# questions

tags: ['Chemistry', 'Electrochemistry', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-cell-reaction-at-2-fe-s-o-2-g-4-h-a-q-2-fe-2-a-q-2-h-2-o-l-e-167-v-fe-2-1-0-3-m-p-o-2-01-atm-an-d-mathrmph3-t-h-ece-llp-o-t-e-n

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

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

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

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

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


tags: ['Biology', 'Plant Kingdom', 'NEET', 'NEET 2011 (Mains)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-four-statements-whether-they-are-correct-of-wrong-i-the-sporophyte-in-liverworts-is-more-elaborate-than-that-in-mosses-ii-salvi

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
consider-the-following-linear-equations-a-x-b-y-cz-0-b-x-cy-a-z-0-c-x-a-y-b-z-0-match-the-conditionsexpressions-in-column-i-with-statements-in-column--1

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-linear-equations-a-x-b-y-cz-0-b-x-cy-a-z-0-c-x-a-y-b-z-0-match-the-conditionsexpressions-in-column-i-with-statements-in-column-

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
consider-the-lines-given-by-l-1-x-3-y-5-0-l-2-3-x-k-y-1-0-l-3-5-x-2-y-12-0-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions--1

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

question:
<div class="question">Consider the lines given by<br/>$$<br/>L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0<br/>$$<br/>Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/t-8SrYw4JSCJlT9NZOgfJHOj9KnGrpZhN-Y1Vj8K5gU.original.fullsize.png"/><br/></div>

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-lines-given-by-l-1-x-3-y-5-0-l-2-3-x-k-y-1-0-l-3-5-x-2-y-12-0-match-the-statementsexpressions-in-column-i-with-the-statementsexpressions-

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

question:
<div class="question">Consider the lines given by<br/>$$<br/>L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0<br/>$$<br/>Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/t-8SrYw4JSCJlT9NZOgfJHOj9KnGrpZhN-Y1Vj8K5gU.original.fullsize.png"/><br/></div>

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

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

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


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-motion-of-a-positive-point-charge-in-a-region-where-there-are-simultaneous-uniform-electric-and-magnetic-fields-e-e-0-j-and-b-b-0-j-at-ti

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

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

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

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

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


tags: ['Mathematics', 'Parabola', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-parabola-y-2-8-x-let-1-be-the-area-of-the-triangle-formed-by-the-end-points-of-its-latusrectum-and-the-point-p-2-1-2-on-the-parabola-and--1

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

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

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


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-parabola-y-2-8-x-let-1-be-the-area-of-the-triangle-formed-by-the-end-points-of-its-latusrectum-and-the-point-p-2-1-2-on-the-parabola-and-

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

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-planes-3-x-6-y-2-z-15-and-2-x-y-2-z-5-statement-i-the-parametric-equations-of-the-line-of-intersection-of-the-given-planes-are-x-3-14-t-y

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

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

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Statement I is false, Statement II is true</span> </div>

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


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-system-of-equations-where-a-b-c-d-0-1-statement-1-the-probability-that-the-system-of-equations-has-a-unique-solution-is-38-statement-2-th

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
consider-the-system-of-equations-x-2-y-3-z-1-x-3-y-4-z-1-and-x-y-2-z-k-statement-1-the-system-of-equations-has-no-solution-for-k-3-statement-2-the-det-1

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-system-of-equations-x-2-y-3-z-1-x-3-y-4-z-1-and-x-y-2-z-k-statement-1-the-system-of-equations-has-no-solution-for-k-3-statement-2-the-det

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

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

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

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

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


title: consider-the-two-curves-c-1-y-2-4-x-c-2-x-2-y-2-6-x-1-0-then
slug: consider-the-two-curves-c-1-y-2-4-x-c-2-x-2-y-2-6-x-1-0-then-25705
tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the two curves<br/>$C_1: y^2=4 x$<br/>$C_2: x^2+y^2-6 x+1=0$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$C_1$ and $C_2$ touch each other only at one point<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$C_1$ and $C_2$ touch each other exactly at two points<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$C_1$ and $C_2$ intersect (but do not touch) at exactly two points<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$C_1$ and $C_2$ neither intersect nor touch each other</span> </li> </ul>

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

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


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
consider-three-planes-p-1-x-y-z-1-p-2-x-y-z-1-an-d-p3-x3-y3-z2-l-e-t-l1-l2-an-d-l3-b-e-t-h-e-l-in-eso-f-in-t-ersec-t-i-o-n-o-f-t-h-e-pl-an-es-p2-an-d-

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

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

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

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

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


title: consider-three-points-p-sin-cos-q-cos-sin-and-r-cos-sin-where-0-4-then
slug: consider-three-points-p-sin-cos-q-cos-sin-and-r-cos-sin-where-0-4-then-11302
tags: ['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider three points $P=(-\sin (\beta-\alpha),-\cos \beta), Q=(\cos (\beta-\alpha), \sin \beta)$ and $R=(\cos (\beta-\alpha+\theta), \sin (\beta-\theta)$, where $0 &lt; \alpha, \beta, \theta &lt; \frac{\pi}{4}$. Then,</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$P, Q, R$ are non- collinear</span> </div>

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


tags: ['Biology', 'Biodiversity and its Conservation', 'NEET', 'NEET 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
cryopreservation-of-gametes-of-threatened-species-in-viable-and-fertile-condition-can-be-referred-to-as-1

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

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

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

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

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


tags: ['Biology', 'Biodiversity and its Conservation', 'NEET']
source_db: MarksBatch1_P2.db
title:
cryopreservation-of-gametes-of-threatened-species-in-viable-and-fertile-condition-can-be-referred-to-as

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

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

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

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

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


title: cuso-4-decolourises-on-addition-of-kcn-the-product-is
slug: cuso-4-decolourises-on-addition-of-kcn-the-product-is-98093
tags: ['Chemistry', 'd and f Block Elements', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch1_P2.db
question:
<div class="question">$\mathrm{CuSO}_4$ decolourises on addition of $\mathrm{KCN}$, the product is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\left[\mathrm{Cu}(\mathrm{CN})_4\right]^{2-}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Cu}^{2+}$ get reduced to form $\left[\mathrm{Cu}(\mathrm{CN})_4\right]^{3-}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Cu}(\mathrm{CN})_2$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{CuCN}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

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


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
cyclohexen-on-ozonolysis-followed-by-reaction-with-zinc-dust-and-water-gives-compound-e-compound-e-on-further-treatment-with-aqueous-koh-yields-compou

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

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

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

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/J3dus6p3LX9F0gdZa1zG9L51PfycDNxiqH1t4Mwq_-o.original.fullsize.png"/><br/></span> </div>

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


title: d-y-2-d-2-x-equals-1
slug: d-y-2-d-2-x-equals-1-98721
question: <div class="question">$\frac{d^2 x}{d y^2}$ equals</div>
tags: ['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{d^2 y}{d x^2}\right)^{-1}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$-\left(\frac{d^2 y}{d x^2}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-2}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

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


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
dissolving-120-g-of-urea-molecular-weight-60-in-1000-g-of-water-gave-a-solution-of-density-115-g-ml-the-molarity-of-the-solution-is

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1.78 \mathrm{M}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$2.00 \mathrm{M}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$2.05 \mathrm{M}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$2.22 \mathrm{M}$</span> </li> </ul>

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

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


tags: ['Chemistry', 'Electrochemistry', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
electrolysis-of-dilute-aq-nacl-solution-was-carried-out-by-passing-10-ma-current-the-time-required-to-liberate-001-mole-of-h-2-gas-at-the-cathode-is-1

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$9.65 \times 10^4 \mathrm{~s}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$19.3 \times 10^4 \mathrm{~s}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$28.95 \times 10^4 \mathrm{~s}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$38.6 \times 10^4 \mathrm{~s}$</span> </li> </ul>

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

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


tags: ['Physics', 'Dual Nature of Matter', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
electrons-with-debroglie-wavelength-fall-on-the-target-in-an-xray-tube-the-cutoff-wavelength-of-the-emitted-xrays-is-1

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\lambda_0=\frac{2 m c \lambda^2}{h}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\lambda_0=\frac{2 h}{m c}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\lambda_0=\frac{2 m^2 c^2 \lambda^3}{h^2}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\lambda_0=\lambda$</span> </li> </ul>

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

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


title: equation-of-common-tangent-of-y-x-2-y-x-2-4-x-4-is
slug: equation-of-common-tangent-of-y-x-2-y-x-2-4-x-4-is-75494
question: <div class="question">Equation of common tangent of $y=x^2, y=-x^2+4 x-4$ is</div>
tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$y=4(x-1)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$y=0$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$y=-4(x-1)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$y=-30 x-50$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
equation-of-the-plane-containing-the-straight-line-2-x-3-y-4-z-and-perpendicular-to-the-plane-containing-the-straight-lines-3-x-4-y-2-z-and-4-x-2-y-3-

