of unit cell becomes $4 \mathrm{m}$. 
<br/>Volume of unit cell $=a^{3}$
<br/>Density $(\rho)=\frac{\text { Mass of unit cell }}{\text { Volume of unit cell }}=\frac{4 \mathrm{m}}{\mathrm{a}^{3}}$</div>


title: tan-4-2-1-cos-1-b-a-tan-4-2-1-cos-1-b-a-is-equal-to
slug: tan-4-2-1-cos-1-b-a-tan-4-2-1-cos-1-b-a-is-equal-to-65252
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">$\tan \left[\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{a}{b}\right)\right]+\tan \left[\frac{\pi}{4}-\frac{1}{2} \cos ^{-1}\left(\frac{a}{b}\right)\right]$ is equal to</div>

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

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

answer:
<div class="solution">Hints : Let $\frac{1}{2} \cos ^{-1}\left(\frac{a}{b}\right)=\theta$, then $\cos 2 \theta=\frac{a}{b}$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \tan \left[\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{a}{b}\right)\right]+\tan \left[\frac{\pi}{4}-\frac{1}{2} \cos ^{-1}\left(\frac{a}{b}\right)\right] \\<br/>&amp; =\tan \left(\frac{\pi}{4}+\theta\right)+\tan \left(\frac{\pi}{4}-\theta\right)=2\left(\frac{1+\tan ^2 \theta}{1-\tan ^2 \theta}\right)=\frac{2}{\cos 2 \theta}=\frac{2}{a / b}=\frac{2 b}{a}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
tangent-is-drawn-at-any-point-p-x-y-on-a-curve-which-passes-through-1-1-the-tangent-cuts-xaxis-and-yaxis-at-a-and-b-respectively-if-apbp-31-then

slug:
tangent-is-drawn-at-any-point-p-x-y-on-a-curve-which-passes-through-1-1-the-tangent-cuts-xaxis-and-yaxis-at-a-and-b-respectively-if-apbp-31-then-29710

question:
<div class="question">Tangent is drawn at any point $\mathrm{P}(\mathrm{x}, \mathrm{y})$ on a curve, which passes through $(1,1)$. The tangent cuts X-axis and Y-axis at A and B respectively. If AP:BP = 3:1, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">the differential equation of the curve is $3 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$, the curve passes through $\left(\frac{1}{8}, 2\right)$</span> </div>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/gXDwAvu2ZJUIH-j_ZB2g7YYww0wi1qEseSw41uBfqRQ.original.fullsize.png"/><br/> <br/>$\frac{P A}{P B}=\frac{3}{1}$
<br/>Equation of tangent $\mathrm{AB}$ is $\mathrm{Y}-\mathrm{y}=\frac{\mathrm{d} \mathrm{y}}{\mathrm{dx}}(\mathrm{X}-\mathrm{x})$
<br/>$\mathrm{A}\left(\frac{\mathrm{xy}^{\prime}-\mathrm{y}}{\mathrm{y}^{\prime}}, 0\right)$ and $\mathrm{B}\left(0, \mathrm{y}-\mathrm{xy}^{\prime}\right)$
<br/>Using section formula : $\mathrm{y}=\frac{1 \times 0+3 \times\left(\mathrm{y}-\mathrm{xy}^{\prime}\right)}{4} \Rightarrow 4 \mathrm{y}=3 \mathrm{y}-3 \mathrm{xy}^{\prime} \Rightarrow 3 \mathrm{xy}^{\prime}=-\mathrm{y} \Rightarrow \frac{3 \mathrm{xdy}}{\mathrm{dx}}+\mathrm{y}=0$
<br/>$\frac{3 d y}{y}+\frac{d x}{x}=0$
<br/>$\mathrm{xy}^{3}=1$
<br/>$\frac{\mathrm{dy}}{\mathrm{dx}_{(1,1)}}=\frac{-1}{3}$
<br/>Slope of Normal $=3$
<br/>Equation of Normal $\Rightarrow y-1=3(x-1) \Rightarrow y-3 x+2=0$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
tangents-are-drawn-to-the-ellipse-9-x-2-5-y-2-1-at-the-ends-of-both-latusrectum-the-area-of-the-quadrilateral-so-formed-is

slug:
tangents-are-drawn-to-the-ellipse-9-x-2-5-y-2-1-at-the-ends-of-both-latusrectum-the-area-of-the-quadrilateral-so-formed-is-91494

question:
<div class="question">Tangents are drawn to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ at the ends of both latusrectum. The area of the quadrilateral, so formed is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">27 sq units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{13}{2}$ sq units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{15}{4} \mathrm{sq}$ units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">45 sq units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">27 sq units</span> </div>

answer:
<div class="solution">We have, $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$
<br/>$\Rightarrow \quad a=3$ and $b=\sqrt{5}$
<br/>$\begin{aligned} \therefore \quad e &amp;=\sqrt{1-\frac{b^{2}}{a^{2}}} \\ &amp;=\sqrt{1-\frac{5}{9}}=\frac{2}{3} \end{aligned}$
<br/>$\therefore$ Foci $=(\pm a e, 0)$
<br/>$=\left(\pm 3 \times \frac{2}{3}, 0\right)=(\pm 2,0)$
<br/>$\therefore$ Ends of latusrectum $=\left(\pm 2, \pm \frac{5}{3}\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a60.png"/> <br/> <br/>Equation of tangent at the end of
<br/>latusrectum $P$ is
<br/>$\frac{x \times 2}{9}+\frac{y \times 5 / 3}{5}=1$
<br/>$\frac{2}{9} x+\frac{1}{3} y=1$
<br/>$2 x+3 y=9$
<br/>$O A=\frac{9}{2}$ and $O B=3$
<br/>$\therefore$ Area of quadrilateral $A B C D$
<br/>$$
<br/>=4 \times \text { Area of } \Delta O A B
<br/>$$
<br/>$=4 \times \frac{1}{2} \times O A \times O B$
<br/>$=4 \times \frac{1}{2} \times \frac{9}{2} \times 3$
<br/>$=27 \mathrm{sq}$ units.</div>


title: tautomerism-is-exhibited-by
slug: tautomerism-is-exhibited-by-96884
question: <div class="question">Tautomerism is exhibited by</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\mathrm{Me}_{3} \mathrm{CCO}\right)_{3} \mathrm{CH}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q58_optb.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q58_optc.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q58_optd.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\left(\mathrm{Me}_{3} \mathrm{CCO}\right)_{3} \mathrm{CH}$, <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q58_optb.png"/>, <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q58_optd.png"/></span> </div>

answer:
<div class="solution">Availability of acidic $\alpha-H$ - atoms at $\left(^{*}\right)$ marked positions, enable the compounds to show ketoenol tautomerism.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a58.png"/></div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
temperature-of-an-ideal-gas-initially-at-2-7-c-is-raised-by-6-c-the-rms-velocity-of-the-gas-molecules-will

slug:
temperature-of-an-ideal-gas-initially-at-2-7-c-is-raised-by-6-c-the-rms-velocity-of-the-gas-molecules-will-91985

question:
<div class="question">Temperature of an ideal gas, initially at $27^{\circ} \mathrm{C}$, is raised by $6^{\circ} \mathrm{C}$. The rms velocity of the gas molecules will</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">increase by nearly $1 \%$</span> </div>

answer:
<div class="solution">Initial temperature of ideal gas,
<br/>$$
<br/>T_{1}=273+27=300 \mathrm{K}
<br/>$$
<br/>when temperature of gas is raised by $6^{\circ} \mathrm{C},$ the final temperature of gas
<br/>$$
<br/>\begin{aligned}
<br/>T_{2} &amp;=273+6+27 \\
<br/>&amp;=306 \mathrm{K}
<br/>\end{aligned}
<br/>$$
<br/>Let initial velocities are $v_{r m_{1}},$ and $v_{r m_{2}}$
<br/>$v_{\mathrm{rms}} \propto \sqrt{T}$
<br/>$\begin{aligned} \frac{v_{\operatorname{mas}_{1}}}{v_{\operatorname{mas}_{2}}} &amp;=\sqrt{\frac{T_{1}}{T_{2}}} \\ v_{\mathrm{rms}_{2}} &amp;=\sqrt{\frac{T_{2}}{T_{1}} \times v_{\mathrm{ms}_{1}}} \\ &amp;=\sqrt{\frac{306}{300}} \times v_{\mathrm{rms}} \\ &amp;=1 \cdot 00 \times v_{\mathrm{rms}_{1}} \end{aligned}$
<br/>So, it will increase by $1 \%$</div>


title: the-4-th-higher-homologue-of-ethane-is
slug: the-4-th-higher-homologue-of-ethane-is-41008
question: <div class="question">The 4 th higher homologue of ethane is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">hexane</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">butane</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">pentane</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">hexane</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">heptane</span> </li> </ul>

answer:
<div class="solution">The molecular formula of ethane is $\mathrm{C}_{2} \mathrm{H}_{\mathrm{6}}$ To get its fourth homologue, we have to add
<br/>$$
<br/>\begin{array}{r}
<br/>4 \times \mathrm{CH}_{2}=\mathrm{C}_{4} \mathrm{H}_{8} \text { to it. } \\
<br/>\mathrm{C}_{2} \mathrm{H}_{6}+\mathrm{C}_4 \mathrm{H}_{8}=\mathrm{C}_6 \mathrm{H}_{14}
<br/>\end{array}
<br/>$$
<br/>$\mathrm{C}_{6} \mathrm{H}_{14}$ is the molecular formula of hexane. Thus, hexane is the fourth homologue of ethane.</div>


title: the-above-conversion-can-be-carried-out-by
slug: the-above-conversion-can-be-carried-out-by-88556
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/LgQVTif2XbjBAXnFayRfKE3JPIuc0vvSqifqbb42rX0.original.fullsize.png"/><br/><br/>The above conversion can be carried out by,</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{Zn}-\mathrm{Hg} /$ Conc. $\mathrm{HCl}$, i. $\mathrm{H}_2 \mathrm{NNH}_2$ ii. $\mathrm{NaOH}$ in ethylene glycol, $\Delta$, i. $\mathrm{HSCH}_2 \mathrm{CH}_2 \mathrm{SH} / \mathrm{H}^{\oplus}$ ii. $\mathrm{H}_2 / \mathrm{Ni}$</span> </div>

answer:
<div class="solution">(A) $\mathrm{Zn}-\mathrm{Hg} /$ Conc. $\mathrm{HCl}-$ Clemmensen Reduction<br/>(B) $\mathrm{NH}_2-\mathrm{NH}_2, \mathrm{NaOH}$ in ethylene glycol, $\Delta-$ Wolf-Kishner reduction<br/>(C) i. $\mathrm{HSCH}_2 \mathrm{CH}_2 \mathrm{SH} / \mathrm{H}^{+}$ii. $\mathrm{H}_2 / \mathrm{Ni}-$ Mozingo method.<br/>These are known to reduce $-\mathrm{CO}-$ to $-\mathrm{CH}_2-$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-acceleration-a-in-ms-2-of-a-body-starting-from-rest-varies-with-time-t-in-s-following-the-equation-a-3-t-4-the-velocity-of-the-body-at-time-t-2-s-

slug:
the-acceleration-a-in-ms-2-of-a-body-starting-from-rest-varies-with-time-t-in-s-following-the-equation-a-3-t-4-the-velocity-of-the-body-at-time-t-2-s-39712

question:
<div class="question">The acceleration ' $a$ ' (in $\mathrm{ms}^{-2}$ ) of a body, starting from rest varies with time $t$ (in $s$ ) following the equation $a=3 t+4$ The velocity of the body at time $t=2 \mathrm{~s}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10 \mathrm{~ms}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$18 \mathrm{~ms}^{-1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$14 \mathrm{~ms}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$26 \mathrm{~ms}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$14 \mathrm{~ms}^{-1}$</span> </div>

answer:
<div class="solution">$a=3 t+4$<br/>$\begin{aligned} &amp; \frac{d V}{d t}=3 t+4 \\ &amp; \int_0^V d V=\int_0^t(3 t+4) d t \\ &amp; V=\frac{3 t^2}{2}+4 t=\frac{12}{2}+8=14 \mathrm{~m} / \mathrm{s}\end{aligned}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-acceleration-of-a-particle-starting-from-rest-moving-in-a-straight-line-with-uniform-acceleration-is-8-mathrmm-mathrmsec2-the-time-taken-by-the-pa

slug:
the-acceleration-of-a-particle-starting-from-rest-moving-in-a-straight-line-with-uniform-acceleration-is-8-mathrmm-mathrmsec2-the-time-taken-by-the-pa-87952

question:
<div class="question">The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is \(8 \mathrm{~m} / \mathrm{sec}^2\). The time taken by the particle to move the second metre is</div>

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

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

answer:
<div class="solution">Hint: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/E5R_ks-SE200Fj3LvzWuwDqUgdhhx4TjPmManXxGM10.original.fullsize.png"/><br/><br/>\(\begin{array}{l|l}<br/>\mathrm{S}=\mathrm{ut}+\frac{1}{2} \mathrm{at}^2 &amp; \mathrm{~S}=\mathrm{uT}+\frac{1}{2} \mathrm{aT}^2 \\<br/>1=\frac{1}{2} \cdot 8 \cdot \mathrm{t}^2 &amp; 2=\frac{1}{2} 8 \cdot \mathrm{T}^2 \\<br/>\frac{1}{4}=\mathrm{t}^2 &amp; \mathrm{~T}^2=\frac{1}{2} \\<br/>\mathrm{t}=\frac{1}{2} &amp; \mathrm{~T}=\frac{1}{\sqrt{2}}<br/>\end{array}\)<br/>\(\text {Time }=\frac{1}{\sqrt{2}}-\frac{1}{2}=\frac{2}{2}=\frac{\sqrt{2}-1}{2}\)</div>


tags: ['Physics', 'Motion In One Dimension', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
the-acceleration-vs-distance-graph-for-a-particle-moving-with-initial-velocity-5-m-s-is-shown-in-the-fiture-the-velocity-of-the-particle-at-x-35-m-wil-1

slug:
the-acceleration-vs-distance-graph-for-a-particle-moving-with-initial-velocity-5-m-s-is-shown-in-the-fiture-the-velocity-of-the-particle-at-x-35-m-wil-1-48781

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2Di6SBMus5o9FfBzz4GkXVvZzm2GM3koZklnzDCmswc.original.fullsize.png"/><br/> <br/>The acceleration vs distance graph for a particle moving with initial velocity $5 \mathrm{~m} / \mathrm{s}$ is shown in the fiture. The velocity of the particle at $x=35 \mathrm{~m}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20.62 \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$20 \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$25 \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$50 \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$25 \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">Area of graph
<br/>$$
<br/>\begin{array}{l}
<br/>=\int \mathrm{adx}=\int \frac{\mathrm{vdv}}{\mathrm{dx}} \cdot \mathrm{d} \mathrm{x}==\int_{\mathrm{V}_{\mathrm{i}}}^{\mathrm{v}_{\mathrm{f}}} \mathrm{v} \mathrm{d} \mathrm{v} \\
<br/>=\left[\frac{\mathrm{v}^{2}}{2}\right]_{\mathrm{v}_{\mathrm{i}}}^{\mathrm{v}_{\mathrm{f}}}, 150+150=\frac{1}{2}\left[\mathrm{~V}_{\mathrm{f}}^{2}-\mathrm{V}_{\mathrm{i}}^{2}\right]
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>300 \times 2=V_{f}^{2}-(5)^{2} \\
<br/>V_{f}^{2}=600+25, V_{f}=25
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Motion In One Dimension', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
the-acceleration-vs-distance-graph-for-a-particle-moving-with-initial-velocity-5-m-s-is-shown-in-the-fiture-the-velocity-of-the-particle-at-x-35-m-wil

slug:
the-acceleration-vs-distance-graph-for-a-particle-moving-with-initial-velocity-5-m-s-is-shown-in-the-fiture-the-velocity-of-the-particle-at-x-35-m-wil-66385

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2Di6SBMus5o9FfBzz4GkXVvZzm2GM3koZklnzDCmswc.original.fullsize.png"/><br/> <br/>The acceleration vs distance graph for a particle moving with initial velocity $5 \mathrm{~m} / \mathrm{s}$ is shown in the fiture. The velocity of the particle at $x=35 \mathrm{~m}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20.62 \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$20 \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$25 \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$50 \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$25 \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">Area of graph
<br/>$$
<br/>\begin{array}{l}
<br/>=\int \mathrm{adx}=\int \frac{\mathrm{vdv}}{\mathrm{dx}} \cdot \mathrm{d} \mathrm{x}==\int_{\mathrm{V}_{\mathrm{i}}}^{\mathrm{v}_{\mathrm{f}}} \mathrm{v} \mathrm{d} \mathrm{v} \\
<br/>=\left[\frac{\mathrm{v}^{2}}{2}\right]_{\mathrm{v}_{\mathrm{i}}}^{\mathrm{v}_{\mathrm{f}}}, 150+150=\frac{1}{2}\left[\mathrm{~V}_{\mathrm{f}}^{2}-\mathrm{V}_{\mathrm{i}}^{2}\right]
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\begin{array}{l}
<br/>300 \times 2=V_{f}^{2}-(5)^{2} \\
<br/>V_{f}^{2}=600+25, V_{f}=25
<br/>\end{array}
<br/>$$</div>


title: the-acid-in-which-o-o-bonding-is-present-is
slug: the-acid-in-which-o-o-bonding-is-present-is-14888
question: <div class="question">The acid in which $\mathrm{O}-\mathrm{O}$ bonding is present is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{6}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{8}$</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">$\mathrm{H}_{2} \mathrm{S}_{4} \mathrm{O}_{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{8}$</span> </div>

answer:
<div class="solution">$\mathrm{H}_{2} \mathrm{S}_{2} \mathrm{O}_{8}$ shows $-\mathrm{O}-\mathrm{O}-$ (peroxy) bonding as follows:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_a14.png"/></div>


title: the-addition-of-hbr-to-2pentene-gives
slug: the-addition-of-hbr-to-2pentene-gives-71737
question: <div class="question">The addition of $\mathrm{HBr}$ to 2-pentene gives</div>
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2-bromopentane only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3-bromopentane only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2-bromopentane and 3-bromopentane</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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-bromopentane and 3-bromopentane</span> </li> </ul>

