# questions

title: what-is-the-solution-of-the-differential-equation-3-e-x-tan-y-d-x-1-e-x-sec-2-y-d-y-0
slug: what-is-the-solution-of-the-differential-equation-3-e-x-tan-y-d-x-1-e-x-sec-2-y-d-y-0-70045
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $3 e^{x} \tan y d x+\left(1+e^{x}\right) \sec ^{2} y d y=0 ? \quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(1+e^{x}\right)^{3} \tan y=c$</span> </div>

answer:
<div class="solution">$3 e^{x} \tan y d x+\left(1+e^{x}\right) \sec ^{2} y d y=0$<br/>By separating the variables, we get<br/>$$<br/>\begin{array}{l}<br/>3 e^{x} d x=\frac{-\left(1+e^{x}\right) \sec ^{2} y}{\tan y} d y \\<br/>\Rightarrow \frac{3 e^{x}}{1+e^{x}} d x=-\frac{\sec ^{2} y}{\tan y} d y<br/>\end{array}<br/>$$<br/>Integrate on both sides,<br/>$\Rightarrow \int \frac{3 e^{x}}{1+e^{x}} d x+\int \frac{\sec ^{2} y}{\tan y} d y=0$<br/>$\Rightarrow 3 \log \left(1+e^{x}\right)+\log \tan y=\log c$<br/>$\Rightarrow \log \left(1+e^{x}\right)^{3} \tan y=\log c$<br/>$(\because \log m+\log n=\log m n)$<br/>$\Rightarrow\left(1+e^{x}\right)^{3} \tan y=c$</div>


title: what-is-the-solution-of-the-differential-equation-a-x-d-x-d-y-2-y-x-y-d-x-d-y
slug: what-is-the-solution-of-the-differential-equation-a-x-d-x-d-y-2-y-x-y-d-x-d-y-92450
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$$<br/>a\left(x \frac{d y}{d x}+2 y\right)=x y \frac{d y}{d x} ?<br/>$$</div>

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


title: what-is-the-solution-of-the-differential-equation-cosec-2-x-y-d-y-d-x
slug: what-is-the-solution-of-the-differential-equation-cosec-2-x-y-d-y-d-x-35285
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $-\operatorname{cosec}^{2}(x+y) d y=d x ? \quad$</div>

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


title: what-is-the-solution-of-the-differential-equation-d-x-d-y-e-x-y-e-y-x-e-y
slug: what-is-the-solution-of-the-differential-equation-d-x-d-y-e-x-y-e-y-x-e-y-42470
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $\frac{d y}{d x}=e^{x-y}\left(e^{y-x}-e^{y}\right) ? \quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=x-e^{x}+c$</span> </div>

answer:
<div class="solution">Given differential equation is<br/>$$<br/>\frac{d y}{d x}=e^{x-y}\left(e^{y-x}-e^{y}\right)=e^{-y} \cdot e^{y}\left(e^{x} \cdot e^{-x}-e^{x}\right)<br/>$$<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \quad \int 1 d y=\int\left(1-e^{x}\right) d x \\<br/>\Rightarrow \quad y=x-e^{x}+c<br/>\end{array}<br/>$$</div>


title: what-is-the-solution-of-the-differential-equation-d-x-d-y-x-2-y-3-y
slug: what-is-the-solution-of-the-differential-equation-d-x-d-y-x-2-y-3-y-70649
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{d y}{d x}=\frac{y}{\left(x+2 y^{3}\right)} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y^{3}-x=c y$</span> </div>

answer:
<div class="solution">$y^{3}-x=c y$<br/>$\Rightarrow 3 y^{2} \frac{d y}{d x}-1=c \frac{d y}{d x}$<br/>$\Rightarrow \frac{d y}{d x}\left(3 y^{2}-c\right)=1$<br/>$\Rightarrow \frac{d y}{d x}\left(3 y^{2}-\frac{y^{3}-x}{y}\right)=1$<br/>$\Rightarrow \frac{d y}{d x}\left(\frac{3 y^{3}-y^{3}+x}{y}\right)=1$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}+2 \mathrm{y}^{3}}$</div>


title: what-is-the-solution-of-the-differential-equation-d-y-d-x-y-x-y-2-0
slug: what-is-the-solution-of-the-differential-equation-d-y-d-x-y-x-y-2-0-85063
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{d x}{d y}+\frac{x}{y}-y^{2}=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 x y=y^{4}+c$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{dx}}{\mathrm{dy}}+\frac{\mathrm{x}}{\mathrm{y}}-\mathrm{y}^{2}=0$<br/>$\frac{d x}{d y}+\frac{x}{y}=y^{2}$<br/>This is a linear differential equation of the form<br/>$\frac{\mathrm{dx}}{\mathrm{dy}}+\mathrm{P}_{1} \mathrm{x}=\mathrm{Q}_{1}$<br/>Here, $\mathrm{P}=\frac{1}{\mathrm{y}}$ and $\mathrm{Q}=\mathrm{y}^{2}$<br/>$\therefore \quad \mathrm{I} . \mathrm{F}=\mathrm{e}^{\int \mathrm{P} \mathrm{dy}}=\mathrm{e}^{\int \frac{1}{\mathrm{y}} \mathrm{dy}}=\mathrm{e}^{\log \mathrm{y}}=\mathrm{y}$<br/>So, required solution is $\mathrm{x} \cdot \mathrm{y}=\int \mathrm{y}^{2} \cdot \mathrm{y} \mathrm{dy}+\mathrm{c}$<br/>$\mathrm{xy}=\int \mathrm{y}^{3} \mathrm{dy}+\mathrm{c}$<br/>$\mathrm{xy}=\frac{\mathrm{y}^{4}}{4}+\mathrm{c}$<br/>$4 x y=y^{4}+c$<br/>$\therefore$ Option (c) is correct.</div>


title: what-is-the-solution-of-the-differential-equation-dx-dy-1-x-2-1-y-2-0
slug: what-is-the-solution-of-the-differential-equation-dx-dy-1-x-2-1-y-2-0-55633
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}+\sqrt{\frac{1-\mathrm{y}^{2}}{1-\mathrm{x}^{2}}}=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sin ^{-1} \mathrm{y}+\sin ^{-1} \mathrm{x}=\mathrm{C}$</span> </div>

answer:
<div class="solution">The differential equation is<br/>$$<br/>\frac{d y}{d x}+\sqrt{\frac{1-y^{2}}{1-x^{2}}}=0<br/>$$<br/>$\Rightarrow \frac{d y}{d x}=-\sqrt{\frac{1-y^{2}}{1-x^{2}}}$<br/>$\Rightarrow \frac{d y}{\sqrt{1-y^{2}}}=\frac{-d x}{\sqrt{1-x^{2}}}$<br/>$\Rightarrow \int \frac{1}{\sqrt{1-y^{2}}} d y+\int \frac{1}{\sqrt{1-x^{2}}} d x=0$<br/>$\Rightarrow \sin ^{-1} y+\sin ^{-1} x=C$</div>


title: what-is-the-solution-of-the-differential-equation-dx-dy-sec-x-y
slug: what-is-the-solution-of-the-differential-equation-dx-dy-sec-x-y-69131
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\sec (\mathrm{x}+\mathrm{y}) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{y}-\tan \left\{\frac{(\mathrm{x}+\mathrm{y})}{2}\right\}=\mathrm{c}$</span> </div>

answer:
<div class="solution">In the equation $\frac{d y}{d x}=\sec (x+y)$<br/>Let $x+y=v$<br/>So, $1+\frac{d y}{d x}=\frac{d v}{d x}$ or $\frac{d y}{d x}=\frac{d v}{d x}-1$<br/>and $\frac{\mathrm{dv}}{\mathrm{dx}}-1=\mathrm{sec} \mathrm{v}$<br/>$\frac{d v}{d x}=1+\sec v=\frac{1+\cos v}{\cos v}$<br/>or, $\frac{\cos \mathrm{v}}{1+\cos \mathrm{v}} \mathrm{dv}=\mathrm{dx}$<br/>$\cos \mathrm{v}=\cos ^{2} \frac{\mathrm{v}}{2}-\sin ^{2} \frac{\mathrm{v}}{2}$<br/>and $1+\cos v=2 \cos ^{2} \frac{\mathrm{v}}{2}$<br/>so, $\left(\frac{2 \cos ^{2} \frac{\mathrm{v}}{2}-2 \sin ^{2} \frac{\mathrm{v}}{2}}{2 \cos ^{2} \frac{\mathrm{v}}{2}}\right) \mathrm{dv}=\mathrm{dx}$<br/>$\left(1-\tan ^{2} \frac{\mathrm{v}}{2}\right) \mathrm{dv}=2 \mathrm{dx}$<br/>or, $\left\{1-\left(\sec ^{2} \frac{\mathrm{v}}{2}-1\right)\right\} \mathrm{dv}=2 \mathrm{dx}$<br/>or, $\left(2-\sec ^{2} \frac{\mathrm{v}}{2}\right) \mathrm{dv}=2 \mathrm{dx}$<br/>Integrating on both the sides<br/>$2 \int \mathrm{dv}-\int \sec ^{2} \frac{\mathrm{v}}{2} \mathrm{dv}=2 \int \mathrm{d} \mathrm{x}+\mathrm{c}_{1}$<br/>where $c_{1}$ is a constant.<br/>$2 \mathrm{v}-2 \tan \frac{\mathrm{v}}{2}=2 \mathrm{x}+\mathrm{c}_{1}$<br/>$2(x+y)-2 \tan \frac{x+y}{2}=2 x+c_{1}$<br/>or, $\quad 2 \mathrm{x}+2 \mathrm{y}-2 \tan \frac{\mathrm{x}+\mathrm{y}}{2}=2 \mathrm{x}+\mathrm{c}_{1}$<br/>$\mathrm{y}-\tan \left(\frac{\mathrm{x}+\mathrm{y}}{2}\right)=\frac{\mathrm{c}_{1}}{2}=\mathrm{c}$<br/>[c is a constant]<br/>so, $y-\tan \frac{x+y}{2}=c$</div>


