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paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-2

paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-2-90426

<div class="question"><strong>Paragraph:</strong><br/>A carbonyl compound $P$, which gives positive iodoform test, undergoes reaction with MeMgBr followed by dehydration to give an olefin $Q$. Ozonolysis of $Q$ leads to a dicarbonyl compound $R$, which undergoes intramolecular aldol reaction to give predominantly $S$.<br/><br/>$$<br/>P \underset{\text { (ii) } \mathrm{H}^{+}, \mathrm{H}_2 \mathrm{O}}{\stackrel{\text { (i) } \mathrm{MeMgBr}}{\longrightarrow}} \mathrm{Q} \underset{\text { Heat }}{\stackrel{\mathrm{O}_3 / \mathrm{Zn}-\mathrm{H}_2 \mathrm{O}}{\longrightarrow}} \mathrm{P}<br/>$$<br/>(iii) $\mathrm{H}_2 \mathrm{SO}_4$ /Heat<strong>Question:</strong><br/>$$<br/>\text { The structures of the products } Q \text { and } R \text {, respectively, are }<br/>$$</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<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/G_Ez0C6KNqIAGZolt9XIaxiUp2i9kIlI-Ewf-7NC9U8.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/JEgaRrlsOeEdgRJIUTbtPeB9tiUSQL5mjXZ2qKlwol4.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/V4lly_EXqdYmoPS3ce1UVjDLMSRYtGYyHI-G-WRAjnA.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/tTgZk0XrQVF2P7sWBDR6WSLAAR0hcm2zWR6QG3E_vnE.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/G_Ez0C6KNqIAGZolt9XIaxiUp2i9kIlI-Ewf-7NC9U8.original.fullsize.png"/><br/></span> </div>

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

MarksBatch2_P2.db

paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-3

paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-3-24509

<div class="question"><strong>Paragraph:</strong><br/>A carbonyl compound $P$, which gives positive iodoform test, undergoes reaction with MeMgBr followed by dehydration to give an olefin $Q$. Ozonolysis of $Q$ leads to a dicarbonyl compound $R$, which undergoes intramolecular aldol reaction to give predominantly $S$.<br/><br/>$$<br/>P \underset{\text { (ii) } \mathrm{H}^{+}, \mathrm{H}_2 \mathrm{O}}{\stackrel{\text { (i) } \mathrm{MeMgBr}}{\longrightarrow}} \mathrm{Q} \underset{\text { Heat }}{\stackrel{\mathrm{O}_3 / \mathrm{Zn}-\mathrm{H}_2 \mathrm{O}}{\longrightarrow}} \mathrm{P}<br/>$$<br/>(iii) $\mathrm{H}_2 \mathrm{SO}_4$ /Heat<strong>Question:</strong><br/>$$<br/>\text { The structures of the products } Q \text { and } R \text {, respectively, are }<br/>$$</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Main']

<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/G_Ez0C6KNqIAGZolt9XIaxiUp2i9kIlI-Ewf-7NC9U8.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/JEgaRrlsOeEdgRJIUTbtPeB9tiUSQL5mjXZ2qKlwol4.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/V4lly_EXqdYmoPS3ce1UVjDLMSRYtGYyHI-G-WRAjnA.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/tTgZk0XrQVF2P7sWBDR6WSLAAR0hcm2zWR6QG3E_vnE.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/G_Ez0C6KNqIAGZolt9XIaxiUp2i9kIlI-Ewf-7NC9U8.original.fullsize.png"/><br/></span> </div>

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

MarksBatch2_P2.db

paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys

paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-97304

<div class="question"><strong>Paragraph:</strong><br/>A carbonyl compound $P$, which gives positive iodoform test, undergoes reaction with MeMgBr followed by dehydration to give an olefin $Q$. Ozonolysis of $Q$ leads to a dicarbonyl compound $R$, which undergoes intramolecular aldol reaction to give predominantly $S$.<br/><br/>$$<br/>P \underset{\text { (ii) } \mathrm{H}^{+}, \mathrm{H}_2 \mathrm{O}}{\stackrel{\text { (i) } \mathrm{MeMgBr}}{\longrightarrow}} \mathrm{Q} \underset{\text { Heat }}{\stackrel{\mathrm{O}_3 / \mathrm{Zn}-\mathrm{H}_2 \mathrm{O}}{\longrightarrow}} \mathrm{P}<br/>$$<br/>(iii) $\mathrm{H}_2 \mathrm{SO}_4$ /Heat<strong>Question:</strong><br/>$$<br/>\text { The structure of the product } S \text {, is }<br/>$$</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/x6KQmR76mR7KqQbxE-vnGE1N0pqEvwbQFCbkqqptpq0.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/12YWVHKZaA-lc0nEcDqWY_5McNW9BgC2NbSmevH7Pho.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RJS7YkTDjKbE0JC7Fx-P990C8yL_WdMjFyx3cFmyx0g.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/1Giw1P5UDTZua-66pXy8zKgPR6rpxUoajvOE2v1oJKA.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/12YWVHKZaA-lc0nEcDqWY_5McNW9BgC2NbSmevH7Pho.original.fullsize.png"/><br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IHXrn5EjOkh5usjWqSwIQSWVV2TCVJQ2Hfk4MkhhAt0.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fLLGF5xFKfkY6kScXTlwRQ6CaiAedofXArjZn1rzZNo.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-1

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-1-91940

<div class="question"><strong>Paragraph:</strong><br/>A circle $C$ of radius 1 is inscribed in an equilateral $\triangle P Q R$. The points of contact of $C$ with the sides $P Q, Q R, R P$ are $D, E, F$ respectively. The line $P Q$ is given by the equation<br/>$\sqrt{3} x+y-6=0$ and the point $D$ is $\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.<strong>Question:</strong><br/>Equations of the sides $Q R, R P$ are</div>

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

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$y=\sqrt{3} x, y=0$</span> </div>

<div class="solution">Clearly, point $E$ and $F$ satisfy the equations given in option (d).</div>

MarksBatch2_P2.db

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-2

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-2-95357

<div class="question"><strong>Paragraph:</strong><br/>A circle $C$ of radius 1 is inscribed in an equilateral $\triangle P Q R$. The points of contact of $C$ with the sides $P Q, Q R, R P$ are $D, E, F$ respectively. The line $P Q$ is given by the equation<br/>$\sqrt{3} x+y-6=0$ and the point $D$ is $\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.<strong>Question:</strong><br/>Points $E$ and $F$ are given by</div>

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

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),(\sqrt{3}, 0)$<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">B</span> <span class="option-data"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right),(\sqrt{3}, 0)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),(\sqrt{3}, 0)$<br/></span> </div>

<div class="solution">Slope of line joining centre of circle to point $D$<br/>$$<br/>=\frac{\frac{3}{2}-1}{\frac{3 \sqrt{3}}{2}-\sqrt{3}}=\frac{1}{\sqrt{3}}<br/>$$<br/>[makes an angle $30^{\circ}$ with $X$-axis]<br/>$\therefore$ Points $E$ and $F$ will make angle $150^{\circ}$ and $-90^{\circ}$ with $X$-axis.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/2MYFwgi_qwiPsepKX_pQpeeqlpaR5ZpeLruv5nwHpbo.original.fullsize.png"/><br/><br/>$\therefore E$ and $F$ are given by and<br/>$\therefore$<br/>$$<br/>\begin{aligned}<br/>\frac{x-\sqrt{3}}{\cos 150^{\circ}} &amp; =\frac{y-1}{\sin 150^{\circ}}=1 \\<br/>\frac{x-\sqrt{3}}{\cos \left(-90^{\circ}\right)} &amp; =\frac{y-1}{\sin \left(-90^{\circ}\right)}=1 \\<br/>E &amp; =\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right) \\<br/>and<br/>F &amp; =(\sqrt{3}, 0)<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-4

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-4-58541

<div class="question"><strong>Paragraph:</strong><br/>A circle $C$ of radius 1 is inscribed in an equilateral $\triangle P Q R$. The points of contact of $C$ with the sides $P Q, Q R, R P$ are $D, E, F$ respectively. The line $P Q$ is given by the equation<br/>$\sqrt{3} x+y-6=0$ and the point $D$ is $\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.<strong>Question:</strong><br/>Points $E$ and $F$ are given by</div>

['Mathematics', 'Straight Lines', 'JEE Main']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),(\sqrt{3}, 0)$<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">B</span> <span class="option-data"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right),(\sqrt{3}, 0)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),(\sqrt{3}, 0)$<br/></span> </div>

<div class="solution">Slope of line joining centre of circle to point $D$<br/>$$<br/>=\frac{\frac{3}{2}-1}{\frac{3 \sqrt{3}}{2}-\sqrt{3}}=\frac{1}{\sqrt{3}}<br/>$$<br/>[makes an angle $30^{\circ}$ with $X$-axis]<br/>$\therefore$ Points $E$ and $F$ will make angle $150^{\circ}$ and $-90^{\circ}$ with $X$-axis.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/2MYFwgi_qwiPsepKX_pQpeeqlpaR5ZpeLruv5nwHpbo.original.fullsize.png"/><br/><br/>$\therefore E$ and $F$ are given by and<br/>$\therefore$<br/>$$<br/>\begin{aligned}<br/>\frac{x-\sqrt{3}}{\cos 150^{\circ}} &amp; =\frac{y-1}{\sin 150^{\circ}}=1 \\<br/>\frac{x-\sqrt{3}}{\cos \left(-90^{\circ}\right)} &amp; =\frac{y-1}{\sin \left(-90^{\circ}\right)}=1 \\<br/>E &amp; =\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right) \\<br/>and<br/>F &amp; =(\sqrt{3}, 0)<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line

paragraph-a-circle-c-of-radius-1-is-inscribed-in-an-equilateral-pqr-the-points-of-contact-of-c-with-the-sides-pq-qr-rp-are-d-e-f-respectively-the-line-96032

<div class="question"><strong>Paragraph:</strong><br/>A circle $C$ of radius 1 is inscribed in an equilateral $\triangle P Q R$. The points of contact of $C$ with the sides $P Q, Q R, R P$ are $D, E, F$ respectively. The line $P Q$ is given by the equation<br/>$\sqrt{3} x+y-6=0$ and the point $D$ is $\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.<strong>Question:</strong><br/>The equation of circle $C$ is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$(x-2 \sqrt{3})^2+(y-1)^2=1$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$(x-2 \sqrt{3})^2+\left(y+\frac{1}{2}\right)^2=1$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$(x-\sqrt{3})^2+(y+1)^2=1$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$(x-\sqrt{3})^2+(y-1)^2=1$</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/>$(x-\sqrt{3})^2+(y-1)^2=1$</span> </div>

<div class="solution">Let centre of circle $C$ be $(h, k)$.<br/>Then, $\left|\frac{\sqrt{3} h+k-6}{\sqrt{3+1}}\right|=1$<br/>$$<br/>\Rightarrow \quad \sqrt{3} h+k-6=2,-2<br/>$$<br/>$$<br/>\Rightarrow \quad \sqrt{3} h+k=4<br/>$$<br/>(Rejecting 2 because origin and centre of $C$ are on the same side of $P Q$ ) The point $(\sqrt{3}, 1)$ satisfies Eq. (i).<br/>$\therefore$ Equation of circle $C$ is $(x-\sqrt{3})^2+(y-1)^2=1$.</div>

MarksBatch2_P2.db

paragraph-a-dense-collection-of-equal-number-of-electrons-and-positive-ions-is-called-neutral-plasma-certain-solids-containing-fixed-positive-ions-sur-1

paragraph-a-dense-collection-of-equal-number-of-electrons-and-positive-ions-is-called-neutral-plasma-certain-solids-containing-fixed-positive-ions-sur-1-15833

<div class="question"><strong>Paragraph:</strong><br/>A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let $N$ be the number density of free electrons, each of mass $m$. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency $\omega_p$, which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega_p$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.<strong>Question:</strong><br/>Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N \approx 4 \times 10^{27} \mathrm{~m}^{-1}$. Take $\varepsilon_0=10^{-11}$ and $m=10^{-30}$, where these quantities are in proper SI units.</div>

['Physics', 'Waves and Sound', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$800 \mathrm{~nm}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$600 \mathrm{~nm}$<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/>$300 \mathrm{~nm}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$200 \mathrm{~nm}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$600 \mathrm{~nm}$<br/></span> </div>

<div class="solution">$$<br/>\begin{gathered}<br/>\omega=2 \pi f=\frac{2 \pi c}{\lambda} \\<br/>\therefore \quad \lambda=\frac{2 \pi c}{\omega}=\frac{2 \pi c}{\sqrt{N e^2 / m \varepsilon_0}}<br/>\end{gathered}<br/>$$<br/>Substituting the values, we get<br/>$$<br/>\lambda=600 \mathrm{~nm}<br/>$$<br/>$\therefore$ Correct option is (b).<br/>Analysis of Question<br/>(i) Paragraph does not make the solution very clear.<br/>(ii) Solution is simple but based on error and trial method.</div>

MarksBatch2_P2.db

paragraph-a-dense-collection-of-equal-number-of-electrons-and-positive-ions-is-called-neutral-plasma-certain-solids-containing-fixed-positive-ions-sur

paragraph-a-dense-collection-of-equal-number-of-electrons-and-positive-ions-is-called-neutral-plasma-certain-solids-containing-fixed-positive-ions-sur-37758

<div class="question"><strong>Paragraph:</strong><br/>A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let $N$ be the number density of free electrons, each of mass $m$. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency $\omega_p$, which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega_p$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.<strong>Question:</strong><br/>Taking the electronic charge as $e$ and the permittivity as $\varepsilon_0$, use dimensional analysis to determine the correct expression for $\omega_p$.</div>

['Physics', 'Units and Dimensions', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\sqrt{\frac{N e}{m \varepsilon_0}}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{\frac{m \varepsilon_0}{N e}}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\sqrt{\frac{N e^2}{m \varepsilon_0}}$<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/>$\sqrt{\frac{m \varepsilon_0}{N e^2}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\sqrt{\frac{N e^2}{m \varepsilon_0}}$<br/></span> </div>

<div class="solution">$N=$ Number of electrons per unit volume<br/>$$<br/>\begin{aligned}<br/>\therefore \quad[N] &amp; =\left[\mathrm{L}^{-3}\right],[e]=[q] \\<br/>&amp; =[I t]=[\mathrm{AT}] \\<br/>{\left[\varepsilon_0\right] } &amp; =\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^4 \mathrm{~A}^2\right]<br/>\end{aligned}<br/>$$<br/>Substituting the dimensions, we can see that,<br/>$$<br/>\left.\sqrt{\frac{N e^2}{m \varepsilon_0}}\right]=\left[\mathrm{T}^{-1}\right]<br/>$$<br/>Angular frequency has also the dimension $\left[\mathrm{T}^{-1}\right]$.<br/>$\therefore$ Correct option is (c).<br/>Analysis of Question<br/>(i) From calculation point of view, question is moderately difficult. Otherwise it is simple.<br/>(ii) Students may commit a mistake in the dimensions of $N$. It is not dimensionless.</div>

MarksBatch2_P2.db

paragraph-a-fair-die-is-tossed-repeatedly-until-a-six-is-obtained-let-x-denotes-the-number-of-tosses-required-question-the-conditional-probability-tha

paragraph-a-fair-die-is-tossed-repeatedly-until-a-six-is-obtained-let-x-denotes-the-number-of-tosses-required-question-the-conditional-probability-tha-28170

<div class="question"><strong>Paragraph:</strong><br/>A fair die is tossed repeatedly until a six is obtained. Let $X$ denotes the number of tosses required.<strong>Question:</strong><br/>The conditional probability that $X \geq 6$ given $X&gt;3$ equals</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{125}{216}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{25}{216}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{5}{36}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{25}{36}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{25}{36}$</span> </div>

<div class="solution">$P((X \geq 6) /(X&gt;3))$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{P((X&gt;3) /(X \geq 6)) \cdot P(X \geq 6)}{P(X&gt;3)} \\<br/>&amp; =\frac{1 \cdot\left[\left(\frac{5}{6}\right)^5 \cdot \frac{1}{6}+\left(\frac{5}{6}\right)^6 \cdot \frac{1}{6}+\ldots \infty\right]}{\left[\left(\frac{5}{6}\right)^3 \cdot \frac{1}{6}+\left(\frac{5}{6}\right)^4 \cdot \frac{1}{6}+\ldots \infty\right]}=\frac{25}{36} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-a-fair-die-is-tossed-repeatedly-until-a-six-is-obtained-let-x-denotes-the-number-of-tosses-required-question-the-probability-that-x-3-equals-1

paragraph-a-fair-die-is-tossed-repeatedly-until-a-six-is-obtained-let-x-denotes-the-number-of-tosses-required-question-the-probability-that-x-3-equals-1-61403

<div class="question"><strong>Paragraph:</strong><br/>A fair die is tossed repeatedly until a six is obtained. Let $X$ denotes the number of tosses required.<strong>Question:</strong><br/>The probability that $X=3$ equals</div>

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

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{25}{216}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{25}{36}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{5}{36}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{125}{216}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{25}{216}$<br/></span> </div>

<div class="solution">$P(X=3)=\frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6}=\frac{25}{216}$</div>

MarksBatch2_P2.db

paragraph-a-fair-die-is-tossed-repeatedly-until-a-six-is-obtained-let-x-denotes-the-number-of-tosses-required-question-the-probability-that-x-3-equals

paragraph-a-fair-die-is-tossed-repeatedly-until-a-six-is-obtained-let-x-denotes-the-number-of-tosses-required-question-the-probability-that-x-3-equals-19963

<div class="question"><strong>Paragraph:</strong><br/>A fair die is tossed repeatedly until a six is obtained. Let $X$ denotes the number of tosses required.<strong>Question:</strong><br/>The probability that $X \geq 3$ equals</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{125}{216}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{25}{36}$<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/>$\frac{5}{36}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{25}{216}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{25}{36}$<br/></span> </div>

<div class="solution">$P(X \geq 3)=\frac{5}{6} \cdot \frac{5}{6} \cdot 1=\frac{25}{36}$</div>

MarksBatch2_P2.db

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-1

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-1-53050

<div class="question"><strong>Paragraph:</strong><br/>A small block of mass $M$ moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/823KW2eHMvGVM1emFpNp7o6zsW6lNXDO8IQ7NsK5ef8.original.fullsize.png"/><br/><strong>Question:</strong><br/>If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point $B$, immediately after it strikes the second incline is</div>

['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\sqrt{30} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{15} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>zero<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/>$-\sqrt{15} \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

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

<div class="solution">In elastic collision, component of $v_1$ parallel to $B C$ will remain unchanged, while component perpendicular to $B C$ will remain unchanged in magnitude but its direction will be reversed.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xIzsRP4HzcJ7OPT5wsEVVrJGT7ByYhltu4ezEse1Wis.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; v_{\|}=v_1 \cos 30^{\circ}=(\sqrt{60})\left(\frac{\sqrt{3}}{2}\right)=\sqrt{45} \mathrm{~ms}^{-1} \\<br/>&amp; v_{\perp}=v_1 \sin 30^{\circ}=(\sqrt{60})\left(\frac{1}{2}\right)=\sqrt{15} \mathrm{~ms}^{-1}<br/>\end{aligned}<br/>$$<br/>Now vertical component of velocity of block :<br/>$$<br/>\begin{aligned}<br/>v &amp; =v_{\perp} \cos 30^{\circ}-v_{\|} \cos 60^{\circ} \\<br/>&amp; =(\sqrt{15})\left(\frac{\sqrt{3}}{2}\right)-(\sqrt{45})\left(\frac{1}{2}\right)=0<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (c).</div>

