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column-ii-shows-five-systems-in-which-two-objects-are-labelled-as-x-and-y-also-in-each-case-a-point-p-is-shown-columni-gives-some-statements-about-x-a

column-ii-shows-five-systems-in-which-two-objects-are-labelled-as-x-and-y-also-in-each-case-a-point-p-is-shown-columni-gives-some-statements-about-x-a-53317

<div class="question">Column II shows five systems in which two objects are labelled as $X$ and $Y$. Also in each case a point $P$ is shown. Column-I gives some statements about $X$ and/or $Y$. Match these statements to the appropriate system(s) from Column-II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Nmjp-6aDDAiZbMGB3BfCFqOpbccuBbUIpfhDHt9E9zY.original.fullsize.png"/><br/></div>

['Physics', 'Work Power Energy', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>(A) p,t, (B) q,s,t, (C) p,r,t, (D) r,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">B</span> <span class="option-data"><br/>(A) q,t, (B) q,s, (C) p,r, (D) r,q,t<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(A) p,t, (B) q,s,t, (C) r,t, (D) r,q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,r, (B) q,s, (C) p,r,t, (D) r,q</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>(A) p,t, (B) q,s,t, (C) p,r,t, (D) r,q,t<br/></span> </div>

<div class="solution">$(\mathrm{A}) \rightarrow(\mathrm{p}, \mathrm{t}) \rightarrow$ Net force on $Y$ is ero.<br/>(B) $\rightarrow(\mathrm{q}, \mathrm{s}, \mathrm{t}) \rightarrow \ln (\mathrm{s})$ and $(\mathrm{t})$ gravitational potential energy of $Y$ is decreasing and that of $X$ is increasing.<br/>(C) $\rightarrow(\mathrm{p}, \mathrm{r}, \mathrm{t}) \rightarrow \operatorname{In}(\mathrm{t}), \mathrm{Y}$ is moving with constant speed.<br/>(D) $\rightarrow$ (q) $\rightarrow$ No explanation is required</div>

MarksBatch1_P2.db

columnii-gives-certain-systems-undergoing-a-process-columni-suggests-changes-in-some-of-the-parameters-related-to-the-system-match-the-statements-in-c

columnii-gives-certain-systems-undergoing-a-process-columni-suggests-changes-in-some-of-the-parameters-related-to-the-system-match-the-statements-in-c-54726

<div class="question">Column-II gives certain systems undergoing a process. Column-I suggests changes in some of the parameters related to the system. Match the statements in Column-I to the appropriate process (es) from Column-II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/G_QqC-7_pg_NzGDr82H8KAsvaf0LgQidVsnjc4WmM3E.original.fullsize.png"/><br/></div>

['Physics', 'Work Power Energy', '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) r, (C) t, (D) r<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p,q,s,t, (B) q, (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">C</span> <span class="option-data"><br/>(A) p,s, (B) r, (C) s, (D) r<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) p,q,s,t, (B) q, (C) t, (D) s</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>(A) p,q,s,t, (B) q, (C) s, (D) s<br/></span> </div>

<div class="solution">No Solution Available</div>

MarksBatch1_P2.db

consider-a-branch-of-the-hyperbola-x-2-2-y-2-2-2-x-4-2-y-6-0-with-vertex-at-the-point-a-let-b-be-one-of-the-end-points-of-its-latus-rectum-if-c-is-the

consider-a-branch-of-the-hyperbola-x-2-2-y-2-2-2-x-4-2-y-6-0-with-vertex-at-the-point-a-let-b-be-one-of-the-end-points-of-its-latus-rectum-if-c-is-the-36102

<div class="question">Consider a branch of the hyperbola $x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the $\triangle A B C$ is</div>

['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1-\sqrt{\frac{2}{3}}$ sq unit<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{\frac{3}{2}}-1$ sq unit<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/>$1+\sqrt{\frac{2}{3}}$ sq unit<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\sqrt{\frac{3}{2}}+1$ sq unit</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\sqrt{\frac{3}{2}}-1$ sq unit<br/></span> </div>

<div class="solution">The given equation can be rewritten as $\frac{(x-\sqrt{2})^2}{1}-\frac{(y+\sqrt{2})^2}{2}=1$ for $A(x, y)$,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/kJiDAgQz4cB8Ql-009_N5DDom8WjagEpxMaE9lZAfMg.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; e=\sqrt{1+\frac{2}{4}}=\sqrt{\frac{3}{2}} \\<br/>&amp; \therefore x-\sqrt{2}=2 \Rightarrow x=2+\sqrt{2} \\<br/>&amp; \text { For } C(x, y), x-\sqrt{2}=a e=\sqrt{6} \\<br/>&amp; \Rightarrow \quad x=\sqrt{6}+\sqrt{2} \\<br/>&amp; \text { Now, } A C=\sqrt{6}+\sqrt{2}-2-\sqrt{2}=\sqrt{6}-2 \\<br/>&amp; \qquad B C=\frac{b^2}{a}=\frac{2}{2}=1 \\<br/>&amp; \text { Area of } \Delta A B C=\frac{1}{2} \times(\sqrt{6}-2) \times B C \\<br/>&amp; =\frac{1}{2} \times(\sqrt{6}-2) \times 1=\sqrt{\frac{3}{2}}-1<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

consider-a-disc-rotating-in-the-horizontal-plane-with-a-constant-angular-speed-about-its-centre-o-the-disc-has-a-shaded-region-on-one-side-of-the-diam

consider-a-disc-rotating-in-the-horizontal-plane-with-a-constant-angular-speed-about-its-centre-o-the-disc-has-a-shaded-region-on-one-side-of-the-diam-85336

<div class="question">Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $\mathrm{O}$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$. The velocity of projection is in the $y$ - $z$ plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed $1 / 8$ rotation, (ii) their range is less than half the disc radius, and (iii) $\omega$ remains constant throughout. Then <br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4Z-V_5_KytE90rgAnNsqUdyYITXRjOw-3D1XemNP20Y.original.fullsize.png"/></div>

['Physics', 'Rotational Motion', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value">Both $P$ and $Q$ land in the unshaded region.</span> </div>

<div class="solution">For $\frac{1}{8}$ of rotation of disc, <br/> <br/>$\begin{aligned} <br/> <br/>t=\frac{1}{8} \times \frac{2 \pi}{\omega}=\frac{\pi}{4 \omega} &amp; \\ <br/> <br/>x \text {-coordinate of } P &amp;=\omega R T \\ <br/> <br/>&amp;=\frac{\pi R}{4}&gt;R \sin 45^{\circ} <br/> <br/>\end{aligned}$ <br/> <br/>To reach the unshaded part, particle$P$ needs to travel horizontal range $&gt;$ $R \sin 45^{\circ} \simeq 0.7 \mathrm{R}$ <br/> <br/>But its range is less than $\mathrm{R} / 2$ so it will land on shaded part. $Q$ is near the origin, its velocity will be nearly along QR so it will land in unshaded part. <br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/nNaoB62ZECYr2P82saF5y7BuBNr7sgHYC9XoG5Dyi1o.original.fullsize.png"/></div>

MarksBatch1_P2.db

consider-a-neutral-conducting-sphere-a-positive-point-charge-is-placed-outside-the-sphere-the-net-charge-on-the-sphere-is-then

consider-a-neutral-conducting-sphere-a-positive-point-charge-is-placed-outside-the-sphere-the-net-charge-on-the-sphere-is-then-92668

<div class="question">Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then</div>

['Physics', 'Electrostatics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

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

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

<div class="solution">$$<br/>\text { Charge will be induced in the conducting sphere, but net charge on it will be zero. }<br/>$$<br/><br/>$$<br/>\therefore \text { Option (d) is correct. }<br/>$$</div>

MarksBatch1_P2.db

consider-a-reaction-a-g-b-h-products-when-concentration-of-both-the-reactants-g-and-h-is-doubled-the-rate-increases-by-eight-times-however-when-concen-1

consider-a-reaction-a-g-b-h-products-when-concentration-of-both-the-reactants-g-and-h-is-doubled-the-rate-increases-by-eight-times-however-when-concen-1-44566

<div class="question">Consider a reaction $a G+b H \rightarrow$ products. When concentration of both the reactants $G$ and $H$ is doubled, the rate increases by eight times. However, when concentration of $G$ is doubled keeping the concentration of $H$ fixed, the rate is doubled. The overall order of the reaction is</div>

['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>0<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>1<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>2<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>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 answer is: <span class="option-value"><br/>3</span> </div>

<div class="solution">$a G+b H \longrightarrow$ Product<br/>Suppose order of reaction $=n$<br/>When concentration of both $G$ and $H$ is double than rate increase by eight times.<br/>$$<br/>\begin{aligned}<br/>\text { rate } &amp; =k(\text { reactants })^n \\<br/>(2)^8 &amp; =k(2)^n \\<br/>(2)^3 &amp; =k(2)^n \\<br/>n &amp; =3<br/>\end{aligned}<br/>$$<br/>when concentration of $G$ is doubled than concentration $H$ is fixed, the rate is double.<br/>than $\quad$ rate $\propto[G]^1[H]^2$</div>

MarksBatch1_P2.db

consider-a-system-of-three-charges-3-q-3-q-and-3-2-q-placed-at-points-a-b-and-c-respectively-as-shown-in-the-figure-take-o-to-be-the-centre-of-the-cir

consider-a-system-of-three-charges-3-q-3-q-and-3-2-q-placed-at-points-a-b-and-c-respectively-as-shown-in-the-figure-take-o-to-be-the-centre-of-the-cir-81349

<div class="question">Consider a system of three charges $\frac{q}{3}, \frac{q}{3}$ and $-\frac{2 q}{3}$ placed at points $A, B$ and $C$, respectively, as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and angle $C A B=60^{\circ}$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/h78g_hb3-iGHvg7H2ZK6jg55DjbqxES3XJLSjyLzlNQ.original.fullsize.png"/><br/></div>

['Physics', 'Electrostatics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>The electric field at point $O$ is $\frac{q}{8 \pi \varepsilon_0 R^2}$ directed along the negative $x$-axis<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>The potential energy of the system is zero<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>The magnitude of the force between the charges at $C$ and $B$ is $\frac{q^2}{54 \pi \varepsilon_0 R^2}$<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/>The potential at point $O$ is $\frac{q}{12 \pi \varepsilon_0 R}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>The magnitude of the force between the charges at $C$ and $B$ is $\frac{q^2}{54 \pi \varepsilon_0 R^2}$<br/></span> </div>

<div class="solution">Distance $B C=A B \sin 60^{\circ}=(2 R) \frac{\sqrt{3}}{2}=\sqrt{3} R$<br/>$\therefore \quad\left|F_{B C}\right|=\frac{1}{4 \pi \varepsilon_0} \frac{(q / 3)(2 q / 3)}{(\sqrt{3} R)^2}=\frac{q^2}{54 \pi \varepsilon_0 R^2}$<br/>$\therefore$ correct option is (c).</div>

MarksBatch1_P2.db

consider-a-thin-spherical-shell-of-radius-r-with-centre-at-the-origin-carrying-uniform-positive-surface-charge-density-the-variation-of-the-magnitude-

consider-a-thin-spherical-shell-of-radius-r-with-centre-at-the-origin-carrying-uniform-positive-surface-charge-density-the-variation-of-the-magnitude-45909

<div class="question">Consider a thin spherical shell of radius $R$ with centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $|\vec{E}(r)|$ and the electric potential $V(r)$ with the distance $r$ from the centre, is best represented by which graph?</div>

['Physics', 'Electrostatics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

<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/E3m10uZEbKDsryfNH6apC8lG9tonBECzbJDocju9JQk.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/ZUWR2PHmFvOetGpbNJ5XeSiX4S1xKyQltlV82kwjSIw.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/etr6C3Rn7FUxZjqQnTXzGHWVwuP02jk7HEZLe3ddo1A.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ikjTnp-ZIgcQMIDCC4C-yNMcYCNlesQGvAePbfXQpaM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ikjTnp-ZIgcQMIDCC4C-yNMcYCNlesQGvAePbfXQpaM.original.fullsize.png"/></span> </div>

<div class="solution">For a thin uniformly positive charged spherical shell <br/> <br/>(i) Inside the shell at any point <br/> <br/>$E=O$ and $V=\frac{1}{4 \pi \in_{0}} \quad \frac{q}{R}=$ constt. <br/> <br/>where $q=$ charge on sphere <br/> <br/>$R=$ Radius of sphere <br/> <br/>(ii) Outside the shell at any point at any distance $r$ from <br/> <br/>the centre $E \propto \frac{1}{r^{2}}$ and $V \propto \frac{1}{r}$</div>

MarksBatch1_P2.db

consider-a-thin-square-sheet-of-side-l-and-thickness-t-made-of-a-material-of-resistivity-the-resistance-between-two-opposite-faces-shown-by-the-shaded-1

consider-a-thin-square-sheet-of-side-l-and-thickness-t-made-of-a-material-of-resistivity-the-resistance-between-two-opposite-faces-shown-by-the-shaded-1-30679

<div class="question">Consider a thin square sheet of side $L$ and thickness $t$, made of a material of resistivity $\rho$. The resistance between two opposite faces, shown by the shaded areas in the figure is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jQWT8TUcKYNLj76kfHeUfDLbThIKMquc23PlGL0NDCw.original.fullsize.png"/><br/></div>

['Physics', 'Current Electricity', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>directly proportional to $L$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>directly proportional to $t$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>independent of $L$<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/>independent of $t$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>independent of $L$<br/></span> </div>

<div class="solution">$R=\frac{\rho(L)}{A}=\frac{\rho L}{t L}=\frac{\rho}{t}$ ie, $R$ is independent of $L$. Hence the correct option is (c).</div>

MarksBatch1_P2.db

consider-a-thin-square-sheet-of-side-l-and-thickness-t-made-of-a-material-of-resistivity-the-resistance-between-two-opposite-faces-shown-by-the-shaded

consider-a-thin-square-sheet-of-side-l-and-thickness-t-made-of-a-material-of-resistivity-the-resistance-between-two-opposite-faces-shown-by-the-shaded-33624

<div class="question">Consider a thin square sheet of side $L$ and thickness $t$, made of a material of resistivity $\rho$. The resistance between two opposite faces, shown by the shaded areas in the figure is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jQWT8TUcKYNLj76kfHeUfDLbThIKMquc23PlGL0NDCw.original.fullsize.png"/><br/></div>

['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>directly proportional to $L$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>directly proportional to $t$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>independent of $L$<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/>independent of $t$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>independent of $L$<br/></span> </div>

<div class="solution">$R=\frac{\rho(L)}{A}=\frac{\rho L}{t L}=\frac{\rho}{t}$ ie, $R$ is independent of $L$. Hence the correct option is (c).</div>

MarksBatch1_P2.db

consider-a-titration-of-potassium-dichromate-solution-with-acidified-mohrs-salt-solution-using-diphenylamine-as-indicator-the-number-of-moles-of-mohrs

consider-a-titration-of-potassium-dichromate-solution-with-acidified-mohrs-salt-solution-using-diphenylamine-as-indicator-the-number-of-moles-of-mohrs-29351

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>6</span> </div>

<div class="solution">Mole ratio is reverse of $x$-factor ratio.</div>

MarksBatch1_P2.db

consider-a-triangle-a-bc-and-let-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-vertices-a-b-and-c-respectively-suppose-a-6-b-10-and-the-area-o

consider-a-triangle-a-bc-and-let-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-vertices-a-b-and-c-respectively-suppose-a-6-b-10-and-the-area-o-24899

<div class="question">Consider a triangle $A B C$ and let $a, b$ and $c$ denote the lengths of the sides opposite to vertices $A, B$ and $C$ respectively. Suppose $a=6, b=10$ and the area of the triangle is $15 \sqrt{3}$. If $\angle A C B$ is obtuse and if $r$ denotes the radius of the incircle of the triangle, then $r^2$ is equal to</div>

['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']

None

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

<div class="solution">$\sin C=\frac{\sqrt{3}}{2}$ and $C$ is given to be obtuse<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad C=\frac{2 \pi}{3}=\sqrt{a^2+b^2-2 a b \cos C} \\<br/>&amp; =\sqrt{6^2+10^2-2 \times 6 \times 10 \times \cos \frac{2 \pi}{3}}=14 \\<br/>&amp; \therefore \quad r=\frac{\Delta}{s} \Rightarrow r^2=\frac{225 \times 3}{\left(\frac{6+10+14}{2}\right)^2}=3<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

consider-all-possible-permutations-of-the-letters-of-the-word-endeanoel-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in--1

consider-all-possible-permutations-of-the-letters-of-the-word-endeanoel-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-1-15009

<div class="question">Consider all possible permutations of the letters of the word ENDEANOEL. 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/1Oj7xgNtMlSTH_bJ2EXv-IV_L6I5OltoIjA2nwL0Yr0.original.fullsize.png"/><br/></div>

['Mathematics', 'Permutation Combination', 'JEE Main']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>(A) p, (B) s, (C) q, (D) q<br/></span> </div>

<div class="solution">(A) If ENDEA is fixed word, then assume this as a single letter.<br/>Total number of letters $=5$ and total number of arrangements $=5$ ! .<br/>(B) If $E$ is at first and last places, then total number of permutations<br/>$$<br/>\frac{7 !}{2 !}=21 \times 5 !<br/>$$<br/>(C) If $\mathrm{D}, \mathrm{L}, \mathrm{N}$ are not in last five positions<br/>$$<br/>\leftarrow \mathrm{D}, \mathrm{L}, \mathrm{N}, \mathrm{N} \rightarrow \leftarrow \text { E, E, E, A, O } \rightarrow<br/>$$<br/>Total number of permutations $=\frac{4 !}{2 !} \times \frac{5 !}{3 !}=2 \times 5$ ! .<br/>(D) Total number of odd positions $=5$<br/>Permutations of AEEEO are $\frac{5 !}{3 !}$<br/>Total number of even positions $=4$<br/>Number of permutations of $\mathrm{N}, \mathrm{N}, \mathrm{D}, \mathrm{L}=\frac{4 !}{2 !}$<br/>Hence, total number of permutations $=\frac{5 !}{3 !} \times \frac{4 !}{2 !}=2 \times 5 !$.</div>

MarksBatch1_P2.db

consider-all-possible-permutations-of-the-letters-of-the-word-endeanoel-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-

consider-all-possible-permutations-of-the-letters-of-the-word-endeanoel-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-37165

<div class="question">Consider all possible permutations of the letters of the word ENDEANOEL. 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/1Oj7xgNtMlSTH_bJ2EXv-IV_L6I5OltoIjA2nwL0Yr0.original.fullsize.png"/><br/></div>

['Mathematics', 'Permutation Combination', '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, (C) s, (D) p<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(A) p, (B) s, (C) q, (D) q<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, (B) r, (C) s, (D) q<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(A) r, (B) s, (C) r, (D) p</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>(A) p, (B) s, (C) q, (D) q<br/></span> </div>

<div class="solution">(A) If ENDEA is fixed word, then assume this as a single letter.<br/>Total number of letters $=5$ and total number of arrangements $=5$ ! .<br/>(B) If $E$ is at first and last places, then total number of permutations<br/>$$<br/>\frac{7 !}{2 !}=21 \times 5 !<br/>$$<br/>(C) If $\mathrm{D}, \mathrm{L}, \mathrm{N}$ are not in last five positions<br/>$$<br/>\leftarrow \mathrm{D}, \mathrm{L}, \mathrm{N}, \mathrm{N} \rightarrow \leftarrow \text { E, E, E, A, O } \rightarrow<br/>$$<br/>Total number of permutations $=\frac{4 !}{2 !} \times \frac{5 !}{3 !}=2 \times 5$ ! .<br/>(D) Total number of odd positions $=5$<br/>Permutations of AEEEO are $\frac{5 !}{3 !}$<br/>Total number of even positions $=4$<br/>Number of permutations of $\mathrm{N}, \mathrm{N}, \mathrm{D}, \mathrm{L}=\frac{4 !}{2 !}$<br/>Hence, total number of permutations $=\frac{5 !}{3 !} \times \frac{4 !}{2 !}=2 \times 5 !$.</div>

MarksBatch1_P2.db

consider-an-electric-field-e-e-0-x-where-e-0-is-a-constant-the-flux-through-the-shaded-area-as-shown-in-the-figure-due-to-this-field-is

consider-an-electric-field-e-e-0-x-where-e-0-is-a-constant-the-flux-through-the-shaded-area-as-shown-in-the-figure-due-to-this-field-is-28226

<div class="question">Consider an electric field $\mathbf{E}=E_0 \hat{\mathbf{x}}$ where $\bar{E}_0$ is a constant. The flux through the shaded area ( as shown in the figure) due to this field is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/E8rJdxbxJvoMYCbK6XQXC69Eq6GFBN_dcj0grzJMvrU.original.fullsize.png"/><br/></div>

['Physics', 'Electrostatics', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$2 E_0 a^2$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{2} E_0 a^2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$E_0 a^2$<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/>$\frac{E_0 a^2}{\sqrt{2}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$E_0 a^2$<br/></span> </div>

<div class="solution">Electric flux, $\mathbf{E} \cdot \mathbf{S}$, or $\phi=E S \cos \theta$ Here, $\theta$ is the angle between $\mathbf{E}$ and $\mathbf{S}$. In this question $\theta=45^{\circ}$, because $\mathbf{S}$ is perpendicular to the surface.<br/>$$<br/>\begin{gathered}<br/>E=E_0 \\<br/>\Rightarrow \quad S=(\sqrt{2 a})(a)=\sqrt{2} a^2 \\<br/>\therefore \phi=\left(E_0\right)\left(\sqrt{2} a^2\right) \cos 45^{\circ}=E_0 a^2<br/>\end{gathered}<br/>$$<br/>$\therefore$ Correct option is (c).<br/>Analysis of Question<br/>(i) Question is moderately tough.<br/>(ii) The given shaded area is a rectangle not a square. One side of this rectangle is $a$ and other side is $\sqrt{2} a$.<br/>(iii) Electric field is uniform, whose magnitude is $E_0$ and direction is positive $x$. In uniform electric field we can use, $\phi=E S \cos \theta$</div>