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

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/75IMYooqGAifqbZWvoBWLKWeuiD7CxK3rOj4DYjNZ1w.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Surface Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-amount-of-electrolytes-required-to-coagulate-a-given-amount-of-agl-colloidal-solution-ve-charge-will-be-in-the-order

slug:
the-amount-of-electrolytes-required-to-coagulate-a-given-amount-of-agl-colloidal-solution-ve-charge-will-be-in-the-order-45919

question:
<div class="question">The amount of electrolytes required to coagulate a given amount of Agl colloidal solution (-ve charge) will be in the order</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$NaNO_{3}&gt;\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}&gt;\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}&gt;\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}&gt;\mathrm{NaNO}_{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}&gt;\mathrm{NaNO}_{3}&gt;\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{NaNO}_{3}&gt;\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}&gt;\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{NaNO}_{3}&gt;\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}&gt;\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}$</span> </div>

answer:
<div class="solution">Since, AgI is a negatively charged sol, so according to Hardy Schulze law, higher the positive charge on the metal atoms, more is the tendency of metal atom to cause coagulation of negatively charged colloid. The order of positive charge is
<br/>$$
<br/>\mathrm{Na}^{+}    &lt; \mathrm{Ba}^{2+} &lt; \mathrm{AI}^{3 +}
<br/>$$
<br/>Thus, the order of amount of electrolytes required is in the order
<br/>$$
<br/>\mathrm{NaNO}_{3}&gt;\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}&gt;\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}
<br/>$$
<br/>Note In paper, the formula of aluminium nitrait is incorrect. Correct formula of aluminium nitrea is $\mathrm{Al}_{2}\left(\mathrm{NO}_{3}\right)_{3}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-amount-of-the-heat-released-when-20-mathrmml-05-mathrmm-mathrmnaoh-is-mixed-with-100-mathrmml-01-mathrmm-mathrmhcl-is-mathrmx-mathrmkj-the-heat-of

slug:
the-amount-of-the-heat-released-when-20-mathrmml-05-mathrmm-mathrmnaoh-is-mixed-with-100-mathrmml-01-mathrmm-mathrmhcl-is-mathrmx-mathrmkj-the-heat-of-33830

question:
<div class="question">The amount of the heat released when \(20 \mathrm{ml} 0.5 \mathrm{M} \mathrm{NaOH}\) is mixed with \(100 \mathrm{ml} 0.1 \mathrm{M} \mathrm{HCl}\) is \(\mathrm{x} \mathrm{kJ}\). The heat of neutralization is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-100 \mathrm{~xkJ} / \mathrm{mol}\)</span> </div>

answer:
<div class="solution">Hints: \(\underset{20 \times 05}{\mathrm{NaOH}}+\underset{100 \times 01}{\mathrm{HCl}} \rightarrow \mathrm{NaCl}+\underset{10 \text { millimole prodiral }}{\mathrm{H}_2 \mathrm{O}}\)<br/>During formation of 10 millimole of \(\mathrm{H}_2 \mathrm{O}\) the heat released is \(\mathrm{xKJ}\). Therefore heat of neutralisation is \(-100 \times \mathrm{KJ} / \mathrm{mol}\) (heat released hence negative)</div>


title: the-angle-between-a-normal-to-the-plane-2-xy2-z10-and-the-xaxis-is
slug: the-angle-between-a-normal-to-the-plane-2-xy2-z10-and-the-xaxis-is-88745
tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The angle between a normal to the plane \(2 x-y+2 z-1=0\) and the X-axis is</div>

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

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

answer:
<div class="solution">Hint: Normal : 2, -1, 2<br/>\(\text {X-axis : } 1,0,0\)<br/>\(\therefore \theta=\cos ^{-1} \frac{2}{3}\)</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-angle-between-a-pair-of-tangents-drawn-from-a-point-p-to-the-circle-x-2-y-2-4-x-6-y-9-sin-2-13-cos-2-0-is-2-the-equation-of-the-locus-of-the-point

slug:
the-angle-between-a-pair-of-tangents-drawn-from-a-point-p-to-the-circle-x-2-y-2-4-x-6-y-9-sin-2-13-cos-2-0-is-2-the-equation-of-the-locus-of-the-point-68557

question:
<div class="question">The angle between a pair of tangents drawn from a point $P$ to the circle
<br/>$x^{2}+y^{2}+4 x-6 y+9 \sin ^{2} \alpha+$
<br/>$13 \cos ^{2} \alpha=0$ is $2 \alpha .$ The equation of the locus of the point $P$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+4 x+6 y+9=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}-4 x+6 y+9=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}-4 x-6 y+9=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}+4 x-6 y+9=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+4 x-6 y+9=0$</span> </div>

answer:
<div class="solution">We have equation of circle $x^{2}+y^{2}+4 x-6 y+9 \sin ^{2} \alpha+13 \cos ^{2} \alpha=0$
<br/>Here, $\begin{aligned} C &amp;=(-2,3) \\ \text { Radius } &amp;=\sqrt{(-2)^{2}+(3)^{2}-\left(9 \sin ^{2} \alpha+13 \cos ^{2} \alpha\right)} \\ &amp;=\sqrt{4+9-9 \sin ^{2} \alpha-13 \cos ^{2} \alpha} \\ &amp;=\sqrt{13-13\left(1-\sin ^{2} \alpha\right)-9 \sin ^{2} \alpha} \\=&amp; \sqrt{13 \sin ^{2} \alpha-9 \sin ^{2} \alpha} \\=&amp; \sqrt{4 \sin ^{2} \alpha}=2 \sin \alpha \\ \text { Here, } &amp; \sin \alpha=\frac{A C}{P C} \end{aligned}$
<br/>$\Rightarrow \quad P C \sin \alpha=A C$
<br/>$\Rightarrow \quad P C^{2} \sin ^{2} \alpha=A C^{2}=(2 \sin \alpha)^{2}$
<br/>$\Rightarrow \quad\left[\left(h+2)^{2}+(k-3)^{2}\right] \sin ^{2} \alpha=4 \sin ^{2} \alpha\right.$
<br/>$\Rightarrow \quad(h+2)^{2}+(k-3)^{2}=4$
<br/>$\Rightarrow \quad h^{2}+4+4 h+k^{2}+9-6 k=4$
<br/>$\Rightarrow \quad h^{2}+k^{2}+4 h-6 k+9=0$
<br/>Hence, locus of a point is $x^{2}+y^{2}+4 x-6 y+9=0$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a40.png"/></div>


tags: ['Mathematics', 'Ellipse', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
the-angle-between-the-lines-joining-the-foci-of-an-ellipse-to-one-particular-extremity-of-the-minor-axis-is-9-0-the-eccentricity-of-the-ellipse-is-1

slug:
the-angle-between-the-lines-joining-the-foci-of-an-ellipse-to-one-particular-extremity-of-the-minor-axis-is-9-0-the-eccentricity-of-the-ellipse-is-1-21674

question:
<div class="question">The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is $90^{\circ}$. The eccentricity of the ellipse is</div>

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

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

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>\text { } \frac{b}{a e}=\tan \frac{\pi}{4} \\<br/>\qquad \begin{array}{c}<br/>a e \Rightarrow \frac{b}{a}=e \\<br/>e^2=1-\frac{b^2}{a^2} \\<br/>e^2=1-e^2<br/>\end{array}<br/>\end{gathered}<br/>$$<br/><br/><br/>$$<br/>e^2=\frac{1}{2} \Rightarrow e=\frac{1}{\sqrt{2}}<br/>$$<br/><br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/jtBAd99n7EcmTx-TLfL5DNC2Occ2iDsJui9-Buwv6dk.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
title:
the-angle-between-the-lines-joining-the-foci-of-an-ellipse-to-one-particular-extremity-of-the-minor-axis-is-9-0-the-eccentricity-of-the-ellipse-is

slug:
the-angle-between-the-lines-joining-the-foci-of-an-ellipse-to-one-particular-extremity-of-the-minor-axis-is-9-0-the-eccentricity-of-the-ellipse-is-75963

question:
<div class="question">The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is $90^{\circ}$. The eccentricity of the ellipse is</div>

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

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

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>\text { } \frac{b}{a e}=\tan \frac{\pi}{4} \\<br/>\qquad \begin{array}{c}<br/>a e \Rightarrow \frac{b}{a}=e \\<br/>e^2=1-\frac{b^2}{a^2} \\<br/>e^2=1-e^2<br/>\end{array}<br/>\end{gathered}<br/>$$<br/><br/><br/>$$<br/>e^2=\frac{1}{2} \Rightarrow e=\frac{1}{\sqrt{2}}<br/>$$<br/><br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/jtBAd99n7EcmTx-TLfL5DNC2Occ2iDsJui9-Buwv6dk.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-angle-of-intersection-between-the-curves-y-sin-x-cos-x-and-x-2-y-2-10-where-x-denotes-the-greatest-integer-x-is

slug:
the-angle-of-intersection-between-the-curves-y-sin-x-cos-x-and-x-2-y-2-10-where-x-denotes-the-greatest-integer-x-is-58259

question:
<div class="question">The angle of intersection between the curves $y=[|\sin x|+|\cos x|]$ and $x^{2}+y^{2}=10,$ where $[x]$ denotes the greatest integer $\leq x,$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\tan ^{-1} 3$</span> </div>

answer:
<div class="solution">Given, $y=\left[|\sin x|+|\cos x|\right.$ land $x^{2}+y^{2}=10$
<br/>We know that $(|\sin x|+|\cos x|) \in[1, \sqrt{2}]$
<br/>$\therefore$
<br/>$$
<br/>y=1
<br/>$$
<br/>The point of intersection of given curve is
<br/>$$
<br/>\begin{aligned}
<br/>&amp; &amp; x^{2}+1^{2} &amp;=10 \\
<br/>\Rightarrow &amp; &amp; x^{2} &amp;=9 \\
<br/>\Rightarrow &amp; &amp; x &amp;=\pm 3
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Point of intersection is (Â±3,1) $\begin{array}{lr}\text { Now, } &amp; x^{2}+y^{2}=10 \\ \Rightarrow \quad 2 x+2 y \frac{d y}{d x} &amp; =0\end{array}$
<br/>$\Rightarrow \quad \frac{d y}{d x}=-\frac{x}{y}$
<br/>At point (-3,1) $\frac{d y}{d x}=\frac{3}{1}=3 \Rightarrow m_{1}=3$
<br/>Slope of line $y=1$ is $m_{2}=0$
<br/>$\therefore$ Angle between two curves is
<br/>$$
<br/>\begin{aligned}
<br/>\tan \theta &amp;=\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}=3 \\
<br/>\Rightarrow \quad \theta &amp;=\tan ^{-1}(3)
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Properties of Triangles', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-angles-of-a-triangle-are-in-the-ratio-2-3-7-and-the-radius-of-the-circumscribed-ircle-is-10-cm-the-length-of-the-smallest-ide-is

slug:
the-angles-of-a-triangle-are-in-the-ratio-2-3-7-and-the-radius-of-the-circumscribed-ircle-is-10-cm-the-length-of-the-smallest-ide-is-95351

question:
<div class="question">The angles of a triangle are in the ratio 2: 3: 7 and the radius of the circumscribed ircle is $10 \mathrm{cm}$. The length of the smallest ide is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$7 \mathrm{cm}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$10 \mathrm{cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, that angles of a triangles are $2 x, 3 x$ and
<br/>$7 x$
<br/>Since, $2 x+3 x+7 x=180$
<br/>$\Rightarrow 12 x=180^{\circ} \Rightarrow x=15^{\circ}$
<br/>So, angles are $30^{\circ}, 45^{\circ}$ and $105^{\circ}$
<br/>Now, $\frac{a}{\sin 30^{\circ}}=2 R$
<br/>$\Rightarrow \quad \frac{a}{\frac{1}{2}}=2 \times 10 \quad[\because R=10 \mathrm{cm}]$
<br/>$\Rightarrow \quad 2 a=20 \Rightarrow a=10 \mathrm{cm}$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-angular-points-of-a-triangle-are-a-1-7-b-5-1-and-c-1-4-the-equation-of-the-bisector-of-the-angle-a-bc-is

slug:
the-angular-points-of-a-triangle-are-a-1-7-b-5-1-and-c-1-4-the-equation-of-the-bisector-of-the-angle-a-bc-is-28379

question:
<div class="question">The angular points of a triangle are $A(-1,-7)$ $B(5,1)$ and $C(1,4)$. The equation of the bisector of the angle $\angle A B C$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=7 y+2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$($ b) $7 y=x+2$</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">$y=7 x+2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7 x=y+2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$($ b) $7 y=x+2$</span> </div>

answer:
<div class="solution">Here, $A B=\sqrt{(5+1)^{2}+(1+7)^{2}}=\sqrt{36+64}=10$
<br/>$B C=\sqrt{\left(1-5)^{2}+(4-1)^{2}\right.}=\sqrt{16+9}=5$
<br/>By angle bisector theorem,
<br/>$$
<br/>A P: C P=10: 5=2: 1
<br/>$$
<br/>$P\left(\frac{2 \times 1+1 \times(-1)}{2+1}, \frac{2 \times 4+1 \times(-7)}{2+1}\right)=P\left(\frac{1}{3}, \frac{1}{3}\right)$
<br/>Required equation of $B P$ is $y-1=\frac{\frac{1}{3}-1}{\frac{1}{3}-5}(x-5)$
<br/>$\Rightarrow \quad y-1=\frac{-2}{-14}(x-5)$
<br/>$\Rightarrow \quad 7 y-7=x-5$
<br/>$\Rightarrow \quad 7 y=x+2$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a42.png"/></div>


title: the-approximate-value-of-sin-3-1-is
slug: the-approximate-value-of-sin-3-1-is-33150
question: <div class="question">The approximate value of $\sin 31^{\circ}$ is</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$&gt;0.5$</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">$&gt;0.6$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$    &lt; 0.5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$    &lt; 0.4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$&gt;0.5$</span> </div>

answer:
<div class="solution">We know that, $\sin 30^{\circ}=\frac{1}{2}=0.5$ In $1^{\text {st }}$ quadrant $\sin x$ is increasing function. $\sin 31^{\circ}&gt;\sin 30^{\circ}$
<br/>$\Rightarrow \quad \sin 31^{\circ}&gt;0.5$</div>


title: the-approximate-value-of-sqrt533-correct-to-4-decimal-places-is
slug: the-approximate-value-of-sqrt533-correct-to-4-decimal-places-is-84417
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2.0125</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The approximate value of \(\sqrt[5]{33}\) correct to 4 decimal places is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2.0000</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2.1001</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2.0125</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">2.0500</span> </li> </ul>

answer:
<div class="solution">Hints: \(y=x^{1 / 5} \Delta y=\left(\frac{d y}{d x}\right) \Delta x=1 / 80 \times 1\)<br/>\(y=2+1 / 80\)</div>


title: the-area-bounded-by-the-parabolas-y-4-x-2-y-9-x-2-and-the-line-y-2-is
slug: the-area-bounded-by-the-parabolas-y-4-x-2-y-9-x-2-and-the-line-y-2-is-19177
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area bounded by the parabolas $y=4 x^2, y=\frac{x^2}{9}$ and the line $y=2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5 \sqrt{2}}{3}$ sq. units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{10 \sqrt{2}}{3}$ sq. units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{15 \sqrt{2}}{3}$ sq. units</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{20 \sqrt{2}}{3}$ sq. units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{20 \sqrt{2}}{3}$ sq. units</span> </div>

answer:
<div class="solution">Hints: $y=4 x^2$<br/>$$<br/>y=\frac{x^2}{4}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp; A=\int_{\mathrm{r}}^2\left[\frac{\sqrt{y}}{2}-3 \sqrt{y}\right] d y=\left(\frac{1}{2}-3\right) \int_0^2 \sqrt{y} d y \\<br/>&amp; =\left(\frac{-\sqrt{\mathrm{y}}}{2}\right) \frac{5}{3}\left(\mathrm{y}^{3 / 2}\right)_0^2=-\frac{5}{3}(2 \sqrt{2}-0) \\<br/>&amp; =\left|-\frac{\sqrt{2}}{3}\right|=\frac{10 \sqrt{2}}{3} \text {, Area of bounded figure }=2 \mathrm{~A}=\frac{20 \sqrt{2}}{3} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


title: the-area-bounded-by-the-parabolas-y-4-x-2-y-9-x-2-and-the-straight-line-y-2-is
slug: the-area-bounded-by-the-parabolas-y-4-x-2-y-9-x-2-and-the-straight-line-y-2-is-48530
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area bounded by the parabolas $y=4 x^{2}, y=\frac{x^{2}}{9}$ and the straight line $y=2$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{20 \sqrt{2}}{3}$ sq. 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">$10 \sqrt{5}$ sq. unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{10 \sqrt{3}}{7}$ sq. unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10 \sqrt{2}$ sq. unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{20 \sqrt{2}}{3}$ sq. unit</span> </div>

answer:
<div class="solution">$y=4 x^{2}, y=\frac{x^{2}}{9}, y=2$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/1II_Cg2ulio41yz_q1rFklkg-BlC4qrI8LxRLMzyBCk.original.fullsize.png"/><br/>$A=2 \int_{0}^{2}\left(3 \sqrt{y}-\frac{\sqrt{y}}{2}\right) d y=2 \int_{0}^{2} \frac{5 \sqrt{y}}{2} d y=\left.5 \frac{y^{3 / 2}}{3 / 2}\right|_{0} ^{2}=\frac{10}{3}\left[2^{3 / 2}\right]=\frac{20 \sqrt{2}}{3}$</div>


title: the-area-bounded-by-y-x-1-and-y-cos-x-and-the-x-axis-is-1
slug: the-area-bounded-by-y-x-1-and-y-cos-x-and-the-x-axis-is-1-76943
question: <div class="question">The area bounded by $y=x+1$ and $y=\cos x$ and the $X$ -axis, is</div>
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 sq unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{3}{2}$ sq unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}$ sq unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{8}$ sq unit</span> </li> </ul>

