# questions

tags: ['Mathematics', 'Differentiation', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly-2

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

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

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

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

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


tags: ['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-functions-defined-implicitly-by-the-equation-y-3-3-y-x-0-on-various-intervals-in-the-real-line-if-x-2-2-the-equation-implicitly

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

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

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

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-distance-of-the-point-1-1-1-from-the-plane-passing-through-the-p

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$2 / \sqrt{75}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$7 / \sqrt{75}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$13 / \sqrt{75}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$23 / \sqrt{75}$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Three Dimensional Geometry', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-shortest-distance-between-l-1-and-l-2-is

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>0<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$17 / \sqrt{3}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$41 / 5 \sqrt{3}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$17 / 5 \sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$17 / 5 \sqrt{3}$</span> </div>

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


tags: ['Mathematics', 'Vector Algebra', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-lines-l-1-3-x-1-1-y-2-2-z-1-l-2-1-x-2-2-y-2-3-z-3-question-the-unit-vector-perpendicular-to-both-l-1-and-l-2-is

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

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

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

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

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


tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-area-bounded-by-th-1

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{3}{4}, 3\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(\frac{21}{64}, \frac{11}{16}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$(9,10)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(0, \frac{21}{64}\right)$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Area Under Curves', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-area-bounded-by-th

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\left(\frac{3}{4}, 3\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(\frac{21}{64}, \frac{11}{16}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$(9,10)$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(0, \frac{21}{64}\right)$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-function-f-x-is

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

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

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

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

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


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-consider-the-polynomial-f-x-1-2-x-3-x-2-4-x-3-let-s-be-the-sum-of-all-distinct-real-roots-of-f-x-and-let-t-s-question-the-real-number-s-lies

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

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

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

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

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


source_db: MarksBatch2_P2.db
title:
paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-1

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

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

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

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

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

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


source_db: MarksBatch2_P2.db
title:
paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan-2

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

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

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

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

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

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


answer: <div class="solution">Iron is removed in the form of slag of $\mathrm{FeSiO}_3$</div>
source_db: MarksBatch2_P2.db
title:
paragraph-copper-is-the-most-noble-of-the-first-row-transition-metals-and-occurs-in-small-deposits-in-several-countries-ores-of-copper-include-chalcan

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

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

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

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{FeSiO}_3$</span> </div>


tags: ['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-1

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

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

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

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

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


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-2

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

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

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

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

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


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo-3

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

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

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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Eq6JNIWXnBI01sTDgigb7jHuZknso7zoWR6BAor3NXI.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_HDsS9GEl6BAgpFSfYdkHMdlHb5znmSRWfOnzHfAyn8.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/iN-Jh5WFNNzafMGkO_5UaQ4zG1cmNJKWBoC5b0k21Jw.original.fullsize.png"/><br/></span> </li> </ul>

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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.original.fullsize.png"/><br/></span> </div>

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


tags: ['Physics', 'Current Electricity', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-electrical-resistance-of-certain-materials-known-as-superconductors-changes-abruptly-from-a-nonzero-value-to-zero-as-their-temperature-is-lo

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

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

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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Eq6JNIWXnBI01sTDgigb7jHuZknso7zoWR6BAor3NXI.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/_HDsS9GEl6BAgpFSfYdkHMdlHb5znmSRWfOnzHfAyn8.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/iN-Jh5WFNNzafMGkO_5UaQ4zG1cmNJKWBoC5b0k21Jw.original.fullsize.png"/><br/></span> </li> </ul>

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/1qP5Xdu-jyN7Q9p7JcB3vR7W0mEGiBfwUL3daciHh5g.original.fullsize.png"/><br/></span> </div>

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


tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-1

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

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

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

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

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


tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if-2

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

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

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

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

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


tags: ['Mathematics', 'Functions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-if-a-continuous-f-defined-on-the-real-line-r-assume-positive-and-negative-values-in-r-then-the-equation-f-x-0-has-a-root-in-r-for-example-if

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

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

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

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

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


tags: ['Physics', 'Mathematics in Physics', 'JEE Advanced', 'JEE Advanced 2018 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-if-the-measurement-errors-in-all-the-independent-quantities-are-known-then-it-is-possible-to-determine-the-error-in-any-dependent-quantity-t-1

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

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

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

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

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


tags: ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2018 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-if-the-measurement-errors-in-all-the-independent-quantities-are-known-then-it-is-possible-to-determine-the-error-in-any-dependent-quantity-t

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math> <mrow> <mn>0.04</mn></mrow> </math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>0.03</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mn>0.02</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>0.01</mn></math></span> </li> </ul>

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

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


tags: ['Physics', 'Atomic Physics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-1

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

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

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

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

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


tags: ['Physics', 'Atomic Physics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h-2

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

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

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

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

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


tags: ['Physics', 'Atomic Physics', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-a-mixture-of-h-h-gas-he-is-singly-ionised-he-atom-h-atoms-and-he-ions-are-excited-to-their-respective-first-excited-states-subsequently-h

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$6.5 \times 10^{-7} \mathrm{~m}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$5.6 \times 10^{-7} \mathrm{~m}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$4.8 \times 10^{-7} \mathrm{~m}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$4.0 \times 10^{-7} \mathrm{~m}$</span> </li> </ul>

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

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


tags: ['Chemistry', 'Solid State', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-1

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$74 \%$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$47.6 \%$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$32 \%$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$26 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$26 \%$</span> </div>

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


tags: ['Chemistry', 'Solid State', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t-2

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>4<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>6<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>12<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>17</span> </li> </ul>

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

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


tags: ['Chemistry', 'Solid State', 'JEE Advanced', 'JEE Advanced 2008 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-hexagonal-system-of-crystals-a-frequently-encountered-arrangement-of-atoms-is-described-as-a-hexagonal-prism-here-the-top-and-bottom-of-t

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$24 \sqrt{2} r^3$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$16 \sqrt{2} r^3$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$12 \sqrt{2} r^3$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{64}{3 \sqrt{3}} r^3$</span> </li> </ul>

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

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


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-product-l-is-1

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

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

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

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/KwVQJ4yEiI650gNQK_2isQuUO3ZR4msjcfvopx0eJQk.original.fullsize.png"/><br/></span> </div>

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


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-product-l-is

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

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

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

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/KwVQJ4yEiI650gNQK_2isQuUO3ZR4msjcfvopx0eJQk.original.fullsize.png"/><br/></span> </div>

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


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structure-of-the-product-i-is

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

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

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

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/83pBEan488VtTmOBa9pYj6l1EqSCSzj_d94U0Xck3ro.original.fullsize.png"/><br/></span> </div>

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


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-in-the-following-reaction-sequence-product-i-j-and-l-are-formed-k-represent-a-reagent-question-the-structures-of-compounds-j-and-k-respectiv

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

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

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/QHrLAdXTmQr321aLV76QNLy-SpXCtNzV3OHdN-oUnDA.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZcwCvzXblmPzNNWA4rCPzMtVtcuKq0QI0dL0tiLqWZU.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Ddt5qpMV8yRxbhjO-bTY3LecFVXB-9iWG-SPI6nutf0.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/kiadc3hAoyJjKTmXQbaPFBxO9QBIGU5bIVk5Y9ER5JM.original.fullsize.png"/><br/></span> </li> </ul>

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/QHrLAdXTmQr321aLV76QNLy-SpXCtNzV3OHdN-oUnDA.original.fullsize.png"/><br/></span> </div>

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


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-1

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{H}_1&gt;\mathrm{H}_2&gt;\mathrm{H}_3&gt;\ldots$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$H_1 &lt; H_2 &lt; H_3 &lt; \ldots$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$H_1&gt;H_3&gt;H_5&gt;\ldots$ and $H_2 &lt; H_4 &lt; H_6 &lt; \ldots$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{H}_1 &lt; \mathrm{H}_3 &lt; \mathrm{H}_5 &lt; \ldots$ and $\mathrm{H}_2&gt;\mathrm{H}_4&gt;\mathrm{H}_6&gt;\ldots$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1-2

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

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$A_1&gt;A_2&gt;\ldots$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$A_1 &lt; A_2 &lt; A_3 &lt; \ldots$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$A_1&gt;A_3&gt;A_5&gt;\ldots$ and $A_2 &lt; A_4 &lt; A_6 &lt; \ldots$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$A_1 &lt; A_3 &lt; A_5 &lt; \ldots$ and $A_2&gt;A_4&gt;A_6&gt;\ldots$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-1-g-1-h-1-denote-the-arithmetic-geometric-and-harmonic-means-respectively-of-two-distinct-positive-numbers-for-n-2-let-a-n-1-and-h-n-1

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$G_1&gt;G_2&gt;G_3&gt;\ldots$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$G_1 &lt; G_2 &lt; G_3 &lt; \ldots$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$G_1=G_2=G_3=\ldots$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$G_1 &lt; G_3 &lt; G_5 &lt; \ldots$ and $G_2&gt;G_4&gt;G_6&gt;\ldots$</span> </li> </ul>

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>6</span> </div>
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-if-the-point-p-a-b-c-with-reference-to-eq-e-lies-on-th

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

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

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-let-b-6-with-a-and-c-satisfying-eq-e-if-and-are-the-ro

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

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

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

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

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


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-and-c-be-three-real-numbers-satisfying-a-b-c-1-8-7-9-2-3-7-7-7-0-0-0-question-let-be-a-solution-of-x-3-1-0-with-im-0-if-a-2-with-b-a

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

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

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

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

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


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-let-z-be-any-point-in-a-b-c-and-l

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

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

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

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

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


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-let-z-be-any-point-in-a-b-c-the-z

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

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>25 and 29<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>30 and 34<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>35 and 39<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>40 and 44</span> </li> </ul>

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

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


tags: ['Mathematics', 'Complex Number', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-the-number-of-elements-in-the-set-1

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

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

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

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

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


tags: ['Mathematics', 'Complex Number', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-b-c-be-three-sets-of-complex-numbers-as-defined-below-a-z-lm-z-1-b-z-z-2-i-3-c-z-re-1-i-z-2-question-the-number-of-elements-in-the-set

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

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

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

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

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


tags: ['Mathematics', 'Permutation Combination', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-1

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que-2

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

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

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

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

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


tags: ['Mathematics', 'Determinants', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-a-be-the-set-of-all-3-3-symmetric-matrices-all-of-whose-entries-are-either-0-or-1-five-of-these-entries-are-1-and-four-of-them-are-0-que

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

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

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

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-1

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

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

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

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g-2

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

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

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

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

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


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2010 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-p-be-an-odd-prime-number-and-t-p-be-the-following-set-of-2-2-matrices-t-p-a-a-c-b-a-a-b-c-0-1-2-p-1-s-t-ro-n-g-q-u-es-t-i-o-n-s-t-ro-n-g

