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501 | let-p-x-1-y-1-and-q-x-2-y-2-y-1-0-y-2-0-be-the-end-points-of-the-latusrectum-of-the-ellipse-x-2-4-y-2-4-the-equations-of-parabolas-with-latusrectum-pq | let-p-x-1-y-1-and-q-x-2-y-2-y-1-0-y-2-0-be-the-end-points-of-the-latusrectum-of-the-ellipse-x-2-4-y-2-4-the-equations-of-parabolas-with-latusrectum-pq-45989 | <div class="question">Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), y_1 < 0, y_2 < 0$, be the end points of the latusrectum of the ellipse $x^2+4 y^2=4$. The equations of parabolas with latusrectum $P Q$ are</div> | ['Mathematics', 'Ellipse', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$x^2+2 \sqrt{3} y=3+\sqrt{3}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$x^2-2 \sqrt{3} y=3+\sqrt{3}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$x^2+2 \sqrt{3} y=3-\sqrt{3}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$x^2-2 \sqrt{3} y=3-\sqrt{3}$</span> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$x^2-2 \sqrt{3} y=3+\sqrt{3}$<br/>, <br/>$x^2+2 \sqrt{3} y=3-\sqrt{3}$<br/></span> </div> | <div class="solution">The equation $x^2+4 y^2=4$ represents an ellipse with 2 and 1 as semi-major and semi-minor axes and eccentricity $\frac{\sqrt{3}}{2}$. Thus, the ends of latusrect are $\left(\sqrt{3}, \frac{1}{2}\right)$ and $\left(\sqrt{3},-\frac{1}{2}\right),\left(-\sqrt{3},-\frac{1}{2}\right)$ and $\left(\sqrt{3},-\frac{1}{2}\right)$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xEg5w8HkzhfUb5VNonqRlTHouyfpQlt8WmCSapQNpQ8.original.fullsize.png"/><br/><br/>According to the question, we consider only $P\left(-\sqrt{3},-\frac{1}{2}\right)$ and $Q\left(\sqrt{3},-\frac{1}{2}\right)$.<br/>Now, $\quad P Q=2 \sqrt{3}$<br/>Thus, the coordinates of the vertex of the parabolas are $A\left(0, \frac{-1+\sqrt{3}}{2}\right)$ and $A^{\prime}\left(0, \frac{-1-\sqrt{3}}{2}\right)$ and corresponding equations are and<br/>$$<br/>(x-0)^2=-4 \cdot \frac{\sqrt{3}}{2}\left(y+\frac{1-\sqrt{3}}{2}\right)<br/>$$<br/>i.e., $(x-0)^2=4 \cdot \frac{\sqrt{3}}{2}\left(y-\frac{-1-\sqrt{3}}{2}\right)$ and<br/>$$<br/>\begin{aligned}<br/>& x^2+2 \sqrt{3} y=3-\sqrt{3} \\<br/>& x^2-2 \sqrt{3} y=3+\sqrt{3}<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
502 | let-p-x-be-a-polynomial-of-degree-4-having-extremum-at-x-1-2-and-lim-x-0-1-x-2-p-x-2-then-the-value-of-p-2-is | let-p-x-be-a-polynomial-of-degree-4-having-extremum-at-x-1-2-and-lim-x-0-1-x-2-p-x-2-then-the-value-of-p-2-is-16434 | <div class="question">Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and $\lim _{x \rightarrow 0}\left[1+\frac{p(x)}{x^2}\right]=2$. Then, the value of $p(2)$ is</div> | ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">0</span> </div> | <div class="solution">Let $p(x)=a x^4+b x^3+c x^2+d x+e$<br/>$$<br/>\Rightarrow \quad p^{\prime}(x)=4 a x^3+3 b x^2+2 c x+d<br/>$$<br/>$$<br/>\therefore \quad p^{\prime}(1)=4 a+3 b+2 c+d=0<br/>$$<br/>and $p^{\prime}(2)=32 a+12 b+4 c+d=0$<br/>Since, $\lim _{x \rightarrow 0}\left(1+\frac{p(x)}{x^2}\right)=2 \quad$ [given]<br/>$$<br/>\lim _{x \rightarrow 0} \frac{a x^4+b x^3+(c+1) x^2+d x+e}{x^2}=2<br/>$$<br/><br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow & & x+1=2, d & =0, e=0 \\<br/>\Rightarrow & c & =1<br/>\end{array}<br/>$$<br/>From Eqs. (i) and (ii)<br/>$$<br/>\begin{array}{rlrl} <br/>& & 4 a+3 b=-2 & \text { and } 32 a+12 b=-4 \\<br/>\Rightarrow & & a=\frac{1}{4} \text { and } b=-1 \\<br/>\therefore & & p(x)=\frac{x^4}{4}-x^3+x^2 \\<br/>\Rightarrow & & p(2)=\frac{16}{4}-8+4<br/>\end{array}<br/>$$</div> | MarksBatch2_P2.db |
503 | let-p-x-be-a-real-polynomial-of-least-degree-which-has-a-local-maximum-at-x-1-and-a-local-minimum-at-x-3-if-p-1-6-and-p-3-2-then-p-0-is | let-p-x-be-a-real-polynomial-of-least-degree-which-has-a-local-maximum-at-x-1-and-a-local-minimum-at-x-3-if-p-1-6-and-p-3-2-then-p-0-is-75729 | <div class="question">Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is</div> | ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">9</span> </div> | <div class="solution">Since, $p(x)$ has a local maximum at $x=1$ and a local minimum at $x=3$ and $p(x)$ is a real polynomial of least degree.
<br/> <br/>Hence, let $p^{\prime}(x)=k(x-1)(x-3)=k\left(x^{2}-4 x+3\right)$
<br/> <br/>$\Rightarrow p(x)=k\left(\frac{x^{3}}{3}-2 x^{2}+3 x\right)+c$
<br/> <br/>Now, $p(1)=6$ and $p(3)=2$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow \frac{4}{3} k+C=6 \text { and } 0+C=2 \Rightarrow k=3 \\
<br/> <br/>\therefore \quad p^{\prime}(x)=3(x-1)(x-3) \Rightarrow p^{\prime}(0)=9
<br/> <br/>\end{array}$</div> | MarksBatch2_P2.db |
504 | let-pqr-be-a-triangle-of-area-with-a-2-b-2-7-and-c-2-5-where-a-b-and-c-are-the-lengths-of-the-sides-of-the-triangle-opposite-to-the-angles-at-p-q-and- | let-pqr-be-a-triangle-of-area-with-a-2-b-2-7-and-c-2-5-where-a-b-and-c-are-the-lengths-of-the-sides-of-the-triangle-opposite-to-the-angles-at-p-q-and-53112 | <div class="question">Let $P Q R$ be a triangle of area $\Delta$ with $a=2, b=\frac{7}{2}$ and $c=\frac{5}{2} ;$ where $a, b$, and $c$ are the lengths of the sides of the triangle opposite to the angles at $P, Q$ and $R$ respectively. Then $\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}$ equals.</div> | ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{4 \Delta}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{45}{4 \Delta}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{3}{4 \Delta}\right)^{2}$</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">$\left(\frac{45}{4 \Lambda}\right)^{2}$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{3}{4 \Delta}\right)^{2}$</span> </div> | <div class="solution">$\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}=\frac{2 \sin P-2 \sin P \cos P}{2 \sin P+2 \sin P \cos P}=\frac{1-\cos P}{1-\sin P}$
<br/> <br/>$=\frac{2 \sin ^{2} \frac{P}{2}}{2 \cos ^{2} \frac{P}{2}}=\tan ^{2} \frac{P}{2}=\frac{(s-b)(s-c)}{s(s-a)}$ where $s=\frac{a+b+c}{2}$
<br/> <br/>$=\frac{(s-b)^{2}(s-c)^{2}}{s(s-a)(s-b)(s-c)}=\frac{(a+c-b)^{2}(a+b-c)^{2}}{16 . \Delta^{2}}$
<br/> <br/>$=\frac{\left(2+\frac{5}{2}-\frac{7}{2}\right)^{2}\left(2+\frac{7}{2}-\frac{5}{2}\right)^{2}}{16 \Delta^{2}}=\frac{1 \times 9}{16 \Delta^{2}}=\left(\frac{3}{4 \Delta}\right)^{2}$</div> | MarksBatch2_P2.db |
505 | let-s-1-2-3-4-the-total-number-of-unordered-pairs-of-disjoint-subsets-of-s-is-equal-to | let-s-1-2-3-4-the-total-number-of-unordered-pairs-of-disjoint-subsets-of-s-is-equal-to-35136 | <div class="question">Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to</div> | ['Mathematics', 'Sets and Relations', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>25<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>34<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>42<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>41</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>41</span> </div> | <div class="solution">Let $A \cap B=\phi, A, B \subset S$<br/>$$<br/>\begin{aligned}<br/>& 3^4=\frac{81+1}{2}=41 \\<br/>\Rightarrow \quad & \frac{3^4+1}{2}=41<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
506 | let-s-be-the-area-of-the-region-enclosed-by-y-e-x-2-y-0-x-0-and-x-1-then | let-s-be-the-area-of-the-region-enclosed-by-y-e-x-2-y-0-x-0-and-x-1-then-66852 | <div class="question">Let $\mathrm{S}$ be the area of the region enclosed by $y=e^{-x^{2}}$, $y=0, x=0$ and $x=1$; then</div> | ['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$S \geq \frac{1}{e}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$S \geq 1-\frac{1}{e}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value">$S \geq \frac{1}{e}$, $S \geq 1-\frac{1}{e}$, $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$</span> </div> | <div class="solution">The given curve $y=e^{-x^{2}}$ Draw a rough sketch of curve at $x=0, y=1$ and at $x=1, y=1 / e$ $\because y=e^{-x^{2}}$ $\Rightarrow \frac{d y}{d x}=-2 x e^{-x^{2}} < 0 \quad \forall x \in(0,1)$
<br/> <br/>$\therefore y=e^{-x^{2}}$ is decreasing on $(0,1)$
<br/> <br/>Hence its graph is as shown in figure given below<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/EsW5pZYCySy0dpsUUr6Q7PG_G-p7oYK6ONlT1YJfaKg.original.fullsize.png"/><br/> <br/> <br/>Now, $\mathrm{S}=$ area exclosed by curve $=\mathrm{XYCO}$
<br/> <br/>and area of rectangle $\mathrm{OCYL}=\frac{1}{e}$
<br/> <br/>Clearly $S>\frac{1}{e} \quad \therefore$ a is true.
<br/> <br/>For $x \in[0,1] \Rightarrow x^{2} < x$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow-x^{2}>-x \quad \Rightarrow e^{-x^{2}} \geq e^{-x} \forall x \in[0,1] \\
<br/> <br/>\Rightarrow \int_{0}^{1} e^{-x^{2}} d x>\int_{0}^{1} e^{-x} d x=1-\frac{1}{e} \\
<br/> <br/>\Rightarrow S>1-\frac{1}{e} \quad \therefore \text { (b) is true. }
<br/> <br/>\end{array}$
<br/> <br/>Now $S < $ area of rectangle $\mathrm{XADO}+$ area of rectangle $\mathrm{ZDCN}$ $\Rightarrow S < \frac{1}{\sqrt{2}} \times 1+\left(1-\frac{1}{\sqrt{2}}\right) \frac{1}{\sqrt{e}}$ $\therefore S < \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right) \quad \because(\mathrm{d})$ is true
<br/> <br/>Also as $\frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right) < 1-\frac{1}{e} \quad \therefore$ (c) is incorrect.</div> | MarksBatch2_P2.db |
507 | let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian-1 | let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian-1-68798 | <div class="question">Let $S$ be the focus of the parabola $y^{2}=8 \mathrm{x}$ and let $P Q$ be the common chord of the circle $x^{2}+y^{2}-2 x-4 y=0$ and the given parabola. The area of the triangle $P Q S$ is</div> | ['Mathematics', 'Parabola', 'JEE Main'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">4</span> </div> | <div class="solution">Given parabola $y^{2}=8 x$
<br/> <br/>and circle $x^{2}+y^{2}-2 x-4 y=0$ pass through the origin
<br/> <br/>$\therefore$ One end of common chord PQ is origin. Say $\mathrm{P}(0,0)$
<br/> <br/>Let $\mathrm{Q}$ be the point $\left(2 t^{2}, 4 t\right)$, then it will satisfy the equation of circle.
<br/> <br/>$\therefore \quad 4 t^{4}+16 t^{2}-4 t^{2}-16 t=0$
<br/> <br/>$\Rightarrow t^{4}+3 t^{2}-4 t=0 \Rightarrow t\left(t^{3}+3 t-4\right)=0$
<br/> <br/>$\Rightarrow t(t-1)\left(t^{2}+t-4\right)=0 \Rightarrow t=0$ or 1
<br/> <br/>For $t=0$, we get point $P$, therefore $t=1$ gives point $Q$ as $(2,4)$.
<br/> <br/>We also observe here that $\mathrm{P}(0,0)$ and $\mathrm{Q}(2,4)$ are end points of diameter of the given circle and focus of the parabola is the point $S(2,0)$.
<br/> <br/>$\therefore \quad$ area $(\Delta \mathrm{PQS})=\frac{1}{2} \times P S \times Q S=\frac{1}{2} \times 2 \times 4=4$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eU4fQJz3adHaoGP7hh6deAA3XZ2LwdvTG4BfAwf634Y.original.fullsize.png"/></div> | MarksBatch2_P2.db |
508 | let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian | let-s-be-the-focus-of-the-parabola-y-2-8-x-and-let-pq-be-the-common-chord-of-the-circle-x-2-y-2-2-x-4-y-0-and-the-given-parabola-the-area-of-the-trian-89208 | <div class="question">Let $S$ be the focus of the parabola $y^{2}=8 \mathrm{x}$ and let $P Q$ be the common chord of the circle $x^{2}+y^{2}-2 x-4 y=0$ and the given parabola. The area of the triangle $P Q S$ is</div> | ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">4</span> </div> | <div class="solution">Given parabola $y^{2}=8 x$
<br/> <br/>and circle $x^{2}+y^{2}-2 x-4 y=0$ pass through the origin
<br/> <br/>$\therefore$ One end of common chord PQ is origin. Say $\mathrm{P}(0,0)$
<br/> <br/>Let $\mathrm{Q}$ be the point $\left(2 t^{2}, 4 t\right)$, then it will satisfy the equation of circle.
<br/> <br/>$\therefore \quad 4 t^{4}+16 t^{2}-4 t^{2}-16 t=0$
<br/> <br/>$\Rightarrow t^{4}+3 t^{2}-4 t=0 \Rightarrow t\left(t^{3}+3 t-4\right)=0$
<br/> <br/>$\Rightarrow t(t-1)\left(t^{2}+t-4\right)=0 \Rightarrow t=0$ or 1
<br/> <br/>For $t=0$, we get point $P$, therefore $t=1$ gives point $Q$ as $(2,4)$.
<br/> <br/>We also observe here that $\mathrm{P}(0,0)$ and $\mathrm{Q}(2,4)$ are end points of diameter of the given circle and focus of the parabola is the point $S(2,0)$.
<br/> <br/>$\therefore \quad$ area $(\Delta \mathrm{PQS})=\frac{1}{2} \times P S \times Q S=\frac{1}{2} \times 2 \times 4=4$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eU4fQJz3adHaoGP7hh6deAA3XZ2LwdvTG4BfAwf634Y.original.fullsize.png"/></div> | MarksBatch2_P2.db |
509 | let-s-k-k-1-2-100-denote-the-sum-of-the-infinite-geometric-series-whose-first-term-is-k-k-1-and-the-common-ratio-is-k-1-then-the-value-of-100-10-0-2-k | let-s-k-k-1-2-100-denote-the-sum-of-the-infinite-geometric-series-whose-first-term-is-k-k-1-and-the-common-ratio-is-k-1-then-the-value-of-100-10-0-2-k-38259 | <div class="question">Let $S_k, k=1,2, \ldots, 100$, denote the sum of the infinite geometric series whose first term is $\frac{k-1}{k !}$ and the common ratio is $\frac{1}{k}$. Then the value of $\frac{100^2}{100 !}+\sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right|$ is</div> | ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">4</span> </div> | <div class="solution">We have, $S_k=\frac{\frac{k-1}{k !}}{1-\frac{1}{k}}=\frac{1}{(k-1) !}$<br/>Now,<br/>$$<br/>\begin{aligned}<br/>& \left(k^2-3 k+1\right) S_k=\{(k-2)(k-1)-1\} \\<br/>& =\frac{1}{(k-3) !}-\frac{1}{(k-1) !} \\<br/>& \Rightarrow \sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right| \\<br/>& =1+1+2-\left(\frac{1}{99 !}+\frac{1}{98 !}\right)=4-\frac{100^2}{100 !}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\Rightarrow \quad \frac{100^2}{100 !}+\sum_{k=1}^{100}\left|\left(k^2-3 k+1\right) S_k\right|=4<br/>$$</div> | MarksBatch2_P2.db |
510 | let-s-n-k-0-n-n-2-kn-k-2-n-and-t-n-k-0-n-1-n-2-kn-k-2-n-for-n-1-2-3-then | let-s-n-k-0-n-n-2-kn-k-2-n-and-t-n-k-0-n-1-n-2-kn-k-2-n-for-n-1-2-3-then-34514 | <div class="question">Let $S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}$ and $T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}$, for $n=1,2,3, \ldots$, then</div> | ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$S_n < \frac{\pi}{3 \sqrt{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$S_n>\frac{\pi}{3 \sqrt{3}}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$T_n < \frac{\pi}{3 \sqrt{3}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$T_n>\frac{\pi}{3 \sqrt{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$S_n < \frac{\pi}{3 \sqrt{3}}$, <br/>$T_n>\frac{\pi}{3 \sqrt{3}}$</span> </div> | <div class="solution">Given,<br/>$$<br/>\begin{aligned}<br/>S_n & =\sum_{k=0}^n \frac{n}{n^2+k n+k^2} \\<br/>& =\sum_{k=0}^n \frac{1}{n} \cdot\left(\frac{1}{1+\frac{k}{n}+\frac{k^2}{n^2}}\right) < \lim _{n \rightarrow \infty} \sum_{k=0}^n \frac{1}{n}\left(\frac{1}{1+\frac{k}{n}+\left(\frac{k}{n}\right)^2}\right) \\<br/>& =\int_0^1 \frac{1}{1+x+x^2} d x=\left[\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2}{\sqrt{3}}\left(x+\frac{1}{2}\right)\right)\right]_0^1 \\<br/>& =\frac{2}{\sqrt{3}} \cdot\left(\frac{\pi}{3}-\frac{\pi}{6}\right)=\frac{\pi}{3 \sqrt{3}}<br/>\end{aligned}<br/>$$<br/>i.e., $\quad S_n < \frac{\pi}{3 \sqrt{3}}$<br/>Similarly, $T_n>\frac{\pi}{3 \sqrt{3}}$.</div> | MarksBatch2_P2.db |
511 | let-the-eccentricity-of-the-hyperbola-a-2-x-2-b-2-y-2-1-be-reciprocal-to-that-of-the-ellipse-x-2-4-y-2-4-if-the-hyperbola-passes-through-a-focus-of-th | let-the-eccentricity-of-the-hyperbola-a-2-x-2-b-2-y-2-1-be-reciprocal-to-that-of-the-ellipse-x-2-4-y-2-4-if-the-hyperbola-passes-through-a-focus-of-th-82215 | <div class="question">Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4 y^2=4$. If the hyperbola passes through a focus of the ellipse, then</div> | ['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>the equation of the hyperbola is $\frac{x^2}{3^2}-\frac{y^2}{2^2}=1$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>a focus of the hyperbola is $(2,0)$<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"><br/>the eccentricity of the hyperbola is $\sqrt{\frac{5}{3}}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>the equation of the hyperbola is $x^2-3 y^2=3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>a focus of the hyperbola is $(2,0)$<br/>, <br/>the equation of the hyperbola is $x^2-3 y^2=3$</span> </div> | <div class="solution">Here, equation of ellipse<br/>$$<br/>\begin{aligned}<br/>& \frac{x^2}{4}+\frac{y^2}{1}=1 \\<br/>& \Rightarrow \quad e^2=1-\frac{b^2}{a^2}=1-\frac{1}{4}=\frac{3}{4} \\<br/>& \therefore \quad e=\frac{\sqrt{3}}{2} \text { and focus }(\pm a e, 0) \\<br/>& =(\pm \sqrt{3}, 0) \\<br/>&<br/>\end{aligned}<br/>$$<br/>For hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, $e_1^2=1+\frac{b^2}{a^2}$ where, $e_1^2=\frac{1}{e^2}=\frac{4}{3}$<br/>$$<br/>\Rightarrow \quad 1+\frac{b^2}{a^2}=\frac{4}{3} \Rightarrow \frac{b^2}{a^2}=\frac{1}{3}<br/>$$<br/>and hyperbola passes through $(\pm \sqrt{3}, 0)$. Now, $\quad \frac{3}{a^2}=1 \Rightarrow a^2=3$<br/>From Eqs. (i) and (ii), we get $b^2=1$<br/>$\therefore$ Equation of hyperbola is<br/>$$<br/>\frac{x^2}{3}-\frac{y^2}{1}=1<br/>$$<br/>Focus is $(\pm a e, 0)$.<br/>Now, $\quad\left(\pm \sqrt{3} \cdot \frac{2}{\sqrt{3}}, 0\right) \Rightarrow(\pm 2,0)$<br/>Hence, both options (b) and (d) are correct.</div> | MarksBatch2_P2.db |
512 | let-the-function-g-2-2-be-given-by-g-u-2-tan-1-e-u-2-then-g-is | let-the-function-g-2-2-be-given-by-g-u-2-tan-1-e-u-2-then-g-is-79924 | <div class="question">Let the function $g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$. Then, $g$ is</div> | ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>even and is strictly increasing in $(0, \infty)$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>odd and is strictly decreasing in $(-\infty, \infty)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>odd and is strictly increasing in $(-\infty, \infty)$<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">D</span> <span class="option-data"><br/>Neither even nor odd, but is strictly increasing in $(-\infty, \infty)$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>odd and is strictly increasing in $(-\infty, \infty)$<br/></span> </div> | <div class="solution">$\therefore \quad g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$<br/>for $\quad u \in(-\infty, \infty)$<br/>and<br/>$$<br/>\begin{aligned}<br/>g(-u) & =2 \tan ^{-1}\left(e^{-u}\right)-\frac{\pi}{2} \\<br/>& =2\left(\cot ^{-1}\left(e^u\right)\right)-\frac{\pi}{2} \\<br/>& =2\left(\frac{\pi}{2}-\tan ^{-1}\left(e^u\right)\right)-\frac{\pi}{2} \\<br/>& =\frac{\pi}{2}-2 \tan ^{-1}\left(e^u\right)=-g(u)<br/>\end{aligned}<br/>$$<br/>$\therefore \quad g(-u)=-g(u)$<br/>$\Rightarrow g(u)$ is an odd function.</div> | MarksBatch2_P2.db |
513 | let-the-straight-line-x-b-divide-the-area-enclosed-by-y-1-x-2-y-0-and-x-0-into-two-parts-r-1-0-x-b-and-r-2-b-x-1-such-that-r-1-r-2-4-1-then-b-equals-t | let-the-straight-line-x-b-divide-the-area-enclosed-by-y-1-x-2-y-0-and-x-0-into-two-parts-r-1-0-x-b-and-r-2-b-x-1-such-that-r-1-r-2-4-1-then-b-equals-t-70961 | <div class="question">Let the straight line $x=b$ divide the area enclosed by $y=(1-x)^2, y=0$ and $x=0$ into two parts $R_1(0 \leq x \leq b)$ and $R_2(b \leq x \leq 1)$ such that $R_1-R_2=\frac{1}{4}$. Then, $b$ equals to</div> | ['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{3}{4}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{1}{2}$<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"><br/>$\frac{1}{3}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{1}{4}$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{1}{2}$<br/></span> </div> | <div class="solution">Here, area between 0 to $b$ is $R_1$ and $b$ to<br/>$$<br/>\begin{aligned}<br/>& 1 \text { is } R_2 \\<br/>& \therefore \int_0^b(1-x)^2 d x-\int_b^1(1-x)^2 d x=\frac{1}{4} \\<br/>& \Rightarrow \quad\left(\frac{(1-x)^3}{-3}\right)_0^b-\left(\frac{(1-x)^3}{-3}\right)_b^1=\frac{1}{4} \\<br/>& \Rightarrow-\frac{1}{3}\left\{(1-b)^3-1\right\}+\frac{1}{3}\left\{0-(1-b)^3\right\} \\<br/>& =\frac{1}{4}<br/>\end{aligned}<br/>$$<br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad-\frac{2}{3}(1-b)^3=-\frac{1}{3}+\frac{1}{4}=-\frac{1}{12} \\<br/>& \Rightarrow \quad(1-b)^3=\frac{1}{8} \\<br/>& \Rightarrow \quad(1-b)=\frac{1}{2} \Rightarrow b=\frac{1}{2} \\<br/>&<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
514 | let-the-vectors-pq-qr-rs-st-tu-and-up-represent-the-sides-of-a-regular-hexagon-statement-i-pq-rs-st-0-statement-ii-pq-rs-0-and-pq-sq-0 | let-the-vectors-pq-qr-rs-st-tu-and-up-represent-the-sides-of-a-regular-hexagon-statement-i-pq-rs-st-0-statement-ii-pq-rs-0-and-pq-sq-0-36339 | <div class="question">Let the vectors $\mathbf{P Q}, \mathbf{Q R}, \mathbf{R S}, \mathbf{S T}, \mathbf{T U}$ and UP represent the sides of a regular hexagon.<br/>Statement I PQ $\times(\mathbf{R S}+\mathbf{S T}) \neq \mathbf{0}$<br/>Statement II $\mathbf{P Q} \times \mathbf{R S}=\mathbf{0}$ and $\mathbf{P Q} \times \mathbf{S Q} \neq \mathbf{0}$</div> | ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>Statement I is true, Statement II is false<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">D</span> <span class="option-data"><br/>Statement I is false, Statement II is true</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is true, Statement II is false<br/></span> </div> | <div class="solution">Since, $\mathbf{P Q}$ is not parallel to $\mathbf{T R}$<br/>$\because$ TR is resultant of $\mathbf{S R}$ and $\mathbf{S T}$ vectors.<br/>$\Rightarrow \quad \mathbf{P Q} \times(\mathbf{R S}+\mathbf{S T}) \neq \mathbf{0}$<br/>But for Statement II, we have $\mathbf{P Q} \times \mathbf{R S}=\mathbf{0}$<br/>which is not possible as $\mathbf{P Q}$ not parallel to $\mathbf{R S}$.<br/>Hence, Statement I is true and Statement II is false.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jQ8Sa0ufMUOmwUe0xgUS2AYCWxDpsiUAFOd6R-w_QmA.original.fullsize.png"/><br/></div> | MarksBatch2_P2.db |