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

answer:
<div class="solution">We have, $y=x+1 \text { and } y=\cos x$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a68.png"/> <br/>$\begin{aligned} \text { Require area } &amp;=\int_{-1}^{0}(x+1) d x+\int_{0}^{\pi / 2} \cos x d x \\ &amp;=\left[\frac{x^{2}}{2}+x\right]_{-1}^{0}+[\sin x]_{0}^{\pi / 2} \\ &amp;=\left[(0)-\left(\frac{1}{2}-1\right)\right]+[1-0] \\ &amp;=\frac{1}{2}+1=\frac{3}{2} \text { sq unit. } \end{aligned}$</div>


title: the-area-bounded-by-y24-x-and-x24-y-is
slug: the-area-bounded-by-y24-x-and-x24-y-is-35334
question: <div class="question">The area bounded by \(y^2=4 x\) and \(x^2=4 y\) is</div>
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{20}{3}\) sq. unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{16}{3}\) sq. unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{14}{3}\) sq. unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{10}{3}\) sq. unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{16}{3}\) sq. unit</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Tw6uZzx7mp-SmIIeQa_NniuEjWawi3ShYvBLlld3QMQ.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \text { Hints: } A=\int_0^4\left(\sqrt{4 \mathrm{x}}-\frac{\mathrm{x}^2}{4}\right) \mathrm{dx} \\ &amp; \left.=2 \cdot \frac{2}{3}\left(\mathrm{x}^{3 / 2}\right)_0^4-\frac{1}{4.3}\right)\left(\mathrm{x}^3\right)_0^4 \\ &amp; =\frac{4}{3}\left[4^{3 / 2}-0\right]-\frac{1}{4.3}\left[4^3-0\right] \\ &amp; =\frac{4^{5 / 2}}{3}-\frac{16}{3}=\frac{32-16}{3}=\frac{16}{3}\end{aligned}\)</div>


title: the-area-enclosed-between-y2x-and-yx-is
slug: the-area-enclosed-between-y2x-and-yx-is-94720
question: <div class="question">The area enclosed between \(y^2=x\) and \(y=x\) is</div>
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{2}{3}\) sq. units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{2}\) units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{1}{3}\) units</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{6}\) units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">\(\frac{1}{6}\) units</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/96pbHyqQFdxA_h3AuaSdf5SbJLyxn5UsSihaQCv5_AU.original.fullsize.png"/><br/><br/>\(\begin{aligned} &amp; \text {Hints: } A=\int_0^1(\sqrt{x}-x) d x \\ &amp; =\frac{2}{3}\left(x^{3 / 2}\right)_0^1-\frac{1}{2}\left(x^2\right)_0^1=\frac{2}{3}[1-0]-\frac{1}{2}[1-0] \\ &amp; =\frac{2}{3}-\frac{1}{2}=\frac{4-3}{6}=\frac{1}{6}\end{aligned}\)</div>


title: the-area-enclosed-by-y-3-x-5-y-0-x-3-and-x-5-is
slug: the-area-enclosed-by-y-3-x-5-y-0-x-3-and-x-5-is-66138
question: <div class="question">The area enclosed by $y=3 x-5, y=0, x=3$ and $x=5$ is</div>
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12 sq. units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">13 sq. units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$13 \frac{1}{2}$ sq. units</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$14 \mathrm{sq}$. units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$14 \mathrm{sq}$. units</span> </div>

answer:
<div class="solution">Hints: $A=\int_3^5(3 x-5) d x$<br/>$=\frac{3}{2}\left(x^2\right)_3^5-5(x)_3^5,=\frac{3}{2}[25-9]-5(5-3)$<br/>$\frac{3}{2} \cdot 16-5(2)=24-10=14$</div>


title: the-area-enclosed-by-y-5-x-2-and-y-x-1-is
slug: the-area-enclosed-by-y-5-x-2-and-y-x-1-is-99298
question: <div class="question">The area enclosed by $y=\sqrt{5-x^{2}}$ and <br/>$y=|x-1|$ is</div>
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\frac{5 \pi}{4}-2\right)$ sq units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{5 \pi-2}{2}\right)$ sq units</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{5 \pi}{4}-\frac{1}{2}\right)$ sq units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{\pi}{2}-5\right)$ sq units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{5 \pi}{4}-\frac{1}{2}\right)$ sq units</span> </div>

answer:
<div class="solution">The graphs of $y=|x-1|$ and $y=\sqrt{5-x^{2}}$ are shown in the figure and the shaded region is the required region bounded by the two curves.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a4.png"/> <br/> <br/>Let $A$ be the area bounded by the given curves. Then
<br/>$$
<br/>\begin{aligned}
<br/>A &amp;=\int_{-1}^{1}\left(\sqrt{5-x^{2}}+x-1\right) d x+\int_{1}^{2}\left(\sqrt{5-x^{2}}-x+1\right) d x \\
<br/>&amp;=\int_{-1}^{1} \sqrt{5-x^{2}} d x+\int_{1}^{2} \sqrt{5-x^{2}} d x +\int_{-1}^{1}(x-1) d x+\int_{1}^{2}(-x+1) d x \\
<br/>&amp;=\int_{-1}^{2} \sqrt{5-x^{2}} d x+\left[\frac{x^{2}}{2}-x\right]^{1}_{-1}+\left[-\frac{x^{2}}{2}+x\right]_{1}^{2} \\
<br/>&amp;=\left[\frac{1}{2} x \sqrt{5-x^{2}}+\frac{5}{2} \sin ^{-1} \frac{x}{\sqrt{5}}\right]_{-1}^{2}-\frac{5}{2} \\
<br/>&amp;=1+\frac{5}{2} \sin ^{-1} \frac{2}{\sqrt{5}}+1+\frac{5}{2} \sin ^{-1} \frac{1}{\sqrt{5}}-\frac{5}{2} \\
<br/>&amp;=-\frac{1}{2}+\frac{5}{2} \sin ^{-1}\left(\frac{2}{\sqrt{5}} \times \sqrt{1-\frac{1}{5}}+\frac{1}{\sqrt{5}} \sqrt{1-\frac{4}{5}}\right) \\
<br/>&amp;=-\frac{1}{2}+\frac{5}{2} \sin ^{-1}(1)=\frac{5 \pi}{4}-\frac{1}{2} \mathrm{squnits}
<br/>\end{aligned}
<br/>$$</div>


title: the-area-of-the-figure-bounded-by-the-parabola-x-2-y-2-x-1-3-y-2-is-1
slug: the-area-of-the-figure-bounded-by-the-parabola-x-2-y-2-x-1-3-y-2-is-1-43416
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area of the figure bounded by the parabola $x=-2 y^{2}, x=1-3 y^{2}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{4}{3}$ square unit</span> </div>

answer:
<div class="solution">Hint:
<br/>$\mathrm{A}=\int_{-1}^{1}\left(1-3 \mathrm{y}^{2}\right)-\left(-2 \mathrm{y}^{2}\right) \mathrm{d} \mathrm{y}$
<br/>$\Rightarrow \mathrm{A}=2 \int_{0}^{1}\left(1-\mathrm{y}^{2}\right) \mathrm{d} \mathrm{y}=\frac{4}{3}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/W7pFc-GiFrJEr0AOi-dpWSzveatl5YswqolP63y1PvY.original.fullsize.png"/></div>


title: the-area-of-the-figure-bounded-by-the-parabolas-x-2-y-2-and-x-1-3-y-2-is
slug: the-area-of-the-figure-bounded-by-the-parabolas-x-2-y-2-and-x-1-3-y-2-is-88826
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area of the figure bounded by the parabolas $x=-2 y^{2}$ and $x=1-3 y^{2}$ is</div>

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

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

answer:
<div class="solution">We have, $x=-2 y^{2}$
<br/>$\Rightarrow \quad y^{2}=-\frac{1}{2} x$ ...(i)
<br/>and $\quad x=1-3 y^{2}$
<br/>$\Rightarrow y^{2}=-\frac{1}{3}(x-1)$  ...(ii)
<br/>From Eqs. (i) and (ii),we get. 
<br/>$\begin{aligned} &amp;-\frac{1}{2} x=-\frac{1}{3}(x-1) \\ \Rightarrow &amp; 3 x=2(x-1) \\ \Rightarrow &amp; 3 x=2 x-2 \\ \Rightarrow &amp; x=-2 \\ \Rightarrow &amp; y^{2}=-\frac{1}{2}(-2)=1 \\ \Rightarrow &amp; y=\pm 1 \end{aligned}$
<br/>$\therefore$ Point of intersection of two curves is $(-2 \pm 1)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a59.png"/> <br/> <br/>$\therefore$ Required Area $=2 \int_{0}^{1} \sin x d x$
<br/>$=2\left[y-\frac{y^{3}}{3}\right]_{0}^{1}$
<br/>$=2\left[1-\frac{1}{3}\right]$
<br/>$=\frac{4}{3} \mathrm{sq}$ units</div>


title: the-area-of-the-region-bounded-by-the-curve-y-x-3-its-tangent-at-11-and-xaxis-is
slug: the-area-of-the-region-bounded-by-the-curve-y-x-3-its-tangent-at-11-and-xaxis-is-21630
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area of the region bounded by the curve $y=x^3$ its tangent at (1,1) and X-axis is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{12} \mathrm{sq}$ 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">$\frac{1}{6} \mathrm{sq}$ unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{17} \mathrm{sq}$ unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{15} \mathrm{sq}$ unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{12} \mathrm{sq}$ unit</span> </div>

answer:
<div class="solution">We have, $y=x^{3}$ and $A(1,1)$
<br/>$\therefore$ $\frac{d y}{d x}=3 x^{2}$
<br/>On putting $x=1$ in $\mathrm{Eq}$. $\frac{d y}{d x}=3(1)^{2}=3$
<br/>$\therefore$ Equation of tangent at $A(1,1)$ is
<br/>$y-1=3(x-1) \Rightarrow y=3 x-2$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_mat_a111.png"/> <br/> <br/>$\therefore$ Required area $=\int_{0}^{1} x^{3} d x-\int_{2 / 3}^{1}(3 x-2) d x$
<br/>$=\left[\frac{x^{4}}{4}\right]_{0}^{1}-\left[\frac{3 x^{2}}{2}-2 x\right]_{2 / 3}^{1}$
<br/>$=\frac{1}{4}-\left[\left(\frac{3}{2}-2\right)-\left(\frac{2}{3}-\frac{4}{3}\right)\right]$
<br/>$=\frac{1}{12} \mathrm{sq}$ unit</div>


tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
title:
the-area-of-the-region-bounded-by-the-curves-y-sin-1-x-x-1-x-and-y-sin-1-x-x-1-x-in-the-first-quadrant-is

slug:
the-area-of-the-region-bounded-by-the-curves-y-sin-1-x-x-1-x-and-y-sin-1-x-x-1-x-in-the-first-quadrant-is-32142

question:
<div class="question">The area of the region, bounded by the curves $y=\sin ^{-1} x+x(1-x)$ and
<br/>$y=\sin ^{-1} x-x(1-x)$ in the first quadrant, is</div>

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

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

answer:
<div class="solution">$\sin ^{-1} x$ is defined, if $-1 \leq x \leq 1$ In first quadrant $0 \leq x \leq 1$ and $x(1-x) \geq 0$
<br/>$\therefore$
<br/>$y=\sin ^{-1} x+x(1-x)$
<br/>Lies above $y=\sin ^{-1} x-x(1-x)$
<br/>On solving, we get $2 x(1-x)=0$
<br/>$\Rightarrow \quad \quad \quad \quad \quad \quad \quad \quad \quad=0,1$
<br/>$\therefore$ Required area $=\int_{0}^{1}\left(y_{1}-y_{2}\right) d x$
<br/>$=\int_{0}^{1}\left[\left\{\sin ^{-1} x+x(1-x)\right\}\right.$
<br/>$\left.-\left\{\sin ^{-1} x-x(1-x)\right\}\right] d x$
<br/>$=2 \int_{0}^{1}\left(x-x^{2}\right) d x$
<br/>$=2\left[\frac{x^{2}}{2}-\frac{x^{3}}{3}\right]_{0}^{1}=2\left(\frac{1}{2}-\frac{1}{3}\right)=\frac{1}{3}$</div>


title: the-area-of-the-region-bounded-by-the-curves-y-x-2-and-x-y-2-is-1
slug: the-area-of-the-region-bounded-by-the-curves-y-x-2-and-x-y-2-is-1-40711
question: <div class="question">The area of the region bounded by the curves $y=x^{2}$ and $x=y^{2}$ is</div>
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 2$</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">3</span> </li> </ul>

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

answer:
<div class="solution">Given curves are $y=x^{2}$ and $x=y^{2}$, which is the form of parabola.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_mat_a9.png"/> <br/>The point of intersection, $x=\left(x^{2}\right)^{2}$$
<br/>\begin{array}{l}
<br/>\Rightarrow \quad x=x^{4} \Rightarrow x\left(1-x^{3}\right)=0 \\
<br/>\Rightarrow \quad x=0 \text { and } 1=x^{3} \\
<br/>\Rightarrow \quad x=0 \text { and } x=1 \\
<br/>\text { When } x=0, \text { then } y=0 \\
<br/>\text { When } x=1, \text { then } y=1^{2}=1
<br/>\end{array}
<br/>$
<br/>$\therefore$ The point of intersection is (0,0) and (1,1)
<br/>$\therefore$ Area of shaded region
<br/>$$
<br/>\begin{array}{l}
<br/>=\int_{0}^{1}\left(y_{2}-y_{1}\right) d x \\
<br/>=\int_{0}^{1}\left[\sqrt{x}-x^{2}\right] d x=\left[\frac{x^{3 / 2}}{3 / 2}-\frac{x^{3}}{3}\right]_{0}^{1} \\
<br/>=\frac{2}{3}(1)^{3 / 2}-\frac{(1)^{3}}{3}-0-0 \\
<br/>=\frac{2}{3}-\frac{1}{3}=\frac{1}{3} \mathrm{sq} \text { units }
<br/>\end{array}
<br/>$$</div>


title: the-area-of-the-region-bounded-by-the-curves-y-x-3-y-x-1-x-2-is-1
slug: the-area-of-the-region-bounded-by-the-curves-y-x-3-y-x-1-x-2-is-1-62506
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area of the region bounded by the curves $y=x^{3}, y=\frac{1}{x}, x=2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4-\log _{e} 2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4}+\log _{e} 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3-\log _{e} 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{15}{4}-\log _{e} 2$</span> </li> </ul>

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

answer:
<div class="solution">First of all we draw the graph $y=x^{3}, y=\frac{1}{x}, x=2$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a70.png"/> <br/> <br/>Required area i.e., (OMPNO) $=\int_{0}^{1} x^{3} d x+\int_{1}^{2} \frac{1}{x} d x$
<br/>$=\left[\frac{x^{4}}{4}\right]_{0}+\left[\log _{4} x\right]_{1}^{2}=\frac{1}{4}+\log _{c} 2$</div>


title: the-area-of-the-region-bounded-by-the-parabola-y-x-2-4-x-5-and-the-straight-line-y-x-1-is-1
slug: the-area-of-the-region-bounded-by-the-parabola-y-x-2-4-x-5-and-the-straight-line-y-x-1-is-1-63051
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area of the region bounded by the parabola $y=x^{2}-4 x+5$ and the straight line $y=x+1$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{9}{2}$</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">$\frac{9}{2}$</span> </div>

answer:
<div class="solution">Given equation of parabola is
<br/>$$
<br/>y=x^{2}-4 x+5
<br/>$$
<br/>$\Rightarrow \quad y=(x-2)^{2}+1$
<br/>$\Rightarrow \quad(x-2)^{2}=(y-1)$
<br/>and equation of line is
<br/>$$
<br/>\begin{array}{r}
<br/>y=x+1 \\
<br/>\Rightarrow \quad x-y=-1
<br/>\end{array}
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a59.png"/> <br/> <br/> <br/>On putting the value of $(y-1)$ from Eqs. (ii) in (1). we get
<br/>$$
<br/>(x-2)^{2}=x \Rightarrow x^{2}+4-4 x=x
<br/>$$
<br/>$\Rightarrow \quad x^{2}-5 x+4=0$
<br/>$\Rightarrow x^{2}-4 x-x+4=0$
<br/>$\Rightarrow \quad x(x-4)-1(x-4)=0$
<br/>$\Rightarrow \quad(x-1)(x-4)=0$ or $x=1,4$
<br/>then from is (ii) $y=2,5$
<br/>$\therefore$ Required area $=\int_{1}^{4}\left\{(x+1)-\left(x^{2}-4 x+5\right)\right\} d x$
<br/>$=\int_{1}^{4}\left(-x^{2}+5 x-4\right) d x=\left[\frac{-x^{3}}{3}+\frac{5 x^{2}}{2}-4 x\right]_{1}^{4}$
<br/>$=\left[-\frac{64}{3}+40-16+\frac{1}{3}-\frac{5}{2}+4\right]$
<br/>$=\left(-21-\frac{3}{2}+28\right)=\frac{9}{2}$</div>


title: the-area-of-the-region-bounded-by-y-2-x-and-y-x-is
slug: the-area-of-the-region-bounded-by-y-2-x-and-y-x-is-81864
question: <div class="question">The area of the region bounded by $y^2=x$ and $y=|x|$ is</div>
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$ sq.unit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{6}$ sq.unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$ sq. unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 sq.unit</span> </li> </ul>

