# questions

tags: ['Chemistry', 'Redox Reactions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-

slug:
match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-65264

question:
<div class="question">Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u3ZyoDHANLps94g_Bgxxf_Fo4PLbZbkKoAmBhvW_0C8.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">A-p,s B-r, C-p,q, D-p,s,<br/>$$<br/>\begin{aligned}<br/>2 \mathrm{CrO}_4^{2-} \longrightarrow \mathrm{Cr}_2 \mathrm{O}_7^{2-} \\<br/>\mathrm{MnO}_4^{-}+\mathrm{NO}_2^{-}+\mathrm{H}^{+} \longrightarrow \mathrm{Mn}^{2+}+\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{O} \\<br/>\mathrm{NO}_3^{-}+\mathrm{H}_2 \mathrm{SO}_4+\mathrm{Fe}^{2+} \longrightarrow \mathrm{Fe}^{3+}+\mathrm{NO}+\mathrm{SO}_4^{2-}+\mathrm{H}_2 \mathrm{O}<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Coordination Compounds', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicates-your-answer-by-darkening-the-appropriate-bubbles-of-4-4-matrix-giv

slug:
match-the-complexes-in-column-i-with-their-properties-listed-in-column-ii-indicates-your-answer-by-darkening-the-appropriate-bubbles-of-4-4-matrix-giv-97122

question:
<div class="question">Match the complexes in Column I with their properties listed in Column II. Indicates your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5dKmHYTGKoudvS18CsnhAI04sI2pOm3IVd68Y3je4K8.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4\left[\left(\mathrm{H}_2 \mathrm{O}\right)_2\right] \mathrm{Cl}_2 p, q, s\right.$<br/>Oxidation state $\mathrm{Co}$ is $+2$<br/>$\Rightarrow$ Paramagnetic, exhibits geometrical isomerism.<br/>(B) $\left[\mathrm{Pt}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right] p, r, s$<br/>$\mathrm{dsp}{ }^2$-hybridisation<br/>$\mathrm{Pt}^{2+}$, diamagnetic square planar, geometrical isomerism<br/>(C) $q, s$ (D) $q, s$</div>


tags: ['Chemistry', 'Hydrocarbons', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-1

slug:
match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-1-73346

question:
<div class="question">Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TJVpaWmgq_MRw31jJHL907USap6KF9NIXYukIKdkNpw.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">A-r; B-p, q; C- p, q, r; D-p<br/>Sodium fusion extract gives Prussian blue when nitrogen is present alongwith carbon. Phenolic group and carboxylate gives positive $\mathrm{FeCl}_3$ test - white precipitate comes with $\mathrm{AgCl}$.<br/>$$<br/>\text { Hydrazone formation occurs effectively at } \mathrm{pH} \approx 4.5 \text {. The reaction proceeds in that }<br/>$$<br/>condition only when $\mathrm{H}^{+}$concentration is merely sufficient to activate. As $\mathrm{H}^{+}$concentration raises, sufficient molecule of hydrazone gets converted into hydrazonium which is not nucleophilic and reaction becomes impossible. Further, low concentration of $\mathrm{H}^{+}$(in the case of hydrolysis of 2, 4-dinitrophenyl hydrazonium bromide) is not effective to proceed elimination (rate determining step).<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/CjI78dy4abclm11LRXCdsAkn5nz4S5BhS3Jyxy1XhUg.original.fullsize.png"/><br/><br/>If it is assumed that the given hydrazonian bromiate is the result as salt main $\mathrm{HBr}$ is used as acid catalyst. Remember! hydrazone formation involves first addition and then elimination (res). In the second step substitution dominates over elimination ( $\mathrm{HBr}$ is not a dehydrating reagent mutually).</div>


tags: ['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles

slug:
match-the-compounds-in-column-i-with-their-characteristic-testsreactionss-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-72456

question:
<div class="question">Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TJVpaWmgq_MRw31jJHL907USap6KF9NIXYukIKdkNpw.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">A-r; B-p, q; C- p, q, r; D-p<br/>Sodium fusion extract gives Prussian blue when nitrogen is present alongwith carbon. Phenolic group and carboxylate gives positive $\mathrm{FeCl}_3$ test - white precipitate comes with $\mathrm{AgCl}$.<br/>$$<br/>\text { Hydrazone formation occurs effectively at } \mathrm{pH} \approx 4.5 \text {. The reaction proceeds in that }<br/>$$<br/>condition only when $\mathrm{H}^{+}$concentration is merely sufficient to activate. As $\mathrm{H}^{+}$concentration raises, sufficient molecule of hydrazone gets converted into hydrazonium which is not nucleophilic and reaction becomes impossible. Further, low concentration of $\mathrm{H}^{+}$(in the case of hydrolysis of 2, 4-dinitrophenyl hydrazonium bromide) is not effective to proceed elimination (rate determining step).<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/CjI78dy4abclm11LRXCdsAkn5nz4S5BhS3Jyxy1XhUg.original.fullsize.png"/><br/><br/>If it is assumed that the given hydrazonian bromiate is the result as salt main $\mathrm{HBr}$ is used as acid catalyst. Remember! hydrazone formation involves first addition and then elimination (res). In the second step substitution dominates over elimination ( $\mathrm{HBr}$ is not a dehydrating reagent mutually).</div>


title: match-the-conics-in-column-i-with-the-statementsexpressions-in-column-ii
slug: match-the-conics-in-column-i-with-the-statementsexpressions-in-column-ii-54802
tags: ['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">$$<br/>\text { Match the conics in Column I with the statements/expressions in Column II. }<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/mttZypB7gORJPl-QC2KlWwOMRUJEI-jGBWeCxz3jUTE.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BSqFu76r11SkdAwZcbkeTMZ0RwfnU1zWrr8QDx9JnTE.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(p) $\frac{1}{\sqrt{h^2+k^2}}=2 \Rightarrow h^2+k^2=\frac{1}{4}$<br/>Hence, locus is a circle.<br/>(q) ||$z+2|-| z-2||=3$<br/>and $\quad 2-(-2)=4&gt;3$<br/>Hence, locus is hyperbola.<br/>(r) Let $x=\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right), y=\frac{2 t}{1+t^2}$<br/>Let $\quad \tan \theta=t$ $\Rightarrow x=\sqrt{3} \cos 2 \theta, y=\sin 2 \theta$<br/>$\therefore \quad \frac{x^2}{3}+y^2=1$<br/>Hence, locus is an ellipse.<br/>(s) Eccentricity $x=1 \Rightarrow$ Parabola, $1 &lt; x &lt; \infty \Rightarrow$ Hyperbola<br/>(t) Let $z=x+i y$<br/>Since, $\operatorname{Re}(z+1)^2=|z|^2+1$<br/>$\Rightarrow \quad(x+1)^2-y^2=x^2+y^2+1$<br/>$\Rightarrow \quad 2 x=2 y^2$<br/>$\Rightarrow \quad x=y^2$<br/>Hence, locus is parabola.</div>


source_db: MarksBatch2_P2.db
title:
match-the-conversions-in-column-i-with-the-types-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-given-

slug:
match-the-conversions-in-column-i-with-the-types-given-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-4-4-matrix-given-26342

question:
<div class="question">Match the conversions in Column I with the type(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ucomvVlKQ_CiBxOlv7mu0IFQfu05rElGpRWCcPrudJI.original.fullsize.png"/><br/></div>

tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RTlN6ixNP8LakoAWNPJ2h8GaTB8etWo-T-TqYSxfK94.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Structure of Atom', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-entries-in-column-i-with-the-correctly-related-quantum-numbers-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the

slug:
match-the-entries-in-column-i-with-the-correctly-related-quantum-numbers-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbles-of-the-67501

question:
<div class="question">Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/2J3KeSRuERC1Xe6fSQAoOrICdqN8QJF--5LumNR_PWQ.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">A-q; B-p, q, r, s; C-p, q, r; D-p, q, r<br/>(B) A hydrogen-like one electron wave function obeying Pauli's principle First, as it is the wave function so it will be characterised by $n, l$ and $m$. Further, it obeys the Pauli's principle, that's why it will be characterised completely by adding spin quantum number also.<br/>(D) Radial probability density of electron at the nucleus basically depends on azimuthal quantum number, but as it is the function of radius depends on principal quantum number, When angular probability density is included, that depends on magnetic quantum number.</div>


title: match-the-extraction-processes-listed-in-column-i-with-metals-listed-in-column-ii
slug: match-the-extraction-processes-listed-in-column-i-with-metals-listed-in-column-ii-64082
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the extraction processes listed in column I with metals listed in column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/T1XuJuazCxU0ENveihRFEqo1WSHuTkcfu_MEsZeXbXA.original.fullsize.png"/><br/></div>

tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2006']

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) P,R, (B) P, (C) Q, (D) S</span> </div>

answer:
<div class="solution">Column I<br/>Column II<br/>(A) Self reduction $\left[\begin{array}{c}P \text { - Lead } \\ R \text { - Copper }\end{array}\right]$<br/>$$<br/>\begin{aligned}<br/>&amp; \text { i.e., } A \rightleftarrows P \text { and } R \\<br/>&amp; \text { In lead metal } \\<br/>&amp; \qquad\left\{\begin{array}{c}<br/>2 \mathrm{PbS}+3 \mathrm{O}_2 \longrightarrow 2 \mathrm{PbO}+2 \mathrm{SO}_2 \\<br/>2 \mathrm{PbO}+\mathrm{PbS} \longrightarrow 3 \mathrm{~Pb}+\mathrm{SO}_2<br/>\end{array}\right.<br/>\end{aligned}<br/>$$<br/>(self reduction)<br/>In copper metal<br/>$$<br/>\begin{aligned}<br/>&amp; 2 \mathrm{Cu}_2 \mathrm{~S}+3 \mathrm{O}_2 \longrightarrow 2 \mathrm{Cu}_2 \mathrm{O}+2 \mathrm{SO}_2 \\<br/>&amp; \mathrm{Cu}_2 \mathrm{~S}+2 \mathrm{Cu}_2 \mathrm{O} \longrightarrow 6 \mathrm{Cu}+\mathrm{SO}_2<br/>\end{aligned}<br/>$$<br/>(self reduction)<br/>(B) Carbon reduction $P$-lead<br/>$$<br/>\begin{aligned}<br/>&amp; \text { i.e., } \mathrm{B}-\mathrm{P} \\<br/>&amp; \text { In lead metal } \\<br/>&amp; \qquad \mathrm{PbO}+\mathrm{C} \longrightarrow \mathrm{Pb}+\mathrm{CO} \\<br/>&amp; \mathrm{PbO}+\mathrm{CO} \longrightarrow \mathrm{Pb}+\mathrm{CO}_2<br/>\end{aligned}<br/>$$<br/>Carbon reduction process is used for the reduction of oxides of electropositive metals like $\mathrm{Fe}, \mathrm{Pb}$ etc. This reduction is carried out with coal or coke on strongly heating.<br/>(C) Complex formation and displacement by metal.<br/>i.e., $C-Q$ (silver)<br/><br/>In silver metal<br/>Argentite mineral of $\mathrm{Ag}\left(\mathrm{Ag}_2 \mathrm{~S}\right)$ is treated with $0.7 \% \mathrm{NaCN}$ solution in presence of air, therefore firstly sodium argento cyanide complex is formed which on heating with zinc metal, therefore by displacement silver is isolated.<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{Ag}_2 \mathrm{~S}+4 \mathrm{NaCN} \longrightarrow 2 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_2\right]+\mathrm{Na}_2 \mathrm{~S} \\<br/>&amp; 2 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_2\right]+\mathrm{Zn} \longrightarrow \mathrm{Na}\left[\mathrm{Zn}(\mathrm{CN})_4\right]+\underset{\text { Soluble }}{2 \mathrm{Ag}} \downarrow<br/>\end{aligned}<br/>$$<br/>(D) Decomposition of iodide-(S) Boron<br/>$$<br/>\text { i.e., }(D)-(S)<br/>$$<br/>By decomposition of iodide boron metal is isolated.</div>