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

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

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

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

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


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-u-1-and-u-2-be-two-urns-such-that-u-1-contains-3-white-and-2-red-balls-and-u-2-contains-only-1-white-ball-a-fair-coin-is-tossed-if-head--1

slug:
paragraph-let-u-1-and-u-2-be-two-urns-such-that-u-1-contains-3-white-and-2-red-balls-and-u-2-contains-only-1-white-ball-a-fair-coin-is-tossed-if-head-1-84793

question:
<div class="question"><strong>Paragraph:</strong><br/>Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 balls are drawn at random from $U_1$ and put into $U_2$. Now, 1 ball is drawn at random from $U_2$.<strong>Question:</strong><br/>Given that the drawn ball from $U_2$ is white, the probability that head appeared on the coin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{17}{23}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{11}{23}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{15}{23}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{12}{23}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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/>$\frac{12}{23}$</span> </div>

answer:
<div class="solution">$P$ (head appeared/white from $V_2$ )<br/>$$<br/>\begin{aligned}<br/>&amp; =P(H) \cdot \frac{\left\{\frac{{ }^3 C_1}{{ }^5 C_1} \times \frac{{ }^2 C_1}{{ }^2 C_1}+\frac{{ }^2 C_1}{{ }^5 C_1} \times \frac{{ }^1 C_1}{{ }^2 C_1}\right\}}{\frac{23}{30}} \\<br/>&amp; =\frac{1}{2} \frac{\left\{\frac{3}{5} \times 1+\frac{2}{5} \times \frac{1}{2}\right\}}{\frac{23}{30}}=\frac{12}{23} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-u-1-and-u-2-be-two-urns-such-that-u-1-contains-3-white-and-2-red-balls-and-u-2-contains-only-1-white-ball-a-fair-coin-is-tossed-if-head-

slug:
paragraph-let-u-1-and-u-2-be-two-urns-such-that-u-1-contains-3-white-and-2-red-balls-and-u-2-contains-only-1-white-ball-a-fair-coin-is-tossed-if-head-30433

question:
<div class="question"><strong>Paragraph:</strong><br/>Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 balls are drawn at random from $U_1$ and put into $U_2$. Now, 1 ball is drawn at random from $U_2$.<strong>Question:</strong><br/>The probability of the drawn ball from $U_2$ being white is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{13}{30}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{23}{30}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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/>$\frac{19}{30}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{11}{30}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{23}{30}$<br/></span> </div>

answer:
<div class="solution">Now, probability of the drawn ball from $\mathrm{V}_2$ being white is<br/>$$<br/>\Rightarrow P \text { (white/ } V_2 \text { ) }<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \left.\Rightarrow P \text { (white/ } V_2\right) \\<br/>&amp; =P(H) \cdot\left\{\frac{{ }^3 C_1}{{ }^5 C_1} \times \frac{{ }^2 C_1}{{ }^2 C_1}+\frac{{ }^2 C_1}{{ }^5 C_1} \times \frac{{ }^1 C_1}{{ }^2 C_1}\right\} \\<br/>&amp; +P(T)\left\{\frac{{ }^3 C_2}{{ }^5 C_2} \times \frac{{ }^3 C_2}{{ }^3 C_2}+\frac{{ }^2 C_2}{{ }^5 C_2} \times \frac{{ }^1 C_1}{{ }^3 C_2}\right. \\<br/>&amp; \left.+\frac{{ }^3 C_1 \cdot{ }^2 C_1}{{ }^5 C_2} \times \frac{{ }^2 C_1}{{ }^3 C_2}\right\} \\<br/>&amp; \text { Now, } P\left(\text { white } / V_2\right)=\frac{1}{2}\left\{\frac{3}{5} \times 1+\frac{2}{5} \times \frac{1}{2}\right\} \\<br/>&amp; +\frac{1}{2}\left\{\frac{3}{10} \times 1+\frac{1}{10} \times \frac{1}{3}+\frac{6}{10} \times \frac{2}{3}\right\}=\frac{23}{30} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-v-r-denotes-the-sum-of-the-first-r-terms-of-an-arithmetic-progression-a-p-whose-first-term-is-r-and-the-common-difference-is-2-r-1-let-t-1

slug:
paragraph-let-v-r-denotes-the-sum-of-the-first-r-terms-of-an-arithmetic-progression-a-p-whose-first-term-is-r-and-the-common-difference-is-2-r-1-let-t-1-53467

question:
<div class="question"><strong>Paragraph:</strong><br/>Let $V_r$ denotes the sum of the first $r$ terms of an arithmetic progression $(A P)$ whose first term is $r$ and the common difference is $(2 r-1)$. Let $T_r=V_{r+1}-V_r-2$ and $Q_r=T_{r+1}-T_r$ for $r=1,2, \ldots$<strong>Question:</strong><br/>$T_r$ is always</div>

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

answer:
<div class="solution">$V_{r+1}-V_r=(r+1)^3-r^3-\frac{1}{2}\left[(r+1)^2-r^2\right]+\frac{1}{2}(1)$<br/>$$<br/>\begin{aligned}<br/>&amp; =3 r^2+2 r+1 \\<br/>\therefore \quad T_r &amp; =3 r^2+2 r-1=(r+1)(3 r-1)<br/>\end{aligned}<br/>$$<br/>which is a composite number.</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-v-r-denotes-the-sum-of-the-first-r-terms-of-an-arithmetic-progression-a-p-whose-first-term-is-r-and-the-common-difference-is-2-r-1-let-t-2

slug:
paragraph-let-v-r-denotes-the-sum-of-the-first-r-terms-of-an-arithmetic-progression-a-p-whose-first-term-is-r-and-the-common-difference-is-2-r-1-let-t-2-11660

question:
<div class="question"><strong>Paragraph:</strong><br/>Let $V_r$ denotes the sum of the first $r$ terms of an arithmetic progression $(A P)$ whose first term is $r$ and the common difference is $(2 r-1)$. Let $T_r=V_{r+1}-V_r-2$ and $Q_r=T_{r+1}-T_r$ for $r=1,2, \ldots$<strong>Question:</strong><br/>Which one of the following is a correct statement?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 5<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 6<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 11<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$Q_1=Q_2=Q_3=\ldots$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 6<br/></span> </div>

answer:
<div class="solution">Since, $T_r=3 r^2+2 r-1$<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; \therefore &amp; T_{r+1} &amp; =3(r+1)^2+2(r+1)-1 \\<br/>&amp; \therefore &amp; Q_r &amp; =T_{r+1}-T_r=3[2 r+1]+2[1] \\<br/>\Rightarrow &amp; Q_r &amp; =6 r+5 \\<br/>\Rightarrow &amp; Q_{r+1} &amp; =6(r+1)+5<br/>\end{array}<br/>$$<br/>Common difference $=Q_{r+1}-Q_r=6$</div>


tags: ['Mathematics', 'Sequences and Series', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-let-v-r-denotes-the-sum-of-the-first-r-terms-of-an-arithmetic-progression-a-p-whose-first-term-is-r-and-the-common-difference-is-2-r-1-let-t

slug:
paragraph-let-v-r-denotes-the-sum-of-the-first-r-terms-of-an-arithmetic-progression-a-p-whose-first-term-is-r-and-the-common-difference-is-2-r-1-let-t-91576

question:
<div class="question"><strong>Paragraph:</strong><br/>Let $V_r$ denotes the sum of the first $r$ terms of an arithmetic progression $(A P)$ whose first term is $r$ and the common difference is $(2 r-1)$. Let $T_r=V_{r+1}-V_r-2$ and $Q_r=T_{r+1}-T_r$ for $r=1,2, \ldots$<strong>Question:</strong><br/>The sum $V_1+V_2+\ldots+V_n$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{1}{12} n(n+1)\left(3 n^2+n+2\right)$<br/></span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; V_r=\frac{r}{2}[2 r+(r-1)(2 r-1)]=\frac{1}{2}\left(2 r^3-r^2+r\right) \\<br/>&amp; \therefore \quad \Sigma V_r=\frac{1}{2}\left[2 \Sigma r^3-\Sigma r^2+\Sigma r\right] \\<br/>&amp; =\frac{1}{2}\left[2\left(\frac{n(n+1)}{2}\right)^2-\frac{n(n+1)(2 n+1)}{6}+\frac{n(n+1)}{2}\right] \\<br/>&amp; =\frac{1}{12} n(n+1)\left(3 n^2+n+2\right) \text {. } \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2006']
answer: <div class="solution">No solution is required.</div>
source_db: MarksBatch2_P2.db
title:
paragraph-modern-trains-are-based-on-maglev-technology-in-which-trains-are-magnetically-leviated-which-runs-its-eds-maglev-system-there-are-coils-on-b-1

slug:
paragraph-modern-trains-are-based-on-maglev-technology-in-which-trains-are-magnetically-leviated-which-runs-its-eds-maglev-system-there-are-coils-on-b-1-79693

question:
<div class="question"><strong>Paragraph:</strong><br/>Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.<br/>There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.<br/>Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.<strong>Question:</strong><br/>What is the disadvantage of this system?</div>

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


tags: ['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2006']
answer: <div class="solution">No solution is required.</div>
source_db: MarksBatch2_P2.db
title:
paragraph-modern-trains-are-based-on-maglev-technology-in-which-trains-are-magnetically-leviated-which-runs-its-eds-maglev-system-there-are-coils-on-b-2

slug:
paragraph-modern-trains-are-based-on-maglev-technology-in-which-trains-are-magnetically-leviated-which-runs-its-eds-maglev-system-there-are-coils-on-b-2-92409

question:
<div class="question"><strong>Paragraph:</strong><br/>Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.<br/>There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.<br/>Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.<strong>Question:</strong><br/>What is the advantage of this system?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>No friction hence no power consumption<br/></span> </div>


tags: ['Physics', 'Electromagnetic Induction', 'JEE Advanced', 'JEE Advanced 2006']
answer: <div class="solution">No solution is required.</div>
source_db: MarksBatch2_P2.db
title:
paragraph-modern-trains-are-based-on-maglev-technology-in-which-trains-are-magnetically-leviated-which-runs-its-eds-maglev-system-there-are-coils-on-b

slug:
paragraph-modern-trains-are-based-on-maglev-technology-in-which-trains-are-magnetically-leviated-which-runs-its-eds-maglev-system-there-are-coils-on-b-44957

question:
<div class="question"><strong>Paragraph:</strong><br/>Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.<br/>There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.<br/>Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.<strong>Question:</strong><br/>Which force causes the train to elevate up?</div>

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

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


tags: ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-1

slug:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-1-66827

question:
<div class="question"><strong>Paragraph:</strong><br/>$P_{14-16}$ : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in the figure. The atmospheric pressure is $p_0$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4uUwDqNsXDagR3met7Ief3ZxT31gi025zIfyrEsgmZk.original.fullsize.png"/><br/><strong>Question:</strong><br/>While the piston is at a distance $2 L$ from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is</div>