515 | let-two-noncollinear-unit-vectors-a-and-b-form-an-acute-angle-a-point-p-moves-so-that-at-any-time-t-the-position-vector-op-where-o-is-the-origin-is-gi | let-two-noncollinear-unit-vectors-a-and-b-form-an-acute-angle-a-point-p-moves-so-that-at-any-time-t-the-position-vector-op-where-o-is-the-origin-is-gi-10387 | <div class="question">Let two non-collinear unit vectors $\mathbf{a}$ and $\hat{\mathbf{b}}$ form an acute angle.<br/>A point $P$ moves so that at any time $t$ the position vector $\mathbf{O P}$ (where, $O$ is the origin) is given by $\mathbf{a} \cos t+\hat{\mathbf{b}} \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\mathbf{O P}$ and $\hat{\mathbf{u}}$ be the unit vector along $\mathbf{O P}$. Then,</div> | ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$<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"><br/>$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}-\hat{\mathbf{b}}}{|\hat{\mathbf{a}}-\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}-\hat{\mathbf{b}}}{|\mathbf{\mathbf { a }}-\hat{\mathbf{b}}|}$ and $M=(1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$<br/></span> </div> | <div class="solution">$\mathbf{O P}=\hat{\mathbf{a}} \cos t+\hat{\mathbf{b}} \sin t$<br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad|\mathbf{O P}|=\sqrt{\left(\hat{\mathbf{a}} \cdot \hat{\mathbf{a}} \cos ^2 t+\hat{\mathbf{b}} \cdot \hat{\mathbf{b}} \sin ^2 t+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \sin t \cos t\right)} \\<br/>& \Rightarrow \quad|\mathbf{O P}|=\sqrt{1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \cdot \sin t \cos t} \\<br/>& \Rightarrow \quad|\mathbf{O P}|=\sqrt{1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \cdot \sin 2 t} \\<br/>& \Rightarrow \quad|\mathbf{O P}|_{\max }=\sqrt{1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}} \text { at } \sin 2 t=1 \Rightarrow t=\frac{\pi}{4} \\<br/>& \Rightarrow \mathbf{O P}\left(\text { at } t=\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}(\hat{\mathbf{a}}+\hat{\mathbf{b}}) \\<br/>&<br/>\end{aligned}<br/>$$<br/>$\therefore$ Unit vector along $\mathbf{O P}$ at $\left(t=\frac{\pi}{4}\right)=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$</div> | MarksBatch2_P2.db |
516 | let-x-0-y-0-be-the-solution-of-the-following-equations-2-x-l-o-g-2-3-y-l-o-g-3-3-l-o-g-x-2-l-o-g-y-then-x-0-is-equal-to | let-x-0-y-0-be-the-solution-of-the-following-equations-2-x-l-o-g-2-3-y-l-o-g-3-3-l-o-g-x-2-l-o-g-y-then-x-0-is-equal-to-82320 | <div class="question">Let $\left(x_0, y_0\right)$ be the solution of the following equations $(2 x)^{\log 2}=(3 y)^{\log 3}$, $3^{\log x}=2^{\log y}$, then $x_0$ is equal to</div> | ['Mathematics', 'Basic of Mathematics', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{1}{6}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{1}{2}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{1}{2}$<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">D</span> <span class="option-data"><br/>6</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{1}{2}$<br/></span> </div> | <div class="solution">Taking log on both sides,<br/>$$<br/>\log 2 \cdot \log (2 x)=\log 3(\log 3 y)<br/>$$<br/><br/><br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow & \log 2\{\log 2+\log x\} \\<br/>& =\log 3\{\log 3+\log y\} \\<br/>\text { and } & \log x \cdot \log 3=\log y \log 2 \\<br/>& \log y=\frac{\log x \cdot \log 3}{\log 2}<br/>\end{array}<br/>$$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>\begin{aligned}<br/>& \log 2\{\log 2+\log x\} \\<br/>&=\log 3 \cdot\left\{\log 3+\frac{\log x \cdot \log 3}{\log 2}\right\} \\<br/>& \Rightarrow \quad(\log 2)^2+\log 2 \cdot \log x \\<br/>&=(\log 3)^2+\frac{(\log 3)^2}{(\log 2)} \cdot \log x \\<br/>& \Rightarrow \quad \log x\left\{\frac{(\log 3)^2}{\log 2}-\log 2\right\} \\<br/>&=(\log 2)^2-(\log 3)^2<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad \log x=-\log 2=\log 2^{-1} \\<br/>& \therefore \quad x=\frac{1}{2}<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
517 | let-x-and-y-be-two-events-such-that-p-x-y-2-1-p-y-x-3-1-and-p-x-y-6-1-which-of-the-following-is-are-correct | let-x-and-y-be-two-events-such-that-p-x-y-2-1-p-y-x-3-1-and-p-x-y-6-1-which-of-the-following-is-are-correct-99603 | <div class="question">Let $X$ and $Y$ be two events such that $P(X \mid Y)=\frac{1}{2}$, $P(Y / X)=\frac{1}{3}$ and $P(X \cap Y)=\frac{1}{6}$. Which of the following is (are) correct?</div> | ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$P(X \cup Y)=\frac{2}{3}$</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="correct"> <span class="option-label">B</span> <span class="option-data">$X$ and $Y$ are independent</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$X$ and $Y$ are not independent</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$P\left(X^{c} \cap Y\right)=\frac{1}{3}$</span> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value">$P(X \cup Y)=\frac{2}{3}$, $X$ and $Y$ are independent</span> </div> | <div class="solution">$\because P(X / Y)=\frac{P(X \cap Y)}{P(Y)} \Rightarrow \frac{1}{2}=\frac{1 / 6}{P(Y)} \Rightarrow P(Y)=\frac{1}{3}$
<br/> <br/>Similarly, $\mathrm{P}(Y / X)=\frac{P(X \cap Y)}{P(X)}$
<br/> <br/>$\Rightarrow \frac{1}{3}=\frac{1 / 6}{P(X)} \Rightarrow P(X)=\frac{1}{2}$
<br/> <br/>(a) $P(X \cup Y)=P(X)+P(Y)-P(X \cap Y)=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}$
<br/> <br/>$\therefore$ (a) is true.
<br/> <br/>(b) $\because P(X \cap Y)=P(X) P(Y)$
<br/> <br/>$\Rightarrow X$ and $Y$ are independent events.
<br/> <br/>$\therefore$ (b) is true.
<br/> <br/>But (c) is not true.
<br/> <br/>(d) $\mathrm{P}\left(X^{C} \cap Y\right)=P\left(X^{C}\right) \times P(Y)=\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$
<br/> <br/>$\therefore$ (d) is not true.</div> | MarksBatch2_P2.db |
518 | let-x-x-2-cos-x-x-0-0-x-0-x-r-then-is | let-x-x-2-cos-x-x-0-0-x-0-x-r-then-is-54281 | <div class="question">Let $\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.$ then $\int$ is</div> | ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">differentiable both at $x=0$ and at $x=2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">differentiable at $x=0$ but not differentiable at $x=2$</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">not differentiable at $x=0$ but differentiable at $x=2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">differentiable neither at $x=0$ nor at $x=2$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">differentiable at $x=0$ but not differentiable at $x=2$</span> </div> | <div class="solution">$f^{\prime}\left(0^{+}\right)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$
<br/> <br/>$\begin{aligned}
<br/> <br/>=\lim _{h \rightarrow 0} \frac{h^{2}\left|\cos \frac{\pi}{h}\right|}{h} &=\lim _{h \rightarrow 0} h\left|\cos \frac{\pi}{h}\right| \\
<br/> <br/>&=0 \times \text { some finite value }=0
<br/> <br/>\end{aligned}$
<br/> <br/>$\begin{array}{c}
<br/> <br/>=0 \times \text { some finite value }=0 \\
<br/> <br/>\text { and } f^{\prime}\left(0^{-}\right)=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0} \frac{h^{2}\left|\cos \frac{\pi}{-h}\right|}{-h} \\
<br/> <br/>=\lim _{h \rightarrow 0}-h\left|\cos \frac{\pi}{h}\right|=0 \times \text { some finite value }=0
<br/> <br/>\end{array}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\because f^{\prime}\left(0^{+}\right)=f^{\prime}\left(0^{-}\right) \therefore f \text { is differentiable at } x=0 \\
<br/> <br/>\text { Now } f^{\prime}\left(2^{+}\right)=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h} \\
<br/> <br/>\quad=\lim _{h \rightarrow 0} \frac{(2+h)^{2}\left|\cos \frac{\pi}{2+h}\right|-4\left|\cos \frac{\pi}{2}\right|}{h}
<br/> <br/>\end{array}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}\left(\cos \frac{\pi}{2+h}\right)}{h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}}{h} \sin \left(\frac{\pi}{2}-\frac{\pi}{2+h}\right)$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}}{h} \sin \left(\frac{\pi h}{2(2+h)}\right)$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2+h)^{2}}{h} \times \frac{\sin \left(\frac{\pi h}{2(2+h)}\right)}{\left(\frac{\pi h}{2(2+h)}\right)} \times \frac{\pi h}{2(2+h)}=\pi$
<br/> <br/>and $f^{\prime}\left(2^{-}\right)=\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2-h)^{2}\left|\cos \left(\frac{\pi}{2-h}\right)\right|-0}{-h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{-(2-h)^{2} \cos \left(\frac{\pi}{2-h}\right)}{-h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2-h)^{2} \sin \left(\frac{\pi}{2}-\frac{\pi}{2-h}\right)}{h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{(2-h)^{2}}{h} \times \frac{\sin \left(\frac{-\pi h}{2(2-h)}\right)}{\left(\frac{-\pi h}{2(2-h)}\right)} \times\left(\frac{-\pi h}{2(2-h)}\right)=-\pi$
<br/> <br/>$\because f^{\prime}\left(2^{+}\right) \neq f^{\prime}\left(2^{-}\right), \therefore f$ is not differentiable at $x=2 .$</div> | MarksBatch2_P2.db |
519 | let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is-1 | let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is-1-36623 | <div class="question">Let $(x, y)$ be any point on the parabola $y^2=4 x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x, y)$ in the ratio $1: 3$. Then, the locus of $P$ is</div> | ['Mathematics', 'Parabola', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$x^2=y$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$y^2=2 x$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$y^2=x$<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">D</span> <span class="option-data"><br/>$x^2=2 y$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$y^2=x$<br/></span> </div> | <div class="solution">$$<br/>\text { By section formula, }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NZeJW46lYjK_Tb4Ljhd5REr04ncqMpNoSecFzLx2gXo.original.fullsize.png"/><br/><br/>$h=\frac{x+0}{4}, k=\frac{y+0}{4}$<br/>$\therefore \quad x=4 h$ and $y=4 k$<br/>Substituting in $y^2=4 x$,<br/>$(4 k)^2=4(4 h) \Rightarrow k^2=h$<br/>or $y^2=x$ is required locus.</div> | MarksBatch2_P2.db |
520 | let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is | let-x-y-be-any-point-on-the-parabola-y-2-4-x-let-p-be-the-point-that-divides-the-line-segment-from-0-0-to-x-y-in-the-ratio-1-3-then-the-locus-of-p-is-57991 | <div class="question">Let $(x, y)$ be any point on the parabola $y^2=4 x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x, y)$ in the ratio $1: 3$. Then, the locus of $P$ is</div> | ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$x^2=y$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$y^2=2 x$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$y^2=x$<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">D</span> <span class="option-data"><br/>$x^2=2 y$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$y^2=x$<br/></span> </div> | <div class="solution">$$<br/>\text { By section formula, }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NZeJW46lYjK_Tb4Ljhd5REr04ncqMpNoSecFzLx2gXo.original.fullsize.png"/><br/><br/>$h=\frac{x+0}{4}, k=\frac{y+0}{4}$<br/>$\therefore \quad x=4 h$ and $y=4 k$<br/>Substituting in $y^2=4 x$,<br/>$(4 k)^2=4(4 h) \Rightarrow k^2=h$<br/>or $y^2=x$ is required locus.</div> | MarksBatch2_P2.db |
521 | let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii-1 | let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii-1-92023 | <div class="question">Let $(x, y)$ be such that $\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}$. Match the statements in Column I with the values in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/-gTv06R5XWK5EKREPVN42FLFgZnm7GEW-trEZSNPICY.original.fullsize.png"/><br/></div> | ['Mathematics', 'Inverse Trigonometric Functions', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-q; D-p<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>A-p; B-q; C-p; D-s<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"><br/>A-s; B-p; C-q; D-p<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>A-r; B-q; C-p; D-r</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p; B-q; C-p; D-s<br/></span> </div> | <div class="solution">(A) If $a=1$ and $b=0$, then $\sin ^{-1} x+\cos ^{-1} y=0$<br/>$$<br/>\Rightarrow \quad \sin ^{-1} x=-\cos ^{-1} y \Rightarrow x^2+y^2=1<br/>$$<br/>(B) If $a=1$ and $b=1$, then<br/>$$<br/>\begin{array}{rlrl} <br/>& & \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1} x y & =\frac{\pi}{2} \\<br/>\Rightarrow & & \cos ^{-1} x-\cos ^{-1} y & =\cos ^{-1} x y \\<br/>\Rightarrow & x y+\sqrt{1-x^2} \sqrt{1-y^2} & =x y \quad \text { [taking sine on both the sides] }<br/>\end{array}<br/>$$<br/>(C) If $a=1$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>& & \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1}(2 x y) & =\frac{\pi}{2} \\<br/>\Rightarrow & & \cos ^{-1} x-\cos ^{-1} y & =\cos ^{-1}(2 x y) \\<br/>\Rightarrow & & x y+\sqrt{1-x^2} \sqrt{1-y^2} & =2 x y \\<br/>\Rightarrow & & x^2+y^2 & =1<br/>\end{aligned}<br/>$$<br/>[on squaring]<br/>(D) If $a=2$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>& & \sin ^{-1}(2 x)+\cos ^{-1}(y)+\cos ^{-1}(2 x y) & =\frac{\pi}{2} \\<br/>\Rightarrow & & 2 x y+\sqrt{1+4 x^2} \sqrt{1-y^2} & =2 x y \\<br/>\Rightarrow & & \left(4 x^2-1\right)\left(y^2-1\right) & =0 .<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
522 | let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii | let-x-y-be-such-that-sin-1-a-x-cos-1-y-cos-1-b-x-y-2-match-the-statements-in-column-i-with-the-values-in-column-ii-13694 | <div class="question">Let $(x, y)$ be such that $\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}$. Match the statements in Column I with the values in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/-gTv06R5XWK5EKREPVN42FLFgZnm7GEW-trEZSNPICY.original.fullsize.png"/><br/></div> | ['Mathematics', 'Inverse Trigonometric Functions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-q; D-p<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>A-p; B-q; C-p; D-s<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"><br/>A-s; B-p; C-q; D-p<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>A-r; B-q; C-p; D-r</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p; B-q; C-p; D-s<br/></span> </div> | <div class="solution">(A) If $a=1$ and $b=0$, then $\sin ^{-1} x+\cos ^{-1} y=0$<br/>$$<br/>\Rightarrow \quad \sin ^{-1} x=-\cos ^{-1} y \Rightarrow x^2+y^2=1<br/>$$<br/>(B) If $a=1$ and $b=1$, then<br/>$$<br/>\begin{array}{rlrl} <br/>& & \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1} x y & =\frac{\pi}{2} \\<br/>\Rightarrow & & \cos ^{-1} x-\cos ^{-1} y & =\cos ^{-1} x y \\<br/>\Rightarrow & x y+\sqrt{1-x^2} \sqrt{1-y^2} & =x y \quad \text { [taking sine on both the sides] }<br/>\end{array}<br/>$$<br/>(C) If $a=1$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>& & \sin ^{-1} x+\cos ^{-1} y+\cos ^{-1}(2 x y) & =\frac{\pi}{2} \\<br/>\Rightarrow & & \cos ^{-1} x-\cos ^{-1} y & =\cos ^{-1}(2 x y) \\<br/>\Rightarrow & & x y+\sqrt{1-x^2} \sqrt{1-y^2} & =2 x y \\<br/>\Rightarrow & & x^2+y^2 & =1<br/>\end{aligned}<br/>$$<br/>[on squaring]<br/>(D) If $a=2$ and $b=2$, then<br/>$$<br/>\begin{aligned}<br/>& & \sin ^{-1}(2 x)+\cos ^{-1}(y)+\cos ^{-1}(2 x y) & =\frac{\pi}{2} \\<br/>\Rightarrow & & 2 x y+\sqrt{1+4 x^2} \sqrt{1-y^2} & =2 x y \\<br/>\Rightarrow & & \left(4 x^2-1\right)\left(y^2-1\right) & =0 .<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
523 | let-x-y-z-be-points-with-integer-coordinates-satisfying-the-system-of-homogeneous-equations-3-x-y-z-0-3-x-z-0-3-x-2-y-z-0-then-the-number-of-such-poin | let-x-y-z-be-points-with-integer-coordinates-satisfying-the-system-of-homogeneous-equations-3-x-y-z-0-3-x-z-0-3-x-2-y-z-0-then-the-number-of-such-poin-25448 | <div class="question">Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations $3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0$. Then, the number of such points for which $x^2+y^2+z^2 \leq 100$ is</div> | ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">7</span> </div> | <div class="solution">Given, $\quad 3 x-y-z=0$<br/>and<br/>$$<br/>\begin{array}{r}<br/>-3 x+2 y+z=0 \\<br/>-3 x+z=0<br/>\end{array}<br/>$$<br/>On adding Eqs. (i) and (ii), we get $y=0$<br/>So, $\quad 3 x=z$<br/>Now, $\quad x^2+y^2+z^2 \leq 100$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow & x^2+(3 x)^2+0 \leq 100 \\<br/>\Rightarrow & 10 x^2 \leq 100 \\<br/>\Rightarrow & x^2 \leq 10 \\<br/>\therefore & x=-3,-2,-1,0,1,2,3<br/>\end{array}<br/>$$<br/>So, number of such 7 points are possible.</div> | MarksBatch2_P2.db |
524 | let-y-x-y-x-g-x-g-x-g-x-y-0-0-x-r-where-f-x-denotes-d-x-df-x-and-g-x-is-a-given-nonconstant-differentiable-function-on-r-with-g-0-g-2-0-then-the-value | let-y-x-y-x-g-x-g-x-g-x-y-0-0-x-r-where-f-x-denotes-d-x-df-x-and-g-x-is-a-given-nonconstant-differentiable-function-on-r-with-g-0-g-2-0-then-the-value-37046 | <div class="question">Let $y^{\prime}(x)+y(x) g^{\prime}(x)=g(x) g^{\prime}(x) y(0)=0, x \in R$, where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $R$ with $g(0)=g(2)=0$. Then, the value of $y$ (2) is...</div> | ['Mathematics', 'Differential Equations', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">0</span> </div> | <div class="solution">$\frac{d y}{d x}+y \cdot g^{\prime}(x)=g(x) g^{\prime}(x)$<br/>$$<br/>\mathrm{IF}=e^{\int g^{\prime}(x) d x}=e^{g(x)}<br/>$$<br/>$\therefore$ Solution is<br/><br/>$$<br/>y\left(e^{g(x)}\right)=\int g(x) \cdot g^{\prime}(x) \cdot e^{g(x)} d x+C<br/>$$<br/>Put $g(x)=t, g^{\prime}(x) d x=d t$<br/>$$<br/>\begin{aligned}<br/>& y\left(e^{g(x)}\right)=\int t \cdot e^t d t+C \\<br/>= & t \cdot e^t-\int 1 \cdot e^t d t+C=t \cdot e^t-e^t+C \\<br/>& y e^{g(x)}=(g(x)-1) e^{g(x)}+C \quad \ldots(\mathrm{i})<br/>\end{aligned}<br/>$$<br/>Given, $y(0)=0, g(0)=g(2)=0$<br/><br/>$\therefore$ Eq. (i) becomes,<br/>$$<br/>\begin{aligned}<br/>& \quad y(0) \cdot e^{g(0)}=(g(0)-1) \cdot e^{g(0)}+C \\<br/>& \Rightarrow \quad 0=(-1) \cdot 1+C \Rightarrow C=1 \\<br/>& \therefore \quad y(x) \cdot e^{g(x)}=(g(x)-1) e^{g(x)}+1 \\<br/>& \Rightarrow \quad y(2) \cdot e^{g(2)}=(g(2)-1) e^{g(2)}+1 \\<br/>& \text { where, } \quad g(2)=0 \\<br/>& \Rightarrow \quad y(2) \cdot 1=(-1) \cdot 1+1 \Rightarrow y(2)=0<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
525 | let-z-1-and-z-2-be-two-distinct-complex-numbers-and-let-z-1-t-z-1-t-z-2-for-some-real-number-t-with-0-t-1-if-ar-g-w-denotes-the-principal-argument-of- | let-z-1-and-z-2-be-two-distinct-complex-numbers-and-let-z-1-t-z-1-t-z-2-for-some-real-number-t-with-0-t-1-if-ar-g-w-denotes-the-principal-argument-of-39811 | <div class="question">Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t) z_1+t z_2$ for some real number $t$ with $0 < t < 1$. If $\arg (w)$ denotes the principal argument of a non-zero complex number $w$, then</div> | ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$<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"><br/>$\arg \left(z-z_1\right)=\arg \left(z-z_2\right)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right|=0$<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="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\arg \left(z-z_1\right)=\arg \left(z_2-z_1\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|$<br/>, <br/>$\left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right|=0$<br/>, <br/>$\arg \left(z-z_1\right)=\arg \left(z_2-z_1\right)$</span> </div> | <div class="solution">Given, $z=\frac{(1-t) z_1+t z_2}{(1-t)+t}$<br/>Clearly, $z$ divides $z_1$ and $z_2$ in the ratio of $t:(1-t), 0 < t < 1$<br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad A P+B P=A B \\<br/>& \text { ie, }\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right| \\<br/>& \Rightarrow \text { Option (a) is true. } \\<br/>& \text { and } \begin{aligned}<br/>\arg \left(z-z_1\right) & =\arg \left(z_2-z\right) \\<br/>& =\arg \left(z_2-z_1\right)<br/>\end{aligned}<br/>\end{aligned}<br/>$$<br/>$\Rightarrow$ (b) is false and (d) is true.<br/>Also, $\arg \left(z-z_1\right)=\arg \left(z_2-z_1\right)$<br/><br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad \arg \left(\frac{z-z_1}{z_2-z_1}\right)=0 \\<br/>& \therefore \frac{z-z_1}{z_2-z_1} \text { is purely real. } \\<br/>& \Rightarrow \quad \frac{z-z_1}{z_2-z_1}=\frac{\bar{z}-\bar{z}_1}{\bar{z}_2-\bar{z}_1} \\<br/>& \text { or }\left|\begin{array}{cc}<br/>z-z_1 & \bar{z}-\bar{z}_1 \\<br/>z_2-z_1 & \bar{z}_2-\bar{z}_1<br/>\end{array}\right|=0 \\<br/>&<br/>\end{aligned}<br/>$$<br/>$\therefore$ Option (c) is correct.<br/>Hence, (a, c, d) is the correct option.</div> | MarksBatch2_P2.db |
526 | let-z-be-a-complex-number-such-that-the-imaginary-part-of-z-is-nonzero-and-a-z-2-z-1-is-real-then-a-cannot-take-the-value | let-z-be-a-complex-number-such-that-the-imaginary-part-of-z-is-nonzero-and-a-z-2-z-1-is-real-then-a-cannot-take-the-value-48085 | <div class="question">Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a=z^{2}+z+1$ is real. Then a cannot take the value</div> | ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)'] | <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">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{3}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{4}$</span> </div> | <div class="solution">$\because \quad \operatorname{Im}(z) \neq 0$ $\Rightarrow \quad z$ is non-real
<br/> <br/>and equation $z^{2}+z+(1-a)=0$
<br/> <br/>will have non-real roots, if $D < 0$
<br/> <br/>$\Rightarrow \quad 1-4(1-a) < 0$ $\Rightarrow \quad 4 a < 3 \Rightarrow a < \frac{3}{4}$
<br/> <br/>$\therefore \quad a$ can not take the value $\frac{3}{4}$.</div> | MarksBatch2_P2.db |
527 | let-z-cos-i-sin-then-the-value-of-m-1-15-im-z-2-m-1-at-2-is | let-z-cos-i-sin-then-the-value-of-m-1-15-im-z-2-m-1-at-2-is-49013 | <div class="question">Let $z=\cos \theta+i \sin \theta$. Then, the value of $\sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right)$ at $\theta=2^{\circ}$ is</div> | ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{1}{\sin 2^{\circ}}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{1}{3 \sin 2^{\circ}}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{1}{2 \sin 2^{\circ}}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{1}{4 \sin 2^{\circ}}$</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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{1}{4 \sin 2^{\circ}}$</span> </div> | <div class="solution">Given that $z=\cos \theta+i \sin \theta=e^{i \theta}$<br/>$$<br/>\begin{aligned}<br/>\therefore \sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right) & =\sum_{m=1}^{15} \operatorname{Im}\left(e^{i \theta}\right)^{2 m-1}=\sum_{m=1}^{15} \operatorname{Im} e^{i(2 m-1) \theta} \\<br/>& =\sin \theta+\sin 3 \theta+\sin 5 \theta+\ldots+\sin 29 \theta \\<br/>& =\frac{\sin \left(\frac{\theta+29 \theta}{2}\right) \sin \left(\frac{15 \times 2 \theta}{2}\right)}{\sin \left(\frac{2 \theta}{2}\right)}=\frac{\sin (15 \theta) \sin (15 \theta)}{\sin \theta}=\frac{1}{4 \sin 2^{\circ}}<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