title: match-the-following-4
slug: match-the-following-4-26393
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/aP6PkZrkpVGT5Ap7bH-wIPaY4lRESHzYikGKWjxXO2w.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) Q,S, (B) S, (C) R, (D) Q,R<br/></span> </div>

answer:
<div class="solution">A match with $\boldsymbol{Q}, \boldsymbol{S} \quad \mathrm{Bi}^{3+}+\mathrm{H}_2 \mathrm{O} \longrightarrow[\mathrm{BiO}]^{+}+2 \mathrm{H}^{+}$ B match with $\boldsymbol{S} \mathrm{NaAlO}_2+2 \mathrm{H}_2 \mathrm{O} \longrightarrow \mathrm{Al}(\mathrm{OH})_3+\mathrm{NaOH}$ C match with $\boldsymbol{R} \quad 2 \mathrm{SiO}_4^{4-}+2 \mathrm{H}^{+} \longrightarrow \mathrm{Si}_2 \mathrm{O}_7^{6-}+\mathrm{H}_2 \mathrm{O}$ Pyrosilicate is formed by treating orthosilicate with acid. D match with $\boldsymbol{Q}, \boldsymbol{R} \mathrm{Na}_2 \mathrm{~B}_4 \mathrm{O}_7 \underset{\text { or } \mathrm{H}_2 \mathrm{SO}_4}{\stackrel{\mathrm{HCl}}{\mathrm{O}_3}} \mathrm{H}_3 \mathrm{BO}_3$<br/>$$<br/>\mathrm{Na}_2 \mathrm{~B}_4 \mathrm{O}_4 \stackrel{\mathrm{H}_2 \mathrm{O}}{\longrightarrow} \mathrm{H}_3 \mathrm{BO}_3<br/>$$</div>


title: match-the-following-5
slug: match-the-following-5-89404
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Xmaa26fH0hR4prHCjmrRrNH-rfprOx-3FtUrzpt4n6w.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) Q, (B) Q, (C) R,S, (D) P,S</span> </div>

answer:
<div class="solution">$$<br/>\text { (A) Match with (Q) }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/pZbxv4ivUF9I7Kp-BSSuA_5S0dEaTlkWLMIkQCP26a4.original.fullsize.png"/><br/><br/>The formation of $\mathrm{CH}_2=\mathrm{CH}-\mathrm{CD}_3$ can be explained on the basis of the fact that $\mathrm{C}-\mathrm{D}$ bond is much stronger than $\mathrm{C}-\mathrm{H}$ bond.<br/>(B) match with (Q)<br/>Reactivity of $\mathrm{PhCHBrCH}_3$ is greater than<br/>$\mathrm{Ph} \mathrm{CHBrCD}_3$ because $\mathrm{C}-\mathrm{D}$ bond is more stronger than $\mathrm{C}-\mathrm{H}$ bond.<br/><br/>$$<br/>\text { (C) match with (R) and (S) }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eunRBn6utbgP22GT6vuptZpANFRSsz0948UmfNNgbvI.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZduTsP4411yF83jMwipCj7LUJoPCBkyfcaz8xXeUkbA.original.fullsize.png"/><br/><br/>In the step (II), a slow unimolecular elimination occurs in the conjugate base of the reactant and hence this mechanism is called $E_1 C B$ or carbanion mechanism. Since step (I) must be reversible, if ethanol containing $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OD}$ is used as solvent, it would be expected that the original bromide would incorporate deuterium (D).<br/><br/>$$<br/>\text { (D) Match with (P) and (S) }<br/>$$<br/><br/>Step I. $\mathrm{PhCH}_2-\mathrm{CH}_2-\mathrm{Br} \stackrel{\text { Slow }}{\longrightarrow} \mathrm{PhCH}_2-\stackrel{+}{\mathrm{C}} \mathrm{H}_2+\mathrm{Br}^{-}$<br/>Step II. $\mathrm{PhCH}_2-\stackrel{+}{\mathrm{C}_2} \mathrm{H}_2 \stackrel{\text { Fast }}{\longrightarrow} \mathrm{Ph}-\mathrm{CH}=\mathrm{CH}_2+\mathrm{H}^{-}$<br/>Rate $\propto\left[\mathrm{PhCH}_2-\mathrm{CH}_2-\mathrm{Br}\right]$<br/>Similarly<br/>Step II. $\mathrm{PhCD}_2 \stackrel{+}{\mathrm{C}_2} \mathrm{H}_2 \stackrel{\text { Fast }}{\longrightarrow} \mathrm{PhCD}=\mathrm{CHD}+\mathrm{H}^{+}$<br/>Rate $\propto\left[\mathrm{PhCD}_2-\mathrm{CH}_2 \mathrm{Br}\right]$<br/>Hence, $E_1$ reaction and first order kinetics.</div>


title: match-the-following-6
slug: match-the-following-6-47862
tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/k7sNssq9bMajNP7pgcEvTBAHN9XF2aDQJpzY1CwNW6A.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (b), (ii) (a), (iii) (c), (iv) (d)</span> </div>

answer:
<div class="solution">(i) Solved the two equations, say, i.e. $x+y=|a|$ and $a x-y=1$, we get $x=\frac{|a|+1}{a+1}&gt;0$ and $y=\frac{|a|-1}{a+1}&gt;0$<br/>when $a+1&gt;0$; we get $a&gt;1$<br/>$\therefore \quad a_0=1$<br/>(ii) We have,<br/>$\mathbf{a}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$<br/>$\Rightarrow \quad \mathbf{a} \cdot \hat{\mathbf{k}}=\gamma$<br/>Now, $\hat{\mathbf{k}} \times(\hat{\mathbf{k}} \times \mathbf{a})=(\hat{\mathbf{k}} \cdot \mathbf{a})-(\hat{\mathbf{k}} \cdot \hat{\mathbf{k}}) \mathbf{a}=\gamma \hat{\mathbf{k}}-(\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}})$ $\Rightarrow \quad \alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}=0 \Rightarrow \alpha=\beta=0$<br/>Also, $\quad \alpha+\beta+\gamma=2 \Rightarrow \gamma=2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/6me9FVDij8eFqKPHPUuH_wTp5Smkwlfj-vCln1v9K74.original.fullsize.png"/><br/><br/>(iv) $\sin A \sin B \sin C+\cos A \cos B \leq \sin A \sin B+\cos A \cos B=\cos (A-B)$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow &amp; \cos (A-B) \geq 1 \Rightarrow \cos (A-B)=1 \\<br/>\Rightarrow &amp; \sin C=1<br/>\end{array}<br/>$$</div>


title: match-the-following-7
slug: match-the-following-7-13447
tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/m1EAsIDafFkvQJfVVNHHUnyzlA5lW-g4xZ8pul4RCq4.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (b), (ii) (a), (iii) (c)<br/></span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { (i) } \sum_{i=1}^{\infty} \tan ^{-1}\left(\frac{1}{2 i^2}\right)=t \Rightarrow \sum_{i=1}^{\infty} \tan ^{-1}\left(\frac{2}{4 i^2-1+1}\right) \\<br/>&amp; \Rightarrow \sum_{i=1}^{\infty} \tan ^{-1}\left\{\frac{(2 i+1)-(2 i-1)}{1+(2 i-1)(2 i+1)}\right\} \\<br/>&amp; \Rightarrow\left(\tan ^{-1} 3-\tan ^{-1} 1\right)+\left(\tan ^{-1} 5-\tan ^{-1} 3\right)+\ldots+\left\{\tan ^{-1}(2 n+1)-\tan ^{-1}(2 n-1)\right\} \\<br/>&amp; \therefore \quad \quad t=\lim _{n \rightarrow \infty}\left(\tan ^{-1}(2 n+1)-\tan ^{-1} 1\right)=\lim _{n \rightarrow \infty} \tan ^{-1}\left(\frac{2 n}{1+2 n+1}\right)=\frac{\pi}{4} \\<br/>&amp; \therefore \quad \tan t=1 .<br/>\end{aligned}<br/>$$<br/>(ii) We have, $\cos \theta_1=\frac{1-\tan ^2 \frac{\theta_1}{2}}{1+\tan ^2 \frac{\theta_2}{2}}=\frac{a}{b+c}$<br/>$$<br/>\Rightarrow \quad \tan ^2 \frac{\theta}{2}=\frac{b+c-a}{b+c+a}<br/>$$<br/>Also, $\quad \cos \theta_3=\frac{1-\tan ^2 \frac{\theta_3}{2}}{1+\tan ^2 \frac{\theta_3}{2}}=\frac{c}{a+b}$<br/>$$<br/>\Rightarrow \quad \tan ^2 \frac{\theta_3}{2}=\frac{a+b-c}{a+b+c}<br/>$$<br/>$\Rightarrow \tan ^2 \frac{\theta_1}{2}+\tan ^2 \frac{\theta_3}{2}=\frac{2 b}{a+b+c}=\frac{2 b}{3 b}=\frac{2}{3}, \quad$ [as, $a, b, c$ are in AP $\Rightarrow 2 b=a+c$ ]<br/>(iii) Line through $(0,1,0)$ and perpendicular to plane $x+2 y+2 z=0$ is given by $\frac{x-0}{1}=\frac{y-1}{2}=\frac{z-0}{2}=r$<br/>$\therefore P(r, 2 r+1,2 r)$ be the foot of perpendicular on the straight line, then<br/>$$<br/>\begin{array}{ll} <br/>&amp; r \cdot 1+(2 r+1) \cdot 2+(2 r) \cdot 2=0 \\<br/>\Rightarrow \quad r= &amp; \quad r\left(-\frac{2}{9}, \frac{5}{9},-\frac{4}{9}\right) \\<br/>\therefore &amp; \quad P(r) \\<br/>\therefore &amp; \text { Required perpendicular distance }=\sqrt{\frac{4+25+16}{81}}=\frac{\sqrt{5}}{3} \text { unit. }<br/>\end{array}<br/>$$</div>


title: match-the-following-8
slug: match-the-following-8-93106
tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UyPhBV62yJ87NM_xRqutVqISAPvRym6ILTSeV_4cma8.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(i) (a), (ii) (d), (iii) (b), (iv) (c)<br/></span> </div>