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

answer:
<div class="solution">Let $p$ be the pressure in equilibrium<br/>Then,<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad p=p_0-\frac{M g}{A}=p_0-\frac{M g}{\pi R^2} \\<br/>&amp; \text { Applying } \quad p_1 V_1=p_2 V_2 \\<br/>&amp; p_0(2 A L)=(p)\left(A L^{\prime}\right) \\<br/>&amp; L^{\prime}=\frac{2 p_0 L}{p}=\left(\frac{p_0}{p_0-\frac{M g}{\pi R^2}}\right) \text { (2L) } \\<br/>&amp; =\left(\frac{p_0 \pi R^2}{\pi R^2 p_0-M g}\right)(2 L) \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>$\therefore$ Option (d) is correct.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-2

slug:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-2-65104

question:
<div class="question"><strong>Paragraph:</strong><br/>$P_{14-16}$ : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in the figure. The atmospheric pressure is $p_0$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4uUwDqNsXDagR3met7Ief3ZxT31gi025zIfyrEsgmZk.original.fullsize.png"/><br/><strong>Question:</strong><br/>The piston is now pulled out slowly and held at a distance $2 L$ from the top. The pressure in the cylinder between its top and the piston will then be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$p_0$<br/></span> </div>

answer:
<div class="solution">Since it is open from top, pressure will be $p_0$. $\therefore$ Option (a) is correct.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-3

slug:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-3-93752

question:
<div class="question"><strong>Paragraph:</strong><br/>$P_{14-16}$ : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in the figure. The atmospheric pressure is $p_0$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4uUwDqNsXDagR3met7Ief3ZxT31gi025zIfyrEsgmZk.original.fullsize.png"/><br/><strong>Question:</strong><br/>The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $\rho$. In equilibrium, the height $H$ of the water column in the cylinder satisfies<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u6L69mqUQ4YOY0tduE71-0P8gZFsbvh9f1xHSgJEuBU.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)+L_0 p_0=0$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\rho g\left(L_0-H\right)^2-p_0\left(L_0-H\right)-L_0 p_0=0$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)-L_0 p_0=0$<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/>$\rho g\left(L_0-H\right)^2-p_0\left(L_0-H\right)+L_0 p_0=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)-L_0 p_0=0$<br/></span> </div>

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>p_1=p_2 \\<br/>p_0+\rho g\left(L_0-H\right)=p<br/>\end{gathered}<br/>$$<br/>Now, applying $p_1 V_1=p_2 V_2$<br/>for the air inside the cylinder, we have<br/>$$<br/>p_0\left(L_0\right)=p\left(L_0-H\right)<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/7oQA8KNhtTUSHFHvt9EQSIUJcVfMbFX9a2U9386X-5c.original.fullsize.png"/><br/><br/>$\therefore \quad p=\frac{p_0 L_0}{L_0-H}$<br/>Substituting in Eq. (i), we have<br/>$$<br/>p_0+\rho g\left(L_0-H\right)=\frac{p_0 L_0}{L_0-H}<br/>$$<br/>or $\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)-p_0 L_0=0$<br/>$\therefore$ Option (c) is correct.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-4

slug:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-4-18500

question:
<div class="question"><strong>Paragraph:</strong><br/>$P_{14-16}$ : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in the figure. The atmospheric pressure is $p_0$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4uUwDqNsXDagR3met7Ief3ZxT31gi025zIfyrEsgmZk.original.fullsize.png"/><br/><strong>Question:</strong><br/>While the piston is at a distance $2 L$ from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is</div>

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

answer:
<div class="solution">Let $p$ be the pressure in equilibrium<br/>Then,<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad p=p_0-\frac{M g}{A}=p_0-\frac{M g}{\pi R^2} \\<br/>&amp; \text { Applying } \quad p_1 V_1=p_2 V_2 \\<br/>&amp; p_0(2 A L)=(p)\left(A L^{\prime}\right) \\<br/>&amp; L^{\prime}=\frac{2 p_0 L}{p}=\left(\frac{p_0}{p_0-\frac{M g}{\pi R^2}}\right) \text { (2L) } \\<br/>&amp; =\left(\frac{p_0 \pi R^2}{\pi R^2 p_0-M g}\right)(2 L) \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>$\therefore$ Option (d) is correct.</div>


tags: ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its

slug:
paragraph-p-14-16-paragraph-for-questions-nos-14-to-16-a-fixed-thermally-conducting-cylinder-has-a-radius-r-and-height-l-0-the-cylinder-is-open-at-its-80281

question:
<div class="question"><strong>Paragraph:</strong><br/>$P_{14-16}$ : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in the figure. The atmospheric pressure is $p_0$.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/4uUwDqNsXDagR3met7Ief3ZxT31gi025zIfyrEsgmZk.original.fullsize.png"/><br/><strong>Question:</strong><br/>The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $\rho$. In equilibrium, the height $H$ of the water column in the cylinder satisfies<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/u6L69mqUQ4YOY0tduE71-0P8gZFsbvh9f1xHSgJEuBU.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)+L_0 p_0=0$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\rho g\left(L_0-H\right)^2-p_0\left(L_0-H\right)-L_0 p_0=0$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)-L_0 p_0=0$<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/>$\rho g\left(L_0-H\right)^2-p_0\left(L_0-H\right)+L_0 p_0=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)-L_0 p_0=0$<br/></span> </div>

answer:
<div class="solution">$$<br/>\begin{gathered}<br/>p_1=p_2 \\<br/>p_0+\rho g\left(L_0-H\right)=p<br/>\end{gathered}<br/>$$<br/>Now, applying $p_1 V_1=p_2 V_2$<br/>for the air inside the cylinder, we have<br/>$$<br/>p_0\left(L_0\right)=p\left(L_0-H\right)<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/7oQA8KNhtTUSHFHvt9EQSIUJcVfMbFX9a2U9386X-5c.original.fullsize.png"/><br/><br/>$\therefore \quad p=\frac{p_0 L_0}{L_0-H}$<br/>Substituting in Eq. (i), we have<br/>$$<br/>p_0+\rho g\left(L_0-H\right)=\frac{p_0 L_0}{L_0-H}<br/>$$<br/>or $\rho g\left(L_0-H\right)^2+p_0\left(L_0-H\right)-p_0 L_0=0$<br/>$\therefore$ Option (c) is correct.</div>


tags: ['Physics', 'Rotational Motion', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-p-17-19-paragraph-for-questions-nos-17-to-19-two-discs-a-and-b-are-mounted-coaxially-on-a-vertical-axle-the-discs-have-moments-of-inertia-i--1

slug:
paragraph-p-17-19-paragraph-for-questions-nos-17-to-19-two-discs-a-and-b-are-mounted-coaxially-on-a-vertical-axle-the-discs-have-moments-of-inertia-i-1-42236

question:
<div class="question"><strong>Paragraph:</strong><br/>$\mathbf{P}_{17 \text { - } 19}$ : Paragraph for Questions Nos. 17 to 19 Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.<strong>Question:</strong><br/>The loss of kinetic energy during the above process is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{I \omega^2}{2}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{i \omega^2}{3}$<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">C</span> <span class="option-data"><br/>$\frac{I \omega^2}{4}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{I \omega^2}{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{i \omega^2}{3}$<br/></span> </div>

answer:
<div class="solution">Loss of kinetic energy $=K_i-K_f$<br/>$$<br/>\begin{aligned}<br/>&amp; =\left\{\frac{1}{2} I(2 \omega)^2+\frac{1}{2}(2 I)(\omega)^2\right\}-\frac{1}{2}(3 I)\left(\frac{4}{3} \omega\right)^2 \\<br/>&amp; =\frac{1}{3} I \omega^2<br/>\end{aligned}<br/>$$<br/>$\therefore$ Option (b) is correct.</div>


tags: ['Physics', 'Rotational Motion', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-p-17-19-paragraph-for-questions-nos-17-to-19-two-discs-a-and-b-are-mounted-coaxially-on-a-vertical-axle-the-discs-have-moments-of-inertia-i--2

slug:
paragraph-p-17-19-paragraph-for-questions-nos-17-to-19-two-discs-a-and-b-are-mounted-coaxially-on-a-vertical-axle-the-discs-have-moments-of-inertia-i-2-66363

question:
<div class="question"><strong>Paragraph:</strong><br/>$\mathbf{P}_{17 \text { - } 19}$ : Paragraph for Questions Nos. 17 to 19 Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.<strong>Question:</strong><br/>The ratio $\frac{x_1}{x_2}$ is</div>

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

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\frac{1}{2} I(2 \omega)^2 &amp; =\frac{1}{2} k x_1^2 \\<br/>\frac{1}{2}(2 I)(\omega)^2 &amp; =\frac{1}{2} k x_2^2<br/>\end{aligned}<br/>$$<br/>From Eqs. (i) and (ii), we have<br/>$$<br/>\frac{x_1}{x_2}=\sqrt{2}<br/>$$<br/>$\therefore$ Option (c) is correct.</div>


tags: ['Physics', 'Rotational Motion', 'JEE Advanced', 'JEE Advanced 2007 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-p-17-19-paragraph-for-questions-nos-17-to-19-two-discs-a-and-b-are-mounted-coaxially-on-a-vertical-axle-the-discs-have-moments-of-inertia-i-

slug:
paragraph-p-17-19-paragraph-for-questions-nos-17-to-19-two-discs-a-and-b-are-mounted-coaxially-on-a-vertical-axle-the-discs-have-moments-of-inertia-i-74108

question:
<div class="question"><strong>Paragraph:</strong><br/>$\mathbf{P}_{17 \text { - } 19}$ : Paragraph for Questions Nos. 17 to 19 Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.<strong>Question:</strong><br/>When disc $B$ is brought in contact with disc $A$, they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\frac{2 I \omega}{3 t}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\frac{9 I \omega}{2 t}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{91 \omega}{4 t}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\frac{3 I \omega}{2 t}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{2 I \omega}{3 t}$<br/></span> </div>

answer:
<div class="solution">Let $\omega^{\prime}$ be the common velocity. Then from conservation of angular momentum, we have<br/>$$<br/>\begin{aligned}<br/>(i+2 I) \omega^{\prime} &amp; =I(2 \omega)+2 I(\omega) \\<br/>\omega^{\prime} &amp; =\frac{4}{3} \omega<br/>\end{aligned}<br/>$$<br/>From the equation,<br/>Angular impulse = change in angular momentum for any of the disc, we have<br/>$$<br/>\begin{aligned}<br/>\tau t &amp; =I(2 \omega)-I\left(\frac{4}{3} \omega\right)=\frac{2 I \omega}{3} \\<br/>\tau &amp; =\frac{2 I \omega}{3 t}<br/>\end{aligned}<br/>$$<br/>$\therefore$ Option (a) is correct.</div>


tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution--1

slug:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution-1-48501

question:
<div class="question"><strong>Paragraph:</strong><br/>p-amino-N, N-dimethylaniline is added to a strongly acidic solution of $X$. The resulting solution is treated with a few drops of aqueous solution of $Y$ to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of $Y$ with the reagent potassium  hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of $Y$ with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of $Z$.<strong>Question:</strong><br/>The compound $Y$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{MgCl}_2$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{FeCl}_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{FeCl}_3$<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/>$\mathrm{ZnCl}_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{FeCl}_3$<br/></span> </div>


tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution--2

slug:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution-2-35554

question:
<div class="question"><strong>Paragraph:</strong><br/>p-amino-N, N-dimethylaniline is added to a strongly acidic solution of $X$. The resulting solution is treated with a few drops of aqueous solution of $Y$ to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of $Y$ with the reagent potassium  hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of $Y$ with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of $Z$.<strong>Question:</strong><br/>The compound $X$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{NaNO}_3$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{NaCl}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Na}_2 \mathrm{SO}_4$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{Na}_2 \mathrm{~S}$</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/>$\mathrm{Na}_2 \mathrm{~S}$</span> </div>


tags: ['Chemistry', 'Amines', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution--3