MarksBatch1_P2.db

consider-an-expanding-sphere-of-instantaneous-radius-r-whose-total-mass-remains-constant-the-expansion-is-such-that-the-instantaneous-density-remains--1

consider-an-expanding-sphere-of-instantaneous-radius-r-whose-total-mass-remains-constant-the-expansion-is-such-that-the-instantaneous-density-remains-1-43548

<div class="question">Consider an expanding sphere of instantaneous radius <math> <mi>R</mi> </math> whose total mass remains constant. The expansion is such that the instantaneous density <math><mi>ρ</mi></math> remains uniform throughout the volume. The rate of fractional change in density <math><mfenced separators="|"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ρ</mi></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>ρ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></math> is constant. The velocity <math><mi>v</mi></math> of any point on the surface of the expanding sphere is proportional to</div>

['Physics', 'Units and Dimensions', 'JEE Advanced', 'JEE Advanced 2017 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>R</mi></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mi>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><msup><mrow><mi>R</mi></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></math></span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><math><mi>R</mi></math></span> </div>

<div class="solution"><p>Density of sphere is <math><mi>ρ</mi><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>v</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi>m</mi></mrow><mrow><mn>4</mn><mi>π</mi><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac></math></p><p><math><mo>⇒</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ρ</mi></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>ρ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi> </mi><mo>-</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mi>R</mi></mrow></mfrac><mi> </mi><mfrac><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math></p><p>Since  <math><mo>⇒</mo><mi> </mi><mi> </mi><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ρ</mi></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>ρ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math>  is constant</p><p><math><mo>∴</mo><mi> </mi><mi> </mi><mi> </mi><mfrac><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi> </mi><mo>∝</mo><mi>R</mi></math></p></div>

MarksBatch1_P2.db

consider-l-1-2-x-3-y-p-3-0-l-2-2-x-3-y-p-3-0-where-p-is-a-real-number-and-c-x-2-y-2-6-x-10-y-30-0-statement-1-if-line-l-1-is-a-chord-of-circle-c-then-

consider-l-1-2-x-3-y-p-3-0-l-2-2-x-3-y-p-3-0-where-p-is-a-real-number-and-c-x-2-y-2-6-x-10-y-30-0-statement-1-if-line-l-1-is-a-chord-of-circle-c-then-73070

<div class="question">Consider $L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0$ where $p$ is a real number and $C: x^2+y^2+6 x-10 y+30=0$.<br/>Statement 1 If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$.<br/>Statement 2 If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$.</div>

['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 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 1 is false, Statement 2 is true</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>Statement 1 is true, Statement 2 is false.<br/></span> </div>

<div class="solution">Equation of circle $C$ is<br/>$$<br/>\begin{aligned}<br/>(x+3)^2+(y-5)^2 &amp; =9+25-30=4 \\<br/>\Rightarrow \quad \quad(x+3)^2+(y-5)^2 &amp; =2^2 \\<br/>\text { Centre } &amp; =(3,-5)<br/>\end{aligned}<br/>$$<br/><br/>If $L_1$ is diameter, then $2(3)+3(-5)+p-3=0 \Rightarrow p=12$<br/>$\therefore \quad L_1$ is $2 x+3 y+9=0$<br/>and<br/>$L_2$ is $2 x+3 y+15=0$<br/>Distance of centre of circle from $L_2=\left|\frac{2(3)+3(-5)+15}{\sqrt{2^2+3^2}}\right|=\frac{6}{\sqrt{13}} &lt; 2 \quad$ [radius of circle]<br/>$\therefore L_2$ is a chord of circle $C$.<br/>Statement 2 is false.</div>

MarksBatch1_P2.db

consider-the-following-cell-reaction-at-2-fe-s-o-2-g-4-h-a-q-2-fe-2-a-q-2-h-2-o-l-e-167-v-fe-2-1-0-3-m-p-o-2-01-atm-an-d-mathrmph3-t-h-ece-llp-o-t-e-n

consider-the-following-cell-reaction-at-2-fe-s-o-2-g-4-h-a-q-2-fe-2-a-q-2-h-2-o-l-e-167-v-fe-2-1-0-3-m-p-o-2-01-atm-an-d-mathrmph3-t-h-ece-llp-o-t-e-n-35542

<div class="question">Consider the following cell reaction,<br/>$$<br/>\begin{array}{cc} <br/>&amp; 2 \mathrm{Fe}(\mathrm{s})+\mathrm{O}_2(g)+4 \mathrm{H}^{+}(a q) \longrightarrow \\<br/>&amp; 2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{H}_2 \mathrm{O}(l), E^{\circ}=1.67 \mathrm{~V} \\<br/>\text { At } &amp; {\left[\mathrm{Fe}^{2+}\right]=10^{-3} \mathrm{M}, \mathrm{P}\left(\mathrm{O}_2\right)=0.1 \mathrm{~atm}}<br/>\end{array}<br/>$$<br/>and $\mathrm{pH}=3$, the cell potential at $25^{\circ} \mathrm{C}$ is</div>

['Chemistry', 'Electrochemistry', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1.47 \mathrm{~V}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$1.77 \mathrm{~V}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$1.87 \mathrm{~V}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$1.57 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$1.57 \mathrm{~V}$</span> </div>

<div class="solution">The half reactions are $\mathrm{Fe}(s) \longrightarrow \mathrm{Fe}^{2+}(\mathrm{aq})+2 e^{-} \times 2$<br/>$$<br/>\begin{array}{r}<br/>\mathrm{O}_2(g)+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O}+2 \mathrm{Fe}(s) \\<br/>+\mathrm{O}_2(g)+4 \mathrm{H}^{+} \longrightarrow 2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{H}{ }_2 \mathrm{O}(l) \\<br/>E=E^{\circ}-\frac{0.059}{4} \log \frac{\left(10^{-3}\right)^2}{\left(10^{-3}\right)^4(0.1)}=1.57 \mathrm{~V}<br/>\end{array}<br/>$$</div>

MarksBatch1_P2.db

consider-the-following-four-statements-whether-they-are-correct-of-wrong-i-the-sporophyte-in-liverworts-is-more-elaborate-than-that-in-mosses-ii-salvi

consider-the-following-four-statements-whether-they-are-correct-of-wrong-i-the-sporophyte-in-liverworts-is-more-elaborate-than-that-in-mosses-ii-salvi-24522

<div class="question">Consider the following four statements whether they are correct of wrong.<br/>I. The sporophyte in liverworts is more elaborate than that in mosses<br/>II. Salvinia is heterosporous<br/>III. The life-cycle in all seed-bearing plants is diplontic<br/>IV. In Pinus male and female cones are borne on different tress</div>

['Biology', 'Plant Kingdom', 'NEET', 'NEET 2011 (Mains)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value">Statements (I) and (IV)</span> </div>

<div class="solution">The sporophyte is more developed in mosses rather than liverwort. The simplest sporophyte is found in Riccia, a liverwort, where it is represented by capsule only. The most complex sporophyte is found in Funaria, a moss. However, the most advanced sporophyte is found in Anthoceros (a hornwort) due to the presence of intercalary meristem.<br/>Pinus (a gymnosperm) is monoecious plant in which male and female cones are borne on different branches of the same plant.</div>

MarksBatch1_P2.db

consider-the-following-linear-equations-a-x-b-y-cz-0-b-x-cy-a-z-0-c-x-a-y-b-z-0-match-the-conditionsexpressions-in-column-i-with-statements-in-column--1

consider-the-following-linear-equations-a-x-b-y-cz-0-b-x-cy-a-z-0-c-x-a-y-b-z-0-match-the-conditionsexpressions-in-column-i-with-statements-in-column-1-70278

<div class="question">Consider the following linear equations<br/>$$<br/>a x+b y+c z=0, b x+c y+a z=0, c x+a y+b z=0<br/>$$<br/>Match the conditions/expressions in Column I with statements in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/67lHKgpE4NWG3yldUeb517taVAiUb-ATGMv_QnXaoBo.original.fullsize.png"/><br/></div>

['Mathematics', 'Determinants', 'JEE Main']

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

<div class="solution">Let $\Delta=\left|\begin{array}{lll}a &amp; b &amp; c \\ b &amp; c &amp; a \\ c &amp; a &amp; b\end{array}\right|=-\frac{1}{2}(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]$<br/>(A) If $a+b+c \neq 0$ and $a^2+b^2+c^2=a b+b c+c a$<br/>$$<br/>\Rightarrow \quad \Delta=0 \text { and } a=b=c \neq 0<br/>$$<br/>The equations represents identical planes.<br/>(B) $a+b+c=0$ and $a^2+b^2+c^2 \neq a b+b c+c a$<br/>$$<br/>\Rightarrow \quad \Delta=0<br/>$$<br/>The equations have infinitely many solutions.<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; a x+b y &amp; =(a+b) z \\<br/>\Rightarrow &amp; b x+c y &amp; =(b+c) z \\<br/>\Rightarrow &amp; &amp; \left(b^2-a c\right) y &amp; =\left(b^2-a c\right) z \Rightarrow y=z \\<br/>\Rightarrow &amp; a x+b y+c y &amp; =0 \\<br/>&amp; a x &amp; =a y \Rightarrow x=y=z<br/>\end{array}<br/>$$<br/>(C) $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq a b+b c+c a$<br/>$$<br/>\Rightarrow \quad \Delta \neq 0<br/>$$<br/>The equation represent planes meeting at only one point.<br/>(D) $a+b+c=0$ and $a^2+b^2+c^2=a b+b c+c a$<br/>$$<br/>\Rightarrow \quad a=b=c=0<br/>$$<br/>The equations represent whole of the three dimensional space.</div>

MarksBatch1_P2.db

consider-the-following-linear-equations-a-x-b-y-cz-0-b-x-cy-a-z-0-c-x-a-y-b-z-0-match-the-conditionsexpressions-in-column-i-with-statements-in-column-

consider-the-following-linear-equations-a-x-b-y-cz-0-b-x-cy-a-z-0-c-x-a-y-b-z-0-match-the-conditionsexpressions-in-column-i-with-statements-in-column-92155

<div class="question">Consider the following linear equations<br/>$$<br/>a x+b y+c z=0, b x+c y+a z=0, c x+a y+b z=0<br/>$$<br/>Match the conditions/expressions in Column I with statements in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/67lHKgpE4NWG3yldUeb517taVAiUb-ATGMv_QnXaoBo.original.fullsize.png"/><br/></div>

['Mathematics', 'Determinants', '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-p, s; C-p; D-q<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>A-r, q; B-r, s; C-r, s; D-r, s<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>A-s; B-p; C-q; D-s<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>A-r; B-q; C-p; 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-r; B-q; C-p; D-s</span> </div>

<div class="solution">Let $\Delta=\left|\begin{array}{lll}a &amp; b &amp; c \\ b &amp; c &amp; a \\ c &amp; a &amp; b\end{array}\right|=-\frac{1}{2}(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]$<br/>(A) If $a+b+c \neq 0$ and $a^2+b^2+c^2=a b+b c+c a$<br/>$$<br/>\Rightarrow \quad \Delta=0 \text { and } a=b=c \neq 0<br/>$$<br/>The equations represents identical planes.<br/>(B) $a+b+c=0$ and $a^2+b^2+c^2 \neq a b+b c+c a$<br/>$$<br/>\Rightarrow \quad \Delta=0<br/>$$<br/>The equations have infinitely many solutions.<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; a x+b y &amp; =(a+b) z \\<br/>\Rightarrow &amp; b x+c y &amp; =(b+c) z \\<br/>\Rightarrow &amp; &amp; \left(b^2-a c\right) y &amp; =\left(b^2-a c\right) z \Rightarrow y=z \\<br/>\Rightarrow &amp; a x+b y+c y &amp; =0 \\<br/>&amp; a x &amp; =a y \Rightarrow x=y=z<br/>\end{array}<br/>$$<br/>(C) $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq a b+b c+c a$<br/>$$<br/>\Rightarrow \quad \Delta \neq 0<br/>$$<br/>The equation represent planes meeting at only one point.<br/>(D) $a+b+c=0$ and $a^2+b^2+c^2=a b+b c+c a$<br/>$$<br/>\Rightarrow \quad a=b=c=0<br/>$$<br/>The equations represent whole of the three dimensional space.</div>

MarksBatch1_P2.db

consider-the-lines-given-by-l-1-x-3-y-5-0-l-2-3-x-k-y-1-0-l-3-5-x-2-y-12-0-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions--1

consider-the-lines-given-by-l-1-x-3-y-5-0-l-2-3-x-k-y-1-0-l-3-5-x-2-y-12-0-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-1-44656

<div class="question">Consider the lines given by<br/>$$<br/>L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0<br/>$$<br/>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/t-8SrYw4JSCJlT9NZOgfJHOj9KnGrpZhN-Y1Vj8K5gU.original.fullsize.png"/><br/></div>

['Mathematics', 'Determinants', 'JEE Main']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>(A) s, (B) p,q, (C) r, (D) p,q,s<br/></span> </div>

<div class="solution">(A) Solving equations $L_1$ and $L_3$,<br/>$$<br/>\begin{aligned}<br/>\frac{x}{-36+10} &amp; =\frac{y}{+12-25}=\frac{1}{2-15} \\<br/>x &amp; =2, y=1<br/>\end{aligned}<br/>$$<br/>$L_1, L_2, L_3$ are concurrent, if point $(2,1)$ lies on $L_2$.<br/>$$<br/>\therefore \quad 6-k-1=0 \Rightarrow k=5<br/>$$<br/>(B) Either $L_1$ is parallel to $L_2$, or $L_3$ is parallel to $L_2$, then<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{1}{3}=\frac{3}{-k} \text { or } \frac{3}{5}=\frac{-k}{2} \\<br/>&amp; k=-9 \quad \text { or } \quad k=-\frac{6}{5}<br/>\end{aligned}<br/>$$<br/>$$<br/>\Rightarrow \quad k=-9 \text { or } k=-\frac{6}{5}<br/>$$<br/>(C) $L_1, L_2, L_3$ form a triangle, if they are not concurrent, or not parallel.<br/>$$<br/>\therefore \quad k \neq 5,-9,-\frac{6}{5} \Rightarrow k=\frac{5}{6}<br/>$$<br/>(D) $L_1, L_2, L_3$ do not form a triangle, if<br/>$$<br/>k=5,-9,-\frac{6}{5}<br/>$$</div>

MarksBatch1_P2.db

consider-the-lines-given-by-l-1-x-3-y-5-0-l-2-3-x-k-y-1-0-l-3-5-x-2-y-12-0-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-

consider-the-lines-given-by-l-1-x-3-y-5-0-l-2-3-x-k-y-1-0-l-3-5-x-2-y-12-0-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-68774

<div class="question">Consider the lines given by<br/>$$<br/>L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0<br/>$$<br/>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/t-8SrYw4JSCJlT9NZOgfJHOj9KnGrpZhN-Y1Vj8K5gU.original.fullsize.png"/><br/></div>

['Mathematics', 'Determinants', '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) p,s, (C) s, (D) p,s<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>(A) s, (B) p,q,r, (C) r, (D) p,q<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>(A) s, (B) p,q, (C) r, (D) p,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, (B) p,q, (C) s, (D) p,q</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>(A) s, (B) p,q, (C) r, (D) p,q,s<br/></span> </div>

<div class="solution">(A) Solving equations $L_1$ and $L_3$,<br/>$$<br/>\begin{aligned}<br/>\frac{x}{-36+10} &amp; =\frac{y}{+12-25}=\frac{1}{2-15} \\<br/>x &amp; =2, y=1<br/>\end{aligned}<br/>$$<br/>$L_1, L_2, L_3$ are concurrent, if point $(2,1)$ lies on $L_2$.<br/>$$<br/>\therefore \quad 6-k-1=0 \Rightarrow k=5<br/>$$<br/>(B) Either $L_1$ is parallel to $L_2$, or $L_3$ is parallel to $L_2$, then<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{1}{3}=\frac{3}{-k} \text { or } \frac{3}{5}=\frac{-k}{2} \\<br/>&amp; k=-9 \quad \text { or } \quad k=-\frac{6}{5}<br/>\end{aligned}<br/>$$<br/>$$<br/>\Rightarrow \quad k=-9 \text { or } k=-\frac{6}{5}<br/>$$<br/>(C) $L_1, L_2, L_3$ form a triangle, if they are not concurrent, or not parallel.<br/>$$<br/>\therefore \quad k \neq 5,-9,-\frac{6}{5} \Rightarrow k=\frac{5}{6}<br/>$$<br/>(D) $L_1, L_2, L_3$ do not form a triangle, if<br/>$$<br/>k=5,-9,-\frac{6}{5}<br/>$$</div>

MarksBatch1_P2.db

consider-the-motion-of-a-positive-point-charge-in-a-region-where-there-are-simultaneous-uniform-electric-and-magnetic-fields-e-e-0-j-and-b-b-0-j-at-ti

consider-the-motion-of-a-positive-point-charge-in-a-region-where-there-are-simultaneous-uniform-electric-and-magnetic-fields-e-e-0-j-and-b-b-0-j-at-ti-76969

<div class="question">Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E}=E_{0} \hat{j}$ and $\vec{B}=B_{0} \hat{j}$. At time $t=0$, this charge has velocity $\vec{v}$ in the in the $x$-y plane, making an angle $\theta$ with the $x$-axis. Which of the following option(s) is (are) correct for time $t&gt;0$ ?</div>

['Physics', 'Magnetic Effects of Current', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">If $\theta=0^{\circ}$, the charge moves in a circular path in the $x$ - plane.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">If $\theta=0^{\circ}$, the charge undergoes helical motion with constant pitch along the $y$-axis.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">If $\theta=10^{\circ}$, the charge undergoes helical motion with its pitch increasing with time, along the $y$-axis.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">If $\theta=90^{\circ}$, the charge undergoes linear but accelerated motion along the $y$-axis.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value">If $\theta=10^{\circ}$, the charge undergoes helical motion with its pitch increasing with time, along the $y$-axis., If $\theta=90^{\circ}$, the charge undergoes linear but accelerated motion along the $y$-axis.</span> </div>

<div class="solution">If $\theta=0^{\circ}$, the charged particle is projected along $x$-axis, due to magnetic field, $\vec{B}$ the charged particle will tend to move in a circular path in $y-z$ plane but due to force of electric field $\vec{E}$, the particle will move in a helical path with increasing pitch. Hence options (A) and (B) are wrong. If $\theta=10^{\circ}$, we can resolve velocity into two rectangular components. One along $x$-axis $\left(v \cos 10^{\circ}\right)$ and one along $y$-axis $\left(v \sin 10^{\circ}\right)$. Due to $v \cos 10^{\circ}$, the particle will move in circular path and due to $v \sin 10^{\circ}$ plus the force due to electric field, the particle will undergo helical motion with its pitch increasing. <br/> <br/>If $\theta=90^{\circ}$, the charge is moving along the magnetic field. Therefore the force due to magnetic field is zero. But the force due to electric field will accelerate the particle along $y$-axis. <br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ESjcDOMtKhQqHP_c9Ylws4MPmEOB0gS0nTW1_sU4eic.original.fullsize.png"/></div>

MarksBatch1_P2.db

consider-the-parabola-y-2-8-x-let-1-be-the-area-of-the-triangle-formed-by-the-end-points-of-its-latusrectum-and-the-point-p-2-1-2-on-the-parabola-and--1

consider-the-parabola-y-2-8-x-let-1-be-the-area-of-the-triangle-formed-by-the-end-points-of-its-latusrectum-and-the-point-p-2-1-2-on-the-parabola-and-1-82906

<div class="question">Consider the parabola $y^2=8 x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latusrectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latusrectum. Then, $\frac{\Delta_1}{\Delta_2}$ is</div>

['Mathematics', 'Parabola', 'JEE Main']

None

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

<div class="solution">As, we know area of triangle formed by three points on parabola is twice the area of triangle formed by corresponding tangents, i.e. area of $\triangle P Q R=2$ area of $\Delta T_1 T_2 T_3$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Tn_oNb9XptnCEq3OqpNXGsGE325gb-xi60MgtbaSZlI.original.fullsize.png"/><br/><br/>$\therefore \quad \Delta_1=2 \Delta_2$ or $\frac{\Delta_1}{\Delta_2}=2$</div>

MarksBatch1_P2.db

consider-the-parabola-y-2-8-x-let-1-be-the-area-of-the-triangle-formed-by-the-end-points-of-its-latusrectum-and-the-point-p-2-1-2-on-the-parabola-and-

consider-the-parabola-y-2-8-x-let-1-be-the-area-of-the-triangle-formed-by-the-end-points-of-its-latusrectum-and-the-point-p-2-1-2-on-the-parabola-and-50750

<div class="question">Consider the parabola $y^2=8 x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latusrectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latusrectum. Then, $\frac{\Delta_1}{\Delta_2}$ is</div>

['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

None

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

<div class="solution">As, we know area of triangle formed by three points on parabola is twice the area of triangle formed by corresponding tangents, i.e. area of $\triangle P Q R=2$ area of $\Delta T_1 T_2 T_3$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Tn_oNb9XptnCEq3OqpNXGsGE325gb-xi60MgtbaSZlI.original.fullsize.png"/><br/><br/>$\therefore \quad \Delta_1=2 \Delta_2$ or $\frac{\Delta_1}{\Delta_2}=2$</div>