title: what-is-the-solution-of-the-differential-equation-dx-dy-x-y-0
slug: what-is-the-solution-of-the-differential-equation-dx-dy-x-y-0-41057
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x y=c$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=0$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}}{\mathrm{x}} \Rightarrow \frac{\mathrm{dy}}{\mathrm{y}}=-\frac{\mathrm{dx}}{\mathrm{x}}$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{y}}+\frac{\mathrm{d} \mathrm{x}}{\mathrm{x}}=0 \Rightarrow \mathrm{O} \frac{\mathrm{dy}}{\mathrm{y}}+\mathrm{O} \frac{\mathrm{d} \mathrm{x}}{\mathrm{x}}=0$<br/>$\Rightarrow \log \mathrm{y}+\log \mathrm{x}=\log \mathrm{c}$<br/>$\Rightarrow \log x y=\log c$<br/>$\Rightarrow x y=c$</div>


title: what-is-the-solution-of-the-differential-equation-dx-dy-xy-x-y-1
slug: what-is-the-solution-of-the-differential-equation-dx-dy-xy-x-y-1-40961
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{xy}+\mathrm{x}+\mathrm{y}+1 ?$</div>

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

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

answer:
<div class="solution">The given differential equation is:<br/>$\frac{d y}{d x}=x y+x+y+1 \Rightarrow \frac{d y}{d x}=(x+1)(y+1)$<br/>Separating variables,<br/>$\Rightarrow \frac{1}{(1+y)} d y=(x+1) d x \Rightarrow \log (1+y)=\frac{x^{2}}{2}+x+c$</div>


title: what-is-the-solution-of-the-differential-equation-dy-dx-x-y-1-x-y-1
slug: what-is-the-solution-of-the-differential-equation-dy-dx-x-y-1-x-y-1-90593
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{\mathrm{x}+\mathrm{y}+1}{\mathrm{x}+\mathrm{y}-1} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{y}-\mathrm{x}+\ln (\mathrm{x}+\mathrm{y})=\mathrm{c}$</span> </div>

answer:
<div class="solution">Let $t=x+y \Rightarrow \frac{d t}{d y}=\frac{d x}{d y}+1$<br/>So, $\frac{d t}{d y}-1=\frac{t+1}{t-1}$<br/>$\Rightarrow \quad \frac{d t}{d y}=\frac{t+1+t-1}{t-1}=\frac{2 t}{t-1}$<br/>$\Rightarrow \quad \int \frac{t-1}{t} d t=2 \int d y$<br/>$\Rightarrow \quad t-\log t=2 y+\mathrm{C}_{1}$<br/>$\Rightarrow \quad x+y-\log (x+y)-2 y=\mathrm{C}_{1}$<br/>$\Rightarrow \quad y-x+\log (x+y)=-\mathrm{C}_{1}=\mathrm{C}$</div>


title: what-is-the-solution-of-the-differential-equation-ln-dx-dy-a-0
slug: what-is-the-solution-of-the-differential-equation-ln-dx-dy-a-0-76719
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation<br/>$\ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{a}=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{y}=\mathrm{xe}^{\mathrm{a}}+\mathrm{c}$</span> </div>

answer:
<div class="solution">$\ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{a}=0$<br/>$\Rightarrow \ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{a} \quad \quad \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{\mathrm{a}}$<br/>$\Rightarrow \int \mathrm{dy}=\int \mathrm{e}^{\mathrm{a}} \cdot \mathrm{dx} \quad \Rightarrow \mathrm{y}=\mathrm{e}^{\mathrm{a}} \cdot \mathrm{x}+\mathrm{c}$</div>


title: what-is-the-solution-of-the-differential-equation-ln-dx-dy-ax-by
slug: what-is-the-solution-of-the-differential-equation-ln-dx-dy-ax-by-50739
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $\ln$ $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{ax}+\mathrm{by} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\mathrm{a}} \mathrm{e}^{\mathrm{ax}}+\frac{1}{\mathrm{~b}} \mathrm{e}^{-\mathrm{by}}=\mathrm{c}$</span> </div>

answer:
<div class="solution">$\frac{d y}{d x}=e^{a x+b y}=e^{a x} \cdot e^{b y}$<br/>Integrating both sides,<br/>$\frac{1}{a} e^{a x}+\frac{1}{b} e^{-b y}=c$</div>


title: what-is-the-solution-of-the-differential-equation-sin-d-x-d-y-a-0
slug: what-is-the-solution-of-the-differential-equation-sin-d-x-d-y-a-0-98134
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation  $\sin \left(\frac{d y}{d x}\right)-a=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=x \sin ^{-1} a+c$</span> </div>

answer:
<div class="solution">$\sin \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{a}=0$<br/>$\sin \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{a}$<br/>$\Rightarrow \frac{d y}{d x}=\sin ^{-1} a$<br/>$\mathrm{dy}=\sin ^{-1} \mathrm{a} \mathrm{dx}$<br/>Now integrating both sides, $\int \mathrm{dy}=\int \sin ^{-1} \mathrm{a} \mathrm{dx}$<br/>$\mathrm{y}=\mathrm{x} \sin ^{-1} \mathrm{a}+\mathrm{c}$<br/>$\therefore$ Option (a) is correct.</div>


title: what-is-the-solution-of-the-differential-equation-x-d-y-y-d-x-0
slug: what-is-the-solution-of-the-differential-equation-x-d-y-y-d-x-0-40026
question: <div class="question">What is the solution of the differential equation $x d y-y d x=0 ?$</div>
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x y=c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{y}=\mathrm{cx}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$x+y=c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y=c$</span> </li> </ul>

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

answer:
<div class="solution">$x d y-y d x=0$<br/>$\Rightarrow x d y=y d x$<br/>$\quad \Rightarrow \frac{d y}{y}=\frac{d x}{x}$<br/>$\Rightarrow \log y=\log x+\log c$<br/>$\Rightarrow \log y-\log x=\log c$<br/>$\Rightarrow \log \left(\frac{y}{x}\right)=\log c$<br/>$\Rightarrow \frac{y}{x}=c \Rightarrow y=c x$</div>


title: what-is-the-solution-of-the-differential-equation-x-d-y-y-d-x-x-y-2-d-x-1
slug: what-is-the-solution-of-the-differential-equation-x-d-y-y-d-x-x-y-2-d-x-1-54072
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $x d y-y d x=x y^{2} d x ? \quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+2 x y^{-1}=c$</span> </div>

answer:
<div class="solution">Given differential equation is $x d y-y d x=x y^{2} d x$<br/>Which can be rewritten as $\frac{x d y-y d x}{y^{2}}=x d x$<br/>$\Rightarrow-d\left(\frac{x}{y}\right)=x d x$<br/>On integrating both sides, we get<br/>$\int-d\left(\frac{x}{y}\right)=\int x d x$<br/>$-\frac{x}{y}=\frac{-c}{2}+\left(\frac{-c}{2}\right) \quad\left(\because \frac{-c}{2}\right.$ is constant $)$<br/>$\Rightarrow \frac{c}{2}=\frac{x^{2}}{2}+\frac{x}{y} \Rightarrow \frac{c}{2}=\frac{x^{2} y+2 x}{2 y}$<br/>$\Rightarrow c=\frac{x^{2} y}{y}+\frac{2 x}{y} \Rightarrow c=x^{2}+\frac{2 x}{y}$<br/>$\Rightarrow x^{2}+2 x y^{-1}=c$</div>