slug:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution-3-95892

question:
<div class="question"><strong>Paragraph:</strong><br/>p-amino-N, N-dimethylaniline is added to a strongly acidic solution of $X$. The resulting solution is treated with a few drops of aqueous solution of $Y$ to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of $Y$ with the reagent potassium  hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of $Y$ with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of $Z$.<strong>Question:</strong><br/>The compound $Z$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Mg}_2\left[\mathrm{Fe}(\mathrm{CN})_6\right]$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Fe}\left[\mathrm{Fe}(\mathrm{CN})_6\right]$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{K}_2 \mathrm{Zn}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{Fe}\left[\mathrm{Fe}(\mathrm{CN})_6\right]$<br/></span> </div>

answer:
<div class="solution">$p$-amino- $\mathrm{N}$, N-dimethylaniline reacts with sulphide ion $\left(\mathrm{S}^{2-}\right)$ in presence of $\mathrm{Fe}^{3+}$ ion to form blue coloured solution of methylene blue :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NZw4_QD8GSLGGMqrQZSpuv_AIQiOAXhksXIRa3V8Zsc.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution-

slug:
paragraph-paminon-ndimethylaniline-is-added-to-a-strongly-acidic-solution-of-x-the-resulting-solution-is-treated-with-a-few-drops-of-aqueous-solution-71735

question:
<div class="question"><strong>Paragraph:</strong><br/>p-amino-N, N-dimethylaniline is added to a strongly acidic solution of $X$. The resulting solution is treated with a few drops of aqueous solution of $Y$ to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of $Y$ with the reagent potassium  hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of $Y$ with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of $Z$.<strong>Question:</strong><br/>The compound $Z$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Mg}_2\left[\mathrm{Fe}(\mathrm{CN})_6\right]$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Fe}\left[\mathrm{Fe}(\mathrm{CN})_6\right]$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{K}_2 \mathrm{Zn}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{Fe}\left[\mathrm{Fe}(\mathrm{CN})_6\right]$<br/></span> </div>

answer:
<div class="solution">$p$-amino- $\mathrm{N}$, N-dimethylaniline reacts with sulphide ion $\left(\mathrm{S}^{2-}\right)$ in presence of $\mathrm{Fe}^{3+}$ ion to form blue coloured solution of methylene blue :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/NZw4_QD8GSLGGMqrQZSpuv_AIQiOAXhksXIRa3V8Zsc.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Motion In One Dimension', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-phase-space-diagrams-are-useful-tools-in-analyzing-all-kinds-of-dynamical-problems-they-are-especially-useful-in-studying-the-changes-in-mot-1

slug:
paragraph-phase-space-diagrams-are-useful-tools-in-analyzing-all-kinds-of-dynamical-problems-they-are-especially-useful-in-studying-the-changes-in-mot-1-33367

question:
<div class="question"><strong>Paragraph:</strong><br/>Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is $x(t) v s p(t)$ curve in this plane. The arrow on the<br/>curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/XUS9yb1oVClHFXXo1R2Su7FpnIAGiALCHaAWRaCkIxo.original.fullsize.png"/><br/><strong>Question:</strong><br/>The phase space diagram for a ball thrown vertically up from ground is</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/iI0Kxu2WnV4NkRO9yY3V7z6whEGe6AvixYB2vbO-hD8.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/DOfzCHyzibJNVnxn0HafBRyOgKJe6uLD1830_7T21VI.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/O1yAEKHii2lM3o6tCdd5OomqXot_8p1UEg0sH2_8QuA.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/h80ZqR_DLfd3NCpT3nZGdB647TUEv-dWcd5q3qSkx1k.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/h80ZqR_DLfd3NCpT3nZGdB647TUEv-dWcd5q3qSkx1k.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">Momentum is first positive but decreasing. Displacement (or say position). is initially zero.<br/>It will first increase. At highest point, momentum is zero and displacement is maximum. After that momentum is downwards (negative) and increasing but displacement is decreasing. Only (d) option satisfies these conditions.<br/>Analysis of Question<br/>(i) Question is simple.<br/>(ii) Since all the graphs given in question are parabolic type, we need not to check the mathematical part of the question.</div>


tags: ['Physics', 'Oscillations', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-phase-space-diagrams-are-useful-tools-in-analyzing-all-kinds-of-dynamical-problems-they-are-especially-useful-in-studying-the-changes-in-mot-2

slug:
paragraph-phase-space-diagrams-are-useful-tools-in-analyzing-all-kinds-of-dynamical-problems-they-are-especially-useful-in-studying-the-changes-in-mot-2-16700

question:
<div class="question"><strong>Paragraph:</strong><br/>Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is $x(t) v s p(t)$ curve in this plane. The arrow on the<br/>curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/XUS9yb1oVClHFXXo1R2Su7FpnIAGiALCHaAWRaCkIxo.original.fullsize.png"/><br/><strong>Question:</strong><br/>Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/53Ml_Imd8sWvUpsp5WkZHm9H8FJc-nQ-pugQxJm97ok.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/ScSj_MoI5Vsl88sva79sD-8B0WCxi3NsGEbXVJF6aSk.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/sxdjVykv7RJmtA94RFRt8YswTOqvFAE_UO9XD0rEZDg.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/1uohWBNlMxXa1WJwPMqyZzuNWvjiQRx4QgWq1YHS0qo.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/1RyOg2mbKfACyAAqKJwVlK_79ZsVcgCMrL5fa53TRSY.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/sxdjVykv7RJmtA94RFRt8YswTOqvFAE_UO9XD0rEZDg.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">In all the given four figures, at mean position the position coordinate is zero. At the same time mass is starting from the extreme position in all fours cases. In figures (c) and (d), extreme position is more than the initial extreme position. But due to viscosity opposite should be the case. Hence, the answer should be either option (a) or option (b).<br/>Correct answer is (b), because mass starts from positive extreme position (from uppermost position). Then, it will move downwards or momentum should be negative.<br/>Analysis of Question<br/>Question is simple but (a) and (b) options are confusing.</div>


tags: ['Physics', 'Oscillations', 'JEE Advanced', 'JEE Advanced 2011 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-phase-space-diagrams-are-useful-tools-in-analyzing-all-kinds-of-dynamical-problems-they-are-especially-useful-in-studying-the-changes-in-mot

slug:
paragraph-phase-space-diagrams-are-useful-tools-in-analyzing-all-kinds-of-dynamical-problems-they-are-especially-useful-in-studying-the-changes-in-mot-15129

question:
<div class="question"><strong>Paragraph:</strong><br/>Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is $x(t) v s p(t)$ curve in this plane. The arrow on the<br/>curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/XUS9yb1oVClHFXXo1R2Su7FpnIAGiALCHaAWRaCkIxo.original.fullsize.png"/><br/><strong>Question:</strong><br/>The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_1$ and $E_2$ are the total mechanical energies respectively. Then,<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/9d_CDTvM8AdXVdLzqYCWwC2MYHrM0IHhSY55oxNI0tQ.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$E_1=\sqrt{2} E_2$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$E_1=2 E_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$E_1=2 E_2$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$E_1=16 E_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$E_1=2 E_2$<br/></span> </div>

answer:
<div class="solution">$E=\frac{1}{2} m \omega^2 A^2$<br/>or<br/>$$<br/>\begin{aligned}<br/>E &amp; \propto A^2 \\<br/>\frac{E_2}{E_1} &amp; =\left(\frac{A_2}{A_1}\right)^2 \\<br/>&amp; =\left(\frac{a}{2 a}\right)^2<br/>\end{aligned}<br/>$$<br/>or $\quad E_1=4 E_2$<br/>$\therefore$ Correct option is (c).<br/>Analysis of Question<br/>(i) Question is simple.<br/>(ii) Since, oscillator is same, $\omega_1=\omega_2$</div>


tags: ['Chemistry', 'Solutions', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-properties-such-as-boiling-point-freezing-point-and-vapour-pressure-of-a-pure-solvent-change-when-solute-molecules-are-added-to-get-homogene-1

slug:
paragraph-properties-such-as-boiling-point-freezing-point-and-vapour-pressure-of-a-pure-solvent-change-when-solute-molecules-are-added-to-get-homogene-1-12719

question:
<div class="question"><strong>Paragraph:</strong><br/>Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.<br/>A solution $M$ is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is $0.9$<br/>Given Freezing point depression constant of water $\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$<br/>Freezing point depression constant of ethanol $\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Boiling point elevation constant of water $\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Boiling point elevation constant of ethanol $\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Standard freezing point of water $=273 \mathrm{~K}$<br/>Standard freezing point of ethanol $=155.7 \mathrm{~K}$<br/>Standard boiling point of water $=373 \mathrm{~K}$<br/>Standard boiling point of ethanol $=351.5 \mathrm{~K}$<br/>Vapour pressure of pure water $=328 \mathrm{~mm}$ of $\mathrm{Hg}$<br/>Vapour pressure of pure ethanol $=40 \mathrm{~mm}$ of $\mathrm{Hg}$<br/>Molecular weight of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$<br/>Molecular weight of ethanol $=46 \mathrm{~g} \mathrm{~mol}^{-1}$<br/>In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.<strong>Question:</strong><br/>The freezing point of the solution $M$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$268.7 \mathrm{~K}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$268.5 \mathrm{~K}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$234.2 \mathrm{~K}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$150.9 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$150.9 \mathrm{~K}$</span> </div>

answer:
<div class="solution">Solution $M$ is mixture of ethanol and water.<br/>Mole fraction of ethanol is $0.9 \Rightarrow$ Solvent is $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}$<br/>Mole fraction of water is $0.1 \Rightarrow \mathrm{H}_2 \mathrm{O}$ is solute<br/>Molality of $\mathrm{H}_2 \mathrm{O}=\frac{n_2}{n_1 M_1}=\frac{0.1}{0.9 \times 46} \times 1000=2.415$<br/>$$<br/>\Delta T_f=k_f \cdot m=2 \times 2.415=4.83<br/>$$<br/>or Freezing point of solution $=155.7-4.83=150.87 \mathrm{~K}$</div>


tags: ['Chemistry', 'Solutions', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-properties-such-as-boiling-point-freezing-point-and-vapour-pressure-of-a-pure-solvent-change-when-solute-molecules-are-added-to-get-homogene-2

slug:
paragraph-properties-such-as-boiling-point-freezing-point-and-vapour-pressure-of-a-pure-solvent-change-when-solute-molecules-are-added-to-get-homogene-2-44306

question:
<div class="question"><strong>Paragraph:</strong><br/>Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.<br/>A solution $M$ is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is $0.9$<br/>Given Freezing point depression constant of water $\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$<br/>Freezing point depression constant of ethanol $\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Boiling point elevation constant of water $\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Boiling point elevation constant of ethanol $\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Standard freezing point of water $=273 \mathrm{~K}$<br/>Standard freezing point of ethanol $=155.7 \mathrm{~K}$<br/>Standard boiling point of water $=373 \mathrm{~K}$<br/>Standard boiling point of ethanol $=351.5 \mathrm{~K}$<br/>Vapour pressure of pure water $=328 \mathrm{~mm}$ of $\mathrm{Hg}$<br/>Vapour pressure of pure ethanol $=40 \mathrm{~mm}$ of $\mathrm{Hg}$<br/>Molecular weight of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$<br/>Molecular weight of ethanol $=46 \mathrm{~g} \mathrm{~mol}^{-1}$<br/>In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.<strong>Question:</strong><br/>Water is added to the solution $M$ such that the mole fraction of water in the solution becomes $0.9$ The boiling point of this solutions is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$380.4 \mathrm{~K}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$376.2 \mathrm{~K}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$375.5 \mathrm{~K}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$354.7 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$376.2 \mathrm{~K}$<br/></span> </div>