MarksBatch2_P2.db

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-2

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-2-56360

<div class="question"><strong>Paragraph:</strong><br/>A small block of mass $M$ moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/823KW2eHMvGVM1emFpNp7o6zsW6lNXDO8IQ7NsK5ef8.original.fullsize.png"/><br/><strong>Question:</strong><br/>The speed of the block at point $B$ immediately after it strikes the second incline is</div>

['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\sqrt{60} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{45} \mathrm{~m} / \mathrm{s}$<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/>$\sqrt{30} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\sqrt{15} \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\sqrt{45} \mathrm{~m} / \mathrm{s}$<br/></span> </div>

<div class="solution">Between $A$ and $B$, height fallen by block $h_1=\sqrt{3} \tan 60^{\circ}=3 \mathrm{~m}$.<br/>$\therefore$ speed of block just before striking the second incline,<br/>$$<br/>v_1=\sqrt{2 g h_1}=\sqrt{2 \times 10 \times 3}=\sqrt{60} \mathrm{~ms}^{-1}<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/EwbZJMMf6q6zuB0UupBlt4sElxLMY-JuAG7DSXN4FNk.original.fullsize.png"/><br/><br/>In perfectly inelastic collision, component of $v_1$ perpendicular to $B C$ will become zero, while component of $v_1$ parallel to $B C$ will remain unchanged.<br/>$\therefore$ speed of block $B$ immediately after it strikes the incline is,<br/>$$<br/>\begin{aligned}<br/>v_2 &amp; =\text { component of } v_1 \text { along } B C \\<br/>&amp; =v_1 \cos 30^{\circ} \\<br/>&amp; =(\sqrt{60})\left(\frac{\sqrt{3}}{2}\right)=\sqrt{45} \mathrm{~ms}^{-1}<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (b)</div>

MarksBatch2_P2.db

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-3

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-3-38219

<div class="question"><strong>Paragraph:</strong><br/>A small block of mass $M$ moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/823KW2eHMvGVM1emFpNp7o6zsW6lNXDO8IQ7NsK5ef8.original.fullsize.png"/><br/><strong>Question:</strong><br/>If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point $B$, immediately after it strikes the second incline is</div>

['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\sqrt{30} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{15} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>zero<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/>$-\sqrt{15} \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

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

<div class="solution">In elastic collision, component of $v_1$ parallel to $B C$ will remain unchanged, while component perpendicular to $B C$ will remain unchanged in magnitude but its direction will be reversed.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xIzsRP4HzcJ7OPT5wsEVVrJGT7ByYhltu4ezEse1Wis.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; v_{\|}=v_1 \cos 30^{\circ}=(\sqrt{60})\left(\frac{\sqrt{3}}{2}\right)=\sqrt{45} \mathrm{~ms}^{-1} \\<br/>&amp; v_{\perp}=v_1 \sin 30^{\circ}=(\sqrt{60})\left(\frac{1}{2}\right)=\sqrt{15} \mathrm{~ms}^{-1}<br/>\end{aligned}<br/>$$<br/>Now vertical component of velocity of block :<br/>$$<br/>\begin{aligned}<br/>v &amp; =v_{\perp} \cos 30^{\circ}-v_{\|} \cos 60^{\circ} \\<br/>&amp; =(\sqrt{15})\left(\frac{\sqrt{3}}{2}\right)-(\sqrt{45})\left(\frac{1}{2}\right)=0<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (c).</div>

MarksBatch2_P2.db

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-4

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-4-24788

<div class="question"><strong>Paragraph:</strong><br/>A small block of mass $M$ moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/823KW2eHMvGVM1emFpNp7o6zsW6lNXDO8IQ7NsK5ef8.original.fullsize.png"/><br/><strong>Question:</strong><br/>The speed of the block at point $B$ immediately after it strikes the second incline is</div>

['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\sqrt{60} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{45} \mathrm{~m} / \mathrm{s}$<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/>$\sqrt{30} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\sqrt{15} \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\sqrt{45} \mathrm{~m} / \mathrm{s}$<br/></span> </div>

<div class="solution">Between $A$ and $B$, height fallen by block $h_1=\sqrt{3} \tan 60^{\circ}=3 \mathrm{~m}$.<br/>$\therefore$ speed of block just before striking the second incline,<br/>$$<br/>v_1=\sqrt{2 g h_1}=\sqrt{2 \times 10 \times 3}=\sqrt{60} \mathrm{~ms}^{-1}<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/EwbZJMMf6q6zuB0UupBlt4sElxLMY-JuAG7DSXN4FNk.original.fullsize.png"/><br/><br/>In perfectly inelastic collision, component of $v_1$ perpendicular to $B C$ will become zero, while component of $v_1$ parallel to $B C$ will remain unchanged.<br/>$\therefore$ speed of block $B$ immediately after it strikes the incline is,<br/>$$<br/>\begin{aligned}<br/>v_2 &amp; =\text { component of } v_1 \text { along } B C \\<br/>&amp; =v_1 \cos 30^{\circ} \\<br/>&amp; =(\sqrt{60})\left(\frac{\sqrt{3}}{2}\right)=\sqrt{45} \mathrm{~ms}^{-1}<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (b)</div>

MarksBatch2_P2.db

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr

paragraph-a-small-block-of-mass-m-moves-on-a-frictionless-surface-of-an-inclined-plane-as-shown-in-figure-the-angle-of-the-incline-suddenly-changes-fr-36125

<div class="question"><strong>Paragraph:</strong><br/>A small block of mass $M$ moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/823KW2eHMvGVM1emFpNp7o6zsW6lNXDO8IQ7NsK5ef8.original.fullsize.png"/><br/><strong>Question:</strong><br/>The speed of the block at point $C$, immediately before it leaves the second incline is</div>

['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\sqrt{120} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\sqrt{105} \mathrm{~m} / \mathrm{s}$<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/>$\sqrt{90} \mathrm{~m} / \mathrm{s}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\sqrt{75} \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\sqrt{105} \mathrm{~m} / \mathrm{s}$<br/></span> </div>

<div class="solution">Height fallen by the block from $B$ to $C$. $h_2=3 \sqrt{3} \tan 30^{\circ}=3 \mathrm{~m}$<br/>Let $v_3$ be the speed of block, at point $C$, just before it leaves the second incline, then<br/>$$<br/>\begin{aligned}<br/>v_3 &amp; =\sqrt{v_2^2+2 g h_2} \\<br/>&amp; =\sqrt{45+2 \times 10 \times 3} \\<br/>&amp; =\sqrt{105} \mathrm{~ms}^{-1}<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (b).</div>

MarksBatch2_P2.db

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-1

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-1-59897

<div class="question"><strong>Paragraph:</strong><br/>A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_1$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains $n$ moles of gas. The temperature of the gas when the bubble is at the bottom is $T_0$, the height of the liquid is $H$ and the atmospheric pressure is $p_0$ (Neglect surface tension).<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6dQeCI9EIjoqs2jkE6KVSXvCiKXoaDTjz52ZnVnPgdw.original.fullsize.png"/><br/><strong>Question:</strong><br/>As the bubble moves upwards, besides the buoyancy force the following forces are acting on it.</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

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

<div class="solution">As the bubble moves upwards, besides the buoyancy force (cause of which is pressure difference) only force of gravity and force of viscosity will act.<br/>$\therefore$ correct option is (d).</div>

MarksBatch2_P2.db

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-2

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-2-88585

<div class="question"><strong>Paragraph:</strong><br/>A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_1$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains $n$ moles of gas. The temperature of the gas when the bubble is at the bottom is $T_0$, the height of the liquid is $H$ and the atmospheric pressure is $p_0$ (Neglect surface tension).<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6dQeCI9EIjoqs2jkE6KVSXvCiKXoaDTjz52ZnVnPgdw.original.fullsize.png"/><br/><strong>Question:</strong><br/>The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\rho_l n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}}{\left(\rho_0+\rho_l g y\right)^{\frac{7}{5}}}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$<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/>$\rho n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}}{\left(\rho_0+\rho_l g y\right)^{\frac{8}{5}}}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{2}{5}}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$<br/></span> </div>

<div class="solution">Buoyancy force<br/>Here, and<br/>$$<br/>\begin{aligned}<br/>F &amp; =(\text { volume of bubble })\left(\rho_l\right) g \\<br/>&amp; =\left(\frac{n R T_2}{p_2}\right) p_l g \\<br/>T_2 &amp; =T_0\left[\frac{p_0+p_l g(H-y)}{p_0+\rho_l g H}\right] \\<br/>p_2 &amp; =p_0+p_l(H-y)<br/>\end{aligned}<br/>$$<br/>Substituting the values we get,<br/>$$<br/>F=\frac{\rho_n n g T_0}{\left(p_0+\rho_l g H\right)^{2 / 5}\left[p_0+\rho_l g(H-y)\right]^{3 / 5}}<br/>$$<br/>$\therefore$ correct option is (b).</div>

MarksBatch2_P2.db

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-3

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-3-91125

<div class="question"><strong>Paragraph:</strong><br/>A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_1$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains $n$ moles of gas. The temperature of the gas when the bubble is at the bottom is $T_0$, the height of the liquid is $H$ and the atmospheric pressure is $p_0$ (Neglect surface tension).<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6dQeCI9EIjoqs2jkE6KVSXvCiKXoaDTjz52ZnVnPgdw.original.fullsize.png"/><br/><strong>Question:</strong><br/>When the gas bubble is at height $y$ from the bottom, its temperature is</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho \lg H}{p_0+\rho \lg y}\right)^{\frac{2}{5}}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{2}{5}}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho l g H}{p_0+\rho \lg y}\right)^{\frac{3}{5}}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{3}{5}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{2}{5}}$<br/></span> </div>

<div class="solution">As there is no exchange of heat. Therefore, process is adiabatic. Applying,<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; T p^{\frac{1-\gamma}{\gamma}}=\text { constant } \\<br/>\therefore \quad &amp; T_2 p_2^{\frac{1-\gamma}{\gamma}}=T_1 p_1^{\frac{1-\gamma}{\gamma}} \\<br/>\text { or } &amp; T_2 &amp; =T_1\left(\frac{p_1}{p_2}\right)^{\frac{1-\gamma}{\gamma}}=T_1\left(\frac{p_2}{p_1}\right)^{\frac{\gamma-1}{\gamma}}<br/>\end{array}<br/>$$<br/><br/>Substituting the values we have,<br/>$$<br/>\begin{aligned}<br/>T_2 &amp; =T_0\left[\frac{p_0+\rho \lg (H-y)^T}{p_0+\rho \lg H}\right]^{\frac{5 / 3-1}{5 / 3}} \\<br/>&amp; =T_0\left[\frac{p_0+\rho \lg (H-y)}{p_0+\rho \lg H}\right]^{2 / 5}<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (b).</div>

MarksBatch2_P2.db

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang

paragraph-a-small-spherical-monoatomic-ideal-gas-bubble-3-5-is-trapped-inside-a-liquid-of-density-1-see-figure-assume-that-the-bubble-does-not-exchang-90914

<div class="question"><strong>Paragraph:</strong><br/>A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_1$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains $n$ moles of gas. The temperature of the gas when the bubble is at the bottom is $T_0$, the height of the liquid is $H$ and the atmospheric pressure is $p_0$ (Neglect surface tension).<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6dQeCI9EIjoqs2jkE6KVSXvCiKXoaDTjz52ZnVnPgdw.original.fullsize.png"/><br/><strong>Question:</strong><br/>When the gas bubble is at height $y$ from the bottom, its temperature is</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho \lg H}{p_0+\rho \lg y}\right)^{\frac{2}{5}}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{2}{5}}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho l g H}{p_0+\rho \lg y}\right)^{\frac{3}{5}}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{3}{5}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{2}{5}}$<br/></span> </div>

<div class="solution">As there is no exchange of heat. Therefore, process is adiabatic. Applying,<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; T p^{\frac{1-\gamma}{\gamma}}=\text { constant } \\<br/>\therefore \quad &amp; T_2 p_2^{\frac{1-\gamma}{\gamma}}=T_1 p_1^{\frac{1-\gamma}{\gamma}} \\<br/>\text { or } &amp; T_2 &amp; =T_1\left(\frac{p_1}{p_2}\right)^{\frac{1-\gamma}{\gamma}}=T_1\left(\frac{p_2}{p_1}\right)^{\frac{\gamma-1}{\gamma}}<br/>\end{array}<br/>$$<br/><br/>Substituting the values we have,<br/>$$<br/>\begin{aligned}<br/>T_2 &amp; =T_0\left[\frac{p_0+\rho \lg (H-y)^T}{p_0+\rho \lg H}\right]^{\frac{5 / 3-1}{5 / 3}} \\<br/>&amp; =T_0\left[\frac{p_0+\rho \lg (H-y)}{p_0+\rho \lg H}\right]^{2 / 5}<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (b).</div>

MarksBatch2_P2.db

paragraph-a-special-metal-s-conducts-electricity-without-any-resistance-a-closed-wire-loop-made-of-s-does-not-allow-any-change-in-flux-through-itself--1

paragraph-a-special-metal-s-conducts-electricity-without-any-resistance-a-closed-wire-loop-made-of-s-does-not-allow-any-change-in-flux-through-itself-1-25447

<div class="question"><strong>Paragraph</strong><br/>A special metal $S$ conducts electricity without any resistance. A closed wire loop, made of $S$, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius $a$, with its center at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(\gg a)$ from the center of the loop with its north pole always facing the loop, as shown in the figure below.<br/><br/>The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}$, where $\mu_{0}$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_{1}$ and $m_{2}$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k m_{1} m_{2}}{r^{4}}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OnwLM4py_dLCUenLLwdFNEy8sa-0i9DHGKEW9cpctzU.original.fullsize.png"/><br/><strong>Question</strong>When the dipole <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is placed at a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:</div>

['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2021 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></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">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mi>r</mi></mfrac></math></span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></span> </div>

<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/571fef32-a285-4778-9cb1-8a36e9213622-image.png" style="width: 230px; height: 84px;"/></p><p>For a super conducting loop, net flux passing through it will remain constant.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><msub><mi>ϕ</mi><mrow><mi>total</mi><mo> </mo></mrow></msub></mfenced><mi>i</mi></msub><mo>=</mo><msub><mfenced><msub><mi>ϕ</mi><mrow><mi>total</mi><mo> </mo></mrow></msub></mfenced><mi>f</mi></msub></math></p><p>As area is same, therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>m</mi></mrow><mrow><mn>2</mn><msup><mi>πr</mi><mn>3</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>I</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>m</mi></mrow><msup><mi>πr</mi><mn>3</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>I</mi><mo>∝</mo><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p></div>

MarksBatch2_P2.db

paragraph-a-special-metal-s-conducts-electricity-without-any-resistance-a-closed-wire-loop-made-of-s-does-not-allow-any-change-in-flux-through-itself--2

paragraph-a-special-metal-s-conducts-electricity-without-any-resistance-a-closed-wire-loop-made-of-s-does-not-allow-any-change-in-flux-through-itself-2-19698

<div class="question"><strong>Paragraph</strong><br/>A special metal $S$ conducts electricity without any resistance. A closed wire loop, made of $S$, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius $a$, with its center at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(\gg a)$ from the center of the loop with its north pole always facing the loop, as shown in the figure below.<br/><br/>The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}$, where $\mu_{0}$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_{1}$ and $m_{2}$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k m_{1} m_{2}}{r^{4}}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OnwLM4py_dLCUenLLwdFNEy8sa-0i9DHGKEW9cpctzU.original.fullsize.png"/><br/><strong>Question</strong>The work done in bringing the dipole from infinity to a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> from the centre of the loop by the given process is proportional to:</div>

['Physics', 'Magnetic Effects of Current', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><msup><mi>r</mi><mn>5</mn></msup></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>5</mn></msup></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>6</mn></msup></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>7</mn></msup></mfrac></math></span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>6</mn></msup></mfrac></math></span> </div>

<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/571fef32-a285-4778-9cb1-8a36e9213622-image.png" style="width: 230px; height: 84px;"/></p><p>For a super conducting loop, net flux passing through it will remain constant.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><msub><mi>ϕ</mi><mrow><mi>total</mi><mo> </mo></mrow></msub></mfenced><mi>i</mi></msub><mo>=</mo><msub><mfenced><msub><mi>ϕ</mi><mrow><mi>total</mi><mo> </mo></mrow></msub></mfenced><mi>f</mi></msub></math></p><p>As area is same, therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>m</mi></mrow><mrow><mn>2</mn><msup><mi>πr</mi><mn>3</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>I</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>m</mi></mrow><msup><mi>πr</mi><mn>3</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>I</mi><mo>∝</mo><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p>The current carrying superconducting loop will also behave like a magnet, whose magnetic dipole moment is given by,<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>1</mn></msub><mo>=</mo><mi>N</mi><mi>I</mi><mi>A</mi><mspace linebreak="newline"></mspace><mo>=</mo><mo>(</mo><mn>1</mn><mo>)</mo><mfenced><mfrac><mrow><mi>a</mi><mi>m</mi></mrow><msup><mi>πr</mi><mn>3</mn></msup></mfrac></mfenced><mo>×</mo><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>a</mi><mn>3</mn></msup></mrow><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><msub><mi>m</mi><mn>1</mn></msub><mo>∝</mo><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p>The repulsive force felt on the magnet will be</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>K</mi><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub></mrow><msup><mi>r</mi><mn>4</mn></msup></mfrac><mo>=</mo><mfrac><mrow><mfenced><msup><mi>K</mi><mo>'</mo></msup></mfenced><mfenced><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></mfenced><mfenced><mi>m</mi></mfenced></mrow><msup><mi>r</mi><mn>4</mn></msup></mfrac><mo>∝</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>7</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>ext</mi></msub><mo>=</mo><mo>-</mo><mo>∫</mo><mi>F</mi><mi mathvariant="normal">d</mi><mi>r</mi><mspace linebreak="newline"></mspace><mo>=</mo><mo>-</mo><mi>K</mi><mo>'</mo><mo>∫</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>7</mn></msup></mfrac><mi mathvariant="normal">d</mi><mi>r</mi></math></p><p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mi>x</mi><mi>n</mi></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></p><p>We get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>ext</mi></msub><mo>∝</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>6</mn></msup></mfrac></math></p></div>

MarksBatch2_P2.db

paragraph-a-special-metal-s-conducts-electricity-without-any-resistance-a-closed-wire-loop-made-of-s-does-not-allow-any-change-in-flux-through-itself-

paragraph-a-special-metal-s-conducts-electricity-without-any-resistance-a-closed-wire-loop-made-of-s-does-not-allow-any-change-in-flux-through-itself-49655

<div class="question"><strong>Paragraph</strong><br/>A special metal $S$ conducts electricity without any resistance. A closed wire loop, made of $S$, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius $a$, with its center at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(\gg a)$ from the center of the loop with its north pole always facing the loop, as shown in the figure below.<br/><br/>The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}$, where $\mu_{0}$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_{1}$ and $m_{2}$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k m_{1} m_{2}}{r^{4}}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OnwLM4py_dLCUenLLwdFNEy8sa-0i9DHGKEW9cpctzU.original.fullsize.png"/><br/><strong>Question</strong>The work done in bringing the dipole from infinity to a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> from the centre of the loop by the given process is proportional to:</div>

['Physics', 'Magnetic Effects of Current', 'JEE Advanced', 'JEE Advanced 2021 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><msup><mi>r</mi><mn>5</mn></msup></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>5</mn></msup></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>6</mn></msup></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>7</mn></msup></mfrac></math></span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>6</mn></msup></mfrac></math></span> </div>