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

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

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

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

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


title: extra-pure-n-2-can-be-obtained-by-heating
slug: extra-pure-n-2-can-be-obtained-by-heating-38165
question: <div class="question">Extra pure $\mathrm{N}_2$ can be obtained by heating</div>
source_db: MarksBatch1_P2.db
tags:
['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{NH}_3$ with $\mathrm{CuO}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{NH}_4 \mathrm{NO}_4$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\mathrm{NH}_4\right)_2 \mathrm{Cr}_2 \mathrm{O}_7$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{Ba}\left(\mathrm{N}_3\right)_2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

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


title: extraction-for-zinc-from-zinc-blende-is-achieved-by
slug: extraction-for-zinc-from-zinc-blende-is-achieved-by-55290
question: <div class="question">Extraction for zinc from zinc blende is achieved by</div>
source_db: MarksBatch1_P2.db
tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']

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

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

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


title: extraction-of-metal-from-the-ore-cassiterite-involves
slug: extraction-of-metal-from-the-ore-cassiterite-involves-36000
question: <div class="question">Extraction of metal from the ore cassiterite involves</div>
source_db: MarksBatch1_P2.db
tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>carbon reduction of an oxide ore<br/>, <br/>removal of iron impurity</span> </div>

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


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
figure-shows-three-resistor-configurations-r-1-r-2-and-r-3-connected-to-3-v-battery-if-the-power-dissipated-by-the-configuration-r-1-r-2-and-r-3-is-p--1

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$P_1&gt;P_2&gt;P_3$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$P_1&gt;P_3&gt;P_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$P_2&gt;P_1&gt;P_3$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$P_3&gt;P_2&gt;P_1$</span> </li> </ul>

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

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


tags: ['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
figure-shows-three-resistor-configurations-r-1-r-2-and-r-3-connected-to-3-v-battery-if-the-power-dissipated-by-the-configuration-r-1-r-2-and-r-3-is-p-

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$P_1&gt;P_2&gt;P_3$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$P_1&gt;P_3&gt;P_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$P_2&gt;P_1&gt;P_3$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$P_3&gt;P_2&gt;P_1$</span> </li> </ul>

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

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


tags: ['Biology', 'Evolution']
options: None
correct_option: None
source_db: MarksBatch1_P2.db
title:
find-out-from-newspapers-and-popular-science-articles-any-new-fossil-discoveries-or-controversies-about-evolution

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

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

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


tags: ['Physics', 'Alternating Current', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch1_P2.db
title:
find-the-time-constant-for-the-given-rc-circuits-in-correct-order-im-g-src-h-ttp-s-c-d-n-q-u-es-t-i-o-n-p-oo-l-g-e-t-ma-r-k-s-a-pp-p-y-q-j-e-e-a-d-v-a

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

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

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

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

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


title: for-0-2-the-solutions-of-m-1-6-cosec-4-m-1-cosec-4-m-4-2-isare
slug: for-0-2-the-solutions-of-m-1-6-cosec-4-m-1-cosec-4-m-4-2-isare-56614
tags: ['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch1_P2.db
question:
<div class="question">For $0 &lt; \theta &lt; \frac{\pi}{2}$, the solution(s) of $\sum_{m=1}^6 \operatorname{cosec}\left[\theta+\frac{(m-1) \pi}{4}\right]$ $\operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right)=4 \sqrt{2}$ is/are</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\frac{\pi}{12}$<br/>, <br/>$\frac{5 \pi}{12}$</span> </div>

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


tags: ['Chemistry', 'Solutions', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">724</span> </div>
source_db: MarksBatch1_P2.db
title:
for-a-dilute-solution-containing-25-g-of-a-nonvolatile-nonelectrolyte-solute-in-100-g-of-water-the-elevation-in-boiling-point-at-1-atm-pressure-is-2-c

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">724</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">740</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">736</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">718</span> </li> </ul>

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


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch1_P2.db
title:
for-a-first-order-reaction-a-p-the-temperature-t-dependent-rate-constant-k-was-found-to-follow-the-equation-lo-g-k-2000-t-60-the-preexponential-factor

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

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

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

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

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


tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch1_P2.db
title:
for-an-ideal-gas-consider-only-pv-work-in-going-from-an-initial-state-x-to-the-final-state-z-the-final-state-z-can-be-reached-by-either-of-the-two-pat

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

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

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

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

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


tags: ['Mathematics', 'Definite Integration', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch1_P2.db
title:
for-any-real-number-x-let-x-denotes-the-largest-integer-less-than-or-equal-to-x-let-f-be-a-real-valued-function-defined-on-the-interval-10-10-by-f-x-x-1

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

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

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


tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch1_P2.db
title:
for-any-real-number-x-let-x-denotes-the-largest-integer-less-than-or-equal-to-x-let-f-be-a-real-valued-function-defined-on-the-interval-10-10-by-f-x-x

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

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

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


title: if-2-s-i-n-4-x-3-c-o-s-4-x-5-1-then
slug: if-2-s-i-n-4-x-3-c-o-s-4-x-5-1-then-17856
question: <div class="question">If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}$, then</div>
tags: ['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\tan ^2 x=\frac{2}{3}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\tan ^2 x=\frac{1}{3}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{2}{125}$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-vectors-in-space-given-by-a-5-i-2-j-and-b-14-2-i-j-3-k-then-the-value-of-2-a-b-a-b-a-2-b-is

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

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

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


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-vectors-such-that-a-b-29-and-a-2-i-3-j-4-k-2-i-3-j-4-k-b-then-a-possible-value-of-a-b-7-i-2-j-3-k-is

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

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

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

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


title: if-a-b-and-c-are-the-sides-of-a-a-bc-such-that-x-2-2-a-b-c-x-3-ab-b-c-c-a-0-has-real-roots-then
tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
slug:
if-a-b-and-c-are-the-sides-of-a-a-bc-such-that-x-2-2-a-b-c-x-3-ab-b-c-c-a-0-has-real-roots-then-82360

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

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

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

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


title: if-a-b-and-c-are-unit-vectors-satisfying-a-b-2-b-c-2-c-a-2-9-then-2-a-5-b-5-c-is
slug: if-a-b-and-c-are-unit-vectors-satisfying-a-b-2-b-c-2-c-a-2-9-then-2-a-5-b-5-c-is-43778
tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9$, then $|2 \vec{a}+5 \vec{b}+5 \vec{c}|$ is</div>

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


title: if-a-b-c-and-d-are-the-unit-vectors-such-that-a-b-c-d-1-and-a-c-2-1-then-1
slug: if-a-b-c-and-d-are-the-unit-vectors-such-that-a-b-c-d-1-and-a-c-2-1-then-1-90395
tags: ['Mathematics', 'Vector Algebra', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are the unit vectors such that $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1$ and $\mathbf{a} \cdot \mathbf{c}=\frac{1}{2}$, then</div>

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

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

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


tags: ['Mathematics', 'Hyperbola', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
if-e-1-is-the-eccentricity-of-the-ellipse-16-x-2-25-y-2-1-and-e-2-is-the-eccentricity-of-the-hyperbola-passing-through-the-focii-of-the-ellipse-and-e-

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{x^2}{9}-\frac{y^2}{16}=1$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{x^2}{16}-\frac{y^2}{9}=-1$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{x^2}{9}-\frac{y^2}{25}=1$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>None of these</span> </li> </ul>

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

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


title: if-f-x-0-x-e-t-2-t-2-t-3-d-t-for-all-x-0-then
slug: if-f-x-0-x-e-t-2-t-2-t-3-d-t-for-all-x-0-then-99657
tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t$ for all $x \in(0, \infty)$, then</div>

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

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

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


tags: ['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
if-f-x-f-x-where-f-x-is-a-continuous-double-differentiable-function-and-g-x-f-x-if-f-x-f-2-x-2-g-2-x-2-and-f-5-5-then-f-10-is

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>0<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>5<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>10<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>25</span> </li> </ul>

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

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


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
if-f-x-is-cubic-polynomial-which-has-local-maximum-at-x-1-if-f-2-18-f-1-1-and-f-x-has-local-minimum-at-x-0-then

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

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

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

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

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


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
title:
if-f-x-is-twice-differentiable-function-such-that-f-a-0-f-b-2-f-c-1-f-d-2-f-e-0-where-a-b-c-d-e-then-the-minimum-number-of-zeros-of-g-x-f-x-2-f-x-f-x-