answer:
<div class="solution">$\chi_{\mathrm{H}_2 \mathrm{O}}=0.9$ (solvent) $\chi_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}}=0.1$ (solute)<br/>$$<br/>\Delta T_b=k_b \times m=0.52 \times \frac{1.0 \times 1000}{0.9 \times 18}=3.2 \mathrm{~K}<br/>$$<br/>Boiling point, $T_b=373+3.2=376.2 \mathrm{~K}$</div>


tags: ['Chemistry', 'Solutions', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-properties-such-as-boiling-point-freezing-point-and-vapour-pressure-of-a-pure-solvent-change-when-solute-molecules-are-added-to-get-homogene

slug:
paragraph-properties-such-as-boiling-point-freezing-point-and-vapour-pressure-of-a-pure-solvent-change-when-solute-molecules-are-added-to-get-homogene-13012

question:
<div class="question"><strong>Paragraph:</strong><br/>Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.<br/>A solution $M$ is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is $0.9$<br/>Given Freezing point depression constant of water $\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$<br/>Freezing point depression constant of ethanol $\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Boiling point elevation constant of water $\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Boiling point elevation constant of ethanol $\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}$<br/>Standard freezing point of water $=273 \mathrm{~K}$<br/>Standard freezing point of ethanol $=155.7 \mathrm{~K}$<br/>Standard boiling point of water $=373 \mathrm{~K}$<br/>Standard boiling point of ethanol $=351.5 \mathrm{~K}$<br/>Vapour pressure of pure water $=328 \mathrm{~mm}$ of $\mathrm{Hg}$<br/>Vapour pressure of pure ethanol $=40 \mathrm{~mm}$ of $\mathrm{Hg}$<br/>Molecular weight of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$<br/>Molecular weight of ethanol $=46 \mathrm{~g} \mathrm{~mol}^{-1}$<br/>In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.<strong>Question:</strong><br/>The vapour pressure of the solution $M$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$39.3 \mathrm{~mm} \mathrm{Hg}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$36.0 \mathrm{~mm} \mathrm{Hg}$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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/>$29.5 \mathrm{~mm} \mathrm{Hg}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$28.8 \mathrm{~mm} \mathrm{Hg}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$36.0 \mathrm{~mm} \mathrm{Hg}$<br/></span> </div>

answer:
<div class="solution">Total vapour pressure, $p=p_A^{\circ} \chi_A$<br/>$$<br/>p=40 \times 0.9=36 \mathrm{~mm} \text { of } \mathrm{Hg}<br/>$$<br/>In the paragraph, it has been directed to take solute as non-volatile, thus $\mathrm{H}_2 \mathrm{O}$ do not contribute in the total vapour pressure.</div>


tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-1

slug:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-1-16630

question:
<div class="question"><strong>Paragraph:</strong><br/>$$<br/>\mathrm{RCONH}_2 \text { is converted into } \mathrm{RNH}_2 \text { by means of Hofmann bromamide degradation. }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/kUvdLVZCCRDn5vZcz3oaK6cUZX7TS8jt4MnQjqIiZ1U.original.fullsize.png"/><br/><br/>In this reaction, $R \mathrm{CONHBr}$ is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.<strong>Question:</strong><br/>What are the constituent amines formed when the mixture of (i) and (ii) undergoes Hofmann bromamide degradation?<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xUil9dp4OC-2i_JpT3k-Ay0mZhy3yc_KoxGe_XOoTO8.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/9oEoYy5QRFL2XT7ezsHdF5ARJDl0lEN4arnP9U2UCIA.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/raLseyC7J4jpeBdxdouJO8i1ntB1EL77OiciIHqKM7A.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/XWvYR9onQwF915KtQ4fuwkz0nopYgxeL-dAMd-1wlrc.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/-f4PlSCzDHo1546w2x7iTwaj1AXy5kUxT2pDP5IEnWc.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/raLseyC7J4jpeBdxdouJO8i1ntB1EL77OiciIHqKM7A.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Oe_E6km97d_rshVwsJnXpBtvih6O1N6XCJ0vI3uSC5I.original.fullsize.png"/><br/><br/>$$<br/>\text { Since the overall reaction is intermolecular, hence there will be no effect on product formation. }<br/>$$</div>


tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-2

slug:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-2-55796

question:
<div class="question"><strong>Paragraph:</strong><br/>$$<br/>\mathrm{RCONH}_2 \text { is converted into } \mathrm{RNH}_2 \text { by means of Hofmann bromamide degradation. }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/kUvdLVZCCRDn5vZcz3oaK6cUZX7TS8jt4MnQjqIiZ1U.original.fullsize.png"/><br/><br/>In this reaction, $R \mathrm{CONHBr}$ is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.<strong>Question:</strong><br/>Which is the rate determining step in Hofmann bromamide degradation?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Formation of (i)<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Formation of (ii)<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Formation of (iii)<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>Formation of (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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Formation of (iv)</span> </div>

answer:
<div class="solution">The rate determining step is probably elimination of $\mathrm{Br}^{-}$to form isocyanate because it is slower step of given conversion.</div>


tags: ['Chemistry', 'Amines', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-3

slug:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-3-22182

question:
<div class="question"><strong>Paragraph:</strong><br/>$$<br/>\mathrm{RCONH}_2 \text { is converted into } \mathrm{RNH}_2 \text { by means of Hofmann bromamide degradation. }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/kUvdLVZCCRDn5vZcz3oaK6cUZX7TS8jt4MnQjqIiZ1U.original.fullsize.png"/><br/><br/>In this reaction, $R \mathrm{CONHBr}$ is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.<strong>Question:</strong><br/>What are the constituent amines formed when the mixture of (i) and (ii) undergoes Hofmann bromamide degradation?<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/xUil9dp4OC-2i_JpT3k-Ay0mZhy3yc_KoxGe_XOoTO8.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/9oEoYy5QRFL2XT7ezsHdF5ARJDl0lEN4arnP9U2UCIA.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/raLseyC7J4jpeBdxdouJO8i1ntB1EL77OiciIHqKM7A.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/XWvYR9onQwF915KtQ4fuwkz0nopYgxeL-dAMd-1wlrc.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/-f4PlSCzDHo1546w2x7iTwaj1AXy5kUxT2pDP5IEnWc.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/raLseyC7J4jpeBdxdouJO8i1ntB1EL77OiciIHqKM7A.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Oe_E6km97d_rshVwsJnXpBtvih6O1N6XCJ0vI3uSC5I.original.fullsize.png"/><br/><br/>$$<br/>\text { Since the overall reaction is intermolecular, hence there will be no effect on product formation. }<br/>$$</div>


tags: ['Chemistry', 'Amines', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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

slug:
paragraph-rconh-2-is-converted-into-rnh-2-by-means-of-hofmann-bromamide-degradation-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-22328

question:
<div class="question"><strong>Paragraph:</strong><br/>$$<br/>\mathrm{RCONH}_2 \text { is converted into } \mathrm{RNH}_2 \text { by means of Hofmann bromamide degradation. }<br/>$$<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/kUvdLVZCCRDn5vZcz3oaK6cUZX7TS8jt4MnQjqIiZ1U.original.fullsize.png"/><br/><br/>In this reaction, $R \mathrm{CONHBr}$ is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.<strong>Question:</strong><br/>How can the conversion of (i) to (ii) be brought about?</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/zDNTmjvVgpklMj_DLaJFK3heYy8d8GQqzVcSCiltWI4.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Application of Derivatives', 'JEE Advanced', 'JEE Advanced 2006']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>0</span> </div>
answer: <div class="solution">$f^{\prime}(c)=0$.</div>
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-for-every-function-f-x-which-is-twice-differentiable-these-will-be-good-approximation-of-1

slug:
paragraph-read-the-following-passage-and-answer-the-questions-for-every-function-f-x-which-is-twice-differentiable-these-will-be-good-approximation-of-1-74881

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>For every function $f(x)$ which is twice differentiable, these will be good approximation of $\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}$. Now, if we take $c=\frac{a+b}{2}$, then using above again, we get $\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}$ and so on.<br/>We get approximation for value of $\int_a^b f(x) d x$.<strong>Question:</strong><br/>If $f^{\prime \prime}(x) &lt; 0, \forall x \in(a, b), c(c, f(c))$ is point of maxima where $c \in(a, b)$, then $f^{\prime}(c)$ is</div>

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


tags: ['Mathematics', 'Limits', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-for-every-function-f-x-which-is-twice-differentiable-these-will-be-good-approximation-of-2

slug:
paragraph-read-the-following-passage-and-answer-the-questions-for-every-function-f-x-which-is-twice-differentiable-these-will-be-good-approximation-of-2-20665

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>For every function $f(x)$ which is twice differentiable, these will be good approximation of $\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}$. Now, if we take $c=\frac{a+b}{2}$, then using above again, we get $\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}$ and so on.<br/>We get approximation for value of $\int_a^b f(x) d x$.<strong>Question:</strong><br/>If $\lim _{t \rightarrow a} \frac{\int_a^t f(x) d x-\frac{(t-a)}{2}\{f(t)+f(a)\}}{(t-a)^3}=0$, then degree of polynomial function $f(x)$ at most 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/>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/>3<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>2</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\lim _{t \rightarrow a} \frac{\int_a^t f(t) d t-\frac{(t-a)}{2}(f(t)+f(a))}{(t-a)^3} &amp; =0 \\<br/>\Rightarrow \quad \lim _{h \rightarrow 0} \frac{\int_a^{a+h} f(t) d t-\frac{h}{2}(f(a+h)+f(a))}{h^3} &amp; =0 \\<br/>\Rightarrow \quad \lim _{h \rightarrow 0} \frac{f(a+h)-\frac{1}{2}(f(a+h)+f(a))-\frac{h}{2}\left(f^{\prime}(a+h)\right)}{3 h^2} &amp; =0<br/>\end{aligned}<br/>$$<br/>$f^{\prime}(a+h)-\frac{1}{2} f^{\prime}(a+h)$<br/>$\Rightarrow \quad \lim _{h \rightarrow 0} \frac{-\frac{1}{2} f^{\prime}(a+h)-\frac{h}{2} f^{\prime \prime}(a+h)}{6 h}=0$<br/>$\Rightarrow \quad \lim _{h \rightarrow 0} \frac{-\frac{h}{2} f^{\prime}(a+h)}{6 h}=0$<br/>$\Rightarrow \quad f^{\prime \prime}(a)=0, \forall a \in R$<br/>$\Rightarrow(x)$ must have maximum degree 1.</div>


tags: ['Mathematics', 'Definite Integration', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-for-every-function-f-x-which-is-twice-differentiable-these-will-be-good-approximation-of

slug:
paragraph-read-the-following-passage-and-answer-the-questions-for-every-function-f-x-which-is-twice-differentiable-these-will-be-good-approximation-of-71334