<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/571fef32-a285-4778-9cb1-8a36e9213622-image.png" style="width: 230px; height: 84px;"/></p><p>For a super conducting loop, net flux passing through it will remain constant.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><msub><mi>ϕ</mi><mrow><mi>total</mi><mo> </mo></mrow></msub></mfenced><mi>i</mi></msub><mo>=</mo><msub><mfenced><msub><mi>ϕ</mi><mrow><mi>total</mi><mo> </mo></mrow></msub></mfenced><mi>f</mi></msub></math></p><p>As area is same, therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>m</mi></mrow><mrow><mn>2</mn><msup><mi>πr</mi><mn>3</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>I</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>m</mi></mrow><msup><mi>πr</mi><mn>3</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>I</mi><mo>∝</mo><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p>The current carrying superconducting loop will also behave like a magnet, whose magnetic dipole moment is given by,<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>1</mn></msub><mo>=</mo><mi>N</mi><mi>I</mi><mi>A</mi><mspace linebreak="newline"></mspace><mo>=</mo><mo>(</mo><mn>1</mn><mo>)</mo><mfenced><mfrac><mrow><mi>a</mi><mi>m</mi></mrow><msup><mi>πr</mi><mn>3</mn></msup></mfrac></mfenced><mo>×</mo><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>a</mi><mn>3</mn></msup></mrow><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><msub><mi>m</mi><mn>1</mn></msub><mo>∝</mo><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p>The repulsive force felt on the magnet will be</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>K</mi><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub></mrow><msup><mi>r</mi><mn>4</mn></msup></mfrac><mo>=</mo><mfrac><mrow><mfenced><msup><mi>K</mi><mo>'</mo></msup></mfenced><mfenced><mfrac><mi>m</mi><msup><mi>r</mi><mn>3</mn></msup></mfrac></mfenced><mfenced><mi>m</mi></mfenced></mrow><msup><mi>r</mi><mn>4</mn></msup></mfrac><mo>∝</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>7</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>ext</mi></msub><mo>=</mo><mo>-</mo><mo>∫</mo><mi>F</mi><mi mathvariant="normal">d</mi><mi>r</mi><mspace linebreak="newline"></mspace><mo>=</mo><mo>-</mo><mi>K</mi><mo>'</mo><mo>∫</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>7</mn></msup></mfrac><mi mathvariant="normal">d</mi><mi>r</mi></math></p><p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mi>x</mi><mi>n</mi></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></p><p>We get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>ext</mi></msub><mo>∝</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><msup><mi>r</mi><mn>6</mn></msup></mfrac></math></p></div>

MarksBatch2_P2.db

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-1

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-1-26508

<div class="question"><strong>Paragraph:</strong><br/>A tertiary alcohol $H$ upon acid catalysed dehydration gives a product $I$. Ozonolysis of I leads to compounds $\mathrm{J}$ and $\mathrm{K}$. Compound $\mathrm{J}$ upon reaction with<br/>$\mathrm{KOH}$ gives benzyl alcohol and a compound $\mathrm{L}$, where $\mathrm{K}$ on reaction with $\mathrm{KOH}$ gives only $M$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fVFs9cvA43gTq2sBJ-pL8aamUV9DPuXd2cFGNHT06I4.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { Compound } H \text { is formed by the reactions of }<br/>$$</div>

['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/LTUNRkZDtofACR2Xbv1KyoA9jEVkDbKp5keaPTzgzCc.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ALprMe-56zf8uy-itnr9bBFu3zOJYvXX-We2fr0iQC8.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/LQv_PT-TcVhYf4LktqxRHtvloH5Q3c5HXgyBLluwkhU.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/zF2KjSfP0_ZPZWZxaiQR7TjB-fVtEcRy89NZJMqbQww.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/ALprMe-56zf8uy-itnr9bBFu3zOJYvXX-We2fr0iQC8.original.fullsize.png"/><br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/1Ag-AN9Naf0n8Nn8tMq1F5qaMExDqsPyhzm8CU_wDqE.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-2

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-2-26929

<div class="question"><strong>Paragraph:</strong><br/>A tertiary alcohol $H$ upon acid catalysed dehydration gives a product $I$. Ozonolysis of I leads to compounds $\mathrm{J}$ and $\mathrm{K}$. Compound $\mathrm{J}$ upon reaction with<br/>$\mathrm{KOH}$ gives benzyl alcohol and a compound $\mathrm{L}$, where $\mathrm{K}$ on reaction with $\mathrm{KOH}$ gives only $M$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fVFs9cvA43gTq2sBJ-pL8aamUV9DPuXd2cFGNHT06I4.original.fullsize.png"/><br/><strong>Question:</strong><br/>The structures of compounds $J, K$ and $L$, respectively, are</div>

['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mathrm{PhCHO}, \mathrm{PhCOCH}_3$ and $\mathrm{PhCOO}^{-} \mathrm{K}^{+}$</span> </div>

<div class="solution">$$<br/>J=\mathrm{PhCH}=\mathrm{O} K=\mathrm{PhCOCH}_3 \quad L=\mathrm{PhCOO}^{-} \mathrm{K}^{+}<br/>$$</div>

MarksBatch2_P2.db

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-3

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-3-63913

<div class="question"><strong>Paragraph:</strong><br/>A tertiary alcohol $H$ upon acid catalysed dehydration gives a product $I$. Ozonolysis of I leads to compounds $\mathrm{J}$ and $\mathrm{K}$. Compound $\mathrm{J}$ upon reaction with<br/>$\mathrm{KOH}$ gives benzyl alcohol and a compound $\mathrm{L}$, where $\mathrm{K}$ on reaction with $\mathrm{KOH}$ gives only $M$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fVFs9cvA43gTq2sBJ-pL8aamUV9DPuXd2cFGNHT06I4.original.fullsize.png"/><br/><strong>Question:</strong><br/>The structures of compounds $J, K$ and $L$, respectively, are</div>

['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mathrm{PhCHO}, \mathrm{PhCOCH}_3$ and $\mathrm{PhCOO}^{-} \mathrm{K}^{+}$</span> </div>

<div class="solution">$$<br/>J=\mathrm{PhCH}=\mathrm{O} K=\mathrm{PhCOCH}_3 \quad L=\mathrm{PhCOO}^{-} \mathrm{K}^{+}<br/>$$</div>

MarksBatch2_P2.db

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w

paragraph-a-tertiary-alcohol-h-upon-acid-catalysed-dehydration-gives-a-product-i-ozonolysis-of-i-leads-to-compounds-j-and-k-compound-j-upon-reaction-w-12032

<div class="question"><strong>Paragraph:</strong><br/>A tertiary alcohol $H$ upon acid catalysed dehydration gives a product $I$. Ozonolysis of I leads to compounds $\mathrm{J}$ and $\mathrm{K}$. Compound $\mathrm{J}$ upon reaction with<br/>$\mathrm{KOH}$ gives benzyl alcohol and a compound $\mathrm{L}$, where $\mathrm{K}$ on reaction with $\mathrm{KOH}$ gives only $M$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fVFs9cvA43gTq2sBJ-pL8aamUV9DPuXd2cFGNHT06I4.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structure of compound } I \text { is }<br/>$$</div>

['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (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/9NA7ujSG7OIEoIvjY2mYDbSqD4c6K1l3PFrr7pRpZPI.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/rGFtDLTU0ZnkCJLhkjnapzCyQVMlrCAcCf5ov1VzPp4.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/XiXd2qohfu5knNXWXGr14Od-34EVgrJshp5DaJqz-jM.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/tk8Q4NxNHj9-HYgG3-EPyfK-sb-TqPEK2hvWnXuGYIs.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/9NA7ujSG7OIEoIvjY2mYDbSqD4c6K1l3PFrr7pRpZPI.original.fullsize.png"/><br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZvgUGm2toOjYicbbbNkb317tpTep6QEYQQWwggoUPUQ.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-1

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-1-36153

<div class="question"><strong>Paragraph:</strong><br/>A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(C M)$ at a distance Lfrom the wall. The disk rolls without slipping with velocity $\mathbf{v}_0=v_0 \hat{\mathbf{i}}$ The coefficient of friction is $\mu$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qu5zp1fIdDHdtmpnI98KAcjT1WguzSHA0bjLfwtOGvg.original.fullsize.png"/><br/><strong>Question:</strong><br/>The maximum value of $v_0$ for which the disk will roll without slipping is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mu g \sqrt{\frac{M}{k}}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mu g \sqrt{\frac{M}{2 k}}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\mu g \sqrt{\frac{3 M}{k}}$<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/>$\mu g \sqrt{\frac{5 M}{2 k}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mu g \sqrt{\frac{3 M}{k}}$<br/></span> </div>

<div class="solution">In case of pure rolling, mechanical energy will remain conserved.<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; \therefore &amp; \frac{1}{2} M v_0^2+\frac{1}{2}\left(\frac{1}{2} M R^2\right)\left(\frac{v_0}{R}\right)^2 &amp; =2\left[\frac{1}{2} k x_{\max }^2\right] \\<br/>&amp; \therefore &amp; x_{\max } &amp; =\sqrt{\frac{3 M}{4 k}} v_0 \\<br/>&amp; \text { As } &amp; &amp; =\frac{2 k x}{3} \\<br/>&amp; f_{\max } &amp; =\mu M g=\frac{2 k x_{\max }}{3}=\frac{2 k}{3} \sqrt{\frac{3 M}{4 k}} v_0 \\<br/>&amp; v_0 &amp; =\mu g \sqrt{\frac{3 M}{k}}<br/>\end{array}<br/>$$<br/>$\therefore$ correct option is (c).</div>

MarksBatch2_P2.db

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-2

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-2-75634

<div class="question"><strong>Paragraph:</strong><br/>A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(C M)$ at a distance Lfrom the wall. The disk rolls without slipping with velocity $\mathbf{v}_0=v_0 \hat{\mathbf{i}}$ The coefficient of friction is $\mu$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qu5zp1fIdDHdtmpnI98KAcjT1WguzSHA0bjLfwtOGvg.original.fullsize.png"/><br/><strong>Question:</strong><br/>The centre of mass of the disk undergoes simple harmonic motion with angular frequency $\omega$ equal to</div>

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

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

<div class="solution">$$<br/>\begin{aligned}<br/>&amp; F_{\text {net }}=-\left(\frac{4 k}{3}\right) \cdot x \\<br/>&amp; \therefore a=\frac{F_{\text {net }}}{M}=-\left(\frac{4 k}{3 M}\right) x=-\omega^2 x \\<br/>&amp; \therefore \omega=\sqrt{\frac{4 k}{3 M}}<br/>\end{aligned}<br/>$$<br/>$\therefore$ correct option is (d).</div>

MarksBatch2_P2.db

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-3

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-3-34419

<div class="question"><strong>Paragraph:</strong><br/>A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(C M)$ at a distance Lfrom the wall. The disk rolls without slipping with velocity $\mathbf{v}_0=v_0 \hat{\mathbf{i}}$ The coefficient of friction is $\mu$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qu5zp1fIdDHdtmpnI98KAcjT1WguzSHA0bjLfwtOGvg.original.fullsize.png"/><br/><strong>Question:</strong><br/>The net external force acting on the disk when its centre of mass is at displacement $x$ with respect to its equilibrium position is</div>

['Physics', 'Oscillations', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$-k x$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$-2 k x$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$-\frac{2 k x}{3}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$-\frac{4 k x}{3}$</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 answer is: <span class="option-value"><br/>$-\frac{4 k x}{3}$</span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/87zdN1DOmFxV1uHxFl1sCju3IHy07z3Cawmy6uu8Uts.original.fullsize.png"/><br/><br/>$\therefore \quad \frac{2 k x-f}{M}=R\left[\frac{f \cdot R}{\frac{1}{2} M R^2}\right]$<br/>Solving this equation, we get $f=\frac{2 k x}{3}$<br/>$$<br/>\therefore\left|F_{\text {net }}\right|=2 k x-f=2 k x-\frac{2 k x}{3}=\frac{4 k x}{3}<br/>$$<br/>This is opposite to displacement.<br/>$$<br/>\therefore \quad F_{\text {net }}=-\frac{4 k x}{3}<br/>$$<br/>$\therefore$ correct option is (d).</div>

MarksBatch2_P2.db

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to

paragraph-a-uniform-thin-cylindrical-disk-of-mass-m-and-radius-r-is-attached-to-two-identical-massless-springs-of-spring-constant-k-which-are-fixed-to-16909

<div class="question"><strong>Paragraph:</strong><br/>A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is $L$. The disk is initially at its equilibrium position with its centre of mass $(C M)$ at a distance Lfrom the wall. The disk rolls without slipping with velocity $\mathbf{v}_0=v_0 \hat{\mathbf{i}}$ The coefficient of friction is $\mu$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qu5zp1fIdDHdtmpnI98KAcjT1WguzSHA0bjLfwtOGvg.original.fullsize.png"/><br/><strong>Question:</strong><br/>The net external force acting on the disk when its centre of mass is at displacement $x$ with respect to its equilibrium position is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$-k x$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$-2 k x$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$-\frac{2 k x}{3}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$-\frac{4 k x}{3}$</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 answer is: <span class="option-value"><br/>$-\frac{4 k x}{3}$</span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/87zdN1DOmFxV1uHxFl1sCju3IHy07z3Cawmy6uu8Uts.original.fullsize.png"/><br/><br/>$\therefore \quad \frac{2 k x-f}{M}=R\left[\frac{f \cdot R}{\frac{1}{2} M R^2}\right]$<br/>Solving this equation, we get $f=\frac{2 k x}{3}$<br/>$$<br/>\therefore\left|F_{\text {net }}\right|=2 k x-f=2 k x-\frac{2 k x}{3}=\frac{4 k x}{3}<br/>$$<br/>This is opposite to displacement.<br/>$$<br/>\therefore \quad F_{\text {net }}=-\frac{4 k x}{3}<br/>$$<br/>$\therefore$ correct option is (d).</div>

MarksBatch2_P2.db

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown--1

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown-1-85978

<div class="question"><strong>Paragraph:</strong><br/>A wooden cylinder of diameter $4 r$, height $H$ and density $\rho / 3$ is kept on a hole of diameter $2 r$ of a tank, filled with liquid of density $\rho$ as shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6MREMJ-UCzca3ZXs06bXwVNpiBYGQd2F5QjMotb1BPM.original.fullsize.png"/><br/><strong>Question:</strong><br/>If height $h_2$ of water level is further decreased, then</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>cylinder will not move up and remains at its original position<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/>for $h_2=h / 3$, cylinder again starts moving up<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>for $h_2=h / 4$, cylinder again starts moving up<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>for $h_2=h / 5$ cylinder again starts moving up</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>cylinder will not move up and remains at its original position<br/></span> </div>

<div class="solution">For $h_2 &lt; \frac{4 h}{9}$, buoyant force will further decrease. Hence, the cylinder remains at its original position.</div>

MarksBatch2_P2.db

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown--2

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown-2-28438

<div class="question"><strong>Paragraph:</strong><br/>A wooden cylinder of diameter $4 r$, height $H$ and density $\rho / 3$ is kept on a hole of diameter $2 r$ of a tank, filled with liquid of density $\rho$ as shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6MREMJ-UCzca3ZXs06bXwVNpiBYGQd2F5QjMotb1BPM.original.fullsize.png"/><br/><strong>Question:</strong><br/>The block in the above question is maintained at the position by external means and the level of liquid is lowered. The height $h_2$ when this external force reduces to zero is<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/3r_E4Wis_imzsqq0DkyQD7L0X7i2x3VvyVHXQ46D1pg.original.fullsize.png"/><br/></div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2006']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{4 h}{9}$<br/></span> </div>

<div class="solution">Again equating the forces, we get<br/>$$<br/>h_2=\frac{4 h}{9}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/O85USzWOgiPdgiiI6AKAZuxlAplRJQO1XTreazUckQA.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown--3

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown-3-11468

<div class="question"><strong>Paragraph:</strong><br/>A wooden cylinder of diameter $4 r$, height $H$ and density $\rho / 3$ is kept on a hole of diameter $2 r$ of a tank, filled with liquid of density $\rho$ as shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6MREMJ-UCzca3ZXs06bXwVNpiBYGQd2F5QjMotb1BPM.original.fullsize.png"/><br/><strong>Question:</strong><br/>Now level of the liquid starts decreasing slowly. When the level of liquid is at a height $h_1$ above the cylinder, the block starts moving up. At what value of $h_1$, will the block rise?</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{4 h}{9}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{5 h}{9}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{5 h}{3}$<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/>remains same</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{5 h}{3}$<br/></span> </div>

<div class="solution">Let $A_1=$ Area of cross-section of cylinder $=4 \pi r^2$<br/>$A_2=$ Area of base of cylinder in air $=\pi r^2$<br/>and $A_3=$ Area of base of cylinder in water<br/>$$<br/>=A_1-A_2=3 \pi r^2<br/>$$<br/>Drawing free body diagram of cylinder.<br/>Equation the net downward forces and net upward forces, we get<br/>$$<br/>h_1=\frac{5}{3} H<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9u78kr0bHzWd5Bs_x7ROAc1N_Js5hNflOjzQsxATUZY.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown-

paragraph-a-wooden-cylinder-of-diameter-4-r-height-h-and-density-3-is-kept-on-a-hole-of-diameter-2-r-of-a-tank-filled-with-liquid-of-density-as-shown-23638

<div class="question"><strong>Paragraph:</strong><br/>A wooden cylinder of diameter $4 r$, height $H$ and density $\rho / 3$ is kept on a hole of diameter $2 r$ of a tank, filled with liquid of density $\rho$ as shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6MREMJ-UCzca3ZXs06bXwVNpiBYGQd2F5QjMotb1BPM.original.fullsize.png"/><br/><strong>Question:</strong><br/>Now level of the liquid starts decreasing slowly. When the level of liquid is at a height $h_1$ above the cylinder, the block starts moving up. At what value of $h_1$, will the block rise?</div>

['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{4 h}{9}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{5 h}{9}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{5 h}{3}$<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/>remains same</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{5 h}{3}$<br/></span> </div>

<div class="solution">Let $A_1=$ Area of cross-section of cylinder $=4 \pi r^2$<br/>$A_2=$ Area of base of cylinder in air $=\pi r^2$<br/>and $A_3=$ Area of base of cylinder in water<br/>$$<br/>=A_1-A_2=3 \pi r^2<br/>$$<br/>Drawing free body diagram of cylinder.<br/>Equation the net downward forces and net upward forces, we get<br/>$$<br/>h_1=\frac{5}{3} H<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9u78kr0bHzWd5Bs_x7ROAc1N_Js5hNflOjzQsxATUZY.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-an-acylic-hydrocarbon-p-having-molecular-formula-c-6-h-10-gave-acetone-as-the-only-organic-product-through-the-following-sequence-of-reactio-1

paragraph-an-acylic-hydrocarbon-p-having-molecular-formula-c-6-h-10-gave-acetone-as-the-only-organic-product-through-the-following-sequence-of-reactio-1-22641

<div class="question"><strong>Paragraph:</strong><br/>An acylic hydrocarbon $P$, having molecular formula $\mathrm{C}_6 \mathrm{H}_{10}$, gave acetone as the only organic product through the following sequence of reactions, in which $Q$ is an intermediate organic compounds.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Awiwyvq-Pf-uzaqEFtX6UV-v10x-aFI91fJ7p7Hgbj0.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structure of the compound } Q \text { is }<br/>$$</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FszO8JGhlRn9hKhRi-juXhBZk5MvOUYBdl9KOTK7p50.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4te81-nwK48ytiAt6tMk4f9vxKcWuHMxr5zRXxvk_JM.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xdJGsTqCvgVKezZ-8HsaCY7c412dcW-YFwYC5gu_y3w.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/EFae35XucuvX6oTMfnEasrSLkI7ss3htCIUXyUpvpwk.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/4te81-nwK48ytiAt6tMk4f9vxKcWuHMxr5zRXxvk_JM.original.fullsize.png"/><br/></span> </div>