MarksBatch1_P2.db

consider-the-planes-3-x-6-y-2-z-15-and-2-x-y-2-z-5-statement-i-the-parametric-equations-of-the-line-of-intersection-of-the-given-planes-are-x-3-14-t-y

consider-the-planes-3-x-6-y-2-z-15-and-2-x-y-2-z-5-statement-i-the-parametric-equations-of-the-line-of-intersection-of-the-given-planes-are-x-3-14-t-y-66157

<div class="question">Consider the planes $3 x-6 y-2 z=15$ and $2 x+y-2 z=5$.<br/>Statement I The parametric equations of the line of intersection of the given planes are $x=3+14 t, y=1+2 t$ and $z=15 t$.<br/>Statement II The vectors $14 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+15 \hat{\mathbf{k}}$ is parallel to the line of intersection of the given planes.</div>

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

<div class="solution">Given planes are $3 x-6 y-2 z=15$ and $2 x+y-2 z=5$<br/>For $z=0$, we get $x=3, y=-1$<br/>Direction ratios of planes are<br/>$$<br/> &lt; 3-6-2&gt;\text { and } &lt; 2 \quad 1-2&gt;<br/>$$<br/>then the dr's of line of intersection of planes is $\langle 14215\rangle$ and line is<br/>$$<br/>\begin{aligned}<br/>\frac{x-3}{14} &amp; =\frac{y+1}{2}=\frac{z-0}{15}=\lambda \text { (say) } \\<br/>\Rightarrow \quad x &amp; =14 \lambda+3, y=2 \lambda-1, z=15 \lambda<br/>\end{aligned}<br/>$$<br/>Hence, Statement I is false.<br/>But Statement II is true.</div>

MarksBatch1_P2.db

consider-the-system-of-equations-where-a-b-c-d-0-1-statement-1-the-probability-that-the-system-of-equations-has-a-unique-solution-is-38-statement-2-th

consider-the-system-of-equations-where-a-b-c-d-0-1-statement-1-the-probability-that-the-system-of-equations-has-a-unique-solution-is-38-statement-2-th-36151

<div class="question">Consider the system of equations<br/>where $\quad a, b, c, d \in\{0,1\}$<br/>Statement 1 The probability that the system of equations has a unique solution, is $3 / 8$.<br/>Statement 2 The probability that the system of equations has a solution, is 1 .</div>

['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.<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 1 is true, Statement 2 is false.<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Statement 1 is false, Statement 2 is true</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.<br/></span> </div>

<div class="solution">The number of all possible deteminants of the form<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/miiF1S9q7HSANsbUETCrPwiWfrn8s-gtqeJ3Wn4bUkI.original.fullsize.png"/><br/><br/><br/>Out of which only 10 determinants given by<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4W-k0G-nLAT2todHF1RbeMvEgzAQgzuWF7dVWZeueic.original.fullsize.png"/><br/><br/><br/>vanish and remaining six determinants have non-zero values.<br/>Hence, the required probability $=\frac{6}{16}=\frac{3}{8}$<br/>$\Rightarrow$ Statement 1 is correct.<br/>Statement 2 is also correct as the homogeneous equations have always a solution and Statement 2 is not a correct explanation of Statement 1.</div>

MarksBatch1_P2.db

consider-the-system-of-equations-x-2-y-3-z-1-x-3-y-4-z-1-and-x-y-2-z-k-statement-1-the-system-of-equations-has-no-solution-for-k-3-statement-2-the-det-1

consider-the-system-of-equations-x-2-y-3-z-1-x-3-y-4-z-1-and-x-y-2-z-k-statement-1-the-system-of-equations-has-no-solution-for-k-3-statement-2-the-det-1-39132

<div class="question">Consider the system of equations $x-2 y+3 z=-1, x-3 y+4 z=1$ and $-x+y-2 z=k$<br/>Statement 1 The system of equations has no solution for $k \neq 3$.<br/>Statement 2 The determinant $\left|\begin{array}{ccc}1 &amp; 3 &amp; -1 \\ -1 &amp; -2 &amp; k \\ 1 &amp; 4 &amp; 1\end{array}\right| \neq 0$, for $k \neq 3$.</div>

['Mathematics', 'Determinants', 'JEE Main']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<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/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is false.<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Statement 1 is false, Statement 2 is true</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<br/></span> </div>

<div class="solution">$$<br/>\text { The given system of equations can be expressed as }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jBbScxHaYhcKuePkpq3WFYXpu4TBDQnvjJ7XbV0GuLg.original.fullsize.png"/><br/><br/>When $k \neq 3$, the given system of equations has no solution.<br/>$\Rightarrow$ Statement 1 is true.Clearly, Statement 2 is also true as it is rearrangement of rows and columns of<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/DWVnYlhNGQEhHijhstXkfpyYNwJkcmw6j9YvbkG-GMo.original.fullsize.png"/><br/><br/>Hence, option (a) is correct.</div>

MarksBatch1_P2.db

consider-the-system-of-equations-x-2-y-3-z-1-x-3-y-4-z-1-and-x-y-2-z-k-statement-1-the-system-of-equations-has-no-solution-for-k-3-statement-2-the-det

consider-the-system-of-equations-x-2-y-3-z-1-x-3-y-4-z-1-and-x-y-2-z-k-statement-1-the-system-of-equations-has-no-solution-for-k-3-statement-2-the-det-83706

<div class="question">Consider the system of equations $x-2 y+3 z=-1, x-3 y+4 z=1$ and $-x+y-2 z=k$<br/>Statement 1 The system of equations has no solution for $k \neq 3$.<br/>Statement 2 The determinant $\left|\begin{array}{ccc}1 &amp; 3 &amp; -1 \\ -1 &amp; -2 &amp; k \\ 1 &amp; 4 &amp; 1\end{array}\right| \neq 0$, for $k \neq 3$.</div>

['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<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/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is false.<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Statement 1 is false, Statement 2 is true</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<br/></span> </div>

<div class="solution">$$<br/>\text { The given system of equations can be expressed as }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jBbScxHaYhcKuePkpq3WFYXpu4TBDQnvjJ7XbV0GuLg.original.fullsize.png"/><br/><br/>When $k \neq 3$, the given system of equations has no solution.<br/>$\Rightarrow$ Statement 1 is true.Clearly, Statement 2 is also true as it is rearrangement of rows and columns of<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/DWVnYlhNGQEhHijhstXkfpyYNwJkcmw6j9YvbkG-GMo.original.fullsize.png"/><br/><br/>Hence, option (a) is correct.</div>

MarksBatch1_P2.db

consider-the-two-curves-c-1-y-2-4-x-c-2-x-2-y-2-6-x-1-0-then

consider-the-two-curves-c-1-y-2-4-x-c-2-x-2-y-2-6-x-1-0-then-25705

<div class="question">Consider the two curves<br/>$C_1: y^2=4 x$<br/>$C_2: x^2+y^2-6 x+1=0$, then</div>

['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$C_1$ and $C_2$ touch each other only at one point<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$C_1$ and $C_2$ touch each other exactly at two points<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/>$C_1$ and $C_2$ intersect (but do not touch) at exactly two points<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$C_1$ and $C_2$ neither intersect nor touch each other</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$C_1$ and $C_2$ touch each other exactly at two points<br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Ka5EvOh33vl1y1JB5YPt0nRaTjg_fhSylQk09GjN5zc.original.fullsize.png"/><br/><br/>For the points of intersection of the two given curves<br/>$$<br/>C_1: y^2=4 x \text { and } C_2: x^2+y^2-6 x+1=0 \text {, }<br/>$$<br/>we have<br/>$$<br/>\begin{aligned}<br/>&amp; x^2+4 x-6 x+1=0 \\<br/>&amp; \Rightarrow \quad x^2-2 x+1=0 \\<br/>&amp; \Rightarrow \quad(x-1)^2=0 \\<br/>&amp; \Rightarrow \quad x=1,1 \\<br/>&amp; \Rightarrow \quad y=2,-2 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>Thus, the given curves touch each other exactly at two points $(1,2)$ and $(1,-2)$.</div>

MarksBatch1_P2.db

consider-three-planes-p-1-x-y-z-1-p-2-x-y-z-1-an-d-p3-x3-y3-z2-l-e-t-l1-l2-an-d-l3-b-e-t-h-e-l-in-eso-f-in-t-ersec-t-i-o-n-o-f-t-h-e-pl-an-es-p2-an-d-

consider-three-planes-p-1-x-y-z-1-p-2-x-y-z-1-an-d-p3-x3-y3-z2-l-e-t-l1-l2-an-d-l3-b-e-t-h-e-l-in-eso-f-in-t-ersec-t-i-o-n-o-f-t-h-e-pl-an-es-p2-an-d-75932

<div class="question">Consider three planes<br/>$$<br/>P_1: x-y+z=1, P_2: x+y-z=-1<br/>$$<br/>and $P_3: x-3 y+3 z=2$<br/>Let $L_1, L_2$ and $L_3$ be the lines of intersection of the planes $P_2$ and $P_3, P_3$ and $P_1, P_1$ and $P_2$, respectively.<br/>Statement 1 Atleast two of the lines $L_1, L_2$ and $L_3$ are non-parallel.<br/>Statement 2 The three planes do not have a common point.</div>

['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Statement 1 is true, Statement 2 is false.<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>Statement 1 is false, Statement 2 is true</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>Statement 1 is false, Statement 2 is true</span> </div>

<div class="solution">The given equations are<br/>$$<br/>\begin{aligned}<br/>&amp; x-y+z=1, \\<br/>&amp; x+y-z=-1 \text { and } x-3 y+3 z=2<br/>\end{aligned}<br/>$$<br/>The system of equations can be put in matrix form as $A X=B$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Z98dskolD9r3E84JgOJxxxWAm-WJoBebFfGiKjaOBfQ.original.fullsize.png"/><br/><br/>which is inconsistent as $\rho(A: B) \neq \rho(A)$.<br/>$\Rightarrow$ The three planes do not have a common point.<br/>$\Rightarrow$ Statement 2 is true.<br/>Since, planes $P_1, P_2$ and $P_3$ are pairwise intersection, their lines of intersection are parallel.<br/>Statement 1 is false.</div>

MarksBatch1_P2.db

consider-three-points-p-sin-cos-q-cos-sin-and-r-cos-sin-where-0-4-then

consider-three-points-p-sin-cos-q-cos-sin-and-r-cos-sin-where-0-4-then-11302

<div class="question">Consider three points $P=(-\sin (\beta-\alpha),-\cos \beta), Q=(\cos (\beta-\alpha), \sin \beta)$ and $R=(\cos (\beta-\alpha+\theta), \sin (\beta-\theta)$, where $0 &lt; \alpha, \beta, \theta &lt; \frac{\pi}{4}$. Then,</div>

['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

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

<div class="solution">For collinear points<br/>$$<br/>\Delta=\left|\begin{array}{ccc}<br/>-\sin (\beta-\alpha) &amp; -\cos \beta &amp; 1 \\<br/>\cos (\beta-\alpha) &amp; \sin \beta &amp; 1 \\<br/>\cos (\beta-\alpha+\theta) &amp; \sin (\beta-\theta) &amp; 1<br/>\end{array}\right|<br/>$$<br/>Clearly, $\Delta \neq 0$ for any value of $\alpha, \beta, \theta$, hence points are non-collinear.</div>

MarksBatch1_P2.db

cryopreservation-of-gametes-of-threatened-species-in-viable-and-fertile-condition-can-be-referred-to-as-1

cryopreservation-of-gametes-of-threatened-species-in-viable-and-fertile-condition-can-be-referred-to-as-1-31378

<div class="question">Cryopreservation of gametes of threatened species in viable and fertile condition can be referred to as:</div>

['Biology', 'Biodiversity and its Conservation', 'NEET', 'NEET 2015 (Phase 1)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value">Advanced <em>Ex-situ</em> conservation of biodiversity</span> </div>

<div class="solution">Cryopreservation of gametes of threatened species in viable and fertile conditions can be referred to as advanced ex-situ conservation of biodiversity because these gametes are stored in liquid nitrogen at a temperature of about - <math><mn>196</mn><mo>°</mo><mi>C</mi></math>.</div>

MarksBatch1_P2.db

cryopreservation-of-gametes-of-threatened-species-in-viable-and-fertile-condition-can-be-referred-to-as

cryopreservation-of-gametes-of-threatened-species-in-viable-and-fertile-condition-can-be-referred-to-as-90209

<div class="question">Cryopreservation of gametes of threatened species in viable and fertile condition can be referred to as:</div>

['Biology', 'Biodiversity and its Conservation', 'NEET']

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

<div class="correct-answer"> The correct answer is: <span class="option-value">Advanced ex-situ conservation of biodiversity</span> </div>

<div class="solution">Cryopreservation of gametes of threatened species in viable and fertile conditions can be referred to as advanced ex-situ conservation of biodiversity because these gametes are stored in liquid nitrogen at a temperature of about - <math><mn>196</mn><mo>°</mo><mi>C</mi></math>.</div>

MarksBatch1_P2.db

cuso-4-decolourises-on-addition-of-kcn-the-product-is

cuso-4-decolourises-on-addition-of-kcn-the-product-is-98093

<div class="question">$\mathrm{CuSO}_4$ decolourises on addition of $\mathrm{KCN}$, the product is</div>

['Chemistry', 'd and f Block Elements', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\left[\mathrm{Cu}(\mathrm{CN})_4\right]^{2-}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Cu}^{2+}$ get reduced to form $\left[\mathrm{Cu}(\mathrm{CN})_4\right]^{3-}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Cu}(\mathrm{CN})_2$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{CuCN}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>$\mathrm{CuCN}$</span> </div>

<div class="solution">$\mathrm{CuSO}_4$ decolourise on addition of $\mathrm{KCN}$, the product is $\mathrm{CuCN}$ along with $(\mathrm{CN})_2$ but in excess of KCN is form colourless potassium tetra cyano cuperate (I)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ygsirUrwGESxx7TTkRG2_eKcb8U2WwPK8vhoyu2t4EI.original.fullsize.png"/><br/></div>

MarksBatch1_P2.db

cyclohexen-on-ozonolysis-followed-by-reaction-with-zinc-dust-and-water-gives-compound-e-compound-e-on-further-treatment-with-aqueous-koh-yields-compou

cyclohexen-on-ozonolysis-followed-by-reaction-with-zinc-dust-and-water-gives-compound-e-compound-e-on-further-treatment-with-aqueous-koh-yields-compou-57703

<div class="question">Cyclohexen on ozonolysis followed by reaction with zinc dust and water gives compound $E$. Compound $E$ on further treatment with aqueous $\mathrm{KOH}$ yields compound $F$. Compound $F$ is :</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/J3dus6p3LX9F0gdZa1zG9L51PfycDNxiqH1t4Mwq_-o.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TNhA39jVZCPOetqn05jZ0z8NpEzzaMuNwJleOcFiPVs.original.fullsize.png"/><br/></span> </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/Qo8TPlOHsVRcW4sjUGR3o1FR6Le31A9N0NWd5YS0K30.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/Xz0f9eA59RfqVT5m4UdwJTnVobqYm4hzbbEkbtqa4D4.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/J3dus6p3LX9F0gdZa1zG9L51PfycDNxiqH1t4Mwq_-o.original.fullsize.png"/><br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/bRM-DP60qw87pjS_oUXRUQniHjJZEAogij57CcxsTRE.original.fullsize.png"/><br/></div>

MarksBatch1_P2.db

d-y-2-d-2-x-equals-1

d-y-2-d-2-x-equals-1-98721

<div class="question">$\frac{d^2 x}{d y^2}$ equals</div>

['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}$</span> </div>

<div class="solution">Since,<br/>$$<br/>\begin{aligned}<br/>\Rightarrow \quad \frac{d}{d y}\left(\frac{d x}{d y}\right) &amp; =\frac{d}{d x}\left(\frac{d y}{d x}\right)^{-1} \frac{d x}{d y} \\<br/>\frac{d^2 x}{d y^2} &amp; =-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-2}\left(\frac{d x}{d y}\right) \\<br/>&amp; =-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

dissolving-120-g-of-urea-molecular-weight-60-in-1000-g-of-water-gave-a-solution-of-density-115-g-ml-the-molarity-of-the-solution-is

dissolving-120-g-of-urea-molecular-weight-60-in-1000-g-of-water-gave-a-solution-of-density-115-g-ml-the-molarity-of-the-solution-is-63634

<div class="question">Dissolving $120 \mathrm{~g}$ of urea (molecular weight 60) in $1000 \mathrm{~g}$ of water gave a solution of density $1.15 \mathrm{~g} / \mathrm{mL}$. The molarity of the solution is</div>

['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1.78 \mathrm{M}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$2.00 \mathrm{M}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$2.05 \mathrm{M}$<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/>$2.22 \mathrm{M}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$2.05 \mathrm{M}$<br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/XDw7zs-Oac0XyZTQ_JVb7ZjLoAxLiZNRWw4dCuV1m8k.original.fullsize.png"/><br/></div>

MarksBatch1_P2.db

electrolysis-of-dilute-aq-nacl-solution-was-carried-out-by-passing-10-ma-current-the-time-required-to-liberate-001-mole-of-h-2-gas-at-the-cathode-is-1

electrolysis-of-dilute-aq-nacl-solution-was-carried-out-by-passing-10-ma-current-the-time-required-to-liberate-001-mole-of-h-2-gas-at-the-cathode-is-1-15735

<div class="question">Electrolysis of dilute aq. $\mathrm{NaCl}$ solution was carried out by passing $10 \mathrm{~mA}$ current. The time required to liberate $0.01$ mole of $\mathrm{H}_2$ gas at the cathode is (1 Faraday $=96500 \mathrm{C} \mathrm{mol}^{-1}$ )</div>

['Chemistry', 'Electrochemistry', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$9.65 \times 10^4 \mathrm{~s}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$19.3 \times 10^4 \mathrm{~s}$<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/>$28.95 \times 10^4 \mathrm{~s}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$38.6 \times 10^4 \mathrm{~s}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$19.3 \times 10^4 \mathrm{~s}$<br/></span> </div>

<div class="solution">$2 \mathrm{H}_2 \mathrm{O}+2 e^{-} \longrightarrow \mathrm{H}_2+2 \mathrm{OH}^{-}$<br/>For $0.01$ mole of $\mathrm{H}_2, 0.02$ mole of electrons are consumed Charge required $=0.02 \times 96500 \mathrm{C}=i \times t$<br/>$$<br/>\text { Time required }=\frac{0.02 \times 96500}{10 \times 10^{-3}}=19.3 \times 10^4 \mathrm{~s}<br/>$$</div>

MarksBatch1_P2.db

electrons-with-debroglie-wavelength-fall-on-the-target-in-an-xray-tube-the-cutoff-wavelength-of-the-emitted-xrays-is-1

electrons-with-debroglie-wavelength-fall-on-the-target-in-an-xray-tube-the-cutoff-wavelength-of-the-emitted-xrays-is-1-54854

<div class="question">Electrons with de-Broglie wavelength $\lambda$ fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is</div>

['Physics', 'Dual Nature of Matter', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\lambda_0=\frac{2 m c \lambda^2}{h}$<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/>$\lambda_0=\frac{2 h}{m c}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\lambda_0=\frac{2 m^2 c^2 \lambda^3}{h^2}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\lambda_0=\lambda$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\lambda_0=\frac{2 m c \lambda^2}{h}$<br/></span> </div>

<div class="solution">Momentum of striking electrons, $p=\frac{h}{\lambda}$<br/>$\therefore$ Kinetic energy of striking electrons, $K=\frac{p^2}{2 m}=\frac{h^2}{2 m \lambda^2}$<br/>This is also, maximum energy of $X$-ray photons.<br/>Therefore, $\quad \frac{h c}{\lambda_0}=\frac{h^2}{2 m \lambda^2}$ or $\lambda_0=\frac{2 m \lambda^2 c}{h}$<br/>$\therefore$ Correct option is (a).</div>

MarksBatch1_P2.db

equation-of-common-tangent-of-y-x-2-y-x-2-4-x-4-is

equation-of-common-tangent-of-y-x-2-y-x-2-4-x-4-is-75494

<div class="question">Equation of common tangent of $y=x^2, y=-x^2+4 x-4$ is</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/>$y=4(x-1)$<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/>$y=0$<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/>$y=-4(x-1)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$y=-30 x-50$</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$y=4(x-1)$<br/>, <br/>$y=0$<br/></span> </div>

<div class="solution">The equation of tangent to $y=x^2$, be $y=m x-\frac{m^2}{4}$, putting in $y=-x^2+4 x-4$, we should only get one value of $x$ i.e. Discriminate must be zero.<br/>$$<br/>\begin{array}{rrr}<br/>\therefore &amp; m x-\frac{m^2}{4} &amp; =-x^2+4 x-4 \\<br/>\Rightarrow &amp; x^2+x(m-4)+4-\frac{m^2}{4} &amp; =0 \\<br/>\Rightarrow &amp; m(m-4) &amp; =0 \\<br/>&amp; \therefore y=0 &amp; \text { or } m=0,4<br/>\end{array}<br/>$$<br/><br/>$$<br/>\therefore y=0 \text { and } y=4(x-1) \text { are tangents. }<br/>$$</div>

MarksBatch1_P2.db

equation-of-the-plane-containing-the-straight-line-2-x-3-y-4-z-and-perpendicular-to-the-plane-containing-the-straight-lines-3-x-4-y-2-z-and-4-x-2-y-3-

equation-of-the-plane-containing-the-straight-line-2-x-3-y-4-z-and-perpendicular-to-the-plane-containing-the-straight-lines-3-x-4-y-2-z-and-4-x-2-y-3-82984

<div class="question">Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is</div>