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

answer:
<div class="solution">Hints: $\mathrm{y}^2=\mathrm{x}$<br/>$$<br/>\left.\int_0^1(\sqrt{x}-x) d x=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}-\frac{x^2}{2}\right]_0^1=\frac{3}{2}-\frac{1}{2}=\frac{4-3}{6}=\frac{1}{6}<br/>$$</div>


tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-area-of-the-region-enclosed-between-parabola-y-2-x-and-the-line-y-m-x-is-48-1-then-the-value-of-m-is

slug:
the-area-of-the-region-enclosed-between-parabola-y-2-x-and-the-line-y-m-x-is-48-1-then-the-value-of-m-is-58777

question:
<div class="question">The area of the region enclosed between parabola $y^{2}=x$ and the line $y=m x$ is $\frac{1}{48}$
<br/>Then, the value of $m$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">-2, 2</span> </div>

answer:
<div class="solution">Equation of parabola is $y^{2}=x$ and line $y=m x$
<br/>For intersection point of both curves put $x=y^{2}$,
<br/>we get
<br/>$$
<br/>y=m y^{2} \Rightarrow y(m y-1)=0
<br/>$$
<br/>$\Rightarrow \quad y=0$ or $y=\frac{1}{m}$
<br/>Then, $\quad x=0$ or $x=\frac{1}{m^{2}}$
<br/>$\therefore$ Intersection points are (0,0) and $P\left(\frac{1}{m^{2}}, \frac{1}{m}\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_mat_a77.png"/> <br/> <br/>$\therefore$ Required area $=\int_{0}^{1 / m}\left(\frac{y}{m}-y^{2}\right)|d y=|\left[\frac{y^{2}}{2 m}-\frac{y^{3}}{3}\right]_{0}^{1 / m} \mid$
<br/>$=\left|\frac{1}{2 m^{3}}-\frac{1}{3 m^{3}}\right|=\left|\frac{1}{6 m^{3}}\right|=\frac{1}{48}$
<br/>$\Rightarrow \quad \frac{1}{6 m^{3}}=\pm \frac{1}{48} \Rightarrow m^{3}=\pm 8$
<br/>(given)
<br/>Now, if $m^{3}=8$
<br/>$\Rightarrow \quad m^{3}=(2)^{3} \Rightarrow m=2$
<br/>If $\quad m^{3}=-8$
<br/>$\Rightarrow \quad m^{3}=(-2)^{3} \Rightarrow m=-2$</div>


tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-area-of-the-region-lying-above-x-axis-and-included-between-the-circle-x-2-y-2-2-a-x-and-the-parabola-y-2-a-x-a-0-is

slug:
the-area-of-the-region-lying-above-x-axis-and-included-between-the-circle-x-2-y-2-2-a-x-and-the-parabola-y-2-a-x-a-0-is-96015

question:
<div class="question">The area of the region lying above $X$ -axis, and included between the circle $x^{2}+y^{2}=2 a x$ and the parabola $y^{2}=a x, a&gt;0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 \pi a^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a^{2}\left(\frac{\pi}{4}-\frac{2}{3}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{16 \pi a^{2}}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi\left(\frac{27}{8}+3 a^{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$a^{2}\left(\frac{\pi}{4}-\frac{2}{3}\right)$</span> </div>

answer:
<div class="solution">Given, equation of circle $x^{2}+y^{2}=2 a x$
<br/>$\Rightarrow$
<br/>$(x-a)^{2}+y^{2}=a^{2}$
<br/>and equation of parabola is $y^{2}=a x, a&gt;0$
<br/>Intersection points of circle and parabola
<br/>$\begin{array}{lr}\Rightarrow &amp; x^{2}+a x=2 a x \\ \Rightarrow &amp; x^{2}=a x\end{array}$
<br/>$\Rightarrow \quad x^{2}-a x=0$
<br/>$\Rightarrow \quad x(x-a)=0$
<br/>$\Rightarrow \quad x=0, a$
<br/>Intersecting points are $(0,0)$ and $(a, a)$
<br/>$\therefore$ Required area $=\frac{\pi a^{2}}{4}-\int_{0}^{a} \sqrt{a x} d x$
<br/>$=\frac{\pi a^{2}}{4}-\sqrt{a}\left(\frac{x^{3 / 2}}{3 / 2}\right)_{0}^{a}$
<br/>$=\frac{\pi a^{2}}{4}-\frac{2 a^{2}}{3}=a^{2}\left(\frac{\pi}{4}-\frac{2}{3}\right)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a68.png"/></div>


title: the-area-of-the-region-x-y-x-2-y-2-1-x-y-is
slug: the-area-of-the-region-x-y-x-2-y-2-1-x-y-is-76473
tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The area of the region $\left\{(x, y): x^{2}+y^{2} \leq 1 \leq x+y\right\}$ is</div>

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

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

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/POYR2TMO0YHWVTml10kHdrDr-2dIbrikx_bEToW8WAE.original.fullsize.png"/><br/> <br/>Area of shaded region $=\frac{\pi}{4}-\frac{1}{2}$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-area-of-the-triangle-formed-by-the-intersection-of-a-line-parallel-to-x-axis-and-passing-through-p-h-k-with-the-lines-y-x-and-x-y-2-is-h-2-the-loc

slug:
the-area-of-the-triangle-formed-by-the-intersection-of-a-line-parallel-to-x-axis-and-passing-through-p-h-k-with-the-lines-y-x-and-x-y-2-is-h-2-the-loc-15560

question:
<div class="question">The area of the triangle formed by the intersection of a line parallel to $X$ -axis and passing through $P(h, k),$ with the lines $y=x$ and $x+y=2$ is $h^{2}$. The locus of the point $P$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$x=y-1$, $x=-(y-1)$</span> </div>

answer:
<div class="solution">Given, ar $(\Delta A B C)=h^{2}$
<br/>$\Rightarrow \quad \frac{1}{2}\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ k &amp; k &amp; 1 \\ 2-k &amp; k &amp; 1\end{array}\right|=\pm h^{2}$
<br/>$\Rightarrow 1(k-k)-1(k-2+k)+1\left(k^{2}-2 k+k^{2}\right)=\pm 2 h^{2}$
<br/>$\Rightarrow \quad-(2 k-2)+\left(2 k^{2}-2 k\right)=\pm 2 h^{2}$
<br/>$\Rightarrow \quad 2-2 k+2 k^{2}-2 k=\pm 2 h^{2}$
<br/>$\Rightarrow \quad 2 k^{2}-4 k+2=\pm 2 h^{2}$
<br/>$\Rightarrow$
<br/>$k^{2}-2 k+1=\pm h^{2}$
<br/>Hence, locus of a point is
<br/>$\Rightarrow$
<br/>$(k-1)^{2}=h^{2}$
<br/>$y-1=\pm x$
<br/>$\Rightarrow \quad x=y-1$ or $x=-(y-1)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a72.png"/></div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: db\MarksBatch2_P2.db
title:
the-atomic-masses-of-helium-and-neon-are-40-and-200-amu-respectively-the-value-of-the-de-broglie-wavelength-of-helium-gas-at-7-3-c-is-m-times-the-de-b

slug:
the-atomic-masses-of-helium-and-neon-are-40-and-200-amu-respectively-the-value-of-the-de-broglie-wavelength-of-helium-gas-at-7-3-c-is-m-times-the-de-b-30209

question:
<div class="question">The atomic masses of helium and neon are $4.0$ and $20.0$ amu respectively. The value of the de Broglie wavelength of helium gas at $-73^{\circ} \mathrm{C}$ is $\mathrm{M}$ times the de Broglie wavelength of neon at $727^{\circ} \mathrm{C}$. The value of $\mathrm{M}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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><li class=""> <span class="option-label">B</span> <span class="option-data">25</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{25}$</span> </li> </ul>

answer:
<div class="solution">$\lambda_{\mathrm{He}}(200 \mathrm{~K})=\mathrm{M} \times \lambda_{\mathrm{Ne}}(1000 \mathrm{~K})$ $\left(\lambda=\frac{\mathrm{h}}{\mathrm{m} \cdot \mathrm{v}}\right)$
<br/>$\frac{\mathrm{h}}{4 \mathrm{amu} \times \mathrm{C}_{\mathrm{mp}}(200)}=\mathrm{M} \times \frac{\mathrm{h}}{20 \mathrm{amu} \times \mathrm{C}_{\mathrm{mp}}(1000)} \quad\left(\mathrm{C}_{\mathrm{mp}}=\sqrt{\frac{2 \mathrm{KT}}{\mathrm{m}}}\right)$
<br/>$\Rightarrow \frac{1}{4 \times \sqrt{\frac{2 \mathrm{~K} \times 200}{4}}}=\frac{\mathrm{M}}{20 \times \sqrt{\frac{2 \mathrm{~K} \times 1000}{20}}} \Rightarrow \frac{1}{4 \times \sqrt{50}}=\frac{\mathrm{M}}{20 \times \sqrt{50}} \Rightarrow \mathrm{M}=5$</div>


title: the-atomic-number-of-cerium-ce-is-58-the-correct-electronic-confipuration-of-ce-3-ion-is
slug: the-atomic-number-of-cerium-ce-is-58-the-correct-electronic-confipuration-of-ce-3-ion-is-59320
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The atomic number of cerium $(\mathrm{Ce})$ is 58 . The correct electronic confipuration of $\mathrm{Ce}^{3+}$ ion is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$[\mathrm{Xe}] 4 f^{1}$</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">$[\mathrm{Kr}] 4 f^{1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[\mathrm{Xe}] 4 f^{13}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$[\mathrm{Kr}] 4 d^{1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$[\mathrm{Xe}] 4 f^{1}$</span> </div>

answer:
<div class="solution">Atomic number of cerium $=58$
<br/>Electronic configuration of Ce
<br/>$$
<br/>=[\mathrm{Xe}] 4 f^{2}, 5 d^{0}, 6 s^{2}
<br/>$$
<br/>Electronic configuration of $\mathrm{Ce}^{3+}$
<br/>$$
<br/>=[\mathrm{Xe}] 4 f^{1}
<br/>$$</div>


title: the-average-length-of-all-vertical-chords-of-the-hyperbola-fracx2a2fracy2b21-a-leq-x-leq-2-a-is
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
slug:
the-average-length-of-all-vertical-chords-of-the-hyperbola-fracx2a2fracy2b21-a-leq-x-leq-2-a-is-14318

question:
<div class="question">The average length of all vertical chords of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, a \leq x \leq 2 a\), is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">None of these</span> </div>

answer:
<div class="solution">Hint : \(A_L=\frac{2 \int_a^{2 a} y d x}{\int_a^{2 a} d x}\)<br/>\(=\frac{2 \int_a^{2 a} \frac{b}{a} \sqrt{x^2-a^2} d x}{\int_a^{2 a} d x}=b[2 \sqrt{3}-\ell \mathrm{n}(2+\sqrt{3})]\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0KGSdKTGxMhCIS-49ohaHOYALDd5Ze1M7Fjie9E32B4.original.fullsize.png"/><br/></div>


title: the-average-ordinate-of-ysin-x-over-0-pi-is
slug: the-average-ordinate-of-ysin-x-over-0-pi-is-74692
question: <div class="question">The average ordinate of \(y=\sin x\) over \([0, \pi]\) is</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{2}{\pi}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{3}{\pi}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{4}{\pi}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\pi\)</span> </li> </ul>

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

answer:
<div class="solution">Hint : As \(\sin x\) is a continuous function in \([0, \pi]\)<br/>Average ordinate will be \(=\frac{1}{\pi-0} \int_0^\pi \sin x d x=\frac{[-\cos x]_0^\pi}{\pi}=\frac{2}{\pi}\)</div>


title: the-average-speed-of-h-2-at-t-1-k-is-equal-to-that-of-o-2-at-t-2-k-the-ratio-t-1-t-2-is
slug: the-average-speed-of-h-2-at-t-1-k-is-equal-to-that-of-o-2-at-t-2-k-the-ratio-t-1-t-2-is-97316
tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The average speed of $\mathrm{H}_2$ at $T_1 \mathrm{~K}$ is equal to that of $\mathrm{O}_2$ at $T_2 \mathrm{~K}$. The ratio $T_1: T_2$ is</div>

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

answer:
<div class="solution">$\left(\mathrm{C}_{\mathrm{av}}\right)_{\mathrm{H}_2}=\left(\mathrm{C}_{\mathrm{av}}\right)_{\mathrm{O}_2}$<br/>$\sqrt{\frac{8 \mathrm{RT}_1}{\pi \mathrm{M}_{\mathrm{H}_2}}}=\sqrt{\frac{8 \mathrm{RT}_2}{\pi \mathrm{M}_{\mathrm{O}_2}}}$<br/>$\therefore \frac{\mathrm{T}_1}{\mathrm{~T}_2}=\frac{\mathrm{M}_{\mathrm{H}_2}}{\mathrm{~N}_{\mathrm{O}_2}}=\frac{2}{32}=\frac{1}{16}$</div>


title: the-axis-of-the-parabola-x-2-2-x-y-y-2-5-x-5-y-5-0-is
slug: the-axis-of-the-parabola-x-2-2-x-y-y-2-5-x-5-y-5-0-is-83883
question: <div class="question">The axis of the parabola $x^{2}+2 x y+y^{2}-5 x+5 y-5=0$ is</div>
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x+y=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+y-1=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x-y+1=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y=\frac{1}{\sqrt{2}}$</span> </li> </ul>

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

answer:
<div class="solution">Given, equation $x^{2}+2 x y+y^{2}-5 x+5 y-5=0$
<br/>$\Rightarrow(x+y)^{2}=5 x-5 y+5$
<br/>$\Rightarrow(x+y)^{2}=5(x-y+1)$
<br/>$\therefore$ Axis of the parabola is $x+y=0$</div>


title: the-basicity-of-aniline-is-weaker-in-comparison-to-that-of-methyl-amine-due-to
slug: the-basicity-of-aniline-is-weaker-in-comparison-to-that-of-methyl-amine-due-to-16778
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The basicity of aniline is weaker in comparison to that of methyl amine due to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">resonance effect of phenyl group in aniline</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a28.png"/> <br/> <br/>The lone pair of electrons, present in N-atom in aniline involve in ring resonance. So it is less basic than methyl amine.</div>


title: the-best-method-for-preparation-of-m-e-3-ccn-is
slug: the-best-method-for-preparation-of-m-e-3-ccn-is-64122
question: <div class="question">The best method for preparation of $Me_{3}\mathrm{CCN}$ is</div>
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">to react $Me_{3} \mathrm{COH}$ with HCN</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">to react $Me_{3}$ CBr with NaCN</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">to react $Me_{3} \mathrm{CMgBr}$ with CICN</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">to react $Me_{3} \mathrm{Cl}$ with $\mathrm{NH}_{2} \mathrm{CN}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">to react $Me_{3} \mathrm{CMgBr}$ with CICN</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a50.png"/></div>


title: the-best-method-for-the-preparation-of-2-2dimethylbutane-is-via-the-reaction-of
slug: the-best-method-for-the-preparation-of-2-2dimethylbutane-is-via-the-reaction-of-28594
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The best method for the preparation of 2, 2-dimethylbutane is via the reaction of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\mathrm{Me}_{3} \mathrm{C}\right)_{2} \mathrm{CuLi}$ and $\mathrm{MeCH}_{2} \mathrm{Br}$</span> </div>

answer:
<div class="solution">2,2 - dimethyl butane is prepared from Corey House alkane synthesis. $\left(\mathrm{Me}_{3} \mathrm{C}\right)_{2} \mathrm{Cul}_{2} \mathrm{Li}+\mathrm{MeCH}_{2} \mathrm{Br} \stackrel{\mathrm{S}_{\mathrm{N}}{2}}{\longrightarrow}$
<br/>$$
<br/>\mathrm{Me}_{3} \mathrm{C}-\mathrm{CH}_{2} \mathrm{CH}_{3}+\mathrm{Me}_{3} \mathrm{C}CuLi
<br/>$$</div>


tags: ['Physics', 'Oscillations', 'JEE Main']
source_db: db\MarksBatch2_P2.db
title:
the-bob-of-a-pendulum-of-mass-m-suspended-by-an-inextensible-string-of-length-l-as-shown-in-the-figure-carries-a-small-charge-a-an-infinite-horizontal-1

slug:
the-bob-of-a-pendulum-of-mass-m-suspended-by-an-inextensible-string-of-length-l-as-shown-in-the-figure-carries-a-small-charge-a-an-infinite-horizontal-1-72863

question:
<div class="question">The bob of a pendulum of mass $m$, suspended by an inextensible string of length $L$ as shown in the figure carries a small charge $a$. An infinite horizontal plane conductor with uniform surface charge density $\sigma$ is placed below it. What will be the time period of the pendulum for small amplitude oscillations?
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q35..png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \pi \sqrt{\frac{\mathrm{L}}{\left(\mathrm{g}-\frac{\mathrm{q} \sigma}{\varepsilon_{0} \mathrm{~m}}\right)}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a35.png"/> <br/> <br/>$\therefore$ Apparent weight of the bob, $w^{\prime}=m g-q E$
<br/>$$
<br/>m y^{\prime}=m g-q E
<br/>$$
<br/>$\therefore$
<br/>$g^{\prime}=g-\frac{q E}{m}$
<br/>Time period of a pendulum. $T=2 \pi \sqrt{\frac{L}{g_{e f f}}}$ $=2 \pi \sqrt{\frac{L}{g-\frac{q E}{m}}}=2 \pi \sqrt{\frac{L}{\left(g-\frac{q \sigma}{\varepsilon_{0} m}\right)}}$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-bob-of-a-pendulum-of-mass-m-suspended-by-an-inextensible-string-of-length-l-as-shown-in-the-figure-carries-a-small-charge-a-an-infinite-horizontal

slug:
the-bob-of-a-pendulum-of-mass-m-suspended-by-an-inextensible-string-of-length-l-as-shown-in-the-figure-carries-a-small-charge-a-an-infinite-horizontal-79790