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>For every function $f(x)$ which is twice differentiable, these will be good approximation of $\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}$. Now, if we take $c=\frac{a+b}{2}$, then using above again, we get $\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}$ and so on.<br/>We get approximation for value of $\int_a^b f(x) d x$.<strong>Question:</strong><br/>Good approximation of $\int_0^{\pi / 2} \sin x d x$, is</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/>$\frac{\pi}{4}(\sqrt{2}+1)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{\pi}{8}(\sqrt{2}+1)$<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/>$\frac{\pi}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\frac{\pi}{8}(\sqrt{2}+1)$<br/></span> </div>

answer:
<div class="solution">$\int_0^{\pi / 2} \sin x d x=\frac{\frac{\pi}{2}-0}{4}\left(\sin 0+\sin \left(\frac{\pi}{2}\right)+2 \sin \left(\frac{0+\frac{\pi}{2}}{2}\right)\right)=\frac{\pi}{8}(1+\sqrt{2})$.</div>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-1

slug:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-1-90397

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>Let $A B C D$ be a square of side length 2 units. $C_2$ is the circle through vertices $A, B, C, D$ and $C_1$ is the circle touching all the sides of square $A B C D$. $L$ is the line through $A$.<strong>Question:</strong><br/>A line $M$ through $A$ is drawn parallel to $B D$. Points $S$ moves such that its distances from the line $B D$ are the vertex $A$ are equal. If locus of $S$ cuts $M$ at $T_2$ and $T_3$ and $A C$ at $T_1$, then area of $\Delta T_1 T_2 T_3$ is</div>

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

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

answer:
<div class="solution">$\because \quad A G=\sqrt{2}$<br/>$$<br/>\therefore \quad A T_1=T_1 G=\frac{1}{\sqrt{2}}<br/>$$<br/>\{ as $A$ is the focus, $T_1$ is the vertex and $B D$ is the directrix of parabola.\}<br/>Also, $T_2 T_3$ is latus rectum<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad T_2 T_3=4 \cdot \frac{1}{\sqrt{2}} \\<br/>&amp; \therefore \text { Area of } \Delta T_1 T_2 T_3=\frac{1}{2} \times \frac{1}{\sqrt{2}} \times \frac{4}{\sqrt{2}}=1 \text { sq unit }<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Pxu686ffJomZ7_Tf427TB2KermtePPSHx--LU89pjaA.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Parabola', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-2

slug:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-2-66029

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>Let $A B C D$ be a square of side length 2 units. $C_2$ is the circle through vertices $A, B, C, D$ and $C_1$ is the circle touching all the sides of square $A B C D$. $L$ is the line through $A$.<strong>Question:</strong><br/>A circle touches the line $L$ and the circle $C_1$ externally such that both the circles are on the same side of the line, then the locus of centre of the circle is</div>

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

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

answer:
<div class="solution">Let $C$ be the centre of the required circle.<br/>Now, draw a line parallel to $L$ at a distance of $r_1$ (radius of $C_1$ ) from it. Now, $C P_1=A C$<br/>$\Rightarrow C$ lies on a parabola.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/T3E8jilQ7fAExrDkNa9zaAaWyOyYLzWpMUcoHIy2QN0.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-3

slug:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-3-94566

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>Let $A B C D$ be a square of side length 2 units. $C_2$ is the circle through vertices $A, B, C, D$ and $C_1$ is the circle touching all the sides of square $A B C D$. $L$ is the line through $A$.<strong>Question:</strong><br/>If $P$ is a point on $C_1$ and $Q$ is a point on $C_2$, then $\frac{P A^2+P B^2+P C^2+P D^2}{Q A^2+Q B^2+Q C^2+Q D^2}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$0.75$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$1.25$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>1<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$0.5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$0.75$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { Here, equation of } C_2:(x-1)^2+(y-1)^2=(\sqrt{2})^2 \text { and } C_1:(x-1)^2+(y-1)^2=(1)^2<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \therefore P(1+\cos \theta, 1+\sin \theta) \\<br/>&amp; \text { and } Q(1+\sqrt{2} \sin \theta, 1+\sqrt{2} \sin \theta) \\<br/>&amp; \therefore P A^2+P B^2+P C^2+P D^2 \\<br/>&amp; =\left\{(1+\cos \theta)^2+(1+\sin \theta)^2\right\} \\<br/>&amp; +\left\{(\cos \theta-1)^2+(1+\sin \theta)^2\right\} \\<br/>&amp; +\left\{(1+\cos \theta)^2+(1-\sin \theta)^2\right\} \\<br/>&amp; +\left\{(1-\cos \theta)^2+(1-\sin \theta)^2\right\} \\<br/>&amp; =16 \\<br/>&amp; \therefore \quad \frac{\Sigma Q A^2}{\Sigma P A^2}=\frac{12}{16}=0.75 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>Hence (a) is the correct answer.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/SEWnUdfopYlFl_vklMsnqLI82c2868zLYyuLSPl5984.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-4

slug:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-4-22987

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>Let $A B C D$ be a square of side length 2 units. $C_2$ is the circle through vertices $A, B, C, D$ and $C_1$ is the circle touching all the sides of square $A B C D$. $L$ is the line through $A$.<strong>Question:</strong><br/>A circle touches the line $L$ and the circle $C_1$ externally such that both the circles are on the same side of the line, then the locus of centre of the circle is</div>

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

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

answer:
<div class="solution">Let $C$ be the centre of the required circle.<br/>Now, draw a line parallel to $L$ at a distance of $r_1$ (radius of $C_1$ ) from it. Now, $C P_1=A C$<br/>$\Rightarrow C$ lies on a parabola.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/T3E8jilQ7fAExrDkNa9zaAaWyOyYLzWpMUcoHIy2QN0.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-5

slug:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-5-18949

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>Let $A B C D$ be a square of side length 2 units. $C_2$ is the circle through vertices $A, B, C, D$ and $C_1$ is the circle touching all the sides of square $A B C D$. $L$ is the line through $A$.<strong>Question:</strong><br/>A line $M$ through $A$ is drawn parallel to $B D$. Points $S$ moves such that its distances from the line $B D$ are the vertex $A$ are equal. If locus of $S$ cuts $M$ at $T_2$ and $T_3$ and $A C$ at $T_1$, then area of $\Delta T_1 T_2 T_3$ is</div>

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

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

answer:
<div class="solution">$\because \quad A G=\sqrt{2}$<br/>$$<br/>\therefore \quad A T_1=T_1 G=\frac{1}{\sqrt{2}}<br/>$$<br/>\{ as $A$ is the focus, $T_1$ is the vertex and $B D$ is the directrix of parabola.\}<br/>Also, $T_2 T_3$ is latus rectum<br/>$$<br/>\begin{aligned}<br/>&amp; \therefore \quad T_2 T_3=4 \cdot \frac{1}{\sqrt{2}} \\<br/>&amp; \therefore \text { Area of } \Delta T_1 T_2 T_3=\frac{1}{2} \times \frac{1}{\sqrt{2}} \times \frac{4}{\sqrt{2}}=1 \text { sq unit }<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Pxu686ffJomZ7_Tf427TB2KermtePPSHx--LU89pjaA.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Circle', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d

slug:
paragraph-read-the-following-passage-and-answer-the-questions-let-a-bc-d-be-a-square-of-side-length-2-units-c-2-is-the-circle-through-vertices-a-b-c-d-87608

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<br/>Let $A B C D$ be a square of side length 2 units. $C_2$ is the circle through vertices $A, B, C, D$ and $C_1$ is the circle touching all the sides of square $A B C D$. $L$ is the line through $A$.<strong>Question:</strong><br/>If $P$ is a point on $C_1$ and $Q$ is a point on $C_2$, then $\frac{P A^2+P B^2+P C^2+P D^2}{Q A^2+Q B^2+Q C^2+Q D^2}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$0.75$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$1.25$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>1<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$0.5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$0.75$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { Here, equation of } C_2:(x-1)^2+(y-1)^2=(\sqrt{2})^2 \text { and } C_1:(x-1)^2+(y-1)^2=(1)^2<br/>$$<br/><br/>$$<br/>\begin{aligned}<br/>&amp; \therefore P(1+\cos \theta, 1+\sin \theta) \\<br/>&amp; \text { and } Q(1+\sqrt{2} \sin \theta, 1+\sqrt{2} \sin \theta) \\<br/>&amp; \therefore P A^2+P B^2+P C^2+P D^2 \\<br/>&amp; =\left\{(1+\cos \theta)^2+(1+\sin \theta)^2\right\} \\<br/>&amp; +\left\{(\cos \theta-1)^2+(1+\sin \theta)^2\right\} \\<br/>&amp; +\left\{(1+\cos \theta)^2+(1-\sin \theta)^2\right\} \\<br/>&amp; +\left\{(1-\cos \theta)^2+(1-\sin \theta)^2\right\} \\<br/>&amp; =16 \\<br/>&amp; \therefore \quad \frac{\Sigma Q A^2}{\Sigma P A^2}=\frac{12}{16}=0.75 \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>Hence (a) is the correct answer.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/SEWnUdfopYlFl_vklMsnqLI82c2868zLYyuLSPl5984.original.fullsize.png"/><br/></div>


title: paragraph-read-the-following-passage-and-answer-the-questions-question-the-value-of-3-2-0-3-2-0-is
tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
slug:
paragraph-read-the-following-passage-and-answer-the-questions-question-the-value-of-3-2-0-3-2-0-is-91894