answer:
<div class="solution">(i) $I=\int_0^{\pi / 2}(\sin x)^{\cos x}\left(\cos x \cot x-\log (\sin x)^{\sin x}\right) d x=\int_0^{\pi / 2} \frac{d}{d x}(\sin x)^{\cos x} d x=1$<br/>(ii) The point of intersection of $-4 y^2=x$ and $x-1=-5 y^2$ is $(-4,-1)$ and $(-4,1)$.<br/>Hence, required area $=2\left\{\int_0^1\left(1-5 y^2\right) d y-\int_0^1-4 y^2 d y\right\}=\frac{4}{3}$.<br/>(iii) The point intersection of $y=3^{x-1} \log x$ and $y=x^x-1$ is $(1,0)$.<br/>Hence, $\quad \frac{d y}{d x}=\frac{3^{x-1}}{x}+3^{x-1} \cdot \log 3 \cdot \log x$<br/>$\therefore \quad\left(\frac{d y}{d x}\right)_{(1,0)}=1$<br/>For<br/>$$<br/>\therefore \quad\left(\frac{d y}{d x}\right)_{(1,0)}=1<br/>$$<br/><br/>If $\theta$ is angle between the curves, then $\tan \theta=0$.<br/>$$<br/>\because \quad \theta=0^{\circ}<br/>$$<br/>(iv)<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{d y}{d x}=\frac{2}{x+y} \Rightarrow \frac{d x}{d y}-\frac{x}{2}=\frac{y}{2} \\<br/>&amp; \Rightarrow \quad x e^{-y / 2}=\frac{1}{2} \cdot \int y \cdot e^{-y / 2} d y \\<br/>&amp; \Rightarrow \quad x+y+2=k e^{y / 2} \\<br/>&amp; \Rightarrow \quad k=3 \\<br/>&amp; \therefore \quad x+y+2=3 e^{y / 2} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>[passing through $(1,0)$ ]</div>


title: match-the-following-columns-1
slug: match-the-following-columns-1-36925
tags: ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following columns<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UWvjzowOsy9s6gkOqAlYesntU2oLVtSYhsrLUqDtv3A.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) p,q, (b) p, r, (c) p,s, (d) p,q,r<br/></span> </div>

answer:
<div class="solution">(a) $\rightarrow$ p, q<br/>(b) $\rightarrow \mathrm{p}, \mathrm{r}$<br/>(c) $\rightarrow$ p, $s$<br/>(d) $\rightarrow$ p, q, r</div>


title: match-the-following-columns-2
slug: match-the-following-columns-2-81096
tags: ['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following columns<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/sMqslWdePeakThlq1yBOr6vLpYxkICZ4jmVngXlMfX8.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) p, (b) p,q,s, (c) p,s, (d) q,r,s<br/></span> </div>

answer:
<div class="solution">(a) $\rightarrow$ p<br/>(b) $\rightarrow$ p, q, s<br/>(c) $\rightarrow$ q, s<br/>(d) $\rightarrow$ q, r, s</div>


tags: ['Biology', 'Animal Kingdom', 'NEET', 'NEET 2020 (Phase 1)']
source_db: MarksBatch2_P2.db
title:
match-the-following-columns-and-select-the-correct-option-columni-columnii-a-gregarious-polyphagous-pest-i-asterias-b-adult-with-radial-symmetry-and-l

slug:
match-the-following-columns-and-select-the-correct-option-columni-columnii-a-gregarious-polyphagous-pest-i-asterias-b-adult-with-radial-symmetry-and-l-89561

question:
<div class="question"><p>Match the following columns and select the correct option.</p><table border="1" cellpadding="1" cellspacing="1" style="width: 400px;"> <tbody> <tr> <td> </td> <td>Column-I</td> <td> </td> <td>Column-II</td> </tr> <tr> <td>(a)</td> <td>Gregarious, polyphagous pest</td> <td>(i)</td> <td><em>Asterias</em></td> </tr> <tr> <td>(b)</td> <td>Adult with radial symmetry and larva with bilateral symmetry</td> <td>(ii)</td> <td>Scorpion</td> </tr> <tr> <td>(c)</td> <td>Book lungs</td> <td>(iii)</td> <td><em>Ctenoplana</em></td> </tr> <tr> <td>(d)</td> <td>Bioluminescence</td> <td>(iv)</td> <td><em>Locusta</em></td> </tr> </tbody></table></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)</span> </div>

answer:
<div class="solution"><p><strong>Gregarious pest:</strong> A pest is an animal that is detrimental or harmful to humans or human concerns. <em>Locusta</em> (locust) is a pest that is dangerous to crops. It causes heavy damage to the vegetation. As it lives in groups of the same individuals but does not help one another, hence called gregarious.</p><p><strong>Body symmetry of Echinoderms:</strong> The adult echinoderms are radially symmetrical, but their larvae are bilaterally symmetrical. The echinoderms are unique in this feature that the larvae are bilaterally symmetrical (an advanced feature) while the adults developing from them are radially symmetrical (a primitive feature). <strong>Examples: <em>Asterias</em> (Starfish), <em>Echinus</em> (Sea urchin), <em>Antedon</em> (sea lily), <em>Cucumaria </em>(sea cucumber), and <em>Ophiura</em> (Brittle star).</strong></p><p><strong>Book-lungs: Present in scorpions, spiders, etc. </strong>These are the modified book-gills.</p><p><strong>Bioluminescence:</strong> Bios means living and lumen means light. Bioluminescence is the property of production and emission of light by a living organism and hence shines in the dark background. It is a naturally occurring phenomenon. This property is well-marked in Ctenophores as they emit light and hence shine against the dark backgrounds. <strong>Examples: <em>Pleurobrachia, Ctenoplana.</em></strong></p></div>


title: match-the-following-for-the-given-process-1
slug: match-the-following-for-the-given-process-1-71120
tags: ['Physics', 'Thermodynamics', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the following for the given process<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/tZNNR6Q2U1U_vB12hZCZHK1J080dytLaYA5IBXtXE3Q.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jSH26CUr5_R-O26KzLkmQbvZWxgIID6T3Mdnt9iL_Ms.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(a) s, (b) p, r, (c) r, (d) q,s<br/></span> </div>

answer:
<div class="solution">(a) $\rightarrow \mathrm{s}$ (b) $\rightarrow \mathrm{p}, \mathrm{r}$, (c) $\rightarrow \mathrm{r}$ (d) $\rightarrow \mathrm{q}, \mathrm{s}$<br/>In process $\mathbf{J} \rightarrow \mathbf{K}: V$ is constant whereas $P$ is decreasing, Therefore, $T$ should also decrease.<br/>$$<br/>\therefore \quad W=0, \Delta U=-\text { ve and } Q &lt; 0<br/>$$<br/>In process $\mathbf{K} \rightarrow \mathbf{L}: P$ is constant while $V$ is increasing. Therefore, temperature should also increase.<br/>$$<br/>\therefore \quad W&gt;0, \Delta U&gt;0 \text { and } Q&gt;0<br/>$$<br/><br/>In process $\mathbf{L} \rightarrow \mathbf{M}$ : This is inverse of process $J \rightarrow K$.<br/>$$<br/>\therefore \quad W=0, \Delta U&gt;0 \text { and } Q&gt;0<br/>$$<br/>In process $\mathbf{M} \rightarrow \mathbf{J}: V$ is decreasing. Therefore, $W &lt; 0$<br/>$$<br/>\begin{array}{lcl} <br/>&amp; (P \mathrm{~V})_J &amp; &lt; (P V)_M \\<br/>\therefore &amp; T_J &amp; &lt; T_M \text { or } \quad \Delta U &lt; 0 \\<br/>\text { Therefore, } &amp; Q &amp; &lt; 0 .<br/>\end{array}<br/>$$<br/>No, solution is required.</div>


tags: ['Chemistry', 'States of Matter', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
match-the-gases-under-specified-conditions-listed-in-column-i-with-their-propertieslaws-in-column-ii-indicate-your-answer-by-darkening-the-appropriate

slug:
match-the-gases-under-specified-conditions-listed-in-column-i-with-their-propertieslaws-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-98087

question:
<div class="question">Match the gases under specified conditions listed in Column I with their properties/laws in Column II. Indicate your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ejgNRpWHs-ac66lvxqM62VDBFoI4F-mmsBx1r-RU0uI.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) $-p, s(\mathrm{~B})-r$ (C) $-p, q$ (D) $-r$<br/>Explanation van der Waals' equation<br/>$$<br/>\left(p+\frac{n^2 a}{V_m^2}\right)\left(V_m-b\right)=n R T<br/>$$<br/><br/>For hydrogen gas $(p=200 \mathrm{~atm}, T=273 \mathrm{~K})$<br/>As pressure is large $V_m$ can be assumed small, thus ' $b$ ' can not be ignored, while due to high pressure $a / V_m^2$ can be considered negligible in comparison to $p$.<br/>$$<br/>p\left(V_m-b\right)=R T \quad \text { and } \quad Z=1+\frac{p b}{R T}<br/>$$<br/>For hydrogen gas $(p \sim 0, T=273 \mathrm{~K})$<br/>when pressure occurs of low about $1 \mathrm{~atm}$ or less and temperature is not very close to the point of liquification $\left[T_c\left(\mathrm{H}_2\right)=33.3 \mathrm{~K}\right]$ gas behaves ideally.<br/>For<br/>$$<br/>P V=n R T<br/>$$<br/>Temperature is close to the point of liquification $\left[T_C\left(\mathrm{CO}_2\right)=304.2\right]$ thus, deviation from ideality appears very high (due to high attractive force of attraction).<br/>For real gas with very large molar volume.<br/>As molar volume is very large $a / V_m^2$ will be negligible and at the same time ' $b$ ' in comparison to $V_m$ is also considered negligible, thus, $p V_m=n R T$</div>


title: match-the-integrals-in-column-i-with-the-values-in-column-ii
slug: match-the-integrals-in-column-i-with-the-values-in-column-ii-38496
tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the integrals in Column I with the values in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/CYjILavSNl1HrZmbzJ5Wzy63qiah4GZwojG_7X4ITSE.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) <img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/cK8Zj8AakkXGyCXyen9p0A6CvTZojRKgVYLM5o4MYug.original.fullsize.png"/><br/><br/>(B)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/uYQamdjeMo3WPD_Hye2LKowKcou_DybIc_gLNbDMuxo.original.fullsize.png"/><br/><br/>(C)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/hkrxgdSxO1eRTvF1q2hhn8A3T7l4JsKxJDYMI0FHwgQ.original.fullsize.png"/><br/><br/>(D)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ncaKmPXp6iUpPInSEkh8-je4ozcxA0l1ewpZH03_PFw.original.fullsize.png"/><br/></div>


title: match-the-reaction-in-column-i-with-appropriate-options-in-column-ii-1
slug: match-the-reaction-in-column-i-with-appropriate-options-in-column-ii-1-21104
tags: ['Chemistry', 'Hydrocarbons', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">$$<br/>\text { Match the reaction in Column I with appropriate options in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/N9Q7iRDy8ErnJ4De4bFeq_7NeMbZAX4r4qqF1C64BJk.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(A) R,S,T, (B) T, (C) P,Q, (D) R</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5z61L57XV5xVQKMEv0xK_JpD_ST6Lh1zTMsxo5I6c2k.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/qCT1dRcjq-40dK0016TKVE0y-1RpWgSe4N6dqXFkvzI.original.fullsize.png"/><br/><br/>Organic chemistry<br/>Conceptual understanding of reaction mechanism IV</div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-reactions-in-column-i-with-appropriate-types-of-stepsreactive-intermediate-involved-in-these-reactions-as-given-in-column-ii