<div class="solution">Explained in the beginning.</div>

MarksBatch2_P2.db

paragraph-an-acylic-hydrocarbon-p-having-molecular-formula-c-6-h-10-gave-acetone-as-the-only-organic-product-through-the-following-sequence-of-reactio-2

paragraph-an-acylic-hydrocarbon-p-having-molecular-formula-c-6-h-10-gave-acetone-as-the-only-organic-product-through-the-following-sequence-of-reactio-2-93339

<div class="question"><strong>Paragraph:</strong><br/>An acylic hydrocarbon $P$, having molecular formula $\mathrm{C}_6 \mathrm{H}_{10}$, gave acetone as the only organic product through the following sequence of reactions, in which $Q$ is an intermediate organic compounds.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Awiwyvq-Pf-uzaqEFtX6UV-v10x-aFI91fJ7p7Hgbj0.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structure of compound } P \text { is }<br/>$$</div>

['Chemistry', 'Hydrocarbons', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/uXH7Hu-nZWeqC8s2APUYqJ0MBi4PsK-DA6KTwA6Aoy4.original.fullsize.png"/><br/></span> </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/yNzVP03dooDto2dqv7Q9UrtwbMrISAcBt6oELRDNPcs.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/qme8Y4cZ32ELzVcQIQ-iRtWAuQ_WN1MhttEE32fB2fk.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/dU-YP_5Ue-JXkSMFpjQ2_G2gm32ukkC-zSZTbEGc57g.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/dU-YP_5Ue-JXkSMFpjQ2_G2gm32ukkC-zSZTbEGc57g.original.fullsize.png"/><br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/geCbU0WqcycZ5B33Vg9LEXRzkKFWuga6A8bb0veispU.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-an-acylic-hydrocarbon-p-having-molecular-formula-c-6-h-10-gave-acetone-as-the-only-organic-product-through-the-following-sequence-of-reactio

paragraph-an-acylic-hydrocarbon-p-having-molecular-formula-c-6-h-10-gave-acetone-as-the-only-organic-product-through-the-following-sequence-of-reactio-69363

<div class="question"><strong>Paragraph:</strong><br/>An acylic hydrocarbon $P$, having molecular formula $\mathrm{C}_6 \mathrm{H}_{10}$, gave acetone as the only organic product through the following sequence of reactions, in which $Q$ is an intermediate organic compounds.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Awiwyvq-Pf-uzaqEFtX6UV-v10x-aFI91fJ7p7Hgbj0.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structure of compound } P \text { is }<br/>$$</div>

['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/uXH7Hu-nZWeqC8s2APUYqJ0MBi4PsK-DA6KTwA6Aoy4.original.fullsize.png"/><br/></span> </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/yNzVP03dooDto2dqv7Q9UrtwbMrISAcBt6oELRDNPcs.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/qme8Y4cZ32ELzVcQIQ-iRtWAuQ_WN1MhttEE32fB2fk.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/dU-YP_5Ue-JXkSMFpjQ2_G2gm32ukkC-zSZTbEGc57g.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/dU-YP_5Ue-JXkSMFpjQ2_G2gm32ukkC-zSZTbEGc57g.original.fullsize.png"/><br/></span> </div>

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/geCbU0WqcycZ5B33Vg9LEXRzkKFWuga6A8bb0veispU.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-carbon14-is-used-to-determine-the-age-of-organic-material-the-procedure-is-based-on-the-formation-of-14-c-by-neutron-capture-in-the-upper-at-1

paragraph-carbon14-is-used-to-determine-the-age-of-organic-material-the-procedure-is-based-on-the-formation-of-14-c-by-neutron-capture-in-the-upper-at-1-85645

<div class="question"><strong>Paragraph:</strong><br/>Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of ${ }^{14} \mathrm{C}$ by neutron capture in the upper atmosphere.<br/>$$<br/>{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1<br/>$$<br/>${ }^{14} \mathrm{C}$ is absorbed by living organisms during photosynthesis. The ${ }^{14} \mathrm{C}$ content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of ${ }^{14} \mathrm{C}$ in the dead being, falls to the decay which ${ }^{14} \mathrm{C}$ undergoes.<br/>$$<br/>{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}<br/>$$<br/>The half life period of ${ }^{14} \mathrm{C}$ is 5770 years. The decay constant $(\lambda)$ can be calculated by using the following formula $\lambda=\frac{0.693}{t_{1 / 2}}$.<br/>The comparison of the $\beta^{-}$activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of ${ }^{14} \mathrm{C}$ to ${ }^{12} \mathrm{C}$ in living matter is $1: 10^{12}$.<strong>Question:</strong><br/>A nuclear explosion has taken place leading to increase in concentration of $C^{14}$ in nearby areas. $\mathrm{C}^{14}$ concentration is $C_1$ in nearby areas and $C_2$ in areas far away. If the age of the fossil is determined to be $T_1$ and $T_2$ at the places respectively then</div>

['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>The age of the fossil will increase at the place where explosion has taken and<br/>$$<br/>T_1-T_2=\frac{1}{\lambda} \ln \frac{C_1}{C_2}<br/>$$<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/>The age of the fossil will decrease at the place where explosion has taken place and $T_1-T_2=\frac{1}{\lambda} \ln \frac{C_1}{C_2}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>The age of fossil will be determined to be same<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{T_1}{T_2}=\frac{C_1}{C_2}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>The age of the fossil will increase at the place where explosion has taken and<br/>$$<br/>T_1-T_2=\frac{1}{\lambda} \ln \frac{C_1}{C_2}<br/>$$<br/></span> </div>

<div class="solution">All radioactive decays are the examples of first order kinetics.<br/>So, decay constant $\lambda=\frac{1}{T_1-T_2} \ln \frac{C_1}{C_2}$<br/>$C_1$ is the concentration at $T_1$ time<br/>$C_2$ is the concentration at $T_2$ time<br/>So, $\quad T_1-T_2=\frac{1}{\lambda} \ln \frac{C_1}{C_2}$</div>

MarksBatch2_P2.db

paragraph-carbon14-is-used-to-determine-the-age-of-organic-material-the-procedure-is-based-on-the-formation-of-14-c-by-neutron-capture-in-the-upper-at-2

paragraph-carbon14-is-used-to-determine-the-age-of-organic-material-the-procedure-is-based-on-the-formation-of-14-c-by-neutron-capture-in-the-upper-at-2-10876

<div class="question"><strong>Paragraph:</strong><br/>Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of ${ }^{14} \mathrm{C}$ by neutron capture in the upper atmosphere.<br/>$$<br/>{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1<br/>$$<br/>${ }^{14} \mathrm{C}$ is absorbed by living organisms during photosynthesis. The ${ }^{14} \mathrm{C}$ content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of ${ }^{14} \mathrm{C}$ in the dead being, falls to the decay which ${ }^{14} \mathrm{C}$ undergoes.<br/>$$<br/>{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}<br/>$$<br/>The half life period of ${ }^{14} \mathrm{C}$ is 5770 years. The decay constant $(\lambda)$ can be calculated by using the following formula $\lambda=\frac{0.693}{t_{1 / 2}}$.<br/>The comparison of the $\beta^{-}$activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of ${ }^{14} \mathrm{C}$ to ${ }^{12} \mathrm{C}$ in living matter is $1: 10^{12}$.<strong>Question:</strong><br/>What should be the age of fossil for meaningful determination of its age?</div>

['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>6 years<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>6000 years<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/>60,000 years<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>It can be used to calculate any age</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>6000 years<br/></span> </div>

<div class="solution">Radio carbon dating method ceases to be accurate over periods longer than 30,000 years.</div>

MarksBatch2_P2.db

paragraph-carbon14-is-used-to-determine-the-age-of-organic-material-the-procedure-is-based-on-the-formation-of-14-c-by-neutron-capture-in-the-upper-at

paragraph-carbon14-is-used-to-determine-the-age-of-organic-material-the-procedure-is-based-on-the-formation-of-14-c-by-neutron-capture-in-the-upper-at-61796

<div class="question"><strong>Paragraph:</strong><br/>Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of ${ }^{14} \mathrm{C}$ by neutron capture in the upper atmosphere.<br/>$$<br/>{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1<br/>$$<br/>${ }^{14} \mathrm{C}$ is absorbed by living organisms during photosynthesis. The ${ }^{14} \mathrm{C}$ content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of ${ }^{14} \mathrm{C}$ in the dead being, falls to the decay which ${ }^{14} \mathrm{C}$ undergoes.<br/>$$<br/>{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}<br/>$$<br/>The half life period of ${ }^{14} \mathrm{C}$ is 5770 years. The decay constant $(\lambda)$ can be calculated by using the following formula $\lambda=\frac{0.693}{t_{1 / 2}}$.<br/>The comparison of the $\beta^{-}$activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of ${ }^{14} \mathrm{C}$ to ${ }^{12} \mathrm{C}$ in living matter is $1: 10^{12}$.<strong>Question:</strong><br/>Which of the following option is correct?</div>

['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2006']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>In living organisms, circulation of ${ }^{14} \mathrm{C}$ from atmosphere is high so the carbon content is constant in organism.<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Carbon dating can be used to find out the age of earth crust and rocks.<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>Radioactive absorption due to cosmic radiation is equal to the rate of radioactive decay, hence the carbon content remains constant in living organisms.<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/>Carbon dating can not be used to determine concentration of ${ }^{14} \mathrm{C}$ in dead beings.</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>Radioactive absorption due to cosmic radiation is equal to the rate of radioactive decay, hence the carbon content remains constant in living organisms.<br/></span> </div>

<div class="solution">Radioactive absorption due to cosmic radiation is equal to the rate of radioactive decay, hence the carbon content as the ratio of $\mathrm{C}^{14}$ to $\mathrm{C}^{12}$ remains constant in living organism.</div>

MarksBatch2_P2.db

paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe-1

paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe-1-54196

<div class="question"><strong>Paragraph:</strong><br/>Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately $6.023 \times 10^{23}$ ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.<br/>A $4.0$ molar aqueous solution of $\mathrm{NaCl}$ is prepared and $500 \mathrm{~mL}$ of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: $\mathrm{Na}=23, \mathrm{Hg}=200 ; 1$ faraday $=96500$ coulombs).<strong>Question:</strong><br/>If the cathode is a Hg electrode, the maximum weight $(g)$ of amalgam formed from this solution is</div>

['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>200<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>225<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>400<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>446</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>446</span> </div>

<div class="solution">$\mathrm{Na}+\mathrm{Hg} \longrightarrow \mathrm{Na}-\mathrm{Hg}$<br/>2 mole $\quad 2$ mole<br/>2 moles of amalgam $=23 \times 2+2 \times 200=446 \mathrm{~g}$</div>

MarksBatch2_P2.db

paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe-2

paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe-2-94191

<div class="question"><strong>Paragraph:</strong><br/>Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately $6.023 \times 10^{23}$ ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.<br/>A $4.0$ molar aqueous solution of $\mathrm{NaCl}$ is prepared and $500 \mathrm{~mL}$ of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: $\mathrm{Na}=23, \mathrm{Hg}=200 ; 1$ faraday $=96500$ coulombs).<strong>Question:</strong><br/>The total charge (coulombs) required for complete electrolysis is</div>

['Chemistry', 'Electrochemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>24125<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>48250<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>96500<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>193000</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>193000</span> </div>

<div class="solution">$2 \mathrm{Cl}^{-} \longrightarrow \mathrm{Cl}_2+2 e^{-}$<br/>Total charge $=2 \times 96500$ Coulamb $=193000$ coulomb</div>

MarksBatch2_P2.db

paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe

paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe-48276

<div class="question"><strong>Paragraph:</strong><br/>Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately $6.023 \times 10^{23}$ ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.<br/>A $4.0$ molar aqueous solution of $\mathrm{NaCl}$ is prepared and $500 \mathrm{~mL}$ of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: $\mathrm{Na}=23, \mathrm{Hg}=200 ; 1$ faraday $=96500$ coulombs).<strong>Question:</strong><br/>The total number of moles of chlorine gas evolved is</div>

['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$0.5$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$1.0$<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/>$2.0$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$3.0$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$1.0$<br/></span> </div>

<div class="solution">$4.0 \mathrm{M} \mathrm{NaCl} .500 \mathrm{~mL}(0.5 \mathrm{~L})$<br/>Moles of $\mathrm{NaCl}=4 \times 0.5=2$<br/>2 moles $\mathrm{NaCl} \equiv 2$ moles $\mathrm{Cl}^{-}$<br/>$$<br/>\underset{2 \text { mole }}{2 \mathrm{Cl}^{-}} \longrightarrow \mathrm{Cl}_2<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-an-evacuated-cylindrical-chamber-of-height-h-having-rigid-conducting-plates-at-the-ends-and-an-insulating-curved-surface-as-shown-i

paragraph-consider-an-evacuated-cylindrical-chamber-of-height-h-having-rigid-conducting-plates-at-the-ends-and-an-insulating-curved-surface-as-shown-i-36120

<div class="question"><strong> Paragraph: </strong> <br/> <br/>Consider an evacuated cylindrical chamber of height $h$ having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius $r \ll h$. Now a high voltage source $(HV)$ is connected across the conducting plates such that the bottom plate is at $+V_{0}$ and the top plate at $-V_{0} .$ Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity) <br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/X0iyOE6ZSfi3sQBAwhZMjz6Hr0rDVbqrliZikAjNnu0.original.fullsize.png"/><br/> <br/> <br/><br/> <strong> Question: </strong> <br/> <br/>The average current in the steady state registered by the ammeter in the circuit will be</div>

['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2016 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Proportional to <math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Proportional to <math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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">C</span> <span class="option-data">Proportional to the potential <math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Zero</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value">Proportional to <math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> </div>

<div class="solution"><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Phy_Papr2_18_Ans_.jpg" style="width: 251px; height: 56px;"/><br/><math><mi>h</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi> </mi><mi>a</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></math> [as u = 0]<br/><math><msqrt><mfrac><mrow><mn>2</mn><mi>h</mi><mi>m</mi></mrow><mrow><mi>q</mi><mi>E</mi></mrow></mfrac></msqrt><mo>=</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>⇒</mo><mi> </mi><mi> </mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>=</mo><mi> </mi><msqrt><mfrac><mrow><mn>2</mn><mi>m</mi></mrow><mrow><mi>q</mi><mo>∆</mo><mi>V</mi></mrow></mfrac></msqrt></math><br/><math><mi>E</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mi>h</mi></mrow></mfrac></math><br/><math><mo>&lt;</mo><mi>c</mi><mi>u</mi><mi>r</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo>&gt;</mo><mi> </mi><mo>=</mo><mfrac><mrow><mi>c</mi><mi>h</mi><mi>a</mi><mi>r</mi><mi>g</mi><mi>e</mi></mrow><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>q</mi><msqrt><mi>q</mi><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></msqrt></mrow><mrow><mn>2</mn><mi>m</mi><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><math><mi>q</mi><mi> </mi><mi>∞</mi><mi> </mi><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math><br/><math><mo>&lt;</mo><mi>I</mi><mo>&gt;</mo><mi>∞</mi><mi> </mi><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></div>

MarksBatch2_P2.db

paragraph-consider-the-circle-x-2-y-2-9-and-the-parabola-y-2-8-x-they-intersect-at-p-and-q-in-the-first-and-the-fourth-quadrants-respectively-tangents-1

paragraph-consider-the-circle-x-2-y-2-9-and-the-parabola-y-2-8-x-they-intersect-at-p-and-q-in-the-first-and-the-fourth-quadrants-respectively-tangents-1-11071

<div class="question"><strong>Paragraph:</strong><br/>Consider the circle $x^2+y^2=9$ and the parabola $y^2=8 x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $\mathrm{X}$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $\mathrm{X}$-axis at $S$.<strong>Question:</strong><br/>The radius of the circumcircle of the $\triangle P R S$ is</div>

['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

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

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

<div class="solution">Equation of circumcircle of $\triangle P R S$ is<br/>$$<br/>(x+1)(x-9)+y^2+\lambda y=0<br/>$$<br/>It will pass through $(1,2 \sqrt{2})$, then<br/>$$<br/>\begin{aligned}<br/>-16+8+\lambda \cdot 2 \sqrt{2} &amp; =0 \\<br/>\lambda \quad \lambda &amp; =\frac{8}{2 \sqrt{2}}=2 \sqrt{2}<br/>\end{aligned}<br/>$$<br/>Equation of circumcircle is<br/>$$<br/>x^2+y^2-8 x+2 \sqrt{2} y-9=0<br/>$$<br/>Hence, its radius is $3 \sqrt{3}$.<br/>ALITER<br/>Let<br/>$$<br/>\begin{array}{ll}<br/>\text { Let } &amp; \angle P S R=\theta \\<br/>\Rightarrow &amp; \sin \theta=\frac{2 \sqrt{2}}{2 \sqrt{3}} \\<br/>\Rightarrow &amp; P R=6 \sqrt{2}=2 R \cdot \sin \theta<br/>\end{array}<br/>$$<br/>$$<br/>\Rightarrow \quad \sin \theta=\frac{2 \sqrt{2}}{2 \sqrt{3}}<br/>$$<br/>$$<br/>\Rightarrow \quad R=3 \sqrt{3} \text {. }<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-circle-x-2-y-2-9-and-the-parabola-y-2-8-x-they-intersect-at-p-and-q-in-the-first-and-the-fourth-quadrants-respectively-tangents-2

paragraph-consider-the-circle-x-2-y-2-9-and-the-parabola-y-2-8-x-they-intersect-at-p-and-q-in-the-first-and-the-fourth-quadrants-respectively-tangents-2-88714

<div class="question"><strong>Paragraph:</strong><br/>Consider the circle $x^2+y^2=9$ and the parabola $y^2=8 x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $\mathrm{X}$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $\mathrm{X}$-axis at $S$.<strong>Question:</strong><br/>The radius of the incircle of the $\triangle P Q R$ is</div>

['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>4<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>3<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{8}{3}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>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/>2</span> </div>

<div class="solution">Radius of incircle is, $r=\frac{\Delta}{s}$<br/>As $\Delta=16 \sqrt{2}$<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad s=\frac{6 \sqrt{2}+6 \sqrt{2}+4 \sqrt{2}}{2}=8 \sqrt{2} \\<br/>&amp; \therefore \quad r=\frac{16 \sqrt{2}}{8 \sqrt{2}}=2 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-circle-x-2-y-2-9-and-the-parabola-y-2-8-x-they-intersect-at-p-and-q-in-the-first-and-the-fourth-quadrants-respectively-tangents

paragraph-consider-the-circle-x-2-y-2-9-and-the-parabola-y-2-8-x-they-intersect-at-p-and-q-in-the-first-and-the-fourth-quadrants-respectively-tangents-39343

<div class="question"><strong>Paragraph:</strong><br/>Consider the circle $x^2+y^2=9$ and the parabola $y^2=8 x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $\mathrm{X}$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $\mathrm{X}$-axis at $S$.<strong>Question:</strong><br/>The ratio of the areas of the $\triangle P Q S$ and $\triangle P Q R$ is</div>

['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1: \sqrt{2}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$1: 2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$1: 4$<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">D</span> <span class="option-data"><br/>$1: 8$</span> </li> </ul>

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

<div class="solution">Coordinates of $P$ and $Q$ are $(1,2 \sqrt{2})$ and $(1,-2 \sqrt{2})$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9dALRJjOh-9hWyHO_nuAnTIRdris7Usk7WPt-roEO5k.original.fullsize.png"/><br/><br/>Area of $\triangle P Q R=\frac{1}{2} \cdot 4 \sqrt{2} \cdot 8=16 \sqrt{2}$<br/>Area of $\triangle P Q S=\frac{1}{2} \cdot 4 \sqrt{2} \cdot 2=4 \sqrt{2}$<br/>Ratio of area of $\triangle P Q S$ and $\triangle P Q R$ is $1: 4$.</div>