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/88iLE50zA6etAHkOmNt74CAbkJliurfrRZp8CzupJk0.original.fullsize.png"/><br/><strong>Question:</strong><br/>The value of $\left[\begin{array}{lll}3 &amp; 2 &amp; 0\end{array}\right]\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data"><br/>$\left[\frac{5}{2}\right]$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>[4]<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left[\frac{3}{2}\right]$</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"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/ZQgvpskrRTIhBmqoFgX30lBrLbKluyTOLvJb7_-5kd4.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
title:
paragraph-read-the-following-passage-and-answer-the-questions-question-the-value-of-the-elements-of-u-1-is

slug:
paragraph-read-the-following-passage-and-answer-the-questions-question-the-value-of-the-elements-of-u-1-is-46543

question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/88iLE50zA6etAHkOmNt74CAbkJliurfrRZp8CzupJk0.original.fullsize.png"/><br/><strong>Question:</strong><br/>The value of the elements of $U^{-1}$ is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/YBHdmRS2Jayt5PH1g06WziEPbMaRY-1B-tsUnQ-177g.original.fullsize.png"/><br/></div>


title: paragraph-read-the-following-passage-and-answer-the-questions-question-the-value-of-u-is
slug: paragraph-read-the-following-passage-and-answer-the-questions-question-the-value-of-u-is-68616
tags: ['Mathematics', 'Matrices', 'JEE Advanced', 'JEE Advanced 2006']
source_db: MarksBatch2_P2.db
question:
<div class="question"><strong>Paragraph:</strong><br/>Read the following passage and answer the questions.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/88iLE50zA6etAHkOmNt74CAbkJliurfrRZp8CzupJk0.original.fullsize.png"/><br/><strong>Question:</strong><br/>The value of $|U|$ is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wgH9BbHmJlVwdCzbsjL7tQqY-vkRnukiLSB1oxMNbTc.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which--1

slug:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which-1-19742

question:
<div class="question"><strong>Paragraph:</strong><br/>Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential $\left(E^{\circ}\right)$ of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their $E^{\circ}(V$ with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.<br/>$$<br/>\begin{array}{cl}<br/>\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} &amp; E^{\circ}=0.54 \\<br/>\mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} &amp; E^{\circ}=1.36 \\<br/>\mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} &amp; E^{\circ}=1.50 \\<br/>\mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} &amp; E^{\circ}=0.77 \\<br/>\mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} &amp; E^{\circ}=1.23<br/>\end{array}<br/>$$<strong>Question:</strong><br/>Among the following, identify the correct statement</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Iodide ion is oxidised by chlorine<br/></span> </div>

answer:
<div class="solution">$2 \mathrm{I}^{-}+\mathrm{Cl}_2 \longrightarrow \mathrm{I}_2+2 \mathrm{Cl}^{-}$<br/>$$<br/>E_{\text {cell }}^{\circ}=1.36-0.54=0.82 \mathrm{~V}<br/>$$<br/>$E_{\text {cell }}^{\circ}$ is $+v e$ cell reactions is feasible.</div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which--2

slug:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which-2-12867

question:
<div class="question"><strong>Paragraph:</strong><br/>Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential $\left(E^{\circ}\right)$ of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their $E^{\circ}(V$ with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.<br/>$$<br/>\begin{array}{cl}<br/>\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} &amp; E^{\circ}=0.54 \\<br/>\mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} &amp; E^{\circ}=1.36 \\<br/>\mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} &amp; E^{\circ}=1.50 \\<br/>\mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} &amp; E^{\circ}=0.77 \\<br/>\mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} &amp; E^{\circ}=1.23<br/>\end{array}<br/>$$<strong>Question:</strong><br/>Sodium fusion extract, obtained from aniline, on treatment with iron (II) sulphate and $\mathrm{H}_2 \mathrm{SO}_4$ in presence of air gives a prussian blue precipitate. The blue colour is due to the formation of</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Fe}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]_2$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_2$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$$<br/>\mathrm{Fe}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3<br/>$$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3$<br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/iMwLX8MdIS0emG5pSZDSPnOA9F7gf0vj-JxCA5ieavY.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which--3

slug:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which-3-25629

question:
<div class="question"><strong>Paragraph:</strong><br/>Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential $\left(E^{\circ}\right)$ of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their $E^{\circ}(V$ with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.<br/>$$<br/>\begin{array}{cl}<br/>\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} &amp; E^{\circ}=0.54 \\<br/>\mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} &amp; E^{\circ}=1.36 \\<br/>\mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} &amp; E^{\circ}=1.50 \\<br/>\mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} &amp; E^{\circ}=0.77 \\<br/>\mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} &amp; E^{\circ}=1.23<br/>\end{array}<br/>$$<strong>Question:</strong><br/>Sodium fusion extract, obtained from aniline, on treatment with iron (II) sulphate and $\mathrm{H}_2 \mathrm{SO}_4$ in presence of air gives a prussian blue precipitate. The blue colour is due to the formation of</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{Fe}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]_2$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_2$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$$<br/>\mathrm{Fe}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3<br/>$$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{Fe}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]_3$<br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/iMwLX8MdIS0emG5pSZDSPnOA9F7gf0vj-JxCA5ieavY.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which-

slug:
paragraph-redox-reaction-play-a-pivotal-role-in-chemistry-and-biology-the-values-of-standard-redox-potential-e-of-two-halfcell-reactions-decide-which-92486

question:
<div class="question"><strong>Paragraph:</strong><br/>Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential $\left(E^{\circ}\right)$ of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their $E^{\circ}(V$ with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.<br/>$$<br/>\begin{array}{cl}<br/>\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} &amp; E^{\circ}=0.54 \\<br/>\mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} &amp; E^{\circ}=1.36 \\<br/>\mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} &amp; E^{\circ}=1.50 \\<br/>\mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} &amp; E^{\circ}=0.77 \\<br/>\mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} &amp; E^{\circ}=1.23<br/>\end{array}<br/>$$<strong>Question:</strong><br/>While $\mathrm{Fe}^{3+}$ is stable, $\mathrm{Mn}^{3+}$ is not stable in acid solution because</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{O}_2$ oxidises $\mathrm{Mn}^{2+}$ to $\mathrm{Mn}^{3+}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{O}_2$ oxidises both $\mathrm{Mn}^{2+}$ and $\mathrm{Fe}^{2+}$ to $\mathrm{Fe}^{3+}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Fe}^{3+}$ oxidises $\mathrm{H}_2 \mathrm{O}$ to $\mathrm{O}_2$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{Mn}^{3+}$ oxidises $\mathrm{H}_2 \mathrm{O}$ to $\mathrm{O}_2$</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/>$\mathrm{Mn}^{3+}$ oxidises $\mathrm{H}_2 \mathrm{O}$ to $\mathrm{O}_2$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; {\left[\mathrm{Mn}^{3+}+e^{-}\right.}\left.\longrightarrow \mathrm{Mn}^{2+}\right] \times 4 \\<br/>&amp; 2 \mathrm{H}_2 \mathrm{O} \longrightarrow 4 \mathrm{H}^{+}+\mathrm{O}_2+4 e^{-} \\<br/>&amp; 4 \mathrm{Mn}^{2+}+2 \mathrm{H}_2 \mathrm{O} \longrightarrow 4 \mathrm{Mn}^{2+}+4 \mathrm{H}^{+}+\mathrm{O}_2 \\<br/>&amp; E_{\text {cell }}^{\circ}= 1.50-1.23=0.27 \mathrm{~V} \\<br/>&amp; E_{\text {cell }}^{\circ} \text { is + ve }<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-1

slug:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-1-18345

question:
<div class="question"><strong>Paragraph:</strong><br/>Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Du2opTwjmV3M-NwcVEb4w_x6hSMB0KtJP7732UoYD-0.original.fullsize.png"/><br/><strong>Question:</strong><br/>Which one of the following reagents is used the above reaction?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>aq. $\mathrm{NaOH}+\mathrm{CH}_3 \mathrm{Cl}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>aq. $\mathrm{NaOH}+\mathrm{CH}_2 \mathrm{Cl}_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>aq. $\mathrm{NaOH}+\mathrm{CHCl}_3$<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/>aq. $\mathrm{NaOH}+\mathrm{CCl}_4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>aq. $\mathrm{NaOH}+\mathrm{CHCl}_3$<br/></span> </div>

answer:
<div class="solution">aq. $\mathrm{NaOH}+\mathrm{CHCl}_3$<br/>$$<br/>\mathrm{Cl}_3 \mathrm{C}-\mathrm{H} \underset{-\mathrm{H}_2 \mathrm{O}}{\stackrel{\mathrm{OH}^{-}}{\longrightarrow}} \mathrm{Cl}_3 \mathrm{C}^{-} \underset{-\mathrm{Cl}^{-}}{\longrightarrow} \mathrm{Cl}_2 \mathrm{C}:<br/>$$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-2

slug:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-2-66537

question:
<div class="question"><strong>Paragraph:</strong><br/>Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Du2opTwjmV3M-NwcVEb4w_x6hSMB0KtJP7732UoYD-0.original.fullsize.png"/><br/><strong>Question:</strong><br/>The electrophile in this reaction is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$: \mathrm{CCl}_2$<br/></span> </div>

answer:
<div class="solution">In Reimer-Tiemann reaction<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/IOBozXMEqhNuMBg0Q6L_JIJOJLYs23aag8wBtoC3OBU.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-3

slug:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-3-33624

question:
<div class="question"><strong>Paragraph:</strong><br/>Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Du2opTwjmV3M-NwcVEb4w_x6hSMB0KtJP7732UoYD-0.original.fullsize.png"/><br/><strong>Question:</strong><br/>The structure of the intermediate $\mathrm{I}$ is</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/ufEUk1j8s6jfvsoJhfg8lv47ygbrWsRJyTN4VSxj5HE.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/d-XGjxXmeveuzoYrGaxk9voyVW1aNHidwq3e9QB66Ow.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/OXLhm9FBFe0UEFkzFWnzJkWIDQBqymao0aq0olvvTCo.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/sGD7l_cZ4zJCRD1c5LAJLAQTuZTxGWsDqfBQ_5VC-Hc.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/d-XGjxXmeveuzoYrGaxk9voyVW1aNHidwq3e9QB66Ow.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wItlqDuGitnTqt9mqZGYafaNfHHbw4_W8zFcogYiWAo.original.fullsize.png"/><br/></div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-4

slug:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-4-70758

question:
<div class="question"><strong>Paragraph:</strong><br/>Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Du2opTwjmV3M-NwcVEb4w_x6hSMB0KtJP7732UoYD-0.original.fullsize.png"/><br/><strong>Question:</strong><br/>Which one of the following reagents is used the above reaction?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>aq. $\mathrm{NaOH}+\mathrm{CH}_3 \mathrm{Cl}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>aq. $\mathrm{NaOH}+\mathrm{CH}_2 \mathrm{Cl}_2$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>aq. $\mathrm{NaOH}+\mathrm{CHCl}_3$<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/>aq. $\mathrm{NaOH}+\mathrm{CCl}_4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>aq. $\mathrm{NaOH}+\mathrm{CHCl}_3$<br/></span> </div>

answer:
<div class="solution">aq. $\mathrm{NaOH}+\mathrm{CHCl}_3$<br/>$$<br/>\mathrm{Cl}_3 \mathrm{C}-\mathrm{H} \underset{-\mathrm{H}_2 \mathrm{O}}{\stackrel{\mathrm{OH}^{-}}{\longrightarrow}} \mathrm{Cl}_3 \mathrm{C}^{-} \underset{-\mathrm{Cl}^{-}}{\longrightarrow} \mathrm{Cl}_2 \mathrm{C}:<br/>$$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'JEE Advanced', 'JEE Advanced 2007 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele

slug:
paragraph-reimertiemann-reaction-introduces-an-aldehyde-group-on-to-the-aromatic-ring-of-phenol-ortho-to-the-hydroxyl-group-this-reaction-involves-ele-55570

question:
<div class="question"><strong>Paragraph:</strong><br/>Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.<img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/Du2opTwjmV3M-NwcVEb4w_x6hSMB0KtJP7732UoYD-0.original.fullsize.png"/><br/><strong>Question:</strong><br/>The structure of the intermediate $\mathrm{I}$ is</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/ufEUk1j8s6jfvsoJhfg8lv47ygbrWsRJyTN4VSxj5HE.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/d-XGjxXmeveuzoYrGaxk9voyVW1aNHidwq3e9QB66Ow.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/OXLhm9FBFe0UEFkzFWnzJkWIDQBqymao0aq0olvvTCo.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/sGD7l_cZ4zJCRD1c5LAJLAQTuZTxGWsDqfBQ_5VC-Hc.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/d-XGjxXmeveuzoYrGaxk9voyVW1aNHidwq3e9QB66Ow.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/wItlqDuGitnTqt9mqZGYafaNfHHbw4_W8zFcogYiWAo.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P2.db
title:
paragraph-scientists-are-working-hard-to-develop-nuclear-fusion-reactor-nuclei-of-heavy-hydrogen-1-2-h-known-as-deuteron-and-denoted-by-d-can-be-thoug-1

slug:
paragraph-scientists-are-working-hard-to-develop-nuclear-fusion-reactor-nuclei-of-heavy-hydrogen-1-2-h-known-as-deuteron-and-denoted-by-d-can-be-thoug-1-28663

question:
<div class="question"><strong>Paragraph:</strong><br/>Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, ${ }_1^2 \mathrm{H}$ known as deuteron and denoted by $D$ can be thought of as a candidate for fusion reactor. The $D$ - $D$ reaction is ${ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+$ energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of ${ }_1^4 \mathrm{H}$ nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $t_0$ before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t $_0$ is called Lawson number. In one of the criteria, $a$ reactor is termed successful if Lawson number is greater than $5 \times 10^{14} \mathrm{scm}^{-3}$.<br/>It may be helpful to use the following : Boltzmann constant $k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}$; $\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}$<strong>Question:</strong><br/>In the core of nuclear fusion reactor, the gas becomes plasma because of</div>