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

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

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


title: if-f-x-min-1-x-2-x-3-then
slug: if-f-x-min-1-x-2-x-3-then-71892
question: <div class="question">If $f(x)=\min \left\{1, x^2, x^3\right\}$, then</div>
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$f(x)$ is continuous everywhere<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$f(x)$ is continuous and differentiable everywhere<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$f(x)$ is not differentiable at two points<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$f(x)$ is not differentiable at one point</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$f(x)$ is continuous everywhere<br/>, <br/>$f(x)$ is not differentiable at one point</span> </div>

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


title: if-f-x-x-2-cos-x-x-1-lo-g-x-x-2-2-x-0-0-x-1-x-1-then
slug: if-f-x-x-2-cos-x-x-1-lo-g-x-x-2-2-x-0-0-x-1-x-1-then-78669
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, &amp; x \leq-\frac{\pi}{2} \\ -\cos x, &amp; -\frac{\pi}{2} &lt; x \leq 0 \\ x-1, &amp; 0 &lt; x \leq 1 \\ \log x, &amp; x&gt;1\end{array}\right.$, then</div>

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

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

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


title: if-i-n-1-x-s-i-n-x-s-i-n-n-x-d-x-n-0-1-2-then
slug: if-i-n-1-x-s-i-n-x-s-i-n-n-x-d-x-n-0-1-2-then-10740
tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $I_n=\int_{-\pi}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x ; n=0,1,2$, $\ldots$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$I_n=I_{n+2}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\sum_{m=1}^{10} I_{2 m+1}=10 \pi$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\sum_{m=1}^{10} I_{2 m}=0$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$I_n=I_{n+1}$</span> </li> </ul>

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

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


title: if-lim-x-0-1-x-lo-g-1-b-2-1-x-2-b-sin-2-b-0-and-then-the-value-of-is
slug: if-lim-x-0-1-x-lo-g-1-b-2-1-x-2-b-sin-2-b-0-and-then-the-value-of-is-36689
tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow 0}\left[1+x \log \left(1+b^2\right)\right]^{1 / x}=2 b \sin ^2 \theta b&gt;0$ and $\theta \in(-\pi, \pi]$, then the value of $\theta$ is</div>

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

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

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


title: if-lim-x-x-1-x-2-x-1-a-x-b-4-then
slug: if-lim-x-x-1-x-2-x-1-a-x-b-4-then-69627
tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4$, then</div>

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

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-p-is-a-3-3-matrix-such-that-p-t-2-p-i-where-p-t-is-the-transpose-of-p-and-i-is-the-3-3-identity-matrix-then-there-exists-a-column-matrix-x-x-y-z-0-

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

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

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

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

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


tags: ['Mathematics', 'Permutation Combination', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-r-s-t-are-prime-numbers-and-p-q-are-the-positive-integers-such-that-lcm-of-p-q-is-r-2-s-4-t-2-then-the-number-of-ordered-pairs-p-q-is-1

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>252<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>254<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>225<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>224</span> </li> </ul>

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

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


tags: ['Mathematics', 'Permutation Combination', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
if-r-s-t-are-prime-numbers-and-p-q-are-the-positive-integers-such-that-lcm-of-p-q-is-r-2-s-4-t-2-then-the-number-of-ordered-pairs-p-q-is

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>252<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>254<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>225<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>224</span> </li> </ul>

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-the-adjoint-of-a-3-3-matrix-p-is-1-2-1-4-1-1-4-7-3-then-the-possible-values-of-the-determinant-of-p-is-are

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

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

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

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

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


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-the-angles-a-b-and-c-of-a-triangle-are-in-an-arithmetic-progression-and-if-a-b-and-c-denote-the-lengths-of-the-sides-opposite-to-a-b-and-c-respecti

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

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

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

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

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


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
if-the-bond-length-of-co-bond-in-carbon-monoxide-is-1128-a-then-what-is-the-value-of-co-bond-length-in-fe-co-5

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$1.15 Ã$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$1.128 Ã$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$1.72 Ã$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$1.118 Ã$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-distance-between-the-plane-a-x-2-y-z-d-and-the-plane-containing-the-lines-2-x-1-3-y-2-4-z-3-and-3-x-2-4-y-3-5-z-4-is-6-then-d-is

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

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-the-distance-of-the-point-p-1-2-1-from-the-plane-x-2-y-2-z-where-0-is-5-then-the-foot-of-the-perpendicular-from-p-to-the-plane-is

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{8}{3}, \frac{4}{3},-\frac{7}{3}\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(\frac{4}{3},-\frac{4}{3}, \frac{1}{3}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{1}{3}, \frac{2}{3}, \frac{10}{3}\right)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(\frac{2}{3},-\frac{1}{3}, \frac{5}{2}\right)$</span> </li> </ul>

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

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


title: if-the-function-f-x-x-3-e-2-x-and-g-x-f-1-x-then-the-value-of-g-1-is
slug: if-the-function-f-x-x-3-e-2-x-and-g-x-f-1-x-then-the-value-of-g-1-is-46015
tags: ['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the function $f(x)=x^3+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is</div>

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


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-the-resultant-of-all-the-external-forces-acting-on-a-system-of-particles-is-zero-then-from-an-inertial-frame-one-can-surely-say-that

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

question:
<div class="question">If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that</div>

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

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-the-straight-lines-2-x-1-k-y-1-2-z-and-5-x-1-2-y-1-k-z-are-coplanar-then-the-plane-s-containing-these-two-lines-is-are

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

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

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

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

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


title: if-the-sum-of-first-n-terms-of-an-ap-is-cn-2-then-the-sum-of-squares-of-these-n-terms-is
slug: if-the-sum-of-first-n-terms-of-an-ap-is-cn-2-then-the-sum-of-squares-of-these-n-terms-is-63010
tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the sum of first $n$ terms of an $\mathrm{AP}$ is $\mathrm{cn}^2$, then the sum of squares of these $n$ terms is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{n\left(4 n^2-1\right) c^2}{6}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{n\left(4 n^2+1\right) c^2}{3}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{n\left(4 n^2-1\right) c^2}{3}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{n\left(4 n^2+1\right) c^2}{6}$</span> </li> </ul>

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

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


tags: ['Physics', 'Atomic Physics', 'JEE Advanced', 'JEE Advanced 2006']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-wavelength-of-the-n-th-line-of-lyman-series-is-equal-to-the-debroglie-wavelength-of-electron-in-initial-orbit-of-a-hydrogen-like-element-z-11-f

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

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

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


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
if-w-i-where-0-and-z-1-satisfies-the-condition-that-1-z-w-w-z-is-purely-real-then-the-set-of-values-of-z-is

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

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

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

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

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


title: if-x-2-10-a-x-11-b-0-have-roots-c-and-d-and-x-2-10-c-x-11-d-0-have-roots-a-and-b-then-find-a-b-c-d
tags: ['Mathematics', 'Quadratic Equation', 'JEE Advanced', 'JEE Advanced 2006']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1210</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-x-2-10-a-x-11-b-0-have-roots-c-and-d-and-x-2-10-c-x-11-d-0-have-roots-a-and-b-then-find-a-b-c-d-33913

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

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


title: if-y-x-satisfies-the-differential-equation-y-y-tan-x-2-x-sec-x-and-y-0-0-then
slug: if-y-x-satisfies-the-differential-equation-y-y-tan-x-2-x-sec-x-and-y-0-0-then-96515
tags: ['Mathematics', 'Differential Equations', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x$ $=2 x \sec x$ and $y(0)=0$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{8 \sqrt{2}}$, $y^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^{2}}{3 \sqrt{3}}$</span> </div>

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


title: if-z-1-and-z-1-then-all-the-values-of-1-z-2-z-lie-on
slug: if-z-1-and-z-1-then-all-the-values-of-1-z-2-z-lie-on-19943
tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|z|=1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on</div>

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

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

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


title: if-z-is-any-complex-number-satisfying-z-3-2-i-2-then-the-maximum-value-of-2-z-6-5-i-is
slug: if-z-is-any-complex-number-satisfying-z-3-2-i-2-then-the-maximum-value-of-2-z-6-5-i-is-19545
tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z$ is any complex number satisfying $|z-3-2 i| \leq 2$, then the maximum value of $|2 z-6+5 i|$ is</div>

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


tags: ['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
image-of-an-object-approaching-a-convex-mirror-of-radius-of-curvature-20-m-along-its-optical-axis-is-observed-to-move-from-3-25-m-to-7-50-m-in-30-s-wh

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

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

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


tags: ['Chemistry', 'Ionic Equilibrium', 'JEE Advanced', 'JEE Advanced 2011 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
title:
in-1-l-saturated-solution-of-agcl-k-sp-agcl-16-1-0-10-01-mole-of-cucl-k-sp-cucl-10-1-0-6-is-added-the-resultant-concentration-of-ag-in-the-solution-is