['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$x+2 y-2 z=0$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$3 x+2 y-2 z=0$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$x-2 y+z=0$<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">D</span> <span class="option-data"><br/>$5 x+2 y-4 z=0$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$x-2 y+z=0$<br/></span> </div>

<div class="solution">The DR's of normal to the plane containing $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$. $\Rightarrow \quad \overrightarrow{\mathbf{n}}_1=\left|\begin{array}{ccc}\hat{\mathbf{i}} &amp; \hat{\mathbf{j}} &amp; \hat{\mathbf{k}} \\ 3 &amp; 4 &amp; 2 \\ 4 &amp; 2 &amp; 3\end{array}\right|=(8 \hat{\mathbf{i}}-\hat{\mathbf{j}}-10 \hat{\mathbf{k}})$<br/>Also, equation of plane containing $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and DR's of normal to be $\overrightarrow{\mathbf{n}}_2=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ESo5D22LF0O07y3bQ2iMsKp6KgLfKJluZJ1-oU31OXI.original.fullsize.png"/><br/><br/><br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; a x+b y+c z=0 \\<br/>\text { where, } \overrightarrow{\mathbf{n}}_1 \cdot \overrightarrow{\mathbf{n}}_2=0 \\<br/>\Rightarrow &amp; 8 a-b-10 c=0 \\<br/>\text { and } &amp; \overrightarrow{\mathbf{n}}_2 \perp(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \\<br/>\Rightarrow &amp; 2 a+3 b+4 c=0<br/>\end{array}<br/>$$<br/>From Eqs. (ii) and (iii), we get<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{a}{-1 \quad-10}=\frac{b}{8}=\frac{c}{-1} \\<br/>\Rightarrow \quad \frac{a}{-4+30} &amp; =\frac{b}{-20-32}=\frac{c}{24+2} \\<br/>\Rightarrow \quad \quad \quad \frac{a}{26} &amp; =\frac{b}{-52}=\frac{c}{26} \\<br/>\Rightarrow \quad \quad \quad \quad \frac{a}{1} &amp; =\frac{b}{-2}=\frac{c}{1}<br/>\end{aligned}<br/>$$<br/>From Eqs. (i) and (iv), required equation of plane, is $x-2 y+z=0$.</div>

MarksBatch1_P2.db

extra-pure-n-2-can-be-obtained-by-heating

extra-pure-n-2-can-be-obtained-by-heating-38165

<div class="question">Extra pure $\mathrm{N}_2$ can be obtained by heating</div>

['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{NH}_3$ with $\mathrm{CuO}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{NH}_4 \mathrm{NO}_4$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\mathrm{NH}_4\right)_2 \mathrm{Cr}_2 \mathrm{O}_7$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{Ba}\left(\mathrm{N}_3\right)_2$</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> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mathrm{Ba}\left(\mathrm{N}_3\right)_2$</span> </div>

<div class="solution">$\mathrm{Ba}\left(\mathrm{N}_3\right)_2 \stackrel{\text { Heat }}{\longrightarrow} \mathrm{Ba}(\mathrm{s})+3 \mathrm{~N}_2(\mathrm{~g})$<br/>Azide salt of barium can be obtained in purest form as well as the decomposition product contain solid $\mathrm{Ba}$ as byproduct along with gaseous nitrogen, hence no additional step of separation is required.<br/>Other reactions are:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/__xySB5n7CNqg_PAIUWe3FIZmDl1RXC6K1XGKtDLPJ4.original.fullsize.png"/><br/></div>

MarksBatch1_P2.db

extraction-for-zinc-from-zinc-blende-is-achieved-by

extraction-for-zinc-from-zinc-blende-is-achieved-by-55290

<div class="question">Extraction for zinc from zinc blende is achieved by</div>

['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>electrolytic reduction<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>roasting followed by reduction with carbon<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/>roasting followed by reduction with another metal<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>roasting followed by self-reduction</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>roasting followed by reduction with carbon<br/></span> </div>

<div class="solution">Zinc blende is roasted and then treated with coke for the reduction.<br/>$$<br/>\begin{gathered}<br/>2 \mathrm{ZnS}+3 \mathrm{O}_2 \stackrel{\Delta}{\longrightarrow} 2 \mathrm{ZnO}+2 \mathrm{SO}_2 \uparrow \\<br/>\mathrm{ZnO}+\mathrm{C} \stackrel{\Delta}{\longrightarrow} \mathrm{Zn}+\mathrm{CO} \uparrow<br/>\end{gathered}<br/>$$</div>

MarksBatch1_P2.db

extraction-of-metal-from-the-ore-cassiterite-involves

extraction-of-metal-from-the-ore-cassiterite-involves-36000

<div class="question">Extraction of metal from the ore cassiterite involves</div>

['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

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

<div class="solution">The important ore of tin is cassiterite $\left(\mathrm{SnO}_2\right)$. Tin is extracted from cassiterite ore by carbon reduction method in a blast furnace. $\mathrm{SnO}_2+2 \mathrm{C} \rightarrow \mathrm{Sn}+2 \mathrm{CO}$<br/>The product often contain traces of iron which is removed by blowing air through the melt to oxidise to $\mathrm{FeO}$ which then floats to the surface.<br/>$2 \mathrm{Fe}+\mathrm{O}_2 \rightarrow 2 \mathrm{FeO}$</div>

MarksBatch1_P2.db

figure-shows-three-resistor-configurations-r-1-r-2-and-r-3-connected-to-3-v-battery-if-the-power-dissipated-by-the-configuration-r-1-r-2-and-r-3-is-p--1

figure-shows-three-resistor-configurations-r-1-r-2-and-r-3-connected-to-3-v-battery-if-the-power-dissipated-by-the-configuration-r-1-r-2-and-r-3-is-p-1-59567

<div class="question">Figure shows three resistor configurations $R_1, R_2$ and $R_3$ connected to $3 \mathrm{~V}$ battery. If the power dissipated by the configuration $R_1, R_2$ and $R_3$ is $P_1, P_2$ and $P_3$, respectively, then<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FaDLcWdj6MIA-JqWZmTiJ6nR8EgzVH5Cszu_B_UC_vs.original.fullsize.png"/><br/></div>

['Physics', 'Current Electricity', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$P_1&gt;P_2&gt;P_3$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$P_1&gt;P_3&gt;P_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$P_2&gt;P_1&gt;P_3$<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/>$P_3&gt;P_2&gt;P_1$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$P_2&gt;P_1&gt;P_3$<br/></span> </div>

<div class="solution">Applying $P=\frac{V^2}{R}$<br/>$R_1=1 \Omega, R_2=0.5 \Omega$ and $R_3=2 \Omega, V_1=V_2=V_3=3$ volt<br/>$\therefore \quad P_1=\frac{(3)^2}{1}=9 \mathrm{~W}$<br/>$\quad P_2=\frac{(3)^2}{0.5}=18 \mathrm{~W}$ and $P_3=\frac{(3)^2}{2}=4.5 \mathrm{~W}$<br/>$\therefore \quad P_2&gt;P_1&gt;P_3$<br/>$\therefore \quad$ correct option is (c).</div>

MarksBatch1_P2.db

figure-shows-three-resistor-configurations-r-1-r-2-and-r-3-connected-to-3-v-battery-if-the-power-dissipated-by-the-configuration-r-1-r-2-and-r-3-is-p-

figure-shows-three-resistor-configurations-r-1-r-2-and-r-3-connected-to-3-v-battery-if-the-power-dissipated-by-the-configuration-r-1-r-2-and-r-3-is-p-68509

<div class="question">Figure shows three resistor configurations $R_1, R_2$ and $R_3$ connected to $3 \mathrm{~V}$ battery. If the power dissipated by the configuration $R_1, R_2$ and $R_3$ is $P_1, P_2$ and $P_3$, respectively, then<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FaDLcWdj6MIA-JqWZmTiJ6nR8EgzVH5Cszu_B_UC_vs.original.fullsize.png"/><br/></div>

['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$P_1&gt;P_2&gt;P_3$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$P_1&gt;P_3&gt;P_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$P_2&gt;P_1&gt;P_3$<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/>$P_3&gt;P_2&gt;P_1$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$P_2&gt;P_1&gt;P_3$<br/></span> </div>

<div class="solution">Applying $P=\frac{V^2}{R}$<br/>$R_1=1 \Omega, R_2=0.5 \Omega$ and $R_3=2 \Omega, V_1=V_2=V_3=3$ volt<br/>$\therefore \quad P_1=\frac{(3)^2}{1}=9 \mathrm{~W}$<br/>$\quad P_2=\frac{(3)^2}{0.5}=18 \mathrm{~W}$ and $P_3=\frac{(3)^2}{2}=4.5 \mathrm{~W}$<br/>$\therefore \quad P_2&gt;P_1&gt;P_3$<br/>$\therefore \quad$ correct option is (c).</div>

MarksBatch1_P2.db

find-out-from-newspapers-and-popular-science-articles-any-new-fossil-discoveries-or-controversies-about-evolution

find-out-from-newspapers-and-popular-science-articles-any-new-fossil-discoveries-or-controversies-about-evolution-63642

<div class="question">Find out from newspapers and popular science articles any new fossil discoveries or controversies about evolution.</div>

['Biology', 'Evolution']

None

None

<div class="solution">Chimps are more evolved than humans (The Times of India):<br/>Chimpanzees are more evolved than humans, a study suggests. There is no doubt that humans are the more advanced species. But a comparison of 14,000 human and chimpanzee genes shows the forces of natural selection have and the greatest impact on our ape cousins.<br/>The researchers' discovery challenges the common assumption that our large brains and high intelligence were the gifts of natural selection. Humans and chimps followed different evolutionary paths from a common ape ancestor about 5 million years ago. Both underwent changes as the fittest survived to pass their genes on to future generations. But the US study shows that humans possess a 'substantially smaller' number of positively-selected genes than chimps.</div>

MarksBatch1_P2.db

find-the-time-constant-for-the-given-rc-circuits-in-correct-order-im-g-src-h-ttp-s-c-d-n-q-u-es-t-i-o-n-p-oo-l-g-e-t-ma-r-k-s-a-pp-p-y-q-j-e-e-a-d-v-a

find-the-time-constant-for-the-given-rc-circuits-in-correct-order-im-g-src-h-ttp-s-c-d-n-q-u-es-t-i-o-n-p-oo-l-g-e-t-ma-r-k-s-a-pp-p-y-q-j-e-e-a-d-v-a-85141

<div class="question">$$<br/>\text { Find the time constant for the given } R C \text { circuits in correct order. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/7aU5aAIeBz3ebRc4ZXsLteRMyxFnx6QQ53YxU94qFDo.original.fullsize.png"/><br/><br/>$R_1=1 \Omega, R_2=2 \Omega, C_1=4 \mu \mathrm{F}, C_2=2 \mu \mathrm{F}$</div>

['Physics', 'Alternating Current', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$18,4,8 / 9$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$18,8 / 9,4$<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/>$4,18,8 / 9$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$4,8 / 9,18$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$18,8 / 9,4$<br/></span> </div>

<div class="solution">$$<br/>\begin{aligned}<br/>\tau &amp; =C R \\<br/>\tau_1 &amp; =\left(C_1+C_2\right)\left(R_1+R_2\right)=18 \mu \mathrm{s} \\<br/>\tau_2 &amp; =\left(\frac{C_1 C_2}{C_1+C_2}\right)\left(\frac{R_1 R_2}{R_1+R_2}\right)=\frac{8}{6} \times \frac{2}{3}=\frac{8}{9} \mu \mathrm{s} \\<br/>\tau_2 &amp; =\left(C_1+C_2\right)\left(\frac{R_1 R_2}{R_1+R_2}\right)=(6)\left(\frac{2}{3}\right)=4 \mu \mathrm{s}<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

for-0-2-the-solutions-of-m-1-6-cosec-4-m-1-cosec-4-m-4-2-isare

for-0-2-the-solutions-of-m-1-6-cosec-4-m-1-cosec-4-m-4-2-isare-56614

<div class="question">For $0 &lt; \theta &lt; \frac{\pi}{2}$, the solution(s) of $\sum_{m=1}^6 \operatorname{cosec}\left[\theta+\frac{(m-1) \pi}{4}\right]$ $\operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right)=4 \sqrt{2}$ is/are</div>

['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

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

<div class="solution">For $0 &lt; \theta &lt; \frac{\pi}{2}$,<br/>$$<br/>\begin{array}{r}<br/>\sum_{m=1}^6 \operatorname{cosec}\left[\theta+\frac{(m-1) \pi}{4}\right] \operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right) \\<br/>=4 \sqrt{2} \\<br/>\Rightarrow \sum_{m=1}^6 \frac{\sin \left[\theta+\frac{m \pi}{4}-\left(\theta+\frac{(m-1) \pi}{4}\right)\right]}{\sin \frac{\pi}{4}\left\{\sin \left(\theta+\frac{(m-1) \pi}{4}\right) \sin \left(\theta+\frac{m \pi}{4}\right)\right\}} \\<br/>=4 \sqrt{2}<br/>\end{array}<br/>$$<br/><br/>$$<br/>\begin{array}{r}<br/>\Rightarrow \sum_{m=1}^6 \frac{\cot \left(\theta+\frac{(m-1) \pi}{4}\right)-\cot \left(\theta+\frac{m \pi}{4}\right)}{1 / 2} \\<br/>=4 \sqrt{2}<br/>\end{array}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad \sum_{m=1}^6\left[\cot \left(\theta+\frac{(m-1) \pi}{4}\right)\right. \\<br/>&amp; \left.-\cot \left(\theta+\frac{m \pi}{4}\right)\right]=4 \\<br/>&amp; \Rightarrow \cot (\theta)-\cot \left(\theta+\frac{\pi}{4}\right)+\cot \left(\theta+\frac{\pi}{4}\right) \\<br/>&amp; -\cot \left(\theta+\frac{2 \pi}{4}\right)+\ldots+\cot \left(\theta+\frac{5 \pi}{4}\right) \\<br/>&amp; -\cot \left(\theta+\frac{6 \pi}{4}\right)=4 \\<br/>&amp; \Rightarrow \quad \cot \theta-\cot \left(\frac{3 \pi}{2}+\theta\right)=4 \\<br/>&amp; \Rightarrow \quad \cot \theta+\tan \theta=4 \\<br/>&amp; \Rightarrow \quad \tan ^2 \theta-4 \tan \theta+1=0 \\<br/>&amp; \Rightarrow \quad(\tan \theta-2)^2-3=0 \\<br/>&amp; \Rightarrow(\tan \theta-2+\sqrt{3})(\tan \theta-2-\sqrt{3})=0 \\<br/>&amp; \Rightarrow \quad \tan \theta=2-\sqrt{3} \text { or } \\<br/>&amp; \tan \theta=2+\sqrt{3} \\<br/>&amp; \Rightarrow \theta=\frac{\pi}{12} ; \theta=\frac{5 \pi}{12} \Rightarrow \theta \in\left(0, \frac{\pi}{2}\right) \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

for-a-dilute-solution-containing-25-g-of-a-nonvolatile-nonelectrolyte-solute-in-100-g-of-water-the-elevation-in-boiling-point-at-1-atm-pressure-is-2-c

for-a-dilute-solution-containing-25-g-of-a-nonvolatile-nonelectrolyte-solute-in-100-g-of-water-the-elevation-in-boiling-point-at-1-atm-pressure-is-2-c-61434

<div class="question">For a dilute solution containing $2.5 \mathrm{~g}$ of a non-volatile non-electrolyte solute in $100 \mathrm{~g}$ of water, the elevation in boiling point at 1 atm pressure is $2^{\circ} \mathrm{C}$. Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure ( $\mathrm{mm}$ of $\mathrm{Hg}$ ) of the solution is (take $K_{b}=0.76 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ )</div>

['Chemistry', 'Solutions', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">724</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">740</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">736</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">718</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value">724</span> </div>

<div class="solution">From Raoult's law, <br/> <br/>$\frac{p^{\circ}-p}{p^{\circ}}=\frac{\text { No. of moles of solute }}{\text { No. of moles of solvent }+\text { No. of moles of solute }}$ <br/> <br/>When the concentration of solute is much lower than the concentration of solvent, <br/> <br/>$\begin{array}{l} <br/> <br/>\frac{p^{\circ}-p}{p^{\circ}}=\frac{\text { No. of moles of solute }}{\text { No. of moles of solvent }} \\ <br/> <br/>\frac{760-p}{760}=\frac{2.5 / m}{100 / 18}...(i) <br/> <br/>\end{array}$ <br/> <br/>From elevation in boiling point, $\Delta T_{b}=K_{b} \times m$ <br/> <br/>$\begin{array}{l} <br/> <br/>2=0.76 \times m \\ <br/> <br/>m=\frac{2}{0.76}...(ii) <br/> <br/>\end{array}$ <br/> <br/>From(i) and (ii), $p=724 \mathrm{~mm}$</div>

MarksBatch1_P2.db

for-a-first-order-reaction-a-p-the-temperature-t-dependent-rate-constant-k-was-found-to-follow-the-equation-lo-g-k-2000-t-60-the-preexponential-factor

for-a-first-order-reaction-a-p-the-temperature-t-dependent-rate-constant-k-was-found-to-follow-the-equation-lo-g-k-2000-t-60-the-preexponential-factor-41301

<div class="question">For a first order reaction, $A \rightarrow P$, the temperature $(T)$ dependent rate constant $(k)$ was found to follow the equation, $\log k=-(2000) / T+6.0$<br/>The pre-exponential factor $A$ and the activation energy $\left(E_a\right)$, respectively, are</div>

['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1.0 \times 10^6 \mathrm{~s}^{-1}$ and $9.2 \mathrm{~kJ} \mathrm{~mol}^{-1}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$6.0 \mathrm{~s}^{-1}$ and $16.6 \mathrm{~kJ} \mathrm{~mol}^{-1}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$1.0 \times 10^6 \mathrm{~s}^{-1}$ and $16.6 \mathrm{~kJ} \mathrm{~mol}^{-1}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$1.0 \times 10^6 \mathrm{~s}^{-1}$ and $38.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$1.0 \times 10^6 \mathrm{~s}^{-1}$ and $38.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$</span> </div>

<div class="solution">Comparing the slope and intercept of the given equation with the following Arrhenius equation :<br/>$$<br/>\log k=-\frac{E_a}{2303 R T}+\log A<br/>$$<br/>Hence, $\log A=6$ i.e. $A=10^6 \mathrm{~s}^{-1}$.<br/>Comparing slope gives $E_a=38.3 \mathrm{~kJ} / \mathrm{mol}$.</div>

MarksBatch1_P2.db

for-an-ideal-gas-consider-only-pv-work-in-going-from-an-initial-state-x-to-the-final-state-z-the-final-state-z-can-be-reached-by-either-of-the-two-pat

for-an-ideal-gas-consider-only-pv-work-in-going-from-an-initial-state-x-to-the-final-state-z-the-final-state-z-can-be-reached-by-either-of-the-two-pat-35339

<div class="question">For an ideal gas, consider only P-V work in going from an initial state $\mathrm{X}$ to the final state $\mathrm{Z}$. The final state $\mathrm{Z}$ can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct? <br/> <br/>[Take $\Delta \mathrm{S}$ as change in entropy and w as work done]. <br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/sUaDDrqK6xaVRvm7zYNxFwLWFJ_-V9oHMI5FoRk7u7Y.original.fullsize.png"/><br/></div>

['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

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

<div class="correct-answer"> The correct answers are: <span class="option-value">$\Delta \mathrm{S}_{x \rightarrow z}=\Delta \mathrm{S}_{x \rightarrow y}+\Delta \mathrm{S}_{y \rightarrow z}$, $\mathrm{w}_{x \rightarrow y \rightarrow z}=\mathrm{w}_{x \rightarrow y}$</span> </div>

<div class="solution">$\Delta S_{X \rightarrow Z}=\Delta S_{X \rightarrow Y}+\Delta S_{Y \rightarrow Z}$ [Entropy is a state function, hence additive] <br/> <br/>$w_{X \rightarrow Y \rightarrow Z}=w_{X \rightarrow Y}$ [Work done in $Y \rightarrow Z$ is zero because it is an isochoric process].</div>

MarksBatch1_P2.db

for-any-real-number-x-let-x-denotes-the-largest-integer-less-than-or-equal-to-x-let-f-be-a-real-valued-function-defined-on-the-interval-10-10-by-f-x-x-1

for-any-real-number-x-let-x-denotes-the-largest-integer-less-than-or-equal-to-x-let-f-be-a-real-valued-function-defined-on-the-interval-10-10-by-f-x-x-1-51112

<div class="question">For any real number $x$, let $[x]$ denotes the largest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by $f(x)=\left\{\begin{array}{cc}x-[x] &amp; \text { if }[x] \text { is odd } \\ 1+[x]-x &amp; \text { if }[x] \text { is even }\end{array}\right.$.<br/>Then the value of $\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos \pi x d x$ is</div>

['Mathematics', 'Definite Integration', 'JEE Main']