528 | let-z-x-i-y-be-a-complex-number-where-x-and-y-are-integers-then-the-area-of-the-rectangle-whose-vertices-are-the-roots-of-the-equation-z-z-3-z-z-3-350 | let-z-x-i-y-be-a-complex-number-where-x-and-y-are-integers-then-the-area-of-the-rectangle-whose-vertices-are-the-roots-of-the-equation-z-z-3-z-z-3-350-55502 | <div class="question">Let $z=x+i y$ be a complex number, where $x$ and $y$ are integers. Then, the area of the rectangle whose vertices are the roots of the equation $z \bar{z}^3+\bar{z} z^3=350$ is</div> | ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>48<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"><br/>32<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>40<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>80</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>48<br/></span> </div> | <div class="solution">Since, $z \bar{z}\left(z^2+\bar{z}^2\right)=250$<br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad 2\left(x^2+y^2\right)\left(x^2-y^2\right)=350 \\<br/>& \Rightarrow \quad\left(x^2+y^2\right)\left(x^2-y^2\right)=175<br/>\end{aligned}<br/>$$<br/>Since, $x, y \in I$, the only possible case which gives integral solution, is and $\quad \begin{aligned} x^2+y^2 & =25 \\ x^2-y^2 & =7\end{aligned}$<br/>From Eqs. (i) and (ii), we get $x^2=16 ; y^2=9 \Rightarrow x=\pm 4 ; y=\pm 3$<br/>$\therefore$ Area of rectangle $=8 \times 6=48$</div> | MarksBatch2_P2.db |
529 | lim-x-4-x-2-16-2-2-s-e-c-2-x-f-t-d-t-equals | lim-x-4-x-2-16-2-2-s-e-c-2-x-f-t-d-t-equals-27793 | <div class="question">$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals</div> | ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{8}{\pi} f(2)$<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"><br/>$\frac{2}{\pi} f(2)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{2}{\pi} f\left(\frac{1}{2}\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$4 f(2)$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{8}{\pi} f(2)$<br/></span> </div> | <div class="solution">$\lim _{x \rightarrow \frac{x}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ $\left[\therefore \frac{0}{0}\right.$ form $]$<br/>Let<br/>$$<br/>\begin{array}{ll}<br/>\text { Let } & L=\lim _{x \rightarrow \frac{x}{4}} \frac{f\left(\sec ^2 x\right) 2 \sec x \sec x \tan x}{2 x} \\<br/>\therefore & L=\frac{2 f(2)}{\pi / 4}=\frac{8 f(2)}{\pi}<br/>\end{array}<br/>$$</div> | MarksBatch2_P2.db |
530 | lines-l-1-y-x-0-and-l-2-2-x-y-0-intersect-the-line-l-3-y-2-0-at-p-and-q-respectively-the-bisector-of-the-acute-angle-between-l-1-and-l-2-intersects-l- | lines-l-1-y-x-0-and-l-2-2-x-y-0-intersect-the-line-l-3-y-2-0-at-p-and-q-respectively-the-bisector-of-the-acute-angle-between-l-1-and-l-2-intersects-l-69591 | <div class="question">Lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.<br/>Statement I The ratio $P R: R Q$ equals $2 \sqrt{2}: \sqrt{5}$.<br/>Statement II In any triangle, bisector of an angle divides the triangle into two similar triangles.</div> | ['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I<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"><br/>Statement I is true, Statement II is false<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Statement I is false, Statement II is true</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I<br/></span> </div> | <div class="solution">In $\triangle O P Q$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/39h5Z-uPmRqp6qXFqjROF92ujcwMsaH5BCQ92r-Xcbk.original.fullsize.png"/><br/><br/>Clearly,<br/>$$<br/>\frac{P R}{R Q}=\frac{O P}{O Q}=\frac{2 \sqrt{2}}{\sqrt{5}}<br/>$$</div> | MarksBatch2_P2.db |
531 | list-i-describes-four-systems-each-with-two-particles-a-and-b-in-relative-motion-as-shown-in-figure-list-ii-gives-possible-magnitudes-of-their-relativ | list-i-describes-four-systems-each-with-two-particles-a-and-b-in-relative-motion-as-shown-in-figure-list-ii-gives-possible-magnitudes-of-their-relativ-99030 | <div class="question"><p>List I describes four systems, each with two particles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>) at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math>.</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"><strong>List-I</strong></td> <td style="text-align: center;"> </td> <td style="text-align: center;"><strong>List-II</strong></td> </tr> <tr> <td>(I)</td> <td> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are moving on a horizontal circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> with uniform angular speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo> </mo><mi>rad</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. The initial angular positions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math> respectively.</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ff049b50-ab5e-454c-8c66-cb6a46d2edb2-image.png" style="width: 140px; height: 128px;"/></p> </td> <td>(P)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></math></td> </tr> <tr> <td>(II)</td> <td> <p>Projectiles <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are fired (in the same vertical plane) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math> respectively, with the same speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math> from the horizontal plane. The initial separation between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is large enough so that they do not collide, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math>.</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/8c0f53b4-2b00-4b3b-b08e-11aa472a0f03-image.png" style="width: 200px; height: 146px;"/></p> </td> <td>(Q)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mfenced><mrow><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced><msqrt><mn>2</mn></msqrt></mfrac></math></td> </tr> <tr> <td>(III)</td> <td> <p>Two harmonic oscillators <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> moving in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction according to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>A</mi></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfrac><mi>t</mi><msub><mi>t</mi><mn>0</mn></msub></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>B</mi></msub><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfenced><mrow><mfrac><mi>t</mi><msub><mi>t</mi><mn>0</mn></msub></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></mfenced></math> respectively, starting from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>. Take <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>,</mo><msub><mi>t</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math>.</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/0e65e96f-c87f-4250-9818-e1145eb3b0a7-image.png" style="width: 280px; height: 145px;"/></p> </td> <td>(R)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>10</mn></msqrt></math></td> </tr> <tr> <td>(IV)</td> <td> <p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is rotating in a horizontal circular path of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math> plane, with constant angular speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo> </mo><mi>rad</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is moving up at a constant speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in the vertical direction as shown in the figure. (Ignore gravity.)</p> <p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/d19263f1-da07-447d-8586-560b6286677b-image.png" style="width: 190px; height: 123px;"/></p> </td> <td>(S)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt></math></td> </tr> <tr> <td> </td> <td> </td> <td>(T)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>25</mn><msup><mi>π</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math></td> </tr> </tbody></table><p>Which one of the following options is correct?</p></div> | ['Physics', 'Work Power Energy', 'JEE Advanced', 'JEE Advanced 2022 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">R</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">S</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">Q</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">R</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">R</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">R</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">S</mi></math></span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi><mo>,</mo><mi>II</mi><mo>→</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>III</mi><mo>→</mo><mi mathvariant="normal">P</mi><mo>,</mo><mi>IV</mi><mo>→</mo><mi mathvariant="normal">R</mi></math></span> </div> | <div class="solution"><blockquote><p>(I)</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/90b87b12-93ae-4b20-a396-6511d7ecc0da-image.png" style="width: 180px; height: 166px;"/></p><p>Angular velocity of both the particles is same. Therefore, angle between them will remain <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>v</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub></mfenced><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>ω</mi><mi>R</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">S</mi></math></p><p>(II)</p><p>The velocity of A at any time t can be written as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac><mi>cos</mi><mn>45</mn><mo>°</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced close="]" open="["><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo>×</mo><mi>sin</mi><mn>45</mn><mo>°</mo><mo>-</mo><mi>g</mi><mo>×</mo><mi>t</mi></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>g</mi><mi>t</mi></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p>Velocity of B at any time t</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><mo>-</mo><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>g</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p>Relative velocity at time t will be,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>v</mi><mo>→</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub><mo>=</mo><mn>5</mn><mi>π</mi><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mi>g</mi><mo>×</mo><mn>0</mn><mo>.</mo><mn>1</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mover><mi>v</mi><mo>→</mo></mover><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub></mfenced><mo>=</mo><msqrt><mn>25</mn><msup><mi>π</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>II</mi><mo>→</mo><mi>T</mi></math></p><p>(III) </p><p>Relative position of A wrt B can be written as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msub><mi>x</mi><mi>A</mi></msub><mo>-</mo><msub><mi>x</mi><mi>B</mi></msub></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mi>t</mi><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfenced><mrow><mi>t</mi><mo>+</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></mfenced></math></p><p>Using trigonometric relation</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><msub><mi>x</mi><mn>0</mn></msub><mi>sin</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p><p>Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>rel</mi></msub><mo>=</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><msub><mi>x</mi><mn>0</mn></msub><mi>cos</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p><p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msub><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mfenced><mrow><mfrac><mi>π</mi><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>×</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>=</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>III</mi><mo>→</mo><mi>P</mi></math></p><p>(IV) </p><p>The velocity of A is in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math> plane and it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>v</mi><mi>A</mi></msub></mfenced><mo>=</mo><mi>ω</mi><mi>r</mi><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and the velocity of B is along <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> axis and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>v</mi><mi>B</mi></msub></mfenced><mo>=</mo><mn>3</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>rel</mi></msub><mo>=</mo><msqrt><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mn>10</mn></msqrt><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>IV</mi><mo>→</mo><mi>R</mi></math></p></blockquote></div> | MarksBatch2_P2.db |
532 | look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-1 | look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-1-67535 | <div class="question">Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xNFfaYiituH7Vz9vRPx1wgTgYD4I1Vbby5q7GRnc9Fs.original.fullsize.png"/><br/><br/>The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6 \mathrm{~m}$. The coordinates of the centres of the different parts are, outer circle 10 , $0)$, left inner circle $(0,0)$, right inner circle $(a, a)$, vertical line $(0,0)$ and horizontal line $(0,-a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is</div> | ['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{a}{10}$<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"><br/>$\frac{a}{8}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{a}{12}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{a}{3}$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{a}{10}$<br/></span> </div> | <div class="solution">$y_{C M}$<br/>$$<br/>\begin{gathered}<br/>=\frac{m_1 y_1+m_2 y_2+m_3 y_3+m_4 y_4+m_5 y_5}{m_1+m_2+m_3+m_4+m_5} \\<br/>=\frac{(6 m)(0)+(m)(a)+m(a)+m(0)}{6 m+m+m+m+m}=\frac{a}{10}<br/>\end{gathered}<br/>$$</div> | MarksBatch2_P2.db |
533 | look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-2 | look-at-the-drawing-given-in-the-figure-which-has-been-drawn-with-ink-of-uniform-linethickness-the-mass-of-ink-used-to-draw-each-of-the-two-inner-circ-2-65763 | <div class="question">Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xNFfaYiituH7Vz9vRPx1wgTgYD4I1Vbby5q7GRnc9Fs.original.fullsize.png"/><br/><br/>The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$. The mass of the ink used to draw the outer circle is $6 \mathrm{~m}$. The coordinates of the centres of the different parts are, outer circle 10 , $0)$, left inner circle $(0,0)$, right inner circle $(a, a)$, vertical line $(0,0)$ and horizontal line $(0,-a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is</div> | ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{a}{10}$<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"><br/>$\frac{a}{8}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{a}{12}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{a}{3}$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{a}{10}$<br/></span> </div> | <div class="solution">$y_{C M}$<br/>$$<br/>\begin{gathered}<br/>=\frac{m_1 y_1+m_2 y_2+m_3 y_3+m_4 y_4+m_5 y_5}{m_1+m_2+m_3+m_4+m_5} \\<br/>=\frac{(6 m)(0)+(m)(a)+m(a)+m(0)}{6 m+m+m+m+m}=\frac{a}{10}<br/>\end{gathered}<br/>$$</div> | MarksBatch2_P2.db |
534 | mark-the-incorrect-option-1 | mark-the-incorrect-option-1-25833 | <div class="question">Mark the incorrect option.</div> | ['Biology', 'Animal Kingdom', 'NEET'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The larva of <em>Scoliodon</em> exists.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Lateral line is absent in <em>Bangarus.</em></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Air bladder is present in <em>Betta.</em></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All chordates are deuterostomes.</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">The larva of <em>Scoliodon</em> exists.</span> </div> | <div class="solution"><p style="text-align: justify;"><em>Scoliodon</em> is a predator of small bony fishes and invertebrates. This species exhibits the most advanced mode of viviparity of any fish, in which the developed embryos form a highly complex placental connection to the mother at a very small size.</p></div> | MarksBatch2_P2.db |
535 | match-each-of-the-compounds-given-in-column-i-with-the-reaction-s-that-they-can-undergo-given-in-column-ii | match-each-of-the-compounds-given-in-column-i-with-the-reaction-s-that-they-can-undergo-given-in-column-ii-19907 | <div class="question">Match each of the compounds given in Column I with the reaction (s) that they can undergo, given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/emrlHMSRSxJBfoD-QedlMy-2bSSQ9SuGiI61DsRkYMY.original.fullsize.png"/><br/></div> | ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q,t, (B) q,s, (C) r,t, (D) q,s<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s, (B) p,q,t, (C) r, (D) q,s<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,s, (B) q,s, (C) r,t, (D) q,t<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">D</span> <span class="option-data"><br/>(A) p,q,s, (B) p,q,t, (C) r, (D) q,t</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,s, (B) q,s, (C) r,t, (D) q,t<br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
536 | match-each-of-the-compounds-is-column-i-with-its-characteristic-reactions-is-column-ii | match-each-of-the-compounds-is-column-i-with-its-characteristic-reactions-is-column-ii-19219 | <div class="question">$$<br/>\text { Match each of the compounds is Column I with its characteristic reaction(s) is Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fjpzYZkIB7A3picoT0unS5p28yTVroAGAOEWZI8JLYs.original.fullsize.png"/><br/></div> | ['Chemistry', 'Carboxylic Acid Derivatives', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q,t, (B) p,s, (C) r,t, (D) s<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s,t, (B) p,s,t, (C) p, (D) r<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"><br/>(A) p,s, (B) p,s, (C) p, (D) s<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,s,t, (B) p,s,t, (C) r, (D) r</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,s,t, (B) p,s,t, (C) p, (D) r<br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
537 | match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii-1 | match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii-1-27838 | <div class="question">Match each of the diatomic molecules in Column I with its property/properties in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FPjqA-UDTF1p1emxoGgSvmMJsFjrHiranmdfbinxMBY.original.fullsize.png"/><br/></div> | ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q,t, (B) s,t, (C) r,t, (D) p,q,s<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s,t, (B) p,s,t, (C) p,q, (D) r,s<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,r,t, (B) s,t, (C) p,q,r, (D) p,r,s<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">D</span> <span class="option-data"><br/>(A) p,q,s,t, (B) p,s,t, (C) p,q, (D) r,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,r,t, (B) s,t, (C) p,q,r, (D) p,r,s<br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
538 | match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii | match-each-of-the-diatomic-molecules-in-column-i-with-its-propertyproperties-in-column-ii-90686 | <div class="question">Match each of the diatomic molecules in Column I with its property/properties in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FPjqA-UDTF1p1emxoGgSvmMJsFjrHiranmdfbinxMBY.original.fullsize.png"/><br/></div> | ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q,t, (B) s,t, (C) r,t, (D) p,q,s<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s,t, (B) p,s,t, (C) p,q, (D) r,s<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,r,t, (B) s,t, (C) p,q,r, (D) p,r,s<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">D</span> <span class="option-data"><br/>(A) p,q,s,t, (B) p,s,t, (C) p,q, (D) r,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,r,t, (B) s,t, (C) p,q,r, (D) p,r,s<br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
539 | match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii-1 | match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii-1-45944 | <div class="question">Match each of the reactions given in Column I with the corresponding products(s) given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/V3GuxJAZ7_VpCvRcy3XmvJlvyEmLatpDGSsk2j_7RAw.original.fullsize.png"/><br/></div> | ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Main'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,t, (B) p,s,t, (C) r,s, (D) p<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"><br/>(A) p,q,s, (B) p,q,t, (C) r, (D) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,q,s, (B) q,s,t, (C) r,s, (D) p<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,t, (B) p,q,t, (C) r, (D) q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,t, (B) p,s,t, (C) r,s, (D) p<br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
540 | match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii | match-each-of-the-reactions-given-in-column-i-with-the-corresponding-productss-given-in-column-ii-49904 | <div class="question">Match each of the reactions given in Column I with the corresponding products(s) given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/V3GuxJAZ7_VpCvRcy3XmvJlvyEmLatpDGSsk2j_7RAw.original.fullsize.png"/><br/></div> | ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,t, (B) p,s,t, (C) r,s, (D) p<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"><br/>(A) p,q,s, (B) p,q,t, (C) r, (D) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,q,s, (B) q,s,t, (C) r,s, (D) p<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,t, (B) p,q,t, (C) r, (D) q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,t, (B) p,s,t, (C) r,s, (D) p<br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