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

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

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


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-a-a-bc-with-fixed-base-bc-the-vertex-a-moves-such-that-cos-b-cos-c-4-sin-2-2-a-if-a-b-and-c-denote-the-lengths-of-the-sides-of-the-triangle-opposit

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

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

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

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

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


tags: ['Physics', 'Alternating Current', 'JEE Advanced', 'JEE Advanced 2021 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">60</span> </div>
source_db: MarksBatch2_P1.db
title:
in-a-circuit-a-metal-filament-lamp-is-connected-in-series-with-a-capacitor-of-capacitance-c-f-across-a-200-v-50-hz-supply-the-power-consumed-by-the-la

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

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

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


tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
title:
in-a-constant-volume-calorimeter-35-g-of-a-gas-with-molecular-weight-28-was-burnt-in-excess-oxygen-at-2980-k-the-temperature-of-the-calorimeter-was-fo-2

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

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

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


tags: ['Physics', 'Wave Optics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-a-youngs-double-slit-experiment-the-separation-between-the-two-slits-is-d-and-the-wavelength-of-the-light-is-the-intensity-of-light-falling-on-slit

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

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

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

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

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


title: in-allene-c-3-h-4-the-types-of-hybridisation-of-the-carbon-atoms-is-are
slug: in-allene-c-3-h-4-the-types-of-hybridisation-of-the-carbon-atoms-is-are-50924
tags: ['Chemistry', 'Hydrocarbons', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In allene $\left(\mathrm{C}_{3} \mathrm{H}_{4}\right)$, the type(s) of hybridisation of the carbon atoms is (are):</div>

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

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

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


tags: ['Physics', 'Ray Optics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-an-experiment-to-determine-the-focal-length-f-of-a-concave-mirror-by-the-u-v-method-a-student-places-the-object-pin-a-on-the-principal-axis-at-a-di

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

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

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

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

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


tags: ['Physics', 'Thermal Properties of Matter', 'JEE Advanced', 'JEE Advanced 2006']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">273</span> </div>
source_db: MarksBatch2_P1.db
title:
in-an-insulated-vessel-005-kg-steam-at-373-k-and-045-kg-of-ice-at-253-k-are-mixed-find-the-final-temperature-of-the-mixture-in-kelvin-given-l-fusion-s

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

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

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


tags: ['Biology', 'Plant - Growth and Development']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
title:
in-animals-there-are-special-glands-secreting-hormones-whereas-there-are-no-glands-in-plants-where-are-plant-hormones-formed-how-are-the-hormones-tran

slug:
in-animals-there-are-special-glands-secreting-hormones-whereas-there-are-no-glands-in-plants-where-are-plant-hormones-formed-how-are-the-hormones-tran-68496

question:
<div class="question">In animals, there are special glands secreting hormones, whereas there are no glands in plants. Where are plant hormones formed? How are the hormones translocated to the site of activity?</div>

answer:
<div class="solution">The plant hormones are synthesised by the plant cells needed. Few hormones are specifically synthesised at a particular part of the plant like auxin synthesised in growing shoot apices and ethylene is secretes by ripened fruits. Cytokinin is found in dividing cells. Unlike plants animal being more advanced, and organised they have proper hormone secreting glands and organs. These are transported through the transport system of their body in both plant and animals. In plants, hormone are translocated via xylem and phloem to the site of activity.</div>


title: in-dicot-stem-the-xylem-is
slug: in-dicot-stem-the-xylem-is-58762
question: <div class="question">In dicot stem, the xylem is</div>
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Exarch</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Mesarch</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Centarch</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Endarch</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Endarch</span> </div>

answer:
<div class="solution">There are four types of primary xylem development: <strong>exarch, endarch, centrarch, and mesarch.</strong><br/>In <strong>endarch </strong>development, the protoxylem begins its development from the innermost procambial cells located adjacent to the pith and development progresses outward. Therefore, the protoxylem is found toward the inside and metaxylem toward the outside of the stem. Endarch development is considered the most highly advanced type of primary xylem development.</div>


source_db: MarksBatch2_P1.db
title:
in-the-cyanide-extraction-process-of-silver-from-argentite-ore-the-oxidising-and-reducing-agents-used-are

slug:
in-the-cyanide-extraction-process-of-silver-from-argentite-ore-the-oxidising-and-reducing-agents-used-are-72603

question:
<div class="question">In the cyanide extraction process of silver from argentite ore, the oxidising and reducing agents used are</div>

tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']

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

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

answer:
<div class="solution">The reactions involved in cyanide extraction process are:
<br/> <br/>$\underset{\text { (argentite) }}{\mathrm{Ag}_{2} \mathrm{~S}}+4 \mathrm{NaCN} \rightarrow 2 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]+\mathrm{Na}_{2} \mathrm{~S}$
<br/> <br/>$4 \mathrm{Na}_{2} \mathrm{~S}+\underset{\text { Oxiding agent }}{5 \mathrm{O}_{2}}+2 \mathrm{H}_{2} \mathrm{O} \rightarrow 2 \mathrm{Na}_{2} \mathrm{SO}_{4}+4 \mathrm{NaOH}+2 \mathrm{~S}$
<br/> <br/>$2 \mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]+\underset{\begin{array}{c}\text { (reducing } \\ \text { agent) }\end{array}}{\mathrm{Zn}} \rightarrow \mathrm{Na}_{2}\left[\mathrm{Zn}(\mathrm{CN})_{4}\right]+2 \mathrm{Ag} \downarrow$</div>


tags: ['Physics', 'Mathematics in Physics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-the-determination-of-youngs-modulus-y-l-d-2-4-m-lg-by-using-searles-method-a-wire-of-length-l-2-m-and-diameter-d-05-mm-is-used-for-a-load-m-25-kg-a

slug:
in-the-determination-of-youngs-modulus-y-l-d-2-4-m-lg-by-using-searles-method-a-wire-of-length-l-2-m-and-diameter-d-05-mm-is-used-for-a-load-m-25-kg-a-99762

question:
<div class="question">In the determination of Young's modulus $\left(Y=\frac{4 M L g}{\pi l d^{2}}\right)$ by using Searle's method, a wire of length $L=2 \mathrm{~m}$ and diameter $d=0.5 \mathrm{~mm}$ is used. For a load $M=2.5 \mathrm{~kg}$, an extension $l=0.25 \mathrm{~mm}$ in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of $0.5 \mathrm{~mm}$. The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the $Y$ measurement</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">due to the errors in the measurements of $d$ and $l$ are the same.</span> </div>

answer:
<div class="solution">The maximum possible error in $Y$ due to $l$ and $d$ $\frac{\Delta Y}{Y}=\frac{\Delta l}{l}+\frac{2 \Delta d}{d}$ Least count $=\frac{\text { Pitch }}{\text { No. of division on circular scale }}$ $=\frac{0.5}{100} \mathrm{~mm}=0.005 \mathrm{~mm}$
<br/> <br/>Here, $\Delta d=\Delta l=0.005 \mathrm{~mm}$
<br/> <br/>Error contribution of $l=\frac{\Delta l}{l}=\frac{0.005 \mathrm{~mm}}{0.25 \mathrm{~mm}}=\frac{1}{50}$
<br/> <br/>Error contribution of
<br/> <br/>$d=\frac{2 \Delta d}{d}=\frac{2 \times 0.005 \mathrm{~mm}}{0.5 \mathrm{~mm}}=\frac{1}{50}$
<br/> <br/>Hence contribution to the maximum possible error in the measurement of $y$ due to $l$ and $d$ is the same.</div>


title: in-the-experiment-to-determine-the-speed-of-sound-using-a-resonance-column
slug: in-the-experiment-to-determine-the-speed-of-sound-using-a-resonance-column-53529
tags: ['Physics', 'Waves and Sound', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the experiment to determine the speed of sound using a resonance column:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Prongs of the tuning fork are kept in a vertical plane<br/></span> </div>

answer:
<div class="solution">Length of air column in resonance is odd integer multiple of $\frac{\lambda}{4}$. Correct option is (a).</div>


tags: ['Chemistry', 'General Organic Chemistry', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-the-following-carbocation-h-ch-3-that-is-most-likely-to-migrate-to-the-positively-charged-carbon-is

slug:
in-the-following-carbocation-h-ch-3-that-is-most-likely-to-migrate-to-the-positively-charged-carbon-is-49327

question:
<div class="question">In the following carbocation, $\mathrm{H} / \mathrm{CH}_3$ that is most likely to migrate to the positively charged carbon is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/SL8Rsu5u8piwlrnZ6cwJsXNsIB9cai7C5FH3HNiqyBU.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{CH}_3$ at $\mathrm{C}-4$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{H}$ at $\mathrm{C}-4$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{CH}_3$ at $\mathrm{C}-2$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{H}$ at $\mathrm{C}-2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{H}$ at $\mathrm{C}-2$</span> </div>