question:
<div class="question">The bob of a pendulum of mass $m$, suspended by an inextensible string of length $L$ as shown in the figure carries a small charge $a$. An infinite horizontal plane conductor with uniform surface charge density $\sigma$ is placed below it. What will be the time period of the pendulum for small amplitude oscillations?
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_q35..png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \pi \sqrt{\frac{\mathrm{L}}{\left(\mathrm{g}-\frac{\mathrm{q} \sigma}{\varepsilon_{0} \mathrm{~m}}\right)}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_phy_a35.png"/> <br/> <br/>$\therefore$ Apparent weight of the bob, $w^{\prime}=m g-q E$
<br/>$$
<br/>m y^{\prime}=m g-q E
<br/>$$
<br/>$\therefore$
<br/>$g^{\prime}=g-\frac{q E}{m}$
<br/>Time period of a pendulum. $T=2 \pi \sqrt{\frac{L}{g_{e f f}}}$ $=2 \pi \sqrt{\frac{L}{g-\frac{q E}{m}}}=2 \pi \sqrt{\frac{L}{\left(g-\frac{q \sigma}{\varepsilon_{0} m}\right)}}$</div>


tags: ['Physics', 'Work Power Energy', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
title:
the-bob-of-a-swinging-seconds-pendulum-one-whose-time-period-is-2-s-has-a-small-speed-v-0-at-its-lowest-point-its-height-from-this-lowest-point-225-s-

slug:
the-bob-of-a-swinging-seconds-pendulum-one-whose-time-period-is-2-s-has-a-small-speed-v-0-at-its-lowest-point-its-height-from-this-lowest-point-225-s-29555

question:
<div class="question">The bob of a swinging seconds pendulum (one whose time period is $2 \mathrm{~s}$ ) has a small speed $\mathrm{v}_{0}$ at its lowest point. Its height from this lowest point $2.25 \mathrm{~s}$ after passing through it is given by</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{v_{0}^{2}}{4 g}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$T=2$ sec.
<br/>at $\mathrm{t}=\mathrm{T}+\mathrm{T} / 8$
<br/>$\mathrm{t}=\mathrm{T}+\mathrm{T} / 8 \quad \mathrm{v}=\mathrm{A} \omega \cos \omega \mathrm{t}=\mathrm{v}_{0} / \sqrt{2}$
<br/>By Mechanical energy conservation:
<br/>$\left(\frac{v_{0}}{\sqrt{2}}\right)^{2}=v_{0}^{2}-2 g h \quad h=\frac{v_{0}^{2}}{4 g}$</div>


title: the-boiling-point-of-the-water-is-higher-than-liquid-hf-the-reason-is-that-1
slug: the-boiling-point-of-the-water-is-higher-than-liquid-hf-the-reason-is-that-1-93780
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The boiling point of the water is higher than liquid $\mathrm{HF}$. The reason is that</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/h9nPftlWG4ZjOIaYSRle02B4XNHrNYFNnYKGpAqOE6k.original.fullsize.png"/></div>


title: the-boiling-points-of-hf-hcl-hbr-and-hi-follow-the-order
slug: the-boiling-points-of-hf-hcl-hbr-and-hi-follow-the-order-35855
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The boiling points of $\mathrm{HF}, \mathrm{HCl}, \mathrm{HBr}$ and $\mathrm{HI}$ follow the order</div>

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

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

answer:
<div class="solution">Due to hydrogen bonding, HF shows highest boiling point.
<br/>Thereafter van der Waals' force decide the boiling point.
<br/>Larger the size or molecules mass, greater is the van der Waals' forces. Hence, higher is the boiling point.
<br/>$\because$ van der Waals' forces is more for HI than HBr, which is in turn is more than HCl.
<br/>Hence, correct order is $\mathrm{HF}&gt;\mathrm{Hl}&gt;\mathrm{HBr}&gt;\mathrm{HCl}$</div>


title: the-bond-angle-in-nf-3-102-3-is-smaller-than-nh-3-107-2-this-is-because-of-1
slug: the-bond-angle-in-nf-3-102-3-is-smaller-than-nh-3-107-2-this-is-because-of-1-82214
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The bond angle in $\mathrm{NF}_{3}\left(102.3^{\circ}\right)$ is smaller than $\mathrm{NH}_{3}\left(107.2^{\circ}\right) .$ This is because of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">opposite polarity of $\mathrm{N}$ in the two molecules</span> </div>

answer:
<div class="solution">In $\mathrm{NF}_{3}$, $\mathrm{N}$ is less electronegative as compared to F but in $\mathrm{NH}_{3}$, it is more electronegative than $\mathrm{H}$. And in case of same central atom. as the electronegativity of other atoms increases, bond angle decreases, Thus, bond angle in $\mathrm{NF}_{3}$ is smaller than that in $\mathrm{NH}_{3}$ because of the difference in the polarity of $\mathrm{N}$ in these molecules.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a53.png"/></div>


title: the-bond-order-of-co-molecule-is
slug: the-bond-order-of-co-molecule-is-74698
question: <div class="question">The bond order of $\mathrm{CO}$ molecule is</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2.5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3.5</span> </li> </ul>

answer:
<div class="solution">Hints : $\mathrm{CO} \rightarrow \sigma(1 \mathrm{~S})^2, \sigma^*(1 \mathrm{~S})^2, \sigma(2 \mathrm{~S})^2, \sigma\left(2 \mathrm{P}_z\right)^2, \pi\left(2 \mathrm{P}_{\mathrm{z}}\right)^2=\pi\left(2 \mathrm{P}_y\right)^2, \sigma^*(2 \mathrm{~S})^2$<br/>$$<br/>\text { B.O }=\frac{\mathrm{N}_{\mathrm{b}}-\mathrm{N}_{\mathrm{o}}}{2}=\frac{10-4}{2}=3<br/>$$</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-calculated-spinonly-magnetic-moment-values-in-mathrmbm-for-leftmathrmfecl4rightand-leftmathrmfemathrmcn6right3-are

slug:
the-calculated-spinonly-magnetic-moment-values-in-mathrmbm-for-leftmathrmfecl4rightand-leftmathrmfemathrmcn6right3-are-83339

question:
<div class="question">The calculated spin-only magnetic moment values in \(\mathrm{BM}\) for \(\left[\mathrm{FeCl}_4\right]^{-}\)and \(\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}\) are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(5.9 \mathrm{~BM},~ 1.732 \mathrm{~BM}~\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4.89 \mathrm{~BM},~ 1.732 \mathrm{~BM}~\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(3.87 \mathrm{~BM},~ 1.732 \mathrm{~BM}~\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(1.732 \mathrm{~BM},~ 2.82 \mathrm{~BM}~\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(5.9 \mathrm{~BM},~ 1.732 \mathrm{~BM}~\)</span> </div>

answer:
<div class="solution">Hint : \(\mathrm{Fe}^{+3}=[\mathrm{Ar}] 3 \mathrm{~d}^5 4 \mathrm{~s}^0\)<br/>\(\left[\mathrm{FeCl}_4\right]^{-}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/7MnXwQDhRinDaP5vuegnXLOZKQBURV1YCfrGhVr4Gbw.original.fullsize.png"/><br/><br/>No pairing as \(\mathrm{Cl}^{-}\)is weak field ligand, hence have five unpaired electron \((n=5)\).<br/>\(\begin{aligned} &amp; \therefore \quad \mu=\sqrt{\mathrm{n}(\mathrm{n}+2)} \text { B.M }=\sqrt{5(5+2)} \text { B.M }=5.9 \text { B.M } \\ &amp; {\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{-3}}\end{aligned}\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Kfxx3jMzL-koCRrQYuSIbtkmeWYRCsWYfvjkQaQXavk.original.fullsize.png"/><br/><br/>Pairing takes place as \(\mathrm{CN}^{-}\)is strong field ligand but has one unpaired electron \((n=1)\)<br/>\(\therefore \mu=\sqrt{n(n+2)} \text { B.M }=\sqrt{1(1+2)}=1.732 \text { B.M }\)</div>


title: the-catalyst-used-for-olefin-polymerization-is
slug: the-catalyst-used-for-olefin-polymerization-is-71291
question: <div class="question">The catalyst used for olefin polymerization is</div>
tags: ['Chemistry', 'Polymers', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">$\mathrm{TiCl}_3+\left(\mathrm{C}_2 \mathrm{H}_5\right)_3 \mathrm{Al}$</div>
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Ziegler-Natta Catalyst</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Wilkinson Catalyst</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Raney nickel catalyst</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Merrifield resin</span> </li> </ul>

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


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-change-in-the-gravitational-potential-energy-when-a-body-of-mass-m-is-raised-to-a-height-n-r-above-the-surface-of-the-earth-is-here-r-is-the-radiu

slug:
the-change-in-the-gravitational-potential-energy-when-a-body-of-mass-m-is-raised-to-a-height-n-r-above-the-surface-of-the-earth-is-here-r-is-the-radiu-95630

question:
<div class="question">The change in the gravitational potential energy when a body of mass $m$ is raised to a height $n \mathrm{R}$ above the surface of the Earth is (here $R$ is the radius of the Earth)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{n}{n+1}\right) m g \mathrm{R}$</span> </div>

answer:
<div class="solution">Hints: $\Delta \mathrm{U}=\frac{m g h}{1+\frac{h}{\mathrm{R}}}=\frac{m g \times n \mathrm{R}}{1+\frac{n \mathrm{R}}{\mathrm{R}}}=\frac{n m g \mathrm{R}}{n+1}$</div>


title: the-change-of-entropy-d-s-is-defined-as
slug: the-change-of-entropy-d-s-is-defined-as-17468
question: <div class="question">The change of entropy $(d S)$ is defined as</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$d S=\frac{\delta q}{T}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$d S=\frac{d H}{T}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$d S=\frac{\delta q_{eqv}}{T}$</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">D</span> <span class="option-data">$d S=\frac{d H-d G}{T}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$d S=\frac{\delta q_{eqv}}{T}$</span> </div>

answer:
<div class="solution">The entropy is the measure of disorder or randomness in a system. When a system changes from one state to another, the change of entropy, $d S$ is given by
<br/>$$
<br/>d S=\frac{\delta q_{\mathrm{rev}}}{T}
<br/>$$</div>


title: the-charge-carried-by-1-millimole-of-m-n-ions-is-193-coulombs-the-value-of-n-is
slug: the-charge-carried-by-1-millimole-of-m-n-ions-is-193-coulombs-the-value-of-n-is-81772
tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The charge carried by 1 millimole of $M^{n+}$ ions is 193 coulombs. The value of $n$ is</div>

answer:
<div class="solution">One millimole $=0.001 \mathrm{mol}$
<br/>i.e. $1 \times 10^{-3} \mathrm{mol}$
<br/>$\because 96500 \mathrm{C}$ is required for $1 \mathrm{mol}$ of electrons.
<br/>$\therefore 193 \mathrm{C}$ give $\frac{193 \times 1}{96500}=0.002 \mathrm{mol},$ i.e. $2 \times 10^{-3} \mathrm{mol}$ of
<br/>electrons.
<br/>Thus, value of $n$ in $M^{n+}$ is $=\frac{0.002}{0.001}=2$
<br/>Hence, option (b) is the correct answer.</div>


title: the-charge-on-the-capacitor-of-capacitance-mathrmc-shown-in-the-figure-below-will-be
slug: the-charge-on-the-capacitor-of-capacitance-mathrmc-shown-in-the-figure-below-will-be-12596
tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The charge on the capacitor of capacitance \(\mathrm{C}\) shown in the figure below will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/g4ETkv0XQ33PgdGmWcybZXMNRpiKj1UPJ9GvROhptnw.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{CE}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\mathrm{CE} \mathrm{R}}{\mathrm{R}_1+r}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{C E R_2}{R_2+r}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{C E R_1}{R_2+r}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{C E R_2}{R_2+r}\)</span> </div>

answer:
<div class="solution">Hints: \(I=\frac{E}{R_2+r}\) (Since finally no current flows through capacitor)<br/>\(\therefore\) Potential difference across \(\mathrm{R}_2, \mathrm{~V}=\mathrm{R}_2=\frac{\mathrm{ER}_2}{\mathrm{R}_2+\mathrm{r}}\)<br/>\(\therefore\) Charge on the capacitor \(\mathrm{Q}=\mathrm{CV}=\frac{\mathrm{CER}_2}{\mathrm{R}_2+\mathrm{r}}\)</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
the-chord-of-the-curve-y-x-2-2-a-x-b-joining-the-points-where-x-and-x-is-parallel-to-the-tangent-to-the-curve-at-abscissa-x-is-equal-to

slug:
the-chord-of-the-curve-y-x-2-2-a-x-b-joining-the-points-where-x-and-x-is-parallel-to-the-tangent-to-the-curve-at-abscissa-x-is-equal-to-18871

question:
<div class="question">The chord of the curve $y=x^{2}+2 a x+b$ joining the points where $x=\alpha$ and $x=\beta,$ is parallel to the tangent to the curve at abscissa $x$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{a+b}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2 a+b}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2 \alpha+\beta}{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\alpha+\beta}{2}$</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">$\frac{\alpha+\beta}{2}$</span> </div>

answer:
<div class="solution">Given curve,
<br/>At $y=x^{2}+2 a x+b$
<br/>$x=a$
<br/>$y=\alpha^{2}+2 a \alpha+b$
<br/>and at $x=\beta$
<br/>$y=\beta^{2}+2 a \beta+b$
<br/>$\therefore$ Slope of line joining $\left(\alpha, \alpha^{2}+2 a \alpha+b\right)$ and
<br/>$\left(\beta, \beta^{2}+2 a \beta+b\right)$ is
<br/>$=\frac{\left(\alpha^{2}+2 a \alpha+b\right)-\left(\beta^{2}+2 a \beta+b\right)}{\alpha-\beta}$
<br/>$=\frac{\alpha^{2}+2 a \alpha+b-\beta^{2}-2 a \beta-b}{\alpha-\beta}$
<br/>$=\frac{\left(\alpha^{2}-\beta^{2}\right)+2 a(\alpha-\beta)}{\alpha-\beta}$
<br/>$\frac{=(\alpha-\beta)(\alpha+\beta)+2 a(\alpha-\beta)}{\alpha-\beta}$
<br/>$=\alpha+\beta+2 a$
<br/>Slope of given curve $=\frac{d y}{d x}$ $=2 x+2 a$
<br/>Now, according to question, tangent is parallel to the chord. Therefore,
<br/>$$
<br/>\begin{array}{r}
<br/>2 x+2 a=\alpha+\beta+2 a \\
<br/>\Rightarrow \quad 2 x=\alpha+\beta \Rightarrow x=\frac{\alpha+\beta}{2}
<br/>\end{array}
<br/>$$</div>


title: the-circles-x-2-y-2-10-x-16-0-and-x-2-y-2-a-2-intersect-at-two-distinct-points-if
slug: the-circles-x-2-y-2-10-x-16-0-and-x-2-y-2-a-2-intersect-at-two-distinct-points-if-10968
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The circles $\mathrm{x}^2+\mathrm{y}^2-10 \mathrm{x}+16=0$ and $\mathrm{x}^2+\mathrm{y}^2=\mathrm{a}^2$ intersect at two distinct points if</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 &lt; $ a $ &lt; 8$</span> </div>

answer:
<div class="solution">Hints : $C_1(5,0) \quad r_1=\sqrt{25-16}=3$<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{C}_2(0,0) \quad \mathrm{r}_2=\mathbf{a} \\<br/>&amp; \mathrm{r}_1 \&amp; \mathrm{r}_2 &lt; \mathrm{C}_1 \mathrm{C}_2 &lt; \mathrm{r}_1+\mathrm{r}_2 \\<br/>&amp; |\mathbf{a}-3| &lt; \sqrt{25} &lt; \mathrm{a}+3 \\<br/>&amp; |\mathbf{a}-3| &lt; 5 &lt; \mathrm{a}+3 \\<br/>&amp; -5 &lt; \mathrm{a}-3 &lt; 5 \\<br/>&amp; -2 &lt; \mathrm{a} &lt; 8 \\<br/>&amp; 2 &lt; \mathrm{a} &lt; 8<br/>\end{aligned}<br/>$$</div>


title: the-coefficient-of-a-3-b-4-c-5-in-the-expansion-of-b-c-c-a-ab-6-is
slug: the-coefficient-of-a-3-b-4-c-5-in-the-expansion-of-b-c-c-a-ab-6-is-97514
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The co-efficient of $a^{3} b^{4} c^{5}$ in the expansion of $(b c+c a+a b)^{6}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{12 !}{3 ! 4 ! 5 !}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{6 !}{3 !}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">33</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">3. $\left(\frac{6 !}{3 ! 3 !}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">3. $\left(\frac{6 !}{3 ! 3 !}\right)$</span> </div>

answer:
<div class="solution">$\left\{\frac{6 !}{p ! q ! r !} a^{q+r} b^{p+r} c^{p+q}\right.$
<br/>$q+r=3, p+r=4, p+q=5, p+q+r=6$
<br/>$\Rightarrow p=3, q=2, r=1$
<br/>co-efficient $=\frac{6 !}{3 ! 2 !}$</div>


title: the-coefficient-of-x-3-in-the-infinite-series-expansion-of-1-x-2-x-2-for-x-1-is
slug: the-coefficient-of-x-3-in-the-infinite-series-expansion-of-1-x-2-x-2-for-x-1-is-14627
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The coefficient of $x^{3}$ in the infinite series expansion of $\frac{2}{(1-x)(2-x)},$ for $|x|    &lt; 1,$ is</div>