MarksBatch2_P2.db

paragraph-consider-the-function-f-defined-by-f-x-x-2-a-x-1-x-2-a-x-1-0-a-2-question-let-g-x-0-e-x-1-t-2-f-t-d-t-which-of-the-following-is-true

paragraph-consider-the-function-f-defined-by-f-x-x-2-a-x-1-x-2-a-x-1-0-a-2-question-let-g-x-0-e-x-1-t-2-f-t-d-t-which-of-the-following-is-true-72355

<div class="question"><strong>Paragraph:</strong><br/>Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ defined by $f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 &lt; a &lt; 2$<strong>Question:</strong><br/>Let $g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t$. Which of the following is true ?</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$<br/></span> </div>

<div class="solution">$\because g^{\prime}(x)=\frac{f^{\prime}\left(e^x\right)}{1+\left(e^x\right)^2} \cdot e^x=2 a\left[\frac{e^{2 x}-1}{\left(e^{2 x}+a e^x+1\right)^2}\right]\left(\frac{e^x}{1+e^{2 x}}\right)$ $\begin{aligned} &amp; g^{\prime}(x)=0 \text {, if } e^{2 x}-1 &amp; =0, \\ \text { i.e., } &amp; &amp; x=0 \\ \text { If } &amp; &amp; x &lt; 0, e^{2 x} &lt; 1 \Rightarrow g^{\prime}(x) &lt; 0 .\end{aligned}$</div>

MarksBatch2_P2.db

paragraph-consider-the-function-f-defined-by-f-x-x-2-a-x-1-x-2-a-x-1-0-a-2-question-which-of-the-following-is-true-1

paragraph-consider-the-function-f-defined-by-f-x-x-2-a-x-1-x-2-a-x-1-0-a-2-question-which-of-the-following-is-true-1-52943

<div class="question"><strong>Paragraph:</strong><br/>Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ defined by $f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 &lt; a &lt; 2$<strong>Question:</strong><br/>Which of the following is true?</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$f(x)$ is decreasing on $(-1,1)$ and has a local minimum 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">B</span> <span class="option-data"><br/>$f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$<br/></span> </div>

<div class="solution">When $x \in(-1,1)$<br/>$$<br/>x^2 &lt; 1 \Rightarrow x^2-1 &lt; 0<br/>$$<br/>$\therefore \quad f^{\prime}(x) &lt; 0 \Rightarrow f(x)$ is decreasing.<br/>Also, at $x=1, f^{\prime \prime}(1)=\frac{4 a}{(a+2)^2&gt;0}$<br/>$[\because 0 &lt; a &lt; 2]$<br/>$\therefore f(x)$ has a local minimum at $x=1$.</div>

MarksBatch2_P2.db

paragraph-consider-the-function-f-defined-by-f-x-x-2-a-x-1-x-2-a-x-1-0-a-2-question-which-of-the-following-is-true

paragraph-consider-the-function-f-defined-by-f-x-x-2-a-x-1-x-2-a-x-1-0-a-2-question-which-of-the-following-is-true-28986

<div class="question"><strong>Paragraph:</strong><br/>Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ defined by $f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 &lt; a &lt; 2$<strong>Question:</strong><br/>Which of the following is true ?</div>

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

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=0$<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">B</span> <span class="option-data"><br/>$(2-a)^2 f^{\prime \prime}(1)-(2+a)^2 f^{\prime \prime}(-1)=0$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$f^{\prime}(1) f^{\prime}(-1)=(2-a)^2$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$f^{\prime}(1) f^{\prime}(-1)=-(2+a)^2$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=0$<br/></span> </div>

<div class="solution">\because f(x)= &amp; \frac{\left(x^2+a x+1\right)-2 a x}{x^2+a x+1}=1-\frac{2 a x}{x^2+a x+1} \\<br/>\therefore \quad f^{\prime}(x) &amp; =-\left[\frac{\left(x^2+a x+1\right) \cdot 2 a-2 a x(2 x+a)}{\left(x^2+a x+1\right)^2}\right] \\<br/>&amp; =-\left[\frac{-2 a x^2+2 a}{\left(x^2+a x+1\right)^2}\right]=2 a\left[\frac{\left(x^2-1\right)}{\left(x^2+a x+1\right)^2}\right]<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \text { and } \begin{aligned}<br/>f^{\prime \prime}(x) &amp; =2 a\left[\frac{\left(x^2+a x+1\right)^2(2 x)-2\left(x^2-1\right)\left(x^2+a x+1\right)(2 x+a)}{\left(x^2+a x+1\right)^4}\right] \\<br/>&amp; =2 a\left[\frac{2 x\left(x^2+a x+1\right)-2\left(x^2-1\right)(2 x+a)}{\left(x^2+a x+1\right)^3}\right]<br/>\end{aligned} \\<br/>&amp; \text { Now, } f^{\prime \prime}(1)=\frac{4 a(a+2)}{(a+2)^3}=\frac{4 a}{(a+2)^2}<br/>\end{aligned}<br/>$$<br/>and $f(-1)=\frac{4 a(a-2)}{(2-a)^3}=-\frac{4 a}{(a-2)^2}$<br/>$$<br/>\therefore(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=4 a-4 a=0<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly-1

paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly-1-58495

<div class="question"><strong>Paragraph:</strong><br/>Consider the functions defined implicitly by the equation $y^3-3 y+x=0$ on various intervals in the real line. If $x \in(-\infty,-2) \cup(2, \infty)$, the equation implicitly defines a unique real valued differentiable function $y=f(x)$. If $x \in(-2,2)$, the equation implicitly defines a unique real valued differentiable function $y=g(x)$, satisfying $g(0)=0$.<strong>Question:</strong><br/>$\int_{-1}^1 g^{\prime}(x) d x$ is equal to</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$2 g(-1)$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>0<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$-2 g(1)$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$2 g(1)$</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/>$2 g(1)$</span> </div>

<div class="solution">Let $I=\int_{-1}^1 g^{\prime}(x) d x=[g(x)]_{-1}^1=g(1)-g(-1)$<br/>Since, $\quad y^3-3 y+x=0$ and $\quad y=g(x)$<br/>$$<br/>\therefore(g(x))^3-3 g(x)+x=0<br/>$$<br/>[by Eq. (i)]<br/>At $\quad x=1$,<br/>$$<br/>\begin{aligned}<br/>(g(1))^3-3 g(1)+1 &amp; =0 \\<br/>\text { At } \quad x &amp; =-1, \\<br/>(g(-1))^3-3 g(-1)-1 &amp; =0<br/>\end{aligned}<br/>$$<br/>On adding Eqs. (i) and (ii), we get<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; (g(1))^3+(g(-1))^3-3(g(1)+g(-1))=0 \\<br/>\Rightarrow &amp; &amp; {[g(1)+g(-1)]\left[(g(1))^2+(g(-1))^2-g(1) g(-1)-3\right]=0} \\<br/>\Rightarrow &amp; g(1)+g(-1)=0 \\<br/>\Rightarrow &amp; g(1) &amp; =-g(-1) \\<br/>&amp; \therefore &amp; I &amp; =g(1)-g(-1)=g(1)-(-g(1))=2 g(1)<br/>\end{array}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly-2

paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly-2-89956

<div class="question"><strong>Paragraph:</strong><br/>Consider the functions defined implicitly by the equation $y^3-3 y+x=0$ on various intervals in the real line. If $x \in(-\infty,-2) \cup(2, \infty)$, the equation implicitly defines a unique real valued differentiable function $y=f(x)$. If $x \in(-2,2)$, the equation implicitly defines a unique real valued differentiable function $y=g(x)$, satisfying $g(0)=0$.<strong>Question:</strong><br/>If $f(-10 \sqrt{2})=2 \sqrt{2}$, then $f^{\prime \prime}(-10 \sqrt{2})$ is equal to</div>

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

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$-\frac{4 \sqrt{2}}{7^3 \cdot 3^2}$<br/></span> </div>

<div class="solution">Given, $y^3-3 y+x=0 \Rightarrow 3 y^2 \frac{d y}{d x}-3 \frac{d y}{d x}+1=0$<br/>$$<br/>\Rightarrow \quad 3 y^2\left(\frac{d^2 y}{d x^2}\right)+6 y\left(\frac{d y}{d x}\right)^2-3 \frac{d^2 y}{d x^2}=0<br/>$$<br/>On substituting $x=-10 \sqrt{2}, y=2 \sqrt{2}$ in Eq. (i), we get<br/>$$<br/>3(2 \sqrt{2})^2 \cdot \frac{d y}{d x}-3 \cdot \frac{d y}{d x}+1=0 \Rightarrow \frac{d y}{d x}=\frac{-1}{21}<br/>$$<br/>Again on substituting $x=-10 \sqrt{2}, y=2 \sqrt{2}$ in Eq. (ii), we get<br/>$$<br/>\begin{aligned}<br/>&amp; 3(2 \sqrt{2})^2 \frac{d^2 y}{d x^2}+6(2 \sqrt{2}) \cdot\left(\frac{-1}{21}\right)^2-3 \cdot \frac{d^2 y}{d x^2}=0 \\<br/>&amp; \Rightarrow \quad 21 \cdot \frac{d^2 y}{d x^2}=-\frac{12 \sqrt{2}}{(21)^2} \Rightarrow \frac{d^2 y}{d x^2}=-\frac{12 \sqrt{2}}{(21)^3}=-\frac{4 \sqrt{2}}{7^3 \cdot 3^2}<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly

paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly-28776

<div class="question"><strong>Paragraph:</strong><br/>Consider the functions defined implicitly by the equation $y^3-3 y+x=0$ on various intervals in the real line. If $x \in(-\infty,-2) \cup(2, \infty)$, the equation implicitly defines a unique real valued differentiable function $y=f(x)$. If $x \in(-2,2)$, the equation implicitly defines a unique real valued differentiable function $y=g(x)$, satisfying $g(0)=0$.<strong>Question:</strong><br/>The area of the region bounded by the curve $y=f(x)$, the $X$-axis and the lines $x=a$ and $x=b$, where $-\infty &lt; a &lt; b &lt; -2$, is</div>

['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\int_a^b \frac{x}{3\left[f(x)^2-1\right]} d x+b f(b)-a f(a)$<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=""> <span class="option-label">B</span> <span class="option-data"><br/>$-\int_a^b \frac{x}{3\left[(f(x))^2-1\right]} d x+b f(b)-a f(a)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\int_a^b \frac{x}{3\left[(f(x))^2-1\right]} d x-b f(b)+a f(a)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$-\int_a^b \frac{x}{3\left[f(x)^2-1\right]} d x-b f(b)-a f(a)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\int_a^b \frac{x}{3\left[f(x)^2-1\right]} d x+b f(b)-a f(a)$<br/></span> </div>

<div class="solution">Required area $=\int_a^b y d x=\int_a^b f(x) d x$<br/>$$<br/>=[f(x) \cdot x]_a^b-\int_a^b f^{\prime}(x) \cdot x d x=b f(b)-a f(a)-\int_a^b f^{\prime}(x) \cdot x d x<br/>$$<br/>$$<br/>=b f(b)-a f(a)+\int_a^b \frac{x}{3\left[(f(x))^2-1\right]} d x<br/>$$<br/>As, $\quad f^{\prime}(x)=\frac{d y}{d x}=-\frac{1}{3\left(y^2-1\right)}=-\frac{1}{3\left[(f(x))^2-1\right]}$</div>

MarksBatch2_P2.db

paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-distance-of-the-point-1-1-1-from-the-plane-passing-through-the-p

paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-distance-of-the-point-1-1-1-from-the-plane-passing-through-the-p-84144

<div class="question"><strong>Paragraph:</strong><br/>Consider the lines: $L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}$<strong>Question:</strong><br/>The distance of the point $(1,1,1)$ from the plane passing through the point $(-1,-2,-1)$ and whose normal is perpendicular to both the lines $L_1$ and $L_2$, is</div>

['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$2 / \sqrt{75}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$7 / \sqrt{75}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$13 / \sqrt{75}$<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/>$23 / \sqrt{75}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$13 / \sqrt{75}$<br/></span> </div>

<div class="solution">The equation of the plane passing through the point $(-1,-2,-1)$ and whose normal is perpendicular to both the given lines $L_1$ and $L_2$ may be written as<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow \quad(x+1)+7(y+2)-5(z+1) &amp; =0 \\<br/>&amp; x+7 y-5 z+10 &amp; =0<br/>\end{array}<br/>$$<br/>The distance of the point $(1,1,1)$ from the plane<br/>$$<br/>=\left|\frac{1+7-5+10}{\sqrt{1+49+25}}\right|=\frac{13}{\sqrt{75}}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-shortest-distance-between-l-1-and-l-2-is

paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-shortest-distance-between-l-1-and-l-2-is-68943

<div class="question"><strong>Paragraph:</strong><br/>Consider the lines: $L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}$<strong>Question:</strong><br/>The shortest distance between $L_1$ and $L_2$ is</div>

['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2008 (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/>$17 / \sqrt{3}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$41 / 5 \sqrt{3}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$17 / 5 \sqrt{3}$</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/>$17 / 5 \sqrt{3}$</span> </div>

<div class="solution">The shortest distance between $L_1$ and $L_2$ is<br/>$$<br/>\begin{array}{r}<br/>\left|\frac{\{(2-(-1)) \hat{\mathbf{i}}+(2-2) \hat{\mathbf{j}}+(3-(-1)) \hat{\mathbf{k}}\} \cdot(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{5 \sqrt{3}}\right| \\<br/>=\left|\frac{(3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}) \cdot(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{5 \sqrt{3}}\right|=\frac{17}{5 \sqrt{3}}<br/>\end{array}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-unit-vector-perpendicular-to-both-l-1-and-l-2-is

paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-unit-vector-perpendicular-to-both-l-1-and-l-2-is-86022

<div class="question"><strong>Paragraph:</strong><br/>Consider the lines: $L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}$<strong>Question:</strong><br/>The unit vector perpendicular to both $L_1$ and $L_2$ is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{-\hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}}{\sqrt{99}}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}}{5 \sqrt{3}}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{-\hat{\mathbf{i}}+7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}}{5 \sqrt{3}}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{7 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}-\hat{\mathbf{k}}}{\sqrt{99}}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\frac{-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}}{5 \sqrt{3}}$<br/></span> </div>

<div class="solution">The equations of given lines in vector form may be written as<br/>$$<br/>L_1: \mathbf{r}(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})<br/>$$<br/>and $L_2: \mathbf{r}=(2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$<br/>$\therefore$ The vector perpendicular to both $L_1$ and $L_2$ is<br/>$$<br/>\left|\begin{array}{rrr}<br/>\hat{\mathbf{i}} &amp; \hat{\mathbf{j}} &amp; \hat{\mathbf{k}} \\<br/>3 &amp; 1 &amp; 2 \\<br/>1 &amp; 2 &amp; 3<br/>\end{array}\right|=-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>\therefore \text { Required unit vector } &amp; =\frac{(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{\sqrt{(-1)^2+(-7)^2+(5)^2}} \\<br/>&amp; =\frac{1}{5 \sqrt{3}}(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-area-bounded-by-th-1

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-area-bounded-by-th-1-20476

<div class="question"><strong>Paragraph:</strong><br/>Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.<strong>Question:</strong><br/>The area bounded by the curve $y=f(x)$ and the lines $x=0, y=0$ and $x=t$, lies in the interval</div>

['Mathematics', 'Area Under Curves', 'JEE Main']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{3}{4}, 3\right)$<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/>$\left(\frac{21}{64}, \frac{11}{16}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$(9,10)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(0, \frac{21}{64}\right)$</span> </li> </ul>

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

<div class="solution">$\int_0^{1 / 2} f(x) d x &lt; \int_0^t f(x) d x &lt; \int_0^{3 / 4} f(x) d x$ Now, $\int f(x) d x$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Y1rOmN-ABGVZnGrMYQXtd39YRuXI_W0Y9AEChDFAJoQ.original.fullsize.png"/><br/><br/><br/>$$<br/>\begin{gathered}<br/>=\int\left(1+2 x+3 x^2+4 x^3\right) d x \\<br/>=x+x^2+x^3+x^4 \\<br/>\Rightarrow \quad \int_0^{1 / 2} f(x) d x=\frac{15}{16}&gt;\frac{3}{4} \\<br/>\int_0^{3 / 4} f(x) d x=\frac{530}{256} &lt; 3<br/>\end{gathered}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-area-bounded-by-th

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-area-bounded-by-th-99530

<div class="question"><strong>Paragraph:</strong><br/>Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.<strong>Question:</strong><br/>The area bounded by the curve $y=f(x)$ and the lines $x=0, y=0$ and $x=t$, lies in the interval</div>

['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{3}{4}, 3\right)$<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/>$\left(\frac{21}{64}, \frac{11}{16}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$(9,10)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(0, \frac{21}{64}\right)$</span> </li> </ul>

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

<div class="solution">$\int_0^{1 / 2} f(x) d x &lt; \int_0^t f(x) d x &lt; \int_0^{3 / 4} f(x) d x$ Now, $\int f(x) d x$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Y1rOmN-ABGVZnGrMYQXtd39YRuXI_W0Y9AEChDFAJoQ.original.fullsize.png"/><br/><br/><br/>$$<br/>\begin{gathered}<br/>=\int\left(1+2 x+3 x^2+4 x^3\right) d x \\<br/>=x+x^2+x^3+x^4 \\<br/>\Rightarrow \quad \int_0^{1 / 2} f(x) d x=\frac{15}{16}&gt;\frac{3}{4} \\<br/>\int_0^{3 / 4} f(x) d x=\frac{530}{256} &lt; 3<br/>\end{gathered}<br/>$$</div>

MarksBatch2_P2.db

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-function-f-x-is

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-function-f-x-is-57748

<div class="question"><strong>Paragraph:</strong><br/>Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.<strong>Question:</strong><br/>The function $f^{\prime}(x)$ is</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>increasing in $\left(-t,-\frac{1}{4}\right)$ and decreasing in $\left(-\frac{1}{4}, t\right)$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>decreasing in $\left(-t,-\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$<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">C</span> <span class="option-data"><br/>increasing in $(-t, t)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>decreasing in $(-t, t)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>decreasing in $\left(-t,-\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$<br/></span> </div>

<div class="solution">Figure is self explanatory<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Yf8F0vDp7puZQLSSihCgSEcJFmQDlRj7xCZcbdgaFhU.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-real-number-s-lies

paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-real-number-s-lies-58199

<div class="question"><strong>Paragraph:</strong><br/>Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.<strong>Question:</strong><br/>The real number $s$ lies in the interval</div>

['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\left(-\frac{1}{4}, 0\right)$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(-11,-\frac{3}{4}\right)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\left(-\frac{3}{4},-\frac{1}{2}\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(0, \frac{1}{4}\right)$</span> </li> </ul>

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

<div class="solution">Given, $f(x)=4 x^3+3 x^2+2 x+1$<br/>$$<br/>\begin{aligned}<br/>&amp; f^{\prime}(x)=2\left(6 x^2+3 x+1\right) \\<br/>&amp; D=9-24 &lt; 0<br/>\end{aligned}<br/>$$<br/>Hence, $f(x)=0$ has only one real root.<br/>$$<br/>\begin{aligned}<br/>f\left(-\frac{1}{2}\right) &amp; =1-1+\frac{3}{4}-\frac{4}{8}&gt;0 \\<br/>f\left(-\frac{3}{4}\right) &amp; =1-\frac{6}{4}+\frac{27}{16}-\frac{108}{64} \\<br/>&amp; =\frac{64-96+108-108}{64} &lt; 0<br/>\end{aligned}<br/>$$<br/>$f(x)$ changes its sign in $\left(-\frac{3}{4}, \frac{-1}{2}\right)$, hence $f(x)=0$ has a root in $\left(\frac{-3}{4}, \frac{-1}{2}\right)$.</div>