answer:
<div class="solution">Hydride shift from $\mathrm{C}-2$ will give most stable resonance stabilised carbocation as Fig.</div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-the-following-reaction-sequence-the-compound-j-is-an-intermediate-j-c-9-h-8-o-2-gives-effervescence-on-treatment-with-nahco-3-and-a-positive-baeyer-1

slug:
in-the-following-reaction-sequence-the-compound-j-is-an-intermediate-j-c-9-h-8-o-2-gives-effervescence-on-treatment-with-nahco-3-and-a-positive-baeyer-1-22814

question:
<div class="question">In the following reaction sequence, the compound $\mathrm{J}$ is an intermediate.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wEQdUNBgA0a1fYiZUUKvCk8wMg-9NCH2X7b_nCIMm3Q.original.fullsize.png"/><br/> <br/> <br/>$\mathbf{J}\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{2}\right)$ gives effervescence on treatment with $\mathrm{NaHCO}_{3}$ and a positive Baeyer's test.<strong> Question:</strong> The compound $\mathbf{K}$ 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/GJpVx3Z5Kjrw3UYhem7itdrTO4TaiMoigUAMz-h4PHE.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/KolWgJGmAxktnW3C3T3iB8xUtV0SZCNdmz5WSzM20ME.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">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/pCWVN9JiSgXxc_KWdW_rXvmW3HAfvAZbJpBNL_cfHCg.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u_T7Cf0hYaMEGcVrJRUWQxTblFcUF2Qaq3ZuiQIhFDY.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/KolWgJGmAxktnW3C3T3iB8xUtV0SZCNdmz5WSzM20ME.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/shdddhtgg76t676g3gh.png"/></div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-following-reaction-sequence-the-compound-j-is-an-intermediate-j-c-9-h-8-o-2-gives-effervescence-on-treatment-with-nahco-3-and-a-positive-baeyer-2

slug:
in-the-following-reaction-sequence-the-compound-j-is-an-intermediate-j-c-9-h-8-o-2-gives-effervescence-on-treatment-with-nahco-3-and-a-positive-baeyer-2-16444

question:
<div class="question">In the following reaction sequence, the compound $\mathrm{J}$ is an intermediate.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wEQdUNBgA0a1fYiZUUKvCk8wMg-9NCH2X7b_nCIMm3Q.original.fullsize.png"/><br/> <br/> <br/>$\mathbf{J}\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{2}\right)$ gives effervescence on treatment with $\mathrm{NaHCO}_{3}$ and a positive Baeyer's test.<strong> Question:</strong> The compound $\mathbf{I}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/DFnR6NI_l1GHvwTdefAymtSlz3fe_-1fJk_APNOF6HI.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">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/cD1NtWzOfi1ZhqVr1ncyJhBdA1gsgHcYOjb94jRxclA.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/LtE_4vaHfn-ocsFI-ffFslfDRKVE7tcPR1WMWf9dv28.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/o7Am4486gKoykHYLsk2LpWs2noPCnZUcna8-qsZp6PE.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/DFnR6NI_l1GHvwTdefAymtSlz3fe_-1fJk_APNOF6HI.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/shdddhtgg76t676g3gh.png"/></div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-the-following-reaction-sequence-the-compound-j-is-an-intermediate-j-c-9-h-8-o-2-gives-effervescence-on-treatment-with-nahco-3-and-a-positive-baeyer

slug:
in-the-following-reaction-sequence-the-compound-j-is-an-intermediate-j-c-9-h-8-o-2-gives-effervescence-on-treatment-with-nahco-3-and-a-positive-baeyer-26143

question:
<div class="question">In the following reaction sequence, the compound $\mathrm{J}$ is an intermediate.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wEQdUNBgA0a1fYiZUUKvCk8wMg-9NCH2X7b_nCIMm3Q.original.fullsize.png"/><br/> <br/> <br/>$\mathbf{J}\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{2}\right)$ gives effervescence on treatment with $\mathrm{NaHCO}_{3}$ and a positive Baeyer's test.<strong> Question:</strong> The compound $\mathbf{I}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/DFnR6NI_l1GHvwTdefAymtSlz3fe_-1fJk_APNOF6HI.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">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/cD1NtWzOfi1ZhqVr1ncyJhBdA1gsgHcYOjb94jRxclA.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/LtE_4vaHfn-ocsFI-ffFslfDRKVE7tcPR1WMWf9dv28.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/o7Am4486gKoykHYLsk2LpWs2noPCnZUcna8-qsZp6PE.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/DFnR6NI_l1GHvwTdefAymtSlz3fe_-1fJk_APNOF6HI.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/shdddhtgg76t676g3gh.png"/></div>


title: in-the-following-reaction-sequence-the-correct-structures-of-e-f-and-g-are-1
slug: in-the-following-reaction-sequence-the-correct-structures-of-e-f-and-g-are-1-34398
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the following reaction sequence, the correct structures of $E, F$ and $G$ are<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/vRmeGTCFSFL0D1alGxSueCfK-zdY5zwtUD9rhX6j-Mg.original.fullsize.png"/><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/k3F3-7je2haDq7_12n1P1m5iFD6n82eA0DYWjE1HQbs.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/nhD7m9Clo_dCMTHogT3HcNGdzB6jIs1g_8t6MetAbwA.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/VnlxbctDCuLlZbYpQySukWy9tIGd9deajMeEqbrS6bQ.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">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OL4wL-dWZlA7CibKKqumxbpIDNibFSKMC_iHMcv5R2A.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/VnlxbctDCuLlZbYpQySukWy9tIGd9deajMeEqbrS6bQ.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/M_hmHQ0sm0mLvJzSz32L6boXFj6Tv5MPfvvVGHq4HVU.original.fullsize.png"/><br/></div>


title: in-the-following-reaction-the-structure-of-the-major-product-x-is-1
slug: in-the-following-reaction-the-structure-of-the-major-product-x-is-1-89851
tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the following reaction,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/BeYLjJnusd9Xzxl0vznAVx5d0UUSSlQE221UaLXUoJc.original.fullsize.png"/><br/><br/>$$<br/>\text { The structure of the major product } X \text { is }<br/>$$</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/26YC4HDKD_i5rzE2S69CDLXPFvNtlToWRf3zX9dT5rs.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/RMr84-hG7z5Kz2mC9SK5vlioGK4LCoSiGYozBfbXyUE.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/0GMRPjdKfsKZ3p2JjvHqXxJU5b7ymc8nbnUpmhTnxG4.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/> &lt; smiles&gt;O=C(Nc1cccc([N+](=O)[O-])c1)c1ccccc1 &lt; /smiles&gt;</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/RMr84-hG7z5Kz2mC9SK5vlioGK4LCoSiGYozBfbXyUE.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/H9cdbc58aJUxa1ExyAOXQsSe72xfyiieY5zntgW4w3g.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wgJoK_F7PbJi-ma8hY9layaKmAHf498O9xvDMW8tGzU.original.fullsize.png"/><br/><br/>Ring 1 is more active, electrophilic substitution takes place over ring 1. - $\mathrm{NH}-\mathrm{C}-\mathrm{Ph}$ is ortho-para directing, para product is predominating.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-following-x-denotes-the-greatest-integer-less-than-or-equal-to-x-match-the-functions-in-column-i-with-the-properties-column-ii-1

slug:
in-the-following-x-denotes-the-greatest-integer-less-than-or-equal-to-x-match-the-functions-in-column-i-with-the-properties-column-ii-1-59536

question:
<div class="question">In the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column I with the properties Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZOJzgUXgnR79Tqmkejx4TZv8JzPFeZlpF6VMfpG-z9A.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, r; B-p, s; C-r, s; D-p, q<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">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, r; B-p, s; C-r, s; D-p, q<br/></span> </div>

answer:
<div class="solution">(A) $x|x|$ is continuous, differentiable and strictly increasing in $(-1,1)$.<br/>(B) $\sqrt{|x|}$ is continuous in $(-1,1)$ and not differentiable at $x=0$.<br/>(C) $x+[x]$ is strictly increasing in $(-1,1)$ and discontinuous at $x=0$ $\Rightarrow$ not differentiable at $x=0$.<br/>(D) $|x-1|+|x+1|=2$ in $(-1,1)$<br/>The function is continuous and differentiable in $(-1,1)$.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-the-following-x-denotes-the-greatest-integer-less-than-or-equal-to-x-match-the-functions-in-column-i-with-the-properties-column-ii

slug:
in-the-following-x-denotes-the-greatest-integer-less-than-or-equal-to-x-match-the-functions-in-column-i-with-the-properties-column-ii-51645