None

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

<div class="solution">Given, $f(x)=\left\{\begin{array}{cc}x-[x] &amp; \text { if }[x] \text { is odd } \\ 1+[x]-x &amp; \text { if }[x] \text { is even }\end{array}\right.$ $f(x)$ and $\cos \pi x$ both are periodic with period 2 and both are even.<br/>$$<br/>\begin{aligned}<br/>\therefore \quad &amp; \int_{-10}^{10} f(x) \cos \pi x d x \\<br/>&amp; =2 \int_0^{10} f(x) \cos \pi x d x<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/38fJZUbYKcCJA8fbhzEPVjPMjiAKOXHpLlVA3ITpe2w.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; =10 \int_0^2 f(x) \cos \pi x d x \\<br/>&amp; \text { Now, } \quad \int_0^1 f(x) \cos \pi x d x \\<br/>&amp; =\int_0^1(1-x) \cos \pi x d x=-\int_0^1 u \cos \pi u d u \\<br/>&amp; \text { and } \int_1^2 f(x) \cos \pi x d x \\<br/>&amp; =\int_1^2(x-1) \cos \pi x d x=-\int_0^1 u \cos \pi u d u \\<br/>&amp; \therefore \quad \int_{-10}^{10} f(x) \cos \pi x d x \\<br/>&amp; =-20 \int_0^1 u \cos \pi u d u=\frac{40}{\pi^2} \\<br/>&amp; \Rightarrow \quad \frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos \pi x d x=4<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

for-any-real-number-x-let-x-denotes-the-largest-integer-less-than-or-equal-to-x-let-f-be-a-real-valued-function-defined-on-the-interval-10-10-by-f-x-x

for-any-real-number-x-let-x-denotes-the-largest-integer-less-than-or-equal-to-x-let-f-be-a-real-valued-function-defined-on-the-interval-10-10-by-f-x-x-28177

<div class="question">For any real number $x$, let $[x]$ denotes the largest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by $f(x)=\left\{\begin{array}{cc}x-[x] &amp; \text { if }[x] \text { is odd } \\ 1+[x]-x &amp; \text { if }[x] \text { is even }\end{array}\right.$.<br/>Then the value of $\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos \pi x d x$ is</div>

['Mathematics', 'Definite Integration', '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">Given, $f(x)=\left\{\begin{array}{cc}x-[x] &amp; \text { if }[x] \text { is odd } \\ 1+[x]-x &amp; \text { if }[x] \text { is even }\end{array}\right.$ $f(x)$ and $\cos \pi x$ both are periodic with period 2 and both are even.<br/>$$<br/>\begin{aligned}<br/>\therefore \quad &amp; \int_{-10}^{10} f(x) \cos \pi x d x \\<br/>&amp; =2 \int_0^{10} f(x) \cos \pi x d x<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/38fJZUbYKcCJA8fbhzEPVjPMjiAKOXHpLlVA3ITpe2w.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; =10 \int_0^2 f(x) \cos \pi x d x \\<br/>&amp; \text { Now, } \quad \int_0^1 f(x) \cos \pi x d x \\<br/>&amp; =\int_0^1(1-x) \cos \pi x d x=-\int_0^1 u \cos \pi u d u \\<br/>&amp; \text { and } \int_1^2 f(x) \cos \pi x d x \\<br/>&amp; =\int_1^2(x-1) \cos \pi x d x=-\int_0^1 u \cos \pi u d u \\<br/>&amp; \therefore \quad \int_{-10}^{10} f(x) \cos \pi x d x \\<br/>&amp; =-20 \int_0^1 u \cos \pi u d u=\frac{40}{\pi^2} \\<br/>&amp; \Rightarrow \quad \frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos \pi x d x=4<br/>\end{aligned}<br/>$$</div>

MarksBatch1_P2.db

if-2-s-i-n-4-x-3-c-o-s-4-x-5-1-then

if-2-s-i-n-4-x-3-c-o-s-4-x-5-1-then-17856

<div class="question">If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}$, then</div>

['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\tan ^2 x=\frac{2}{3}$<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">B</span> <span class="option-data"><br/>$\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$<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/>$\tan ^2 x=\frac{1}{3}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{2}{125}$</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$\tan ^2 x=\frac{2}{3}$<br/>, <br/>$\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$<br/></span> </div>

<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5} \\<br/>&amp; \Rightarrow \quad \frac{\sin ^4 x}{2}+\frac{\left(1-\sin ^2 x\right)^2}{3}=\frac{1}{5} \\<br/>&amp; \Rightarrow \frac{\sin ^4 x}{2}+\frac{1+\sin ^4 x-2 \sin ^2 x}{3}=\frac{1}{5} \\<br/>&amp; \Rightarrow \quad 5 \sin ^4 x-4 \sin ^2 x+2=\frac{6}{5} \\<br/>&amp; \Rightarrow \quad 25 \sin ^4 x-20 \sin ^2 x+4=0 \\<br/>&amp; \Rightarrow \quad\left(5 \sin ^2 x-2\right)^2=0 \\<br/>&amp; \Rightarrow \sin ^2 x=\frac{2}{5}, \cos ^2 x=\frac{3}{5}, \tan ^2 x=\frac{2}{3} \\<br/>&amp; \therefore \quad \frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-a-and-b-are-vectors-in-space-given-by-a-5-i-2-j-and-b-14-2-i-j-3-k-then-the-value-of-2-a-b-a-b-a-2-b-is

if-a-and-b-are-vectors-in-space-given-by-a-5-i-2-j-and-b-14-2-i-j-3-k-then-the-value-of-2-a-b-a-b-a-2-b-is-86610

<div class="question">If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are vectors in space given by $\overrightarrow{\mathbf{a}}=\frac{\hat{\mathbf{i}}-2 \hat{\mathbf{j}}}{\sqrt{5}}$ and $\overrightarrow{\mathbf{b}}=\frac{2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}}{\sqrt{14}}$, then the value of $(2 \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot[(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}-2 \overrightarrow{\mathbf{b}})]$ is</div>

['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>

<div class="solution">From the given information, it is clear that<br/>$$<br/>\begin{gathered}<br/>\overrightarrow{\mathbf{a}}=\frac{\hat{\mathbf{i}}-2 \hat{\mathbf{j}}}{\sqrt{5}} \\<br/>\Rightarrow \quad|\overrightarrow{\mathbf{a}}|=1,|\overrightarrow{\mathbf{b}}|=1, \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=0 \\<br/>\text { Now, }(2 \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot[(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}-2 \overrightarrow{\mathbf{b}})]<br/>\end{gathered}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; =(2 \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot\left[a^2 \overrightarrow{\mathbf{b}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \cdot \overrightarrow{\mathbf{a}}+2 b^2 \cdot \overrightarrow{\mathbf{a}}\right. \\<br/>&amp; =[2 \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}] \cdot[\overrightarrow{\mathbf{b}}+2 \overrightarrow{\mathbf{a}}]=4 \overrightarrow{\mathbf{a}}^2+\overrightarrow{\mathbf{b}}^2 \\<br/>&amp; =4 \cdot 1+1=5 \quad \text { [as } a \cdot b=0]<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-a-and-b-are-vectors-such-that-a-b-29-and-a-2-i-3-j-4-k-2-i-3-j-4-k-b-then-a-possible-value-of-a-b-7-i-2-j-3-k-is

if-a-and-b-are-vectors-such-that-a-b-29-and-a-2-i-3-j-4-k-2-i-3-j-4-k-b-then-a-possible-value-of-a-b-7-i-2-j-3-k-is-63468

<div class="question">If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}+\vec{b}|=\sqrt{29}$ and $\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}$, then a possible value of $(\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ is</div>

['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

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

<div class="solution">Given that $\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}$ $\Rightarrow(\vec{a}+\vec{b}) \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=\overrightarrow{0}$ <br/> <br/>But $\vec{a}+\vec{b} \neq 0$ and $2 \hat{i}+3 \hat{j}+4 \hat{k} \neq 0$ <br/> <br/>$\therefore(\vec{a}+\vec{b}) \|(2 \hat{i}+3 \hat{j}+4 \hat{k})$. <br/> <br/>Let $\vec{a}+\vec{b}=\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ <br/> <br/>Also given that $|\vec{a}+\vec{b}|=\sqrt{29} \Rightarrow \lambda=\pm 1$ <br/> <br/>$\therefore \vec{a}+\vec{b}=\pm(2 \hat{i}+3 \hat{j}+4 \hat{k})$ <br/> <br/>So, $(\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{i}+3 \hat{k})=\pm 4$</div>

MarksBatch2_P1.db

if-a-b-and-c-are-the-sides-of-a-a-bc-such-that-x-2-2-a-b-c-x-3-ab-b-c-c-a-0-has-real-roots-then

if-a-b-and-c-are-the-sides-of-a-a-bc-such-that-x-2-2-a-b-c-x-3-ab-b-c-c-a-0-has-real-roots-then-82360

<div class="question">If $a, b$ and $c$ are the sides of a $\triangle A B C$ such that $x^2-2(a+b+c) x+3 \lambda(a b+b c+c a)=0$ has real roots, then</div>

['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\lambda &lt; \frac{4}{3}$</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/>$\lambda&gt;\frac{5}{3}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\lambda \in\left(\frac{4}{3}, \frac{5}{3}\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\lambda \in\left(\frac{1}{3}, \frac{5}{3}\right)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\lambda &lt; \frac{4}{3}$</span> </div>

<div class="solution">Since, roots are real, $D \geq 0$<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; 4(a+b+c)^2-12 \lambda(a b+b c+c a) \geq 0 \\<br/>\Rightarrow &amp; (a+b+c)^2 \geq 3 \lambda(a b+b c+c a) \\<br/>\Rightarrow &amp; a^2+b^2+c^2 \geq(a b+b c+c a)(3 \lambda-2) \\<br/>\Rightarrow &amp; 3 \lambda-2 \leq \frac{a^2+b^2+c^2}{a b+b c+c a} \\<br/>\text { where, } a^2+b^2 \geq 2 a b, b^2+c^2 \geq 2 b c \text { and } c^2+a^2 \geq 2 c a \\<br/>\Rightarrow &amp; a^2+b^2+c^2 \geq(a b+b c+c a) \\<br/>\therefore &amp; \frac{a^2+b^2+c^2}{a b+b c+c a} \geq 1<br/>\end{array}<br/>$$<br/>Also, $\quad \cos A=\frac{b^2+c^2-a^2}{2 b c} &lt; 1$<br/>$$<br/>\Rightarrow \quad b^2+c^2-a^2 &lt; 2 b c<br/>$$<br/>Similarly, $c^2+a^2-b^2 &lt; 2 c a$ and $a^2+b^2-c^2 &lt; 2 a b$<br/>$$<br/>\Rightarrow a^2+b^2+c^2 &lt; 2(a b+b c+c a) \text { or } \frac{a^2+b^2+c^2}{a b+b c+c a} &lt; 2<br/>$$<br/>From Eqs. (i), (ii) and (iii)<br/>$$<br/>\begin{array}{lrl}<br/>\therefore &amp; 3 \lambda-2 &amp; &lt; 2 \\<br/>\Rightarrow &amp; \lambda &lt; \frac{4}{3}<br/>\end{array}<br/>$$</div>

MarksBatch2_P1.db

if-a-b-and-c-are-unit-vectors-satisfying-a-b-2-b-c-2-c-a-2-9-then-2-a-5-b-5-c-is

if-a-b-and-c-are-unit-vectors-satisfying-a-b-2-b-c-2-c-a-2-9-then-2-a-5-b-5-c-is-43778

<div class="question">If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9$, then $|2 \vec{a}+5 \vec{b}+5 \vec{c}|$ is</div>

['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

None

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

<div class="solution">$\begin{array}{l} <br/> <br/>\because|\vec{a}|=|\vec{b}|=|\vec{c}|=1 \\ <br/> <br/>|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9 \\ <br/> <br/>\Rightarrow 2\left(|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}\right)-2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=9 \\ <br/> <br/>\Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=\frac{-3}{2} <br/> <br/>\end{array}$ <br/> <br/>Also, <br/> <br/>$\begin{array}{l} <br/> <br/>|\vec{a}+\vec{b}+\vec{c}|^{2}=|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}) \\ <br/> <br/>=1+1+1+2 \times\left(-\frac{3}{2}\right)=0 \\ <br/> <br/>\Rightarrow \vec{a}+\vec{b}+\vec{c}=0 \Rightarrow(\vec{b}+\vec{c})=-\vec{a} \\ <br/> <br/>\therefore|2 \vec{a}+5(\vec{b}+\vec{c})|=|2 \vec{a}-5 \vec{a}|=|-3 \vec{a}|=3 <br/> <br/>\end{array}$</div>

MarksBatch2_P1.db

if-a-b-c-and-d-are-the-unit-vectors-such-that-a-b-c-d-1-and-a-c-2-1-then-1

if-a-b-c-and-d-are-the-unit-vectors-such-that-a-b-c-d-1-and-a-c-2-1-then-1-90395

<div class="question">If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are the unit vectors such that $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$ and $\mathbf{a} \cdot \mathbf{c}=\frac{1}{2}$, then</div>

['Mathematics', 'Vector Algebra', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathbf{b}, \mathbf{d}$ are non-parallel<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\mathbf{a}, \mathbf{b}, \mathbf{d}$ are non-coplanar<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, d are parallel and $\mathbf{b}, \mathbf{c}$ are parallel</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mathbf{a}, \mathbf{b}, \mathbf{d}$ are non-coplanar<br/></span> </div>

<div class="solution">Let angle between $\mathbf{a}$ and $\mathbf{b}$ be $\theta_1, \mathbf{c}$ and $\mathbf{d}$ be $\theta_2$ and $\mathbf{a} \times \mathbf{b}$ and $\mathbf{c} \times \mathbf{d}$ be $\theta$.<br/>Since, $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$<br/>$\Rightarrow \quad \sin \theta_1 \cdot \sin \theta_2 \cdot \cos \theta=1$<br/>$\Rightarrow \theta_1=90^{\circ}, \theta_2=90^{\circ}, \theta=0^{\circ}$<br/>$\Rightarrow \mathbf{a} \perp \mathbf{b}, \mathbf{c} \perp \mathbf{d},(\mathbf{a} \times \mathbf{b}) \|(\mathbf{c} \times \mathbf{d})$<br/>So, $\mathbf{a} \times \mathbf{b}=k(\mathbf{c} \times \mathbf{d})$<br/>and $\mathbf{a} \times \mathbf{b}=k(\mathbf{c} \times \mathbf{d})$<br/>$\Rightarrow \quad(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}=k(\mathbf{c} \times \mathbf{d}) \cdot \mathbf{c}$<br/>and $(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{d}=k(\mathbf{c} \times \mathbf{d}) \cdot \mathbf{d}$<br/>$\Rightarrow\left[\begin{array}{lll}\mathbf{a} &amp; \mathbf{b} &amp; \mathbf{c}\end{array}\right]=0$ and $\left[\begin{array}{lll}\mathbf{a} &amp; \mathbf{b} &amp; \mathbf{d}\end{array}\right]=0$<br/><br/>$\Rightarrow \mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{a}, \mathbf{b}, \mathbf{d}$ are coplanar vectors, so options (a) and (b) are incorrect.<br/>Let $\quad \mathbf{b} \| \mathbf{d} \Rightarrow \mathbf{b}=\pm \mathbf{d}$<br/>As $\quad(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$<br/>$(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{b})=\pm 1$<br/>$\Rightarrow \quad[\mathbf{a} \times \mathbf{b} \mathbf{c} \mathbf{b}]=\pm 1$<br/>$\Rightarrow \quad[\mathbf{c} \mathbf{b} \mathbf{a} \times \mathbf{b}]=\pm 1$<br/>$\Rightarrow \quad \mathbf{c} \cdot[\mathbf{b} \times(\mathbf{a} \times \mathbf{b})]=\pm 1$<br/>$\Rightarrow \quad \mathbf{c} \cdot[\mathbf{a}-(\mathbf{b} \cdot \mathbf{a}) \mathbf{b}]=\pm 1$<br/>$\Rightarrow \quad \mathbf{c} \cdot \mathbf{a}=\pm 1 \quad[\because \mathbf{a} \cdot \mathbf{b}=0]$<br/>which is a contradiction, so option (c) is correct.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/bsRyUwQUt7tJ1oXqJm47bT41RaPfrDvEeKzQHNgf_r4.original.fullsize.png"/><br/><br/>Let option (d) be correct.<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; \mathbf{d}=\pm \mathbf{a} \text { and } \mathbf{c}=\pm \mathbf{b} \\<br/>\text { As } &amp; (\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1 \\<br/>\Rightarrow &amp; (\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{b} \times \mathbf{a})=\pm 1<br/>\end{array}<br/>$$<br/>which is a contradiction, so option (d) is incorrect.<br/>Alternatively option (c) and (d) may be observed from the given figure.</div>

MarksBatch2_P1.db

if-a-b-c-and-d-are-the-unit-vectors-such-that-a-b-c-d-1-and-a-c-2-1-then

if-a-b-c-and-d-are-the-unit-vectors-such-that-a-b-c-d-1-and-a-c-2-1-then-91974

<div class="question">If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are the unit vectors such that $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$ and $\mathbf{a} \cdot \mathbf{c}=\frac{1}{2}$, then</div>

['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathbf{b}, \mathbf{d}$ are non-parallel<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\mathbf{a}, \mathbf{b}, \mathbf{d}$ are non-coplanar<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, d are parallel and $\mathbf{b}, \mathbf{c}$ are parallel</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mathbf{a}, \mathbf{b}, \mathbf{d}$ are non-coplanar<br/></span> </div>

<div class="solution">Let angle between $\mathbf{a}$ and $\mathbf{b}$ be $\theta_1, \mathbf{c}$ and $\mathbf{d}$ be $\theta_2$ and $\mathbf{a} \times \mathbf{b}$ and $\mathbf{c} \times \mathbf{d}$ be $\theta$.<br/>Since, $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$<br/>$\Rightarrow \quad \sin \theta_1 \cdot \sin \theta_2 \cdot \cos \theta=1$<br/>$\Rightarrow \theta_1=90^{\circ}, \theta_2=90^{\circ}, \theta=0^{\circ}$<br/>$\Rightarrow \mathbf{a} \perp \mathbf{b}, \mathbf{c} \perp \mathbf{d},(\mathbf{a} \times \mathbf{b}) \|(\mathbf{c} \times \mathbf{d})$<br/>So, $\mathbf{a} \times \mathbf{b}=k(\mathbf{c} \times \mathbf{d})$<br/>and $\mathbf{a} \times \mathbf{b}=k(\mathbf{c} \times \mathbf{d})$<br/>$\Rightarrow \quad(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}=k(\mathbf{c} \times \mathbf{d}) \cdot \mathbf{c}$<br/>and $(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{d}=k(\mathbf{c} \times \mathbf{d}) \cdot \mathbf{d}$<br/>$\Rightarrow\left[\begin{array}{lll}\mathbf{a} &amp; \mathbf{b} &amp; \mathbf{c}\end{array}\right]=0$ and $\left[\begin{array}{lll}\mathbf{a} &amp; \mathbf{b} &amp; \mathbf{d}\end{array}\right]=0$<br/><br/>$\Rightarrow \mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{a}, \mathbf{b}, \mathbf{d}$ are coplanar vectors, so options (a) and (b) are incorrect.<br/>Let $\quad \mathbf{b} \| \mathbf{d} \Rightarrow \mathbf{b}=\pm \mathbf{d}$<br/>As $\quad(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$<br/>$(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{b})=\pm 1$<br/>$\Rightarrow \quad[\mathbf{a} \times \mathbf{b} \mathbf{c} \mathbf{b}]=\pm 1$<br/>$\Rightarrow \quad[\mathbf{c} \mathbf{b} \mathbf{a} \times \mathbf{b}]=\pm 1$<br/>$\Rightarrow \quad \mathbf{c} \cdot[\mathbf{b} \times(\mathbf{a} \times \mathbf{b})]=\pm 1$<br/>$\Rightarrow \quad \mathbf{c} \cdot[\mathbf{a}-(\mathbf{b} \cdot \mathbf{a}) \mathbf{b}]=\pm 1$<br/>$\Rightarrow \quad \mathbf{c} \cdot \mathbf{a}=\pm 1 \quad[\because \mathbf{a} \cdot \mathbf{b}=0]$<br/>which is a contradiction, so option (c) is correct.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/bsRyUwQUt7tJ1oXqJm47bT41RaPfrDvEeKzQHNgf_r4.original.fullsize.png"/><br/><br/>Let option (d) be correct.<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; \mathbf{d}=\pm \mathbf{a} \text { and } \mathbf{c}=\pm \mathbf{b} \\<br/>\text { As } &amp; (\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1 \\<br/>\Rightarrow &amp; (\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{b} \times \mathbf{a})=\pm 1<br/>\end{array}<br/>$$<br/>which is a contradiction, so option (d) is incorrect.<br/>Alternatively option (c) and (d) may be observed from the given figure.</div>

MarksBatch2_P1.db

if-e-1-is-the-eccentricity-of-the-ellipse-16-x-2-25-y-2-1-and-e-2-is-the-eccentricity-of-the-hyperbola-passing-through-the-focii-of-the-ellipse-and-e-

if-e-1-is-the-eccentricity-of-the-ellipse-16-x-2-25-y-2-1-and-e-2-is-the-eccentricity-of-the-hyperbola-passing-through-the-focii-of-the-ellipse-and-e-40826

<div class="question">If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of the hyperbola passing through the focii of the ellipse and $e_1 e_2=1$, then equation of the hyperbola is</div>

['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{x^2}{9}-\frac{y^2}{16}=1$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{x^2}{16}-\frac{y^2}{9}=-1$<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/>$\frac{x^2}{9}-\frac{y^2}{25}=1$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>None of these</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$\frac{x^2}{16}-\frac{y^2}{9}=-1$<br/></span> </div>

<div class="solution">The eccentricity of $\frac{x^2}{16}+\frac{y^2}{25}=1$ is $e_1=\sqrt{1-\frac{16}{25}}=\frac{3}{5}$ $\therefore \quad e_2=\frac{5}{3}$<br/>$\Rightarrow$ Foci of ellipse $(0, \pm 3)$<br/>$\therefore$ Equation of hyperbola is $\frac{x^2}{16}-\frac{y^2}{9}=-1$<br/>Hence (b) is the correct answer.</div>

MarksBatch2_P1.db

if-f-x-0-x-e-t-2-t-2-t-3-d-t-for-all-x-0-then

if-f-x-0-x-e-t-2-t-2-t-3-d-t-for-all-x-0-then-99657

<div class="question">If $f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t$ for all $x \in(0, \infty)$, then</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