MarksBatch2_P2.db

paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-1

paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-1-70547

<div class="question"><strong>Paragraph:</strong><br/>Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite $\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)$, atacamite $\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)$, cuprite $\left(\mathrm{Cu}_2 \mathrm{O}\right)$, copper glance $\left(\mathrm{Cu}_2 \mathrm{~S}\right)$ and malachite $\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)$. However, 80\% of the world copper production comes from the ore chalcopyrite $\left(\mathrm{CuFeS}_2\right)$. The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.<strong>Question:</strong><br/>Partial roasting of chalcopyrite produces</div>

['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\mathrm{Cu}_2 \mathrm{~S}$ and $\mathrm{FeO}$<br/></span> </div>

<div class="solution">$2 \mathrm{CuFeS}_2+4 \mathrm{O}_2 \longrightarrow \mathrm{Cu}_2 \mathrm{~S}$<br/>$$<br/>+2 \mathrm{FeO}+3 \mathrm{SO}_2<br/>$$</div>

MarksBatch2_P2.db

paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-2

paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-2-65008

<div class="question"><strong>Paragraph:</strong><br/>Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite $\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)$, atacamite $\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)$, cuprite $\left(\mathrm{Cu}_2 \mathrm{O}\right)$, copper glance $\left(\mathrm{Cu}_2 \mathrm{~S}\right)$ and malachite $\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)$. However, 80\% of the world copper production comes from the ore chalcopyrite $\left(\mathrm{CuFeS}_2\right)$. The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.<strong>Question:</strong><br/>In self-reduction, the reducing species is</div>

['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{S}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{O}^{2-}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{S}^{2-}$<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/>$\mathrm{SO}_2$</span> </li> </ul>

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

<div class="solution">$\mathrm{S}^{2-}$ acts as reducing species $2 \mathrm{Cu}_2 \mathrm{O}+\mathrm{Cu}_2 \mathrm{~S}^{2-} \longrightarrow 6 \mathrm{Cu}+\mathrm{SO}_2$<br/>Metallurgy<br/>Conceptual<br/>III, II, II</div>

MarksBatch2_P2.db

paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan

paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-72783

<div class="question"><strong>Paragraph:</strong><br/>Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite $\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)$, atacamite $\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)$, cuprite $\left(\mathrm{Cu}_2 \mathrm{O}\right)$, copper glance $\left(\mathrm{Cu}_2 \mathrm{~S}\right)$ and malachite $\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)$. However, 80\% of the world copper production comes from the ore chalcopyrite $\left(\mathrm{CuFeS}_2\right)$. The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.<strong>Question:</strong><br/>Iron is removed from chalcopyrite as</div>

['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{FeO}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{FeS}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Fe}_2 \mathrm{O}_3$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{FeSiO}_3$</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/>$\mathrm{FeSiO}_3$</span> </div>

<div class="solution">Iron is removed in the form of slag of $\mathrm{FeSiO}_3$</div>

MarksBatch2_P2.db

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-1

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-1-65179

<div class="question"><strong>Paragraph:</strong><br/>Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature $T_C(0)$. An interesting property of superconductors is that their critical temperature becomes smaller than $T_C(0)$ if they are placed in a magnetic field i.e., the critical temperature $T_C(B)$ is a function of the magnetic field strength $B$. The dependence of $T_C(B)$ on $B$ is shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_FYhQfph9ZU_3U1GDvqFWoqKuV9LURP8FOgdidn3QG8.original.fullsize.png"/><br/><strong>Question:</strong><br/>A superconductor has $T_C(0)=100 \mathrm{~K}$. When a magnetic field of $7.5$ Tesla is applied, its $T_C$ decreases to $75 \mathrm{~K}$. For this material one can definitely say that when (Note : $\mathrm{T}$ = Tesla)</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/>$B=5 \mathrm{~T}, T_C(B)=80 \mathrm{~K}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$B=5 \mathrm{~T}, 75 \mathrm{~K} &lt; T_C(B) &lt; 100 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$B=10 \mathrm{~T}, 75 \mathrm{~K} &lt; T_C(B) &lt; 100 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$B=10 \mathrm{~T}, T_C(B)=70 \mathrm{~K}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$B=5 \mathrm{~T}, 75 \mathrm{~K} &lt; T_C(B) &lt; 100 \mathrm{~K}$</span> </div>

<div class="solution">With increase in temperature, $T_C$ is decreasing.<br/>$$<br/>\begin{aligned}<br/>T_C(0) &amp; =100 \mathrm{~K} \\<br/>T_C &amp; =75 \mathrm{~K} \text { at } B=7.5 \mathrm{~T}<br/>\end{aligned}<br/>$$<br/>Hence, at $B=5 \mathrm{~T}, \quad T_C$ should lie between $75 \mathrm{~K}$ and $100 \mathrm{~K}$. Hence, the correct option should be (b).</div>

MarksBatch2_P2.db

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-2

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-2-63278

<div class="question"><strong>Paragraph:</strong><br/>Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature $T_C(0)$. An interesting property of superconductors is that their critical temperature becomes smaller than $T_C(0)$ if they are placed in a magnetic field i.e., the critical temperature $T_C(B)$ is a function of the magnetic field strength $B$. The dependence of $T_C(B)$ on $B$ is shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_FYhQfph9ZU_3U1GDvqFWoqKuV9LURP8FOgdidn3QG8.original.fullsize.png"/><br/><strong>Question:</strong><br/>A superconductor has $T_C(0)=100 \mathrm{~K}$. When a magnetic field of $7.5$ Tesla is applied, its $T_C$ decreases to $75 \mathrm{~K}$. For this material one can definitely say that when (Note : $\mathrm{T}$ = Tesla)</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$B=5 \mathrm{~T}, T_C(B)=80 \mathrm{~K}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$B=5 \mathrm{~T}, 75 \mathrm{~K} &lt; T_C(B) &lt; 100 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$B=10 \mathrm{~T}, 75 \mathrm{~K} &lt; T_C(B) &lt; 100 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$B=10 \mathrm{~T}, T_C(B)=70 \mathrm{~K}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$B=5 \mathrm{~T}, 75 \mathrm{~K} &lt; T_C(B) &lt; 100 \mathrm{~K}$</span> </div>

<div class="solution">With increase in temperature, $T_C$ is decreasing.<br/>$$<br/>\begin{aligned}<br/>T_C(0) &amp; =100 \mathrm{~K} \\<br/>T_C &amp; =75 \mathrm{~K} \text { at } B=7.5 \mathrm{~T}<br/>\end{aligned}<br/>$$<br/>Hence, at $B=5 \mathrm{~T}, \quad T_C$ should lie between $75 \mathrm{~K}$ and $100 \mathrm{~K}$. Hence, the correct option should be (b).</div>

MarksBatch2_P2.db

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-3

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-3-76182

<div class="question"><strong>Paragraph:</strong><br/>Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature $T_C(0)$. An interesting property of superconductors is that their critical temperature becomes smaller than $T_C(0)$ if they are placed in a magnetic field i.e., the critical temperature $T_C(B)$ is a function of the magnetic field strength $B$. The dependence of $T_C(B)$ on $B$ is shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_FYhQfph9ZU_3U1GDvqFWoqKuV9LURP8FOgdidn3QG8.original.fullsize.png"/><br/><strong>Question:</strong><br/>In the graphs below, the resistance $R$ of a superconductor is shown as a function of its temperature $T$ for two different magnetic fields $B_1$ (solid line) and $B_2$ (dashed line). If $B_2$ is larger than $B_1$, which of the following graphs shows the correct variation of $R$ with $T$ in these fields?</div>

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

<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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.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/Eq6JNIWXnBI01sTDgigb7jHuZknso7zoWR6BAor3NXI.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/_HDsS9GEl6BAgpFSfYdkHMdlHb5znmSRWfOnzHfAyn8.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/iN-Jh5WFNNzafMGkO_5UaQ4zG1cmNJKWBoC5b0k21Jw.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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.original.fullsize.png"/><br/></span> </div>

<div class="solution">If $B_2&gt;B_1$, critical temperature, (at which resistance of semiconductors abruptly becomes zero) in case-2 will be less than compared to case- 1 .<br/>$\therefore$ The correct option is (a).</div>

MarksBatch2_P2.db

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo

paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-31186

<div class="question"><strong>Paragraph:</strong><br/>Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature $T_C(0)$. An interesting property of superconductors is that their critical temperature becomes smaller than $T_C(0)$ if they are placed in a magnetic field i.e., the critical temperature $T_C(B)$ is a function of the magnetic field strength $B$. The dependence of $T_C(B)$ on $B$ is shown in the figure.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_FYhQfph9ZU_3U1GDvqFWoqKuV9LURP8FOgdidn3QG8.original.fullsize.png"/><br/><strong>Question:</strong><br/>In the graphs below, the resistance $R$ of a superconductor is shown as a function of its temperature $T$ for two different magnetic fields $B_1$ (solid line) and $B_2$ (dashed line). If $B_2$ is larger than $B_1$, which of the following graphs shows the correct variation of $R$ with $T$ in these fields?</div>

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

<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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.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/Eq6JNIWXnBI01sTDgigb7jHuZknso7zoWR6BAor3NXI.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/_HDsS9GEl6BAgpFSfYdkHMdlHb5znmSRWfOnzHfAyn8.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/iN-Jh5WFNNzafMGkO_5UaQ4zG1cmNJKWBoC5b0k21Jw.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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.original.fullsize.png"/><br/></span> </div>

<div class="solution">If $B_2&gt;B_1$, critical temperature, (at which resistance of semiconductors abruptly becomes zero) in case-2 will be less than compared to case- 1 .<br/>$\therefore$ The correct option is (a).</div>

MarksBatch2_P2.db

paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-1

paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-1-94809

<div class="question"><strong>Paragraph:</strong><br/>If a continuous $f$ defined on the real line $R$, assume positive and negative values in $R$, then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum values is negative, then the equation $f(x)=0$ has a root in $R$.<br/>Consider $f(x)=k e^x-x$ for all real $x$, where $k$ is real constant.<strong>Question:</strong><br/>The positive value of $k$ for which $k e^x-x=0$ has only one root is</div>

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

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

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

<div class="solution">Let $f(x)=k e^x-x$<br/>$\begin{aligned} &amp; &amp; f^{\prime}(x) &amp; =k e^x-1=0 \\ &amp; &amp; x &amp; =-\ln k \\ &amp; \therefore &amp; f^{\prime \prime}(x) &amp; =k e^x \\ &amp; \text { Hence, } &amp; {\left[f^{\prime \prime}(x)\right]_{x=-\ln k} } &amp; =1&gt;0 \\ &amp; &amp; f(-\ln k) &amp; =1+\ln k\end{aligned}$<br/>For one root of given equation<br/>$$<br/>\begin{aligned}<br/>1+\ln k &amp; =0 \\<br/>Hence,<br/>k &amp; =\frac{1}{e} .<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-2

paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-2-33286

<div class="question"><strong>Paragraph:</strong><br/>If a continuous $f$ defined on the real line $R$, assume positive and negative values in $R$, then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum values is negative, then the equation $f(x)=0$ has a root in $R$.<br/>Consider $f(x)=k e^x-x$ for all real $x$, where $k$ is real constant.<strong>Question:</strong><br/>The line $y=x$ meets $y=k e^x$ for $k \leq 0$ at</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>no point<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>one point<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/>two points<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>more than two points</span> </li> </ul>

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

<div class="solution">$$<br/>\text { Let } y=x \text { intersect the curve } y=k e^x \text { at exactly one point when } k \leq 0 \text {. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/uKO5RYlQIaKgxfRDXqdX_fJqPi1BzWjfwyABk87f3Xk.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if

paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-91655

<div class="question"><strong>Paragraph:</strong><br/>If a continuous $f$ defined on the real line $R$, assume positive and negative values in $R$, then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum values is negative, then the equation $f(x)=0$ has a root in $R$.<br/>Consider $f(x)=k e^x-x$ for all real $x$, where $k$ is real constant.<strong>Question:</strong><br/>For $k&gt;0$, the set of all values of $k$ for which $k e^x-x=0$ has two distinct roots, is</div>

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

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(0, \frac{1}{e}\right)$<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/>$\left(\frac{1}{e}, 1\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{1}{e}, \infty\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$(0,1)$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$\left(0, \frac{1}{e}\right)$<br/></span> </div>

<div class="solution">For two distinct roots $1+\ln k &lt; 0(k&gt;0)$<br/>$\ln k &lt; -1 \quad k &lt; \frac{1}{e}$<br/>Hence, $k \in\left(0, \frac{1}{e}\right)$</div>

MarksBatch2_P2.db

paragraph-if-the-measurement-errors-in-all-the-independent-quantities-are-known-then-it-is-possible-to-determine-the-error-in-any-dependent-quantity-t-1

paragraph-if-the-measurement-errors-in-all-the-independent-quantities-are-known-then-it-is-possible-to-determine-the-error-in-any-dependent-quantity-t-1-33821

<div class="question"><strong> Paragraph: </strong> If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation <math><mi>z</mi><mi> </mi><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>y</mi></mrow></mfrac></math>. If the errors in <math><mi>x</mi><mo>,</mo><mi> </mi><mi>y</mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi> </mi><mi>z</mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi> </mi><mo>∆</mo><mi>x</mi><mo>,</mo><mi> </mi><mo>∆</mo><mi>y</mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi> </mi><mo>∆</mo><mi>z</mi></math> , respectively, then<br/><br/><math><mi>z</mi><mo>±</mo><mo>∆</mo><mi>z</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>±</mo><mo>∆</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>±</mo><mo>∆</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>y</mi></mrow></mfrac><mfenced separators="|"><mrow><mn>1</mn><mo>±</mo><mfrac><mrow><mo>∆</mo><mi>x</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>±</mo><mfrac><mrow><mo>∆</mo><mi>y</mi></mrow><mrow><mi>y</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><br/>The series expansion for <math><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>±</mo><mfrac><mrow><mo>∆</mo><mi>y</mi></mrow><mrow><mi>y</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, to first power in <math><mrow><mrow><mo>∆</mo><mi>y</mi></mrow><mo>/</mo><mrow><mi>y</mi></mrow></mrow><mo>,</mo><mi> </mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi> </mi><mn>1</mn><mo>∓</mo><mfenced separators="|"><mrow><mrow><mrow><mo>∆</mo><mi>y</mi></mrow><mo>/</mo><mrow><mi>y</mi></mrow></mrow></mrow></mfenced></math>. The relative errors in independent variables are always added. So the error in <math> <mi>z</mi> </math> will be<br/><br/><math> <mrow> <mtext>Δ</mtext><mi>z</mi><mo>=</mo><mi>z</mi><mrow><mo>(</mo> <mrow> <mfrac> <mrow> <mtext>Δ</mtext><mi>x</mi></mrow> <mi>x</mi> </mfrac> <mo>+</mo><mfrac> <mrow> <mtext>Δ</mtext><mi>y</mi></mrow> <mi>y</mi> </mfrac> </mrow> <mo>)</mo></mrow><mo>.</mo></mrow> </math><br/>The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y &lt; 1\). Therefore, the higher powers of these quantities are neglected.<br/> <strong> Question : </strong> Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}&lt;&lt;1)\), then what is the error \(\Delta r\) in determining \(r\) ?</div>

['Physics', 'Mathematics in Physics', 'JEE Advanced', 'JEE Advanced 2018 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mo>∆</mo><mi>a</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mn>2</mn><mo>∆</mo><mi>a</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></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">C</span> <span class="option-data"><math><mfrac><mrow><mn>2</mn><mo>∆</mo><mi>a</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>2</mn><mi>a</mi><mo>∆</mo><mi>a</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><math><mfrac><mrow><mn>2</mn><mo>∆</mo><mi>a</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span> </div>

<div class="solution"><math><mi>r</mi><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfrac></mrow></mfenced></math><br/><math><mfrac><mrow><mo>∆</mo><mi>r</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∆</mo><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mi> </mi></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∆</mo><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow></mfrac></math><br/><math><mo>=</mo><mi> </mi><mfrac><mrow><mo>∆</mo><mi>a</mi></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∆</mo><mi>a</mi></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow></mfrac></math><br/><math><mo>=</mo><mi> </mi><mfrac><mrow><mo>∆</mo><mi>a</mi><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow></mfrac></math><br/><math><mo>∴</mo><mi> </mi><mi> </mi><mi> </mi><mo>∆</mo><mi>r</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mo>∆</mo><mi>a</mi></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfrac><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>∆</mo><mi>a</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></div>

MarksBatch2_P2.db

paragraph-if-the-measurement-errors-in-all-the-independent-quantities-are-known-then-it-is-possible-to-determine-the-error-in-any-dependent-quantity-t

paragraph-if-the-measurement-errors-in-all-the-independent-quantities-are-known-then-it-is-possible-to-determine-the-error-in-any-dependent-quantity-t-25278

<div class="question"><strong> Paragraph: </strong> If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation <math><mi>z</mi><mi> </mi><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>y</mi></mrow></mfrac></math>. If the errors in <math><mi>x</mi><mo>,</mo><mi> </mi><mi>y</mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi> </mi><mi>z</mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi> </mi><mo>∆</mo><mi>x</mi><mo>,</mo><mi> </mi><mo>∆</mo><mi>y</mi><mi> </mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi> </mi><mo>∆</mo><mi>z</mi></math> , respectively, then<br/><br/><math><mi>z</mi><mo>±</mo><mo>∆</mo><mi>z</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>±</mo><mo>∆</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>±</mo><mo>∆</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>y</mi></mrow></mfrac><mfenced separators="|"><mrow><mn>1</mn><mo>±</mo><mfrac><mrow><mo>∆</mo><mi>x</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>±</mo><mfrac><mrow><mo>∆</mo><mi>y</mi></mrow><mrow><mi>y</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><br/>The series expansion for <math><msup><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>±</mo><mfrac><mrow><mo>∆</mo><mi>y</mi></mrow><mrow><mi>y</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, to first power in <math><mrow><mrow><mo>∆</mo><mi>y</mi></mrow><mo>/</mo><mrow><mi>y</mi></mrow></mrow><mo>,</mo><mi> </mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi> </mi><mn>1</mn><mo>∓</mo><mfenced separators="|"><mrow><mrow><mrow><mo>∆</mo><mi>y</mi></mrow><mo>/</mo><mrow><mi>y</mi></mrow></mrow></mrow></mfenced></math>. The relative errors in independent variables are always added. So the error in <math> <mi>z</mi> </math> will be<br/><br/><math> <mrow> <mtext>Δ</mtext><mi>z</mi><mo>=</mo><mi>z</mi><mrow><mo>(</mo> <mrow> <mfrac> <mrow> <mtext>Δ</mtext><mi>x</mi></mrow> <mi>x</mi> </mfrac> <mo>+</mo><mfrac> <mrow> <mtext>Δ</mtext><mi>y</mi></mrow> <mi>y</mi> </mfrac> </mrow> <mo>)</mo></mrow><mo>.</mo></mrow> </math><br/><br/> <strong> Question : </strong>The above derivation makes the assumption that <math> <mrow> <mfrac> <mrow> <mtext>Δ</mtext><mi>r</mi></mrow> <mi>x</mi> </mfrac> <mo>≪</mo><mn>1</mn><mo>,</mo><mfrac> <mrow> <mtext>Δ</mtext><mi>y</mi></mrow> <mi>y</mi> </mfrac> <mo>≪</mo><mn>1</mn></mrow> </math>. Therefore, the higher powers of these quantities are neglected.</div>

['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2018 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math> <mrow> <mn>0.04</mn></mrow> </math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>0.03</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mn>0.02</mn></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><mn>0.01</mn></math></span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><math><mn>0.02</mn></math></span> </div>