question:
<div class="question">In the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column I with the properties Column II.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZOJzgUXgnR79Tqmkejx4TZv8JzPFeZlpF6VMfpG-z9A.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, r; B-p, s; C-r, s; D-p, q<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">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, r; B-p, s; C-r, s; D-p, q<br/></span> </div>

answer:
<div class="solution">(A) $x|x|$ is continuous, differentiable and strictly increasing in $(-1,1)$.<br/>(B) $\sqrt{|x|}$ is continuous in $(-1,1)$ and not differentiable at $x=0$.<br/>(C) $x+[x]$ is strictly increasing in $(-1,1)$ and discontinuous at $x=0$ $\Rightarrow$ not differentiable at $x=0$.<br/>(D) $|x-1|+|x+1|=2$ in $(-1,1)$<br/>The function is continuous and differentiable in $(-1,1)$.</div>


tags: ['Physics', 'Capacitance', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-given-circuit-a-charge-of-80-c-is-given-to-the-upper-plate-of-the-4-f-capacitor-then-in-the-steady-state-the-charge-on-the-upper-plate-of-the-3-1

slug:
in-the-given-circuit-a-charge-of-80-c-is-given-to-the-upper-plate-of-the-4-f-capacitor-then-in-the-steady-state-the-charge-on-the-upper-plate-of-the-3-1-50593

question:
<div class="question">In the given circuit, a charge of $+80 \mu C$ is given to the upper plate of the $4 \mu F$ capacitor. Then in the steady state, the charge on the upper plate of the $3 \mu F$ capacitor is
<br/> <br/><img src=" data:image/png;base64,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"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$+32 \mu \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$+40 \mu \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$+48 \mu \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$+80 \mu \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$+48 \mu \mathrm{C}$</span> </div>

answer:
<div class="solution">The total charge on plate $A$ will be $80 \mu \mathrm{C}$.
<br/> <br/>$2 \mu F$ and $3 \mu F$ capacitors are in parallel. Therefore, $C_{e q}=2+3=5 \mathrm{HF}$
<br/> <br/>Charge on capacitor of $3 \mu \mathrm{F}$ capacitance $q=\frac{3}{5} \times 80=48 \mu C$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/hg2rttPavpB446uZyDbOiN4HL3hyjh-OD7VgG6EegZM.original.fullsize.png"/></div>


tags: ['Physics', 'Capacitance', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-the-given-circuit-a-charge-of-80-c-is-given-to-the-upper-plate-of-the-4-f-capacitor-then-in-the-steady-state-the-charge-on-the-upper-plate-of-the-3

slug:
in-the-given-circuit-a-charge-of-80-c-is-given-to-the-upper-plate-of-the-4-f-capacitor-then-in-the-steady-state-the-charge-on-the-upper-plate-of-the-3-24198

question:
<div class="question">In the given circuit, a charge of $+80 \mu C$ is given to the upper plate of the $4 \mu F$ capacitor. Then in the steady state, the charge on the upper plate of the $3 \mu F$ capacitor is
<br/> <br/><img src=" 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"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$+32 \mu \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$+40 \mu \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$+48 \mu \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$+80 \mu \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$+48 \mu \mathrm{C}$</span> </div>

answer:
<div class="solution">The total charge on plate $A$ will be $80 \mu \mathrm{C}$.
<br/> <br/>$2 \mu F$ and $3 \mu F$ capacitors are in parallel. Therefore, $C_{e q}=2+3=5 \mathrm{HF}$
<br/> <br/>Charge on capacitor of $3 \mu \mathrm{F}$ capacitance $q=\frac{3}{5} \times 80=48 \mu C$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/hg2rttPavpB446uZyDbOiN4HL3hyjh-OD7VgG6EegZM.original.fullsize.png"/></div>


tags: ['Physics', 'Alternating Current', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-the-given-circuit-the-ac-source-has-100-rad-s-considering-the-inductor-and-capacitor-to-be-ideal-the-correct-choices-is-are

slug:
in-the-given-circuit-the-ac-source-has-100-rad-s-considering-the-inductor-and-capacitor-to-be-ideal-the-correct-choices-is-are-78356

question:
<div class="question">In the given circuit, the $\mathrm{AC}$ source has $\omega=100 \mathrm{rad} / \mathrm{s}$. Considering the inductor and capacitor to be ideal, the correct choice(s) is (are)
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/OyGpPSLZMw0huKfOCnZEjjxWU3IFQF8cPowE_JIDs6E.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The current through the circuit, $I$ is $0.3 \mathrm{~A}$.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The current through the circuit, $I$ is $0.3 \sqrt{2} A$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The voltage across $100 \Omega$ resistor $=10 \sqrt{2} \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The voltage across $50 \Omega$ resistor $=10 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">The current through the circuit, $I$ is $0.3 \mathrm{~A}$., The voltage across $100 \Omega$ resistor $=10 \sqrt{2} \mathrm{~V}$</span> </div>

answer:
<div class="solution">Impedance across $A B, R C$ part of the circuit
<br/> <br/>$\begin{aligned} \mathrm{Z}_{1}=&amp; \sqrt{X_{c}^{2}+R_{1}^{2}}=\sqrt{\left(\frac{1}{\omega C}\right)^{2}+R_{1}^{2}} \\=&amp; \sqrt{(100)^{2}+(100)^{2}}=100 \sqrt{2} \end{aligned}$
<br/> <br/>$\therefore \quad I_{1}=\frac{V}{Z_{1}}=\frac{20}{100 \sqrt{2}}$
<br/> <br/>$\quad\left[\right.$ leads emf by $\left.\phi_{1}\right]$
<br/> <br/>where $\cos \phi_{1}=\frac{R}{Z_{1}}=\frac{100}{100 \sqrt{2}}=\frac{1}{\sqrt{2}} \Rightarrow \theta=45^{\circ}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ed64m2hdoG2W80ioJw89XZcyqG4WBGJtaNWRyTdBbrs.original.fullsize.png"/><br/> <br/> <br/>Impedance across $C D, L R$ part of the circuit.
<br/> <br/>$\mathrm{Z}_{2}=\sqrt{X_{L}^{2}+R_{2}^{2}}=\sqrt{(\omega L)^{2}+R_{2}^{2}}$
<br/> <br/>$=\sqrt{(0.5 \times 100)^{2}+(50)^{2}}=50 \sqrt{2} \Omega$
<br/> <br/>$\therefore \quad I_{2}=\frac{V}{Z_{2}}=\frac{20}{50 \sqrt{2}}$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/srQHHHyf8YGl8W5QJp-lTf3vffyvgV-w4nhnlGsmfq0.original.fullsize.png"/><br/> <br/> <br/>where $\cos \phi_{2}=\frac{R}{Z_{2}}=\frac{50}{50 \sqrt{2}}=\frac{1}{\sqrt{2}} \Rightarrow \phi_{2}=45^{\circ}$ $\therefore \quad$ Current $I$ from the circuit
<br/> <br/>$I=\frac{20}{100 \sqrt{2}}+\frac{20}{50 \sqrt{2}}=\mathrm{I}_{1}+\mathrm{I}_{2} \simeq 0.3 \mathrm{~A}$</div>


title: in-the-newman-projection-for-22dimethylbutane-x-and-y-can-respectively-be
slug: in-the-newman-projection-for-22dimethylbutane-x-and-y-can-respectively-be-16190
tags: ['Chemistry', 'General Organic Chemistry', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the Newman projection for 2,2-dimethylbutane<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jyUNg_6mD-YRdCFk2RndTgNILG1dMpSe27gLekPA9TA.original.fullsize.png"/><br/><br/>$X$ and $Y$ can respectively be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{H}$ and $\mathrm{H}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{H}$ and $\mathrm{C}_2 \mathrm{H}_5$<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/>$\mathrm{C}_2 \mathrm{H}_5$ and $\mathrm{H}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{CH}_3$ and $\mathrm{CH}_3$</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 answers are:
<span class="option-value"><br/>$\mathrm{H}$ and $\mathrm{C}_2 \mathrm{H}_5$<br/>, <br/>$\mathrm{CH}_3$ and $\mathrm{CH}_3$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/LRcYxFJ8pgtTu7gJBL1lLnFmiLfSD0N9djoHPP9yG8U.original.fullsize.png"/><br/><br/><br/>Conformation projection along $\mathrm{C}_1-\mathrm{C}_2$<br/>$\mathrm{C}_1$ contains all three $\mathrm{Hs}$<br/>So, $\quad X=\mathrm{H}$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/UmKWGE4IzP_J8gN7hc4PzbG2cHLO0s06uQ6HUgtSrR8.original.fullsize.png"/><br/><br/><br/>$\mathrm{C}_2$ contains two methyl and one ethyl group<br/>So, $\quad Y=\mathrm{C}_2 \mathrm{H}_5$<br/>Conformational projection along<br/>$$<br/>\mathrm{C}_2-\mathrm{C}_3<br/>$$<br/>$\mathrm{C}_2$ contains three methyl groups<br/>$\left(\mathrm{C}_2\right.$ form back carbon in the given structure)<br/>So, $\quad Y=\mathrm{CH}_3$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/99XNPwbz6laYONQD43P2IwseZwQ1EWIPEfxCtXPMZA8.original.fullsize.png"/><br/><br/>$\mathrm{C}_3$ contains two Hs and one methyl group<br/>$\left(\mathrm{C}_3\right.$ form front carbon in the given structure)<br/>So, $\quad X=\mathrm{CH}_3$<br/>$\mathrm{C}_3$ contains two Hs and one methyl group<br/>$\left(\mathrm{C}_3\right.$ form front carbon in the given structure)<br/>So, $\quad X=\mathrm{CH}_3$<br/>Isomerism (Stereochemistry)<br/>Conceptual (Structural visualisation) III</div>