title: what-is-the-solution-of-the-differential-equation-x-d-y-y-d-x-x-y-2-d-x
slug: what-is-the-solution-of-the-differential-equation-x-d-y-y-d-x-x-y-2-d-x-64320
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $x d y-y d x=x y^{2} d x ? \quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y x^{2}+2 x=2 c y$</span> </div>

answer:
<div class="solution">Given, $x d y-y d x=x y^{2} d x$<br/>$\Rightarrow \frac{x d y-y d x}{y^{2}}=x d x \Rightarrow \frac{y d x-x d y}{y^{2}}=-x d x$<br/>$\Rightarrow \int d\left(\frac{x}{y}\right)=-\int x d x \Rightarrow \frac{x}{y}=\frac{-x^{2}}{2}+c=\frac{-x^{2}+2 c}{2}$<br/>$\Rightarrow \quad y x^{2}+2 x=2 c y$</div>


title: what-is-the-solution-of-the-differential-equation-x-y-dx-dy-dx-dy-1
slug: what-is-the-solution-of-the-differential-equation-x-y-dx-dy-dx-dy-1-81609
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $(\mathrm{x}+\mathrm{y})(\mathrm{dx}-\mathrm{dy})=\mathrm{dx}+\mathrm{dy} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(y-x)+\log (x+y)=c$</span> </div>

answer:
<div class="solution">Given differential equation is:<br/>$(x+y)(d x-d y)=d x+d y$<br/>$\Rightarrow(\mathrm{x}+\mathrm{y}) \mathrm{dx}-(\mathrm{x}+\mathrm{y}) \mathrm{dy}=\mathrm{dx}+\mathrm{dy}$<br/>$\Rightarrow(x+y-1) d x=(x+y+1) d y$<br/>$\Rightarrow \frac{d y}{d x}=\frac{x+y-1}{x+y+1}$<br/>Let $x+y=v$ and $\frac{d y}{d x}=\frac{d v}{d x}-1$<br/>$\therefore \quad \frac{d v}{d x}-1=\frac{v-1}{v+1}$<br/>$\Rightarrow \frac{d v}{d x}=\frac{v-1}{v+1}+1$<br/>$\Rightarrow \frac{d v}{d x}=\frac{v-1+v+1}{v+1}$<br/>$\Rightarrow \frac{v+1}{2 v} d v=d x$<br/>$\Rightarrow \frac{1}{2} \int 1 d v+\frac{1}{2} \int \frac{1}{v} d v=\int 1 d x$<br/>$\Rightarrow \frac{1}{2} v+\frac{1}{2} \log v=x+c_{1}$<br/>$\Rightarrow x+y+\log (x+y)=2 x+c \quad\left(\because 2 c_{1}=c=\right.$ costant $)$<br/>$\Rightarrow(y-x)+\log (x+y)=c$</div>


title: what-is-the-solution-of-the-differential-equation-x-y-dx-dy-dx-dy
slug: what-is-the-solution-of-the-differential-equation-x-y-dx-dy-dx-dy-94409
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation $(\mathrm{x}+\mathrm{y})$ $(\mathrm{dx}-\mathrm{dy})=\mathrm{dx}+\mathrm{dy} ? \quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{y}-\mathrm{x}+\ln (\mathrm{x}+\mathrm{y})=\mathrm{c}$</span> </div>

answer:
<div class="solution">Differential equation is $(x+y)(d x-d y)=d x+d y$<br/>dividing by $\mathrm{dx}$ on both the sides<br/>$(x+y)\left(1-\frac{d y}{d x}\right)=1+\frac{d y}{d x}$<br/>Putting $\mathrm{x}+\mathrm{y}=\mathrm{v}$<br/>$1+\frac{d y}{d x}=\frac{d v}{d x}$ and $\frac{d y}{d x}=\frac{d v}{d x}-1$<br/>The equation changes to<br/>$\mathrm{v}\left\{1-\left(\frac{\mathrm{dv}}{\mathrm{dx}}-1\right)\right\}=\frac{\mathrm{dv}}{\mathrm{dx}}$<br/>$v\left(2-\frac{d v}{d x}\right)=\frac{d v}{d x}$<br/>$2 \mathrm{v}-\mathrm{v} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{dv}}{\mathrm{dx}}$<br/>$2 \mathrm{v}=(1+\mathrm{v}) \frac{\mathrm{d} \mathrm{v}}{\mathrm{d} \mathrm{x}}$<br/>$\left(\frac{1+\mathrm{v}}{\mathrm{v}}\right) \mathrm{d} \mathrm{v}=2 \mathrm{dx}$<br/>or,$\left(\frac{1}{\mathrm{v}}+1\right) \mathrm{dv}=2 \mathrm{dx}$<br/>Integrating on both the sides,<br/>$\int \frac{\mathrm{dv}}{\mathrm{v}}+\int \mathrm{dv}=2 \int \mathrm{d} \mathrm{x}+\mathrm{c}$<br/>$\log _{v}+v=2 x+c$<br/>Putting $\mathrm{v}=\mathrm{x}+\mathrm{y}$<br/>$\log (x+y)+x+y=2 x+c$<br/>or, $\log (x+y)+y-x=c$<br/>or, $y-x+\log (x+y)=c$</div>


title: what-is-the-solution-of-the-differential-equation-y-2-y-d-x-x-d-y-0
slug: what-is-the-solution-of-the-differential-equation-y-2-y-d-x-x-d-y-0-42018
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the differential equation <br/>$\frac{y d x-x d y}{y^{2}}=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=c x$</span> </div>

answer:
<div class="solution">$\frac{y \mathrm{dx}-\mathrm{x} \mathrm{dy}}{\mathrm{y}^{2}}=0$<br/>$\therefore \quad y d x-x d y=0$<br/>$\therefore \quad \frac{d y}{y}=\frac{d x}{x}$<br/>Now integrating both sides,<br/>$\int \frac{\mathrm{dy}}{\mathrm{y}}=\int \frac{\mathrm{dx}}{\mathrm{x}}$<br/>$\Rightarrow \log y=\log x+\log c$<br/>$\Rightarrow \log \mathrm{y}=\log \mathrm{c} \mathrm{x}$<br/>$\mathrm{y}=\mathrm{cx}$<br/>$\therefore$ Option (b) is correct.</div>


title: what-is-the-solution-of-the-equation-ln-d-x-d-y-x-0
slug: what-is-the-solution-of-the-equation-ln-d-x-d-y-x-0-36387
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the solution of the equation $\quad$<br/>$\ln \left(\frac{d y}{d x}\right)+x=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y+e^{-x}=c$</span> </div>

answer:
<div class="solution">Consider the given differential equation<br/>$\ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+\mathrm{x}=0$<br/>$\Rightarrow \ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=-\mathrm{x}$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{-\mathrm{x}}$<br/>on separating the variables, we get $\mathrm{dy}=\mathrm{e}^{-\mathrm{x}} \mathrm{dx}$<br/>on integrating both sides, we get<br/>$\int \mathrm{dy}=\int \mathrm{e}^{-\mathrm{x}} \mathrm{dx}$<br/>$\Rightarrow \mathrm{y}=\frac{\mathrm{e}^{-\mathrm{x}}}{-1}+\mathrm{C}=-\mathrm{e}^{-\mathrm{x}}+\mathrm{C}$</div>


title: what-is-the-solution-of-y-1-x-y-2-x-y-2-y-0-0
slug: what-is-the-solution-of-y-1-x-y-2-x-y-2-y-0-0-76791
question: <div class="question">What is the solution of $y^{\prime}=1+x+y^{2}+x y^{2}, y(0)=0$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=\tan ^{2}\left(\frac{x^{2}}{2}+x\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{y}=\tan ^{2}\left(\mathrm{x}^{2}+\mathrm{x}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y=\tan \left(x^{2}+x\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\quad y=\tan \left(\frac{x^{2}}{2}+x\right)$</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">$\quad y=\tan \left(\frac{x^{2}}{2}+x\right)$</span> </div>

answer:
<div class="solution">Given differential equation is<br/>$\frac{d y}{d x}=1+x+y^{2}+x y^{2}$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=(1+\mathrm{x})\left(1+\mathrm{y}^{2}\right)$<br/>$\Rightarrow \frac{d y}{1+y^{2}}=(1+x) d x$<br/>$\Rightarrow \tan ^{-1} y=\frac{x^{2}}{2}+x+c$<br/>Given that when $\mathrm{x}=0, \mathrm{y}(0)=0 .$ Hence, $\mathrm{c}=0$<br/>$\Rightarrow \quad y=\tan \left(\frac{x^{2}}{2}+x\right)$</div>


title: what-is-the-solution-set-for-the-equation-x-4-26-x-2-25-0
slug: what-is-the-solution-set-for-the-equation-x-4-26-x-2-25-0-16891
question: <div class="question">What is the solution set for the equation $x^{4}-26 x^{2}+25=0$</div>
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\{-5,-1,1,5\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\{-5,-1\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\{1,5\}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\{-5,0,1,5\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\{-5,-1,1,5\}$</span> </div>