slug:
match-the-reactions-in-column-i-with-appropriate-types-of-stepsreactive-intermediate-involved-in-these-reactions-as-given-in-column-ii-40077

question:
<div class="question">Match the reactions in Column I with appropriate types of steps/reactive intermediate involved in these reactions as given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/poSsUnlRvHd6dET79r9AVpuWiLjj9Sp2MvScv-cFARQ.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RXewfmccGPifySWpcS4turpHvZqdqba01NXVmjCknvs.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(a) <img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IFjuTZURdU5JiHtTb_LQEiwlFpURZ2fynnPHhznWqDM.original.fullsize.png"/><br/><br/>(b)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eApzd4UV9vZUVc3ktP7NhEyeOfodco9EfwZFDq2f-9Y.original.fullsize.png"/><br/><br/>(c)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/tZ9C1KMNShw79HBFaereEVSeF8mheA8Dr_wEbScZESI.original.fullsize.png"/><br/><br/>(d)<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_nhmZi9HFG2paBcmYxYqAQ-vIbmLrI8-mda2GVLICRc.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Solid State', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-specified-conditions-in-solids-listed-in-column-i-with-their-properties-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbl

slug:
match-the-specified-conditions-in-solids-listed-in-column-i-with-their-properties-in-column-ii-indicate-your-answer-by-darkening-the-appropriate-bubbl-76131

question:
<div class="question">Match the specified conditions in solids listed in Column I with their properties in Column II. Indicate your answer by darkening the appropriate bubbles of the $4 \times 4$ matrix given in the ORS.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/FOTt4rDNn0Mz5UEqBBqMEkmbIqJQz5GwTJtTKCsEoiI.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">$\mathrm{A}-p, s, \quad \mathrm{~B}-p, q, \quad \mathrm{C}-q, \quad \mathrm{D}-q, r$</div>


title: match-the-statements-given-in-column-i-with-the-intervalsunion-of-intervals-given-in-column-ii
tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
slug:
match-the-statements-given-in-column-i-with-the-intervalsunion-of-intervals-given-in-column-ii-34527

question:
<div class="question">Match the statements given in Column I with the intervals/union of intervals given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/1If8DbPHHmiKRCE2AbXYbzNrmcvvJu2t-0TjT049750.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) Given, $|z|=1 \Rightarrow z \cdot \bar{z}=1$<br/>$$<br/>\therefore \quad \frac{2 i z}{1-z^2}=\frac{2 i z}{z \cdot \bar{z}-z^2}=\frac{2 i}{\bar{z}-z} \text {, }<br/>$$<br/>Let $\quad z=x+i y$<br/>$$<br/>\therefore z-\bar{z}=2 i y=\frac{2 i}{-2 i y}=-\frac{1}{y}<br/>$$<br/>where, $y=\sqrt{1-x^2}$<br/>$\therefore \quad-1 \leq y \leq 1 \Rightarrow-1 \leq y$<br/>and $y \leq 1 \Rightarrow-1 \geq \frac{1}{y}$ and $\frac{1}{y} \geq 1$<br/>$\Rightarrow \operatorname{Re}\left(\frac{2 i z}{1-z^2}\right) \in(-\infty,-1] \cup[1, \infty)$<br/>(B) $f(x)=\sin ^{-1}\left(\frac{8\left(3^{x-2}\right)}{1-3^{2(x-1)}}\right)$,<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \text { For domain, }-1 \leq \frac{8\left(3^{x-2}\right)}{1-3^{2(x-1)}} \leq 1 \\<br/>&amp; \Rightarrow \quad-1 \leq \frac{9 \cdot\left(3^{x-2}\right)-\left(3^{x-2}\right)}{1-3^{2(x-1)}} \leq 1 \\<br/>&amp; \therefore \quad-1 \leq \frac{3^x-3^{x-2}}{1-3^x \cdot 3^{(x-2)}} \leq 1 \\<br/>&amp; \frac{3^x-3^{x-2}}{1-3^x \cdot 3^{x-2}} \geq-1 \\<br/>&amp; \Rightarrow \frac{\left(3^x-1\right)\left(3^{x-2}-1\right)}{\left(3^{x-1}+1\right)\left(3^{x-1}-1\right)} \geq 0 \\<br/>&amp; \Rightarrow \quad x \in(-\infty, 0] \cup(1, \infty) \\<br/>&amp; \text { and } \frac{3^x-3^{x-2}}{1-3^x \cdot 3^{x-2}} \leq 1 \\<br/>&amp; \Rightarrow \frac{\left(3^{x-2}-1\right)\left(3^x+1\right)}{\left(3^{x-1}+1\right)\left(3^{x-1}-1\right)} \geq 0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wbvtTD4CyAPEJNBYNn5_bGlNIbQ1-mTtQArtAlYtJz0.original.fullsize.png"/><br/><br/>and $x \in(-\infty, 1) \cup[2, \infty)$<br/>$\therefore \quad x \in(-\infty, 0] \cup[2, \infty)$<br/>(C) $f(\theta)=\left|\begin{array}{ccc}1 &amp; \tan \theta &amp; 1 \\ -\tan \theta &amp; 1 &amp; \tan \theta \\ -1 &amp; -\tan \theta &amp; 1\end{array}\right|$<br/>$$<br/>\begin{aligned}<br/>R_1 &amp; \rightarrow R_1+R_3, \\<br/>f(\theta) &amp; =\left|\begin{array}{ccc}<br/>0 &amp; 0 &amp; 2 \\<br/>-\tan \theta &amp; 1 &amp; \tan \theta \\<br/>-1 &amp; -\tan \theta &amp; 1<br/>\end{array}\right| \\<br/>&amp; =2\left(\tan ^2 \theta+1\right)=2 \sec ^2 \theta \geq 2 \\<br/>f(\theta) &amp; \in[2, \infty)<br/>\end{aligned}<br/>$$<br/>$$<br/>\text { (D) } \begin{aligned}<br/>f(x) &amp; =x^{3 / 2}(3 x-10) ; x \geq 0 \\<br/>f^{\prime}(x) &amp; =x^{3 / 2} \cdot 3+\frac{3}{2} \cdot x^{1 / 2}(3 x-10) \\<br/>&amp; =3 x^{1 / 2}\left\{x+\frac{1}{2}(3 x-10)\right\}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{3}{2} x^{1 / 2}\{2 x+3 x-10\} \\<br/>&amp; =\frac{15}{2} x^{1 / 2}(x-2) \\<br/>&amp; \\<br/>\therefore \quad &amp; x \geq 2<br/>\end{aligned}<br/>$$</div>


title: match-the-statements-given-in-column-i-with-the-values-given-in-column-ii
slug: match-the-statements-given-in-column-i-with-the-values-given-in-column-ii-45449
tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">$$<br/>\text { Match the statements given in Column I with the values given in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/whLuoe5en1AhyDekxTeAe0GHJFttO-ylB8P6jVIZtEg.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) $\therefore|\mathbf{a}|=\sqrt{1+3}=2$ $|\mathbf{b}|=\sqrt{1+3}=2$ $|\mathbf{c}|=\sqrt{12}=2 \sqrt{3}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/RGdCdtVHJH1U8RZNrjguZqUUsjiCKu2UqCzCxQh8vWA.original.fullsize.png"/><br/><br/>Using cosine law,<br/>$$<br/>\begin{aligned}<br/>\cos C &amp; =\frac{|\mathbf{a}|^2+|\mathbf{b}|^2-|\mathbf{c}|^2}{2|\mathbf{a}||\mathbf{b}|} \\<br/>&amp; =\frac{4+4-12}{2 \times 2 \times 2}=\frac{-4}{8}=\frac{-1}{2} \\<br/>\Rightarrow \quad \angle C &amp; =120^{\circ}=\frac{2 \pi}{3}<br/>\end{aligned}<br/>$$<br/>(B)<br/>$$<br/>\begin{aligned}<br/>&amp; \int_a^b f(x) d x-3\left(\frac{x^2}{2}\right)_a^b=\left(a^2-b^2\right) \\<br/>&amp; \Rightarrow \int_a^b f(x) d x-\frac{3}{2}\left(b^2-a^2\right)=\left(a^2-b^2\right) \\<br/>&amp; \Rightarrow \int_a^b f(x) d x=\left(a^2-b^2\right)+\frac{3}{2}\left(b^2-a^2\right) \\<br/>&amp; =\frac{b^2-a^2}{2} \\<br/>&amp; \Rightarrow \int_a^b f(x) d x=\frac{b^2-a^2}{2} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>f(x)=x \Rightarrow f\left(\frac{\pi}{6}\right)=\frac{\pi}{6}<br/>$$<br/>(C)<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\pi^2}{\log _e 3} \int_{7 / 6}^{5 / 6} \sec (\pi x) d x \\<br/>&amp; \Rightarrow \frac{\pi^2}{\log _e 3}\left\{\frac{\log |\sec \pi x+\tan \pi x|}{\pi}\right\}_{7 / 6}^{5 / 6} \\<br/>&amp; \Rightarrow \frac{\pi}{\log 3}\left\{\log \left|\sec \frac{5 \pi}{6}+\tan \frac{5 \pi}{6}\right|\right. \\<br/>&amp; \left.\quad-\log \left|\sec \frac{7 \pi}{6}+\tan \frac{7 \pi}{6}\right|\right\} \\<br/>&amp; \Rightarrow \frac{\pi}{\log 3}\left\{\log |\sqrt{3}|-\log \left|\frac{1}{\sqrt{3}}\right|\right\} \\<br/>&amp; \Rightarrow \frac{\pi}{\log 3}\{\log 3\}=\pi<br/>\end{aligned}<br/>$$<br/><br/>(D) $$<br/>\begin{aligned}<br/>&amp; \left|\arg \frac{1}{(1-z)}\right|, \text { for }|z|=1 \\<br/>&amp; \Rightarrow\left|\arg (1-z)^{-1}\right| \\<br/>&amp; \Rightarrow|-\arg (1-z)| \Rightarrow|\arg (1-z)|<br/>\end{aligned}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/HSJMw5sIZCbFDqS37hzW91eRYY39SwiowxyUXRxah4U.original.fullsize.png"/><br/><br/>From figure, $\arg (z-1)$ is maximum $=\pi$.</div>


title: match-the-statements-in-column-i-with-the-properties-column-ii
slug: match-the-statements-in-column-i-with-the-properties-column-ii-19904
tags: ['Mathematics', 'Circle', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the statements in Column I with the properties Column II. <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/WqEVKx8z2jQTeJCQ2HtKkkAVDBSMLgrVvfEfm8t3x-M.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) When two circles are intersecting they have a common normal and common tangent.<br/>(B) Two mutually extemal circles have a common normal and common tangent.<br/>(C) When one circle lies inside of other, then they have a common normal but no common tangent.<br/>(D) Two branches of a hyperbola have a common normal but no common tangent.</div>