<div class="correct-answer"> The correct answers are: <span class="option-value">$f$ has a local maximum at $x=2$, $f$ is decreasing on $(2,3)$, there exists some $c \in(0, \infty)$, such that $f^{\prime \prime}(c)=0$, $f$ has a local minimum at $x=3$</span> </div>

<div class="solution">$f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t$ <br/> <br/>$\Rightarrow f^{\prime}(x)=e^{x^{2}} \cdot(x-2)(x-3)$ <br/> <br/>Put $f^{\prime}(x)=0 \Rightarrow x=2,3$ <br/> <br/>$f^{\prime \prime}(x)=e^{x^{2}} \cdot 2 x\left(x^{2}-5 x+6\right)+e^{x^{2}}(2 x-5)$ <br/> <br/>$f^{\prime \prime}(2)=-$ ve and $f^{\prime \prime}(3)=+$ ve <br/> <br/>$\therefore \quad x=2$ is a point of local maxima. <br/> <br/>and $x=3$ is a point of local minima. <br/> <br/>Also for $x \in(2,3), f^{\prime}(x) &lt; 0$ <br/> <br/>$\Rightarrow \quad f$ is decreasing on $(2,3)$. <br/> <br/>Also we observe $f^{\prime \prime}(0) &lt; 0$ and $f^{\prime \prime}(1)&gt;0$ <br/> <br/>$\therefore \quad$ There exists some $C \in(0,1)$ such that $f^{\prime \prime}(C)=0$ <br/> <br/>Hence all the options are correct.</div>

MarksBatch2_P1.db

if-f-x-f-x-where-f-x-is-a-continuous-double-differentiable-function-and-g-x-f-x-if-f-x-f-2-x-2-g-2-x-2-and-f-5-5-then-f-10-is

if-f-x-f-x-where-f-x-is-a-continuous-double-differentiable-function-and-g-x-f-x-if-f-x-f-2-x-2-g-2-x-2-and-f-5-5-then-f-10-is-26784

<div class="question">If $f^{\prime \prime}(x)=-f(x)$, where $f(x)$ is a continuous double differentiable function and $g(x)=f^{\prime}(x)$. If $F(x)=\left(f\left(\frac{x}{2}\right)\right)^2+\left(g\left(\frac{x}{2}\right)\right)^2$ and $F(5)=5$, then $F(10)$ is</div>

['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>0<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>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">C</span> <span class="option-data"><br/>10<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>25</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>5<br/></span> </div>

<div class="solution">As, $f(x)=-f(x) \Rightarrow \frac{d}{d x}\left(f^{\prime}(x)\right)=-f(x)$ $\Rightarrow \quad g^{\prime}(x)=-f(x)$ and $f^{\prime}(x)=g(x)$ where, $\therefore \quad F^{\prime}(x)=2\left(f\left(\frac{x}{2}\right)\right) \cdot f^{\prime}\left(\frac{x}{2}\right) \cdot \frac{1}{2}+2\left(g\left(\frac{x}{2}\right)\right) \cdot g^{\prime}\left(\frac{x}{2}\right) \cdot \frac{1}{2}=0$ [using Eq. (1)] $\therefore F(x)$ is constant $\Rightarrow F(10)=F(5)=5$</div>

MarksBatch2_P1.db

if-f-x-is-cubic-polynomial-which-has-local-maximum-at-x-1-if-f-2-18-f-1-1-and-f-x-has-local-minimum-at-x-0-then

if-f-x-is-cubic-polynomial-which-has-local-maximum-at-x-1-if-f-2-18-f-1-1-and-f-x-has-local-minimum-at-x-0-then-91259

<div class="question">If $f(x)$ is cubic polynomial which has local maximum at $x=-1$. If $f(2)=18, f(1)=-1$ and $f^{\prime}(x)$ has local minimum at $x=0$, then</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>the distance between $(-1,2)$ and $(a, f(a))$ where $x=a$ is the point of local minima is $2 \sqrt{5}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$f(x)$ is increasing for $x \in[1,2 \sqrt{5}]$<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="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$f(x)$ has local minima at $x=1$<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/>the value of $f(0)=5$</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$f(x)$ is increasing for $x \in[1,2 \sqrt{5}]$<br/>, <br/>$f(x)$ has local minima at $x=1$<br/></span> </div>

<div class="solution">Since $f(x)$ has local maxima at $x=-1$ and $f^{\prime}(x)$ has local minima at $x=0$.<br/>$$<br/>\begin{aligned}<br/>f \wedge(x) &amp; =\lambda x \\<br/>f^{\prime}(x) &amp; =\lambda \frac{x^2}{2}+c \\<br/>\frac{\lambda}{2}+c &amp; =0 \Rightarrow \lambda=-2 c<br/>\end{aligned}<br/>$$<br/>$$<br/>\left[f^{\prime}(-1)=0\right]<br/>$$<br/>Again, On integrating both sides, we get<br/>$$<br/>\begin{aligned}<br/>&amp; f(x)=\lambda \frac{x^3}{6}+c x+d \\<br/>&amp; f(2)=\lambda\left(\frac{8}{6}\right)+2 c+d=18 \\<br/>&amp; f(1)=\frac{\lambda}{6}+c+d=-1<br/>\end{aligned}<br/>$$<br/>and<br/>$\therefore$ Using Eqs. (i), (ii) and (iii), we get<br/>$$<br/>\begin{aligned}<br/>f(x) &amp; =\frac{1}{4}\left(19 x^3-57 x+34\right) \\<br/>\therefore \quad f^{\prime}(x) &amp; =\frac{1}{4}\left(57 x^2-57\right) \\<br/>&amp; =\frac{57}{4}(x-1)(x+1), \text { using number line rule }<br/>\end{aligned}<br/>$$<br/>$\therefore f(x)$ is increasing for $[1,2 \sqrt{5}]$ and $f(x)$ is increasing for $[1,2 \sqrt{5}]$ and $f(x)$ has local maximum at $x=-1$ and $f(x)$ has local minimum at $C x=1$.<br/>Also,<br/>$$<br/>f(0)=\frac{34}{4} \text {. }<br/>$$</div>

MarksBatch2_P1.db

if-f-x-is-twice-differentiable-function-such-that-f-a-0-f-b-2-f-c-1-f-d-2-f-e-0-where-a-b-c-d-e-then-the-minimum-number-of-zeros-of-g-x-f-x-2-f-x-f-x-

if-f-x-is-twice-differentiable-function-such-that-f-a-0-f-b-2-f-c-1-f-d-2-f-e-0-where-a-b-c-d-e-then-the-minimum-number-of-zeros-of-g-x-f-x-2-f-x-f-x-36402

<div class="question">If $f(x)$ is twice differentiable function such that $f(a)=0, f(b)=2, f(c)=1, f(d)=2$, $f(e)=0$, where $a &lt; b &lt; c &lt; d &lt; e$, then the minimum number of zeros of $g(x)=\left\{f^{\prime}(x)\right\}^2+f^{\prime \prime}(x) \cdot f(x)$ in the interval $[a, e]$ is</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006']

None

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

<div class="solution">Let, $g(x)=\frac{d}{d x}\left[f(x) \cdot f^{\prime}(x)\right]$ to get the zero of $g(x)$ we take function $h(x)=f(x) \cdot f^{\prime}(x)$ between any two roots of $h(x)$ there lies atleast one root of $h^{\prime}(x)=0$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad g(x)=0 \\<br/>&amp; \Rightarrow \quad h(x)=0 \\<br/>&amp; \Rightarrow \quad f(x)=0 \text { or } f^{\prime}(x)=0 \\<br/>&amp; \text { If } f(x)=0 \text { has } 4 \text { minimum solutions, } \\<br/>&amp; f^{\prime}(x)=0 \text { has } 3 \text { minimum solutions, } \\<br/>&amp; h(x)=0 \text { has } 7 \text { minimum solutions, then } \\<br/>&amp; h^{\prime}(x)=g(x)=0 \text { has } 6 \text { minimum solutions. } \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-f-x-min-1-x-2-x-3-then

if-f-x-min-1-x-2-x-3-then-71892

<div class="question">If $f(x)=\min \left\{1, x^2, x^3\right\}$, then</div>

['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2006']

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

<div class="solution">Here, $f(x)=\min .\left\{1, x^2, x^3\right\}$ which could be graphically shown as<br/>$$<br/>\therefore \quad f(x)=\left\{\begin{array}{c}<br/>1, x \geq 1 \\<br/>x^3, x &lt; 1<br/>\end{array}\right.<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/aMjTHolEozuYG-SqnJHOZWv-BUVKNUy9PvmW4-6Za-k.original.fullsize.png"/><br/><br/>$$<br/>\Rightarrow f(x) \text { is continuous for } x \in R \text { and not differentiable at } x=1 \text { due to sharp edge. }<br/>$$</div>

MarksBatch2_P1.db

if-f-x-x-2-cos-x-x-1-lo-g-x-x-2-2-x-0-0-x-1-x-1-then

if-f-x-x-2-cos-x-x-1-lo-g-x-x-2-2-x-0-0-x-1-x-1-then-78669

<div class="question">If $f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, &amp; x \leq-\frac{\pi}{2} \\ -\cos x, &amp; -\frac{\pi}{2} &lt; x \leq 0 \\ x-1, &amp; 0 &lt; x \leq 1 \\ \log x, &amp; x&gt;1\end{array}\right.$, then</div>

['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']

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

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$f(x)$ is continuous at $x=-\frac{\pi}{2}$<br/>, <br/>$f(x)$ is not differentiable at $x=0$<br/>, <br/>$f(x)$ is differentiable at $x=1$<br/>, <br/>$f(x)$ is differentiable at $x=-\frac{3}{2}$</span> </div>

<div class="solution">$f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, &amp; x \leq-\frac{\pi}{2} \\ -\cos x, &amp; -\frac{\pi}{2} &lt; x \leq 0 \\ x-1, &amp; 0 &lt; x \leq 1 \\ \log x, &amp; x&gt;1\end{array}\right.$<br/>Continuity at $x=-\frac{\pi}{2}$,<br/>$$<br/>\begin{aligned}<br/>&amp; f\left(-\frac{\pi}{2}\right)=-\left(-\frac{\pi}{2}\right)-\frac{\pi}{2}=0 \\<br/>&amp; \mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0}-\cos \left(-\frac{\pi}{2}+h\right)=0<br/>\end{aligned}<br/>$$<br/><br/>$\therefore$ Continuous at $x=0$<br/>Continuity at $x=0 \Rightarrow f(0)=-1$ $\mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0}(0+h)-1=-1$<br/>$\therefore$ Continuous at $x=0$<br/>Continuity at $x=1 ; f(1)=0$ $\mathrm{RHL} \Rightarrow \lim _{h \rightarrow 0} \log (1+h)=0$<br/>$\therefore$ Continuous at $x=1$<br/>Here, $f^{\prime}(x)=\left\{\begin{array}{cc}-1, &amp; x \leq-\frac{\pi}{2} \\ \sin x, &amp; -\frac{\pi}{2} &lt; x \leq 0 \\ 1, &amp; 0 &lt; x \leq 1 \\ \frac{1}{x}, &amp; x&gt;1\end{array}\right.$<br/>Differentiable at $x=0$, $\mathrm{LHD}=0, \mathrm{RHD}=1$<br/>$\therefore \quad$ Not differentiable at $x=0$<br/><br/>Differentiable at $x=1$, $\mathrm{LHD}=1, \mathrm{RHD}=1$<br/>$\therefore$ Differentiable at $x=1$<br/>Also, for $x=-\frac{3}{2} \Rightarrow f(x)=-x-\frac{3}{2}$<br/>$\therefore$ Differentiable at $x=-\frac{3}{2}$</div>

MarksBatch2_P1.db

if-i-n-1-x-s-i-n-x-s-i-n-n-x-d-x-n-0-1-2-then

if-i-n-1-x-s-i-n-x-s-i-n-n-x-d-x-n-0-1-2-then-10740

<div class="question">If $I_n=\int_{-\pi}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x ; n=0,1,2$, $\ldots$, then</div>

['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$I_n=I_{n+2}$<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="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\sum_{m=1}^{10} I_{2 m+1}=10 \pi$<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/>$\sum_{m=1}^{10} I_{2 m}=0$<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">D</span> <span class="option-data"><br/>$I_n=I_{n+1}$</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$I_n=I_{n+2}$<br/>, <br/>$\sum_{m=1}^{10} I_{2 m+1}=10 \pi$<br/>, <br/>$\sum_{m=1}^{10} I_{2 m}=0$<br/></span> </div>

<div class="solution">(a)<br/>$$<br/>\begin{aligned}<br/>&amp; I_n=\int_{-\pi}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x \\<br/>&amp; I_n=\int_{-\pi}^\pi \frac{\pi^x \sin n x}{\left(1+\pi^x\right) \sin x} d x \\<br/>&amp; {\left[\because \int_a^b f(x) d x=\int_a^b f(a+b-x) d x\right]} \\<br/>&amp; 2 I_n=\int_{-\pi}^\pi \frac{\sin n x}{\sin x} d x<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad 2 I_n=2 \int_0^\pi \frac{\sin n x}{\sin x} d x \\<br/>&amp; \Rightarrow \quad I_n=\int_0^\pi \frac{\sin n x}{\sin x} d x \\<br/>&amp; \text { Now, } \quad I_{n+2}-I_n \\<br/>&amp; =\int_0^\pi \frac{\sin (n+2) x-\sin n x}{\sin x} d x \\<br/>&amp; =\int_0^\pi \frac{2 \cos (n+1) x \sin x}{\sin x} d x \\<br/>&amp; =2\left[\frac{\sin (n+1) x}{(n+1)}\right]_0^\pi=0 \\<br/>&amp; \Rightarrow \quad I_{n+2}=I_n \\<br/>&amp; \text { (b) As } I_3=I_5=\ldots=I_{21} \\<br/>&amp; \therefore \sum_{m=1}^{10} I_{2 m+1}=10 I_3=10 \int_0^\pi \frac{\sin 3 x}{\sin x} d x \\<br/>&amp; =10 \int_0^\pi\left(3-4 \sin ^2 x\right) d x \\<br/>&amp; =10[3 x-2 x+\sin 2 x]_0^\pi=10 \pi \\<br/>&amp; \text { (c) As } \quad I_2=I_4=\ldots=I_{20} \\<br/>&amp; \therefore \quad \sum_{m=1}^{10} I_{2 m}=10 \int_0^\pi \frac{\sin 2 x}{\sin x} d x \\<br/>&amp; =20[\sin x]_0^\pi=0 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-lim-x-0-1-x-lo-g-1-b-2-1-x-2-b-sin-2-b-0-and-then-the-value-of-is

if-lim-x-0-1-x-lo-g-1-b-2-1-x-2-b-sin-2-b-0-and-then-the-value-of-is-36689

<div class="question">If $\lim _{x \rightarrow 0}\left[1+x \log \left(1+b^2\right)\right]^{1 / x}=2 b \sin ^2 \theta b&gt;0$ and $\theta \in(-\pi, \pi]$, then the value of $\theta$ is</div>

['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']

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

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

<div class="solution">Here, $\lim _{x \rightarrow 0}\left\{1+x \log \left(1+b^2\right)\right\}^{1 / x}$<br/>$$<br/>\begin{aligned}<br/>&amp; \text { Given, } \quad\left[1^{\infty} \text { from }\right] \\<br/>&amp; \Rightarrow \quad e^{\lim _{x \rightarrow 0}\left\{x \log \left(1+b^2\right)\right\} \cdot \frac{1}{x}} \\<br/>&amp; \Rightarrow \quad e^{\log \left(1+b^2\right)}=(1+b)^2 \\<br/>&amp; \lim _{x \rightarrow 0}\left\{1+x \log \left(1+b^2\right)\right\}^{1 / x}=2 b \sin ^2 \theta \\<br/>&amp; \Rightarrow \quad\left(1+b^2\right)=2 b \sin ^2 \theta \\<br/>&amp; \therefore \quad \sin ^2 \theta=\frac{1+b^2}{2 b} \quad \ldots \text { (ii) }<br/>\end{aligned}<br/>$$<br/>By $A M \geq G M$,<br/>$$<br/>\frac{b+\frac{1}{b}}{2} \geq\left(b \cdot \frac{1}{b}\right)^{1 / 2} \Rightarrow \frac{b^2+1}{2 b} \geq 1<br/>$$<br/>From Eqs. (ii) and (iii), we get $\sin ^2 \theta=1$<br/>$$<br/>\Rightarrow \quad \theta=\pm \frac{\pi}{2} \text { as } \theta \in(-\pi, \pi]<br/>$$</div>

MarksBatch2_P1.db

if-lim-x-x-1-x-2-x-1-a-x-b-4-then

if-lim-x-x-1-x-2-x-1-a-x-b-4-then-69627

<div class="question">If $\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4$, then</div>

['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value">$a=1, b=-4$</span> </div>

<div class="solution">Given: $\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4$ <br/> <br/>$\begin{array}{l} <br/> <br/>\Rightarrow \lim _{x \rightarrow \infty} \frac{x^{2}+x+1-a x^{2}-a x-b x-b}{x+1}=4 \\ <br/> <br/>\Rightarrow \lim _{x \rightarrow \infty} \frac{(1-a) x^{2}+(1-a-b) x+(1-b)}{x+1}=4 <br/> <br/>\end{array}$ <br/> <br/>For this limit to be finite $1-a=0 \Rightarrow a=1$ then given limit reduces to <br/> <br/>$\lim _{x \rightarrow \infty} \frac{-b x+(1-b)}{x+1}=4 \Rightarrow \lim _{x \rightarrow \infty} \frac{-b+\frac{(1-b)}{x}}{1+\frac{1}{x}}=4$ <br/> <br/>$\Rightarrow-b=4$ or $b=-4, \therefore a=1, b=-4$</div>

MarksBatch2_P1.db

if-p-is-a-3-3-matrix-such-that-p-t-2-p-i-where-p-t-is-the-transpose-of-p-and-i-is-the-3-3-identity-matrix-then-there-exists-a-column-matrix-x-x-y-z-0-

if-p-is-a-3-3-matrix-such-that-p-t-2-p-i-where-p-t-is-the-transpose-of-p-and-i-is-the-3-3-identity-matrix-then-there-exists-a-column-matrix-x-x-y-z-0-60539

<div class="question">If $P$ is a $3 \times 3$ matrix such that $P^{T}=2 P+I$, where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $\quad X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that</div>

['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value">$P X=-X$</span> </div>

<div class="solution">$P^{T}=2 P+I$ <br/> <br/>$\begin{array}{l} <br/> <br/>\Rightarrow P=2 P^{T}+I \Rightarrow P=2(2 P+I)+I \\ <br/> <br/>\Rightarrow P=4 P+3 I \Rightarrow P+I=0 \\ <br/> <br/>\Rightarrow P X+X=0 \Rightarrow P X=-X <br/> <br/>\end{array}$</div>

MarksBatch2_P1.db

if-r-s-t-are-prime-numbers-and-p-q-are-the-positive-integers-such-that-lcm-of-p-q-is-r-2-s-4-t-2-then-the-number-of-ordered-pairs-p-q-is-1

if-r-s-t-are-prime-numbers-and-p-q-are-the-positive-integers-such-that-lcm-of-p-q-is-r-2-s-4-t-2-then-the-number-of-ordered-pairs-p-q-is-1-15079

<div class="question">If $r, s, t$ are prime numbers and $p, q$ are the positive integers such that LCM of $p, q$ is $r^2 s^4 t^2$, then the number of ordered pairs $(p, q)$ is</div>

['Mathematics', 'Permutation Combination', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>252<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>254<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>225<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/>224</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>225<br/></span> </div>

<div class="solution">Since, $r, s, t$ are prime numbers.<br/>$\therefore$ Selection of $p$ and $q$ are as under<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/PK48_ScZ9Uo3f0yPK15Dvens-4AMMDMQM64oOlmGudc.original.fullsize.png"/><br/><br/>$\therefore$ Total number of ways to select $s=9$.<br/>Similarly, the number of ways to select $t=5$.<br/>$\therefore$ Total number of ways $=5 \times 9 \times 5=225$.</div>

MarksBatch2_P1.db

if-r-s-t-are-prime-numbers-and-p-q-are-the-positive-integers-such-that-lcm-of-p-q-is-r-2-s-4-t-2-then-the-number-of-ordered-pairs-p-q-is

if-r-s-t-are-prime-numbers-and-p-q-are-the-positive-integers-such-that-lcm-of-p-q-is-r-2-s-4-t-2-then-the-number-of-ordered-pairs-p-q-is-57452

<div class="question">If $r, s, t$ are prime numbers and $p, q$ are the positive integers such that LCM of $p, q$ is $r^2 s^4 t^2$, then the number of ordered pairs $(p, q)$ is</div>

['Mathematics', 'Permutation Combination', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>252<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>254<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>225<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/>224</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>225<br/></span> </div>

<div class="solution">Since, $r, s, t$ are prime numbers.<br/>$\therefore$ Selection of $p$ and $q$ are as under<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/PK48_ScZ9Uo3f0yPK15Dvens-4AMMDMQM64oOlmGudc.original.fullsize.png"/><br/><br/>$\therefore$ Total number of ways to select $s=9$.<br/>Similarly, the number of ways to select $t=5$.<br/>$\therefore$ Total number of ways $=5 \times 9 \times 5=225$.</div>

MarksBatch2_P1.db

if-the-adjoint-of-a-3-3-matrix-p-is-1-2-1-4-1-1-4-7-3-then-the-possible-values-of-the-determinant-of-p-is-are

if-the-adjoint-of-a-3-3-matrix-p-is-1-2-1-4-1-1-4-7-3-then-the-possible-values-of-the-determinant-of-p-is-are-90249