answer:
<div class="solution">Consider, $x^{4}-26 x^{2}+25=0$ $\Rightarrow \quad x^{4}-25 x^{2}-x^{2}+25=0$<br/>$\quad=x^{2}\left(x^{2}-25\right)-\left(x^{2}-25\right)=0$<br/>$\Rightarrow \quad\left(x^{2}-25\right)\left(x^{2}-1\right)=0$<br/>$\Rightarrow \quad(x-5)(x+5)(x-1)(x+1)=0$<br/>$\Rightarrow \quad x=5,-5,1,-1$<br/>$\therefore$ Solution set for given equation is $\{5,-5,-1,1\}$</div>


tags: ['Chemistry', 'Waves and Sound', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">79 m/s</span> </div>
answer: <div class="solution">Correct Option is : (A)<br/>79 m/s</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-speed-of-a-transverse-wave-on-the-string-stretched-between-two-rigid-supports-given-that-it-vibrates-in-its-fundamental-mode-with-a-freque

slug:
what-is-the-speed-of-a-transverse-wave-on-the-string-stretched-between-two-rigid-supports-given-that-it-vibrates-in-its-fundamental-mode-with-a-freque-29612

question:
<div class="question">What is the speed of a transverse wave on the string stretched between two rigid supports, given that it vibrates in its fundamental mode with a frequency of 45 Hz, a mass of 3.5 Ã ${{10}^{-2}}$kg, and a linear mass density of 4.0 Ã ${{10}^{-2}}$kg${{m}^{-1}}$?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">79 m/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><li class=""> <span class="option-label">B</span> <span class="option-data">81 m/s</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">66m/s</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">60 m/s</span> </li> </ul>


tags: ['Chemistry', 'Waves and Sound', 'NEET']
answer: <div class="solution">Correct Option is : (A)<br/>280 m/s</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-speed-of-sound-in-air-at-standard-temperature-and-pressure-where-the-mass-of-1-mole-of-air-is-290-10-3-kg

slug:
what-is-the-speed-of-sound-in-air-at-standard-temperature-and-pressure-where-the-mass-of-1-mole-of-air-is-290-10-3-kg-43808

question:
<div class="question">What is the speed of sound in air at standard temperature and pressure, where the mass of 1 mole of air is 29.0 Ã${{10}^{-3}}$kg?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">280 m/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><li class=""> <span class="option-label">B</span> <span class="option-data">274m/s</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">331m/s</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">319m/s</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">280 m/s</span> </div>


tags: ['Chemistry', 'Waves and Sound', 'NEET']
answer: <div class="solution">Correct Option is : (A)<br/>5.06 km/s</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-speed-of-sound-in-steel-if-a-steel-rod-100-cm-long-clamped-at-its-middle-has-a-fundamental-frequency-of-longitudinal-vibrations-of-253-khz

slug:
what-is-the-speed-of-sound-in-steel-if-a-steel-rod-100-cm-long-clamped-at-its-middle-has-a-fundamental-frequency-of-longitudinal-vibrations-of-253-khz-48961

question:
<div class="question">What is the speed of sound in steel if a steel rod 100 cm long, clamped at its middle, has a fundamental frequency of longitudinal vibrations of 2.53 kHz?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">5.06 km/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><li class=""> <span class="option-label">B</span> <span class="option-data">1.01 km/s</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2.52 km/s</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5.91 km/s</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">5.06 km/s</span> </div>


title: what-is-the-square-root-of-2-1-i-2-3
slug: what-is-the-square-root-of-2-1-i-2-3-54083
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the square root of $\frac{1}{2}-\mathrm{i} \frac{\sqrt{3}}{2}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pm\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)$</span> </div>

answer:
<div class="solution">Let $\sqrt{\frac{1}{2}-\frac{\mathrm{i} \sqrt{3}}{2}}=\mathrm{x}-\mathrm{iy}$<br/>Squaring both the sides,<br/>$\Rightarrow \frac{1}{2}-\frac{i \sqrt{3}}{2}=(x-i y)^{2} \Rightarrow \frac{1}{2}-\frac{i \sqrt{3}}{2}=x^{2}-y^{2}-2 i x y$<br/>Comparing real and imaginary parts, we get<br/>$\mathrm{x}^{2}-\mathrm{y}^{2}=\frac{1}{2}$<br/>and $2 x y=\frac{\sqrt{3}}{2}$<br/>and $\left(x^{2}+y^{2}\right)^{2}=\left(x^{2}-y^{2}\right)^{2}+4 x^{2} y^{2}=\frac{1}{4}+\frac{3}{4}=1$<br/>$\Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}=1$<br/>On solving eqs. (1) and (3) we get<br/>$\mathrm{x}^{2}=\frac{3}{4}$ and $\mathrm{y}^{2}=\frac{1}{4}$<br/>$\Rightarrow x=\pm \frac{\sqrt{3}}{2}, y=\pm \frac{1}{2}$<br/>$\therefore \sqrt{\frac{1}{2}-\frac{\mathrm{i} \sqrt{3}}{2}}=\pm\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)$</div>


title: what-is-the-square-root-of-i-where-i-1
slug: what-is-the-square-root-of-i-where-i-1-28504
question: <div class="question">What is the square root of $i$, where $i=\sqrt{-1}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1+i}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1-i}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1+i}{\sqrt{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

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

answer:
<div class="solution">Let $\sqrt{\mathrm{i}}=\mathrm{x}+\mathrm{iy}$<br/>$\mathrm{i}=(\mathrm{x}+\mathrm{iy})^{2}$<br/>$x^{2}-y^{2}+2 x y i=0+i$<br/>$x^{2}-y^{2}=0 ; 2 x y=1$<br/>Now, $\left(x^{2}+y^{2}\right)^{2}=\left(x^{2}-y^{2}\right)^{2}+4 x^{2} y^{2}$<br/>$\left(x^{2}+y^{2}\right)^{2}=0+1$<br/>$x^{2}+y^{2}=1$<br/>$x^{2}-y^{2}=0$<br/>$x^{2}+y^{2}=1$<br/>$2 \mathrm{x}^{2}=1$<br/>$\mathrm{x}^{2}=\frac{1}{2} \Rightarrow \mathrm{x}=\pm \frac{1}{\sqrt{2}}$<br/>$\mathrm{y}^{2}=\frac{1}{2} \Rightarrow \mathrm{y}=\pm \frac{1}{\sqrt{2}}$<br/>$\therefore \sqrt{\mathrm{i}}=\pm\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}} \mathrm{i}\right)$<br/>$=\frac{1}{\sqrt{2}}(1+\mathrm{i})$ or $\frac{-1}{\sqrt{2}}(1+\mathrm{i}) .$</div>


title: what-is-the-square-root-of-the-complex-number-5-12-i
slug: what-is-the-square-root-of-the-complex-number-5-12-i-24709
question: <div class="question">What is the square root of the complex number $-5+12 \mathrm{i}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2-3 \mathrm{i}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2+3 \mathrm{i}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$-2+3 \mathrm{i}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{-5}+\sqrt{12 \mathrm{i}}$</span> </li> </ul>

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

answer:
<div class="solution">Let $\sqrt{-5+12 \mathrm{i}}=\mathrm{x}+\mathrm{iy}$<br/>$\Rightarrow(\mathrm{x}+\mathrm{iy})^{2}=-5+12 \mathrm{i}$<br/>$\Rightarrow \mathrm{x}^{2}-\mathrm{y}^{2}+\mathrm{i} 2 \mathrm{xy}=-5+12 \mathrm{i}$<br/>$\Rightarrow \mathrm{x}^{2}-\mathrm{y}^{2}=-5 \mathrm{and} 2 \mathrm{xy}=12 \Rightarrow \mathrm{xy}=6$<br/>$\quad\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right)^{2}+4 \mathrm{x}^{2} \mathrm{y}^{2}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{2}$<br/>$\Rightarrow(-5)^{2}+4 \times(6)^{2}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{2}$<br/>$\Rightarrow\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{2}=25+144=169$<br/>$\Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}=\pm 13$ and $\mathrm{x}^{2}-\mathrm{y}^{2}=-5$<br/>Adding both $2 \mathrm{x}^{2}=\pm 13-5=8$ or $-18$<br/>$\Rightarrow \mathrm{x}^{2}=4($ -ve discarded $)$<br/>$\Rightarrow \mathrm{x}=\pm 2$ and $\mathrm{y}^{2}=\pm 13-4=9$ or $-13$<br/>Discard $\mathrm{y}^{2}=-13, \mathrm{y}^{2}=9$<br/>$\Rightarrow \mathrm{y}=\pm 3$<br/>$\Rightarrow \mathrm{x}+\mathrm{iy}=\pm(2+3 \mathrm{i})$<br/>$\Rightarrow \sqrt{-5+12 \mathrm{i}}=(2+3 \mathrm{i})$</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-cell-potential-e-cell-o-for-following-cell-reaction-2fes-o-2-g-2h-2-ol-2fe-2-aq-4oh-aq-given-e-o-fe-2-aq-fe-0-44-v-e-o-o-2-g-h-2-