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

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

answer:
<div class="solution">Given expression can be rewritten as
<br/>$$
<br/>\begin{array}{l}
<br/>2(1-x)^{-1} \times 2^{-1}\left(1-\frac{x}{2}\right)^{-1} \\
<br/>\quad=\left[1+x+x^{2}+x^{3}+\ldots\right] \\
<br/>\quad\left[1+\left(\frac{x}{2}\right)+\left(\frac{x}{2}\right)^{2}+\left(\frac{x}{2}\right)^{3}+\ldots\right.
<br/>\end{array}
<br/>$$
<br/>$\therefore$ Coefficient of $x^{3}$ in the above expression is
<br/>$$
<br/>\begin{array}{l}
<br/>=\left[\left(\frac{1}{2}\right)^{3}+\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)+1\right] \\
<br/>=\left[\frac{1}{8}+\frac{1}{4}+\frac{1}{2}+1\right] \\
<br/>=\left[\frac{1+2+4+8}{8}\right]=\frac{15}{8}
<br/>\end{array}
<br/>$$</div>


title: the-coefficient-of-x-10-in-the-expansion-of-1-1-x-1-x-20
slug: the-coefficient-of-x-10-in-the-expansion-of-1-1-x-1-x-20-15276
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The coefficient of $x^{10}$ in the expansion of $1+(1+x)+\dots+(1+x)^{20}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{19} C_{9}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">${ }^{20} C_{10}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">${ }^{21} C_{11}$</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">${ }^{22} C_{12}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{21} C_{11}$</span> </div>

answer:
<div class="solution">The given series is in GP. Hence, its sum $S=\frac{1\left\{(1+x)^{20+1}-1\right\}}{(1+x)-1}=\frac{(1+x)^{21}-1}{x}$
<br/>Therefore, the required coefficient of $x^{10}$ in
<br/>the expansion of $(1+x)^{21}-1$
<br/>$=$ Coefficient of $x^{11}$ in the expansion of
<br/>$$
<br/>={ }^{21} C_{11}
<br/>$$</div>


title: the-coefficient-of-x-n-where-n-is-any-positive-integer-in-the-expansion-of-1-2-x-3-x-2-12-is
slug: the-coefficient-of-x-n-where-n-is-any-positive-integer-in-the-expansion-of-1-2-x-3-x-2-12-is-93409
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The coefficient of $x^n$, where $n$ is any positive integer, in the expansion of $\left(1+2 x+3 x^2+\ldots \ldots \ldots\right)^{1 / 2}$ 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">$\frac{n+1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 n+1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{n}+1$</span> </li> </ul>

answer:
<div class="solution">$$<br/>s=1+2 x+3 x^2 .<br/>$$<br/>.$\infty$<br/>Hints : $\frac{x S=x+2 x^2+\ldots \ldots \ldots \ldots}{s(1-x)=1+x+x^2+\ldots \ldots \ldots . \infty}$<br/>$$<br/>s=\frac{1}{(1-x)^2}<br/>$$<br/>$$<br/>f(x)=\frac{1}{1-x}, f(x)=(1-x)^{-1}=1+x+x^2+x^3 \ldots \ldots \ldots \ldots . . .0=1<br/>$$</div>


title: the-coefficient-of-xn-in-the-expansion-of-frace7-xexe3-x-is
slug: the-coefficient-of-xn-in-the-expansion-of-frace7-xexe3-x-is-88184
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The coefficient of \(x^n\) in the expansion of \(\frac{e^{7 x}+e^x}{e^{3 x}}\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{4^{\mathrm{n}}+(-2)^{\mathrm{n}}}{\underline{n}}\)</span> </div>

answer:
<div class="solution">Hints: \(\frac{e^{7 x}+e^x}{e^{3 x}}=e^{4 x}+e^{-2 x}\)<br/>Co-efficient of \(x^n\)<br/>\(\frac{(4)^{\mathrm{n}}}{\mathrm{n} !}+\frac{(2)^{\mathrm{n}}}{\mathrm{n} !}(-1)^{\mathrm{n}}=\frac{4^{\mathrm{n}}+(-2)^{\mathrm{n}}}{\mathrm{n} !}\)</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
the-common-chord-of-the-circles-x-2-y-2-4-x-4-y-0-and-2-x-2-2-y-2-32-subtends-at-the-origin-an-angle-equal-to

slug:
the-common-chord-of-the-circles-x-2-y-2-4-x-4-y-0-and-2-x-2-2-y-2-32-subtends-at-the-origin-an-angle-equal-to-37902

question:
<div class="question">The common chord of the circles $x^{2}+y^{2}-4 x-4 y=0 \quad$ and $\quad 2 x^{2}+2 y^{2}=32$ subtends at the origin an angle equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{6}$</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">Given, equation of circles are $x^{2}+y^{2}-4 x-4 y=0$ and $2 x^{2}+2 y^{2}=32$
<br/>or $x^{2}+y^{2}-4 x-4 y=0$
<br/>and $x^{2}+y^{2}=16$
<br/>$\therefore$ Equation of common chord is
<br/>$\left(x^{2}+y^{2}-4 x-4 y\right)-\left(x^{2}+y^{2}-16\right)=0$
<br/>$\Rightarrow \quad-4 x-4 y+16=0$
<br/>$\Rightarrow \quad x+y=4$
<br/>This common chord passes through (2,2)
<br/>i.e. centre of first circle. Also, (0,0) is at the circumference of the first circle.
<br/>$\therefore$ Common chord will subtent $\frac{\pi}{2}$ angle at (0,0)</div>


title: the-complex-number-z-satisfying-the-equation-z-i-z-1-1-is
slug: the-complex-number-z-satisfying-the-equation-z-i-z-1-1-is-95251
question: <div class="question">The complex number $z$ satisfying the equation $|z-i|=|z+1|=1$ is</div>
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1+i$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-1+i$</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">D</span> <span class="option-data">$1-i$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">0, $-1+i$</span> </div>

answer:
<div class="solution">We have, $|z-i|=|z+1|=1$
<br/>Let, $\quad z=x+iy$
<br/>$\therefore \quad|z-i|=1$
<br/>$\Rightarrow \quad|x+iy-i \mid=1$
<br/>$\Rightarrow|x+(y-1) i|=1$
<br/>$\Rightarrow \quad x^{2}+(y-1)^{2}=1$
<br/>Also, $\quad|z+1|=1$
<br/>$\Rightarrow \quad|x+ iy+1 \mid=1$
<br/>$\Rightarrow \quad|(x+1)+i y|=1$
<br/>$\Rightarrow \quad(x+1)^{2}+y^{2}=1$
<br/>From Eqs. (i) and (ii), we get $x=y=0$ and $x=-1, y=1$
<br/>$\therefore \quad z=0,-1+i$</div>


title: the-compound-that-will-have-a-permanent-dipole-moment-among-the-following-is
slug: the-compound-that-will-have-a-permanent-dipole-moment-among-the-following-is-74269
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">I</span> </div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The compound that will have a permanent dipole moment among the following is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q13.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">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">B</span> <span class="option-data">II</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">III</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">IV</span> </li> </ul>

answer:
<div class="solution">Symmetrical molecules and symmetrical trans-alkene have a net dipole moment zero. $\mathrm{CH}_{2} \mathrm{Cl}_{2}$ is not a symmetrical molecule, thus it will have a permanent dipole.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a13.png"/></div>


tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
the-compound-that-would-produce-a-nauseating-smellodour-with-a-hot-mixture-of-chloroform-and-ethanolic-potassium-hydroxide-is

slug:
the-compound-that-would-produce-a-nauseating-smellodour-with-a-hot-mixture-of-chloroform-and-ethanolic-potassium-hydroxide-is-14741

question:
<div class="question">The compound that would produce a nauseating smell/odour with a hot mixture of chloroform and ethanolic potassium hydroxide is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">PhCONH $_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">PhNHCH $_{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{PhNH}_{2}$</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">PhOH</span> </li> </ul>

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

answer:
<div class="solution">When primary amine react with chloroform and ethanolic potassium hydroxide, they produce - isocyamides as main product. The reaction is also known as carbyl amine reaction. $\mathrm{Ph}-\mathrm{NH}_{2} \frac{\mathrm{CHCl}_{3}}{\mathrm{KOH} \text { (ethanolic) }} \mathrm{Ph} \cdot \mathrm{N} \longrightarrow \mathrm{C}$
<br/>(Nauseating smell)</div>


tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-compound-which-evolves-carbon-dioxide-on-treatment-with-aqueous-solution-of-sodium-bicarbonate-at-2-5-c-is

slug:
the-compound-which-evolves-carbon-dioxide-on-treatment-with-aqueous-solution-of-sodium-bicarbonate-at-2-5-c-is-29325

question:
<div class="question">The compound, which evolves carbon dioxide on treatment with aqueous solution of sodium bicarbonate at $25^{\circ} \mathrm{C},$ is</div>

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

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

answer:
<div class="solution">$\mathrm{CH}_{3}-\mathrm{C}O-\mathrm{Cl}$ hydrolyses to form $\mathrm{CH}_{3} \mathrm{COOH}$
<br/>even at $25^{\circ} \mathrm{C},$ which subsequently reacts with $NaHCO_{3}$ (sodium bicarbonate) present in the same medium to form carbon dioxide $\left(\mathrm{CO}_{2}\right)$.
<br/>The reaction involved is given below. $\mathrm{CH}_{3} \mathrm{COCl}+\mathrm{NaHCO}_{3} \longrightarrow \mathrm{CH}_{3} \mathrm{COONa}+\mathrm{HCl}+\mathrm{CO}_{2}$
<br/>Thus, the option (b) is correct answer.</div>


title: the-compounds-a-and-b-are-respectively
slug: the-compounds-a-and-b-are-respectively-17749
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lTCT445xb60uiXsVXRNZlDTT6yD4YyrI8qIsNpW25aI.original.fullsize.png"/><br/> <br/>The compounds A and B are respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lAqkN2OudU832upODyRQBPGkVs3nmX9ZSC4-9lfLcG0.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lTz3NtdU_2lYM0MDwOcvs6ExSMTphvSmF4V_y4Nh-IQ.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Qi8sNXTldKlBzhxaUMk5kr9-5n22wYXIg8cdtVqBUGI.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ygGltzJ5FIs2LNx-0Zvdou4rCjS_ufuGeNTlgIvvbHM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ygGltzJ5FIs2LNx-0Zvdou4rCjS_ufuGeNTlgIvvbHM.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/cyq8TCDao-NYjx_83UjUdzWkar2_xvlB1-S_7kwZ3Zg.original.fullsize.png"/></div>


title: the-compounds-capable-of-producing-achiral-compound-on-heating-at-10-0-c-isare
slug: the-compounds-capable-of-producing-achiral-compound-on-heating-at-10-0-c-isare-23988
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The compound(s), capable of producing achiral compound on heating at $100^{\circ} \mathrm{C}$ is/are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q80_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q80_optb.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q80_optc.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q80_optd.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q80_optd.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a80.png"/> <br/>Thus, compound (d), capable of producing achiral compound on heating at $100^{\circ} \mathrm{C}$.</div>


title: the-compounds-x-and-y-are-respectively
slug: the-compounds-x-and-y-are-respectively-59513
tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The compounds $\mathrm{X}$ and $\mathrm{Y}$ are respectively
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/zQme5Js8DBIQgcJSWD5u4EpLGmdqHSm-uzvBdzZLOqc.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mx2ukqE4B7vmdl6FDZ6dPMm6JaXwDlaL4t1ZjIPH3x4.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9eu3VuCn2zvvaV9vdl0AF4ei7-Itd_gZLLDjlpsbFLE.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6fxrHjkYLgmrPoBauAUb3Ndd4FSnakokw_hVi9eZPCA.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/BJ5RUzLeLAE3LixTlbXvC_ImlM9e7rgBm5j7TWVM1D0.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/BJ5RUzLeLAE3LixTlbXvC_ImlM9e7rgBm5j7TWVM1D0.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/PfMHj1B7VMWHR12AsVf3z6W3nTS0z7GqfuGHtLDpq_M.original.fullsize.png"/></div>


title: the-compressibility-factor-z-of-one-mole-of-a-van-der-waals-gas-of-negligible-a-value-is
slug: the-compressibility-factor-z-of-one-mole-of-a-van-der-waals-gas-of-negligible-a-value-is-74650
tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The compressibility factor $(Z)$ of one mole of a van der Waals' gas of negligible ' $a$ ' value is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{b p}{R T}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1+\frac{b p}{R T}$</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">D</span> <span class="option-data">$1-\frac{b p}{R T}$</span> </li> </ul>

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

answer:
<div class="solution">van der Waals' equation is $\left(p+\frac{a}{V^{2}}\right)(V-b)=R T$
<br/>(for 1 mole)
<br/>If $a$ is negligible, $p+\frac{a}{V^{2}} \approx p$
<br/>$$
<br/>p(V-b)=R T \Rightarrow p V-p b=R T
<br/>$$
<br/>or $\frac{p V}{R T}-\frac{p b}{R T}=1 \Rightarrow \frac{p V}{R T}=1+\frac{p b}{R T}$</div>


title: the-condition-for-a-reaction-to-occur-spontaneously-is
slug: the-condition-for-a-reaction-to-occur-spontaneously-is-96656
question: <div class="question">The condition for a reaction to occur spontaneously is</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\Delta H$ must be negative</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\Delta S$ must be negative</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(\Delta H-T \Delta S)$ must be negative</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$(\Delta H+T \Delta S)$ must be negative</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\Delta H-T \Delta S)$ must be negative</span> </div>

answer:
<div class="solution">For a spontaneous reaction, $\Delta G$ = negative and $\Delta S=$ positive
<br/>$$
<br/>\Delta G=\Delta H-T \Delta S=-v e
<br/>$$
<br/>Hence, $\Delta H-T \Delta S=-$ ve only when $\Delta S=+$ ve and
<br/>$$
<br/>\Delta H=-v e
<br/>$$</div>


title: the-condition-of-spontaneity-of-a-process-is
slug: the-condition-of-spontaneity-of-a-process-is-32751
question: <div class="question">The condition of spontaneity of a process is</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">lowering of enthropy at constant temperature and pressure</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">lowering of Gibbs free energy of system at constant temperature and pressure</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">increase of entropy of system at constant temperature and pressure</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">increase of Gibbs free energy of the universe at constant temperature and pressure</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">lowering of Gibbs free energy of system at constant temperature and pressure</span> </div>

answer:
<div class="solution">From Gibbs Helmholtz equation,
<br/>$$
<br/>\Delta G=\Delta H-T \Delta S
<br/>$$
<br/>A process will be spontaneous, if
<br/>$$
<br/>\Delta G=-\text { ve }
<br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P2.db
title:
the-conducting-loop-in-the-form-of-a-circle-is-placed-in-a-uniform-magnetic-field-with-its-plane-perpendicular-to-the-direction-of-the-field-an-emf-wi

slug:
the-conducting-loop-in-the-form-of-a-circle-is-placed-in-a-uniform-magnetic-field-with-its-plane-perpendicular-to-the-direction-of-the-field-an-emf-wi-20264

question:
<div class="question">The conducting loop in the form of a circle is placed in a uniform magnetic field with its plane perpendicular to the direction of the field. An emf will be induced in the loop, if</div>

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

answer:
<div class="solution">Whenever, there is a change in the magnetic flux crossing through the loop, an emf induced in it and hence there is an amount of induced current through the loop. If loop is rotated about one of its diameter or the loop is deformed the flux through it varies and causing the emf induced in it.</div>


tags: ['Chemistry', 'Coordination Compounds', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-conductivity-measurement-of-a-coordination-compound-of-cobalt-iii-shows-that-it-dissociates-into-3-ions-in-solution-the-compound-is

slug:
the-conductivity-measurement-of-a-coordination-compound-of-cobalt-iii-shows-that-it-dissociates-into-3-ions-in-solution-the-compound-is-34857

question:
<div class="question">The conductivity measurement of a coordination compound of cobalt (III) shows that it dissociates into 3 ions in solution. The compound is</div>

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

answer:
<div class="solution">Co-ordinate compound of Co (III) dissociates into 3 ions in the solution means it has two ionisable-ions out side the coordination sphere with oxidation state of cobalt $(\mathrm{Co})=+3$
<br/>In option Name : Hexaammine cobalt (III) chloride Formula : $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right] \mathrm{Cl}_{3}$
<br/>$\because$ Oxidation state of $\mathrm{Co}=(+) 3$
<br/>In option (b) Name : Pentaammine sulphatocobalt (III) chloride Formula : $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}\left(\mathrm{SO}_{4}\right)\right] \mathrm{Cl}$
<br/>$\because$ Oxidation state of $\mathrm{Co}=+3$
<br/>In option (c) Name : Pentaamminechloridocobalt
<br/>(III) sulphate Formula : $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right] \mathrm{SO}_{4}$
<br/>$\because$ Oxidation state of $\mathrm{Co}=+3$
<br/>In option (d) Name : Pentaamminechloridocobalt
<br/>(III) chloride Formula: $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right] \mathrm{Cl}_{2}$
<br/>$\because$ Oxidation state of $\mathrm{Co}=+3$
<br/>Dissociation of this compound is shown below:
<br/>$\left[\mathrm{CO}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right] \mathrm{Cl}_{2} \rightarrow \mathrm{CO} \underbrace{\left.\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right]^{2+}+2 \mathrm{Cl}^{-}}_{3 \text { ions }}$
<br/>This compound can give total 3 -ions, as follows. one $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right]^{2+}$ and two $\mathrm{Cl}^{-}$ ions.
<br/>Hence, option (d) is the correct answer.</div>


tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
title:
the-conformations-of-n-butane-commonly-known-as-eclipsed-gauche-and-anticonformations-can-be-interconverted-by

slug:
the-conformations-of-n-butane-commonly-known-as-eclipsed-gauche-and-anticonformations-can-be-interconverted-by-73959

question:
<div class="question">The conformations of $n$ -butane, commonly known as eclipsed, gauche and anti-conformations can be interconverted by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">rotation around $C-H$ bond of a methyl group</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">rotation around $\mathrm{C}-\mathrm{H}$ bond of a methylene group</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">rotation around C1-C2 linkage</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">rotation around C2-C3 linkage</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">rotation around C2-C3 linkage</span> </div>

answer:
<div class="solution">The conformation of $n$ -butane, commonly known as eclipsed, gauche and anti-conformations can be interconverted by rotation around C2-C3 linkage.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a69.png"/> <br/>Hence, the option (d) is correct answer.</div>


title: the-conversion-of-ch-3-ch-2-cooh-to-can-be-accomplished-by
slug: the-conversion-of-ch-3-ch-2-cooh-to-can-be-accomplished-by-60261
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The conversion of $\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{COOH}$ to <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q80.png"/> <br/>can be accomplished by</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a80.png"/> <br/>Hence, (a) and (d) are correct answers.</div>


title: the-conversionss-that-can-be-carried-out-by-bromine-in-carbon-tetrachloride-solvent-isare
slug: the-conversionss-that-can-be-carried-out-by-bromine-in-carbon-tetrachloride-solvent-isare-85371
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'WBJEE', 'WBJEE 2023']
answer: <div class="solution">Hint : (A) Addition Reaction<br/>(D) Borodine Hunsdiecker reaction</div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The conversions(s) that can be carried out by bromine in carbon tetrachloride solvent is/are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\mathrm{PhCH}=\mathrm{CHCH}_3 \rightarrow \mathrm{PhCHBrCHBrCH}{ }_3\)</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"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/wy0Dasjj2_zET4oeJuk7O9n6NUjY2d9i1z1egxokXDM.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{COOH} \rightarrow \mathrm{CH}_3 \mathrm{CHBrCOOH}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/N6yIxYPos9vjeCMX9oskguNs-MpZtLtWE4hHRXZubCI.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">\(\mathrm{PhCH}=\mathrm{CHCH}_3 \rightarrow \mathrm{PhCHBrCHBrCH}{ }_3\), <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/N6yIxYPos9vjeCMX9oskguNs-MpZtLtWE4hHRXZubCI.original.fullsize.png"/><br/></span> </div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P2.db
title:
the-coordinate-of-a-point-on-the-auxiliary-circle-of-the-ellipse-x-2-2-y-2-4-corresponding-to-the-point-on-the-ellipse-whose-eccentric-angle-is-6-0-wi

slug:
the-coordinate-of-a-point-on-the-auxiliary-circle-of-the-ellipse-x-2-2-y-2-4-corresponding-to-the-point-on-the-ellipse-whose-eccentric-angle-is-6-0-wi-62680

question:
<div class="question">The co-ordinate of a point on the auxiliary circle of the ellipse $x^{2}+2 y^{2}=4$ corresponding to the point on the ellipse whose eccentric angle is $60^{\circ}$ will be</div>