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
match-the-statements-of-column-i-with-these-in-column-ii-note-here-z-takes-values-in-the-complex-plane-and-im-z-and-re-z-denote-respectively-the-imagi

slug:
match-the-statements-of-column-i-with-these-in-column-ii-note-here-z-takes-values-in-the-complex-plane-and-im-z-and-re-z-denote-respectively-the-imagi-35042

question:
<div class="question">Match the statements of Column I with these in Column II.<br/>[Note : Here $z$ takes values in the complex plane and $\operatorname{Im}(z)$ and $\operatorname{Re}(z)$ denote respectively, the imaginary part and the real part of $z$ ]<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/QRq94L5Vcpqg1gnxhabTl1RleIIK8yieHzlu7WztlPU.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) $z$ is equidistant from the points $i|z|$ and $-i|z|$, whose perpendicular bisector is $\operatorname{Im}(z)=0$.<br/>(B) Sum of distance of $z$ from $(4,0)$ and $(-4,0)$ is a constant 10 , hence locus of $z$ is ellipse with semi-major axis 5 and focus at $(\pm 4,0), a e=4$.<br/>$$<br/>\therefore \quad e=\frac{4}{5}<br/>$$<br/>(C) $|z| \leq|w|+\left|\frac{1}{w}\right|=\frac{5}{2} &lt; 3$<br/>(D) $|z| \leq|w|+\left|\frac{1}{w}\right|=2$<br/>$$<br/>\therefore \quad \operatorname{Re}(z) \leq|z| \leq 2<br/>$$</div>


title: match-the-statements-of-column-i-with-values-of-column-ii
slug: match-the-statements-of-column-i-with-values-of-column-ii-82791
tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">$$<br/>\text { Match the statements of Column I with values of Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Kor3d9nZEDMiXJlMc071VJTT0cxHG3RIi9V6AwRthlk.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) Equation of the line passing through origin is<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{x}{a}=\frac{y}{b}=\frac{z}{c} \\<br/>&amp; \therefore \quad\left|\begin{array}{ccc}<br/>2 &amp; 1 &amp; -1 \\<br/>1 &amp; -2 &amp; 1 \\<br/>a &amp; b &amp; c<br/>\end{array}\right|=0 \\<br/>&amp; \Rightarrow \quad a(-1)-b(3)+c(-5)=0 \\<br/>&amp; \Rightarrow \quad-a-3 b-5 c=0 \\<br/>&amp; \Rightarrow \quad a+3 b+5 c=0 \\<br/>&amp; \text { Also, } \quad\left|\begin{array}{ccc}<br/>\frac{8}{3} &amp; -3 &amp; 1 \\<br/>2 &amp; -1 &amp; 1 \\<br/>a &amp; b &amp; c<br/>\end{array}\right|=0<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad a(-2)-b\left(\frac{2}{3}\right)+c\left(\frac{10}{3}\right)=0 \\<br/>&amp; \Rightarrow \quad 2 a+\frac{2 b}{3}-\frac{10 c}{3}=0 \\<br/>&amp; 3 a+b-5 c=0 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>From Eqs. (i) and (ii),<br/>$$<br/>\begin{aligned}<br/>\frac{a}{-20} &amp; =\frac{b}{20}=\frac{c}{-8} \\<br/>\frac{a}{5} &amp; =\frac{b}{-5}=\frac{c}{4}<br/>\end{aligned}<br/>$$<br/>Equation of line is<br/>$$<br/>\frac{x}{5}=\frac{y}{-5}=\frac{z}{4}=\lambda \text { (say) }<br/>$$<br/>Also, $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+1}{1}=k_1$ (say)<br/>Now, $\frac{x-\frac{8}{3}}{2}=\frac{y+3}{-1}=\frac{z-1}{1}=k_2$ (say)<br/>Point on (iii) is $(5 \lambda,-5 \lambda,+4 \lambda)$ Point on (iv) is<br/><br/>$$<br/>\left(2+k_1, 1-2 k_1,-1+k_1\right)<br/>$$<br/>Point on $(\mathrm{v})$ is<br/>$$<br/>\left(\frac{8}{3}+2 k_2,-3-k_2, 1+k_2\right)<br/>$$<br/>On solving, $2+k_1+1-2 k_1=0$<br/>$$<br/>-k_1+3=0<br/>$$<br/>$$<br/>\begin{aligned}<br/>k_1 &amp; =3 \\<br/>P &amp; \equiv(5,-5,2)<br/>\end{aligned}<br/>$$<br/>Again, for $Q$<br/>$$<br/>\begin{array}{r}<br/>\frac{8}{3}+2 k_2-3-k_2=0 \\<br/>k_2-\frac{1}{3}=0 \\<br/>k_2=\frac{1}{3} \\<br/>Q \equiv\left(\frac{10}{3}, \frac{-10}{3}, \frac{4}{3}\right)<br/>\end{array}<br/>$$<br/>$$<br/>\text { Now, } \begin{aligned}<br/>P Q &amp; =\sqrt{\left(\frac{5}{3}\right)^2+\left(\frac{5}{3}\right)^2+\left(\frac{2}{3}\right)^2} \\<br/>&amp; =\frac{\sqrt{54}}{3}<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>P Q^2=d^2=\frac{54}{9}=6<br/>$$<br/>(B) $\tan ^{-1}\left(\frac{x+3-x+3}{1+\left(x^2-9\right)}\right)=\tan ^{-1}\left(\frac{3}{4}\right)$<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow &amp; &amp; \frac{6}{x^2-8} &amp; =\frac{3}{4} \\<br/>\Rightarrow &amp; &amp; 3 x^2 &amp; =48 \\<br/>\Rightarrow &amp; x &amp; =\pm 4<br/>\end{array}<br/>$$<br/>(C)<br/>$$<br/>\begin{aligned}<br/>&amp; (\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}) \cdot\left(\overrightarrow{\mathbf{b}}+\frac{\overrightarrow{\mathbf{a}}-\mu \overrightarrow{\mathbf{b}}}{4}\right)=0 \\<br/>&amp; \Rightarrow \quad(\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}) \cdot(4 \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{a}}-\mu \overrightarrow{\mathbf{b}})=0 \\<br/>&amp; \text { Also, } 2\left|\overrightarrow{\mathbf{b}}+\frac{\overrightarrow{\mathbf{a}}-\mu \overrightarrow{\mathbf{b}}}{4}\right|=|\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}| \\<br/>&amp; \Rightarrow \quad 2\left|\frac{(4-\mu) \overrightarrow{\mathbf{b}}^2-\overrightarrow{\mathbf{a}}^2=0}{4}\right|=|\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}|<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \frac{(4-\mu)^2 \overrightarrow{\mathbf{b}}^2}{4}+\frac{\overrightarrow{\mathbf{a}}^2}{4}=\overrightarrow{\mathbf{b}}^2+\overrightarrow{\mathbf{a}}^2 \\<br/>&amp; \Rightarrow \quad \frac{3 \overrightarrow{\mathbf{a}}^2}{4}=\frac{(4-\mu)^2-4}{4} \cdot \overrightarrow{\mathbf{a}}^2 \\<br/>&amp; 3 \overrightarrow{\mathbf{a}}^2=C(4-\mu)^2-4 \overrightarrow{\mathbf{b}}^2<br/>\end{aligned}<br/>$$<br/>From Eqs. (i) and (ii)<br/>$$<br/>\begin{aligned}<br/>&amp; 3(4-\mu)=(4-\mu)^2-4 \\<br/>&amp; (4-\mu)^2-3(4-\mu)-4=0 \\<br/>&amp; \Rightarrow \quad \mu=0,5 \\<br/>&amp; \mu=5 \text { is not admissible. }<br/>\end{aligned}<br/>$$<br/>(D) $f(0)=9$,<br/>$$<br/>\begin{aligned}<br/>&amp; f(x)=\frac{\sin \left(\frac{9 x}{2}\right)}{\sin \frac{x}{2}} \\<br/>&amp; =\left(3-4 \sin ^2 \frac{x}{2}\right)\left(3-4 \sin ^2 \frac{3 x}{2}\right)<br/>\end{aligned}<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; =9-12 \sin ^2 \frac{x}{2}-12 \sin ^2 \frac{3 x}{2} \\<br/>&amp; \quad+16 \sin ^2 \frac{x}{2} \cdot \sin ^2 \frac{3 x}{2} \\<br/>&amp; =9-6(1-\cos x)-6(1-\cos 3 x) \\<br/>&amp; \quad+4(1-\cos x)(1-\cos 3 x) \\<br/>&amp; =1+6 \cos x+6 \cos 3 x-4 \cos x \\<br/>&amp; \quad-4 \cos 3 x+4 \cos x \cos 3 x \\<br/>&amp; \text { Let } \quad I=\frac{2}{\pi} \int_{-\pi}^\pi \frac{\sin \frac{9 x}{2}}{2} d x \\<br/>&amp; =\frac{4}{\pi} \int_0^\pi 1+2 \cos x+2 \cos 3 x \\<br/>&amp; =\frac{4}{\pi} \times \pi \quad+2(\cos 4 x+\cos x) d x \\<br/>&amp; =4<br/>\end{aligned}<br/>$$</div>


title: match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-1
slug: match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-1-50872
tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the statements/expressions given in Column I with the values given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OiiFrlYsfzglw0nz5y3f0pvY8jShZJrQWZJzU2zh_l0.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(a) Let $f(x)=x e^{\sin x}-\cos x$ $\overrightarrow{f^{\prime}}(x)=e^{\sin x}+x e^{\sin x} \cos x+\sin x \geq 0$ For interval $x \in\left(0, \frac{\pi}{2}\right)$, $f$ is strictly increasing.<br/>$\therefore \quad f(0)=-1$ and $f\left(\frac{\pi}{2}\right)=\frac{\pi}{2} e \Rightarrow$ one solution<br/>(b) Since, $\left|\begin{array}{lll}k &amp; 4 &amp; 1 \\ 4 &amp; k &amp; 2 \\ 2 &amp; 2 &amp; 1\end{array}\right|=0$ $\Rightarrow k(k-4)-4(0)+1(8-2 k)=0$ $\Rightarrow \quad k^2-6 k+8=0$ $\Rightarrow \quad k=2,4$<br/>(c) Let $y=|x-1|+|x-2|+|x+1|$ $+|x+2|$<br/>For solutions, $4 k \geq 6 \Rightarrow k \geq 3 / 2$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9-zLMBos4B1Ewe3LfmvgASEnGCtnhtCQTIuDD_3VqJk.original.fullsize.png"/><br/><br/>$\therefore$ Integer values of $k$ are $2,3,4,5$.<br/>(d) Given, $\frac{d y}{d x}=y+1$<br/>$\Rightarrow \quad \ln |(y+1)|=x+C$ $\Rightarrow \ln 2=C \Rightarrow \ln |y+1|=x+\ln 2$<br/>Put $x=\ln 2$<br/>$\therefore \quad \ln (y+1)=\ln 2+\ln 2=\ln 4$<br/>$y+1=4 \Rightarrow y=3$</div>