slug:
what-is-the-standard-cell-potential-e-cell-o-for-following-cell-reaction-2fes-o-2-g-2h-2-ol-2fe-2-aq-4oh-aq-given-e-o-fe-2-aq-fe-0-44-v-e-o-o-2-g-h-2-59615

question:
<div class="question">What is the standard cell Potential <math> <mfenced> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>Cell</mtext></mrow> <mtext>o</mtext> </msubsup> </mrow> </mfenced> </math> for following cell reaction?<br/> <math> <mrow> <mtext>2Fe(s)</mtext><mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msub> <mtext>O</mtext> <mtext>2</mtext> </msub> <mtext>(g)</mtext><mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msub> <mrow> <mtext>2H</mtext></mrow> <mtext>2</mtext> </msub> <mtext>O(l)</mtext><mtext>â</mtext><mo>â</mo><mtext>â</mtext><msup> <mrow> <mtext>2Fe</mtext></mrow> <mrow> <mtext>2+</mtext></mrow> </msup> <mtext>(aq)</mtext><mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>4OH</mtext></mrow> <mtext>-</mtext> </msup> <mtext>(aq)</mtext></mrow> </math><br/> <br/>
Given<br/> <math> <mrow> <msup> <mtext>E</mtext> <mtext>o</mtext> </msup> <msub> <mrow></mrow> <mrow> <msup> <mrow> <mtext>Fe</mtext></mrow> <mrow> <mtext>2+</mtext></mrow> </msup> <mtext>(aq)</mtext><mtext>â</mtext><mtext>|</mtext><mtext>â</mtext><mtext>Fe</mtext><mtext>â</mtext><mtext>=</mtext><mtext>â</mtext><mo>â</mo><mtext>â</mtext><mtext>0</mtext><mtext>.44</mtext><mtext>â</mtext><mtext>V</mtext></mrow> </msub> </mrow> </math><br/> <math> <mrow> <msup> <mtext>E</mtext> <mtext>o</mtext> </msup> <msub> <mrow></mrow> <mrow> <msub> <mtext>O</mtext> <mtext>2</mtext> </msub> <mtext>(g)</mtext><mtext>â</mtext><mtext>|</mtext><mtext>â</mtext><msub> <mtext>H</mtext> <mtext>2</mtext> </msub> <mtext>O</mtext><mtext>â</mtext><mtext>|</mtext><mtext>â</mtext><mtext>OH</mtext><mtext>â</mtext><mtext>=</mtext><mtext>â</mtext><mtext>0</mtext><mtext>.4</mtext><mtext>â</mtext><mtext>V</mtext></mrow> </msub> </mrow> </math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mtext>= </mtext><mo>−</mo><mn>0.</mn><mtext>48</mtext><mtext> </mtext><mtext>V</mtext></mrow></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mtext>= </mtext><mo>−</mo><mtext>0</mtext><mtext>.04</mtext><mtext> </mtext><mtext>V</mtext></mrow></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mtext>= +</mtext><mtext> </mtext><mtext>0</mtext><mtext>.84</mtext><mtext> </mtext><mtext>V</mtext></mrow></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><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">l</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mi mathvariant="normal">Â </mi><mo>+</mo><mi mathvariant="normal">Â </mi><mn>1.28</mn><mi mathvariant="normal">V</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mtext>= +</mtext><mtext> </mtext><mtext>0</mtext><mtext>.84</mtext><mtext> </mtext><mtext>V</mtext></mrow></math></span> </div>

answer:
<div class="solution"><math> <mrow> <mtext>2Fe(s)</mtext><mtext> </mtext><mtext>+</mtext><mtext> </mtext><msub> <mtext>O</mtext> <mtext>2</mtext> </msub> <mtext>(g)</mtext><mtext> </mtext><mtext>+</mtext><mtext> </mtext><msub> <mrow> <mtext>2H</mtext></mrow> <mtext>2</mtext> </msub> <mtext>O(l)</mtext><mo>⇌</mo><msup> <mrow> <mtext>2Fe</mtext></mrow> <mrow> <mtext>2+</mtext></mrow> </msup> <mtext>(aq)</mtext><mtext> </mtext><mtext>+</mtext><mtext> </mtext><msup> <mrow> <mtext>4OH</mtext></mrow> <mo>−</mo> </msup> <mtext>(aq)</mtext></mrow></math><br/>For the given cell reaction<br/><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mtext> </mtext><mtext>=</mtext><mtext> </mtext><msubsup> <mtext>E</mtext> <mrow> <mtext>cathode</mtext></mrow> <mtext>o</mtext> </msubsup> <mo>−</mo><msubsup> <mtext>E</mtext> <mrow> <mtext>anode</mtext></mrow> <mtext>o</mtext> </msubsup> </mrow></math><br/><math> <mrow> <mtext>=</mtext><mtext> </mtext><mtext>0</mtext><mtext>.40</mtext><mo>−</mo><mfenced> <mrow> <mo>−</mo><mtext>0</mtext><mtext>.44</mtext></mrow> </mfenced><mtext> </mtext><mtext>=</mtext><mtext> </mtext><mtext>0</mtext><mtext>.84</mtext><mtext> </mtext><mtext>V</mtext></mrow></math></div>


title: what-is-the-standard-cell-potential-for-the-reaction-with-k-1-equilibrium-constant-1
slug: what-is-the-standard-cell-potential-for-the-reaction-with-k-1-equilibrium-constant-1-17664
tags: ['Chemistry', 'Electrochemistry', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Zero</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the standard cell potential for the reaction with $K=1$ (equilibrium constant)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">One</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2.303</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Infinity</span> </li> </ul>

answer:
<div class="solution">The relationship between cell potential and the equilibrium constant. We know.<br/>$\Delta G=-R T \ln k=-n F E^{\circ}$<br/>$\Delta G=-2.303 \mathrm{RT} \log k=-n F E^{\circ}$<br/>$\log K_f=\frac{n F \times E^{\circ}}{R T \times 2.303}$<br/>$\log (\mathrm{l})=\frac{n}{0.059} \times E^{\circ} \quad(n=1)$<br/>$\Rightarrow \quad 0 \times 0.059=E^{\circ} \Rightarrow E^{\circ}=$ zero</div>


title: what-is-the-standard-deviation-of-7-9-11-13-15
slug: what-is-the-standard-deviation-of-7-9-11-13-15-80096
question: <div class="question">What is the standard deviation of $7,9,11,13,15 ?$</div>
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2.8$</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.7$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2.8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Given observations are $7,9,11,13,15$<br/>$\bar{x}=\frac{7+9+11+13+15}{5}=\frac{55}{5}=11$<br/>Now, Variance $=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$<br/>$=\frac{(7-11)^{2}+(9-11)^{2}+(11-11)^{2}+(13-11)^{2}+(15-11)}{2}$<br/>$\frac{16+4+4+16}{5}=8$<br/>$\mathrm{SD}=\sqrt{\mathrm{v}}=\sqrt{8}=2 \sqrt{2}=2.8$<br/>$(\because \sqrt{2}=1.414)$</div>


title: what-is-the-standard-deviation-of-numbers-7-9-11-13-15
slug: what-is-the-standard-deviation-of-numbers-7-9-11-13-15-31750
question: <div class="question">What is the standard deviation of numbers $7,9,11,13,15$</div>
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2.8$</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">22</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">26</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.4$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2.8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Mean of given numbers $=\frac{7+9+11+13+15}{5}=\frac{55}{5}=11$<br/>Now<br/>$\mathrm{SD}=\sqrt{\frac{(7-11)^{2}+(9-11)^{2}+(11-11)^{2}+(13-11)^{2}+(15-11)^{2}}{5}}$<br/>$=\sqrt{\frac{16+4+0+4+16}{5}}=\sqrt{8}=2.8($ aprox $)$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$9$</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-deviation-of-the-following-series-beginarraylcccc-hline-text-measurements-010-1020-2030-3040-hline-text-frequency-1-3-4-2-hline-e

slug:
what-is-the-standard-deviation-of-the-following-series-beginarraylcccc-hline-text-measurements-010-1020-2030-3040-hline-text-frequency-1-3-4-2-hline-e-77257

question:
<div class="question">What is the standard deviation of the following series<br/>\(\begin{array}{|l|c|c|c|c|} \hline \text { Measurements } &amp; 0-10 &amp; 10-20 &amp; 20-30 &amp; 30-40 \\ \hline \text { Frequency } &amp; 1 &amp; 3 &amp; 4 &amp; 2 \\ \hline \end{array}\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$81$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$7.6$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$9$</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><li class=""> <span class="option-label">D</span> <span class="option-data">$2.26$</span> </li> </ul>