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

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

answer:
<div class="solution">$\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$
<br/>Equation of auxiliary circle is
<br/>$x^{2}+y^{2}=4$
<br/>$\therefore$ Point on the auxiliary circle is
<br/>$\begin{array}{l}
<br/>p\left(2 \cos 60^{\circ}, 2 \sin 60^{\circ}\right) \\
<br/>p(1, \sqrt{3})
<br/>\end{array}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/nfCjUhotOOM8Mb6Tih9Uke8GcDyN2Db2TF1KWjuRwOo.original.fullsize.png"/></div>


title: the-coordinates-of-a-moving-point-p-are-left2-t244-t6right-then-its-locus-will-be-a
slug: the-coordinates-of-a-moving-point-p-are-left2-t244-t6right-then-its-locus-will-be-a-62837
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The coordinates of a moving point \(p\) are \(\left(2 t^2+4,4 t+6\right)\). Then its locus will be a</div>

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

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

answer:
<div class="solution">Hints: \(x=2 t^2+4, y=4 t+6, \quad y=4 t+6 \rightarrow t=\left(\frac{y-6}{4}\right)\)<br/>\(\begin{aligned}<br/>&amp; x=2\left(\frac{y-6}{4}\right)^2+4 \Rightarrow \frac{(y-6)^2}{8}=x-4 \\<br/>&amp; (y-6)^2=4(2)(x-4)<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
title:
the-coordinates-of-a-point-on-the-line-x-y-1-0-which-is-at-a-distance-5-1-unit-from-the-line-3-x-4-y-2-0-are

slug:
the-coordinates-of-a-point-on-the-line-x-y-1-0-which-is-at-a-distance-5-1-unit-from-the-line-3-x-4-y-2-0-are-38798

question:
<div class="question">The coordinates of a point on the line $x+y+1=0,$ which is at a distance $\frac{1}{5}$ unit from the line $3 x+4 y+2=0,$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">(-3,2), (-1,0)</span> </div>

answer:
<div class="solution">Let $(h, k)$ be the point on line $x+y+1=0$
<br/>So.
<br/>$$
<br/>h+k+1=0
<br/>$$
<br/>$$
<br/>h=-1-k
<br/>$$
<br/>Given, distance between $(h, k)$ and line $3 x+4 y+2=0$ is $\frac{1}{5}$
<br/>$\therefore$
<br/>$\frac{1}{5}=\left|\frac{(3 h)+(4 k)+2}{\sqrt{3^{2}+4^{2}}}\right|$
<br/>$\Rightarrow \quad \frac{1}{5}=\left|\frac{3(-1-k)+4 k+2}{\sqrt{25}}\right|$
<br/>$\Rightarrow \quad \frac{1}{5}=\pm\left(\frac{-3-3 k+4 k+2}{5}\right)$
<br/>$\Rightarrow$
<br/>$\frac{1}{5}=\pm\left(\frac{k-1}{5}\right)$
<br/>$\Rightarrow \quad \pm(k-1)=1$
<br/>$\Rightarrow \quad k=0$ and 2
<br/>$\therefore \quad h=-1, -3$
<br/>Hence, the required points are (-1,0) and (-3,2)</div>


title: the-coordinates-of-the-focus-of-the-parabola-described-parametrically-by-x-5-t-2-2-y-10-t-4-are-1
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
slug:
the-coordinates-of-the-focus-of-the-parabola-described-parametrically-by-x-5-t-2-2-y-10-t-4-are-1-46096

question:
<div class="question">The co-ordinates of the focus of the parabola described parametrically by $x=5 t^2+2, y=10 t+4$ are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(7,4)$</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">$(3,4)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(3,-4)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(-7,4)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(7,4)$</span> </div>

answer:
<div class="solution">Hints: $\mathrm{x}=5 \mathrm{t}^2+2 ; \mathrm{y}=10 \mathrm{t}+4,\left(\frac{y-4}{10}\right)^2=\left(\frac{x-2}{5}\right)$ or, $(\mathrm{y}-4)^2=20(\mathrm{x}-2)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/yjcw3qwuBOX3fDGW_AZ-bWB46NIqM3-bYpADbz7mDbI.original.fullsize.png"/><br/></div>


title: the-coordinates-of-the-focus-of-the-parabola-described-parametrically-by-x-5-t-2-2-y-10-t-4-are-2
tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: db\MarksBatch2_P2.db
slug:
the-coordinates-of-the-focus-of-the-parabola-described-parametrically-by-x-5-t-2-2-y-10-t-4-are-2-74910

question:
<div class="question">The co-ordinates of the focus of the parabola described parametrically by $x=5 t^2+2, y=10 t+4$ are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(7,4)$</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">$(3,4)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(3,-4)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(-7,4)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(7,4)$</span> </div>

answer:
<div class="solution">Hints: $\mathrm{x}=5 \mathrm{t}^2+2 ; \mathrm{y}=10 \mathrm{t}+4,\left(\frac{y-4}{10}\right)^2=\left(\frac{x-2}{5}\right)$ or, $(\mathrm{y}-4)^2=20(\mathrm{x}-2)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/yjcw3qwuBOX3fDGW_AZ-bWB46NIqM3-bYpADbz7mDbI.original.fullsize.png"/><br/></div>


title: the-coordinates-of-the-foot-of-the-perpendicular-from-0-0-upon-the-line-x-y-2-are
slug: the-coordinates-of-the-foot-of-the-perpendicular-from-0-0-upon-the-line-x-y-2-are-97241
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The co-ordinates of the foot of the perpendicular from $(0,0)$ upon the line $x+y=2$ are</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">$(-2,1)$</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">$(1,2)$</span> </li> </ul>

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

answer:
<div class="solution">Hints : Let $\mathrm{P}$ be the foot of the perpendicular. P lies on a line perpendicular to $x+y=2$.<br/>$\therefore$ Equation of the line on which P lies is of the form : $x-y+k=0$<br/>But this line passes through $(0,0)$.<br/>$$<br/>\therefore k=0<br/>$$<br/>Hence, co-ordinates of P may be obtained by solving $x+y=2$ and $y=x$<br/>$$<br/>\therefore x=1, y=1<br/>$$<br/>Hence, $\mathrm{P} \equiv(1,1)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/0hOJ1Huz-Lim9xOWC0DqV8xC3L7cbkJfY9oZvx2Uagg.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P2.db
title:
the-coordinates-of-the-point-on-the-curve-y-x-2-3-x-2-where-the-tangent-is-perpendicular-to-the-straight-line-y-x-are

slug:
the-coordinates-of-the-point-on-the-curve-y-x-2-3-x-2-where-the-tangent-is-perpendicular-to-the-straight-line-y-x-are-91078

question:
<div class="question">The co-ordinates of the point on the curve $y=x^2-3 x+2$ where the tangent is perpendicular to the straight line $y=x$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(1,0)$</span> </div>

answer:
<div class="solution">Hints: $y=x^2-3 x+2$<br/>$\frac{d y}{d x}=2 x-3=-1 \Rightarrow x=1$ at $x=1, y=0$<br/>$\therefore$ Point is $(1,0)$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
title:
the-coordinates-of-the-two-points-lying-on-xy4-and-at-a-unit-distance-from-the-straight-line-4-x3-y10-are

slug:
the-coordinates-of-the-two-points-lying-on-xy4-and-at-a-unit-distance-from-the-straight-line-4-x3-y10-are-85646

question:
<div class="question">The coordinates of the two points lying on \(x+y=4\) and at a unit distance from the straight line \(4 x+3 y=10\) are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((-3,1),(7,11)\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\((3,1),(-7,11)\)</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">C</span> <span class="option-data">\((3,1),(7,11)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\((5,3),(-1,2)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((3,1),(-7,11)\)</span> </div>

answer:
<div class="solution">Hints : Let \(p(h, 4-h)\)<br/>\(\begin{aligned}<br/>&amp; \left|\frac{4 h+3(4-h)-10}{5}\right|=1 \\<br/>&amp; |h+2|=5 \\<br/>&amp; h=3,-7 ; p=1,1 \\<br/>&amp; (3,1),(-7,11)<br/>\end{aligned}\)</div>


title: the-correct-basicity-order-of-the-following-lanthanide-ions-is
slug: the-correct-basicity-order-of-the-following-lanthanide-ions-is-37533
question: <div class="question">The correct basicity order of the following lanthanide ions is</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{La}^{3+}&gt;\mathrm{Lu}^{3 +}&gt;\mathrm{Ce}^{3+}&gt;\mathrm{Eu}^{3+}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Ce}^{3}&gt;\mathrm{Lu}^{3+}&gt;\mathrm{La}^{3+} &gt; \mathrm{Eu}^{3 \mathrm{e}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Lu}^{3 \cdot}&gt;\mathrm{Ce}^{3}&gt;\mathrm{Eu}^{3+}&gt;\mathrm{La}^{3+}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{La}^{3+}&gt;\mathrm{Ce}^{3+}&gt;\mathrm{Eu}^{3+}&gt;\mathrm{Lu}^{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{La}^{3+}&gt;\mathrm{Ce}^{3+}&gt;\mathrm{Eu}^{3+}&gt;\mathrm{Lu}^{3}$</span> </div>

answer:
<div class="solution">As size of cation decreases, ionic character decreases, thus basicity also decreases Hence, correct order is $\mathrm{La}^{3+}&gt;\mathrm{Ce}^{3+}&gt;\mathrm{Eu}^{3+}&gt;\mathrm{Lu}^{3+}$</div>


title: the-correct-bond-order-of-b-f-bond-in-bf-3-molecule-is
slug: the-correct-bond-order-of-b-f-bond-in-bf-3-molecule-is-23820
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct bond order of $\mathrm{B}-\mathrm{F}$ bond in $\mathrm{BF}_3$ molecule is</div>

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

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

answer:
<div class="solution">B.O $=\frac{2+1+1}{3}=1 \frac{1}{3}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dZttLNYae7O_1Qc2ONbP65vgVKNqwUIgu0BlOUtVS4Q.original.fullsize.png"/></div>


title: the-correct-dimensional-formula-for-impulse-is-given-by
slug: the-correct-dimensional-formula-for-impulse-is-given-by-43120
question: <div class="question">The correct dimensional formula for impulse is given by</div>
tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{ML}^{2} \mathrm{T}^{-2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">MLT $^{-1}$</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">$\mathrm{ML}^{2} \mathrm{T}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{MLT}^{-2}$</span> </li> </ul>

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

answer:
<div class="solution">As we know that,
<br/>Impulse, $I=$ force $(F) \times$ small time interval
<br/>$\begin{array}{ll}
<br/>I=m a \times t &amp; {[\because F=m a]} \\
<br/>I=[\mathrm{M}]\left[\mathrm{LT}^{-2}\right] \times[\mathrm{T}] \\
<br/>I=\left[\mathrm{MLT}^{-1}\right]
<br/>\end{array}$
<br/>Hence, the correct dimensional formula for impulse is given by $\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-1} 1 .\right.$</div>


title: the-correct-option-for-the-above-reaction-is-1
slug: the-correct-option-for-the-above-reaction-is-1-25732
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/PXqXM7kQaCQywWOsN9L5NwtwJ-QJr6SH3BN-xJIR5Sw.original.fullsize.png"/><br/><br/>The correct option for the above reaction is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/MNyYI1_Ijgx8UfPt31oFBt8r_oF6i7nCYaKelNDcVyA.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/8mQj0BZLfZahupdwBx11O0WpXEdGY9kQWn7wCuRVLCA.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/vAHX4AcoDqOz-H9ysqSg18LND7vryQb0hKbPLQjn-3s.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mbICjfLPOvgpI4lFfJey1sDsgH_-oub1QJKgmEbIWrc.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/MNyYI1_Ijgx8UfPt31oFBt8r_oF6i7nCYaKelNDcVyA.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/1YRM04QplBY4b3rsBSWfs18FRenHdwyfj3CSyAyOUgM.original.fullsize.png"/><br/><br/>In acidic medium, only monohalogenation takes place.<br/>\(\begin{aligned}<br/>&amp; \mathrm{Y}: \mathrm{CHBr}_3 \\<br/>&amp; \mathrm{Z}: \mathrm{CH}_3 \mathrm{COONa}<br/>\end{aligned}\)<br/>Second step is bromoform reaction that takes place here on \(\mathrm{CH}_3 \mathrm{COCH}_2-\mathrm{Br}\)</div>


title: the-correct-order-of-acid-strength-of-the-following-substituted-phenols-in-water-at-2-8-c-is-1
tags: ['Chemistry', 'General Organic Chemistry', 'JEE Main']
source_db: db\MarksBatch2_P2.db
slug:
the-correct-order-of-acid-strength-of-the-following-substituted-phenols-in-water-at-2-8-c-is-1-90903

question:
<div class="question">The correct order of acid strength of the following substituted phenols in water at $28^{\circ} \mathrm{C}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">p-fluorophenol $     &lt; $ p-chlorophenol $ &lt; $ p-nitrophenol</span> </div>

answer:
<div class="solution">The order of electron withdrawing tendency from benzene ring i.e..
<br/>$$
<br/>-F     &lt; -C l &lt; -N O_{2}
<br/>$$
<br/>$\therefore$ Correct order of acidic strength of substituted phenols will be<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a34.png"/></div>


tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P2.db
title:
the-correct-order-of-acid-strengths-of-benzoic-acid-x-peroxybenzoic-acid-y-and-p-nitrobenzoic-acid-z-is

slug:
the-correct-order-of-acid-strengths-of-benzoic-acid-x-peroxybenzoic-acid-y-and-p-nitrobenzoic-acid-z-is-99276

question:
<div class="question">The correct order of acid strengths of benzoic acid $(X),$ peroxybenzoic acid $(Y)$ and $p$-nitrobenzoic acid $(Z)$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$Z&gt;X&gt;Y$</span> </div>

answer:
<div class="solution">$\because$ In $p$ -nitrobenzoic acid $(Z)-\mathrm{NO}_{2}$ present as electron withdrawing group, thus is most acidic
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a60.png"/> <br/> <br/>$(X) \because$ Benzoic acid is stabilised via resonance, thus is more acidic than per oxybenzoic acid.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a60a.png"/> <br/> <br/>(Y) peroxy benzoic acid does not show any such resonance.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a60b.png"/> <br/> <br/>Thus is least acidic.
<br/>Hence, correct order is $Z&gt;X&gt;Y$</div>


title: the-correct-order-of-acidity-for-the-following-compounds-is-3
slug: the-correct-order-of-acidity-for-the-following-compounds-is-3-66921
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of acidity for the following compounds is:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/1DrY3lejnslvrZAYEUkuGH8nQL-Bo1FeqLTr8Xst7K8.original.fullsize.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$II    &lt; I &lt; III &lt; IV$</span> </div>

answer:
<div class="solution">Hint:
<br/>Compound IV is a carboxylic acid whereas others are alcohols. So IV is most acidic, amongst the alcohols. Presence of $-\mathrm{NO}_{2}$ increases acidic strength while the presence of $-\mathrm{CH}_{3}$ lowers. Amongst I and III, more acidic is $\mathrm{I}$ as $-\mathrm{NO}_{2}$ exerts it's $-\mathrm{R}$ influence in para position.</div>


title: the-correct-order-of-acidity-of-above-compounds-is
slug: the-correct-order-of-acidity-of-above-compounds-is-53860
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XxXWwc5fNpdfC4mqIV6kQ2p_OthQKLNbdZuKQh9RIJE.original.fullsize.png"/><br/><br/>The correct order of acidity of above compounds is</div>