title: match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii
slug: match-the-statementsexpressions-given-in-column-i-with-the-values-given-in-column-ii-83406
tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the statements/expressions given in Column I with the values given in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/X5zrl_D_5t94xmdVslBxnMcxIhCVjgRpxOOFvHdKn0o.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(a) $2 \sin ^2 \theta+\sin ^2 2 \theta=2$<br/>$\Rightarrow \quad \sin ^2 2 \theta=2 \cos ^2 \theta$<br/>$\Rightarrow \quad 4 \sin ^2 \theta \cos ^2 \theta=2 \cos ^2 \theta$<br/>$\Rightarrow \quad \cos ^2 \theta=0$ or $\sin ^2 \theta=\frac{1}{2}$<br/>$\Rightarrow \quad \cos \theta=0$ or $\sin \theta=\pm \frac{1}{\sqrt{2}}$<br/>$\Rightarrow \quad \theta=\pm \frac{\pi}{4}$ or $\frac{\pi}{2}$<br/>(b) $f(x)=\left[\frac{6 x}{\pi}\right] \cos \left[\frac{3 x}{\pi}\right]$<br/>Possible points of discontinuity of $\left[\frac{6 x}{\pi}\right]$ are<br/>$$<br/>\begin{gathered}<br/>\frac{6 x}{\pi}=n, n \in I \\<br/>\Rightarrow \quad x=\frac{n \pi}{6} \Rightarrow x=\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \pi \\<br/>\lim _{x \rightarrow \pi^{-} / 6} f(x)=0 \cos 0=0 \\<br/>\lim _{x \rightarrow \pi^{+} / 6} f(x)=1 \cos 0=1<br/>\end{gathered}<br/>$$<br/>$\therefore$ Discontinuous at $x=\frac{\pi}{6}$.<br/>Similarly, discontinuous at $x=\frac{\pi}{3}, \frac{\pi}{2}, \pi$.<br/>(c) Here, $V=\left\|\begin{array}{lll}1 &amp; 1 &amp; 0 \\ 1 &amp; 2 &amp; 0 \\ 1 &amp; 1 &amp; \pi\end{array}\right\|=\pi$ cubic unit<br/>(d) Given, $\mathbf{a}+\mathbf{b}+\sqrt{3} \mathbf{c}=\mathbf{0}$<br/>$$<br/>\begin{array}{ll}<br/>\Rightarrow &amp; \mathbf{a}+\mathbf{b}=-\sqrt{3} \mathbf{c} \\<br/>\Rightarrow &amp; |\mathbf{a}+\mathbf{b}|^2=\mid \sqrt{3} \mathbf{c}^2<br/>\end{array}<br/>$$<br/><br/>$$<br/>\begin{array}{lrl}<br/>\Rightarrow &amp; a^2+b^2+2 \mathbf{a} \cdot \mathbf{b}=3 c^2 \\<br/>\Rightarrow &amp; 2+2 \cos \theta=3 \\<br/>\Rightarrow &amp; \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}<br/>\end{array}<br/>$$</div>


title: match-the-statementsexpressions-in-column-i-with-the-open-intervals-in-column-ii-1
slug: match-the-statementsexpressions-in-column-i-with-the-open-intervals-in-column-ii-1-45958
tags: ['Mathematics', 'Definite Integration', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">$$<br/>\text { Match the statements/expressions in Column I with the open intervals in Column II. }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YSS353722UU7V-kPU9gVFl_hn_aTbFqHT6L7rH7xHFg.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(a) Given, $(x-3)^2 \cdot y^{\prime}+y=0$<br/>$$<br/>\begin{array}{rlrl}<br/>\Rightarrow &amp; \frac{d y}{d x} &amp; =-\frac{y}{(x-3)^2} \\<br/>\Rightarrow &amp; &amp; \int \frac{d y}{y} &amp; =-\int \frac{d x}{(x-3)^3} \\<br/>\Rightarrow &amp; &amp; \ln y &amp; =\frac{1}{(x-3)}+\ln C \\<br/>\Rightarrow &amp; &amp; y &amp; =C e^{\frac{1}{x-3}}, C \neq 0<br/>\end{array}<br/>$$<br/>$\therefore$ Domain of $y$ is $x \in R-\{3\}$<br/>Aliter Given differential equation is homogeneous linear differential equation and has $x=3$ as a singular point, hence $x=3$ cannot be in domain of solution.<br/>(b) Let $I=\int_1^5(x-1)(x-2)(x-3)(x-4)$ $(x-5) d x$<br/>Let $x-3=t \Rightarrow d x=d t$<br/>$\therefore I=\int_{-2}^2(t+2)(t+1) t(t-1)(t-2) d t$<br/>$\because$ Integrand is an odd function.<br/>$\therefore \quad I=0$<br/>Aliter Let<br/>$I=\int_1^5(x-1)(x-2)(x-3)(x-4)$<br/>$(x-5) d x$<br/>Using, $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$ $I=\int_1^5(5-x)(4-x)(3-x)(2-x)$ $(1-x) d x$<br/>On adding Eqs. (i) and (ii) we get 2I $=0 \Rightarrow I=0$<br/>(c) Let $f(x)=\cos ^2 x+\sin x$<br/>$\Rightarrow f^{\prime}(x)=-2 \cos x \sin x+\cos x$ $=\cos x(1-2 \sin x)=0$ (say)<br/>Sign scheme for first derivative<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TLmS4uc2BFsrzGpUJe2DJ2LMHFf6yxl77ASJG4WjUWY.original.fullsize.png"/><br/><br/>Points of local maxima are $\frac{\pi}{6}, \frac{5 \pi}{6}$.<br/>Aliter $y=\cos ^2 x+\sin x$<br/>$$<br/>\Rightarrow \quad y=\frac{5}{4}-\left(\sin x-\frac{1}{2}\right)^2<br/>$$<br/>For $y$ to be maximum,<br/>$$<br/>\begin{aligned}<br/>&amp; \left(\sin x-\frac{1}{2}\right)^2=0 \Rightarrow \sin x=\frac{1}{2} \\<br/>\Rightarrow \quad &amp; x=n \pi+(-1)^n \frac{\pi}{6}, n \in I<br/>\end{aligned}<br/>$$<br/>(d) Let $y=\tan ^{-1}(\sin x+\cos x)$<br/>$$<br/>\Rightarrow \quad \frac{d y}{d x}=\frac{\cos x-\sin x}{1+(\sin x+\cos x)^2}<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/QXVtSrn6Dj_MC10OOrVKBH-PujNlc9ggm4Rte2x4G1M.original.fullsize.png"/><br/><br/>Clearly, by graph, $\cos x&gt;\sin x$ is true for option(s).</div>


title: match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-column-ii
slug: match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-in-column-ii-59788
tags: ['Mathematics', 'Basic of Mathematics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NV6MxxlCrdeWHLs9r4Y0EOApcJx3jyAt7MA2okb20P0.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) Let $y=\frac{x^2+2 x+4}{x+2}$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow x^2+(2-y) x+(4-2 y)=0 \\<br/>&amp; \Rightarrow \quad(2-y)^2-4(4-2 y) \geq 0 \\<br/>&amp; \Rightarrow \quad y^2+4 y-12 \geq 0 \\<br/>&amp; \Rightarrow \quad y \leq-6, y \geq 2 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>$\therefore$ Minimum value of $y$ is 2 .<br/>(B) Since, $(A+B)(A-B)=(A-B)(A+B)$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \quad A^2-A B+B A-B^2=A^2+A B-B A-B^2 \\<br/>&amp; \Rightarrow \quad A B=B A \\<br/>&amp; \text { and } \quad(A B)^t=(-1)^k A B \\<br/>&amp; \Rightarrow \quad B^t A^t=(-1)^k A B \\<br/>&amp; \Rightarrow \quad-B A=(-1)^k A B \quad\left[\because B^t=-B, A^t=A\right] \\<br/>&amp; \Rightarrow \quad B A=(-1)^{k+1} A B \\<br/>&amp; \Rightarrow \quad(-1)^{k+1}=1 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/><br/>$\therefore k+1$ is even or $k$ is odd.<br/>(C) $1 &lt; 2^{\left(-k+3^{-a}\right)} &lt; 2 \Rightarrow 0 &lt; -k+3^{-a} &lt; 1$<br/>Given, $a=\log _3 \log _3 2 \Rightarrow 3^a=\log _3 2$<br/>$$<br/>\begin{array}{lc}<br/>\Rightarrow &amp; 3^{-a}=\log _2 3 \\<br/>\therefore &amp; k &lt; \log _2 3 &lt; 2 \\<br/>\text { and } &amp; 1+k&gt;\log _2 3&gt;1 \Rightarrow k&gt;0<br/>\end{array}<br/>$$<br/>From Eqs. (ii) and (iii), $0 &lt; k &lt; 2 \Rightarrow k=1$<br/>$[\because k$ is an integer]<br/>$$<br/>\begin{array}{rlrl}<br/>\text { (D) } &amp; \sin \theta &amp; =\cos \phi \\<br/>\Rightarrow &amp; \cos \left(\frac{\pi}{2}-\theta\right) &amp; =\cos \phi \\<br/>\Rightarrow &amp; \frac{\pi}{2}-\theta &amp; =2 n \pi \pm \phi, n \in Z \\<br/>\Rightarrow &amp; &amp; \theta \pm \phi-\frac{\pi}{2} &amp; =-2 n \pi, n \in Z \\<br/>\Rightarrow &amp; &amp; \frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right) &amp; =-2 n, n \in Z<br/>\end{array}<br/>$$</div>


title: match-the-transformation-in-column-i-with-appropriate-options-in-column-ii
slug: match-the-transformation-in-column-i-with-appropriate-options-in-column-ii-61290
tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">Match the transformation in Column I with appropriate options in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/TQElWvj8uVhqy8L6g1_0H549TaOc7nepzl4wiNxtBnw.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">(A) $\mathrm{CO}_2(\mathrm{~s}) \longrightarrow \mathrm{CO}_2(\mathrm{~g})$<br/>It is just a phase transition (sublimation) as no chemical change has occurred. Sublimation is always endothermic. Product is gas, more disordered, hence $\Delta s$ is positive.<br/>(B) $\mathrm{CaCO}_3(\mathrm{~s}) \longrightarrow \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_2(\mathrm{~g})$<br/>It is a chemical decomposition, not a phase change. Thermal decomposition occur at the expense of energy, hence endothermic. Product contain a gaseous species, hence, $\Delta S&gt;0$.<br/>(C) $2 \mathrm{H} \longrightarrow \mathrm{H}_2(\mathrm{~g})$<br/>A new $\mathrm{H}-\mathrm{H}$ covalent bond is being formed, hence, $\Delta H &lt; 0$.<br/>Also, product is less disordered than reactant, $\Delta S &lt; 0$.<br/>(D) Allotropes are considered as different phase, hence $\mathrm{P}_{\text {(white, solid) }} \rightarrow \mathrm{P}_{\text {(red, solid) }}$ is a phase transition as well as allotropic change.<br/>Also, red phosphorus is more ordered than white phosphorus, $\Delta S &lt; 0$.</div>


title: mathematics-application-of-derivatives-jee-advanced-64b169cef23ac7d519f9aea2
slug: mathematics-application-of-derivatives-jee-advanced-64b169cef23ac7d519f9aea2-17204
tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5J9jOFz5SPHXym-2E7yNW6FpxEcbDxNYH9Z-gh2VeMk.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$g(x)$ has local maxima at $x=1+\log _e 2$ and local minima at $x=e$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$f(x)$ has local maxima at $x=1$, and local minima at $x=2$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$g(x)$ has no local minima<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$f(x)$ has no local maxima</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$g(x)$ has local maxima at $x=1+\log _e 2$ and local minima at $x=e$<br/>, <br/>$f(x)$ has local maxima at $x=1$, and local minima at $x=2$<br/></span> </div>