answer:
<div class="solution">\(\begin{array}{|l|c|c|c|c|c|} \hline \text { Class } &amp; f_1 &amp; u_i &amp; \begin{array}{c} d=y_i-A, \\ A=25 \end{array} &amp; f_{i d i} &amp; f_{i d i}{ }^2 \\ \hline 0-10 &amp; 1 &amp; 5 &amp; -20 &amp; -20 &amp; 400 \\ \hline 10-20 &amp; 3 &amp; 15 &amp; -10 &amp; -30 &amp; 300 \\ \hline 20-30 &amp; 4 &amp; 25 &amp; 0 &amp; 0 &amp; 0 \\ \hline 30-40 &amp; 2 &amp; 35 &amp; 10 &amp; 20 &amp; 200 \\ \hline \text { Total } &amp; 10 &amp; &amp; &amp; -30 &amp; 900 \\ \hline \end{array}\)<br/>$\sigma^2=\frac{\sum f_i d_i^2}{\sum f_i}-\left(\frac{\sum f_i d_i}{\sum f_i}\right)^2=\frac{900}{10}-\left(\frac{-30}{10}\right)^2$<br/>$\sigma^2=90-9=81 \Rightarrow \sigma=9$.</div>


title: what-is-the-standard-electrode-potential-for-the-halfcell-reaction-involving-fluorine
slug: what-is-the-standard-electrode-potential-for-the-halfcell-reaction-involving-fluorine-44529
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the standard electrode potential for the half-cell reaction involving fluorine?</div>

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

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

answer:
<div class="solution">The paragraph mentions that the standard electrode potential for fluorine is the highest indicating that fluorine gas (F<sub>2</sub> ) has the maximum tendency to get reduced to fluoride ions (F- ) and therefore fluorine gas is the strongest oxidising agent and fluoride ion is the weakest reducing agent.</div>


title: what-is-the-standard-electrode-potential-of-copper-ion-cu2-compared-to-hydrogen-ion-h
slug: what-is-the-standard-electrode-potential-of-copper-ion-cu2-compared-to-hydrogen-ion-h-42567
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the standard electrode potential of copper ion (Cu2+) compared to hydrogen ion (H+)?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Cu2+ has a higher standard electrode potential.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Cu2+ has a lower standard electrode potential.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both have the same standard electrode potential.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Cu2+ has a higher standard electrode potential.</span> </div>

answer:
<div class="solution">The standard electrode potential of copper ion (Cu2+) is positive, indicating that it has a higher potential compared to hydrogen ion (H+).</div>


title: what-is-the-standard-enthalpy-of-combustion-defined-as
slug: what-is-the-standard-enthalpy-of-combustion-defined-as-86874
question: <div class="question">What is the standard enthalpy of combustion defined as?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The enthalpy change when a substance undergoes combustion in any state</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The enthalpy change when a substance undergoes combustion in its standard state at any temperature</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The enthalpy change when a substance undergoes combustion in its standard state at a specified temperature</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The enthalpy change when a substance undergoes combustion in its gaseous state</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The enthalpy change when a substance undergoes combustion in its standard state at a specified temperature</span> </div>

answer:
<div class="solution">The standard enthalpy of combustion is defined as the enthalpy change per mole (or per unit amount) of a substance when it undergoes combustion, with all the reactants and products being in their standard states at a specified temperature.</div>


title: what-is-the-standard-enthalpy-of-formation-fh-a-special-case-of
slug: what-is-the-standard-enthalpy-of-formation-fh-a-special-case-of-21882
question: <div class="question">What is the standard enthalpy of formation, âfH , a special case of?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Standard molar enthalpy of fusion</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Standard molar enthalpy of vaporization</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Standard molar enthalpy of sublimation</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Standard molar enthalpy of reaction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Standard molar enthalpy of reaction</span> </div>

answer:
<div class="solution">The standard enthalpy of  formation, âfH , is a special case of the standard  molar enthalpy of reaction, ârH, where one mole of a compound is formed from its constituent elements.</div>


title: what-is-the-standard-enthalpy-of-formation-of-an-element-in-its-reference-state
slug: what-is-the-standard-enthalpy-of-formation-of-an-element-in-its-reference-state-54238
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Zero</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the standard enthalpy of formation of an element in its reference state?</div>

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

answer:
<div class="solution">By convention, the standard enthalpy of formation of an element in its reference state is taken as zero.</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'TS EAMCET', 'TS EAMCET 2019 (06 May Shift 1)']
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-enthalpy-of-reaction-in-kj-when-two-moles-of-fe-2-o-3-s-reacts-with-h-2-gas-to-give-fe-metal-h-f-of-fe-2-o-3-s-and-h-2-o-l-are-82

slug:
what-is-the-standard-enthalpy-of-reaction-in-kj-when-two-moles-of-fe-2-o-3-s-reacts-with-h-2-gas-to-give-fe-metal-h-f-of-fe-2-o-3-s-and-h-2-o-l-are-82-27834

question:
<div class="question">What is the standard enthalpy of reaction (in kJ) when two moles of $\mathrm{Fe}_2 \mathrm{O}_3(s)$ reacts with $\mathrm{H}_2$ gas to give Fe metal?<br/>$\Delta H_f^{\circ}$ of $\mathrm{Fe}_2 \mathrm{O}_3(s)$ and $\mathrm{H}_2 \mathrm{O}(l)$ are -824.2 and $-285.83 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-66.58$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$-33.3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-538.37$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-1110.03$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-66.58$</span> </div>

answer:
<div class="solution">When $\mathrm{Fe}_2 \mathrm{O}_3(s)$ reacts with $\mathrm{H}_2(g)$, it will give $\mathrm{Fe}$<br/>and $\mathrm{H}_2 \mathrm{O}$ as follows: $\mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \rightarrow 2 \mathrm{Fe}+3 \mathrm{H}_2 \mathrm{O}$ When 2 moles of $\mathrm{Fe}_2 \mathrm{O}_3$ are used, the reaction becomes as follows:<br/>$2\left[\mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \rightarrow 2 \mathrm{Fe}+3 \mathrm{H}_2 \mathrm{O}\right]$<br/>Given, $\Delta H_f^{\circ}$ for $\mathrm{Fe}_2 \mathrm{O}_3(s)=-824.2 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta H_f^{\circ}$ for $\mathrm{H}_2 \mathrm{O}(l)=-285.83 \mathrm{~kJ} \mathrm{~mol}^{-1}$<br/>$\because$ Standard enthalpy of reaction<br/>$\left(\Delta H_R^{\circ}\right)=\sum \Delta H_f^{\circ}$ (Product) $-\Sigma \Delta H_f^{\circ}$ (Reacant $)$<br/>and $\left.\begin{array}{c}H_f^{\circ} \text { for } \mathrm{H}_2=0 \\ H_f^{\circ} \text { for } \mathrm{Fe}=0\end{array}\right]$ (as are present in free state)<br/>Thus,<br/>$\begin{aligned} \Delta H_R^{\circ} &amp; =[6 \times-285.83]-[2 \times(-824.2)] \\ \Delta H_R^{\circ} &amp; =(-1714.98)+(1648.4) \\ \Delta H_R^{\circ} &amp; =-66.58 \mathrm{~kJ} / \mathrm{mol}\end{aligned}$<br/>Hence, option (a) is the correct answer.</div>


title: what-is-the-standard-enthalpy-of-reaction
slug: what-is-the-standard-enthalpy-of-reaction-31127
question: <div class="question">What is the standard enthalpy of reaction?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The enthalpy change for a reaction under non-standard conditions</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The enthalpy change for a reaction when all substances are in their standard states</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The enthalpy change for a reaction carried out at high pressure</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The enthalpy change for a reaction at any given temperature</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The enthalpy change for a reaction when all substances are in their standard states</span> </div>

answer:
<div class="solution">The standard enthalpy of reaction is the enthalpy change for a reaction when all the participating substances are in their standard states.</div>


title: what-is-the-standard-enthalpy-of-sublimation
slug: what-is-the-standard-enthalpy-of-sublimation-39776
question: <div class="question">What is the standard enthalpy of sublimation?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The heat change when a solid melts at a constant temperature</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The heat change when a solid vaporizes at a constant temperature</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The heat change when a solid sublimes at a constant temperature</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The heat change when a liquid vaporizes at a constant temperature</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The heat change when a solid sublimes at a constant temperature</span> </div>

answer:
<div class="solution">The standard enthalpy of  sublimation, â<sub>sub</sub>H, is the heat change when one mole of a solid substance sublimes at a constant temperature and under standard pressure (1 bar).</div>