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

answer:
<div class="solution">Hint : \(-\mathrm{NO}_2\) shows strong \(-\mathrm{R}\)<br/>\(-\mathrm{CH}=\mathrm{CH}_2\) shows \(-\mathrm{R}\)<br/>\(-\mathrm{CH}_3\) shows hyperconjugation<br/>\(-\mathrm{NMe}_2\) shows \(+\mathrm{R}\)<br/>Donating groups lowers acidity while withdrawing groups raise acidity</div>


title: the-correct-order-of-acidity-of-the-following-hydra-acids-is
slug: the-correct-order-of-acidity-of-the-following-hydra-acids-is-21416
question: <div class="question">The correct order of acidity of the following hydra acids is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{HF}&gt;\mathrm{HCl}&gt;\mathrm{HBr}&gt;\mathrm{HI}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{HF} &lt; \mathrm{HCl} &lt; \mathrm{HBr} &lt; \mathrm{HI}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{HF} &lt; \mathrm{HCl}&gt;\mathrm{HBr}&gt;\mathrm{HI}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{HF}&gt;\mathrm{HCl} &lt; \mathrm{HBr}&gt;\mathrm{HI}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{HF} &lt; \mathrm{HCl} &lt; \mathrm{HBr} &lt; \mathrm{HI}$</span> </div>

answer:
<div class="solution">$\mathrm{HI}&gt;\mathrm{HBr}&gt;\mathrm{HCl}&gt;\mathrm{HF}$<br/>Down the group acidic strength increases.</div>


title: the-correct-order-of-basicity-of-the-following-compounds-is-1
slug: the-correct-order-of-basicity-of-the-following-compounds-is-1-63175
tags: ['Chemistry', 'Amines', 'JEE Main']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of basicity of the following compounds is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_q5.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1    &lt; 2 &lt; 3 &lt; 4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1    &lt; 2 &lt; 4 &lt; 3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2    &lt; 1 &lt; 3 &lt; 4$</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">$4    &lt; 3 &lt; 2 &lt; 1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2    &lt; 1 &lt; 3 &lt; 4$</span> </div>

answer:
<div class="solution">The correct order is $2    &lt; 1 &lt; 3 &lt; 4$. Since, more the number of $2^{\circ}$ - NH- group, more is the basic nature.
<br/>(1) Thus, 3 and 4 are more basic than 1 and 2 .
<br/>(ii) Between 3 and 4,4 has one two - $\mathrm{NH}_{2}$ group while 3 has only one $-\mathrm{NH}_{2}$ group. $\therefore 4$ is more basic than 3
<br/>(iii) Between 1 and 2,1 has one $-\mathrm{NH}_{2}$ group with no resonance, thus is more basic than 2.</div>


title: the-correct-order-of-boiling-points-of-nethylethanamine-i-ethoxyethane-ii-and-butan2ol-iii-is
slug: the-correct-order-of-boiling-points-of-nethylethanamine-i-ethoxyethane-ii-and-butan2ol-iii-is-43010
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of boiling points of N-ethylethanamine (I), ethoxyethane (II) and butan-2-ol (III) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">II \(  &lt; \) I \( &lt; \) III</span> </div>

answer:
<div class="solution">Hint : Butan-2-ol shows stronger H-bonding than N-ethylenthanamine. Ethoxyethane involves dipole association, weaker than \(\mathrm{H}\)-bonding</div>


title: the-correct-order-of-decreasing-acidity-of-nitrophenols-will-be
slug: the-correct-order-of-decreasing-acidity-of-nitrophenols-will-be-18352
question: <div class="question">The correct order of decreasing acidity of nitrophenols will be</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">m-Nitrophenol \( &gt; \) p-Nitrophenol \( &gt; \) o-Nitrophenol</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">o-Nitrophenol \( &gt; \) m- Nitrophenol \( &gt; \) p-Nitrophenol</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">p-Nitrophenol \( &gt; \) m- Nitrophenol \( &gt; \) o-Nitrophenol</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">p-Nitrophenol \( &gt; \) o-nitrophenol \( &gt; \) m-Nitrophenol</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">p-Nitrophenol \( &gt; \) o-nitrophenol \( &gt; \) m-Nitrophenol</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/wEV86Eh7pCJW3HXCzJ5UkjVE9Kfk1DYxoqgkgl5UaNg.original.fullsize.png"/><br/><br/>Due to \(-\mathrm{I}\) and \(-\mathrm{R}\) influence, \(\mathrm{NO}_2\) in ortho-postion should have raised the acidity to the maximum extent. But it is due to intramolecular \(\mathrm{H}\) - bonding, ortho-nitrophenol is less acidic than para - nitrophenol.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kyFSdBc4cTv2AyixTimrpFvLCDbaQYnoJOxLnX9CHgE.original.fullsize.png"/><br/></div>


title: the-correct-order-of-decreasing-h-c-h-angle-in-the-following-molecule-is
slug: the-correct-order-of-decreasing-h-c-h-angle-in-the-following-molecule-is-23890
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of decreasing $\mathrm{H}-\mathrm{C}-\mathrm{H}$ angle in the following molecule is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q8.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">II&gt;I&gt;III</span> </div>

answer:
<div class="solution">$\mathrm{H}-\mathrm{C}-\mathrm{H}$ bond angle is about $120^{\circ}$, when $\mathrm{C}$ is bonded through a double bond but in case of $\mathrm{CH}_{4}$ the $\mathrm{H}-\mathrm{C}-\mathrm{H}$ bond angle is only $109^{\circ} \mathrm{2B}^{\prime}$
<br/>Cyclic compounds have a $\mathrm{H}-\mathrm{C}-\mathrm{H}$ bond angle Feater than $109^{\circ} 28^{\prime}$ due to hindrance. Thus, the correct order of $1 \mathrm{H}-\mathrm{C}-\mathrm{H}$ bond angle is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a8.png"/></div>


tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P2.db
title:
the-correct-order-of-decreasing-length-of-the-bond-as-indicated-by-the-arrow-in-the-following-structures-is

slug:
the-correct-order-of-decreasing-length-of-the-bond-as-indicated-by-the-arrow-in-the-following-structures-is-18131

question:
<div class="question">The correct order of decreasing length of the bond as indicated by the arrow in the following structures is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q14.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I&gt;II&gt;III</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II&gt;I&gt;III</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">III&gt;II&gt;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">D</span> <span class="option-data">I&gt;III&gt;II</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">III&gt;II&gt;I</span> </div>

answer:
<div class="solution">The length of carbon-carbon single bond is always greater than that of the carbon-carbon double bond.
<br/>In III, positive charge is not in conjugation of double bond, so the bond does not acquire a double bond character. However, chances of acquiring double bond character fof the indicated bond) is much more in I than that in case of II.
<br/>Thus, the correct order of decreasing bond length is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a14.png"/></div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P2.db
title:
the-correct-order-of-equivalent-conductances-at-infinite-dilution-in-water-at-room-temperature-for-h-k-ch-3-coo-and-ho-ions-is

slug:
the-correct-order-of-equivalent-conductances-at-infinite-dilution-in-water-at-room-temperature-for-h-k-ch-3-coo-and-ho-ions-is-76125

question:
<div class="question">The correct order of equivalent conductances at infinite dilution in water at room temperature for $\mathrm{H}^{+}, \mathrm{K}^{+}, \mathrm{CH}_{3} \mathrm{COO}^{-}$ and $\mathrm{HO}^{-}$ ions is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$H^{+}&gt;\mathrm{HO}^{-}&gt;\mathrm{K}^{+}&gt;\mathrm{CH}_{3} \mathrm{COO}^{-}$</span> </div>

answer:
<div class="solution">Equivalent conductance $\left(\Lambda_{\mathrm{eq}}\right)$ is defined as the conducting power of all the ions produced by one gram equivalent of an electrolyte in a given solution.in case of weak electrolytes (as $\mathrm{CH}_{3} \mathrm{COOH}, \mathrm{NH}_{4} \mathrm{OH}, \mathrm{AgCl}$ etc. $),$ ionisation is very small compared to strong electrolytes (as KCl, NaOH etc.), hence $\Lambda_{\text {oq }}$ of weak electrolyte is low. Moreover, ionisation of $\mathrm{H}^{*}$ is maximum among given ions. Thus, correct of order of $\Lambda_{\mathrm{eq}}$ is $\mathrm{H}^{+}&gt;\mathrm{OH}^{-}&gt;\mathrm{K}^{+}&gt;\mathrm{CH}_{3} \mathrm{COO}^{-}$</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
title:
the-correct-order-of-mathrmcmathrmo-bond-length-in-ethyl-propanoate-i-ethyl-propenoate-ii-and-ethenyl-propanoate-iii-is

slug:
the-correct-order-of-mathrmcmathrmo-bond-length-in-ethyl-propanoate-i-ethyl-propenoate-ii-and-ethenyl-propanoate-iii-is-69585

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/JOHwl2vLQhRo2OpNCwPZEruNfarYEHN1abgQknQNLEc.original.fullsize.png"/><br/><br/>The correct order of \(\mathrm{C}=\mathrm{O}\) bond length in ethyl propanoate (I), ethyl propenoate (II) and ethenyl propanoate (III) is</div>

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/9vPWVl6zurgpdylpFNSspBc52GzkXIdchgM0Fm3Jo5I.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sKwtAONtQ7wo0Jg9MuslpxGLX9gpwmUDzkS8w__BAVE.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/L5eNQ4XQ8lNviWhM5HSinmf5XAlAi5-woiD5rvXPzoc.original.fullsize.png"/><br/><br/>\(\mathrm{C}=\mathrm{O}\) bond length<br/>\(\text {II } &gt; \text { I } &gt; \text { III }\)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/BeTHb-ND8nryRu3mKpqZ-hCbeSMzcfvltQ56cQ2gpsk.original.fullsize.png"/><br/> bond has the most single bond character in compound II and the least single bond character in compound III.</div>


title: the-correct-order-of-o-o-bond-length-in-o-2-h-2-o-2-and-o-3-is
slug: the-correct-order-of-o-o-bond-length-in-o-2-h-2-o-2-and-o-3-is-65243
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of $\mathrm{O}-\mathrm{O}$ bond length in $\mathrm{O}_{2}, \mathrm{H}_{2} \mathrm{O}_{2}$ and $\mathrm{O}_{3}$ is</div>

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

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

answer:
<div class="solution">Correct order of $O-O$ bond length is
<br/>$$
<br/>\mathrm{H}_{2} \mathrm{O}_{2}&gt;\mathrm{O}_{3}&gt;\mathrm{O}_{2}
<br/>$$
<br/>(i) Between $\mathrm{O}_{3}$ and $\mathrm{O}_{2}$
<br/>$\because$ Bond order of $\mathrm{O}_{2}&gt;\mathrm{O}_{3}$
<br/>Thus, bond length of $\mathrm{O}_{3}&gt;\mathrm{O}_{2}$
<br/>$$
<br/>\left ( \because \text { Bond order } \propto \frac{1}{\text { Bond length }}\right)
<br/>$$
<br/>(ii) Between $\mathrm{O}_{3}$ and $\mathrm{H}_{2} \mathrm{O}_{2}$
<br/>$\because \mathrm{O}_{3}$ has $\pi$ -bonds in resonance so, has shorter bond length than $O-O$ in $H_{2} O_{2}$
<br/>$\therefore$ Hence, (b) is the answer.</div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P2.db
title:
the-correct-order-of-reactivity-for-the-addition-reaction-of-the-following-carbonyl-compounds-with-ethylmagnesium-iodide-is

slug:
the-correct-order-of-reactivity-for-the-addition-reaction-of-the-following-carbonyl-compounds-with-ethylmagnesium-iodide-is-79457

question:
<div class="question">The correct order of reactivity for the addition reaction of the following carbonyl compounds with ethylmagnesium iodide is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q50.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">I &gt; III &gt; II &gt; IV</span> </div>

answer:
<div class="solution">The correct order of reactivity for the addition reaction of the given carbonyl compounds with ethylmagnesium iodide is $1&gt;111&gt;11&gt;\mathrm{IV}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a50.png"/> <br/>This order can be explained on the basis of following two factors:
<br/>(i) Inductive effect Greater the number of alkyl (electron releasing) groups attached to carbonyl group, greater will be the electron density on carbonyl carbon. Thus, it lowers the attack of nucleophile and hence, reactivity decreases.
<br/>(ii) Steric effect As the number of alkyl group attached to carbonyl carbon increases, the attack of nucleophile on carbonyl group becomes more and more difficult due to steric hinderance.</div>


title: the-correct-order-of-relative-stability-for-the-given-free-radicals-is
slug: the-correct-order-of-relative-stability-for-the-given-free-radicals-is-47414
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of relative stability for the given free radicals is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/y0S6qXNK-EH4wkf1HK0g5HUeHUHA20PhuhJyCu94doE.original.fullsize.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">III &gt; II &gt; I</span> </div>

answer:
<div class="solution">III&gt;II&gt;I <br/>Adjacent functional group appear to weaken $\mathrm{C}-\mathrm{H}$ Bonds, hence making free radical more stable <br/>( All the three are molecules - not radicals. We have assumed them as radicals)</div>


title: the-correct-order-of-relative-stability-of-the-given-conformers-of-n-butane-is
slug: the-correct-order-of-relative-stability-of-the-given-conformers-of-n-butane-is-14481
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">II &gt; I = III<br/>Anti form is more stable than Gauche form.</div>
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of relative stability of the given conformers of $n$-butane is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/A62ZSesLTqMxUyZxvfJpl11GwSpSp1l4QqIneQhmYYo.original.fullsize.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">II &gt; I = III</span> </div>


title: the-correct-order-of-the-addition-reaction-rates-of-halogen-acids-with-ethylene-is
slug: the-correct-order-of-the-addition-reaction-rates-of-halogen-acids-with-ethylene-is-16719
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct order of the addition reaction rates of halogen acids with ethylene is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">hydrogen iodide $&gt;$ hydrogen bromide &gt; hydrogen chloride</span> </div>

answer:
<div class="solution">In halogen acids, as the size of halogen atom increases, the bond between halogen and hydrogen atom weakens. Therefore, the case with which the bond can be broken increases.
<br/>The correct order of the addition reaction rates of halogen acids with ethylene is hydrogen iodide (HI) $&gt;$ Hydrogen bromide (HBr)
<br/>Hydrogen chloride (HCl) So, the option (b) is correct answer.</div>


title: the-correct-pair-of-electron-affinity-order-is
slug: the-correct-pair-of-electron-affinity-order-is-52044
question: <div class="question">The correct pair of electron affinity order is</div>
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{O}&gt;\mathrm{S}, \mathrm{F}&gt;\mathrm{Cl}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{O} &lt; \mathrm{S}, \mathrm{Cl}&gt;\mathrm{F}$</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">$\mathrm{S}&gt;\mathrm{O}, \mathrm{F}&gt;\mathrm{Cl}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{S} &lt; \mathrm{O}, \mathrm{Cl}&gt;\mathrm{F}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{O} &lt; \mathrm{S}, \mathrm{Cl}&gt;\mathrm{F}$</span> </div>

answer:
<div class="solution">Second period elements of p-block have lower values of electron affinity than expected due to unusually smaller size.</div>


title: the-correct-relationship-between-molecules-i-and-ii-is
slug: the-correct-relationship-between-molecules-i-and-ii-is-42656
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/pMgC6j9BWle38AFv5yGmGCiL0oXsVfkf43QCOPrx-Y4.original.fullsize.png"/><br/><br/>The correct relationship between molecules I and II is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/MNQ0Nw-r6A05AMhRGglHCDcTfPUTNQ860vG3tXOuoCw.original.fullsize.png"/><br/><br/>They are homomers</div>


title: the-correct-sets-of-reactions-to-synthesize-benzoic-acid-starting-from-benzene-isare
slug: the-correct-sets-of-reactions-to-synthesize-benzoic-acid-starting-from-benzene-isare-14469
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P2.db
question:
<div class="question">The correct set(s) of reactions to synthesize benzoic acid starting from benzene is/are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">(i) \(\mathrm{Br}_2 / \mathrm{Fe}\)<br/>(ii) Mg / dry ether<br/>(iii) \(\mathrm{CO}_2\)<br/>(iv) \(\mathrm{H}_3 \mathrm{O}^{\oplus}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">(i) \(\mathrm{Br}_2 / \mathrm{Fe}\)<br/>(ii) \(\mathrm{NH}_3, 25^{\circ} \mathrm{C}\)<br/>(iii) \(\mathrm{NaNO}_2\), dil. \(\mathrm{HCl}, 0^{\circ}\) to \(5^{\circ} \mathrm{C}\)<br/>(iv) \(\mathrm{CuCN} / \mathrm{KCN}\)<br/>(v) dil. \(\mathrm{HCl}, \Delta\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">(i) \(\mathrm{CH}_3 \mathrm{Cl}\), Anhydrous \(\mathrm{AlCl}_3\)<br/>(ii) \(\mathrm{KMnO}_4 \mid \mathrm{OH}^{\ominus}, \Delta\)<br/>(iii) \(\mathrm{H}_3 \mathrm{O}^{\oplus}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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">(i) \(\mathrm{CH}_3 \mathrm{COCl}\), Anhydrous \(\mathrm{AlCl}_3\)<br/>(ii) \(\mathrm{Br}_2, \mathrm{NaOH}\),<br/>(iii) \(\mathrm{H}_3 \mathrm{O}^{\oplus}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">(i) \(\mathrm{Br}_2 / \mathrm{Fe}\)<br/>(ii) Mg / dry ether<br/>(iii) \(\mathrm{CO}_2\)<br/>(iv) \(\mathrm{H}_3 \mathrm{O}^{\oplus}\), (i) \(\mathrm{CH}_3 \mathrm{Cl}\), Anhydrous \(\mathrm{AlCl}_3\)<br/>(ii) \(\mathrm{KMnO}_4 \mid \mathrm{OH}^{\ominus}, \Delta\)<br/>(iii) \(\mathrm{H}_3 \mathrm{O}^{\oplus}\), (i) \(\mathrm{CH}_3 \mathrm{COCl}\), Anhydrous \(\mathrm{AlCl}_3\)<br/>(ii) \(\mathrm{Br}_2, \mathrm{NaOH}\),<br/>(iii) \(\mathrm{H}_3 \mathrm{O}^{\oplus}\)</span> </div>

answer:
<div class="solution">Hint : (A) <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ENwGM37H0NWp6TPQjfdEJ0PVr-sKGohpMomskbUcF6Q.original.fullsize.png"/><br/><br/>(C)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/34JIsF1M45wGKqQsrIi_57QdxmTf2NjlbIb5msUm5Io.original.fullsize.png"/><br/><br/>(D)<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/YoYHa72-kmxRUpCBBvZgZh4vMZDejK2WUcncmrcR5Js.original.fullsize.png"/><br/></div>