<div class="question">If the adjoint of a $3 \times 3$ matrix $P$ is $\left[\begin{array}{lll}1 &amp; 4 &amp; 4 \\ 2 &amp; 1 &amp; 7 \\ 1 &amp; 1 &amp; 3\end{array}\right]$, then the possible value(s) of the determinant of $P$ is (are)</div>

['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

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

<div class="solution">We know for a third order matrix $P$, <br/> <br/>$|\operatorname{Adj} P|=|P|^{2}$ <br/> <br/>Where <br/> <br/>$\begin{aligned} <br/> <br/>&amp;|\operatorname{Adj} P|=1(3-7)-4(6-7)+4(2-1)=4 \\ <br/> <br/>\therefore \quad &amp;|P|^{2}=4 \mathrm{P}|P|=2 \text { or }-2 <br/> <br/>\end{aligned}$</div>

MarksBatch2_P1.db

if-the-angles-a-b-and-c-of-a-triangle-are-in-an-arithmetic-progression-and-if-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-a-b-and-c-respecti

if-the-angles-a-b-and-c-of-a-triangle-are-in-an-arithmetic-progression-and-if-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-a-b-and-c-respecti-70870

<div class="question">If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $\frac{a}{c} \sin 2 C+\frac{c}{a} \sin 2 A$ is</div>

['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2010 (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/>$\frac{\sqrt{3}}{2}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>1<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\sqrt{3}$</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 answer is: <span class="option-value"><br/>$\sqrt{3}$</span> </div>

<div class="solution">Since, $A, B, C$ are in $\mathrm{AP}$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad 2 B=A+C \text { ie, } \angle B=60^{\circ} \\<br/>&amp; \therefore \frac{a}{c}(2 \sin C \cos C)+\frac{c}{a}(2 \sin A \cos A) \\<br/>&amp; =2 k(a \cos C+c \cos A) \\<br/>&amp; \quad\left[\begin{array}{l}<br/>\text { using, } \\<br/>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{1}{k}<br/>\end{array}\right] \\<br/>&amp; =2 k(b) \\<br/>&amp; =2 \sin B<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>=\sqrt{3}<br/>$$<br/>$$<br/>\text { [using, } b=a \cos C+c \cos A]<br/>$$</div>

MarksBatch2_P1.db

if-the-bond-length-of-co-bond-in-carbon-monoxide-is-1128-a-then-what-is-the-value-of-co-bond-length-in-fe-co-5

if-the-bond-length-of-co-bond-in-carbon-monoxide-is-1128-a-then-what-is-the-value-of-co-bond-length-in-fe-co-5-24121

<div class="question">If the bond length of $\mathrm{CO}$ bond in carbon monoxide is $1.128 \mathrm{~A}^{\circ}$, then what is the value of $\mathrm{CO}$ bond length in $\mathrm{Fe}(\mathrm{CO})_5$ ?</div>

['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$1.15 Å$<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/>$1.128 Å$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$1.72 Å$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$1.118 Å$</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$1.15 Å$<br/></span> </div>

<div class="solution">The metal (Fe) makes the back bonding to $\mathrm{CO}$. The metal to ligand bonding creates a synergic effect which strengthens the bond $\mathrm{CO}$ and the metal ( $\mathrm{Fe}$ ). Which results in the contraction of $\mathrm{CO}$ bond length (Synergic means working together towards a common end. This is a bond formed by a ligand to metal ion which has a $\sigma$-bond from ligand to metal ion and the back donation of electron density from the metal to the $\pi^*$ orbitals on the ligand. Thus enhancing the acceptor strength of the p-orbital. Hence up to a point the effect of $\sigma$-bond formation strengthens the $\pi$-bonding or vice-versa. Due to this synergic effect bond order decreases.</div>

MarksBatch2_P1.db

if-the-distance-between-the-plane-a-x-2-y-z-d-and-the-plane-containing-the-lines-2-x-1-3-y-2-4-z-3-and-3-x-2-4-y-3-5-z-4-is-6-then-d-is

if-the-distance-between-the-plane-a-x-2-y-z-d-and-the-plane-containing-the-lines-2-x-1-3-y-2-4-z-3-and-3-x-2-4-y-3-5-z-4-is-6-then-d-is-65701

<div class="question">If the distance between the plane $A x-2 y+z=d$ and the plane containing the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad$ and $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ is $\sqrt{6}$, then $|d|$ is</div>

['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

None

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

<div class="solution">Equation of plane containing the given lines is $\left|\begin{array}{ccc}x-1 &amp; y-2 &amp; z-3 \\ 2 &amp; 3 &amp; 4 \\ 3 &amp; 4 &amp; 5\end{array}\right|=0$<br/>$$<br/>\begin{array}{rr}<br/>\Rightarrow &amp; (x-1)(-1)-(y-2)(-2) \\<br/>&amp; +(z-3)(-1)=0 \\<br/>\Rightarrow &amp; -x+1+2 y-4-z+3=0 \\<br/>\Rightarrow &amp; -x+2 y-z=0 \quad \ldots(i)<br/>\end{array}<br/>$$<br/>Given plane is<br/>$$<br/>x-2 y+z=d<br/>$$<br/><br/>Eqs. (i) and (ii) are parallel.<br/>Clearly, $\quad A=1$<br/>Now, distance between plane<br/>$$<br/>\begin{aligned}<br/>&amp; =\left|\frac{d}{\sqrt{1+4+1}}\right|=\sqrt{6} \\<br/>&amp; \Rightarrow \quad|d|=6 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-the-distance-of-the-point-p-1-2-1-from-the-plane-x-2-y-2-z-where-0-is-5-then-the-foot-of-the-perpendicular-from-p-to-the-plane-is

if-the-distance-of-the-point-p-1-2-1-from-the-plane-x-2-y-2-z-where-0-is-5-then-the-foot-of-the-perpendicular-from-p-to-the-plane-is-69527

<div class="question">If the distance of the point $P(1,-2,1)$ from the plane $x+2 y-2 z=\alpha$, where $\alpha&gt;0$, is 5 , then the foot of the perpendicular from $P$ to the plane is</div>

['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{8}{3}, \frac{4}{3},-\frac{7}{3}\right)$<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/>$\left(\frac{4}{3},-\frac{4}{3}, \frac{1}{3}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{1}{3}, \frac{2}{3}, \frac{10}{3}\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(\frac{2}{3},-\frac{1}{3}, \frac{5}{2}\right)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\left(\frac{8}{3}, \frac{4}{3},-\frac{7}{3}\right)$<br/></span> </div>

<div class="solution">$$<br/>\text { Distance of point } P \text { from plane }=5<br/>$$<br/><br/>$$<br/>\begin{gathered}<br/>\therefore=\left|\frac{1-4-2-\alpha}{3}\right| \\<br/>\alpha=10<br/>\end{gathered}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/MOxuxyIje_ig706wFu1-BnuXYO0fzZdi-AYRS4GoBC8.original.fullsize.png"/><br/><br/><br/>Foot of perpendicular<br/>$$<br/>\begin{aligned}<br/>\quad \frac{x-1}{1} &amp; =\frac{y+2}{2}=\frac{z-1}{-2}=\frac{5}{3} \\<br/>\Rightarrow \quad \quad \quad x &amp; =\frac{8}{3}, y=\frac{4}{3}, z=-\frac{7}{3}<br/>\end{aligned}<br/>$$<br/>Thus, the foot of the perpendicular is<br/>$$<br/>A\left(\frac{8}{3}, \frac{4}{3},-\frac{7}{3}\right)<br/>$$</div>

MarksBatch2_P1.db

if-the-function-f-x-x-3-e-2-x-and-g-x-f-1-x-then-the-value-of-g-1-is

if-the-function-f-x-x-3-e-2-x-and-g-x-f-1-x-then-the-value-of-g-1-is-46015

<div class="question">If the function $f(x)=x^3+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is</div>

['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

None

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

<div class="solution">Given, $g\{f(x)\}=x$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow &amp; g^{\prime}\{f(x)\} f^{\prime}(x)=1 \\<br/>\text { If } &amp; f(x)=1 \Rightarrow x=0, f(0)=1<br/>\end{array}<br/>$$<br/>Substitute $x=0$ in Eq. (i), we get<br/>$$<br/>\begin{aligned}<br/>&amp; g^{\prime}(1)=\frac{1}{f^{\prime}(0)} \\<br/>&amp; \Rightarrow \quad g^{\prime}(1)=2 \\<br/>&amp; {\left[\because f^{\prime}(x)=3 x^2+\frac{1}{2} e^{x / 2} \Rightarrow f^{\prime}(0)=\frac{1}{2}\right]} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-the-resultant-of-all-the-external-forces-acting-on-a-system-of-particles-is-zero-then-from-an-inertial-frame-one-can-surely-say-that

if-the-resultant-of-all-the-external-forces-acting-on-a-system-of-particles-is-zero-then-from-an-inertial-frame-one-can-surely-say-that-64667

<div class="question">If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that</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/>linear momentum of the system does not change in time<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/>kinetic energy of the system does not change in time<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>angular momentum of the system does not change in time<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>potential energy of the system does not change in time</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>linear momentum of the system does not change in time<br/></span> </div>

<div class="solution">On a system of particles if,<br/>$$<br/>\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{ext} .}=0<br/>$$<br/>then $\quad \overrightarrow{\mathbf{P}}_{\text {system }}=$ constant<br/>No other conclusions can be drawn.</div>

MarksBatch2_P1.db

if-the-straight-lines-2-x-1-k-y-1-2-z-and-5-x-1-2-y-1-k-z-are-coplanar-then-the-plane-s-containing-these-two-lines-is-are

if-the-straight-lines-2-x-1-k-y-1-2-z-and-5-x-1-2-y-1-k-z-are-coplanar-then-the-plane-s-containing-these-two-lines-is-are-33372

<div class="question">If the straight lines $\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}$ and $\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}$ are coplanar, then the plane (s) containing these two lines is (are)</div>

['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

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

<div class="solution">Given that lines are coplanar. <br/> <br/>$\therefore\left|\begin{array}{ccc} <br/> <br/>x_{2}-x_{1} &amp; y_{2}-y_{1} &amp; z_{2}-z_{1} \\ <br/> <br/>a_{1} &amp; b_{1} &amp; c_{1} \\ <br/> <br/>a_{2} &amp; b_{2} &amp; c_{2} <br/> <br/>\end{array}\right|=\left|\begin{array}{ccc} <br/> <br/>2 &amp; 0 &amp; 0 \\ <br/> <br/>2 &amp; k &amp; 2 \\ <br/> <br/>5 &amp; 2 &amp; k <br/> <br/>\end{array}\right|=0 \Rightarrow k=\pm 2$ <br/> <br/>For $k=2$, equation of the plane is given by $\left|\begin{array}{ccc}x-1 &amp; y+1 &amp; z \\ 2 &amp; 2 &amp; 2 \\ 5 &amp; 2 &amp; 2\end{array}\right|=0 \Rightarrow y-z+1=0$ <br/> <br/>For $k=-2$, equation of the plane is given by $\left|\begin{array}{ccc}x-1 &amp; y+1 &amp; z \\ 2 &amp; -2 &amp; 2 \\ 5 &amp; 2 &amp; -2\end{array}\right|=0 \Rightarrow y+z+1=0$</div>

MarksBatch2_P1.db

if-the-sum-of-first-n-terms-of-an-ap-is-cn-2-then-the-sum-of-squares-of-these-n-terms-is

if-the-sum-of-first-n-terms-of-an-ap-is-cn-2-then-the-sum-of-squares-of-these-n-terms-is-63010

<div class="question">If the sum of first $n$ terms of an $\mathrm{AP}$ is $\mathrm{cn}^2$, then the sum of squares of these $n$ terms is</div>

['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{n\left(4 n^2-1\right) c^2}{6}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{n\left(4 n^2+1\right) c^2}{3}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{n\left(4 n^2-1\right) c^2}{3}$<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/>$\frac{n\left(4 n^2+1\right) c^2}{6}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{n\left(4 n^2-1\right) c^2}{3}$<br/></span> </div>

<div class="solution">Let $S_n=c^2$<br/>$$<br/>\begin{aligned}<br/>&amp; S_{n-1}=c(n-1)^2=c n^2+c-2 c n \\<br/>&amp; \therefore T_n=2 c n-c \quad\left(\because T_n=S_n-S_{n-1}\right) \\<br/>&amp; T_n^2=(2 c n-c)^2=4 c^2 n^2+c^2-4 c^2 n \\<br/>&amp; \therefore \text { Sum }=\Sigma T_n^2=\frac{4 c^2 \cdot n(n+1)(2 n+1)}{6} \\<br/>&amp; =\frac{2 c^2 n(n+1)(2 n+1)+3 n c^2-6 c^2 n(n+1)}{3} \\<br/>&amp; =\frac{n c^2\left[4 n^2+6 n+2+3-6 n-6\right]}{3} \\<br/>&amp; =\frac{n c^2\left(4 n^2-1\right)}{3}<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-the-wavelength-of-the-n-th-line-of-lyman-series-is-equal-to-the-debroglie-wavelength-of-electron-in-initial-orbit-of-a-hydrogen-like-element-z-11-f

if-the-wavelength-of-the-n-th-line-of-lyman-series-is-equal-to-the-debroglie-wavelength-of-electron-in-initial-orbit-of-a-hydrogen-like-element-z-11-f-92561

<div class="question">If the wavelength of the $n^{\text {th }}$ line of Lyman series is equal to the de-Broglie wavelength of electron in initial orbit of a hydrogen like element $(Z=11)$. Find the value of $n$.</div>

['Physics', 'Atomic Physics', 'JEE Advanced', 'JEE Advanced 2006']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>

<div class="solution">$n^{\text {th }}$ line of Lyman series means transition from $(n+1)$ th state to first state.<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{1}{\lambda}=R Z^2\left[1-\frac{1}{(n+1)^2}\right] \\<br/>&amp; \text { de-Broglie wavelength, } \quad \lambda=\frac{h}{m v}=\frac{h r}{m v r}=\frac{(2 \pi)(h r)}{(n+1) h}=\frac{2 \pi r}{(n+1)} \\<br/>&amp; \frac{1}{\lambda}=\frac{(n+1)}{2 \pi r} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>or<br/>Equating (i) and (ii), we get<br/>$$<br/>\left(\frac{n+1}{2 \pi r}\right)=R Z^2\left[\frac{n(n+2)}{(n+1)^2}\right]<br/>$$<br/>Now, as<br/>$$<br/>\begin{aligned}<br/>&amp; r \propto \frac{n^2}{Z} \\<br/>&amp; r=\frac{(n+1)^2}{11} r_0<br/>\end{aligned}<br/>$$<br/>$$<br/>\therefore \quad r=\frac{(n+1)^2}{11} r_0<br/>$$<br/>Substituting in equations (iii), we get<br/>$$<br/>\frac{11}{2 \pi r_0}=\frac{R(11)^2(n)(n+2)}{(n+1)}<br/>$$<br/>or<br/>$$<br/>(n+1)=\left(1.09 \times 10^7\right)(11)(2 \pi) \times\left(0.529 \times 10^{-10}\right)\left(n^2+2 n\right)<br/>$$<br/>Solving this equation we get, $n=24$</div>

MarksBatch2_P1.db

if-w-i-where-0-and-z-1-satisfies-the-condition-that-1-z-w-w-z-is-purely-real-then-the-set-of-values-of-z-is

if-w-i-where-0-and-z-1-satisfies-the-condition-that-1-z-w-w-z-is-purely-real-then-the-set-of-values-of-z-is-74440

<div class="question">If $w=\alpha+i \beta$, where $\beta \neq 0$ and $z \neq 1$ satisfies the condition that $\left(\frac{w-\bar{w} z}{1-z}\right)$ is purely real, then the set of values of $z$ is</div>

['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$|z|=1$ and $z \neq 2$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$|z|=1$ and $z \neq 1$<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/>$z=\bar{Z}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>None of these</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$|z|=1$ and $z \neq 1$<br/></span> </div>

<div class="solution">Let $z_1=\frac{w-\bar{w}_z}{1-z}$, be purely real.<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow &amp; z_1 &amp; =\bar{z}_1 \\<br/>&amp; \therefore &amp; \frac{w-\bar{w} z}{1-z} &amp; =\frac{\bar{w}-w \bar{z}}{1-\bar{z}} \\<br/>\Rightarrow &amp; &amp; w-w \bar{z}-\bar{w} z+\bar{w} z \cdot \bar{z} &amp; =\bar{w}-z \bar{w}-w \bar{z}+w z \cdot \bar{z} \\<br/>\Rightarrow &amp; &amp; (w-\bar{w})+(\bar{w}-w)|z|^2 &amp; =0 \\<br/>\Rightarrow &amp; &amp; (w-\bar{w})\left(1-|z|^2\right) &amp; =0 \\<br/>\Rightarrow &amp; &amp; |z|^2 &amp; =1 \\<br/>\Rightarrow &amp; &amp; |z| &amp; =1 \text { and } z \neq 1 .<br/>\end{array}<br/>$$<br/>$$<br/>\text { [as, } w-\bar{w} \neq 0 \text {, since } \beta \neq 0 \text { ] }<br/>$$</div>

MarksBatch2_P1.db

if-x-2-10-a-x-11-b-0-have-roots-c-and-d-and-x-2-10-c-x-11-d-0-have-roots-a-and-b-then-find-a-b-c-d

if-x-2-10-a-x-11-b-0-have-roots-c-and-d-and-x-2-10-c-x-11-d-0-have-roots-a-and-b-then-find-a-b-c-d-33913

<div class="question">If $x^2-10 a x-11 b=0$ have roots $c$ and $d$ and $x^2-10 c x-11 d=0$ have roots $a$ and $b$, then, find $a+b+c+d$.</div>

['Mathematics', 'Quadratic Equation', 'JEE Advanced', 'JEE Advanced 2006']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">1210</span> </div>

<div class="solution">Here, $a+b=10 c$<br/>$$<br/>c+d=10 a<br/>$$<br/>On subtracting Eq. (ii) from Eq. (i), we get, $\quad(a-c)+(b-d)=10(c-a)$ $\Rightarrow \quad(b-d)=11(c-a)$<br/>Since, $c$ is the root of $x^2-10 a x+11 b=0$<br/>$$<br/>\Rightarrow \quad c^2-10 a c-11 b=0<br/>$$<br/>Similarly, $a$ is the root of<br/>$$<br/>\begin{aligned}<br/>\Rightarrow &amp; x^2-10 c x+11 d=0 \\<br/>&amp; a^2-10 c a-11 d=0<br/>\end{aligned}<br/>$$<br/>On subtracting Eq. (v) from Eq. (iv), we get<br/>$$<br/>\begin{aligned}<br/>&amp; \left(c^2-a^2\right)=11(b-d) \\<br/>&amp; \therefore \quad(c+a)(c-a)=11 \times 11(c-a) \quad\{\text { from (i) \&amp; (ii) } \\<br/>&amp; \Rightarrow \quad c+a=121 . \\<br/>&amp; \therefore \quad a+b+c+d=10 c+10 a \\<br/>&amp; =10(c+a)=1210 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P1.db

if-y-x-satisfies-the-differential-equation-y-y-tan-x-2-x-sec-x-and-y-0-0-then

if-y-x-satisfies-the-differential-equation-y-y-tan-x-2-x-sec-x-and-y-0-0-then-96515

<div class="question">If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x$ $=2 x \sec x$ and $y(0)=0$, then</div>

['Mathematics', 'Differential Equations', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{8 \sqrt{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y^{\prime}\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{18}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y\left(\frac{\pi}{3}\right)=\frac{\pi^{2}}{9}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$y^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^{2}}{3 \sqrt{3}}$</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">$y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{8 \sqrt{2}}$, $y^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^{2}}{3 \sqrt{3}}$</span> </div>

<div class="solution">$\frac{d y}{d x}-y \tan x=2 x \sec x$ <br/> <br/>I.F. $=e^{-\int \tan x d x}=\cos x$ <br/> <br/>$\therefore \quad y \cdot \cos x=\int 2 x d x=x^{2}+c$ <br/> <br/>Now, $y(0)=0 \Rightarrow c=0, \therefore y=x^{2} \sec x$ <br/> <br/>$\Rightarrow y^{\prime}=2 x \sec x+x^{2} \sec x \tan x$ <br/> <br/>Now, $y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{16} \times \sqrt{2}=\frac{\pi^{2}}{8 \sqrt{2}}$ <br/> <br/>$y\left(\frac{\pi}{3}\right)=\frac{\pi^{2}}{9} \times 2=\frac{2 \pi^{2}}{9}$ <br/> <br/>$y^{\prime}\left(\frac{\pi}{4}\right)=\frac{2 \pi}{4} \times \sqrt{2}+\frac{\pi^{2}}{8 \sqrt{2}} \times 1=\frac{\pi^{2}}{8 \sqrt{2}}+\frac{\pi}{\sqrt{2}}$ <br/> <br/>$y^{\prime}\left(\frac{\pi}{3}\right)=\frac{2 \pi}{3} \times 2+\frac{2 \pi^{2}}{9} \times \sqrt{3}=\frac{2 \pi^{2}}{3 \sqrt{3}}+\frac{4 \pi}{3}$</div>

MarksBatch2_P1.db

if-z-1-and-z-1-then-all-the-values-of-1-z-2-z-lie-on

if-z-1-and-z-1-then-all-the-values-of-1-z-2-z-lie-on-19943

<div class="question">If $|z|=1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on</div>