tags: ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-the-options-given-below-let-e-denote-the-rest-mass-energy-of-a-nucleus-and-n-a-neutron-the-correct-option-is-2

slug:
in-the-options-given-below-let-e-denote-the-rest-mass-energy-of-a-nucleus-and-n-a-neutron-the-correct-option-is-2-49560

question:
<div class="question">In the options given below, let $E$ denote the rest mass energy of a nucleus and $n$ a neutron. The correct option is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$E\left({ }_{92}^{236} \mathrm{U}\right)&gt;E\left({ }_{53}^{137} \mathrm{I}\right)+E\left({ }_{36}^{97} Y\right)+2 E(n)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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/>$\left.E{ }_{92}^{236} \mathrm{U}\right) &lt; E\left({ }_{53}^{137} \mathrm{I}\right)+E\left({ }_{39}^{97} Y\right)+2 E(n)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$E\left({ }_{92}^{236} \mathrm{U}\right) &lt; E\left(5_{56}^{140} \mathrm{Ba}\right)+E\left({ }_{36}^{94} \mathrm{Kr}\right)+2 E(n)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$E\left(9_{92}^{236} \mathrm{U}\right) &lt; E\left(5_{56}^{140} \mathrm{Ba}\right)+E\left({ }_{36}^{94} \mathrm{Kr}\right)+2 E(n)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$E\left({ }_{92}^{236} \mathrm{U}\right)&gt;E\left({ }_{53}^{137} \mathrm{I}\right)+E\left({ }_{36}^{97} Y\right)+2 E(n)$<br/></span> </div>

answer:
<div class="solution">Rest mass of parent nucleus should be greater than the rest mass of daughter nuclei. Therefore, option (a) will be correct.</div>


title: in-the-reaction-2-x-b-2-h-6-bh-2-x-2-bh-4-t-h-e-amin-e-s-x-is-are
slug: in-the-reaction-2-x-b-2-h-6-bh-2-x-2-bh-4-t-h-e-amin-e-s-x-is-are-10748
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'JEE Advanced', 'JEE Advanced 2009 (Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction,<br/>$$<br/>2 \mathrm{X}+\mathrm{B}_2 \mathrm{H}_6 \longrightarrow\left[\mathrm{BH}_2(\mathrm{X})_2\right]^{+}\left[\mathrm{BH}_4\right]^{-}<br/>$$<br/>The amine (s) $X$, is (are)</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$\mathrm{NH}_3$<br/>, <br/>$\mathrm{CH}_3 \mathrm{NH}_2$<br/>, <br/>$\left(\mathrm{CH}_3\right)_2 \mathrm{NH}$<br/></span> </div>

answer:
<div class="solution">Small amines such as $\mathrm{NH}_3 \cdot \mathrm{CH}_3 \mathrm{NH}_2$ and $\left(\mathrm{CH}_3\right)_2 \mathrm{NH}$ give unsymmetrical cleavage of diborane according to following reaction,<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{B}_2 \mathrm{H}_6+2 \mathrm{NH}_3 \longrightarrow \\<br/>&amp; {\left[\mathrm{H}_2 \mathrm{~B}\left(\mathrm{NH}_3\right)_2\right]^{+}\left[\mathrm{BH}_4\right]^{-}} \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>Large amines, such as $\left(\mathrm{CH}_3\right)_3 \mathrm{~N}$ give symmetrical cleavage of diborane according to following reaction,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YhfSvFekusS1ekfppnWA9ksQ_OT9BCLkCB9xM2Mq8yE.rendered.fullsize.png"/></div>


title: in-the-reaction-hbr-the-products-are
slug: in-the-reaction-hbr-the-products-are-56247
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/WpcSmf9mSruKQidMpbn1ZkSwknUTJW66ijOz6VU1J7s.original.fullsize.png"/><br/><br/>$\stackrel{\mathrm{HBr}}{\longrightarrow}$<br/>the products are,</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/FCC6s1TCUwlVXhDdrao6sdSUvSPNT92S_AwtN33AbTQ.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/eWQApNuBIFyrGtkKrexTVAXUOi-DpUX5TlEBJOE1sXo.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/fCSpZTT3bifF7LTiLb9bcf1RWs2S4KRJkAhqBldx4kE.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/pRzZgsoOl-UImBOA0IhkHwBv8rKdO35XQk13rMQb8hk.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/pRzZgsoOl-UImBOA0IhkHwBv8rKdO35XQk13rMQb8hk.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/02XAyKlCumqJiH_vIJ0TSA_ZowYSW4yJVsuZjF_Svbg.original.fullsize.png"/><br/><br/>Alcohol-phenol-ether<br/>Conceptual, understanding of mechanism<br/>II (Given options raised the difficulty level)</div>


title: in-the-reaction-the-intermediates-is-are
slug: in-the-reaction-the-intermediates-is-are-26288
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction, the intermediate(s) is (are)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/7Ig40VgdgL1GnJfo-MIrPzx591YZTtZidoA2jAweZ0I.original.fullsize.png"/><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/QKOeqjn-5kWFb3dgBVLOMMNvejmzW0S0PkCly4CI_vk.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/Csqt09PIiI4kqtYRh-Ggeo0-p636NxhX1iwx2q4muV0.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4i8s67LE1UC5KfnYcyWN7MgpC6A7Y5k8bP4LYSjLhuc.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">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/neorBxWHY9M_V6zZV-hua3ZZnPVLPSkNKDN6jDjcGgE.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/QKOeqjn-5kWFb3dgBVLOMMNvejmzW0S0PkCly4CI_vk.original.fullsize.png"/><br/>, <br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4i8s67LE1UC5KfnYcyWN7MgpC6A7Y5k8bP4LYSjLhuc.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5RdSSwS0JrRF8Uf9VPicMs07VYx1jh9ckx_ouXuPQGw.original.fullsize.png"/><br/><br/>- $\mathrm{OH}$ and in aqueous in alkaline medium $-\overline{\mathrm{O}}$ are ortho and para directing. We expect the main intermediate products as mono ortho/para substituted product. Remember! if we consider (b), which could be there in small amount then we need to consider even (d) because $-\overline{\mathrm{O}}$ is strongest activating group that facilitates electrophilic substitution reaction overall on the ring. $-\overline{\mathrm{O}}$<br/>activates ortho and para position more but doesn't deactivate the meta position. We get a small amount of even the meta substituted product. But, as the question appears to be directly concerned answer should be only (a) and (c).<br/>Reaction mechanism (Electrophilic substitution)<br/>Conceptual<br/>II</div>


title: in-the-reaction-the-structure-of-the-product-t-is-1
slug: in-the-reaction-the-structure-of-the-product-t-is-1-89697
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction the structure of the product $T$ is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ty1ZkhKwuyldG1DQNcjd0AT3HPZOA43ocCLBqaMwhlg.original.fullsize.png"/><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/TSu69oPn4xTuC50RwgJPSGnSOC6XJVJZ-mowin5UJg0.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/QKC3ihcIr9XuXXsAK3kUGuAJy7UWULhCP56OhPlrx5A.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/JSMfI_RJhvcjmYjkePSDh0TGHLSxAyjrjbspJDmZuLc.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/WGmILMBmvsnyTghZRpx_lM0tY_NIiaNyxEG-TAi0vTM.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/QKC3ihcIr9XuXXsAK3kUGuAJy7UWULhCP56OhPlrx5A.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/eUfPSVZUxXHhPXPRNB41PEarJhi5jLOvwDpPGp4wf0U.original.fullsize.png"/><br/><br/>Acids and their derivatives Conceptual (Reaction understanding) II</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
in-the-scheme-given-below-the-total-number-of-intramolecular-aldol-condensation-products-formed-from-y-is-1