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


tags: ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-scientists-are-working-hard-to-develop-nuclear-fusion-reactor-nuclei-of-heavy-hydrogen-1-2-h-known-as-deuteron-and-denoted-by-d-can-be-thoug-2

slug:
paragraph-scientists-are-working-hard-to-develop-nuclear-fusion-reactor-nuclei-of-heavy-hydrogen-1-2-h-known-as-deuteron-and-denoted-by-d-can-be-thoug-2-40676

question:
<div class="question"><strong>Paragraph:</strong><br/>Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, ${ }_1^2 \mathrm{H}$ known as deuteron and denoted by $D$ can be thought of as a candidate for fusion reactor. The $D$ - $D$ reaction is ${ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+$ energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of ${ }_1^4 \mathrm{H}$ nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $t_0$ before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t $_0$ is called Lawson number. In one of the criteria, $a$ reactor is termed successful if Lawson number is greater than $5 \times 10^{14} \mathrm{scm}^{-3}$.<br/>It may be helpful to use the following : Boltzmann constant $k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}$; $\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}$<strong>Question:</strong><br/>Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Deuteron density $=20 \times 10^{12} \mathrm{~cm}^{-3}$, confinement time $=5.0 \times 10^{-3} \mathrm{~s}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Deuteron density $=8.0 \times 10^{14} \mathrm{~cm}^{-3}$, confinement time $=9.0 \times 10^{-1} \mathrm{~s}$<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/>Deuteron density $=4.0 \times 10^{23} \mathrm{~cm}^{-3}$, confinement time $=1.0 \times 10^{-11} \mathrm{~s}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Deuteron density $=1.0 \times 10^{24} \mathrm{~cm}^{-3}$, confinement time $=4.0 \times 10^{-12} \mathrm{~s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Deuteron density $=8.0 \times 10^{14} \mathrm{~cm}^{-3}$, confinement time $=9.0 \times 10^{-1} \mathrm{~s}$<br/></span> </div>

answer:
<div class="solution">As given in the paragraph, a reactor is termed successful, if<br/>$$<br/>n t_0&gt;5 \times 10^{14} \mathrm{scm}^{-3}<br/>$$</div>


tags: ['Physics', 'Nuclear Physics', 'JEE Advanced', 'JEE Advanced 2009 (Paper 1)']
source_db: MarksBatch2_P2.db
title:
paragraph-scientists-are-working-hard-to-develop-nuclear-fusion-reactor-nuclei-of-heavy-hydrogen-1-2-h-known-as-deuteron-and-denoted-by-d-can-be-thoug

slug:
paragraph-scientists-are-working-hard-to-develop-nuclear-fusion-reactor-nuclei-of-heavy-hydrogen-1-2-h-known-as-deuteron-and-denoted-by-d-can-be-thoug-54985

question:
<div class="question"><strong>Paragraph:</strong><br/>Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, ${ }_1^2 \mathrm{H}$ known as deuteron and denoted by $D$ can be thought of as a candidate for fusion reactor. The $D$ - $D$ reaction is ${ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+$ energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of ${ }_1^4 \mathrm{H}$ nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $t_0$ before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t $_0$ is called Lawson number. In one of the criteria, $a$ reactor is termed successful if Lawson number is greater than $5 \times 10^{14} \mathrm{scm}^{-3}$.<br/>It may be helpful to use the following : Boltzmann constant $k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}$; $\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}$<strong>Question:</strong><br/>Assume that two deuteron nuclei in the core of fusion reactor at temperature $T$ are moving towards each other, each with kinetic energy $1.5 \mathrm{kT}$, when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature $T$ required for them to reach a separation of $4 \times 10^{-15} \mathrm{~m}$ is in the range</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$1.0 \times 10^9 \mathrm{~K} &lt; T &lt; 2.0 \times 10^9 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$20 \times 10^9 \mathrm{~K} &lt; T &lt; 3.0 \times 10^9 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$3.0 \times 10^9 \mathrm{~K} &lt; T &lt; 4.0 \times 10^9 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$4.0 \times 10^9 \mathrm{~K} &lt; T &lt; 5.0 \times 10^9 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$1.0 \times 10^9 \mathrm{~K} &lt; T &lt; 2.0 \times 10^9 \mathrm{~K}$</span> </div>

answer:
<div class="solution">From conservation of mechanical energy, we have<br/>$$<br/>\begin{gathered}<br/>U_i+K_i=U_f+U_f \\<br/>0+2(1.5 k T)=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{(e)(e)}{d}+0<br/>\end{gathered}<br/>$$<br/>Substituting the values, we get<br/>$$<br/>T=1.4 \times 10^9 \mathrm{~K}<br/>$$</div>


tags: ['Mathematics', 'Ellipse', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-coordinates-of-a-an

slug:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-coordinates-of-a-an-60424

question:
<div class="question"><strong>Paragraph:</strong><br/>Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$.<strong>Question:</strong><br/>The coordinates of $A$ and $B$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$(3,0)$ and $(0,2)$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(-\frac{8}{5}, \frac{2 \sqrt{161}}{15}\right)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\left(-\frac{8}{5}, \frac{2 \sqrt{161}}{15}\right)$ and $(0,2)$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$(3,0)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$(3,0)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$</span> </div>

answer:
<div class="solution">$$<br/>\text { Figure is self explanatory }<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/jDX6GkEhtNdY7zYO8pXOklILC_QwTFTGY_kNUs2S2r4.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Ellipse', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-equation-of-the-loc

slug:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-equation-of-the-loc-63661

question:
<div class="question"><strong>Paragraph:</strong><br/>Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$.<strong>Question:</strong><br/>The equation of the locus of the point whose distance from the point $P$ and the line $A B$ are equal, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$9 x^2+y^2-6 x y-54 x-62 y$<br/>$$<br/>+241=0<br/>$$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$x^2+9 y^2+6 x y-54 x+62 y$<br/>$$<br/>-241=0<br/>$$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$9 x^2+9 y^2-6 x y-54 x-62 y$<br/>$$<br/>-241=0<br/>$$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$x^2+y^2-2 x y+27 x+31 y$<br/>$$<br/>-120=0<br/>$$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$9 x^2+y^2-6 x y-54 x-62 y$<br/>$$<br/>+241=0<br/>$$<br/></span> </div>

answer:
<div class="solution">Equation of $A B$ is $y-0=-\frac{1}{3}(x-3)$<br/>$$<br/>\begin{gathered}<br/>x+3 y-3=0 \\<br/>|x+3 y-3|^2=10\left[(x-3)^2+(y-4)^2\right]<br/>\end{gathered}<br/>$$<br/>(Look at coefficient of $x^2$ and $y^2$ in the<br/>answers)</div>


tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-orthocentre-of-the--1

slug:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-orthocentre-of-the-1-38147

question:
<div class="question"><strong>Paragraph:</strong><br/>Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$.<strong>Question:</strong><br/>The orthocentre of the triangle $P A B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\left(5, \frac{8}{7}\right)$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(\frac{7}{5}, \frac{25}{8}\right)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{11}{5}, \frac{8}{5}\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(\frac{8}{25}, \frac{7}{5}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\left(\frac{11}{5}, \frac{8}{5}\right)$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { Equation of } A B \text { is }<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/vGNDL6PX8Om2ubizKilLEdDE0M3SMgzemqgZ4ITooP0.original.fullsize.png"/><br/><br/><br/>$$<br/>\begin{aligned}<br/>y-0 &amp; =\frac{\frac{8}{5}}{-\frac{9}{5}-3}(x-3)=\frac{8}{-24}(x-3) \\<br/>\Rightarrow \quad y &amp; =-\frac{1}{3}(x-3) \\<br/>\Rightarrow \quad x &amp; +3 y=3<br/>\end{aligned}<br/>$$<br/>Equation of the straight line perpendicular to $A B$ through $P$ is $3 x-y=5$.<br/>Equation of $P A$ is $x-3=0$.<br/>The equation of straight line perpendicular to $P A$ through $B\left(\frac{-9}{5}, \frac{8}{5}\right)$. is $y=\frac{8}{5}$.<br/>Hence, the orthocentre is $\left(\frac{11}{5}, \frac{8}{5}\right)$.</div>


tags: ['Mathematics', 'Straight Lines', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch2_P2.db
title:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-orthocentre-of-the-

slug:
paragraph-tangents-are-drawn-from-the-point-p-3-4-to-the-ellipse-9-x-2-4-y-2-1-touching-the-ellipse-at-points-a-and-b-question-the-orthocentre-of-the-21076

question:
<div class="question"><strong>Paragraph:</strong><br/>Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$.<strong>Question:</strong><br/>The orthocentre of the triangle $P A B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\left(5, \frac{8}{7}\right)$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\left(\frac{7}{5}, \frac{25}{8}\right)$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$\left(\frac{11}{5}, \frac{8}{5}\right)$<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$\left(\frac{8}{25}, \frac{7}{5}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\left(\frac{11}{5}, \frac{8}{5}\right)$<br/></span> </div>

answer:
<div class="solution">$$<br/>\text { Equation of } A B \text { is }<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/vGNDL6PX8Om2ubizKilLEdDE0M3SMgzemqgZ4ITooP0.original.fullsize.png"/><br/><br/><br/>$$<br/>\begin{aligned}<br/>y-0 &amp; =\frac{\frac{8}{5}}{-\frac{9}{5}-3}(x-3)=\frac{8}{-24}(x-3) \\<br/>\Rightarrow \quad y &amp; =-\frac{1}{3}(x-3) \\<br/>\Rightarrow \quad x &amp; +3 y=3<br/>\end{aligned}<br/>$$<br/>Equation of the straight line perpendicular to $A B$ through $P$ is $3 x-y=5$.<br/>Equation of $P A$ is $x-3=0$.<br/>The equation of straight line perpendicular to $P A$ through $B\left(\frac{-9}{5}, \frac{8}{5}\right)$. is $y=\frac{8}{5}$.<br/>Hence, the orthocentre is $\left(\frac{11}{5}, \frac{8}{5}\right)$.</div>