answer:
<div class="solution">Here, $f(x)=\left\{\begin{array}{cc}e^x, &amp; 0 \leq x \leq 1 \\ 2-e^{x-1}, &amp; 1 &lt; x \leq 2 \\ x-e, &amp; 2 &lt; x \leq 3\end{array}\right.$ and $\quad g(x)=\int_0^x f(t) d t$<br/>$$<br/>\Rightarrow \quad g^{\prime}(x)=f(x)<br/>$$<br/>where, $g^{\prime}(x)=0$<br/>$\Rightarrow \quad x=1+\log _e 2$<br/>and $\quad x=e$.<br/>Also, $\quad g(x)=\left\{\begin{array}{cc}-e^{x-1}, &amp; 1 &lt; x \leq 2 \\ 1, &amp; 2 &lt; x \leq 3\end{array}\right.$<br/>$g\left(1+\log _e 2\right)=-e^{\log _e 2} &lt; 0$, hence at $x=1+\log _e 2, g(x)$ has a local maximum.<br/>Also, $g(e)=1&gt;0$, hence at $x=e, g(x)$ has a local minima.<br/>$\because f(x)$ is discontinuous at $x=1$, then we get<br/>Local maxima at $x=1$ and local minima at $x=2$.</div>


tags: ['Chemistry', 's Block Elements', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
mgso-4-on-reaction-with-nh-4-oh-and-na-2-hpo-4-forms-a-white-crystalline-precipitate-what-is-its-formula

slug:
mgso-4-on-reaction-with-nh-4-oh-and-na-2-hpo-4-forms-a-white-crystalline-precipitate-what-is-its-formula-97515

question:
<div class="question">$\mathrm{MgSO}_4$ on reaction with $\mathrm{NH}_4 \mathrm{OH}$ and $\mathrm{Na}_2 \mathrm{HPO}_4$ forms a white crystalline precipitate. What is its formula?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Mg}_3\left(\mathrm{PO}_4\right)_2$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{MgCl}_2 \cdot \mathrm{MgSO}_4$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{MgSO}_4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4$<br/></span> </div>

answer:
<div class="solution">$\mathrm{MgSO}_4$ on reaction with $\mathrm{Na}_2 \mathrm{HPO}_4$ (disodium hydrogen phosphate) in presence of $\mathrm{NH}_4 \mathrm{OH}$ to give white precipitate of magnesium ammonium phosphate $\left(\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4\right)$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/MpRey9k-Y2s158WLfD5sHyIYsJf5OFY_P-DHgLZbHkM.original.fullsize.png"/><br/></div>


title: most-advanced-invertebrates-are
slug: most-advanced-invertebrates-are-36788
question: <div class="question">Most advanced invertebrates are</div>
tags: ['Biology', 'Animal Kingdom', 'JIPMER', 'JIPMER 2019']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Arthropoda</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Annelida</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Mollusca</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Cephalopoda</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Cephalopods such as octopus, cuttlefish and squid (coleoids) are the most advanced invertebrates among the given options. They belong to class-Cephalopoda of phylum - Mollusca. These organisms have a prominent head and a set arms or tentacles. They are widely regarded as the most intelligent of invertebrates and have well-developed senses and larger brains. Their nervous system is the most complex of all invertebrates.</div>


tags: ['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1-1

slug:
most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1-1-56498

question:
<div class="question">Most materials have the refractive index, $n&gt;1$. So, when a light ray from air enters a naturally occurring material, then by Snell's
<br/> <br/>law, $\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}$, it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, $\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}$, where $c$ is the speed of electromagnetic waves in vacuum, $v$ its speed in the medium, $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability of the medium respectively. In normal materials, both $\varepsilon_{r}$ and $\mu_{r}$, are positive, implying positive $n$ for the medium. When both $\varepsilon_{r}$ and $\mu_{r}$ are negative, one must choose the negative root of $n$. Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since $n$ is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
<br/> <br/><strong> Question:</strong> Choose the correct statement.</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The speed of light in the meta-material is $v=\frac{c}{|n|}$</span> </div>

answer:
<div class="solution">Speed of light in a medium, $V=\frac{C}{n}$ $n$ for meta-material $=|n|$
<br/> <br/>$\therefore \quad$ Speed of light in the meta-material $V=\frac{C}{|n|}$.</div>


tags: ['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1

slug:
most-materials-have-the-refractive-index-n-1-so-when-a-light-ray-from-air-enters-a-naturally-occurring-material-then-by-snells-law-s-i-n-2-s-i-n-1-n-1-19098

question:
<div class="question">Most materials have the refractive index, $n&gt;1$. So, when a light ray from air enters a naturally occurring material, then by Snell's
<br/> <br/>law, $\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}$, it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, $\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}$, where $c$ is the speed of electromagnetic waves in vacuum, $v$ its speed in the medium, $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability of the medium respectively. In normal materials, both $\varepsilon_{r}$ and $\mu_{r}$, are positive, implying positive $n$ for the medium. When both $\varepsilon_{r}$ and $\mu_{r}$ are negative, one must choose the negative root of $n$. Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since $n$ is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
<br/> <br/><strong> Question:</strong> For light incident from air on a meta-material, the appropriate ray diagram is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fSliPo124VmuV8rBTNz1ynTcNkB_PxMLqSjUjyTw938.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4dqL3uaNGI97arahJ7edochFqZbIrEY5uXyLQq7Hs74.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ldjy3BnC7u5EUqgGGiUss9jHaX9uYJHrNPVHSadDD-M.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jzYvCJqcyvPF0qxXJbmYQaWxYTEp97QbOjNL57kZFcc.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ldjy3BnC7u5EUqgGGiUss9jHaX9uYJHrNPVHSadDD-M.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Let $n_{1}=$ refractive index of air and $n_{2}=$ refractive index of meta material which is negative
<br/> <br/>From Snell's law, $\frac{n_{2}}{n_{1}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$
<br/> <br/>$\because \quad n_{2}$ negative $\therefore \theta_{2}$ isalso negative hence graph (c) is correct.</div>


source_db: MarksBatch2_P2.db
title:
native-silver-metal-forms-a-water-soluble-complex-with-a-dilute-aqueous-solution-of-nacn-in-the-presence-of-1

slug:
native-silver-metal-forms-a-water-soluble-complex-with-a-dilute-aqueous-solution-of-nacn-in-the-presence-of-1-33128

question:
<div class="question">Native silver metal forms a water soluble complex with a dilute aqueous solution of $\mathrm{NaCN}$ in the presence of</div>

tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']

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

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

answer:
<div class="solution">A water soluble complex of silver with a dilute aqueous solution of $\mathrm{NaCN}$ is sodium argentocyanide. In the cyanide process, the native form is crushed and treated with $0.1-0.2 \%$ solution of $\mathrm{NaCN}$ and aerated.<br/>$$<br/>4 \mathrm{Ag}+8 \mathrm{NaCN}+2 \mathrm{H}_2 \mathrm{O}+\mathrm{O}_2 \longrightarrow 4 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_2\right]+4 \mathrm{NaOH}<br/>$$<br/>Argentocyanide is soluble. Further, metals are recovered from the complex by reduction with zinc</div>


tags: ['Chemistry', 'Coordination Compounds', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
nicl-2-p-c-2-h-5-2-c-6-h-5-2-exhibits-temperature-dependent-magnetic-behaviour-paramagneticdiamagnetic-the-coordination-geometries-of-ni-2-in-the-para

slug:
nicl-2-p-c-2-h-5-2-c-6-h-5-2-exhibits-temperature-dependent-magnetic-behaviour-paramagneticdiamagnetic-the-coordination-geometries-of-ni-2-in-the-para-82493

question:
<div class="question">$\mathrm{NiCl}_{2}\left\{\mathrm{P}\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)\right\}_{2}$ exhibits temperature depend-ent magnetic behaviour (paramagnetic/diamagnetic). The coordination geometries of $\mathrm{Ni}^{2+}$ in the paramagnetic and diamagnetic states are respectively</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">tetrahedral and square planar</span> </div>

answer:
<div class="solution">In both states (paramagnetic and diamagnetic) of the given complex, $\mathrm{Ni}$ exists as $\mathrm{Ni}^{2+}$ whose electronic configuration is $[\mathrm{Ar}] 3 d^{8} 4 s^{0}$.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Y0qPb6T-E96_GJJk5j2b10GNt-stG3VzU8NSyV44cF0.original.fullsize.png"/><br/> <br/> <br/>In the above paramagnetic state the geometry of the complex is $s p^{3}$ giving tetrahedral geometry.
<br/> <br/>The diamagnetic state is achieved by pairing of electrons in $3 d$ orbital.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IY3jCGJMIjP0XjucA1Crn0M1Un_UO6SeFvtnewMMv0o.original.fullsize.png"/><br/> <br/> <br/>Thus, the geometry of the complex will be $d s p^{2}$ giving square planar geometry.</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
one-indian-and-four-american-men-and-their-wives-are-to-be-seated-randomly-around-a-circular-table-then-the-conditional-probability-that-the-indian-ma

slug:
one-indian-and-four-american-men-and-their-wives-are-to-be-seated-randomly-around-a-circular-table-then-the-conditional-probability-that-the-indian-ma-40154

question:
<div class="question">One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{2}{5}$<br/></span> </div>

answer:
<div class="solution">Let $E=$ event when each American man is seated adjacent to his wife and $A=$ event when Indian man is seated adjacent to his wife.<br/>Now, $n(A \cap E)=(4 !) \times(2 !)^5$<br/>Even when each American man is seated adjacent to his wife.<br/>Again<br/>$$<br/>\begin{aligned}<br/>n(E) &amp; =(5 !) \times(2 !)^4 \\<br/>P\left(\frac{A}{E}\right) &amp; =\frac{n(A \cap E)}{n(E)} \\<br/>&amp; =\frac{(4 !) \times(2 !)^5}{(5 !) \times(2 !)^4}=\frac{2}{5}<br/>\end{aligned}<br/>$$<br/>$$<br/>\Rightarrow \quad P\left(\frac{A}{E}\right)=\frac{n(A \cap E)}{n(E)}<br/>$$<br/>ALITER<br/>Fixing four American couples and one Indian man in between any two couples, we have 5 different ways in which his wife can be seated, of which 2 cases are favourable. $\therefore$ Required probability $=\frac{2}{5}$.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy-1

slug:
one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy-1-14367

question:
<div class="question">One mole of a monatomic ideal gas is taken through a cycle $A B C D A$ as shown in the $p$-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5ZEJlNpluXbfGVD34o37DgTSv2X9t_fYuXv-lWGpHTg.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BtZmk11u2LxVWqhXGJSwF68nRVZSvKCI2ONt9M52A6I.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">Internal energy $\propto T \propto p V$<br/>This is because<br/>$$<br/>U=\frac{n f}{2} R T=\frac{f}{2} p V<br/>$$<br/>Here, $n=$ number of moles<br/>$f=$ degree of freedom<br/>If the product $p V$ increases, then internal energy will increase and if product decreases, the internal energy will decrease.<br/>Further, work is done on the gas, if volume of gas decreases. For heat exchange.<br/>$$<br/>Q=W+\Delta U<br/>$$<br/>Work done is area under $p-V$ graph. If volume increases work done by gas is positive and if volume decreases work done by gas is negative. Further $\Delta U$ is positive if product of $p V$ is increasing and $N U$ is negative, if product of $p V$ is decreasing. If heat is taken by the gas $Q$ is positive and if heat is lost by the gas $Q$ is negative.<br/>Keeping the above points in mind the answer to this question is as under.<br/>(A) $\rightarrow(\mathrm{p}, \mathrm{r}, \mathrm{t})$<br/>(B) $\rightarrow(\mathrm{p}, \mathrm{r})$<br/>(C) $\rightarrow$ (q, s)<br/>(D) $\rightarrow(\mathrm{r}, \mathrm{t})$<br/>Analysis of Question<br/>(i) Calculation wise, question is slightly lengthy. Otherwise question is theory based and simple.<br/>(ii) In process $D A$,<br/>$\begin{array}{rlrl} &amp; &amp; p_A V_A &amp; =p_D V_D \\ \therefore &amp; &amp; T_A &amp; =T_D \\ \text { or } &amp; \Delta U &amp; =0\end{array}$<br/>Further, volume of gas is decreasing. Therefore, work is done on the gas or work done by gas is negative. Therefore, $Q$ is negative or heat is lost.<br/>(iii) This question covers almost all the concepts of first law of thermodynamics.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy

slug:
one-mole-of-a-monatomic-ideal-gas-is-taken-through-a-cycle-a-bc-d-a-as-shown-in-the-p-v-diagram-column-ii-gives-the-characteristics-involved-in-the-cy-19620

question:
<div class="question">One mole of a monatomic ideal gas is taken through a cycle $A B C D A$ as shown in the $p$-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5ZEJlNpluXbfGVD34o37DgTSv2X9t_fYuXv-lWGpHTg.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BtZmk11u2LxVWqhXGJSwF68nRVZSvKCI2ONt9M52A6I.original.fullsize.png"/><br/></div>