541 | match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bond-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-4 | match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bond-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-4-22019 | <div class="question">Match the chemical substance in Column I with type of polymers/type of bond in Column II. Indicate your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/aBcz_-koYhRdo9Zs0JRtXZYX4tgTBARB13Q2LAWna7c.original.fullsize.png"/><br/></div> | ['Chemistry', 'Polymers', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-p; D-q, r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-q, r; C-p, r; D-s<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"><br/>A-s; B-p; C-p; D-q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>A-p, s; B-r; C-p, q; D-s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-q, r; C-p, r; D-s<br/></span> </div> | <div class="solution">(A) $p, s$<br/>(B) $q, r$<br/>(C) $p, r$<br/>(D) $s$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/0u0UXSHYX2s-C2PZe6qS4EkMFVTL9LWZNY5ja52juUQ.original.fullsize.png"/><br/><br/>Sucrose (diasaccharide) < smiles>OCC1OC(CO)C(O)C(O)C1O < /smiles><br/>Sucrose/glucosidic as well as fructosidic linkage</div> | MarksBatch2_P2.db |
542 | match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of--1 | match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-1-75635 | <div class="question">Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/w4x9XLhiCV8vUdID4JoLSlbgktj2IlXjSR4y-LbxBJs.original.fullsize.png"/><br/></div> | ['Chemistry', 'General Organic Chemistry', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p, r; B-p, q; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-p; C-p, s; D-q, r<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>A-p, q, s; B-q; C-r, s, s; D-r<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">D</span> <span class="option-data"><br/>A-r, s; B-q; C-p; D-p, q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, q, s; B-q; C-r, s, s; D-r<br/></span> </div> | <div class="solution">A-p, $q, s, \quad \mathrm{~B}-q, \quad \mathrm{C}-r, s, \quad \mathrm{D}-r$</div> | MarksBatch2_P2.db |
543 | match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of- | match-the-chemical-substance-in-column-i-with-type-of-polymerstype-of-bonds-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-76065 | <div class="question">Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/w4x9XLhiCV8vUdID4JoLSlbgktj2IlXjSR4y-LbxBJs.original.fullsize.png"/><br/></div> | ['Chemistry', 'General Organic Chemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p, r; B-p, q; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-p; C-p, s; D-q, r<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>A-p, q, s; B-q; C-r, s, s; D-r<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">D</span> <span class="option-data"><br/>A-r, s; B-q; C-p; D-p, q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, q, s; B-q; C-r, s, s; D-r<br/></span> </div> | <div class="solution">A-p, $q, s, \quad \mathrm{~B}-q, \quad \mathrm{C}-r, s, \quad \mathrm{D}-r$</div> | MarksBatch2_P2.db |
544 | match-the-column-normals-at-p-q-r-are-drawn-to-y-2-4-x-which-intersect-at-3-0-then | match-the-column-normals-at-p-q-r-are-drawn-to-y-2-4-x-which-intersect-at-3-0-then-58100 | <div class="question">Match the column.<br/>Normals at $P, Q, R$ are drawn to $y^2=4 x$ which intersect at $(3,0)$. Then,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/V8pddult167dBNpS7Z8V5Cw_zSQ4Lu0oQfWh1qEbXzw.original.fullsize.png"/><br/></div> | ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(i) (a), (ii) (b), (iii) (d), (iv) (c)<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"><br/>(i) (b), (ii) (a), (iii) (d), (iv) (c)<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(i) (a), (ii) (b), (iii) (c), (iv) (d)<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(i) (b), (ii) (a), (iii) (c), (iv) (d)</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (a), (ii) (b), (iii) (d), (iv) (c)<br/></span> </div> | <div class="solution">Since, equation of normal is $y+x t=2 a t+a t^3$ passes through $(3,0)$.<br/>$$<br/>\Rightarrow \quad 3 t=2 t+t^3 \Rightarrow t=0,1,-1 \text {. }<br/>$$<br/>$\therefore P(1,2), Q(0,0), R(1,-2)$. Thus,<br/>(i) Area of $\triangle P Q R=\frac{1}{2} \times 1 \times 4=2$<br/>(ii) Centroid of $\triangle P Q R=\left(\frac{2}{3}, 0\right)$<br/>Equation of circle passing through $P, Q, R$ is<br/>$$<br/>\begin{aligned}<br/>& \quad(x-1)(x-1)+(y-2)(y+2)+\lambda(x-1)=0 \\<br/>& \Rightarrow \quad 1-4-\lambda=0 \Rightarrow \lambda=-3 \\<br/>& \therefore \text { Required equation of circle is } x^2+y^2-5 x-1=0 \\<br/>& \therefore \text { Centre }\left(\frac{5}{2}, 0\right) \text { and radius } \frac{5}{2} .<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
545 | match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix--1 | match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-1-89933 | <div class="question">Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u3ZyoDHANLps94g_Bgxxf_Fo4PLbZbkKoAmBhvW_0C8.original.fullsize.png"/><br/></div> | ['Chemistry', 'Redox Reactions', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-q, r; C-p, r; D-r, s<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>A-s; B-p; C-p; D-q<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>A-p, s; B-r; C-p, q; D-p, 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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-r; C-p, q; D-p, s</span> </div> | <div class="solution">A-p,s B-r, C-p,q, D-p,s,<br/>$$<br/>\begin{aligned}<br/>2 \mathrm{CrO}_4^{2-} \longrightarrow \mathrm{Cr}_2 \mathrm{O}_7^{2-} \\<br/>\mathrm{MnO}_4^{-}+\mathrm{NO}_2^{-}+\mathrm{H}^{+} \longrightarrow \mathrm{Mn}^{2+}+\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{O} \\<br/>\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{SO}_4+\mathrm{Fe}^{2+} \longrightarrow \mathrm{Fe}^{3+}+\mathrm{NO}+\mathrm{SO}_4^{2-}+\mathrm{H}_2 \mathrm{O}<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
546 | match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix- | match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-65264 | <div class="question">Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u3ZyoDHANLps94g_Bgxxf_Fo4PLbZbkKoAmBhvW_0C8.original.fullsize.png"/><br/></div> | ['Chemistry', 'Redox Reactions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-q, r; C-p, r; D-r, s<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>A-s; B-p; C-p; D-q<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>A-p, s; B-r; C-p, q; D-p, 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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-r; C-p, q; D-p, s</span> </div> | <div class="solution">A-p,s B-r, C-p,q, D-p,s,<br/>$$<br/>\begin{aligned}<br/>2 \mathrm{CrO}_4^{2-} \longrightarrow \mathrm{Cr}_2 \mathrm{O}_7^{2-} \\<br/>\mathrm{MnO}_4^{-}+\mathrm{NO}_2^{-}+\mathrm{H}^{+} \longrightarrow \mathrm{Mn}^{2+}+\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{O} \\<br/>\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{SO}_4+\mathrm{Fe}^{2+} \longrightarrow \mathrm{Fe}^{3+}+\mathrm{NO}+\mathrm{SO}_4^{2-}+\mathrm{H}_2 \mathrm{O}<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
547 | match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicates-your-answer-by-darkening-the-appropriate-bubbles-of-4-4-matrix-giv | match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicates-your-answer-by-darkening-the-appropriate-bubbles-of-4-4-matrix-giv-97122 | <div class="question">Match the complexes in Column I with their properties listed in Column II. Indicates your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5dKmHYTGKoudvS18CsnhAI04sI2pOm3IVd68Y3je4K8.original.fullsize.png"/><br/></div> | ['Chemistry', 'Coordination Compounds', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p, r; B-p, q, s; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-p, r, s; C-p, s; D-q, r<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>A-p, q, s; B-p, r, s; C-q, s; D-q, s<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">D</span> <span class="option-data"><br/>A-r, s; B-q; C-p; D-p, q, s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, q, s; B-p, r, s; C-q, s; D-q, s<br/></span> </div> | <div class="solution">(A) $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4\left[\left(\mathrm{H}_2 \mathrm{O}\right)_2\right] \mathrm{Cl}_2 p, q, s\right.$<br/>Oxidation state $\mathrm{Co}$ is $+2$<br/>$\Rightarrow$ Paramagnetic, exhibits geometrical isomerism.<br/>(B) $\left[\mathrm{Pt}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right] p, r, s$<br/>$\mathrm{dsp}{ }^2$-hybridisation<br/>$\mathrm{Pt}^{2+}$, diamagnetic square planar, geometrical isomerism<br/>(C) $q, s$ (D) $q, s$</div> | MarksBatch2_P2.db |
548 | match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-1 | match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-1-73346 | <div class="question">Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TJVpaWmgq_MRw31jJHL907USap6KF9NIXYukIKdkNpw.original.fullsize.png"/><br/></div> | ['Chemistry', 'Hydrocarbons', 'JEE Main'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) r,s, (B) p,q, (C) p,q,r, (D) p<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"><br/>(A) r, (B) p,q,s, (C) p,q,s, (D) p,q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) s, (B) q,r,s, (C) p,q,r,s, (D) p,q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) r,s, (B) p,q,s, (C) p,q,r, (D) p</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) r,s, (B) p,q, (C) p,q,r, (D) p<br/></span> </div> | <div class="solution">A-r; B-p, q; C- p, q, r; D-p<br/>Sodium fusion extract gives Prussian blue when nitrogen is present alongwith carbon. Phenolic group and carboxylate gives positive $\mathrm{FeCl}_3$ test - white precipitate comes with $\mathrm{AgCl}$.<br/>$$<br/>\text { Hydrazone formation occurs effectively at } \mathrm{pH} \approx 4.5 \text {. The reaction proceeds in that }<br/>$$<br/>condition only when $\mathrm{H}^{+}$concentration is merely sufficient to activate. As $\mathrm{H}^{+}$concentration raises, sufficient molecule of hydrazone gets converted into hydrazonium which is not nucleophilic and reaction becomes impossible. Further, low concentration of $\mathrm{H}^{+}$(in the case of hydrolysis of 2, 4-dinitrophenyl hydrazonium bromide) is not effective to proceed elimination (rate determining step).<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/CjI78dy4abclm11LRXCdsAkn5nz4S5BhS3Jyxy1XhUg.original.fullsize.png"/><br/><br/>If it is assumed that the given hydrazonian bromiate is the result as salt main $\mathrm{HBr}$ is used as acid catalyst. Remember! hydrazone formation involves first addition and then elimination (res). In the second step substitution dominates over elimination ( $\mathrm{HBr}$ is not a dehydrating reagent mutually).</div> | MarksBatch2_P2.db |
549 | match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles | match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-72456 | <div class="question">Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TJVpaWmgq_MRw31jJHL907USap6KF9NIXYukIKdkNpw.original.fullsize.png"/><br/></div> | ['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) r,s, (B) p,q, (C) p,q,r, (D) p<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"><br/>(A) r, (B) p,q,s, (C) p,q,s, (D) p,q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) s, (B) q,r,s, (C) p,q,r,s, (D) p,q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) r,s, (B) p,q,s, (C) p,q,r, (D) p</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) r,s, (B) p,q, (C) p,q,r, (D) p<br/></span> </div> | <div class="solution">A-r; B-p, q; C- p, q, r; D-p<br/>Sodium fusion extract gives Prussian blue when nitrogen is present alongwith carbon. Phenolic group and carboxylate gives positive $\mathrm{FeCl}_3$ test - white precipitate comes with $\mathrm{AgCl}$.<br/>$$<br/>\text { Hydrazone formation occurs effectively at } \mathrm{pH} \approx 4.5 \text {. The reaction proceeds in that }<br/>$$<br/>condition only when $\mathrm{H}^{+}$concentration is merely sufficient to activate. As $\mathrm{H}^{+}$concentration raises, sufficient molecule of hydrazone gets converted into hydrazonium which is not nucleophilic and reaction becomes impossible. Further, low concentration of $\mathrm{H}^{+}$(in the case of hydrolysis of 2, 4-dinitrophenyl hydrazonium bromide) is not effective to proceed elimination (rate determining step).<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/CjI78dy4abclm11LRXCdsAkn5nz4S5BhS3Jyxy1XhUg.original.fullsize.png"/><br/><br/>If it is assumed that the given hydrazonian bromiate is the result as salt main $\mathrm{HBr}$ is used as acid catalyst. Remember! hydrazone formation involves first addition and then elimination (res). In the second step substitution dominates over elimination ( $\mathrm{HBr}$ is not a dehydrating reagent mutually).</div> | MarksBatch2_P2.db |
550 | match-the-conics-in-column-i-with-the-statementsexpressions-in-column-ii | match-the-conics-in-column-i-with-the-statementsexpressions-in-column-ii-54802 | <div class="question">$$<br/>\text { Match the conics in Column I with the statements/expressions in Column II. }<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/mttZypB7gORJPl-QC2KlWwOMRUJEI-jGBWeCxz3jUTE.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BSqFu76r11SkdAwZcbkeTMZ0RwfnU1zWrr8QDx9JnTE.original.fullsize.png"/><br/></div> | ['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) s, (B) q,s, (C) s, (D) p,s<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) p, (B) q,s, (C) r, (D) q,s<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) s, (B) s,t, (C) s, (D) p,s<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) p, (B) s,t, (C) r, (D) q,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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p, (B) s,t, (C) r, (D) q,s</span> </div> | <div class="solution">(p) $\frac{1}{\sqrt{h^2+k^2}}=2 \Rightarrow h^2+k^2=\frac{1}{4}$<br/>Hence, locus is a circle.<br/>(q) ||$z+2|-| z-2||=3$<br/>and $\quad 2-(-2)=4>3$<br/>Hence, locus is hyperbola.<br/>(r) Let $x=\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right), y=\frac{2 t}{1+t^2}$<br/>Let $\quad \tan \theta=t$ $\Rightarrow x=\sqrt{3} \cos 2 \theta, y=\sin 2 \theta$<br/>$\therefore \quad \frac{x^2}{3}+y^2=1$<br/>Hence, locus is an ellipse.<br/>(s) Eccentricity $x=1 \Rightarrow$ Parabola, $1 < x < \infty \Rightarrow$ Hyperbola<br/>(t) Let $z=x+i y$<br/>Since, $\operatorname{Re}(z+1)^2=|z|^2+1$<br/>$\Rightarrow \quad(x+1)^2-y^2=x^2+y^2+1$<br/>$\Rightarrow \quad 2 x=2 y^2$<br/>$\Rightarrow \quad x=y^2$<br/>Hence, locus is parabola.</div> | MarksBatch2_P2.db |
551 | match-the-conversions-in-column-i-with-the-types-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-given- | match-the-conversions-in-column-i-with-the-types-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-given-26342 | <div class="question">Match the conversions in Column I with the type(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ucomvVlKQ_CiBxOlv7mu0IFQfu05rElGpRWCcPrudJI.original.fullsize.png"/><br/></div> | ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) p, (B) q, (C) p,r, (D) p,r,s<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"><br/>(A) q,r, (B) p,q, (C) s, (D) q,r,s<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) p, (B) q,s, (C) r, (D) q,r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q, (B) q, (C) p,q, (D) q,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p, (B) q, (C) p,r, (D) p,r,s<br/></span> </div> | <div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RTlN6ixNP8LakoAWNPJ2h8GaTB8etWo-T-TqYSxfK94.original.fullsize.png"/><br/></div> | MarksBatch2_P2.db |
552 | match-the-entries-in-column-i-with-the-correctly-related-quantum-numbers-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the | match-the-entries-in-column-i-with-the-correctly-related-quantum-numbers-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-67501 | <div class="question">Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/2J3KeSRuERC1Xe6fSQAoOrICdqN8QJF--5LumNR_PWQ.original.fullsize.png"/><br/></div> | ['Chemistry', 'Structure of Atom', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q, (B) q,r,s, (C) p,q,r,s, (D) p,q<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) p, (B) p,q,s, (C) p,q,s, (D) p<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) p, (B) q,r,s, (C) p,q,r,s, (D) p,q<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) q, (B) p,q,r,s, (C) p,q,r, (D) p</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) q, (B) p,q,r,s, (C) p,q,r, (D) p</span> </div> | <div class="solution">A-q; B-p, q, r, s; C-p, q, r; D-p, q, r<br/>(B) A hydrogen-like one electron wave function obeying Pauli's principle First, as it is the wave function so it will be characterised by $n, l$ and $m$. Further, it obeys the Pauli's principle, that's why it will be characterised completely by adding spin quantum number also.<br/>(D) Radial probability density of electron at the nucleus basically depends on azimuthal quantum number, but as it is the function of radius depends on principal quantum number, When angular probability density is included, that depends on magnetic quantum number.</div> | MarksBatch2_P2.db |
553 | match-the-extraction-processes-listed-in-column-i-with-metals-listed-in-column-ii | match-the-extraction-processes-listed-in-column-i-with-metals-listed-in-column-ii-64082 | <div class="question">Match the extraction processes listed in column I with metals listed in column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/T1XuJuazCxU0ENveihRFEqo1WSHuTkcfu_MEsZeXbXA.original.fullsize.png"/><br/></div> | ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) P,S, (B) R, (C) P,Q, (D) P,S<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) Q,S, (B) R, (C) S, (D) S<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) P,R, (B) P, R, (C) Q, (D) R<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) P,R, (B) P, (C) Q, (D) 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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) P,R, (B) P, (C) Q, (D) S</span> </div> | <div class="solution">Column I<br/>Column II<br/>(A) Self reduction $\left[\begin{array}{c}P \text { - Lead } \\ R \text { - Copper }\end{array}\right]$<br/>$$<br/>\begin{aligned}<br/>& \text { i.e., } A \rightleftarrows P \text { and } R \\<br/>& \text { In lead metal } \\<br/>& \qquad\left\{\begin{array}{c}<br/>2 \mathrm{PbS}+3 \mathrm{O}_2 \longrightarrow 2 \mathrm{PbO}+2 \mathrm{SO}_2 \\<br/>2 \mathrm{PbO}+\mathrm{PbS} \longrightarrow 3 \mathrm{~Pb}+\mathrm{SO}_2<br/>\end{array}\right.<br/>\end{aligned}<br/>$$<br/>(self reduction)<br/>In copper metal<br/>$$<br/>\begin{aligned}<br/>& 2 \mathrm{Cu}_2 \mathrm{~S}+3 \mathrm{O}_2 \longrightarrow 2 \mathrm{Cu}_2 \mathrm{O}+2 \mathrm{SO}_2 \\<br/>& \mathrm{Cu}_2 \mathrm{~S}+2 \mathrm{Cu}_2 \mathrm{O} \longrightarrow 6 \mathrm{Cu}+\mathrm{SO}_2<br/>\end{aligned}<br/>$$<br/>(self reduction)<br/>(B) Carbon reduction $P$-lead<br/>$$<br/>\begin{aligned}<br/>& \text { i.e., } \mathrm{B}-\mathrm{P} \\<br/>& \text { In lead metal } \\<br/>& \qquad \mathrm{PbO}+\mathrm{C} \longrightarrow \mathrm{Pb}+\mathrm{CO} \\<br/>& \mathrm{PbO}+\mathrm{CO} \longrightarrow \mathrm{Pb}+\mathrm{CO}_2<br/>\end{aligned}<br/>$$<br/>Carbon reduction process is used for the reduction of oxides of electropositive metals like $\mathrm{Fe}, \mathrm{Pb}$ etc. This reduction is carried out with coal or coke on strongly heating.<br/>(C) Complex formation and displacement by metal.<br/>i.e., $C-Q$ (silver)<br/><br/>In silver metal<br/>Argentite mineral of $\mathrm{Ag}\left(\mathrm{Ag}_2 \mathrm{~S}\right)$ is treated with $0.7 \% \mathrm{NaCN}$ solution in presence of air, therefore firstly sodium argento cyanide complex is formed which on heating with zinc metal, therefore by displacement silver is isolated.<br/>$$<br/>\begin{aligned}<br/>& \mathrm{Ag}_2 \mathrm{~S}+4 \mathrm{NaCN} \longrightarrow 2 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_2\right]+\mathrm{Na}_2 \mathrm{~S} \\<br/>& 2 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_2\right]+\mathrm{Zn} \longrightarrow \mathrm{Na}\left[\mathrm{Zn}(\mathrm{CN})_4\right]+\underset{\text { Soluble }}{2 \mathrm{Ag}} \downarrow<br/>\end{aligned}<br/>$$<br/>(D) Decomposition of iodide-(S) Boron<br/>$$<br/>\text { i.e., }(D)-(S)<br/>$$<br/>By decomposition of iodide boron metal is isolated.</div> | MarksBatch2_P2.db |
554 | match-the-following-4 | match-the-following-4-26393 | <div class="question">Match the following<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/aP6PkZrkpVGT5Ap7bH-wIPaY4lRESHzYikGKWjxXO2w.original.fullsize.png"/><br/></div> | ['Chemistry', 'p Block Elements (Group 13 & 14)', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) Q,S, (B) S, (C) R, (D) Q,R<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"><br/>(A) Q,S, (B) R, (C) S, (D) Q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) P,R, (B) P, R, (C) R, (D) Q,R<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) P,R, (B) P, (C) R, (D) Q,R</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) Q,S, (B) S, (C) R, (D) Q,R<br/></span> </div> | <div class="solution">A match with $\boldsymbol{Q}, \boldsymbol{S} \quad \mathrm{Bi}^{3+}+\mathrm{H}_2 \mathrm{O} \longrightarrow[\mathrm{BiO}]^{+}+2 \mathrm{H}^{+}$ B match with $\boldsymbol{S} \mathrm{NaAlO}_2+2 \mathrm{H}_2 \mathrm{O} \longrightarrow \mathrm{Al}(\mathrm{OH})_3+\mathrm{NaOH}$ C match with $\boldsymbol{R} \quad 2 \mathrm{SiO}_4^{4-}+2 \mathrm{H}^{+} \longrightarrow \mathrm{Si}_2 \mathrm{O}_7^{6-}+\mathrm{H}_2 \mathrm{O}$ Pyrosilicate is formed by treating orthosilicate with acid. D match with $\boldsymbol{Q}, \boldsymbol{R} \mathrm{Na}_2 \mathrm{~B}_4 \mathrm{O}_7 \underset{\text { or } \mathrm{H}_2 \mathrm{SO}_4}{\stackrel{\mathrm{HCl}}{\mathrm{O}_3}} \mathrm{H}_3 \mathrm{BO}_3$<br/>$$<br/>\mathrm{Na}_2 \mathrm{~B}_4 \mathrm{O}_4 \stackrel{\mathrm{H}_2 \mathrm{O}}{\longrightarrow} \mathrm{H}_3 \mathrm{BO}_3<br/>$$</div> | MarksBatch2_P2.db |