title: what-is-the-standard-enthalpy-of-vaporization-vap-h-for-water
slug: what-is-the-standard-enthalpy-of-vaporization-vap-h-for-water-87362
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the standard enthalpy  of vaporization (â<sub>vap</sub>H) for water?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">â<sub>vap</sub>H= + 40.79 kJ molâ1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">â<sub>vap</sub>H= â 393.5 kJ molâ1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">â<sub>vap</sub>H= â 2802.0 kJ molâ1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">â<sub>vap</sub>H= 435.0 kJ molâ1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">â<sub>vap</sub>H= + 40.79 kJ molâ1</span> </div>

answer:
<div class="solution">The standard enthalpy of  vaporization (â<sub>vap</sub>H) for water is  â<sub>vap</sub>H = + 40.79 kJ molâ1. This represents the heat required to vaporize one mole of water at its boiling point under standard pressure.</div>


tags: ['Chemistry', 'Electrochemistry', 'MHT CET', 'MHT CET 2020 (15 Oct Shift 2)']
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-free-energy-change-for-the-cell-having-following-cell-reaction-2-ag-aq-cd-s-2-ag-s-cd-aq-2-e-cell-1-20-v

slug:
what-is-the-standard-free-energy-change-for-the-cell-having-following-cell-reaction-2-ag-aq-cd-s-2-ag-s-cd-aq-2-e-cell-1-20-v-33132

question:
<div class="question">What is the standard free energy change for the cell, having following cell reaction? $2 \mathrm{Ag}_{(\mathrm{aq} .)}^{+}+\mathrm{Cd}_{(\mathrm{s})} \longrightarrow 2 \mathrm{Ag}_{(\mathrm{s})}+\mathrm{Cd}_{(\mathrm{aq})}^{2+}, \mathrm{E}^{\circ} \mathrm{cell}=1 \cdot 20 \mathrm{~V}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-231 \cdot 6 \mathrm{~kJ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$-160 \cdot 8 \mathrm{~kJ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-115 \cdot 8 \mathrm{~kJ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-260 \cdot 8 \mathrm{K} \mathrm{j}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-231 \cdot 6 \mathrm{~kJ}$</span> </div>

answer:
<div class="solution">$2 \mathrm{Ag}_{(\mathrm{aq})}^{+}+\mathrm{Cd}_{(\mathrm{s})} \longrightarrow 2 \mathrm{Ag}_{(\mathrm{s})}+\mathrm{Cd}_{(\mathrm{aq})}^{2+}$<br/>$\therefore \mathrm{n}=2$<br/>$\Delta \mathrm{G}^{0}=-\mathrm{nFE}_{\mathrm{cell}}^{0}=-2 \times 96500 \times 1.20=-231600 \mathrm{~J}=-231.6 \mathrm{~kJ}$</div>


title: what-is-the-standard-molar-enthalpy-of-formation
slug: what-is-the-standard-molar-enthalpy-of-formation-25663
question: <div class="question">What is the standard molar enthalpy of formation?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The heat change when a compound is formed from its elements in their most stable states</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The heat change when a compound is dissolved in water</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The heat change when a compound is vaporized</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The heat change when a compound undergoes sublimation</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The heat change when a compound is formed from its elements in their most stable states</span> </div>

answer:
<div class="solution">The standard molar enthalpy of formation is the heat change when one mole of a compound is formed from its elements in their most stable states of aggregation.</div>


title: what-is-the-standard-potential-of-cell-ni-ni-2-1-m-cu-2-1-m-cu-if-e-cu-0-337-v-and-e-ni-0-236-v
tags: ['Chemistry', 'Electrochemistry', 'MHT CET', 'MHT CET 2020 (13 Oct Shift 1)']
source_db: MarksBatch2_P4.db
slug:
what-is-the-standard-potential-of-cell-ni-ni-2-1-m-cu-2-1-m-cu-if-e-cu-0-337-v-and-e-ni-0-236-v-44905

question:
<div class="question">What is the standard potential of cell $\mathrm{Ni}\left|\mathrm{Ni}^{2+}{ }_{(1 \mathrm{M})} \| \mathrm{Cu}^{2+}{ }_{(1 \mathrm{M})}\right| \mathrm{Cu}$ If $\mathrm{E}^{\circ} \mathrm{Cu}=0 \cdot 337 \mathrm{~V}$ and $\mathrm{E}^{\circ} \mathrm{Ni}=-0 \cdot 236 \mathrm{~V}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0 \cdot 101 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-0 \cdot 136 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.573 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-0.753 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.573 \mathrm{~V}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{E}_{\text {ceil }}^{0} &amp;=\mathrm{E}_{\text {cuthode }}^{0}-\mathrm{E}_{\text {anode }}^{0}=\mathrm{E}_{\mathrm{Cu}}^{0}-\mathrm{E}_{\mathrm{Ni}_{\mathrm{i}}}^{0} \\ &amp;=0.337 \mathrm{~V}-(-0.236 \mathrm{~V})=0.573 \mathrm{~V} \end{aligned}$</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-reduction-potential-e-o-for-f-e-3-f-e-given-that-f-e-2-2-e-f-e-e-f-e-2-f-e-o-047-v-f-e-3-e-f-e-2-e-f-e-3-f-e-2-o-077-v

slug:
what-is-the-standard-reduction-potential-e-o-for-f-e-3-f-e-given-that-f-e-2-2-e-f-e-e-f-e-2-f-e-o-047-v-f-e-3-e-f-e-2-e-f-e-3-f-e-2-o-077-v-85621

question:
<div class="question">What is the standard reduction potential <math><mfenced separators="|"><mrow><msup><mrow><mtext>E</mtext></mrow><mrow><mtext>o</mtext></mrow></msup></mrow></mfenced></math> for <math><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>â</mo><mtext>F</mtext><mtext>e</mtext><mo>?</mo></math><br/> <br/>
Given that:<br/> <br/> <math><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>+</mo><mn>2</mn><msup><mrow><mtext>e</mtext></mrow><mrow><mo>-</mo></mrow></msup><mo>â</mo><mtext>F</mtext><mtext>e</mtext><mo>;</mo><msubsup><mrow><mtext>E</mtext></mrow><mrow><mrow><mrow><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup></mrow><mo>/</mo><mrow><mtext>F</mtext><mtext>e</mtext></mrow></mrow></mrow><mrow><mtext>o</mtext></mrow></msubsup><mo>=</mo><mtext> </mtext><mo>-</mo><mn>0.47</mn><mtext>V</mtext></math><br/> <br/> <math><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>+</mo><msup><mrow><mtext>e</mtext></mrow><mrow><mo>-</mo></mrow></msup><mo>â</mo><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>;</mo><msubsup><mrow><mtext>E</mtext></mrow><mrow><mrow><mrow><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></mrow><mo>/</mo><mrow><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup></mrow></mrow></mrow><mrow><mtext>o</mtext></mrow></msubsup><mo>=</mo><mo>+</mo><mn>0.77</mn><mtext>V</mtext></math></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mo>-</mo><mn>0.057</mn><mtext> </mtext><mtext>V</mtext></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mo>+</mo><mn>0.30</mn><mtext>Â </mtext><mtext>V</mtext></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mo>-</mo><mn>0.30</mn><mtext>Â </mtext><mtext>V</mtext></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mo>+</mo><mn>0.057</mn><mtext>Â </mtext><mtext>V</mtext></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mo>-</mo><mn>0.057</mn><mtext> </mtext><mtext>V</mtext></math></span> </div>