<div class="solution"><math><mi>N</mi><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>λ</mi><mi>t</mi></mrow></msup></math><br/><math><mrow><mrow><mi mathvariant="normal">ln</mi></mrow><mo>⁡</mo><mrow><mi>N</mi></mrow></mrow><mo>=</mo><mrow><mrow><mi mathvariant="normal">ln</mi></mrow><mo>⁡</mo><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mrow><mo>-</mo><mi>λ</mi><mi>t</mi></math><br/><math><mfrac><mrow><mn>d</mn><mi>N</mi></mrow><mrow><mi>N</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>d</mn><mi>λ</mi><mi>t</mi></math><br/>Converting to error,<br/><math><mfrac><mrow><mo>∆</mo><mi>N</mi></mrow><mrow><mi>N</mi></mrow></mfrac><mo>=</mo><mo>∆</mo><mi>λ</mi><mi>t</mi></math><br/><math><mo>∴</mo><mi> </mi><mi> </mi><mo>∆</mo><mi>λ</mi><mo>=</mo><mfrac><mrow><mn>40</mn></mrow><mrow><mn>2000</mn><mo>×</mo><mi>L</mi></mrow></mfrac><mo>=</mo><mn>0.02</mn></math> (<math> <mi>N</mi> </math> is number of nuclei left undecayed)</div>

MarksBatch2_P2.db

paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-1

paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-1-86600

<div class="question"><strong>Paragraph:</strong><br/>In a mixture of $\mathrm{H}-\mathrm{H}^{+}$gas $\left(\mathrm{He}^{+}\right.$is singly ionised $\mathrm{He}$ atom), $\mathrm{H}$ atoms and $\mathrm{He}^{+}$ions are excited to their respective first excited states. Subsequently, $\mathrm{H}$ atoms transfer their total excitation energy to $\mathrm{He}^{+}$ions (by collisions). Assume that the Bohr model of atom is exactly valid.<strong>Question:</strong><br/>The quantum number $n$ of the state finally populated in $\mathrm{He}^{+}$ions is</div>

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

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

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

<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/r6y4Y5YL5_LJaLt87AYv_LVqSQ1_jK42VKAGQxJKFOQ.original.fullsize.png"/><br/><br/>Energy given by $\mathrm{H}$-atom in transition from $n=2$ to $n=1$ is equal to energy taken by $\mathrm{He}^{+}$ atom in transition from $n=2$ to $n=4$.<br/>$\therefore$ correct option is (c).</div>

MarksBatch2_P2.db

paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-2

paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-2-40567

<div class="question"><strong>Paragraph:</strong><br/>In a mixture of $\mathrm{H}-\mathrm{H}^{+}$gas $\left(\mathrm{He}^{+}\right.$is singly ionised $\mathrm{He}$ atom), $\mathrm{H}$ atoms and $\mathrm{He}^{+}$ions are excited to their respective first excited states. Subsequently, $\mathrm{H}$ atoms transfer their total excitation energy to $\mathrm{He}^{+}$ions (by collisions). Assume that the Bohr model of atom is exactly valid.<strong>Question:</strong><br/>The ratio of the kinetic energy of the $n=2$ electron for the $\mathrm{H}$ atom to that of $\mathrm{He}^{+}$ ion is</div>

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

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

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

<div class="solution">Kinetic energy $K \propto Z^2$<br/>$$<br/>\frac{K_H}{K_{\mathrm{He}^{+}}}=\left(\frac{1}{2}\right)^2=\frac{1}{4}<br/>$$<br/>$\therefore$ correct option is (a).</div>

MarksBatch2_P2.db

paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h

paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-75826

<div class="question"><strong>Paragraph:</strong><br/>In a mixture of $\mathrm{H}-\mathrm{H}^{+}$gas $\left(\mathrm{He}^{+}\right.$is singly ionised $\mathrm{He}$ atom), $\mathrm{H}$ atoms and $\mathrm{He}^{+}$ions are excited to their respective first excited states. Subsequently, $\mathrm{H}$ atoms transfer their total excitation energy to $\mathrm{He}^{+}$ions (by collisions). Assume that the Bohr model of atom is exactly valid.<strong>Question:</strong><br/>The wavelength of light emitted in the visible region by $\mathrm{He}^{+}$ions after collisions with $\mathrm{H}$ atoms is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$6.5 \times 10^{-7} \mathrm{~m}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$5.6 \times 10^{-7} \mathrm{~m}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$4.8 \times 10^{-7} \mathrm{~m}$<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/>$4.0 \times 10^{-7} \mathrm{~m}$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$4.8 \times 10^{-7} \mathrm{~m}$<br/></span> </div>

<div class="solution">Visible light lies in the range, $\lambda_1=4000 Å$ to $\lambda_2=7000 Å$.<br/>Energy of photons corresponding to these wavelengths (in $\mathrm{eV}$ ) would be;<br/>$$<br/>\begin{aligned}<br/>&amp; E_1=\frac{12375}{4000}=3.09 \mathrm{eV} \\<br/>&amp; E_2=\frac{12375}{7000}=1.77 \mathrm{eV}<br/>\end{aligned}<br/>$$<br/>From energy level diagram of $\mathrm{He}^{+}$atom we can see that in transition from $n=4$ to $n=3$, energy of photon released will lie between $E_1$ and $E_2$.<br/>$$<br/>\Delta E_{43}=-3.4-(-6.04)=2.64 \mathrm{eV}<br/>$$<br/>Wavelength of photon corresponding to this energy.<br/>$$<br/>\lambda=\frac{12375}{2.64} Å=4687.5 Å=4.68 \times 10^{-7} \mathrm{~m}<br/>$$<br/>Therefore, (c) is the most correct option.</div>

MarksBatch2_P2.db

paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-1

paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-1-44050

<div class="question"><strong>Paragraph:</strong><br/>In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' $r$ '.<strong>Question:</strong><br/>The empty space in this $\mathrm{HCP}$ unit cell is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$74 \%$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$47.6 \%$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$32 \%$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$26 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>$26 \%$</span> </div>

<div class="solution">Packing fraction $=\frac{\text { Volume of the atoms in one unit cell }}{\text { Volume of one unit cell }}=74 \%$ Empty space $=26 \%$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OnsGaUN15fScF3Q0tueA8hVAg9RDtQFJmWn4vJYC4Xw.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TD8j_GOcJoybP1EWu5xu2fMT_oi3wE3GVJirHqfr4AI.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-2

paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-2-18651

<div class="question"><strong>Paragraph:</strong><br/>In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' $r$ '.<strong>Question:</strong><br/>The number of atoms in this HCP unit cell is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>4<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>6<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/>12<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>17</span> </li> </ul>

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

<div class="solution">The effective number of atoms in the hcp is $=12 \times \frac{1}{6}+2 \times \frac{1}{2}+3=6$</div>

MarksBatch2_P2.db

paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t

paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-39153

<div class="question"><strong>Paragraph:</strong><br/>In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' $r$ '.<strong>Question:</strong><br/>The volume of this $\mathrm{HCP}$ unit cell is</div>

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

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$24 \sqrt{2} r^3$<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">B</span> <span class="option-data"><br/>$16 \sqrt{2} r^3$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$12 \sqrt{2} r^3$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{64}{3 \sqrt{3}} r^3$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$24 \sqrt{2} r^3$<br/></span> </div>

<div class="solution">Height of unit cell $=\sqrt{\frac{2}{4}} 4 r$<br/>$$<br/>\begin{aligned}<br/>&amp; \text { Base area }=6 \times \frac{\sqrt{3}}{4}(2 r)^2 \\<br/>&amp; \text { Volume }=\frac{6 \sqrt{3}}{4}(2 r)^2 \cdot \sqrt{\frac{2}{3}} 4 r=24 \sqrt{2} r^3<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-product-l-is-1

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-product-l-is-1-14403

<div class="question"><strong>Paragraph:</strong><br/>In the following reaction sequence, product $I . J$ and $L$ are formed. K represent a reagent<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IFqau304_fmyEucY6QplMeb2z8xAY65xBAV6E0EMBIk.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structure of product } L \text { is }<br/>$$</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Main']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/duFxiRueBOFPAmKgDThaPCGfc6lYU-SAApcC3MfZCNk.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/KwVQJ4yEiI650gNQK_2isQuUO3ZR4msjcfvopx0eJQk.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/JYkNfUHgdXPayjCyTuKsYc3Q7lavR-XhK5mKYbw_pF8.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/J_Al3RGF6yJENWMf35n2vb9yw3p2q8nxiM03Z63BrE8.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/KwVQJ4yEiI650gNQK_2isQuUO3ZR4msjcfvopx0eJQk.original.fullsize.png"/><br/></span> </div>

<div class="solution">$\mathrm{H}_2, \mathrm{Pd} / \mathrm{BaSO}_4$, quinoline is well known reagent.<br/>Rosenmund's reaction (conversion of acyl halide to aldehyde) $\mathrm{H}_2, \mathrm{Pd}, \mathrm{BaSO}_4$ is known as Lindlar's reagent and reduces triple bond to double bond (syn addition) giving cis product. Reagent has simultaneous action because usually reaction is not selective-follows radical mechanism.</div>

MarksBatch2_P2.db

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-product-l-is

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-product-l-is-25845

<div class="question"><strong>Paragraph:</strong><br/>In the following reaction sequence, product $I . J$ and $L$ are formed. K represent a reagent<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IFqau304_fmyEucY6QplMeb2z8xAY65xBAV6E0EMBIk.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structure of product } L \text { is }<br/>$$</div>

['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/duFxiRueBOFPAmKgDThaPCGfc6lYU-SAApcC3MfZCNk.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/KwVQJ4yEiI650gNQK_2isQuUO3ZR4msjcfvopx0eJQk.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/JYkNfUHgdXPayjCyTuKsYc3Q7lavR-XhK5mKYbw_pF8.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/J_Al3RGF6yJENWMf35n2vb9yw3p2q8nxiM03Z63BrE8.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/KwVQJ4yEiI650gNQK_2isQuUO3ZR4msjcfvopx0eJQk.original.fullsize.png"/><br/></span> </div>

<div class="solution">$\mathrm{H}_2, \mathrm{Pd} / \mathrm{BaSO}_4$, quinoline is well known reagent.<br/>Rosenmund's reaction (conversion of acyl halide to aldehyde) $\mathrm{H}_2, \mathrm{Pd}, \mathrm{BaSO}_4$ is known as Lindlar's reagent and reduces triple bond to double bond (syn addition) giving cis product. Reagent has simultaneous action because usually reaction is not selective-follows radical mechanism.</div>

MarksBatch2_P2.db

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-the-product-i-is

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-the-product-i-is-39295

<div class="question"><strong>Paragraph:</strong><br/>In the following reaction sequence, product $I . J$ and $L$ are formed. K represent a reagent<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IFqau304_fmyEucY6QplMeb2z8xAY65xBAV6E0EMBIk.original.fullsize.png"/><br/><strong>Question:</strong><br/>The structure of the product $I$ is<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/oWm2oQJwKwLNa4e0g-OjTfWF9dRGvrWTBr2WchBfWPk.original.fullsize.png"/><br/></div>

['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/X708kGVEweO4sJwCou2GEkKLpd42sKTfoLW6q4HxaCE.original.fullsize.png"/><br/></span> </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/dN4oalhkudT1utssEFzTduUL6vHiHsiXiclsmnfbrFI.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/9XX9p4lHim5ozcpKGYnRKgME7Rh5rhExY4fy94c5LgA.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/83pBEan488VtTmOBa9pYj6l1EqSCSzj_d94U0Xck3ro.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/83pBEan488VtTmOBa9pYj6l1EqSCSzj_d94U0Xck3ro.original.fullsize.png"/><br/></span> </div>

<div class="solution">$\mathrm{NaBH}_4$ reduces $-\mathrm{CHO}$ selectively into $-\mathrm{CH}_2 \mathrm{OH}$, which is further converted into corresponding bromide by $\mathrm{PBr}_3$ without affecting the triple bond.</div>

MarksBatch2_P2.db

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structures-of-compounds-j-and-k-respectiv

paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structures-of-compounds-j-and-k-respectiv-78418

<div class="question"><strong>Paragraph:</strong><br/>In the following reaction sequence, product $I . J$ and $L$ are formed. K represent a reagent<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IFqau304_fmyEucY6QplMeb2z8xAY65xBAV6E0EMBIk.original.fullsize.png"/><br/><strong>Question:</strong><br/>$$<br/>\text { The structures of compounds } J \text { and } K \text {, respectively, are }<br/>$$</div>

['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

<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/QHrLAdXTmQr321aLV76QNLy-SpXCtNzV3OHdN-oUnDA.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/ZcwCvzXblmPzNNWA4rCPzMtVtcuKq0QI0dL0tiLqWZU.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/Ddt5qpMV8yRxbhjO-bTY3LecFVXB-9iWG-SPI6nutf0.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/kiadc3hAoyJjKTmXQbaPFBxO9QBIGU5bIVk5Y9ER5JM.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/QHrLAdXTmQr321aLV76QNLy-SpXCtNzV3OHdN-oUnDA.original.fullsize.png"/><br/></span> </div>

<div class="solution">Bromide converted into Grignard reagent and further carbonation (I. $\mathrm{CO}_2, \mathrm{II} \mathrm{H}_3 \mathrm{O}^{+}$) produces carboxylic acid $(J)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/7pUH2gQK2pt8LZgZBAUOLLz8eAQXUxSWW1yAQUWhv9o.original.fullsize.png"/><br/></div>

MarksBatch2_P2.db

paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-1

paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-1-85822

<div class="question"><strong>Paragraph:</strong><br/>Let $A_1, G_1, H_1$ denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For $n \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n, G_n, H_n$, respectively.<strong>Question:</strong><br/>Which of the following statements is correct?</div>

['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{H}_1&gt;\mathrm{H}_2&gt;\mathrm{H}_3&gt;\ldots$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$H_1 &lt; H_2 &lt; H_3 &lt; \ldots$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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/>$H_1&gt;H_3&gt;H_5&gt;\ldots$ and $H_2 &lt; H_4 &lt; H_6 &lt; \ldots$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{H}_1 &lt; \mathrm{H}_3 &lt; \mathrm{H}_5 &lt; \ldots$ and $\mathrm{H}_2&gt;\mathrm{H}_4&gt;\mathrm{H}_6&gt;\ldots$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$H_1 &lt; H_2 &lt; H_3 &lt; \ldots$</span> </div>

<div class="solution">As above $A_1&gt;H_2&gt;H_1, A_2&gt;H_3&gt;H_2$ $\therefore \quad H_1 &lt; H_2 &lt; H_3 &lt; \ldots$</div>

MarksBatch2_P2.db

paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-2

paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-2-10149

<div class="question"><strong>Paragraph:</strong><br/>Let $A_1, G_1, H_1$ denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For $n \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n, G_n, H_n$, respectively.<strong>Question:</strong><br/>Which of the following statements is correct?</div>

['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$A_1&gt;A_2&gt;\ldots$<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/>$A_1 &lt; A_2 &lt; A_3 &lt; \ldots$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$A_1&gt;A_3&gt;A_5&gt;\ldots$ and $A_2 &lt; A_4 &lt; A_6 &lt; \ldots$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$A_1 &lt; A_3 &lt; A_5 &lt; \ldots$ and $A_2&gt;A_4&gt;A_6&gt;\ldots$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$A_1&gt;A_2&gt;\ldots$<br/></span> </div>

<div class="solution">Let $f(x)=k e^x-x$<br/>$\begin{aligned} &amp; &amp; f^{\prime}(x) &amp; =k e^x-1=0 \\ &amp; &amp; x &amp; =-\ln k \\ &amp; \therefore &amp; f^{\prime \prime}(x) &amp; =k e^x \\ &amp; \text { Hence, } &amp; {\left[f^{\prime \prime}(x)\right]_{x=-\ln k} } &amp; =1&gt;0 \\ &amp; &amp; f(-\ln k) &amp; =1+\ln k\end{aligned}$<br/>For one root of given equation<br/>$$<br/>\begin{aligned}<br/>1+\ln k &amp; =0 \\<br/>Hence,<br/>k &amp; =\frac{1}{e} .<br/>\end{aligned}<br/>$$<br/>$A_2$ is $A M$ of $A_1$ and $H_1$ and $A_1&gt;H_1$<br/>$$<br/>\Rightarrow \quad A_1&gt;A_2&gt;H_1<br/>$$<br/>$A_3$ is $\mathrm{AM}$ of $\mathrm{A}_2$ and $\mathrm{H}_2$ and $\mathrm{A}_2&gt;\mathrm{H}_2$<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; A_2&gt;A_3&gt;H_2 \\<br/>\therefore &amp; A_1&gt;A_2&gt;A_3&gt;\ldots<br/>\end{array}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1

paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-44012

<div class="question"><strong>Paragraph:</strong><br/>Let $A_1, G_1, H_1$ denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For $n \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n, G_n, H_n$, respectively.<strong>Question:</strong><br/>Which one of the following statements is correct?</div>

['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$G_1&gt;G_2&gt;G_3&gt;\ldots$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$G_1 &lt; G_2 &lt; G_3 &lt; \ldots$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$G_1=G_2=G_3=\ldots$<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/>$G_1 &lt; G_3 &lt; G_5 &lt; \ldots$ and $G_2&gt;G_4&gt;G_6&gt;\ldots$</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$G_1=G_2=G_3=\ldots$<br/></span> </div>

<div class="solution">$A_1=\frac{a+b}{2}, G_1=\sqrt{a b}$ and $H_1=\frac{2 a b}{a+b}$ $\begin{aligned} A_n=\frac{A_{n-1}+H_{n-1}}{2}, G_n &amp; =\sqrt{A_{n-1} H_{n-1}}, \\ H_n &amp; =\frac{2 A_{n-1} H_{n-1}}{A_{n-1}+H_{n-1}}\end{aligned}$<br/>Clearly, $\quad G_1=G_2=G_3=\ldots=\sqrt{a b}$.</div>

MarksBatch2_P2.db

paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-if-the-point-p-a-b-c-with-reference-to-eq-e-lies-on-th

paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-if-the-point-p-a-b-c-with-reference-to-eq-e-lies-on-th-31019

<div class="question"><strong>Paragraph:</strong><br/>Let $a$, b and $c$ be three real numbers satisfying $\left[\begin{array}{lll}a &amp; b &amp; c\end{array}\right]\left[\begin{array}{lll}1 &amp; 9 &amp; 7 \\ 8 &amp; 2 &amp; 7 \\ 7 &amp; 3 &amp; 7\end{array}\right]=\left[\begin{array}{lll}0 &amp; 0 &amp; 0\end{array}\right]$<strong>Question:</strong><br/>If the point $P(a, b, c)$, with reference to Eq. (E), lies on the plane $2 x+y+z=1$, then the value of $7 a+b+c$ is</div>