555 | match-the-following-5 | match-the-following-5-89404 | <div class="question">Match the following <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Xmaa26fH0hR4prHCjmrRrNH-rfprOx-3FtUrzpt4n6w.original.fullsize.png"/><br/></div> | ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) Q, (B) P, (C) R,S, (D) Q,S<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) S, (B) R, (C) R, (D) P,S<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) R, (B) Q, (C) S, (D) Q,S<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) Q, (B) Q, (C) R,S, (D) P,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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) Q, (B) Q, (C) R,S, (D) P,S</span> </div> | <div class="solution">$$<br/>\text { (A) Match with (Q) }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/pZbxv4ivUF9I7Kp-BSSuA_5S0dEaTlkWLMIkQCP26a4.original.fullsize.png"/><br/><br/>The formation of $\mathrm{CH}_2=\mathrm{CH}-\mathrm{CD}_3$ can be explained on the basis of the fact that $\mathrm{C}-\mathrm{D}$ bond is much stronger than $\mathrm{C}-\mathrm{H}$ bond.<br/>(B) match with (Q)<br/>Reactivity of $\mathrm{PhCHBrCH}_3$ is greater than<br/>$\mathrm{Ph} \mathrm{CHBrCD}_3$ because $\mathrm{C}-\mathrm{D}$ bond is more stronger than $\mathrm{C}-\mathrm{H}$ bond.<br/><br/>$$<br/>\text { (C) match with (R) and (S) }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eunRBn6utbgP22GT6vuptZpANFRSsz0948UmfNNgbvI.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZduTsP4411yF83jMwipCj7LUJoPCBkyfcaz8xXeUkbA.original.fullsize.png"/><br/><br/>In the step (II), a slow unimolecular elimination occurs in the conjugate base of the reactant and hence this mechanism is called $E_1 C B$ or carbanion mechanism. Since step (I) must be reversible, if ethanol containing $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OD}$ is used as solvent, it would be expected that the original bromide would incorporate deuterium (D).<br/><br/>$$<br/>\text { (D) Match with (P) and (S) }<br/>$$<br/><br/>Step I. $\mathrm{PhCH}_2-\mathrm{CH}_2-\mathrm{Br} \stackrel{\text { Slow }}{\longrightarrow} \mathrm{PhCH}_2-\stackrel{+}{\mathrm{C}} \mathrm{H}_2+\mathrm{Br}^{-}$<br/>Step II. $\mathrm{PhCH}_2-\stackrel{+}{\mathrm{C}_2} \mathrm{H}_2 \stackrel{\text { Fast }}{\longrightarrow} \mathrm{Ph}-\mathrm{CH}=\mathrm{CH}_2+\mathrm{H}^{-}$<br/>Rate $\propto\left[\mathrm{PhCH}_2-\mathrm{CH}_2-\mathrm{Br}\right]$<br/>Similarly<br/>Step II. $\mathrm{PhCD}_2 \stackrel{+}{\mathrm{C}_2} \mathrm{H}_2 \stackrel{\text { Fast }}{\longrightarrow} \mathrm{PhCD}=\mathrm{CHD}+\mathrm{H}^{+}$<br/>Rate $\propto\left[\mathrm{PhCD}_2-\mathrm{CH}_2 \mathrm{Br}\right]$<br/>Hence, $E_1$ reaction and first order kinetics.</div> | MarksBatch2_P2.db |
556 | match-the-following-6 | match-the-following-6-47862 | <div class="question">Match the following<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/k7sNssq9bMajNP7pgcEvTBAHN9XF2aDQJpzY1CwNW6A.original.fullsize.png"/><br/></div> | ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(i) (a), (ii) (b), (iii) (d), (iv) (c)<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(i) (c), (ii) (a), (iii) (b), (iv) (d)<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(i) (a), (ii) (d), (iii) (b), (iv) (c)<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(i) (b), (ii) (a), (iii) (c), (iv) (d)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (b), (ii) (a), (iii) (c), (iv) (d)</span> </div> | <div class="solution">(i) Solved the two equations, say, i.e. $x+y=|a|$ and $a x-y=1$, we get $x=\frac{|a|+1}{a+1}>0$ and $y=\frac{|a|-1}{a+1}>0$<br/>when $a+1>0$; we get $a>1$<br/>$\therefore \quad a_0=1$<br/>(ii) We have,<br/>$\mathbf{a}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$<br/>$\Rightarrow \quad \mathbf{a} \cdot \hat{\mathbf{k}}=\gamma$<br/>Now, $\hat{\mathbf{k}} \times(\hat{\mathbf{k}} \times \mathbf{a})=(\hat{\mathbf{k}} \cdot \mathbf{a})-(\hat{\mathbf{k}} \cdot \hat{\mathbf{k}}) \mathbf{a}=\gamma \hat{\mathbf{k}}-(\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}})$ $\Rightarrow \quad \alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}=0 \Rightarrow \alpha=\beta=0$<br/>Also, $\quad \alpha+\beta+\gamma=2 \Rightarrow \gamma=2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6me9FVDij8eFqKPHPUuH_wTp5Smkwlfj-vCln1v9K74.original.fullsize.png"/><br/><br/>(iv) $\sin A \sin B \sin C+\cos A \cos B \leq \sin A \sin B+\cos A \cos B=\cos (A-B)$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow & \cos (A-B) \geq 1 \Rightarrow \cos (A-B)=1 \\<br/>\Rightarrow & \sin C=1<br/>\end{array}<br/>$$</div> | MarksBatch2_P2.db |
557 | match-the-following-7 | match-the-following-7-13447 | <div class="question">Match the following.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/m1EAsIDafFkvQJfVVNHHUnyzlA5lW-g4xZ8pul4RCq4.original.fullsize.png"/><br/></div> | ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(i) (a), (ii) (b), (iii) (c)<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(i) (b), (ii) (a), (iii) (c)<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"><br/>(i) (a), (ii) (c), (iii) (b)<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(i) (b), (ii) (c), (iii) (a)</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (b), (ii) (a), (iii) (c)<br/></span> </div> | <div class="solution">$$<br/>\begin{aligned}<br/>& \text { (i) } \sum_{i=1}^{\infty} \tan ^{-1}\left(\frac{1}{2 i^2}\right)=t \Rightarrow \sum_{i=1}^{\infty} \tan ^{-1}\left(\frac{2}{4 i^2-1+1}\right) \\<br/>& \Rightarrow \sum_{i=1}^{\infty} \tan ^{-1}\left\{\frac{(2 i+1)-(2 i-1)}{1+(2 i-1)(2 i+1)}\right\} \\<br/>& \Rightarrow\left(\tan ^{-1} 3-\tan ^{-1} 1\right)+\left(\tan ^{-1} 5-\tan ^{-1} 3\right)+\ldots+\left\{\tan ^{-1}(2 n+1)-\tan ^{-1}(2 n-1)\right\} \\<br/>& \therefore \quad \quad t=\lim _{n \rightarrow \infty}\left(\tan ^{-1}(2 n+1)-\tan ^{-1} 1\right)=\lim _{n \rightarrow \infty} \tan ^{-1}\left(\frac{2 n}{1+2 n+1}\right)=\frac{\pi}{4} \\<br/>& \therefore \quad \tan t=1 .<br/>\end{aligned}<br/>$$<br/>(ii) We have, $\cos \theta_1=\frac{1-\tan ^2 \frac{\theta_1}{2}}{1+\tan ^2 \frac{\theta_2}{2}}=\frac{a}{b+c}$<br/>$$<br/>\Rightarrow \quad \tan ^2 \frac{\theta}{2}=\frac{b+c-a}{b+c+a}<br/>$$<br/>Also, $\quad \cos \theta_3=\frac{1-\tan ^2 \frac{\theta_3}{2}}{1+\tan ^2 \frac{\theta_3}{2}}=\frac{c}{a+b}$<br/>$$<br/>\Rightarrow \quad \tan ^2 \frac{\theta_3}{2}=\frac{a+b-c}{a+b+c}<br/>$$<br/>$\Rightarrow \tan ^2 \frac{\theta_1}{2}+\tan ^2 \frac{\theta_3}{2}=\frac{2 b}{a+b+c}=\frac{2 b}{3 b}=\frac{2}{3}, \quad$ [as, $a, b, c$ are in AP $\Rightarrow 2 b=a+c$ ]<br/>(iii) Line through $(0,1,0)$ and perpendicular to plane $x+2 y+2 z=0$ is given by $\frac{x-0}{1}=\frac{y-1}{2}=\frac{z-0}{2}=r$<br/>$\therefore P(r, 2 r+1,2 r)$ be the foot of perpendicular on the straight line, then<br/>$$<br/>\begin{array}{ll} <br/>& r \cdot 1+(2 r+1) \cdot 2+(2 r) \cdot 2=0 \\<br/>\Rightarrow \quad r= & \quad r\left(-\frac{2}{9}, \frac{5}{9},-\frac{4}{9}\right) \\<br/>\therefore & \quad P(r) \\<br/>\therefore & \text { Required perpendicular distance }=\sqrt{\frac{4+25+16}{81}}=\frac{\sqrt{5}}{3} \text { unit. }<br/>\end{array}<br/>$$</div> | MarksBatch2_P2.db |
558 | match-the-following-8 | match-the-following-8-93106 | <div class="question">Match the following.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UyPhBV62yJ87NM_xRqutVqISAPvRym6ILTSeV_4cma8.original.fullsize.png"/><br/></div> | ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(i) (a), (ii) (b), (iii) (d), (iv) (c)<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(i) (c), (ii) (a), (iii) (b), (iv) (d)<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(i) (a), (ii) (d), (iii) (b), (iv) (c)<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">D</span> <span class="option-data"><br/>(i) (b), (ii) (a), (iii) (c), (iv) (d)</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (a), (ii) (d), (iii) (b), (iv) (c)<br/></span> </div> | <div class="solution">(i) $I=\int_0^{\pi / 2}(\sin x)^{\cos x}\left(\cos x \cot x-\log (\sin x)^{\sin x}\right) d x=\int_0^{\pi / 2} \frac{d}{d x}(\sin x)^{\cos x} d x=1$<br/>(ii) The point of intersection of $-4 y^2=x$ and $x-1=-5 y^2$ is $(-4,-1)$ and $(-4,1)$.<br/>Hence, required area $=2\left\{\int_0^1\left(1-5 y^2\right) d y-\int_0^1-4 y^2 d y\right\}=\frac{4}{3}$.<br/>(iii) The point intersection of $y=3^{x-1} \log x$ and $y=x^x-1$ is $(1,0)$.<br/>Hence, $\quad \frac{d y}{d x}=\frac{3^{x-1}}{x}+3^{x-1} \cdot \log 3 \cdot \log x$<br/>$\therefore \quad\left(\frac{d y}{d x}\right)_{(1,0)}=1$<br/>For<br/>$$<br/>\therefore \quad\left(\frac{d y}{d x}\right)_{(1,0)}=1<br/>$$<br/><br/>If $\theta$ is angle between the curves, then $\tan \theta=0$.<br/>$$<br/>\because \quad \theta=0^{\circ}<br/>$$<br/>(iv)<br/>$$<br/>\begin{aligned}<br/>& \frac{d y}{d x}=\frac{2}{x+y} \Rightarrow \frac{d x}{d y}-\frac{x}{2}=\frac{y}{2} \\<br/>& \Rightarrow \quad x e^{-y / 2}=\frac{1}{2} \cdot \int y \cdot e^{-y / 2} d y \\<br/>& \Rightarrow \quad x+y+2=k e^{y / 2} \\<br/>& \Rightarrow \quad k=3 \\<br/>& \therefore \quad x+y+2=3 e^{y / 2} \\<br/>&<br/>\end{aligned}<br/>$$<br/>[passing through $(1,0)$ ]</div> | MarksBatch2_P2.db |
559 | match-the-following-45 | match-the-following-45-44575 | <div class="question">Match the following <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Xmaa26fH0hR4prHCjmrRrNH-rfprOx-3FtUrzpt4n6w.original.fullsize.png"/><br/></div> | ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) Q, (B) P, (C) R,S, (D) Q,S<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) S, (B) R, (C) R, (D) P,S<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) R, (B) Q, (C) S, (D) Q,S<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) Q, (B) Q, (C) R,S, (D) P,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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) Q, (B) Q, (C) R,S, (D) P,S</span> </div> | <div class="solution">$$<br/>\text { (A) Match with (Q) }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/pZbxv4ivUF9I7Kp-BSSuA_5S0dEaTlkWLMIkQCP26a4.original.fullsize.png"/><br/><br/>The formation of $\mathrm{CH}_2=\mathrm{CH}-\mathrm{CD}_3$ can be explained on the basis of the fact that $\mathrm{C}-\mathrm{D}$ bond is much stronger than $\mathrm{C}-\mathrm{H}$ bond.<br/>(B) match with (Q)<br/>Reactivity of $\mathrm{PhCHBrCH}_3$ is greater than<br/>$\mathrm{Ph} \mathrm{CHBrCD}_3$ because $\mathrm{C}-\mathrm{D}$ bond is more stronger than $\mathrm{C}-\mathrm{H}$ bond.<br/><br/>$$<br/>\text { (C) match with (R) and (S) }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eunRBn6utbgP22GT6vuptZpANFRSsz0948UmfNNgbvI.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZduTsP4411yF83jMwipCj7LUJoPCBkyfcaz8xXeUkbA.original.fullsize.png"/><br/><br/>In the step (II), a slow unimolecular elimination occurs in the conjugate base of the reactant and hence this mechanism is called $E_1 C B$ or carbanion mechanism. Since step (I) must be reversible, if ethanol containing $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OD}$ is used as solvent, it would be expected that the original bromide would incorporate deuterium (D).<br/><br/>$$<br/>\text { (D) Match with (P) and (S) }<br/>$$<br/><br/>Step I. $\mathrm{PhCH}_2-\mathrm{CH}_2-\mathrm{Br} \stackrel{\text { Slow }}{\longrightarrow} \mathrm{PhCH}_2-\stackrel{+}{\mathrm{C}} \mathrm{H}_2+\mathrm{Br}^{-}$<br/>Step II. $\mathrm{PhCH}_2-\stackrel{+}{\mathrm{C}_2} \mathrm{H}_2 \stackrel{\text { Fast }}{\longrightarrow} \mathrm{Ph}-\mathrm{CH}=\mathrm{CH}_2+\mathrm{H}^{-}$<br/>Rate $\propto\left[\mathrm{PhCH}_2-\mathrm{CH}_2-\mathrm{Br}\right]$<br/>Similarly<br/>Step II. $\mathrm{PhCD}_2 \stackrel{+}{\mathrm{C}_2} \mathrm{H}_2 \stackrel{\text { Fast }}{\longrightarrow} \mathrm{PhCD}=\mathrm{CHD}+\mathrm{H}^{+}$<br/>Rate $\propto\left[\mathrm{PhCD}_2-\mathrm{CH}_2 \mathrm{Br}\right]$<br/>Hence, $E_1$ reaction and first order kinetics.</div> | MarksBatch2_P2.db |
560 | match-the-following-50 | match-the-following-50-89056 | <div class="question">Match the following.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UyPhBV62yJ87NM_xRqutVqISAPvRym6ILTSeV_4cma8.original.fullsize.png"/><br/></div> | ['Mathematics', 'Definite Integration', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(i) (a), (ii) (b), (iii) (d), (iv) (c)<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(i) (c), (ii) (a), (iii) (b), (iv) (d)<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(i) (a), (ii) (d), (iii) (b), (iv) (c)<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">D</span> <span class="option-data"><br/>(i) (b), (ii) (a), (iii) (c), (iv) (d)</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (a), (ii) (d), (iii) (b), (iv) (c)<br/></span> </div> | <div class="solution">(i) $I=\int_0^{\pi / 2}(\sin x)^{\cos x}\left(\cos x \cot x-\log (\sin x)^{\sin x}\right) d x=\int_0^{\pi / 2} \frac{d}{d x}(\sin x)^{\cos x} d x=1$<br/>(ii) The point of intersection of $-4 y^2=x$ and $x-1=-5 y^2$ is $(-4,-1)$ and $(-4,1)$.<br/>Hence, required area $=2\left\{\int_0^1\left(1-5 y^2\right) d y-\int_0^1-4 y^2 d y\right\}=\frac{4}{3}$.<br/>(iii) The point intersection of $y=3^{x-1} \log x$ and $y=x^x-1$ is $(1,0)$.<br/>Hence, $\quad \frac{d y}{d x}=\frac{3^{x-1}}{x}+3^{x-1} \cdot \log 3 \cdot \log x$<br/>$\therefore \quad\left(\frac{d y}{d x}\right)_{(1,0)}=1$<br/>For<br/>$$<br/>\therefore \quad\left(\frac{d y}{d x}\right)_{(1,0)}=1<br/>$$<br/><br/>If $\theta$ is angle between the curves, then $\tan \theta=0$.<br/>$$<br/>\because \quad \theta=0^{\circ}<br/>$$<br/>(iv)<br/>$$<br/>\begin{aligned}<br/>& \frac{d y}{d x}=\frac{2}{x+y} \Rightarrow \frac{d x}{d y}-\frac{x}{2}=\frac{y}{2} \\<br/>& \Rightarrow \quad x e^{-y / 2}=\frac{1}{2} \cdot \int y \cdot e^{-y / 2} d y \\<br/>& \Rightarrow \quad x+y+2=k e^{y / 2} \\<br/>& \Rightarrow \quad k=3 \\<br/>& \therefore \quad x+y+2=3 e^{y / 2} \\<br/>&<br/>\end{aligned}<br/>$$<br/>[passing through $(1,0)$ ]</div> | MarksBatch2_P2.db |
561 | match-the-following-columns-1 | match-the-following-columns-1-36925 | <div class="question">Match the following columns<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UWvjzowOsy9s6gkOqAlYesntU2oLVtSYhsrLUqDtv3A.original.fullsize.png"/><br/></div> | ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(a) p,q, (b) p, (c) p, (d) p,q<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(a) p,q,r, (b) p, s, (c) p,r, (d) p,q,r<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(a) p,q, (b) p, r, (c) p,s, (d) p,q,r<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">D</span> <span class="option-data"><br/>(a) p,s, (b) p, s, (c) p, (d) p,q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) p,q, (b) p, r, (c) p,s, (d) p,q,r<br/></span> </div> | <div class="solution">(a) $\rightarrow$ p, q<br/>(b) $\rightarrow \mathrm{p}, \mathrm{r}$<br/>(c) $\rightarrow$ p, $s$<br/>(d) $\rightarrow$ p, q, r</div> | MarksBatch2_P2.db |
562 | match-the-following-columns-2 | match-the-following-columns-2-81096 | <div class="question">Match the following columns<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/sMqslWdePeakThlq1yBOr6vLpYxkICZ4jmVngXlMfX8.original.fullsize.png"/><br/></div> | ['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(a) p,q, (b) p,q, (c) p, (d) q,r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(a) p, (b) p,q,s, (c) p,s, (d) q,r,s<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"><br/>(a) p,q, (b) p, q, (c) p,s, (d) p,q,r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(a) p, (b) p, s, (c) p, (d) p,q,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) p, (b) p,q,s, (c) p,s, (d) q,r,s<br/></span> </div> | <div class="solution">(a) $\rightarrow$ p<br/>(b) $\rightarrow$ p, q, s<br/>(c) $\rightarrow$ q, s<br/>(d) $\rightarrow$ q, r, s</div> | MarksBatch2_P2.db |
563 | match-the-following-columns-7 | match-the-following-columns-7-99594 | <div class="question">Match the following columns<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/sMqslWdePeakThlq1yBOr6vLpYxkICZ4jmVngXlMfX8.original.fullsize.png"/><br/></div> | ['Physics', 'Electromagnetic Induction', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(a) p,q, (b) p,q, (c) p, (d) q,r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(a) p, (b) p,q,s, (c) p,s, (d) q,r,s<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"><br/>(a) p,q, (b) p, q, (c) p,s, (d) p,q,r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(a) p, (b) p, s, (c) p, (d) p,q,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) p, (b) p,q,s, (c) p,s, (d) q,r,s<br/></span> </div> | <div class="solution">(a) $\rightarrow$ p<br/>(b) $\rightarrow$ p, q, s<br/>(c) $\rightarrow$ q, s<br/>(d) $\rightarrow$ q, r, s</div> | MarksBatch2_P2.db |
564 | match-the-following-columns-8 | match-the-following-columns-8-18368 | <div class="question">Match the following columns<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UWvjzowOsy9s6gkOqAlYesntU2oLVtSYhsrLUqDtv3A.original.fullsize.png"/><br/></div> | ['Physics', 'Nuclear Physics', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(a) p,q, (b) p, (c) p, (d) p,q<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(a) p,q,r, (b) p, s, (c) p,r, (d) p,q,r<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(a) p,q, (b) p, r, (c) p,s, (d) p,q,r<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">D</span> <span class="option-data"><br/>(a) p,s, (b) p, s, (c) p, (d) p,q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) p,q, (b) p, r, (c) p,s, (d) p,q,r<br/></span> </div> | <div class="solution">(a) $\rightarrow$ p, q<br/>(b) $\rightarrow \mathrm{p}, \mathrm{r}$<br/>(c) $\rightarrow$ p, $s$<br/>(d) $\rightarrow$ p, q, r</div> | MarksBatch2_P2.db |
565 | match-the-following-columns-and-select-the-correct-option-columni-columnii-a-gregarious-polyphagous-pest-i-asterias-b-adult-with-radial-symmetry-and-l | match-the-following-columns-and-select-the-correct-option-columni-columnii-a-gregarious-polyphagous-pest-i-asterias-b-adult-with-radial-symmetry-and-l-89561 | <div class="question"><p>Match the following columns and select the correct option.</p><table border="1" cellpadding="1" cellspacing="1" style="width: 400px;"> <tbody> <tr> <td> </td> <td>Column-I</td> <td> </td> <td>Column-II</td> </tr> <tr> <td>(a)</td> <td>Gregarious, polyphagous pest</td> <td>(i)</td> <td><em>Asterias</em></td> </tr> <tr> <td>(b)</td> <td>Adult with radial symmetry and larva with bilateral symmetry</td> <td>(ii)</td> <td>Scorpion</td> </tr> <tr> <td>(c)</td> <td>Book lungs</td> <td>(iii)</td> <td><em>Ctenoplana</em></td> </tr> <tr> <td>(d)</td> <td>Bioluminescence</td> <td>(iv)</td> <td><em>Locusta</em></td> </tr> </tbody></table></div> | ['Biology', 'Animal Kingdom', 'NEET', 'NEET 2020 (Phase 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">(a)-(iv), (b)-(i), (c)-(ii), (d)-(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">(a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(a)-(i), (b)-(iii), (c)-(ii), (d)-(iv)</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">(a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)</span> </div> | <div class="solution"><p><strong>Gregarious pest:</strong> A pest is an animal that is detrimental or harmful to humans or human concerns. <em>Locusta</em> (locust) is a pest that is dangerous to crops. It causes heavy damage to the vegetation. As it lives in groups of the same individuals but does not help one another, hence called gregarious.</p><p><strong>Body symmetry of Echinoderms:</strong> The adult echinoderms are radially symmetrical, but their larvae are bilaterally symmetrical. The echinoderms are unique in this feature that the larvae are bilaterally symmetrical (an advanced feature) while the adults developing from them are radially symmetrical (a primitive feature). <strong>Examples: <em>Asterias</em> (Starfish), <em>Echinus</em> (Sea urchin), <em>Antedon</em> (sea lily), <em>Cucumaria </em>(sea cucumber), and <em>Ophiura</em> (Brittle star).</strong></p><p><strong>Book-lungs: Present in scorpions, spiders, etc. </strong>These are the modified book-gills.</p><p><strong>Bioluminescence:</strong> Bios means living and lumen means light. Bioluminescence is the property of production and emission of light by a living organism and hence shines in the dark background. It is a naturally occurring phenomenon. This property is well-marked in Ctenophores as they emit light and hence shine against the dark backgrounds. <strong>Examples: <em>Pleurobrachia, Ctenoplana.</em></strong></p></div> | MarksBatch2_P2.db |
566 | match-the-following-for-the-given-process-1 | match-the-following-for-the-given-process-1-71120 | <div class="question">Match the following for the given process<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/tZNNR6Q2U1U_vB12hZCZHK1J080dytLaYA5IBXtXE3Q.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jSH26CUr5_R-O26KzLkmQbvZWxgIID6T3Mdnt9iL_Ms.original.fullsize.png"/><br/></div> | ['Physics', 'Thermodynamics', 'JEE Main'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(a) s, (b) p, r, (c) r, (d) q,s<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"><br/>(a) r, (b) p, s, (c) q, (d) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(a) s, (b) p, r, (c) q, (d) q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(a) r, (b) p, s, (c) r, (d) q,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) s, (b) p, r, (c) r, (d) q,s<br/></span> </div> | <div class="solution">(a) $\rightarrow \mathrm{s}$ (b) $\rightarrow \mathrm{p}, \mathrm{r}$, (c) $\rightarrow \mathrm{r}$ (d) $\rightarrow \mathrm{q}, \mathrm{s}$<br/>In process $\mathbf{J} \rightarrow \mathbf{K}: V$ is constant whereas $P$ is decreasing, Therefore, $T$ should also decrease.<br/>$$<br/>\therefore \quad W=0, \Delta U=-\text { ve and } Q < 0<br/>$$<br/>In process $\mathbf{K} \rightarrow \mathbf{L}: P$ is constant while $V$ is increasing. Therefore, temperature should also increase.<br/>$$<br/>\therefore \quad W>0, \Delta U>0 \text { and } Q>0<br/>$$<br/><br/>In process $\mathbf{L} \rightarrow \mathbf{M}$ : This is inverse of process $J \rightarrow K$.<br/>$$<br/>\therefore \quad W=0, \Delta U>0 \text { and } Q>0<br/>$$<br/>In process $\mathbf{M} \rightarrow \mathbf{J}: V$ is decreasing. Therefore, $W < 0$<br/>$$<br/>\begin{array}{lcl} <br/>& (P \mathrm{~V})_J & < (P V)_M \\<br/>\therefore & T_J & < T_M \text { or } \quad \Delta U < 0 \\<br/>\text { Therefore, } & Q & < 0 .<br/>\end{array}<br/>$$<br/>No, solution is required.</div> | MarksBatch2_P2.db |