slug:
in-the-scheme-given-below-the-total-number-of-intramolecular-aldol-condensation-products-formed-from-y-is-1-41519

question:
<div class="question">In the scheme given below, the total number of intramolecular aldol condensation products formed from ' $Y$ ' is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/5H7_nrPU-8mhsPUhRpuQ421CGQImWgh3nDByt7s54gI.original.fullsize.png"/><br/></div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/2Zrp04_0tXQ4SVhpaVe-xuXBOlFDWBEbgSoc8krhlZg.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/-OIeOWmBWVaTrc0z1CbgH4SQuxniqCdxlYbyqMYVSwA.original.fullsize.png"/><br/><br/>For aldol condensation $\mathrm{C}-5$ and $\mathrm{C}-7$ can attack to C-1 similarly C-2 and C-10 can attack to C-6 but all give same product.<br/>Carbonyl compounds<br/>Conceptual<br/>III</div>


tags: ['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P1.db
title:
incandescent-bulbs-are-designed-by-keeping-in-mind-that-the-resistance-of-their-filament-increases-with-the-increase-in-temperature-if-at-room-tempera

slug:
incandescent-bulbs-are-designed-by-keeping-in-mind-that-the-resistance-of-their-filament-increases-with-the-increase-in-temperature-if-at-room-tempera-80357

question:
<div class="question">Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, $100 \mathrm{~W}, 60 \mathrm{~W}$ and $40 \mathrm{~W}$ bulbs have filament resistances $R_{100}, R_{60}$ and $R_{40}$, respectively, the relation between these resistances is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{1}{R_{100}}=\frac{1}{R_{40}}+\frac{1}{R_{60}}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$R_{100}=R_{40}+R_{60}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$R_{100}&gt;R_{60}&gt;R_{40}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{1}{R_{100}}&gt;\frac{1}{R_{60}}&gt;\frac{1}{R_{40}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>$\frac{1}{R_{100}}&gt;\frac{1}{R_{60}}&gt;\frac{1}{R_{40}}$</span> </div>

answer:
<div class="solution">$R=\frac{V^2}{P}$ or $R \propto \frac{1}{P}$ $\therefore \quad \frac{1}{R_{100}}&gt;\frac{1}{R_{60}}&gt;\frac{1}{R_{40}}$<br/>Hence, the correct option is (d).</div>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
inradius-of-a-circle-which-is-inscribed-in-an-isosceles-triangle-one-of-whose-angle-is-2-3-is-3-then-area-of-triangle-is

slug:
inradius-of-a-circle-which-is-inscribed-in-an-isosceles-triangle-one-of-whose-angle-is-2-3-is-3-then-area-of-triangle-is-76117

question:
<div class="question">Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is $2 \pi / 3$, is $\sqrt{3}$, then area of triangle is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$12+7 \sqrt{3}$<br/></span> </div>

answer:
<div class="solution">Let $A B=A C$ and $\angle A=120^{\circ}$<br/>$\therefore$ Area of $\Delta=\frac{1}{2} a^2 \sin 120^{\circ}$<br/>where, $a=A D+B D$<br/>$=\sqrt{3} \tan 30^{\circ}+\sqrt{3} \cot 15^{\circ}$<br/>$=1+\sqrt{3}\left(\frac{1+\tan 45^{\circ} \tan 30^{\circ}}{\tan 45^{\circ}-\tan 30^{\circ}}\right)$<br/>$=1+\sqrt{3}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)$<br/>$\therefore \quad a=4+2 \sqrt{3}$<br/>$\therefore$ Area of $\Delta=\frac{1}{2}(4+2 \sqrt{3})^2\left(\frac{\sqrt{3}}{2}\right)=12+7 \sqrt{3}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/PXl03TAYLnGL7W1m-BtgA8nH4CI9IIu9JAkQGn0XaRE.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Properties of Triangles', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P1.db
title:
internal-bisector-of-a-of-a-bc-meets-side-bc-at-d-a-line-drawn-through-d-perpendicular-to-a-d-intersects-the-side-a-c-at-e-and-side-a-b-at-f-if-a-b-an

slug:
internal-bisector-of-a-of-a-bc-meets-side-bc-at-d-a-line-drawn-through-d-perpendicular-to-a-d-intersects-the-side-a-c-at-e-and-side-a-b-at-f-if-a-b-an-37619

question:
<div class="question">Internal bisector of $\angle A$ of $\triangle A B C$ meets side $B C$ at $D$. A line drawn through $D$ perpendicular to $A D$ intersects the side $A C$ at $E$ and side $A B$ at $F$. If $a, b$ and $c$ represent sides of $\triangle A B C$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>$A E$ is $\mathrm{HM}$ of $b$ and $c$<br/>, <br/>$A D=\frac{2 b c}{b+c} \cos \frac{A}{2}$<br/>, <br/>$E F=\frac{4 b c}{b+c} \sin \frac{A}{2}$<br/>, <br/>the $\triangle A E F$ is isosceles</span> </div>

answer:
<div class="solution">We have, $\triangle A B C=\triangle A B D+\triangle A C D$<br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \frac{1}{2} b c \sin A=\frac{1}{2} c A D \sin \frac{A}{2}+\frac{1}{2} b A D \sin \frac{A}{2} \\<br/>&amp; \Rightarrow \quad A D=\frac{2 b c}{b+c} \cos \frac{A}{2}<br/>\end{aligned}<br/>$$<br/>Again, $\quad A E=A D \sec \frac{A}{2}=\frac{2 b c}{b+c}$<br/>$\therefore A E$ is $\mathrm{HM}$ of $b$ and $c$.<br/>$$<br/>\begin{aligned}<br/>E F &amp; =E D+D F=2 D E=2 A D \tan \frac{A}{2} \\<br/>&amp; =2 \frac{2 b c}{b+c} \cos \frac{A}{2} \tan \frac{A}{2}=\frac{4 b c}{b+c} \sin \frac{A}{2}<br/>\end{aligned}<br/>$$<br/>As $A D \perp E F$ and $D E=D F$ and $A D$ is bisector.<br/>$\Rightarrow \triangle A E F$ is isosceles.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_Xl9hjzWwQRk4E-I6GNC2M5Xwg65OHYLXnkwwQZg4pM.original.fullsize.png"/><br/></div>


title: let-0-2-be-such-that-2-cos-1-sin-sin-2-tan-2-cot-2-cos-1-tan-2-0-and-1-sin-2-3-then-cannot-satisfy
tags: ['Mathematics', 'Trigonometric Equations', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch2_P1.db
slug:
let-0-2-be-such-that-2-cos-1-sin-sin-2-tan-2-cot-2-cos-1-tan-2-0-and-1-sin-2-3-then-cannot-satisfy-27871

question:
<div class="question">Let $\theta, \varphi \in[0,2 \pi]$ be such that $2 \cos \theta(1-\sin \varphi)=\sin ^{2} \theta \times$ $\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1, \tan (2 \pi-\theta)&gt;0$ and
<br/> <br/>$-1 &lt; \sin \theta &lt; -\frac{\sqrt{3}}{2}$, then $\varphi$ cannot satisfy</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$0 &lt; \varphi &lt; \frac{\pi}{2}$, $\frac{4 \pi}{3} &lt; \varphi &lt; \frac{3 \pi}{2}$, $\frac{3 \pi}{2} &lt; \varphi &lt; 2 \pi$</span> </div>

answer:
<div class="solution">As $\tan (2 \pi-\theta)&gt;0$ and $-1 &lt; \sin \theta &lt; -\frac{\sqrt{3}}{2}, \theta \in[0,2 \pi]$
<br/> <br/>Hence $\frac{3 \pi}{2} &lt; \theta &lt; \frac{5 \pi}{3}$
<br/> <br/>Now $2 \cos \theta(1-\sin \varphi)=\sin ^{2} \theta\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1$
<br/> <br/>$\Rightarrow 2 \cos \theta(1-\sin \varphi)=2 \sin \theta \cos \varphi-1$
<br/> <br/>$\Rightarrow 2 \cos \theta+1=2 \sin (\theta+\varphi)$
<br/> <br/>As $\quad \theta \in\left(\frac{3 \pi}{2}, \frac{5 \pi}{3}\right), 1 &lt; 2 \sin (\theta+\varphi) &lt; 2$
<br/> <br/>As $\theta+\varphi \in\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$ or $(\theta+\varphi) \in\left(\frac{13 \pi}{6}, \frac{17 \pi}{6}\right)$
<br/> <br/>We have $\varphi \in\left(-\frac{3 \pi}{2},-\frac{2 \pi}{3}\right) \cup\left(\frac{2 \pi}{3}, \frac{7 \pi}{6}\right)$</div>