['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>the Y-axis</span> </div>

<div class="solution">Let $z=\cos \theta+i \sin \theta$<br/>$$<br/>\begin{aligned}<br/>\frac{z}{1-z^2} &amp; =\frac{\cos \theta+i \sin \theta}{1-(\cos 2 \theta+i \sin 2 \theta)} \\<br/>&amp; =\frac{\cos \theta+i \sin \theta}{2 \sin ^2 \theta-2 i \sin \theta \cos \theta} \\<br/>&amp; =\frac{\cos \theta+i \sin \theta}{-2 i \sin \theta(\cos \theta+i \sin \theta)} \\<br/>&amp; =\frac{i}{2 \sin \theta}<br/>\end{aligned}<br/>$$<br/>Hence, $\frac{z}{1-z^2}$ lies on the imaginary axis, i.e. $x=0$.<br/>ALITER<br/>Let<br/>$E=\frac{z}{1-z^2}=\frac{z}{z \bar{z}-z^2}=\frac{1}{\bar{z}-z}$ which is imaginary.</div>

MarksBatch2_P1.db

if-z-is-any-complex-number-satisfying-z-3-2-i-2-then-the-maximum-value-of-2-z-6-5-i-is

if-z-is-any-complex-number-satisfying-z-3-2-i-2-then-the-maximum-value-of-2-z-6-5-i-is-19545

<div class="question">If $z$ is any complex number satisfying $|z-3-2 i| \leq 2$, then the maximum value of $|2 z-6+5 i|$ is</div>

['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>

<div class="solution">Given, $|z-3-2 i| \leq 2$<br/>To find minimum of $|2 z-6+5 i|$ or $\quad 2\left|z-3+\frac{5}{2} i\right|$<br/>Using triangle inequality, i.e. ||$z_1|-| z_2|| \leq\left|z_1+z_2\right|$ $\therefore\left|z-3+\frac{5}{2} i\right|$ $=\left|z-3-2 i+2 i+\frac{5}{2} i\right|$ $=\left|(z-3-2 i)+\frac{9}{2} i\right| \geq|| z-3-2 i\left|-\frac{9}{2}\right|$ $\geq\left|2-\frac{9}{2}\right| \geq \frac{5}{2} \Rightarrow\left|z-3+\frac{5}{2} i\right| \geq \frac{5}{2}$ or $\quad|2 z-6+5 i| \geq 5$</div>

MarksBatch2_P1.db

image-of-an-object-approaching-a-convex-mirror-of-radius-of-curvature-20-m-along-its-optical-axis-is-observed-to-move-from-3-25-m-to-7-50-m-in-30-s-wh

image-of-an-object-approaching-a-convex-mirror-of-radius-of-curvature-20-m-along-its-optical-axis-is-observed-to-move-from-3-25-m-to-7-50-m-in-30-s-wh-40875

<div class="question">Image of an object approaching a convex mirror of radius of curvature $20 \mathrm{~m}$ along its optical axis is observed to move from $\frac{25}{3} \mathrm{~m}$ to $\frac{50}{7} \mathrm{~m}$ in $30 \mathrm{~s}$. What is the speed of the object in $\mathrm{km} \mathrm{h}^{-1}$ ?</div>

['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']

None

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

<div class="solution">Using mirror formula twice,<br/>$\frac{1}{+25 / 3}+\frac{1}{-u_1}=\frac{1}{+10}$<br/>or $\quad \frac{1}{u_1}=\frac{3}{25}-\frac{1}{10}$ or $u_1=50 \mathrm{~m}$<br/>and, $\quad \frac{1}{(+50 / 7)}+\frac{1}{-u_2}=\frac{1}{+10}$<br/>$\therefore \quad \frac{1}{u_2}=\frac{7}{50}-\frac{1}{10}$ or $u_2=25 \mathrm{~m}$<br/>Speed of object $=\frac{u_1-u_2}{\text { time }}$<br/>$=\frac{25}{30} \mathrm{~ms}^{-1}$<br/>$=3 \mathrm{kmh}^{-1}$<br/>$\therefore$ The answer is 3 .</div>

MarksBatch2_P1.db

in-1-l-saturated-solution-of-agcl-k-sp-agcl-16-1-0-10-01-mole-of-cucl-k-sp-cucl-10-1-0-6-is-added-the-resultant-concentration-of-ag-in-the-solution-is

in-1-l-saturated-solution-of-agcl-k-sp-agcl-16-1-0-10-01-mole-of-cucl-k-sp-cucl-10-1-0-6-is-added-the-resultant-concentration-of-ag-in-the-solution-is-57581

<div class="question">In $1 \mathrm{~L}$ saturated solution of $\mathrm{AgCl}\left[K_{\mathrm{sp}} \mathrm{AgCl}=1.6 \times 10^{-10}\right], 0.1$ mole of $\mathrm{CuCl}\left[K_{\mathrm{sp}}(\mathrm{CuCl})=1.0 \times 10^{-6}\right]$ is added. The resultant concentration of $\mathrm{Ag}^{+}$in the solution is $1.6 \times 10^{-x}$. The value of $x$ is</div>

['Chemistry', 'Ionic Equilibrium', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>

<div class="solution">It is a case of simultaneous solubility of salts with a common ion. Here solubility product of $\mathrm{CuCl}$ is much greater than that of $\mathrm{AgCl}$, it can be assumed that $\mathrm{Cl}^{-}$in solution comes mainly from $\mathrm{CuCl}$.<br/>$$<br/>\Rightarrow \quad\left[\mathrm{Cl}^{-}\right]=\sqrt{K_{s p}(\mathrm{CuCl})}=10^{-3} \mathrm{M}<br/>$$<br/>Now for $\mathrm{AgCl}: K_{s p}=1.6 \times 10^{-10}=\left[\mathrm{Ag}^{+}\right]\left[\mathrm{Cl}^{-}\right]=\left[\mathrm{Ag}^{+}\right] \times 10^{-3} \Rightarrow\left[\mathrm{Ag}^{+}\right]=1.6 \times 10^{-7}$</div>

MarksBatch2_P1.db

in-a-a-bc-with-fixed-base-bc-the-vertex-a-moves-such-that-cos-b-cos-c-4-sin-2-2-a-if-a-b-and-c-denote-the-lengths-of-the-sides-of-the-triangle-opposit

in-a-a-bc-with-fixed-base-bc-the-vertex-a-moves-such-that-cos-b-cos-c-4-sin-2-2-a-if-a-b-and-c-denote-the-lengths-of-the-sides-of-the-triangle-opposit-15702

<div class="question">In a $\triangle A B C$ with fixed base $B C$, the vertex $A$ moves such that $\cos B+\cos C=4 \sin ^2 \frac{A}{2}$. If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$ respectively, then</div>

['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$b+c=4 a$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$b+c=2 a$<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/>locus of point $A$ is an ellipse<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/>locus of point $A$ is a pair of straight line</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>$b+c=2 a$<br/>, <br/>locus of point $A$ is an ellipse<br/></span> </div>

<div class="solution">Given, $\cos B+\cos C=4 \sin ^2 \frac{A}{2}$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow 2 \cos \left(\frac{B+C}{2}\right) \cos \left(\frac{B-C}{2}\right)=4 \sin ^2 \frac{A}{2} \\<br/>&amp; \Rightarrow 2 \sin \frac{A}{2}\left[\cos \left(\frac{B-C}{2}\right)-2 \sin \frac{A}{2}\right]=0 \\<br/>&amp; \Rightarrow \cos \left(\frac{B-C}{2}\right)-2 \cos \left(\frac{B+C}{2}\right)=0 \\<br/>&amp; \Rightarrow-\cos \frac{B}{2} \cos \frac{C}{2}+3 \sin \frac{B}{2} \sin \frac{C}{2}=0 \\<br/>&amp; \Rightarrow \quad \tan \frac{B}{2} \tan \frac{C}{2}=\frac{1}{3} \\<br/>&amp; \Rightarrow \quad \sqrt{\frac{(s-a)(s-c)}{s(s-b)} \cdot \frac{(s-b)(s-a)}{s(s-c)}}=\frac{1}{3} \\<br/>&amp; \Rightarrow \quad \frac{s-a}{s}=\frac{1}{3} \Rightarrow 2 s=3 a \\<br/>&amp; \Rightarrow \quad b+c=2 a<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/yyaF-wntrMyQyNp7xROYMA8aGwXqkF61bQHF6flrPxg.original.fullsize.png"/><br/><br/>$\therefore$ Locus of $A$ is an ellipse.</div>

MarksBatch2_P1.db

in-a-circuit-a-metal-filament-lamp-is-connected-in-series-with-a-capacitor-of-capacitance-c-f-across-a-200-v-50-hz-supply-the-power-consumed-by-the-la

in-a-circuit-a-metal-filament-lamp-is-connected-in-series-with-a-capacitor-of-capacitance-c-f-across-a-200-v-50-hz-supply-the-power-consumed-by-the-la-92226

<div class="question">In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu F$ across a $200 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. The power consumed by the lamp is $500 \mathrm{~W}$ while the voltage drop across it is $100 \mathrm{~V}$. Assume that there is no inductive load in the circuit. Take $r m s$ values of the voltages. The magnitude of the phaseangle (in degrees) between the current and the supply voltage is $\varphi$. Assume, $\pi \sqrt{3} \approx 5$.<br/>The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math> is _____.</div>

['Physics', 'Alternating Current', 'JEE Advanced', 'JEE Advanced 2021 (Paper 2)']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">60</span> </div>

<div class="solution"><p>Given, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>500</mn><mo> </mo><mi mathvariant="normal">W</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>R</mi></msub><mo>=</mo><mn>100</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>50</mn><mo> </mo><mi>Hz</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow><mi>r</mi><mi>m</mi><mi>s</mi></mrow></msub><mo>=</mo><mn>200</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/a9534dee-8bf1-4528-9e41-ca0b7e175386-image.png" style="width: 250px; height: 172px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><msup><msub><mi>V</mi><mi>R</mi></msub><mn>2</mn></msup><mi>R</mi></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>R</mi><mo>=</mo><mfrac><msup><msub><mi>V</mi><mi>R</mi></msub><mn>2</mn></msup><mi>P</mi></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>R</mi><mo>=</mo><mfrac><msup><mfenced><mn>100</mn></mfenced><mn>2</mn></msup><mn>500</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo> </mo><mi>R</mi><mo>=</mo><mn>20</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p>Now current in the circuit,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><msub><mi>i</mi><mrow><mi>r</mi><mi>m</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>voltage</mi><mo> </mo><mi>across</mi><mo> </mo><mi>resistor</mi></mrow><mi>resistance</mi></mfrac><mspace linebreak="newline"></mspace><mi>i</mi><mo>=</mo><mfrac><mn>100</mn><mn>20</mn></mfrac><mspace linebreak="newline"></mspace><mi>i</mi><mo>=</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p><p>Phasor diagram of the given circuit for voltage,</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/74da4edb-8ed0-4431-8111-49c799cec0a8-image.png" style="width: 274px; height: 174px;"/> </p><p>Now, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>V</mi><mn>2</mn></msup><mo>=</mo><msup><msub><mi>V</mi><mi>C</mi></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>V</mi><mi>R</mi></msub><mn>2</mn></msup><mspace linebreak="newline"></mspace><mo>⇒</mo><msup><mn>200</mn><mn>2</mn></msup><mo>=</mo><msup><msub><mi>V</mi><mi>C</mi></msub><mn>2</mn></msup><mo>+</mo><msup><mn>100</mn><mn>2</mn></msup><mspace linebreak="newline"></mspace><mo>⇒</mo><msub><mi>V</mi><mi>C</mi></msub><mo>=</mo><mn>100</mn><msqrt><mn>3</mn></msqrt><mo> </mo><mi mathvariant="normal">V</mi></math></p><p>Now, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>C</mi></msub><mo>=</mo><mn>100</mn><msqrt><mn>3</mn></msqrt><mo> </mo><mi mathvariant="normal">V</mi><mo>=</mo><mi>I</mi><msub><mi>X</mi><mi>C</mi></msub><mspace linebreak="newline"></mspace><mo>⇒</mo><msub><mi>X</mi><mi>C</mi></msub><mo>=</mo><mfrac><mrow><mn>100</mn><msqrt><mn>3</mn></msqrt></mrow><mn>5</mn></mfrac><mo>=</mo><mn>20</mn><msqrt><mn>3</mn></msqrt><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p>and </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>C</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mn>2</mn><mi>π</mi><mi>f</mi></mrow></mfenced><mi>C</mi></mrow></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mn>20</mn><msqrt><mn>3</mn></msqrt><mo>=</mo><mfrac><mn>1</mn><mrow><mfenced><mrow><mn>2</mn><mi>π</mi><mo>×</mo><mn>100</mn></mrow></mfenced><mi>C</mi></mrow></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>C</mi><mo>=</mo><mn>100</mn><mo> </mo><mi>μF</mi></math></p><p>From the phasor diagram,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mi>ϕ</mi></mfenced><mo>=</mo><mfrac><msub><mi>X</mi><mi>C</mi></msub><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><mn>20</mn><msqrt><mn>3</mn></msqrt></mrow><mn>20</mn></mfrac><mo>=</mo><msqrt><mn>3</mn></msqrt><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>ϕ</mi><mo>=</mo><mn>60</mn><mo>°</mo></math></p></div>

MarksBatch2_P1.db

in-a-constant-volume-calorimeter-35-g-of-a-gas-with-molecular-weight-28-was-burnt-in-excess-oxygen-at-2980-k-the-temperature-of-the-calorimeter-was-fo-2

in-a-constant-volume-calorimeter-35-g-of-a-gas-with-molecular-weight-28-was-burnt-in-excess-oxygen-at-2980-k-the-temperature-of-the-calorimeter-was-fo-2-23881

<div class="question">In a constant volume calorimeter, $3.5 \mathrm{~g}$ of a gas with molecular weight $=28$ was burnt in excess oxygen at $298.0 \mathrm{~K}$. The temperature of the calorimeter was found to increases from $298.0 \mathrm{~K}$ to $298.45 \mathrm{~K}$ due to the combustion process. Given, that the heat capacity of the calorimeter is $2.5$ $\mathrm{kJ} \mathrm{K}^{-1}$, the numerical value for the enthalpy of combustion of the gas in $\mathrm{kJ} \mathrm{mol}^{-1}$ is</div>

['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']

None

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

<div class="solution">The temperature rise is : $\Delta T=T_2-T_1=298.45-298=0.45 \mathrm{~K}$<br/>This indicates that heat produced from combustion of $3.5 \mathrm{~g}$ of compound rises temperature of calorimeter by $0.45 \mathrm{~K}$.<br/>Heat produced $=0.45 \mathrm{~K} \times 2.5 \mathrm{k} \mathrm{JK}^{-1}=1.125 \mathrm{~kJ}$<br/>$\Rightarrow$ Heat produced from $28 \mathrm{~g}$ of compound $(1.0 \mathrm{~mol})=\frac{1.125}{3.5} \times 28=9 \mathrm{~kJ}$</div>

MarksBatch2_P1.db

in-a-youngs-double-slit-experiment-the-separation-between-the-two-slits-is-d-and-the-wavelength-of-the-light-is-the-intensity-of-light-falling-on-slit

in-a-youngs-double-slit-experiment-the-separation-between-the-two-slits-is-d-and-the-wavelength-of-the-light-is-the-intensity-of-light-falling-on-slit-99049

<div class="question">In a Young's double slit experiment, the separation between the two slits is $d$ and the wavelength of the light is $\lambda$. The intensity of light falling on slit 1 is four times the intensity of light falling on slit 2 . Choose the correct choice(s),</div>

['Physics', 'Wave Optics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>If $d=\lambda$, the screen will contain only one maximum<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/>If $\lambda &lt; d &lt; 2 \lambda$, at least one more maximum (besides the central maximum) will be observed on the screen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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/>If the intensity of light falling on slit 1 is reduced so that it becomes equal to that of slit 2 , the intensities of the observed dark and bright fringes will increase<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>If the intensity of light falling on slit 2 is increased so that it becomes equal to that of slit 1 , the intensities of the observed dark and bright fringes will increase</span> </li> </ul>

<div class="correct-answer"> The correct answers are: <span class="option-value"><br/>If $d=\lambda$, the screen will contain only one maximum<br/>, <br/>If $\lambda &lt; d &lt; 2 \lambda$, at least one more maximum (besides the central maximum) will be observed on the screen</span> </div>

<div class="solution">For $d=\lambda$, there will be only one, acentral maxima.<br/>For $\lambda &lt; d &lt; 2 \lambda$, there will be three maximas on the screen corresponding to path difference, $\Delta x=0$ and $\Delta x=\pm \lambda$ correct options are (a) and (b).</div>

MarksBatch2_P1.db

in-allene-c-3-h-4-the-types-of-hybridisation-of-the-carbon-atoms-is-are

in-allene-c-3-h-4-the-types-of-hybridisation-of-the-carbon-atoms-is-are-50924

<div class="question">In allene $\left(\mathrm{C}_{3} \mathrm{H}_{4}\right)$, the type(s) of hybridisation of the carbon atoms is (are):</div>

['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']

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

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

<div class="solution">Allene $\left(\mathbf{C}_{3} \mathbf{H}_{4}\right)$ is $\stackrel{s p^{2}}{\mathrm{H}_{2}} \mathrm{C}=\stackrel{s p}{\mathrm{C}}=\stackrel{s p^{2}}{\mathrm{CH}_{2}}$</div>

MarksBatch2_P1.db

in-an-experiment-to-determine-the-focal-length-f-of-a-concave-mirror-by-the-u-v-method-a-student-places-the-object-pin-a-on-the-principal-axis-at-a-di

in-an-experiment-to-determine-the-focal-length-f-of-a-concave-mirror-by-the-u-v-method-a-student-places-the-object-pin-a-on-the-principal-axis-at-a-di-86546

<div class="question">In an experiment to determine the focal length $(f)$ of a concave mirror by the $u-v$ method, a student places the object pin $A$ on the principal axis at a distance $x$ from the pole $P$. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then</div>

['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$f &lt; x &lt; 2 f$</span> </div>

<div class="solution">Since object and image move in opposite directions, the positioning should be as shown in the figure. Object lies between focus and centre of curvature<br/>$$<br/>f &lt; x &lt; 2 f \text {. }<br/>$$<br/>$\therefore$ Correct option is (b).<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/bqW8pAFMsT8csiwUNi0inuhJaEUpXBN3jFUTp_1l5iQ.original.fullsize.png"/><br/></div>

MarksBatch2_P1.db

in-an-insulated-vessel-005-kg-steam-at-373-k-and-045-kg-of-ice-at-253-k-are-mixed-find-the-final-temperature-of-the-mixture-in-kelvin-given-l-fusion-s

in-an-insulated-vessel-005-kg-steam-at-373-k-and-045-kg-of-ice-at-253-k-are-mixed-find-the-final-temperature-of-the-mixture-in-kelvin-given-l-fusion-s-34750

<div class="question">In an insulated vessel, $0.05 \mathrm{~kg}$ steam at $373 \mathrm{~K}$ and $0.45 \mathrm{~kg}$ of ice at $253 \mathrm{~K}$ are mixed. Find the final temperature of the mixture (in kelvin).<br/>$$<br/>\text { Given, } \begin{aligned}<br/>L_{\text {fusion }} &amp; =80 \mathrm{cal} / \mathrm{g}=336 \mathrm{~J} / \mathrm{g}, L_{\text {vaporization }}=540 \mathrm{cal} / \mathrm{g}=2268 \mathrm{~J} / \mathrm{g}, \\<br/>S_{\text {ice }} &amp; =2100 \mathrm{~J} / \mathrm{kg}, K=0.5 \mathrm{cal} / \mathrm{gK} \text { and } S_{\text {water }}=4200 \mathrm{~J} / \mathrm{kg}, K=1 \mathrm{cal} / \mathrm{gK} .<br/>\end{aligned}<br/>$$</div>

['Physics', 'Thermal Properties of Matter', 'JEE Advanced', 'JEE Advanced 2006']

None

<div class="correct-answer"> The correct answer is: <span class="option-value">273</span> </div>

<div class="solution">$0.05 \mathrm{~kg}$ steam at $373 \mathrm{~K} \stackrel{Q_1}{\longrightarrow} 0.05 \mathrm{~kg}$ water at $373 \mathrm{~K}$<br/>$0.05 \mathrm{~kg}$ water at $373 \mathrm{~K} \stackrel{Q_2}{\longrightarrow} 0.05$ kg water at $273 \mathrm{~K}$<br/>$0.45 \mathrm{~kg}$ ice at $253 \mathrm{~K} \stackrel{Q_3}{\longrightarrow} 0.45 \mathrm{~kg}$ ice at $273 \mathrm{~K}$<br/>$0.45 \mathrm{~kg}$ ice at $273 \mathrm{~K} \stackrel{Q_4}{\longrightarrow} 0.45 \mathrm{~kg}$ water at $273 \mathrm{~K}$<br/>$$<br/>\begin{aligned}<br/>&amp; Q_1=(50)(540)=27,000 \mathrm{cal}=27 \mathrm{kcal} \\<br/>&amp; Q_2=(50)(1)(100)=5000 \mathrm{cal}=5 \mathrm{kcal} \\<br/>&amp; Q_3=(450)(0.5)(20)=4500 \mathrm{cal}=4.5 \mathrm{kcal} \\<br/>&amp; Q_4=(450)(80)=36000 \mathrm{cal}=36 \mathrm{kcal}<br/>\end{aligned}<br/>$$<br/>Now since $Q_1+Q_2&gt;Q_3$ but $Q_1+Q_2 &lt; Q_3+Q_4$ ice will come to $273 \mathrm{~K}$ from $253 \mathrm{~K}$, but whole ice will not melt. Therefore temperature of the mixture is $273 \mathrm{~K}$.</div>

MarksBatch2_P1.db