567 | match-the-following-for-the-given-process | match-the-following-for-the-given-process-49961 | <div class="question">Match the following for the given process<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/tZNNR6Q2U1U_vB12hZCZHK1J080dytLaYA5IBXtXE3Q.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jSH26CUr5_R-O26KzLkmQbvZWxgIID6T3Mdnt9iL_Ms.original.fullsize.png"/><br/></div> | ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(a) s, (b) p, r, (c) r, (d) q,s<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"><br/>(a) r, (b) p, s, (c) q, (d) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(a) s, (b) p, r, (c) q, (d) q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(a) r, (b) p, s, (c) r, (d) q,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) s, (b) p, r, (c) r, (d) q,s<br/></span> </div> | <div class="solution">(a) $\rightarrow \mathrm{s}$ (b) $\rightarrow \mathrm{p}, \mathrm{r}$, (c) $\rightarrow \mathrm{r}$ (d) $\rightarrow \mathrm{q}, \mathrm{s}$<br/>In process $\mathbf{J} \rightarrow \mathbf{K}: V$ is constant whereas $P$ is decreasing, Therefore, $T$ should also decrease.<br/>$$<br/>\therefore \quad W=0, \Delta U=-\text { ve and } Q < 0<br/>$$<br/>In process $\mathbf{K} \rightarrow \mathbf{L}: P$ is constant while $V$ is increasing. Therefore, temperature should also increase.<br/>$$<br/>\therefore \quad W>0, \Delta U>0 \text { and } Q>0<br/>$$<br/><br/>In process $\mathbf{L} \rightarrow \mathbf{M}$ : This is inverse of process $J \rightarrow K$.<br/>$$<br/>\therefore \quad W=0, \Delta U>0 \text { and } Q>0<br/>$$<br/>In process $\mathbf{M} \rightarrow \mathbf{J}: V$ is decreasing. Therefore, $W < 0$<br/>$$<br/>\begin{array}{lcl} <br/>& (P \mathrm{~V})_J & < (P V)_M \\<br/>\therefore & T_J & < T_M \text { or } \quad \Delta U < 0 \\<br/>\text { Therefore, } & Q & < 0 .<br/>\end{array}<br/>$$<br/>No, solution is required.</div> | MarksBatch2_P2.db |
568 | match-the-gases-under-specified-conditions-listed-in-column-i-with-their-propertieslaws-in-column-ii-indicate-your-answer-by-darkening-the-appropriate | match-the-gases-under-specified-conditions-listed-in-column-i-with-their-propertieslaws-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-98087 | <div class="question">Match the gases under specified conditions listed in Column I with their properties/laws in Column II. Indicate your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ejgNRpWHs-ac66lvxqM62VDBFoI4F-mmsBx1r-RU0uI.original.fullsize.png"/><br/></div> | ['Chemistry', 'States of Matter', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p, q; B-r; C-r, s; D-p, q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>A-s; B-p; C-p; D-q<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>A-p, s; B-r; C-p, q; D-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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-r; C-p, q; D-r</span> </div> | <div class="solution">(A) $-p, s(\mathrm{~B})-r$ (C) $-p, q$ (D) $-r$<br/>Explanation van der Waals' equation<br/>$$<br/>\left(p+\frac{n^2 a}{V_m^2}\right)\left(V_m-b\right)=n R T<br/>$$<br/><br/>For hydrogen gas $(p=200 \mathrm{~atm}, T=273 \mathrm{~K})$<br/>As pressure is large $V_m$ can be assumed small, thus ' $b$ ' can not be ignored, while due to high pressure $a / V_m^2$ can be considered negligible in comparison to $p$.<br/>$$<br/>p\left(V_m-b\right)=R T \quad \text { and } \quad Z=1+\frac{p b}{R T}<br/>$$<br/>For hydrogen gas $(p \sim 0, T=273 \mathrm{~K})$<br/>when pressure occurs of low about $1 \mathrm{~atm}$ or less and temperature is not very close to the point of liquification $\left[T_c\left(\mathrm{H}_2\right)=33.3 \mathrm{~K}\right]$ gas behaves ideally.<br/>For<br/>$$<br/>P V=n R T<br/>$$<br/>Temperature is close to the point of liquification $\left[T_C\left(\mathrm{CO}_2\right)=304.2\right]$ thus, deviation from ideality appears very high (due to high attractive force of attraction).<br/>For real gas with very large molar volume.<br/>As molar volume is very large $a / V_m^2$ will be negligible and at the same time ' $b$ ' in comparison to $V_m$ is also considered negligible, thus, $p V_m=n R T$</div> | MarksBatch2_P2.db |
569 | match-the-integrals-in-column-i-with-the-values-in-column-ii | match-the-integrals-in-column-i-with-the-values-in-column-ii-38496 | <div class="question">Match the integrals in Column I with the values in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/CYjILavSNl1HrZmbzJ5Wzy63qiah4GZwojG_7X4ITSE.original.fullsize.png"/><br/></div> | ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-s; B-p; C-p; D-q<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-r; B-s; C-q; D-r<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>A-s; B-s; C-p; D-r<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">D</span> <span class="option-data"><br/>A-r; B-q; C-p; D-s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-s; B-s; C-p; D-r<br/></span> </div> | <div class="solution">(A) <img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/cK8Zj8AakkXGyCXyen9p0A6CvTZojRKgVYLM5o4MYug.original.fullsize.png"/><br/><br/>(B)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/uYQamdjeMo3WPD_Hye2LKowKcou_DybIc_gLNbDMuxo.original.fullsize.png"/><br/><br/>(C)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/hkrxgdSxO1eRTvF1q2hhn8A3T7l4JsKxJDYMI0FHwgQ.original.fullsize.png"/><br/><br/>(D)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ncaKmPXp6iUpPInSEkh8-je4ozcxA0l1ewpZH03_PFw.original.fullsize.png"/><br/></div> | MarksBatch2_P2.db |
570 | match-the-reaction-in-column-i-with-appropriate-options-in-column-ii-1 | match-the-reaction-in-column-i-with-appropriate-options-in-column-ii-1-21104 | <div class="question">$$<br/>\text { Match the reaction in Column I with appropriate options in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/N9Q7iRDy8ErnJ4De4bFeq_7NeMbZAX4r4qqF1C64BJk.original.fullsize.png"/><br/></div> | ['Chemistry', 'Hydrocarbons', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) R,S, (B) R,T, (C) P, (D) T<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) R,S,T, (B) R,T, (C) P,Q, (D) R<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) R,S, (B) T, (C) P, (D) T<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) R,S,T, (B) T, (C) P,Q, (D) 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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) R,S,T, (B) T, (C) P,Q, (D) R</span> </div> | <div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5z61L57XV5xVQKMEv0xK_JpD_ST6Lh1zTMsxo5I6c2k.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qCT1dRcjq-40dK0016TKVE0y-1RpWgSe4N6dqXFkvzI.original.fullsize.png"/><br/><br/>Organic chemistry<br/>Conceptual understanding of reaction mechanism IV</div> | MarksBatch2_P2.db |
571 | match-the-reaction-in-column-i-with-appropriate-options-in-column-ii | match-the-reaction-in-column-i-with-appropriate-options-in-column-ii-18377 | <div class="question">$$<br/>\text { Match the reaction in Column I with appropriate options in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/N9Q7iRDy8ErnJ4De4bFeq_7NeMbZAX4r4qqF1C64BJk.original.fullsize.png"/><br/></div> | ['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) R,S, (B) R,T, (C) P, (D) T<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) R,S,T, (B) R,T, (C) P,Q, (D) R<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) R,S, (B) T, (C) P, (D) T<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) R,S,T, (B) T, (C) P,Q, (D) 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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) R,S,T, (B) T, (C) P,Q, (D) R</span> </div> | <div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5z61L57XV5xVQKMEv0xK_JpD_ST6Lh1zTMsxo5I6c2k.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qCT1dRcjq-40dK0016TKVE0y-1RpWgSe4N6dqXFkvzI.original.fullsize.png"/><br/><br/>Organic chemistry<br/>Conceptual understanding of reaction mechanism IV</div> | MarksBatch2_P2.db |
572 | match-the-reactions-in-column-i-with-appropriate-types-of-stepsreactive-intermediate-involved-in-these-reactions-as-given-in-column-ii | match-the-reactions-in-column-i-with-appropriate-types-of-stepsreactive-intermediate-involved-in-these-reactions-as-given-in-column-ii-40077 | <div class="question">Match the reactions in Column I with appropriate types of steps/reactive intermediate involved in these reactions as given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/poSsUnlRvHd6dET79r9AVpuWiLjj9Sp2MvScv-cFARQ.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RXewfmccGPifySWpcS4turpHvZqdqba01NXVmjCknvs.original.fullsize.png"/><br/></div> | ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) p, (B) q, (C) p,s, (D) p,r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) q,r, (B) p,q, (C) r,s, (D) q,r,s<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(A) r,s,t, (B) p,s, (C) r,s, (D) q,r<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">D</span> <span class="option-data"><br/>(A) p,q, (B) p,r, (C) p,q, (D) q,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) r,s,t, (B) p,s, (C) r,s, (D) q,r<br/></span> </div> | <div class="solution">(a) <img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IFjuTZURdU5JiHtTb_LQEiwlFpURZ2fynnPHhznWqDM.original.fullsize.png"/><br/><br/>(b)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eApzd4UV9vZUVc3ktP7NhEyeOfodco9EfwZFDq2f-9Y.original.fullsize.png"/><br/><br/>(c)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/tZ9C1KMNShw79HBFaereEVSeF8mheA8Dr_wEbScZESI.original.fullsize.png"/><br/><br/>(d)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_nhmZi9HFG2paBcmYxYqAQ-vIbmLrI8-mda2GVLICRc.original.fullsize.png"/><br/></div> | MarksBatch2_P2.db |
573 | match-the-specified-conditions-in-solids-listed-in-column-i-with-their-properties-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbl | match-the-specified-conditions-in-solids-listed-in-column-i-with-their-properties-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbl-76131 | <div class="question">Match the specified conditions in solids listed in Column I with their properties in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FOTt4rDNn0Mz5UEqBBqMEkmbIqJQz5GwTJtTKCsEoiI.original.fullsize.png"/><br/></div> | ['Chemistry', 'Solid State', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p; B-q; C-p; D-p, r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>A-p, s; B-p, q; C-q; D-p, r<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"><br/>A-s; B-p; C-p; D-q, r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>A-p, s; B-p; C-p, q; D-p, s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, s; B-p, q; C-q; D-p, r<br/></span> </div> | <div class="solution">$\mathrm{A}-p, s, \quad \mathrm{~B}-p, q, \quad \mathrm{C}-q, \quad \mathrm{D}-q, r$</div> | MarksBatch2_P2.db |
574 | match-the-statements-given-in-column-i-with-the-intervalsunion-of-intervals-given-in-column-ii | match-the-statements-given-in-column-i-with-the-intervalsunion-of-intervals-given-in-column-ii-34527 | <div class="question">Match the statements given in Column I with the intervals/union of intervals given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/1If8DbPHHmiKRCE2AbXYbzNrmcvvJu2t-0TjT049750.original.fullsize.png"/><br/></div> | ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q, (B) r, (C) s, (D) t<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) s, (B) t, (C) r, (D) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) q, (B) r, (C) s, (D) r<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) s, (B) t, (C) r, (D) 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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) s, (B) t, (C) r, (D) r</span> </div> | <div class="solution">(A) Given, $|z|=1 \Rightarrow z \cdot \bar{z}=1$<br/>$$<br/>\therefore \quad \frac{2 i z}{1-z^2}=\frac{2 i z}{z \cdot \bar{z}-z^2}=\frac{2 i}{\bar{z}-z} \text {, }<br/>$$<br/>Let $\quad z=x+i y$<br/>$$<br/>\therefore z-\bar{z}=2 i y=\frac{2 i}{-2 i y}=-\frac{1}{y}<br/>$$<br/>where, $y=\sqrt{1-x^2}$<br/>$\therefore \quad-1 \leq y \leq 1 \Rightarrow-1 \leq y$<br/>and $y \leq 1 \Rightarrow-1 \geq \frac{1}{y}$ and $\frac{1}{y} \geq 1$<br/>$\Rightarrow \operatorname{Re}\left(\frac{2 i z}{1-z^2}\right) \in(-\infty,-1] \cup[1, \infty)$<br/>(B) $f(x)=\sin ^{-1}\left(\frac{8\left(3^{x-2}\right)}{1-3^{2(x-1)}}\right)$,<br/><br/>$$<br/>\begin{aligned}<br/>& \text { For domain, }-1 \leq \frac{8\left(3^{x-2}\right)}{1-3^{2(x-1)}} \leq 1 \\<br/>& \Rightarrow \quad-1 \leq \frac{9 \cdot\left(3^{x-2}\right)-\left(3^{x-2}\right)}{1-3^{2(x-1)}} \leq 1 \\<br/>& \therefore \quad-1 \leq \frac{3^x-3^{x-2}}{1-3^x \cdot 3^{(x-2)}} \leq 1 \\<br/>& \frac{3^x-3^{x-2}}{1-3^x \cdot 3^{x-2}} \geq-1 \\<br/>& \Rightarrow \frac{\left(3^x-1\right)\left(3^{x-2}-1\right)}{\left(3^{x-1}+1\right)\left(3^{x-1}-1\right)} \geq 0 \\<br/>& \Rightarrow \quad x \in(-\infty, 0] \cup(1, \infty) \\<br/>& \text { and } \frac{3^x-3^{x-2}}{1-3^x \cdot 3^{x-2}} \leq 1 \\<br/>& \Rightarrow \frac{\left(3^{x-2}-1\right)\left(3^x+1\right)}{\left(3^{x-1}+1\right)\left(3^{x-1}-1\right)} \geq 0 \\<br/>&<br/>\end{aligned}<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wbvtTD4CyAPEJNBYNn5_bGlNIbQ1-mTtQArtAlYtJz0.original.fullsize.png"/><br/><br/>and $x \in(-\infty, 1) \cup[2, \infty)$<br/>$\therefore \quad x \in(-\infty, 0] \cup[2, \infty)$<br/>(C) $f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$<br/>$$<br/>\begin{aligned}<br/>R_1 & \rightarrow R_1+R_3, \\<br/>f(\theta) & =\left|\begin{array}{ccc}<br/>0 & 0 & 2 \\<br/>-\tan \theta & 1 & \tan \theta \\<br/>-1 & -\tan \theta & 1<br/>\end{array}\right| \\<br/>& =2\left(\tan ^2 \theta+1\right)=2 \sec ^2 \theta \geq 2 \\<br/>f(\theta) & \in[2, \infty)<br/>\end{aligned}<br/>$$<br/>$$<br/>\text { (D) } \begin{aligned}<br/>f(x) & =x^{3 / 2}(3 x-10) ; x \geq 0 \\<br/>f^{\prime}(x) & =x^{3 / 2} \cdot 3+\frac{3}{2} \cdot x^{1 / 2}(3 x-10) \\<br/>& =3 x^{1 / 2}\left\{x+\frac{1}{2}(3 x-10)\right\}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>& =\frac{3}{2} x^{1 / 2}\{2 x+3 x-10\} \\<br/>& =\frac{15}{2} x^{1 / 2}(x-2) \\<br/>& \\<br/>\therefore \quad & x \geq 2<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
575 | match-the-statements-given-in-column-i-with-the-values-given-in-column-ii | match-the-statements-given-in-column-i-with-the-values-given-in-column-ii-45449 | <div class="question">$$<br/>\text { Match the statements given in Column I with the values given in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/whLuoe5en1AhyDekxTeAe0GHJFttO-ylB8P6jVIZtEg.original.fullsize.png"/><br/></div> | ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) q, (B) p, (C) s, (D) s<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"><br/>(A) p, (B) p, (C) r, (D) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) q, (B) r, (C) s, (D) q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p, (B) s, (C) r, (D) s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) q, (B) p, (C) s, (D) s<br/></span> </div> | <div class="solution">(A) $\therefore|\mathbf{a}|=\sqrt{1+3}=2$ $|\mathbf{b}|=\sqrt{1+3}=2$ $|\mathbf{c}|=\sqrt{12}=2 \sqrt{3}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RGdCdtVHJH1U8RZNrjguZqUUsjiCKu2UqCzCxQh8vWA.original.fullsize.png"/><br/><br/>Using cosine law,<br/>$$<br/>\begin{aligned}<br/>\cos C & =\frac{|\mathbf{a}|^2+|\mathbf{b}|^2-|\mathbf{c}|^2}{2|\mathbf{a}||\mathbf{b}|} \\<br/>& =\frac{4+4-12}{2 \times 2 \times 2}=\frac{-4}{8}=\frac{-1}{2} \\<br/>\Rightarrow \quad \angle C & =120^{\circ}=\frac{2 \pi}{3}<br/>\end{aligned}<br/>$$<br/>(B)<br/>$$<br/>\begin{aligned}<br/>& \int_a^b f(x) d x-3\left(\frac{x^2}{2}\right)_a^b=\left(a^2-b^2\right) \\<br/>& \Rightarrow \int_a^b f(x) d x-\frac{3}{2}\left(b^2-a^2\right)=\left(a^2-b^2\right) \\<br/>& \Rightarrow \int_a^b f(x) d x=\left(a^2-b^2\right)+\frac{3}{2}\left(b^2-a^2\right) \\<br/>& =\frac{b^2-a^2}{2} \\<br/>& \Rightarrow \int_a^b f(x) d x=\frac{b^2-a^2}{2} \\<br/>&<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>f(x)=x \Rightarrow f\left(\frac{\pi}{6}\right)=\frac{\pi}{6}<br/>$$<br/>(C)<br/>$$<br/>\begin{aligned}<br/>& \frac{\pi^2}{\log _e 3} \int_{7 / 6}^{5 / 6} \sec (\pi x) d x \\<br/>& \Rightarrow \frac{\pi^2}{\log _e 3}\left\{\frac{\log |\sec \pi x+\tan \pi x|}{\pi}\right\}_{7 / 6}^{5 / 6} \\<br/>& \Rightarrow \frac{\pi}{\log 3}\left\{\log \left|\sec \frac{5 \pi}{6}+\tan \frac{5 \pi}{6}\right|\right. \\<br/>& \left.\quad-\log \left|\sec \frac{7 \pi}{6}+\tan \frac{7 \pi}{6}\right|\right\} \\<br/>& \Rightarrow \frac{\pi}{\log 3}\left\{\log |\sqrt{3}|-\log \left|\frac{1}{\sqrt{3}}\right|\right\} \\<br/>& \Rightarrow \frac{\pi}{\log 3}\{\log 3\}=\pi<br/>\end{aligned}<br/>$$<br/><br/>(D) $$<br/>\begin{aligned}<br/>& \left|\arg \frac{1}{(1-z)}\right|, \text { for }|z|=1 \\<br/>& \Rightarrow\left|\arg (1-z)^{-1}\right| \\<br/>& \Rightarrow|-\arg (1-z)| \Rightarrow|\arg (1-z)|<br/>\end{aligned}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/HSJMw5sIZCbFDqS37hzW91eRYY39SwiowxyUXRxah4U.original.fullsize.png"/><br/><br/>From figure, $\arg (z-1)$ is maximum $=\pi$.</div> | MarksBatch2_P2.db |
576 | match-the-statements-in-column-i-with-the-properties-column-ii | match-the-statements-in-column-i-with-the-properties-column-ii-19904 | <div class="question">Match the statements in Column I with the properties Column II. <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/WqEVKx8z2jQTeJCQ2HtKkkAVDBSMLgrVvfEfm8t3x-M.original.fullsize.png"/><br/></div> | ['Mathematics', 'Circle', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>A-p, r; B-p, q, s; C-q, r; D-q, s<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-p; B-p, s; C-p; D-q, s<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>A-p, q; B-p, q; C-q, r; D-q, r<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">D</span> <span class="option-data"><br/>A-r, s; B-q; C-p; D-p, q, s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>A-p, q; B-p, q; C-q, r; D-q, r<br/></span> </div> | <div class="solution">(A) When two circles are intersecting they have a common normal and common tangent.<br/>(B) Two mutually extemal circles have a common normal and common tangent.<br/>(C) When one circle lies inside of other, then they have a common normal but no common tangent.<br/>(D) Two branches of a hyperbola have a common normal but no common tangent.</div> | MarksBatch2_P2.db |
577 | match-the-statements-of-column-i-with-these-in-column-ii-note-here-z-takes-values-in-the-complex-plane-and-im-z-and-re-z-denote-respectively-the-imagi | match-the-statements-of-column-i-with-these-in-column-ii-note-here-z-takes-values-in-the-complex-plane-and-im-z-and-re-z-denote-respectively-the-imagi-35042 | <div class="question">Match the statements of Column I with these in Column II.<br/>[Note : Here $z$ takes values in the complex plane and $\operatorname{Im}(z)$ and $\operatorname{Re}(z)$ denote respectively, the imaginary part and the real part of $z$ ]<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/QRq94L5Vcpqg1gnxhabTl1RleIIK8yieHzlu7WztlPU.original.fullsize.png"/><br/></div> | ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) q, (B) p, (C) p,s, (D) q,r,s<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) q,r, (B) p, (C) p,s,t, (D) q,r,s,t<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"><br/>(A) q, (B) q, (C) p,s,t, (D) q,r,s,t<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) q,r, (B) q, (C) p,s, (D) q,r,s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) q,r, (B) p, (C) p,s,t, (D) q,r,s,t<br/></span> </div> | <div class="solution">(A) $z$ is equidistant from the points $i|z|$ and $-i|z|$, whose perpendicular bisector is $\operatorname{Im}(z)=0$.<br/>(B) Sum of distance of $z$ from $(4,0)$ and $(-4,0)$ is a constant 10 , hence locus of $z$ is ellipse with semi-major axis 5 and focus at $(\pm 4,0), a e=4$.<br/>$$<br/>\therefore \quad e=\frac{4}{5}<br/>$$<br/>(C) $|z| \leq|w|+\left|\frac{1}{w}\right|=\frac{5}{2} < 3$<br/>(D) $|z| \leq|w|+\left|\frac{1}{w}\right|=2$<br/>$$<br/>\therefore \quad \operatorname{Re}(z) \leq|z| \leq 2<br/>$$</div> | MarksBatch2_P2.db |
578 | match-the-statements-of-column-i-with-values-of-column-ii | match-the-statements-of-column-i-with-values-of-column-ii-82791 | <div class="question">$$<br/>\text { Match the statements of Column I with values of Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Kor3d9nZEDMiXJlMc071VJTT0cxHG3RIi9V6AwRthlk.original.fullsize.png"/><br/></div> | ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) t, (B) p,r, (C) p, (D) r<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"><br/>(A) q,r,t, (B) p,r, (C) p,s, (D) q,r<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) t, (B) q,r, (C) p,s, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) q,r,t, (B) q,r, (C) p, (D) q,r</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) t, (B) p,r, (C) p, (D) r<br/></span> </div> | <div class="solution">(A) Equation of the line passing through origin is<br/>$$<br/>\begin{aligned}<br/>& \frac{x}{a}=\frac{y}{b}=\frac{z}{c} \\<br/>& \therefore \quad\left|\begin{array}{ccc}<br/>2 & 1 & -1 \\<br/>1 & -2 & 1 \\<br/>a & b & c<br/>\end{array}\right|=0 \\<br/>& \Rightarrow \quad a(-1)-b(3)+c(-5)=0 \\<br/>& \Rightarrow \quad-a-3 b-5 c=0 \\<br/>& \Rightarrow \quad a+3 b+5 c=0 \\<br/>& \text { Also, } \quad\left|\begin{array}{ccc}<br/>\frac{8}{3} & -3 & 1 \\<br/>2 & -1 & 1 \\<br/>a & b & c<br/>\end{array}\right|=0<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>& \therefore \quad a(-2)-b\left(\frac{2}{3}\right)+c\left(\frac{10}{3}\right)=0 \\<br/>& \Rightarrow \quad 2 a+\frac{2 b}{3}-\frac{10 c}{3}=0 \\<br/>& 3 a+b-5 c=0 \\<br/>&<br/>\end{aligned}<br/>$$<br/>From Eqs. (i) and (ii),<br/>$$<br/>\begin{aligned}<br/>\frac{a}{-20} & =\frac{b}{20}=\frac{c}{-8} \\<br/>\frac{a}{5} & =\frac{b}{-5}=\frac{c}{4}<br/>\end{aligned}<br/>$$<br/>Equation of line is<br/>$$<br/>\frac{x}{5}=\frac{y}{-5}=\frac{z}{4}=\lambda \text { (say) }<br/>$$<br/>Also, $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+1}{1}=k_1$ (say)<br/>Now, $\frac{x-\frac{8}{3}}{2}=\frac{y+3}{-1}=\frac{z-1}{1}=k_2$ (say)<br/>Point on (iii) is $(5 \lambda,-5 \lambda,+4 \lambda)$ Point on (iv) is<br/><br/>$$<br/>\left(2+k_1, 1-2 k_1,-1+k_1\right)<br/>$$<br/>Point on $(\mathrm{v})$ is<br/>$$<br/>\left(\frac{8}{3}+2 k_2,-3-k_2, 1+k_2\right)<br/>$$<br/>On solving, $2+k_1+1-2 k_1=0$<br/>$$<br/>-k_1+3=0<br/>$$<br/>$$<br/>\begin{aligned}<br/>k_1 & =3 \\<br/>P & \equiv(5,-5,2)<br/>\end{aligned}<br/>$$<br/>Again, for $Q$<br/>$$<br/>\begin{array}{r}<br/>\frac{8}{3}+2 k_2-3-k_2=0 \\<br/>k_2-\frac{1}{3}=0 \\<br/>k_2=\frac{1}{3} \\<br/>Q \equiv\left(\frac{10}{3}, \frac{-10}{3}, \frac{4}{3}\right)<br/>\end{array}<br/>$$<br/>$$<br/>\text { Now, } \begin{aligned}<br/>P Q & =\sqrt{\left(\frac{5}{3}\right)^2+\left(\frac{5}{3}\right)^2+\left(\frac{2}{3}\right)^2} \\<br/>& =\frac{\sqrt{54}}{3}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>P Q^2=d^2=\frac{54}{9}=6<br/>$$<br/>(B) $\tan ^{-1}\left(\frac{x+3-x+3}{1+\left(x^2-9\right)}\right)=\tan ^{-1}\left(\frac{3}{4}\right)$<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow & & \frac{6}{x^2-8} & =\frac{3}{4} \\<br/>\Rightarrow & & 3 x^2 & =48 \\<br/>\Rightarrow & x & =\pm 4<br/>\end{array}<br/>$$<br/>(C)<br/>$$<br/>\begin{aligned}<br/>& (\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}) \cdot\left(\overrightarrow{\mathbf{b}}+\frac{\overrightarrow{\mathbf{a}}-\mu \overrightarrow{\mathbf{b}}}{4}\right)=0 \\<br/>& \Rightarrow \quad(\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}) \cdot(4 \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{a}}-\mu \overrightarrow{\mathbf{b}})=0 \\<br/>& \text { Also, } 2\left|\overrightarrow{\mathbf{b}}+\frac{\overrightarrow{\mathbf{a}}-\mu \overrightarrow{\mathbf{b}}}{4}\right|=|\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}| \\<br/>& \Rightarrow \quad 2\left|\frac{(4-\mu) \overrightarrow{\mathbf{b}}^2-\overrightarrow{\mathbf{a}}^2=0}{4}\right|=|\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}|<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \frac{(4-\mu)^2 \overrightarrow{\mathbf{b}}^2}{4}+\frac{\overrightarrow{\mathbf{a}}^2}{4}=\overrightarrow{\mathbf{b}}^2+\overrightarrow{\mathbf{a}}^2 \\<br/>& \Rightarrow \quad \frac{3 \overrightarrow{\mathbf{a}}^2}{4}=\frac{(4-\mu)^2-4}{4} \cdot \overrightarrow{\mathbf{a}}^2 \\<br/>& 3 \overrightarrow{\mathbf{a}}^2=C(4-\mu)^2-4 \overrightarrow{\mathbf{b}}^2<br/>\end{aligned}<br/>$$<br/>From Eqs. (i) and (ii)<br/>$$<br/>\begin{aligned}<br/>& 3(4-\mu)=(4-\mu)^2-4 \\<br/>& (4-\mu)^2-3(4-\mu)-4=0 \\<br/>& \Rightarrow \quad \mu=0,5 \\<br/>& \mu=5 \text { is not admissible. }<br/>\end{aligned}<br/>$$<br/>(D) $f(0)=9$,<br/>$$<br/>\begin{aligned}<br/>& f(x)=\frac{\sin \left(\frac{9 x}{2}\right)}{\sin \frac{x}{2}} \\<br/>& =\left(3-4 \sin ^2 \frac{x}{2}\right)\left(3-4 \sin ^2 \frac{3 x}{2}\right)<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>& =9-12 \sin ^2 \frac{x}{2}-12 \sin ^2 \frac{3 x}{2} \\<br/>& \quad+16 \sin ^2 \frac{x}{2} \cdot \sin ^2 \frac{3 x}{2} \\<br/>& =9-6(1-\cos x)-6(1-\cos 3 x) \\<br/>& \quad+4(1-\cos x)(1-\cos 3 x) \\<br/>& =1+6 \cos x+6 \cos 3 x-4 \cos x \\<br/>& \quad-4 \cos 3 x+4 \cos x \cos 3 x \\<br/>& \text { Let } \quad I=\frac{2}{\pi} \int_{-\pi}^\pi \frac{\sin \frac{9 x}{2}}{2} d x \\<br/>& =\frac{4}{\pi} \int_0^\pi 1+2 \cos x+2 \cos 3 x \\<br/>& =\frac{4}{\pi} \times \pi \quad+2(\cos 4 x+\cos x) d x \\<br/>& =4<br/>\end{aligned}<br/>$$</div> | MarksBatch2_P2.db |