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

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

answer:
<div class="solution">Internal energy $\propto T \propto p V$<br/>This is because<br/>$$<br/>U=\frac{n f}{2} R T=\frac{f}{2} p V<br/>$$<br/>Here, $n=$ number of moles<br/>$f=$ degree of freedom<br/>If the product $p V$ increases, then internal energy will increase and if product decreases, the internal energy will decrease.<br/>Further, work is done on the gas, if volume of gas decreases. For heat exchange.<br/>$$<br/>Q=W+\Delta U<br/>$$<br/>Work done is area under $p-V$ graph. If volume increases work done by gas is positive and if volume decreases work done by gas is negative. Further $\Delta U$ is positive if product of $p V$ is increasing and $N U$ is negative, if product of $p V$ is decreasing. If heat is taken by the gas $Q$ is positive and if heat is lost by the gas $Q$ is negative.<br/>Keeping the above points in mind the answer to this question is as under.<br/>(A) $\rightarrow(\mathrm{p}, \mathrm{r}, \mathrm{t})$<br/>(B) $\rightarrow(\mathrm{p}, \mathrm{r})$<br/>(C) $\rightarrow$ (q, s)<br/>(D) $\rightarrow(\mathrm{r}, \mathrm{t})$<br/>Analysis of Question<br/>(i) Calculation wise, question is slightly lengthy. Otherwise question is theory based and simple.<br/>(ii) In process $D A$,<br/>$\begin{array}{rlrl} &amp; &amp; p_A V_A &amp; =p_D V_D \\ \therefore &amp; &amp; T_A &amp; =T_D \\ \text { or } &amp; \Delta U &amp; =0\end{array}$<br/>Further, volume of gas is decreasing. Therefore, work is done on the gas or work done by gas is negative. Therefore, $Q$ is negative or heat is lost.<br/>(iii) This question covers almost all the concepts of first law of thermodynamics.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
one-mole-of-an-ideal-gas-in-initial-state-a-undergoes-a-cyclic-process-a-bc-a-as-shown-in-the-figure-its-pressure-at-a-is-p-0-choose-the-correct-optio

slug:
one-mole-of-an-ideal-gas-in-initial-state-a-undergoes-a-cyclic-process-a-bc-a-as-shown-in-the-figure-its-pressure-at-a-is-p-0-choose-the-correct-optio-86909

question:
<div class="question">One mole of an ideal gas in initial state A undergoes a cyclic process $A B C A$, as shown in the figure. Its pressure at $A$ is $p_0$. Choose the correct option(s) from the following<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YOIwtV0RRE5ULb6-11MzP9TjZOlhttuirHvaljEJBUY.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>Internal energies at $A$ and $B$ are the same<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Work done by the gas in process $A B$ is $p_0 V_0 \ln 4$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Pressure at $C$ is $\frac{p_0}{4}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Temperature at $C$ is $\frac{T_0}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>Internal energies at $A$ and $B$ are the same<br/>, <br/>Work done by the gas in process $A B$ is $p_0 V_0 \ln 4$<br/></span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; T_A=T_B \quad \therefore U_A=U_B \\<br/>&amp; W_{A B}=(1)(R) T_0 \ln \left(\frac{V_f}{V_i}\right) \\<br/>&amp; =R T_0 \ln \left(\frac{4 V_0}{V_0}\right) \\<br/>&amp; =p_0 V_0 \ln (4)<br/>\end{aligned}<br/>$$<br/>Information regarding $p$ and $T$ at $C$ can not be obtained from the given graph. Unless it is mentioned that line $B C$ passes through origin or not.<br/>Hence, the correct options are (a) and (b).</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P2.db
title:
one-mole-of-an-ideal-gas-is-taken-from-a-to-b-along-two-paths-denoted-by-the-solid-and-the-dashed-lines-as-shown-in-the-graph-below-if-the-work-done-a

slug:
one-mole-of-an-ideal-gas-is-taken-from-a-to-b-along-two-paths-denoted-by-the-solid-and-the-dashed-lines-as-shown-in-the-graph-below-if-the-work-done-a-36079

question:
<div class="question">One mole of an ideal gas is taken from $a$ to $b$ along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is $W_s$ and that along the dotted line path is $W_d$, then the integer closest to the ratio $\frac{W_d}{W_s}$ is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/v5OJsPNE5bM9jw0nhHQppgggg6_GeY4sEcPU19pQ4wg.original.fullsize.png"/><br/></div>

answer:
<div class="solution">Solid line represents reversible isothermal process.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/sH7FLf3CnNk6SLizs2CpYGsL1ts_CnefDrRK8phgEt0.original.fullsize.png"/><br/><br/>So, work $W_s=-4 \times 0.5 \ln \left(\frac{5.5}{0.5}\right)=-2 \ln 11 \mathrm{~L}-\mathrm{atm}$<br/>Dotted line represents irreversible process<br/>So, work<br/>$$<br/>W_d=-\left\{4 \times 1.5+1 \times 1+\frac{2}{3} \times 2.5\right\}=-\left\{6+1+\frac{5}{3}\right\} \mathrm{L}-\mathrm{atm}=-\frac{26}{3} \mathrm{~L}-\mathrm{atm}<br/>$$<br/>So $\quad \frac{W_d}{W_s}=\frac{26}{3 \times 2 \ln 11} \approx 2$<br/>Energetics<br/>Conceptual<br/>III</div>


title: oogamous-type-of-sexual-reproduction-is-found-in
slug: oogamous-type-of-sexual-reproduction-is-found-in-28796
question: <div class="question">Oogamous type of sexual reproduction is found in</div>
tags: ['Biology', 'Plant Kingdom', 'NEET']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><em>Chlamydomonas</em></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><em>Volvox</em></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><em>Spirogyra</em></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Both (A) and (B)</span> </li> </ul>

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

answer:
<div class="solution">Oogamy is a form of anisogamy (heterogamy) in which the female gamete (e.g. egg cell) is significantly larger than the male gamete and is non-motile. Oogamy is the most advanced form of reproduction, with the egg cell or oogonium retained and fertilized on the parent plant.</div>


title: oxidation-states-of-the-metal-in-the-minerals-haematite-and-magnetite-respectively-are-2
slug: oxidation-states-of-the-metal-in-the-minerals-haematite-and-magnetite-respectively-are-2-13202
source_db: MarksBatch2_P2.db
question:
<div class="question">Oxidation states of the metal in the minerals haematite and magnetite, respectively, are</div>

tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>III in haematite and II, III in magnetite</span> </div>

answer:
<div class="solution">Haematite is $\mathrm{Fe}_2 \mathrm{O}_3$, in which oxidation number of iron is III. Magnetite is $\mathrm{Fe}_3 \mathrm{O}_4$ which is infact a mixed oxide (FeO $\cdot \mathrm{Fe}_2 \mathrm{O}_3$ ) hence, iron is present in both II and III oxidation state.</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-1

slug:
paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-1-21115

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/fDDj5QZVFCnhb-qmvfxHFAXCtadF-DLEJ6XXvXoBisE.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZC_3cEPYQINFO5Cg7nqDwPUu6gX5mXEbH9PTs8RhQ8c.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/zaY9cN1FZlaG5fXsOxFu9ViyfTWdk1k2YGaMr9bs6-I.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OZW0o_xQ6wXMVp-vyDqjXvJW_mxLO6sacRFZxJA9IZM.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZC_3cEPYQINFO5Cg7nqDwPUu6gX5mXEbH9PTs8RhQ8c.original.fullsize.png"/><br/></span> </div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-2

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Main']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys-3

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-a-carbonyl-compound-p-which-gives-positive-iodoform-test-undergoes-reaction-with-memgbr-followed-by-dehydration-to-give-an-olefin-q-ozonolys

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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
answer: <div class="solution">Clearly, point $E$ and $F$ satisfy the equations given in option (d).</div>
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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


tags: ['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Circle', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$(x-\sqrt{3})^2+(y-1)^2=1$</span> </div>

answer:
<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>


tags: ['Physics', 'Waves and Sound', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-a-dense-collection-of-equal-number-of-electrons-and-positive-ions-is-called-neutral-plasma-certain-solids-containing-fixed-positive-ions-sur-1

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Units and Dimensions', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-a-dense-collection-of-equal-number-of-electrons-and-positive-ions-is-called-neutral-plasma-certain-solids-containing-fixed-positive-ions-sur

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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">$P(X=3)=\frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6}=\frac{25}{216}$</div>
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">$P(X \geq 3)=\frac{5}{6} \cdot \frac{5}{6} \cdot 1=\frac{25}{36}$</div>
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2021 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Advanced', 'JEE Advanced 2021 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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-

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

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


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

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


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

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


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

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


tags: ['Physics', 'Oscillations', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Oscillations', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\sqrt{\frac{4 k}{3 M}}$</span> </div>

answer:
<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>


tags: ['Physics', 'Oscillations', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Oscillations', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Mechanical Properties of Fluids', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
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-

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
answer: <div class="solution">Explained in the beginning.</div>
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>


tags: ['Chemistry', 'Hydrocarbons', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

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


tags: ['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

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


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

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


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>446</span> </div>

answer:
<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>


tags: ['Chemistry', 'Electrochemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>193000</span> </div>

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


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-chemical-reactions-involve-interaction-of-atoms-and-molecules-a-large-number-of-atomsmolecules-approximately-6023-1-0-23-are-present-in-a-fe

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2016 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-an-evacuated-cylindrical-chamber-of-height-h-having-rigid-conducting-plates-at-the-ends-and-an-insulating-curved-surface-as-shown-i

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>2</span> </div>
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

answer:
<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>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

correct_option:
<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>

answer:
<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>


tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
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

slug:
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

question:
<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>

options:
<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>

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

answer:
<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>