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

<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/>12<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>7<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">$$<br/>\begin{aligned}<br/>&amp; \text { Given, }[a b c]_{1 \times 3}\left[\begin{array}{lll}<br/>1 &amp; 9 &amp; 7 \\<br/>8 &amp; 2 &amp; 7 \\<br/>7 &amp; 3 &amp; 7<br/>\end{array}\right]_{3 \times 3}=\left[\begin{array}{lll}<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right] \\<br/>&amp; \Rightarrow \quad\left[\begin{array}{c}<br/>a+8 b+7 c \\<br/>9 a+2 b+3 c \\<br/>7 a+7 b+7 c<br/>\end{array}\right]=\left[\begin{array}{l}<br/>0 \\<br/>0 \\<br/>0<br/>\end{array}\right] \\<br/>&amp; \Rightarrow \quad a+8 b+7 c=0 \\<br/>&amp; \Rightarrow 9 a+2 b+3 c=0 \\<br/>&amp; \Rightarrow \quad a+b+c=0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get<br/>$$<br/>7 a+c=0<br/>$$<br/>Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get<br/>$$<br/>\begin{array}{ll} <br/>&amp; 6 a-b=0 \\<br/>\therefore \quad &amp; b=6 a \text { and } c=-7 a<br/>\end{array}<br/>$$<br/>As $(a, b, c)$ lies on $2 x+y+z=1$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow &amp; 2 a+b+c=1 \\<br/>\Rightarrow &amp; 2 a+6 a-7 a=1 \\<br/>\Rightarrow &amp; a=1, b=6 \text { and } c=-7 \\<br/>\therefore &amp; 7 a+b+c=7+6-7=6<br/>\end{array}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-let-b-6-with-a-and-c-satisfying-eq-e-if-and-are-the-ro

paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-let-b-6-with-a-and-c-satisfying-eq-e-if-and-are-the-ro-96523

<div class="question"><strong>Paragraph:</strong><br/>Let $a$, b and $c$ be three real numbers satisfying $\left[\begin{array}{lll}a &amp; b &amp; c\end{array}\right]\left[\begin{array}{lll}1 &amp; 9 &amp; 7 \\ 8 &amp; 2 &amp; 7 \\ 7 &amp; 3 &amp; 7\end{array}\right]=\left[\begin{array}{lll}0 &amp; 0 &amp; 0\end{array}\right]$<strong>Question:</strong><br/>Let $b=6$, with $a$ and $c$ satisfying Eq. (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x^2+b x+c=0$, then $\sum_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n$ is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>6<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>7<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/>$\frac{6}{7}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\infty$</span> </li> </ul>

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

<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Given, }[a b c]_{1 \times 3}\left[\begin{array}{lll}<br/>1 &amp; 9 &amp; 7 \\<br/>8 &amp; 2 &amp; 7 \\<br/>7 &amp; 3 &amp; 7<br/>\end{array}\right]_{3 \times 3}=\left[\begin{array}{lll}<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right] \\<br/>&amp; \Rightarrow \quad\left[\begin{array}{c}<br/>a+8 b+7 c \\<br/>9 a+2 b+3 c \\<br/>7 a+7 b+7 c<br/>\end{array}\right]=\left[\begin{array}{l}<br/>0 \\<br/>0 \\<br/>0<br/>\end{array}\right] \\<br/>&amp; \Rightarrow \quad a+8 b+7 c=0 \\<br/>&amp; \Rightarrow 9 a+2 b+3 c=0 \\<br/>&amp; \Rightarrow \quad a+b+c=0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get<br/>$$<br/>7 a+c=0<br/>$$<br/>Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get<br/>$$<br/>\begin{array}{ll} <br/>&amp; 6 a-b=0 \\<br/>\therefore \quad &amp; b=6 a \text { and } c=-7 a<br/>\end{array}<br/>$$<br/>If $b=6, a=1$ and $c=-7$<br/>$$<br/>\begin{array}{ll}<br/>\therefore &amp; a x^2+b x+c=0 \\<br/>\Rightarrow &amp; x^2+6 x-7=0 \\<br/>\Rightarrow &amp; (x+7)(x-1)=0 \\<br/>\therefore &amp; x=1,-7 \\<br/>\Rightarrow &amp; \quad \sum_{n=0}^{\infty}\left(\frac{1}{1}-\frac{1}{7}\right)^n \Rightarrow \sum_{n=0}^{\infty}\left(\frac{6}{7}\right)^n \\<br/>\Rightarrow &amp; 1+\frac{6}{7}+\left(\frac{6}{7}\right)^n+\ldots \infty \\<br/>&amp; =\frac{1}{1-\frac{6}{7}}=\frac{1}{1 / 7}=7<br/>\end{array}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-let-be-a-solution-of-x-3-1-0-with-im-0-if-a-2-with-b-a

paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-let-be-a-solution-of-x-3-1-0-with-im-0-if-a-2-with-b-a-59084

<div class="question"><strong>Paragraph:</strong><br/>Let $a$, b and $c$ be three real numbers satisfying $\left[\begin{array}{lll}a &amp; b &amp; c\end{array}\right]\left[\begin{array}{lll}1 &amp; 9 &amp; 7 \\ 8 &amp; 2 &amp; 7 \\ 7 &amp; 3 &amp; 7\end{array}\right]=\left[\begin{array}{lll}0 &amp; 0 &amp; 0\end{array}\right]$<strong>Question:</strong><br/>Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)&gt;0$. If $a=2$, with $b$ and $c$ satisfying Eq. (E), then the value of $\frac{3}{\omega^a}+\frac{1}{\omega^b}+\frac{3}{\omega^c}$</div>

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

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

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

<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Given, }[a b c]_{1 \times 3}\left[\begin{array}{lll}<br/>1 &amp; 9 &amp; 7 \\<br/>8 &amp; 2 &amp; 7 \\<br/>7 &amp; 3 &amp; 7<br/>\end{array}\right]_{3 \times 3}=\left[\begin{array}{lll}<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right] \\<br/>&amp; \Rightarrow \quad\left[\begin{array}{c}<br/>a+8 b+7 c \\<br/>9 a+2 b+3 c \\<br/>7 a+7 b+7 c<br/>\end{array}\right]=\left[\begin{array}{l}<br/>0 \\<br/>0 \\<br/>0<br/>\end{array}\right] \\<br/>&amp; \Rightarrow \quad a+8 b+7 c=0 \\<br/>&amp; \Rightarrow 9 a+2 b+3 c=0 \\<br/>&amp; \Rightarrow \quad a+b+c=0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get<br/>$$<br/>7 a+c=0<br/>$$<br/>Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get<br/>$$<br/>\begin{array}{ll} <br/>&amp; 6 a-b=0 \\<br/>\therefore \quad &amp; b=6 a \text { and } c=-7 a<br/>\end{array}<br/>$$<br/>If $a=2, b=12$ and $c=-14$<br/>$$<br/>\begin{aligned}<br/>\therefore &amp; \frac{3}{\omega^a}+\frac{1}{\omega^b}+\frac{3}{\omega^c} \\<br/>\Rightarrow &amp; \frac{3}{\omega^2}+\frac{1}{\omega^{12}}+\frac{3}{\omega^{-14}}=\frac{3}{\omega^2}+1+3 \omega^2 \\<br/>&amp; =3 \omega+1+3 \omega^2 \\<br/>&amp; =1+3\left(\omega+\omega^2\right) \\<br/>&amp; =1-3=-2<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-let-z-be-any-point-in-a-b-c-and-l

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-let-z-be-any-point-in-a-b-c-and-l-72610

<div class="question"><strong>Paragraph:</strong><br/>Let $A, B, C$ be three sets of complex numbers as defined below.<br/>$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.<strong>Question:</strong><br/>Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $|w-2-i| &lt; 3$. Then, $|z|-|w|+3$ lies between</div>

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

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$-3$ and 9</span> </div>

<div class="solution">$$<br/>\begin{array}{ll} <br/>&amp; |w-(2+i)| &lt; 3 \\<br/>\Rightarrow &amp; \quad|| w|-| 2+i|| &lt; 3 \\<br/>\Rightarrow &amp; -3+\sqrt{5} &lt; |w| &lt; 3+\sqrt{5} \\<br/>\Rightarrow &amp; -3-\sqrt{5} &lt; -|w| &lt; 3-\sqrt{5}<br/>\end{array}<br/>$$<br/>Also, $|z-(2+i)|=3$<br/>$$<br/>\begin{aligned}<br/>\Rightarrow \quad-3+\sqrt{5} &amp; \leq|z| \leq 3+\sqrt{5} \\<br/>-3 &amp; &lt; |z|-|w|+3 &lt; 9<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-let-z-be-any-point-in-a-b-c-the-z

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-let-z-be-any-point-in-a-b-c-the-z-52624

<div class="question"><strong>Paragraph:</strong><br/>Let $A, B, C$ be three sets of complex numbers as defined below.<br/>$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.<strong>Question:</strong><br/>Let $z$ be any point in $A \cap B \cap C$. The $|z+1-i|^2+|z-5-i|^2$ lies between</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>25 and 29<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>30 and 34<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>35 and 39<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/>40 and 44</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>35 and 39<br/></span> </div>

<div class="solution">$|z+1-i|^2+|z-5-i|^2=(x+1)^2+(y-1)^2+(x-5)^2+(y-1)^2$<br/>$$<br/>\begin{aligned}<br/>&amp; =2\left(x^2+y^2-4 x-2 y\right)+28 \\<br/>&amp; =2(4)+28 \\<br/>&amp; {\left[\because x^2+y^2-4 x-2 y=4\right]} \\<br/>&amp; =36 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-the-number-of-elements-in-the-set-1

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-the-number-of-elements-in-the-set-1-54107

<div class="question"><strong>Paragraph:</strong><br/>Let $A, B, C$ be three sets of complex numbers as defined below.<br/>$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.<strong>Question:</strong><br/>The number of elements in the set $A \cap B \cap C$ is</div>

['Mathematics', 'Complex Number', 'JEE Main']

<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/>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/>2<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\infty$</span> </li> </ul>

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

<div class="solution">Let $z=x+i y$<br/>Set $A$ corresponds to the region $y \geq 1$.<br/>Set $B$ consists of points lying on the circle, centred at $(2,1)$ and radius 3 , i.e., $x^2+y^2-4 x-2 y=4$<br/><br/>$$<br/>\text { Set } C \text { consists of points lying on the } x+y=\sqrt{2}<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/3NScNru6X2gqkLN9tmqN6mzwDUF8CODMBd5aCcEBqAU.original.fullsize.png"/><br/><br/>Clearly, there is only one point of intersection of the line<br/>$$<br/>x+y=\sqrt{2} \text { and circle } x^2+y^2-4 x-2 y=4 .<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-the-number-of-elements-in-the-set

paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-the-number-of-elements-in-the-set-35639

<div class="question"><strong>Paragraph:</strong><br/>Let $A, B, C$ be three sets of complex numbers as defined below.<br/>$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.<strong>Question:</strong><br/>The number of elements in the set $A \cap B \cap C$ is</div>

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

<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/>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/>2<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\infty$</span> </li> </ul>

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

<div class="solution">Let $z=x+i y$<br/>Set $A$ corresponds to the region $y \geq 1$.<br/>Set $B$ consists of points lying on the circle, centred at $(2,1)$ and radius 3 , i.e., $x^2+y^2-4 x-2 y=4$<br/><br/>$$<br/>\text { Set } C \text { consists of points lying on the } x+y=\sqrt{2}<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/3NScNru6X2gqkLN9tmqN6mzwDUF8CODMBd5aCcEBqAU.original.fullsize.png"/><br/><br/>Clearly, there is only one point of intersection of the line<br/>$$<br/>x+y=\sqrt{2} \text { and circle } x^2+y^2-4 x-2 y=4 .<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-1

paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-1-75049

<div class="question"><strong>Paragraph:</strong><br/>Let $A$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .<strong>Question:</strong><br/>The number of matrices in $A$ is</div>

['Mathematics', 'Permutation Combination', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']

<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>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=""> <span class="option-label">B</span> <span class="option-data"><br/>6<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>9<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>3</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>12<br/></span> </div>

<div class="solution">Case $I$ When all three diagonal elements are 1 , then<br/>Number of matrices $={ }^3 C_1=3$<br/>Case II When two diagonal elements are zero and one element is one no, then Number of matrices $={ }^3 C_1 \cdot{ }^3 C_1=9$<br/>$\therefore$ Total matrices $=3+9=12$</div>

MarksBatch2_P2.db

paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-2

paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-2-71555

<div class="question"><strong>Paragraph:</strong><br/>Let $A$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .<strong>Question:</strong><br/>The number of matrices $A$ in $A$ for which the system of linear equations $A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has a unique solution, is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>less than 4<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>atleast 4 but less than 7<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/>atleast 7 but less than 10<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>at least 10</span> </li> </ul>

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>atleast 4 but less than 7<br/></span> </div>

<div class="solution">Given, $A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$<br/>For unique solution, $\operatorname{det}(A) \neq 0$<br/>$$<br/>\begin{gathered}<br/>\text { Case I } \operatorname{det}(A)=\left|\begin{array}{lll}<br/>1 &amp; a &amp; b \\<br/>a &amp; 1 &amp; c \\<br/>b &amp; c &amp; 1<br/>\end{array}\right| \\<br/>=1-a^2-b^2-c^2+2 a b c \neq 0<br/>\end{gathered}<br/>$$<br/>Here $a, b, c$ is selected from $1,0,0$. (No case is possible)<br/>Case II<br/>(i) $\operatorname{det}(A)=\left|\begin{array}{lll}1 &amp; a &amp; b \\ a &amp; 0 &amp; c \\ b &amp; c &amp; 0\end{array}\right|=2 a b c-c^2 \neq 0$<br/>Here $a, b, c$ are selected from $1,1,0$. (2 cases are possible)<br/>(ii) $\operatorname{det}(A)=\left|\begin{array}{lll}0 &amp; a &amp; b \\ a &amp; 1 &amp; c \\ b &amp; c &amp; 0\end{array}\right|=2 a b c-b^2 \neq 0$<br/>Here $a, b, c$ are selected from $1,1,0$. (2 cases are possible)<br/>(iii) $\operatorname{det}(A)=\left|\begin{array}{lll}0 &amp; a &amp; b \\ a &amp; 0 &amp; c \\ b &amp; c &amp; 1\end{array}\right|=2 a b c-a^2 \neq 0$<br/>Here $a, b, c$ are selected from $1,1,0$. (2 cases are possible)<br/>Hence, there are exactly 6 matrices for unique solution. Hence, option (b) is correct.</div>

MarksBatch2_P2.db

paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que

paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-46237

<div class="question"><strong>Paragraph:</strong><br/>Let $A$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .<strong>Question:</strong><br/>The number of matrices $A$ in $A$ for which the system of linear equations $A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ is is inconsistent, is</div>

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

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

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

<div class="solution">Given, $A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$<br/>Case I $\left[\begin{array}{lll}1 &amp; a &amp; b \\ a &amp; 1 &amp; c \\ b &amp; c &amp; 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$<br/>$a, b, c$ are selected from $1,0,0$.<br/>$\Rightarrow x+a y+b z=1 \Rightarrow a x+y+c z=0$ $b x+c y+z=0$<br/>(i) If $a=1, b=c=0$, then $x+y=1$ Inconsistent system of equation $x+y=0$<br/>(ii) If $a=0=c, b=1$, then $x+z=1$, $y=0$<br/>Inconsistent system of equation<br/>$$<br/>x+z=0<br/>$$<br/>(iii) If $c=1, a=b=0$, then $x=1, z=0$, $y=0$<br/>Case II<br/>(i) $\left[\begin{array}{lll}1 &amp; a &amp; b \\ a &amp; 0 &amp; c \\ b &amp; c &amp; 0\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$<br/>$a, b, c$ are selected from $1,1,0$.<br/>$$<br/>\begin{aligned}<br/>\Rightarrow x+a y+b z &amp; =1 \Rightarrow a x+c z=0 \\<br/>b x+c y &amp; =0<br/>\end{aligned}<br/>$$<br/>Clearly, in all three cases, solutions are possible, so system is consistent.<br/>$$<br/>\begin{gathered}<br/>\text { (ii) }\left[\begin{array}{lll}<br/>0 &amp; a &amp; b \\<br/>a &amp; 1 &amp; c \\<br/>b &amp; c &amp; 0<br/>\end{array}\right]\left[\begin{array}{l}<br/>x \\<br/>y \\<br/>z<br/>\end{array}\right]=\left[\begin{array}{l}<br/>1 \\<br/>0 \\<br/>0<br/>\end{array}\right] \\<br/>\Rightarrow a y+b z=1 \Rightarrow a x+y+c z=0 \\<br/>b x+c y=0<br/>\end{gathered}<br/>$$<br/>Clearly, $b=0, a=c=1$ gives<br/>$$<br/>y=1 ; x+y+z=0<br/>$$<br/>Inconsistent system $y=0$<br/>More than 2 matrices are possible.<br/>Hence, option (b) is correct.</div>

MarksBatch2_P2.db

paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-1

paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-1-25149

<div class="question"><strong>Paragraph:</strong><br/>Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices<br/>$$<br/>T_p=\left\{A=\left[\begin{array}{ll}<br/>a &amp; b \\<br/>c &amp; a<br/>\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}<br/>$$<strong>Question:</strong><br/>The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and det $(A)$ is divisible by $p$ is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$(p-1)^2$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$2(p-1)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$(p-1)^2+1$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$2 p-1$</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/>$2 p-1$</span> </div>

<div class="solution">Given, $A=\left[\begin{array}{ll}a &amp; b \\ c &amp; a\end{array}\right]$,<br/>$a, b, c \in\{0,1,2, \ldots, p-1\}$<br/>If $A$ is skew-symmetric matrix, then $a=0, b=-c$<br/>$\therefore \quad|A|=-b^2$.<br/>Thus, $P$ divides $|A|$ only when $b=0$...(i)<br/>Again, if $A$ is symmetric matrix, then $b=c$ and $|A|=a^2-b^2$.<br/>Thus, $p$ divides $|A|$ if either $p$ divides $(a-b)$ or $p$ divides $(a+b)$. $p$ divides $(a-b)$, only when $a=b$ ie, $a=b \in\{0,1,2, \ldots,(p-1)\}$<br/>ie, pchoices<br/>$p$ divides $(a+b)$.<br/>$\Rightarrow p$ choices, including $a=b=0$ included in (i)<br/>$\therefore$ Total number of choices are $(p+p-1)=2 p-1$.<br/>Hence, (c) is the correct option.</div>

MarksBatch2_P2.db

paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-2

paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-2-62824

<div class="question"><strong>Paragraph:</strong><br/>Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices<br/>$$<br/>T_p=\left\{A=\left[\begin{array}{ll}<br/>a &amp; b \\<br/>c &amp; a<br/>\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}<br/>$$<strong>Question:</strong><br/>The number of $A$ in $T_p$ such that $\operatorname{det}(A)$ is not divisible by $p$, is</div>

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

<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$2 p^2$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$p^3-5 p$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$p^3-3 p$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$p^3-p^2$</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> </ul>

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

<div class="solution">The number of matrices for which $p$ does not divide $\operatorname{Tr}(A)=(p-1) p^2$ of these $(p-1)^2$ are such that $p$ divides $|A|$. The number of matrices for which $p$ divides $\operatorname{Tr}(A)$ and $p$ does not divides $|A|$ are $(p-1)^2$<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \text { Required number } \\<br/>&amp; =(p-1) p^2-(p-1)^2+(p-1)^2 \\<br/>&amp; =p^3-p^2<br/>\end{aligned}<br/>$$</div>

MarksBatch2_P2.db

paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g

paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-81764

<div class="question"><strong>Paragraph:</strong><br/>Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices<br/>$$<br/>T_p=\left\{A=\left[\begin{array}{ll}<br/>a &amp; b \\<br/>c &amp; a<br/>\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}<br/>$$<strong>Question:</strong><br/>The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but det $(A)$ is divisible by $p$ is<br/>[Note : The trace of a matrix is the sum of its diagonal entries.]</div>

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

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

<div class="correct-answer"> The correct answer is: <span class="option-value"><br/>$(p-1)^2$<br/></span> </div>

<div class="solution">Trace of $A=2 a$, will be divisible by $p$ iff $a=0$.<br/>$|A|=a^2-b c$, for $\left(a^2-b c\right)$ to be divisible<br/>by $p$. There are exactly $(p-1)$ ordered pairs $(b, c)$ for any value of $a$.<br/>$\therefore$ Required number is $(p-1)^2$.<br/>Hence, (c) is the correct option.</div>

MarksBatch2_P2.db