579 | match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-1 | match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-1-50872 | <div class="question">Match the statements/expressions given in Column I with the values given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OiiFrlYsfzglw0nz5y3f0pvY8jShZJrQWZJzU2zh_l0.original.fullsize.png"/><br/></div> | ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) s, (B) q,s, (C) p,r,s,t, (D) q<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p, (B) q,s, (C) q,r,s,t, (D) r<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"><br/>(A) s, (B) s,t, (C) q,r,s, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p, (B) s,t, (C) q,r,s,t, (D) q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p, (B) q,s, (C) q,r,s,t, (D) r<br/></span> </div> | <div class="solution">(a) Let $f(x)=x e^{\sin x}-\cos x$ $\overrightarrow{f^{\prime}}(x)=e^{\sin x}+x e^{\sin x} \cos x+\sin x \geq 0$ For interval $x \in\left(0, \frac{\pi}{2}\right)$, $f$ is strictly increasing.<br/>$\therefore \quad f(0)=-1$ and $f\left(\frac{\pi}{2}\right)=\frac{\pi}{2} e \Rightarrow$ one solution<br/>(b) Since, $\left|\begin{array}{lll}k & 4 & 1 \\ 4 & k & 2 \\ 2 & 2 & 1\end{array}\right|=0$ $\Rightarrow k(k-4)-4(0)+1(8-2 k)=0$ $\Rightarrow \quad k^2-6 k+8=0$ $\Rightarrow \quad k=2,4$<br/>(c) Let $y=|x-1|+|x-2|+|x+1|$ $+|x+2|$<br/>For solutions, $4 k \geq 6 \Rightarrow k \geq 3 / 2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9-zLMBos4B1Ewe3LfmvgASEnGCtnhtCQTIuDD_3VqJk.original.fullsize.png"/><br/><br/>$\therefore$ Integer values of $k$ are $2,3,4,5$.<br/>(d) Given, $\frac{d y}{d x}=y+1$<br/>$\Rightarrow \quad \ln |(y+1)|=x+C$ $\Rightarrow \ln 2=C \Rightarrow \ln |y+1|=x+\ln 2$<br/>Put $x=\ln 2$<br/>$\therefore \quad \ln (y+1)=\ln 2+\ln 2=\ln 4$<br/>$y+1=4 \Rightarrow y=3$</div> | MarksBatch2_P2.db |
580 | match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-2 | match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-2-52479 | <div class="question">Match the statements/expressions given in Column I with the values given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OiiFrlYsfzglw0nz5y3f0pvY8jShZJrQWZJzU2zh_l0.original.fullsize.png"/><br/></div> | ['Mathematics', 'Functions', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) s, (B) q,s, (C) p,r,s,t, (D) q<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p, (B) q,s, (C) q,r,s,t, (D) r<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"><br/>(A) s, (B) s,t, (C) q,r,s, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p, (B) s,t, (C) q,r,s,t, (D) q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p, (B) q,s, (C) q,r,s,t, (D) r<br/></span> </div> | <div class="solution">(a) Let $f(x)=x e^{\sin x}-\cos x$ $\overrightarrow{f^{\prime}}(x)=e^{\sin x}+x e^{\sin x} \cos x+\sin x \geq 0$ For interval $x \in\left(0, \frac{\pi}{2}\right)$, $f$ is strictly increasing.<br/>$\therefore \quad f(0)=-1$ and $f\left(\frac{\pi}{2}\right)=\frac{\pi}{2} e \Rightarrow$ one solution<br/>(b) Since, $\left|\begin{array}{lll}k & 4 & 1 \\ 4 & k & 2 \\ 2 & 2 & 1\end{array}\right|=0$ $\Rightarrow k(k-4)-4(0)+1(8-2 k)=0$ $\Rightarrow \quad k^2-6 k+8=0$ $\Rightarrow \quad k=2,4$<br/>(c) Let $y=|x-1|+|x-2|+|x+1|$ $+|x+2|$<br/>For solutions, $4 k \geq 6 \Rightarrow k \geq 3 / 2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9-zLMBos4B1Ewe3LfmvgASEnGCtnhtCQTIuDD_3VqJk.original.fullsize.png"/><br/><br/>$\therefore$ Integer values of $k$ are $2,3,4,5$.<br/>(d) Given, $\frac{d y}{d x}=y+1$<br/>$\Rightarrow \quad \ln |(y+1)|=x+C$ $\Rightarrow \ln 2=C \Rightarrow \ln |y+1|=x+\ln 2$<br/>Put $x=\ln 2$<br/>$\therefore \quad \ln (y+1)=\ln 2+\ln 2=\ln 4$<br/>$y+1=4 \Rightarrow y=3$</div> | MarksBatch2_P2.db |
581 | match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii | match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-83406 | <div class="question">Match the statements/expressions given in Column I with the values given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/X5zrl_D_5t94xmdVslBxnMcxIhCVjgRpxOOFvHdKn0o.original.fullsize.png"/><br/></div> | ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) q,s, (B) p,r,s,t, (C) t, (D) r<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"><br/>(A) p,s, (B) q,r,s, (C) s, (D) q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) q,s, (B) q,r,s, (C) t, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,s, (B) p,r,s,t, (C) t, (D) q</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) q,s, (B) p,r,s,t, (C) t, (D) r<br/></span> </div> | <div class="solution">(a) $2 \sin ^2 \theta+\sin ^2 2 \theta=2$<br/>$\Rightarrow \quad \sin ^2 2 \theta=2 \cos ^2 \theta$<br/>$\Rightarrow \quad 4 \sin ^2 \theta \cos ^2 \theta=2 \cos ^2 \theta$<br/>$\Rightarrow \quad \cos ^2 \theta=0$ or $\sin ^2 \theta=\frac{1}{2}$<br/>$\Rightarrow \quad \cos \theta=0$ or $\sin \theta=\pm \frac{1}{\sqrt{2}}$<br/>$\Rightarrow \quad \theta=\pm \frac{\pi}{4}$ or $\frac{\pi}{2}$<br/>(b) $f(x)=\left[\frac{6 x}{\pi}\right] \cos \left[\frac{3 x}{\pi}\right]$<br/>Possible points of discontinuity of $\left[\frac{6 x}{\pi}\right]$ are<br/>$$<br/>\begin{gathered}<br/>\frac{6 x}{\pi}=n, n \in I \\<br/>\Rightarrow \quad x=\frac{n \pi}{6} \Rightarrow x=\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \pi \\<br/>\lim _{x \rightarrow \pi^{-} / 6} f(x)=0 \cos 0=0 \\<br/>\lim _{x \rightarrow \pi^{+} / 6} f(x)=1 \cos 0=1<br/>\end{gathered}<br/>$$<br/>$\therefore$ Discontinuous at $x=\frac{\pi}{6}$.<br/>Similarly, discontinuous at $x=\frac{\pi}{3}, \frac{\pi}{2}, \pi$.<br/>(c) Here, $V=\left\|\begin{array}{lll}1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & \pi\end{array}\right\|=\pi$ cubic unit<br/>(d) Given, $\mathbf{a}+\mathbf{b}+\sqrt{3} \mathbf{c}=\mathbf{0}$<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow & \mathbf{a}+\mathbf{b}=-\sqrt{3} \mathbf{c} \\<br/>\Rightarrow & |\mathbf{a}+\mathbf{b}|^2=\mid \sqrt{3} \mathbf{c}^2<br/>\end{array}<br/>$$<br/><br/>$$<br/>\begin{array}{lrl}<br/>\Rightarrow & a^2+b^2+2 \mathbf{a} \cdot \mathbf{b}=3 c^2 \\<br/>\Rightarrow & 2+2 \cos \theta=3 \\<br/>\Rightarrow & \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}<br/>\end{array}<br/>$$</div> | MarksBatch2_P2.db |
582 | match-the-statementsexpressions-in-column-i-with-the-open-intervals-in-column-ii-1 | match-the-statementsexpressions-in-column-i-with-the-open-intervals-in-column-ii-1-45958 | <div class="question">$$<br/>\text { Match the statements/expressions in Column I with the open intervals in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YSS353722UU7V-kPU9gVFl_hn_aTbFqHT6L7rH7xHFg.original.fullsize.png"/><br/></div> | ['Mathematics', 'Definite Integration', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,t, (B) q,t, (C) r,s, (D) p,s<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s, (B) p,t, (C) p,q,r,t, (D) s<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"><br/>(A) p,q,s, (B) q,t, (C) r,s,t, (D) p,s<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,t, (B) p,t, (C) p,q,r,t, (D) s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,s, (B) p,t, (C) p,q,r,t, (D) s<br/></span> </div> | <div class="solution">(a) Given, $(x-3)^2 \cdot y^{\prime}+y=0$<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow & \frac{d y}{d x} & =-\frac{y}{(x-3)^2} \\<br/>\Rightarrow & & \int \frac{d y}{y} & =-\int \frac{d x}{(x-3)^3} \\<br/>\Rightarrow & & \ln y & =\frac{1}{(x-3)}+\ln C \\<br/>\Rightarrow & & y & =C e^{\frac{1}{x-3}}, C \neq 0<br/>\end{array}<br/>$$<br/>$\therefore$ Domain of $y$ is $x \in R-\{3\}$<br/>Aliter Given differential equation is homogeneous linear differential equation and has $x=3$ as a singular point, hence $x=3$ cannot be in domain of solution.<br/>(b) Let $I=\int_1^5(x-1)(x-2)(x-3)(x-4)$ $(x-5) d x$<br/>Let $x-3=t \Rightarrow d x=d t$<br/>$\therefore I=\int_{-2}^2(t+2)(t+1) t(t-1)(t-2) d t$<br/>$\because$ Integrand is an odd function.<br/>$\therefore \quad I=0$<br/>Aliter Let<br/>$I=\int_1^5(x-1)(x-2)(x-3)(x-4)$<br/>$(x-5) d x$<br/>Using, $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$ $I=\int_1^5(5-x)(4-x)(3-x)(2-x)$ $(1-x) d x$<br/>On adding Eqs. (i) and (ii) we get 2I $=0 \Rightarrow I=0$<br/>(c) Let $f(x)=\cos ^2 x+\sin x$<br/>$\Rightarrow f^{\prime}(x)=-2 \cos x \sin x+\cos x$ $=\cos x(1-2 \sin x)=0$ (say)<br/>Sign scheme for first derivative<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TLmS4uc2BFsrzGpUJe2DJ2LMHFf6yxl77ASJG4WjUWY.original.fullsize.png"/><br/><br/>Points of local maxima are $\frac{\pi}{6}, \frac{5 \pi}{6}$.<br/>Aliter $y=\cos ^2 x+\sin x$<br/>$$<br/>\Rightarrow \quad y=\frac{5}{4}-\left(\sin x-\frac{1}{2}\right)^2<br/>$$<br/>For $y$ to be maximum,<br/>$$<br/>\begin{aligned}<br/>& \left(\sin x-\frac{1}{2}\right)^2=0 \Rightarrow \sin x=\frac{1}{2} \\<br/>\Rightarrow \quad & x=n \pi+(-1)^n \frac{\pi}{6}, n \in I<br/>\end{aligned}<br/>$$<br/>(d) Let $y=\tan ^{-1}(\sin x+\cos x)$<br/>$$<br/>\Rightarrow \quad \frac{d y}{d x}=\frac{\cos x-\sin x}{1+(\sin x+\cos x)^2}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/QXVtSrn6Dj_MC10OOrVKBH-PujNlc9ggm4Rte2x4G1M.original.fullsize.png"/><br/><br/>Clearly, by graph, $\cos x>\sin x$ is true for option(s).</div> | MarksBatch2_P2.db |
583 | match-the-statementsexpressions-in-column-i-with-the-open-intervals-in-column-ii | match-the-statementsexpressions-in-column-i-with-the-open-intervals-in-column-ii-11131 | <div class="question">$$<br/>\text { Match the statements/expressions in Column I with the open intervals in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YSS353722UU7V-kPU9gVFl_hn_aTbFqHT6L7rH7xHFg.original.fullsize.png"/><br/></div> | ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,t, (B) q,t, (C) r,s, (D) p,s<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s, (B) p,t, (C) p,q,r,t, (D) s<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"><br/>(A) p,q,s, (B) q,t, (C) r,s,t, (D) p,s<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,t, (B) p,t, (C) p,q,r,t, (D) s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,s, (B) p,t, (C) p,q,r,t, (D) s<br/></span> </div> | <div class="solution">(a) Given, $(x-3)^2 \cdot y^{\prime}+y=0$<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow & \frac{d y}{d x} & =-\frac{y}{(x-3)^2} \\<br/>\Rightarrow & & \int \frac{d y}{y} & =-\int \frac{d x}{(x-3)^3} \\<br/>\Rightarrow & & \ln y & =\frac{1}{(x-3)}+\ln C \\<br/>\Rightarrow & & y & =C e^{\frac{1}{x-3}}, C \neq 0<br/>\end{array}<br/>$$<br/>$\therefore$ Domain of $y$ is $x \in R-\{3\}$<br/>Aliter Given differential equation is homogeneous linear differential equation and has $x=3$ as a singular point, hence $x=3$ cannot be in domain of solution.<br/>(b) Let $I=\int_1^5(x-1)(x-2)(x-3)(x-4)$ $(x-5) d x$<br/>Let $x-3=t \Rightarrow d x=d t$<br/>$\therefore I=\int_{-2}^2(t+2)(t+1) t(t-1)(t-2) d t$<br/>$\because$ Integrand is an odd function.<br/>$\therefore \quad I=0$<br/>Aliter Let<br/>$I=\int_1^5(x-1)(x-2)(x-3)(x-4)$<br/>$(x-5) d x$<br/>Using, $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$ $I=\int_1^5(5-x)(4-x)(3-x)(2-x)$ $(1-x) d x$<br/>On adding Eqs. (i) and (ii) we get 2I $=0 \Rightarrow I=0$<br/>(c) Let $f(x)=\cos ^2 x+\sin x$<br/>$\Rightarrow f^{\prime}(x)=-2 \cos x \sin x+\cos x$ $=\cos x(1-2 \sin x)=0$ (say)<br/>Sign scheme for first derivative<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TLmS4uc2BFsrzGpUJe2DJ2LMHFf6yxl77ASJG4WjUWY.original.fullsize.png"/><br/><br/>Points of local maxima are $\frac{\pi}{6}, \frac{5 \pi}{6}$.<br/>Aliter $y=\cos ^2 x+\sin x$<br/>$$<br/>\Rightarrow \quad y=\frac{5}{4}-\left(\sin x-\frac{1}{2}\right)^2<br/>$$<br/>For $y$ to be maximum,<br/>$$<br/>\begin{aligned}<br/>& \left(\sin x-\frac{1}{2}\right)^2=0 \Rightarrow \sin x=\frac{1}{2} \\<br/>\Rightarrow \quad & x=n \pi+(-1)^n \frac{\pi}{6}, n \in I<br/>\end{aligned}<br/>$$<br/>(d) Let $y=\tan ^{-1}(\sin x+\cos x)$<br/>$$<br/>\Rightarrow \quad \frac{d y}{d x}=\frac{\cos x-\sin x}{1+(\sin x+\cos x)^2}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/QXVtSrn6Dj_MC10OOrVKBH-PujNlc9ggm4Rte2x4G1M.original.fullsize.png"/><br/><br/>Clearly, by graph, $\cos x>\sin x$ is true for option(s).</div> | MarksBatch2_P2.db |
584 | match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-column-ii | match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-column-ii-59788 | <div class="question">Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NV6MxxlCrdeWHLs9r4Y0EOApcJx3jyAt7MA2okb20P0.original.fullsize.png"/><br/></div> | ['Mathematics', 'Basic of Mathematics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) r, (B) q,r, (C) s, (D) p,r<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) s, (B) q,s, (C) r, (D) p,q<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) s, (B) q,r, (C) s, (D) p,q<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) r, (B) q,s, (C) r, (D) p,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> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) r, (B) q,s, (C) r, (D) p,r</span> </div> | <div class="solution">(A) Let $y=\frac{x^2+2 x+4}{x+2}$<br/>$$<br/>\begin{aligned}<br/>& \Rightarrow x^2+(2-y) x+(4-2 y)=0 \\<br/>& \Rightarrow \quad(2-y)^2-4(4-2 y) \geq 0 \\<br/>& \Rightarrow \quad y^2+4 y-12 \geq 0 \\<br/>& \Rightarrow \quad y \leq-6, y \geq 2 \\<br/>&<br/>\end{aligned}<br/>$$<br/>$\therefore$ Minimum value of $y$ is 2 .<br/>(B) Since, $(A+B)(A-B)=(A-B)(A+B)$<br/>$$<br/>\begin{aligned}<br/>& \Rightarrow \quad A^2-A B+B A-B^2=A^2+A B-B A-B^2 \\<br/>& \Rightarrow \quad A B=B A \\<br/>& \text { and } \quad(A B)^t=(-1)^k A B \\<br/>& \Rightarrow \quad B^t A^t=(-1)^k A B \\<br/>& \Rightarrow \quad-B A=(-1)^k A B \quad\left[\because B^t=-B, A^t=A\right] \\<br/>& \Rightarrow \quad B A=(-1)^{k+1} A B \\<br/>& \Rightarrow \quad(-1)^{k+1}=1 \\<br/>&<br/>\end{aligned}<br/>$$<br/><br/>$\therefore k+1$ is even or $k$ is odd.<br/>(C) $1 < 2^{\left(-k+3^{-a}\right)} < 2 \Rightarrow 0 < -k+3^{-a} < 1$<br/>Given, $a=\log _3 \log _3 2 \Rightarrow 3^a=\log _3 2$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow & 3^{-a}=\log _2 3 \\<br/>\therefore & k < \log _2 3 < 2 \\<br/>\text { and } & 1+k>\log _2 3>1 \Rightarrow k>0<br/>\end{array}<br/>$$<br/>From Eqs. (ii) and (iii), $0 < k < 2 \Rightarrow k=1$<br/>$[\because k$ is an integer]<br/>$$<br/>\begin{array}{rlrl}<br/>\text { (D) } & \sin \theta & =\cos \phi \\<br/>\Rightarrow & \cos \left(\frac{\pi}{2}-\theta\right) & =\cos \phi \\<br/>\Rightarrow & \frac{\pi}{2}-\theta & =2 n \pi \pm \phi, n \in Z \\<br/>\Rightarrow & & \theta \pm \phi-\frac{\pi}{2} & =-2 n \pi, n \in Z \\<br/>\Rightarrow & & \frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right) & =-2 n, n \in Z<br/>\end{array}<br/>$$</div> | MarksBatch2_P2.db |
585 | match-the-transformation-in-column-i-with-appropriate-options-in-column-ii | match-the-transformation-in-column-i-with-appropriate-options-in-column-ii-61290 | <div class="question">Match the transformation in Column I with appropriate options in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TQElWvj8uVhqy8L6g1_0H549TaOc7nepzl4wiNxtBnw.original.fullsize.png"/><br/></div> | ['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,r,s, (B) r,s, (C) t, (D) p,q<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,s, (B) p,s, (C) s, (D) p,q,s<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,s, (B) q,r, (C) s, (D) p,q<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,r,s, (B) r,s, (C) t, (D) p,q,t</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,r,s, (B) r,s, (C) t, (D) p,q,t</span> </div> | <div class="solution">(A) $\mathrm{CO}_2(\mathrm{~s}) \longrightarrow \mathrm{CO}_2(\mathrm{~g})$<br/>It is just a phase transition (sublimation) as no chemical change has occurred. Sublimation is always endothermic. Product is gas, more disordered, hence $\Delta s$ is positive.<br/>(B) $\mathrm{CaCO}_3(\mathrm{~s}) \longrightarrow \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_2(\mathrm{~g})$<br/>It is a chemical decomposition, not a phase change. Thermal decomposition occur at the expense of energy, hence endothermic. Product contain a gaseous species, hence, $\Delta S>0$.<br/>(C) $2 \mathrm{H} \longrightarrow \mathrm{H}_2(\mathrm{~g})$<br/>A new $\mathrm{H}-\mathrm{H}$ covalent bond is being formed, hence, $\Delta H < 0$.<br/>Also, product is less disordered than reactant, $\Delta S < 0$.<br/>(D) Allotropes are considered as different phase, hence $\mathrm{P}_{\text {(white, solid) }} \rightarrow \mathrm{P}_{\text {(red, solid) }}$ is a phase transition as well as allotropic change.<br/>Also, red phosphorus is more ordered than white phosphorus, $\Delta S < 0$.</div> | MarksBatch2_P2.db |
586 | mathematics-application-of-derivatives-jee-advanced-64b169cef23ac7d519f9aea2 | mathematics-application-of-derivatives-jee-advanced-64b169cef23ac7d519f9aea2-17204 | <div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5J9jOFz5SPHXym-2E7yNW6FpxEcbDxNYH9Z-gh2VeMk.original.fullsize.png"/><br/></div> | ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$g(x)$ has local maxima at $x=1+\log _e 2$ and local minima at $x=e$<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="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$f(x)$ has local maxima at $x=1$, and local minima at $x=2$<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"><br/>$g(x)$ has no local minima<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$f(x)$ has no local maxima</span> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$g(x)$ has local maxima at $x=1+\log _e 2$ and local minima at $x=e$<br/>, <br/>$f(x)$ has local maxima at $x=1$, and local minima at $x=2$<br/></span> </div> | <div class="solution">Here, $f(x)=\left\{\begin{array}{cc}e^x, & 0 \leq x \leq 1 \\ 2-e^{x-1}, & 1 < x \leq 2 \\ x-e, & 2 < x \leq 3\end{array}\right.$ and $\quad g(x)=\int_0^x f(t) d t$<br/>$$<br/>\Rightarrow \quad g^{\prime}(x)=f(x)<br/>$$<br/>where, $g^{\prime}(x)=0$<br/>$\Rightarrow \quad x=1+\log _e 2$<br/>and $\quad x=e$.<br/>Also, $\quad g(x)=\left\{\begin{array}{cc}-e^{x-1}, & 1 < x \leq 2 \\ 1, & 2 < x \leq 3\end{array}\right.$<br/>$g\left(1+\log _e 2\right)=-e^{\log _e 2} < 0$, hence at $x=1+\log _e 2, g(x)$ has a local maximum.<br/>Also, $g(e)=1>0$, hence at $x=e, g(x)$ has a local minima.<br/>$\because f(x)$ is discontinuous at $x=1$, then we get<br/>Local maxima at $x=1$ and local minima at $x=2$.</div> | MarksBatch2_P2.db |
587 | mgso-4-on-reaction-with-nh-4-oh-and-na-2-hpo-4-forms-a-white-crystalline-precipitate-what-is-its-formula | mgso-4-on-reaction-with-nh-4-oh-and-na-2-hpo-4-forms-a-white-crystalline-precipitate-what-is-its-formula-97515 | <div class="question">$\mathrm{MgSO}_4$ on reaction with $\mathrm{NH}_4 \mathrm{OH}$ and $\mathrm{Na}_2 \mathrm{HPO}_4$ forms a white crystalline precipitate. What is its formula?</div> | ['Chemistry', 's Block Elements', 'JEE Advanced', 'JEE Advanced 2006'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4$<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"><br/>$\mathrm{Mg}_3\left(\mathrm{PO}_4\right)_2$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{MgCl}_2 \cdot \mathrm{MgSO}_4$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{MgSO}_4$</span> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4$<br/></span> </div> | <div class="solution">$\mathrm{MgSO}_4$ on reaction with $\mathrm{Na}_2 \mathrm{HPO}_4$ (disodium hydrogen phosphate) in presence of $\mathrm{NH}_4 \mathrm{OH}$ to give white precipitate of magnesium ammonium phosphate $\left(\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4\right)$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/MpRey9k-Y2s158WLfD5sHyIYsJf5OFY_P-DHgLZbHkM.original.fullsize.png"/><br/></div> | MarksBatch2_P2.db |
588 | most-advanced-invertebrates-are | most-advanced-invertebrates-are-36788 | <div class="question">Most advanced invertebrates are</div> | ['Biology', 'Animal Kingdom', 'JIPMER', 'JIPMER 2019'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Arthropoda</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Annelida</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Mollusca</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Cephalopoda</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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> | <div class="correct-answer">
The correct answer is:
<span class="option-value">Cephalopoda</span> </div> | <div class="solution">Cephalopods such as octopus, cuttlefish and squid (coleoids) are the most advanced invertebrates among the given options. They belong to class-Cephalopoda of phylum - Mollusca. These organisms have a prominent head and a set arms or tentacles. They are widely regarded as the most intelligent of invertebrates and have well-developed senses and larger brains. Their nervous system is the most complex of all invertebrates.</div> | MarksBatch2_P2.db |
589 | most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1-1 | most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1-1-56498 | <div class="question">Most materials have the refractive index, $n>1$. So, when a light ray from air enters a naturally occurring material, then by Snell's
<br/> <br/>law, $\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}$, it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, $\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}$, where $c$ is the speed of electromagnetic waves in vacuum, $v$ its speed in the medium, $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability of the medium respectively. In normal materials, both $\varepsilon_{r}$ and $\mu_{r}$, are positive, implying positive $n$ for the medium. When both $\varepsilon_{r}$ and $\mu_{r}$ are negative, one must choose the negative root of $n$. Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since $n$ is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
<br/> <br/><strong> Question:</strong> Choose the correct statement.</div> | ['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The speed of light in the meta-material is $v=c|n|$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The speed of light in the meta-material is $v=\frac{c}{|n|}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The speed of light in the meta-material is $v=c$.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The wavelength of the light in the meta-material $\left(\lambda_{m}\right)$ is given by $\lambda_{m}=\lambda_{\text {air }}|n|$, where $\lambda_{\text {air }}$ is wavelength of the light in air.</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">The speed of light in the meta-material is $v=\frac{c}{|n|}$</span> </div> | <div class="solution">Speed of light in a medium, $V=\frac{C}{n}$ $n$ for meta-material $=|n|$
<br/> <br/>$\therefore \quad$ Speed of light in the meta-material $V=\frac{C}{|n|}$.</div> | MarksBatch2_P2.db |
590 | most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1 | most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1-19098 | <div class="question">Most materials have the refractive index, $n>1$. So, when a light ray from air enters a naturally occurring material, then by Snell's
<br/> <br/>law, $\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}$, it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, $\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}$, where $c$ is the speed of electromagnetic waves in vacuum, $v$ its speed in the medium, $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability of the medium respectively. In normal materials, both $\varepsilon_{r}$ and $\mu_{r}$, are positive, implying positive $n$ for the medium. When both $\varepsilon_{r}$ and $\mu_{r}$ are negative, one must choose the negative root of $n$. Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since $n$ is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
<br/> <br/><strong> Question:</strong> For light incident from air on a meta-material, the appropriate ray diagram is</div> | ['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fSliPo124VmuV8rBTNz1ynTcNkB_PxMLqSjUjyTw938.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/jee_advanced/4dqL3uaNGI97arahJ7edochFqZbIrEY5uXyLQq7Hs74.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ldjy3BnC7u5EUqgGGiUss9jHaX9uYJHrNPVHSadDD-M.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jzYvCJqcyvPF0qxXJbmYQaWxYTEp97QbOjNL57kZFcc.original.fullsize.png"/></span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ldjy3BnC7u5EUqgGGiUss9jHaX9uYJHrNPVHSadDD-M.original.fullsize.png"/></span> </div> | <div class="solution">Let $n_{1}=$ refractive index of air and $n_{2}=$ refractive index of meta material which is negative
<br/> <br/>From Snell's law, $\frac{n_{2}}{n_{1}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$
<br/> <br/>$\because \quad n_{2}$ negative $\therefore \theta_{2}$ isalso negative hence graph (c) is correct.</div> | MarksBatch2_P2.db |
591 | native-silver-metal-forms-a-water-soluble-complex-with-a-dilute-aqueous-solution-of-nacn-in-the-presence-of-1 | native-silver-metal-forms-a-water-soluble-complex-with-a-dilute-aqueous-solution-of-nacn-in-the-presence-of-1-33128 | <div class="question">Native silver metal forms a water soluble complex with a dilute aqueous solution of $\mathrm{NaCN}$ in the presence of</div> | ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>nitrogen<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>oxygen<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"><br/>carbon dioxide<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>argon</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>oxygen<br/></span> </div> | <div class="solution">A water soluble complex of silver with a dilute aqueous solution of $\mathrm{NaCN}$ is sodium argentocyanide. In the cyanide process, the native form is crushed and treated with $0.1-0.2 \%$ solution of $\mathrm{NaCN}$ and aerated.<br/>$$<br/>4 \mathrm{Ag}+8 \mathrm{NaCN}+2 \mathrm{H}_2 \mathrm{O}+\mathrm{O}_2 \longrightarrow 4 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_2\right]+4 \mathrm{NaOH}<br/>$$<br/>Argentocyanide is soluble. Further, metals are recovered from the complex by reduction with zinc</div> | MarksBatch2_P2.db |
592 | nicl-2-p-c-2-h-5-2-c-6-h-5-2-exhibits-temperature-dependent-magnetic-behaviour-paramagneticdiamagnetic-the-coordination-geometries-of-ni-2-in-the-para | nicl-2-p-c-2-h-5-2-c-6-h-5-2-exhibits-temperature-dependent-magnetic-behaviour-paramagneticdiamagnetic-the-coordination-geometries-of-ni-2-in-the-para-82493 | <div class="question">$\mathrm{NiCl}_{2}\left\{\mathrm{P}\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)\right\}_{2}$ exhibits temperature depend-ent magnetic behaviour (paramagnetic/diamagnetic). The coordination geometries of $\mathrm{Ni}^{2+}$ in the paramagnetic and diamagnetic states are respectively</div> | ['Chemistry', 'Coordination Compounds', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">tetrahedral and tetrahedral</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">square planar and square planar</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">tetrahedral and square planar</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">square planar and tetrahedral</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value">tetrahedral and square planar</span> </div> | <div class="solution">In both states (paramagnetic and diamagnetic) of the given complex, $\mathrm{Ni}$ exists as $\mathrm{Ni}^{2+}$ whose electronic configuration is $[\mathrm{Ar}] 3 d^{8} 4 s^{0}$.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Y0qPb6T-E96_GJJk5j2b10GNt-stG3VzU8NSyV44cF0.original.fullsize.png"/><br/> <br/> <br/>In the above paramagnetic state the geometry of the complex is $s p^{3}$ giving tetrahedral geometry.
<br/> <br/>The diamagnetic state is achieved by pairing of electrons in $3 d$ orbital.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IY3jCGJMIjP0XjucA1Crn0M1Un_UO6SeFvtnewMMv0o.original.fullsize.png"/><br/> <br/> <br/>Thus, the geometry of the complex will be $d s p^{2}$ giving square planar geometry.</div> | MarksBatch2_P2.db |
593 | one-indian-and-four-american-men-and-their-wives-are-to-be-seated-randomly-around-a-circular-table-then-the-conditional-probability-that-the-indian-ma | one-indian-and-four-american-men-and-their-wives-are-to-be-seated-randomly-around-a-circular-table-then-the-conditional-probability-that-the-indian-ma-40154 | <div class="question">One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is</div> | ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{1}{2}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{1}{3}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{2}{5}$<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">D</span> <span class="option-data"><br/>$\frac{1}{5}$</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{2}{5}$<br/></span> </div> | <div class="solution">Let $E=$ event when each American man is seated adjacent to his wife and $A=$ event when Indian man is seated adjacent to his wife.<br/>Now, $n(A \cap E)=(4 !) \times(2 !)^5$<br/>Even when each American man is seated adjacent to his wife.<br/>Again<br/>$$<br/>\begin{aligned}<br/>n(E) & =(5 !) \times(2 !)^4 \\<br/>P\left(\frac{A}{E}\right) & =\frac{n(A \cap E)}{n(E)} \\<br/>& =\frac{(4 !) \times(2 !)^5}{(5 !) \times(2 !)^4}=\frac{2}{5}<br/>\end{aligned}<br/>$$<br/>$$<br/>\Rightarrow \quad P\left(\frac{A}{E}\right)=\frac{n(A \cap E)}{n(E)}<br/>$$<br/>ALITER<br/>Fixing four American couples and one Indian man in between any two couples, we have 5 different ways in which his wife can be seated, of which 2 cases are favourable. $\therefore$ Required probability $=\frac{2}{5}$.</div> | MarksBatch2_P2.db |
594 | one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy-1 | one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy-1-14367 | <div class="question">One mole of a monatomic ideal gas is taken through a cycle $A B C D A$ as shown in the $p$-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5ZEJlNpluXbfGVD34o37DgTSv2X9t_fYuXv-lWGpHTg.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BtZmk11u2LxVWqhXGJSwF68nRVZSvKCI2ONt9M52A6I.original.fullsize.png"/><br/></div> | ['Physics', 'Thermodynamics', 'JEE Main'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,r,t, (B) q,r, (C) q,s, (D) r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,t, (B) p,r, (C) q,r, (D) s<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"><br/>(A) p,s, (B) q,r, (C) q,s, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,r,t, (B) p,q,r, (C) q,r, (D) s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,t, (B) p,r, (C) q,r, (D) s<br/></span> </div> | <div class="solution">Internal energy $\propto T \propto p V$<br/>This is because<br/>$$<br/>U=\frac{n f}{2} R T=\frac{f}{2} p V<br/>$$<br/>Here, $n=$ number of moles<br/>$f=$ degree of freedom<br/>If the product $p V$ increases, then internal energy will increase and if product decreases, the internal energy will decrease.<br/>Further, work is done on the gas, if volume of gas decreases. For heat exchange.<br/>$$<br/>Q=W+\Delta U<br/>$$<br/>Work done is area under $p-V$ graph. If volume increases work done by gas is positive and if volume decreases work done by gas is negative. Further $\Delta U$ is positive if product of $p V$ is increasing and $N U$ is negative, if product of $p V$ is decreasing. If heat is taken by the gas $Q$ is positive and if heat is lost by the gas $Q$ is negative.<br/>Keeping the above points in mind the answer to this question is as under.<br/>(A) $\rightarrow(\mathrm{p}, \mathrm{r}, \mathrm{t})$<br/>(B) $\rightarrow(\mathrm{p}, \mathrm{r})$<br/>(C) $\rightarrow$ (q, s)<br/>(D) $\rightarrow(\mathrm{r}, \mathrm{t})$<br/>Analysis of Question<br/>(i) Calculation wise, question is slightly lengthy. Otherwise question is theory based and simple.<br/>(ii) In process $D A$,<br/>$\begin{array}{rlrl} & & p_A V_A & =p_D V_D \\ \therefore & & T_A & =T_D \\ \text { or } & \Delta U & =0\end{array}$<br/>Further, volume of gas is decreasing. Therefore, work is done on the gas or work done by gas is negative. Therefore, $Q$ is negative or heat is lost.<br/>(iii) This question covers almost all the concepts of first law of thermodynamics.</div> | MarksBatch2_P2.db |
595 | one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy | one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy-19620 | <div class="question">One mole of a monatomic ideal gas is taken through a cycle $A B C D A$ as shown in the $p$-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5ZEJlNpluXbfGVD34o37DgTSv2X9t_fYuXv-lWGpHTg.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BtZmk11u2LxVWqhXGJSwF68nRVZSvKCI2ONt9M52A6I.original.fullsize.png"/><br/></div> | ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,q,r,t, (B) q,r, (C) q,s, (D) r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,t, (B) p,r, (C) q,r, (D) s<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"><br/>(A) p,s, (B) q,r, (C) q,s, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,r,t, (B) p,q,r, (C) q,r, (D) s</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) p,q,t, (B) p,r, (C) q,r, (D) s<br/></span> </div> | <div class="solution">Internal energy $\propto T \propto p V$<br/>This is because<br/>$$<br/>U=\frac{n f}{2} R T=\frac{f}{2} p V<br/>$$<br/>Here, $n=$ number of moles<br/>$f=$ degree of freedom<br/>If the product $p V$ increases, then internal energy will increase and if product decreases, the internal energy will decrease.<br/>Further, work is done on the gas, if volume of gas decreases. For heat exchange.<br/>$$<br/>Q=W+\Delta U<br/>$$<br/>Work done is area under $p-V$ graph. If volume increases work done by gas is positive and if volume decreases work done by gas is negative. Further $\Delta U$ is positive if product of $p V$ is increasing and $N U$ is negative, if product of $p V$ is decreasing. If heat is taken by the gas $Q$ is positive and if heat is lost by the gas $Q$ is negative.<br/>Keeping the above points in mind the answer to this question is as under.<br/>(A) $\rightarrow(\mathrm{p}, \mathrm{r}, \mathrm{t})$<br/>(B) $\rightarrow(\mathrm{p}, \mathrm{r})$<br/>(C) $\rightarrow$ (q, s)<br/>(D) $\rightarrow(\mathrm{r}, \mathrm{t})$<br/>Analysis of Question<br/>(i) Calculation wise, question is slightly lengthy. Otherwise question is theory based and simple.<br/>(ii) In process $D A$,<br/>$\begin{array}{rlrl} & & p_A V_A & =p_D V_D \\ \therefore & & T_A & =T_D \\ \text { or } & \Delta U & =0\end{array}$<br/>Further, volume of gas is decreasing. Therefore, work is done on the gas or work done by gas is negative. Therefore, $Q$ is negative or heat is lost.<br/>(iii) This question covers almost all the concepts of first law of thermodynamics.</div> | MarksBatch2_P2.db |
596 | one-mole-of-an-ideal-gas-in-initial-state-a-undergoes-a-cyclic-process-a-bc-a-as-shown-in-the-figure-its-pressure-at-a-is-p-0-choose-the-correct-optio | one-mole-of-an-ideal-gas-in-initial-state-a-undergoes-a-cyclic-process-a-bc-a-as-shown-in-the-figure-its-pressure-at-a-is-p-0-choose-the-correct-optio-86909 | <div class="question">One mole of an ideal gas in initial state A undergoes a cyclic process $A B C A$, as shown in the figure. Its pressure at $A$ is $p_0$. Choose the correct option(s) from the following<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YOIwtV0RRE5ULb6-11MzP9TjZOlhttuirHvaljEJBUY.original.fullsize.png"/><br/></div> | ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)'] | <ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>Internal energies at $A$ and $B$ are the same<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="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Work done by the gas in process $A B$ is $p_0 V_0 \ln 4$<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"><br/>Pressure at $C$ is $\frac{p_0}{4}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Temperature at $C$ is $\frac{T_0}{4}$</span> </li> </ul> | <div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>Internal energies at $A$ and $B$ are the same<br/>, <br/>Work done by the gas in process $A B$ is $p_0 V_0 \ln 4$<br/></span> </div> | <div class="solution">$$<br/>\begin{aligned}<br/>& T_A=T_B \quad \therefore U_A=U_B \\<br/>& W_{A B}=(1)(R) T_0 \ln \left(\frac{V_f}{V_i}\right) \\<br/>& =R T_0 \ln \left(\frac{4 V_0}{V_0}\right) \\<br/>& =p_0 V_0 \ln (4)<br/>\end{aligned}<br/>$$<br/>Information regarding $p$ and $T$ at $C$ can not be obtained from the given graph. Unless it is mentioned that line $B C$ passes through origin or not.<br/>Hence, the correct options are (a) and (b).</div> | MarksBatch2_P2.db |
597 | one-mole-of-an-ideal-gas-is-taken-from-a-to-b-along-two-paths-denoted-by-the-solid-and-the-dashed-lines-as-shown-in-the-graph-below-if-the-work-done-a | one-mole-of-an-ideal-gas-is-taken-from-a-to-b-along-two-paths-denoted-by-the-solid-and-the-dashed-lines-as-shown-in-the-graph-below-if-the-work-done-a-36079 | <div class="question">One mole of an ideal gas is taken from $a$ to $b$ along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is $W_s$ and that along the dotted line path is $W_d$, then the integer closest to the ratio $\frac{W_d}{W_s}$ is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/v5OJsPNE5bM9jw0nhHQppgggg6_GeY4sEcPU19pQ4wg.original.fullsize.png"/><br/></div> | ['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)'] | None | <div class="correct-answer">
The correct answer is:
<span class="option-value">2</span> </div> | <div class="solution">Solid line represents reversible isothermal process.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/sH7FLf3CnNk6SLizs2CpYGsL1ts_CnefDrRK8phgEt0.original.fullsize.png"/><br/><br/>So, work $W_s=-4 \times 0.5 \ln \left(\frac{5.5}{0.5}\right)=-2 \ln 11 \mathrm{~L}-\mathrm{atm}$<br/>Dotted line represents irreversible process<br/>So, work<br/>$$<br/>W_d=-\left\{4 \times 1.5+1 \times 1+\frac{2}{3} \times 2.5\right\}=-\left\{6+1+\frac{5}{3}\right\} \mathrm{L}-\mathrm{atm}=-\frac{26}{3} \mathrm{~L}-\mathrm{atm}<br/>$$<br/>So $\quad \frac{W_d}{W_s}=\frac{26}{3 \times 2 \ln 11} \approx 2$<br/>Energetics<br/>Conceptual<br/>III</div> | MarksBatch2_P2.db |
598 | oogamous-type-of-sexual-reproduction-is-found-in | oogamous-type-of-sexual-reproduction-is-found-in-28796 | <div class="question">Oogamous type of sexual reproduction is found in</div> | ['Biology', 'Plant Kingdom', 'NEET'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><em>Chlamydomonas</em></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><em>Volvox</em></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><em>Spirogyra</em></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Both (A) and (B)</span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><em>Volvox</em></span> </div> | <div class="solution">Oogamy is a form of anisogamy (heterogamy) in which the female gamete (e.g. egg cell) is significantly larger than the male gamete and is non-motile. Oogamy is the most advanced form of reproduction, with the egg cell or oogonium retained and fertilized on the parent plant.</div> | MarksBatch2_P2.db |
599 | oxidation-states-of-the-metal-in-the-minerals-haematite-and-magnetite-respectively-are-2 | oxidation-states-of-the-metal-in-the-minerals-haematite-and-magnetite-respectively-are-2-13202 | <div class="question">Oxidation states of the metal in the minerals haematite and magnetite, respectively, are</div> | ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>II, III in haematite and III in magnetite<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>II, III in haematite and II in magnetite<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>II in haematite and II, III in magnetite<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>III in haematite and II, III in magnetite</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>III in haematite and II, III in magnetite</span> </div> | <div class="solution">Haematite is $\mathrm{Fe}_2 \mathrm{O}_3$, in which oxidation number of iron is III. Magnetite is $\mathrm{Fe}_3 \mathrm{O}_4$ which is infact a mixed oxide (FeO $\cdot \mathrm{Fe}_2 \mathrm{O}_3$ ) hence, iron is present in both II and III oxidation state.</div> | MarksBatch2_P2.db |
600 | paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-1 | paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-1-21115 | <div class="question"><strong>Paragraph:</strong><br/>A carbonyl compound $P$, which gives positive iodoform test, undergoes reaction with MeMgBr followed by dehydration to give an olefin $Q$. Ozonolysis of $Q$ leads to a dicarbonyl compound $R$, which undergoes intramolecular aldol reaction to give predominantly $S$.<br/><br/>$$<br/>P \underset{\text { (ii) } \mathrm{H}^{+}, \mathrm{H}_2 \mathrm{O}}{\stackrel{\text { (i) } \mathrm{MeMgBr}}{\longrightarrow}} \mathrm{Q} \underset{\text { Heat }}{\stackrel{\mathrm{O}_3 / \mathrm{Zn}-\mathrm{H}_2 \mathrm{O}}{\longrightarrow}} \mathrm{P}<br/>$$<br/>(iii) $\mathrm{H}_2 \mathrm{SO}_4$ /Heat<strong>Question:</strong><br/>$$<br/>\text { The structure of the carbonyl compound } P \text {, is }<br/>$$</div> | ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)'] | <ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fDDj5QZVFCnhb-qmvfxHFAXCtadF-DLEJ6XXvXoBisE.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZC_3cEPYQINFO5Cg7nqDwPUu6gX5mXEbH9PTs8RhQ8c.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">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/zaY9cN1FZlaG5fXsOxFu9ViyfTWdk1k2YGaMr9bs6-I.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OZW0o_xQ6wXMVp-vyDqjXvJW_mxLO6sacRFZxJA9IZM.original.fullsize.png"/><br/></span> </li> </ul> | <div class="correct-answer">
The correct answer is:
<span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZC_3cEPYQINFO5Cg7nqDwPUu6gX5mXEbH9PTs8RhQ8c.original.fullsize.png"/><br/></span> </div> | <div class="solution">No Solution Available</div> | MarksBatch2_P2.db |
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