answer:
<div class="solution">â(i)<math><mtext>F</mtext><msup><mrow><mtext>e</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>â</mo><mtext>F</mtext><mtext>e </mtext></math> <math> <mrow> <msubsup> <mtext>E</mtext> <mn>1</mn> <mtext>o</mtext> </msubsup> </mrow> </math> ;<br/><br/>'n' factor = 3<br/><br/><math> <mrow> <mtext>Î</mtext><msub> <mtext>G</mtext> <mn>1</mn> </msub> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><mo>â</mo><mn>3</mn><mtext>â</mtext><mo>Ã</mo><mtext>F</mtext><mo>Ã</mo><msubsup> <mtext>E</mtext> <mn>1</mn> <mtext>o</mtext> </msubsup> </mrow> </math><br/><br/>(ii)<math> <mrow> <mtext>F</mtext><msup> <mtext>e</mtext> <mrow> <mn>3</mn><mo>+</mo></mrow> </msup> <mo>+</mo><msup> <mtext>e</mtext> <mo>â</mo> </msup> <mo>â</mo><mtext>F</mtext><msup> <mtext>e</mtext> <mrow> <mn>2</mn><mo>+</mo></mrow> </msup> </mrow> </math><math> <mrow> <msubsup> <mtext>E</mtext> <mn>2</mn> <mtext>o</mtext> </msubsup> </mrow> </math> ;0.77<br/><br/>'n' factor = 1<br/><br/><math> <mrow> <mtext>Î</mtext><msub> <mtext>G</mtext> <mn>2</mn> </msub> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><mo>â</mo><mn>1</mn><mtext>â</mtext><mo>Ã</mo><mtext>0.77</mtext><mo>Ã</mo><msubsup> <mtext>f</mtext></msubsup></mrow></math><br/><br/>(iii)<math> <mrow> <mtext>F</mtext><msup> <mtext>e</mtext> <mrow> <mn>2</mn><mo>+</mo></mrow> </msup> <mo>+</mo><mn>2</mn><msup> <mtext>e</mtext> <mo>â</mo> </msup> <mo>â</mo><mtext>F</mtext><mtext>e</mtext></mrow> </math> <math> <mrow> <msubsup> <mtext>E</mtext> <mn>3</mn> <mtext>o</mtext> </msubsup> </mrow> </math> ;0.77<br/><br/>'n' factor = 2<br/><br/><math> <mrow> <mtext>Î</mtext><msub> <mtext>G</mtext> <mn>3</mn> </msub> <mtext>â</mtext><mo>=</mo><mtext>â0.47</mtext><mtext>â</mtext><mo>Ã</mo><mtext>2</mtext><mo>Ã</mo><msubsup> <mtext>f</mtext></msubsup></mrow></math><br/><br/><math> <mrow> <mtext>Î</mtext><msub> <mtext>G</mtext> <mn>1</mn> </msub> <mo>=</mo><mtext>â</mtext><mtext>Î</mtext><msub> <mtext>G</mtext> <mn>2</mn> </msub> <mtext>â</mtext><mo>+</mo><mtext>â</mtext><mtext>Î</mtext><msub> <mtext>G</mtext> <mn>3</mn> </msub> </mrow> </math><br/><br/><math> <mrow> <mo>â</mo><mn>3</mn><mtext>f</mtext><msubsup> <mtext>E</mtext> <mrow> <msup> <mrow></mrow> <mn>1</mn> </msup> </mrow> <mtext>o</mtext> </msubsup> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><mo>â</mo><mn>1</mn><mo>Ã</mo><mn>0.77</mn><mo>Ã</mo><mtext>f</mtext><mo>+</mo><mn>2</mn><mo>Ã</mo><mn>0.47</mn><mo>Ã</mo><mtext>f</mtext></mrow> </math><br/><br/><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <msup> <mrow></mrow> <mn>1</mn> </msup> </mrow> <mtext>o</mtext> </msubsup> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac> <mrow> <mo>â</mo><mn>2</mn><mo>Ã</mo><mn>0.47</mn><mo>+</mo><mn>0.77</mn></mrow> <mn>3</mn> </mfrac> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><mo>â</mo><mn>0.057</mn><mo> </mo><mtext>V</mtext></mrow> </math></div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2017 (08 Apr Online)']
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-reduction-potential-e-o-for-fe-3-fe-given-that-fe-2-2-e-fe-e-fe-2-fe-o-0-47-v-fe-3-e-fe-2-e-fe-3-fe-2-o-0-77-v

slug:
what-is-the-standard-reduction-potential-e-o-for-fe-3-fe-given-that-fe-2-2-e-fe-e-fe-2-fe-o-0-47-v-fe-3-e-fe-2-e-fe-3-fe-2-o-0-77-v-79579

question:
<div class="question">What is the standard reduction potential <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Fe</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>→</mo><mi>Fe</mi><mo>?</mo></math><br/><br/>Given that:<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>+</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mo>-</mo></msup><mo>→</mo><mi>Fe</mi><mo>;</mo><msubsup><mi mathvariant="normal">E</mi><mrow><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>/</mo><mi>Fe</mi></mrow><mi mathvariant="normal">o</mi></msubsup><mo>=</mo><mo>-</mo><mn>0</mn><mo>.47</mo><mi mathvariant="normal">V</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Fe</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>+</mo><msup><mi mathvariant="normal">e</mi><mo>-</mo></msup><mo>→</mo><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>;</mo><msubsup><mi mathvariant="normal">E</mi><mrow><msup><mi>Fe</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup></mrow><mi mathvariant="normal">o</mi></msubsup><mo>=</mo><mo>+</mo><mn>0</mn><mo>.77</mo><mi mathvariant="normal">V</mi></math></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.057</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mn>0</mn><mo>.30</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.30</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mn>0</mn><mo>.057</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.057</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced><mo> </mo><msup><mi>Fe</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>→</mo><mi>Fe</mi><mo> </mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi mathvariant="normal">E</mi><mn>1</mn><mi mathvariant="normal">o</mi></msubsup></math>;<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi mathvariant="normal">n</mi><mo>'</mo></math> factor<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mtext>ΔG</mtext><mn>1</mn><mo>∘</mo></msubsup><mo>=</mo><mo>−</mo><mn>3</mn><mi mathvariant="normal"> </mi><mo>×</mo><mi mathvariant="normal">F</mi><mo>×</mo><msubsup><mi mathvariant="normal">E</mi><mn>1</mn><mi mathvariant="normal">o</mi></msubsup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced><mo> </mo><msup><mi>Fe</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>+</mo><msup><mi mathvariant="normal">e</mi><mo>−</mo></msup><mo>→</mo><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup></math><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi mathvariant="normal">E</mi><mn>2</mn><mi mathvariant="normal">o</mi></msubsup></math>; <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>77</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi mathvariant="normal">n</mi><mo>'</mo></math> factor<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><msubsup><mi mathvariant="normal">G</mi><mn>2</mn><mo>∘</mo></msubsup><mo>=</mo><mo>-</mo><mn>1</mn><mo>×</mo><mi mathvariant="normal">F</mi><mo>×</mo><mn>0</mn><mo>.</mo><mn>77</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>iii</mi></mfenced><mo> </mo><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>+</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mo>−</mo></msup><mo>→</mo><mi>Fe</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi mathvariant="normal">E</mi><mn>3</mn><mi mathvariant="normal">o</mi></msubsup><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>47</mn></math> <br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi mathvariant="normal">n</mi><mo>'</mo></math> factor<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><msubsup><mi mathvariant="normal">G</mi><mn>3</mn><mo>∘</mo></msubsup><mo>=</mo><mo>-</mo><mn>2</mn><mo>×</mo><mi mathvariant="normal">F</mi><mo>×</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>47</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ΔG</mi><mn>1</mn></msub><mo>=</mo><msub><mi>ΔG</mi><mn>2</mn></msub><mo>+</mo><msub><mi>ΔG</mi><mn>3</mn></msub></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>3</mn><msubsup><mi>FE</mi><msup><mrow></mrow><mn>1</mn></msup><mi mathvariant="normal">o</mi></msubsup><mtext> </mtext><mo>=</mo><mtext> </mtext><mfenced><mrow><mo>−</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>.77</mo><mo>×</mo><mi mathvariant="normal">F</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mo>×</mo><mn>0</mn><mo>.47</mo><mo>×</mo><mi mathvariant="normal">F</mi></mrow></mfenced></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mtext>-E</mtext><msup><mrow></mrow><mn>1</mn></msup><mi mathvariant="normal">o</mi></msubsup><mo>=</mo><mfrac><mrow><mfenced close="]" open="["><mrow><mfenced><mrow><mn>2</mn><mo>×</mo><mn>0</mn><mo>.47</mo></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.77</mo></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">F</mi></mrow><mrow><mn>3</mn><mi mathvariant="normal">F</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.057</mo><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mtext>E</mtext><msup><mrow></mrow><mn>1</mn></msup><mi mathvariant="normal">o</mi></msubsup><mo>=</mo><mo>-</mo><mi mathvariant="normal"> </mi><mn>0</mn><mo>.057</mo><mo> </mo><mi mathvariant="normal">V</mi></math></p></div>


title: what-is-the-standard-state-c-0-for-a-solute-when-measuring-concentrations-with-respect-to-it
slug: what-is-the-standard-state-c-0-for-a-solute-when-measuring-concentrations-with-respect-to-it-90360
tags: ['Chemistry', 'Chemical Equilibrium', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the standard state (c<sub>0</sub>) for a solute when measuring concentrations with respect to it?</div>

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

answer:
<div class="solution">The standard state (c<sub>0</sub>) for a solute is 1 molar solution, and all concentrations can be measured with respect to it.</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 bar</span> </div>
answer: <div class="solution">For a pure gas, the standard state is 1 bar.</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-standard-state-for-a-pure-gas-when-expressing-equilibrium-constants-as-dimensionless-quantities

slug:
what-is-the-standard-state-for-a-pure-gas-when-expressing-equilibrium-constants-as-dimensionless-quantities-57623

question:
<div class="question">What is the standard state for a pure gas when expressing equilibrium constants as dimensionless quantities?</div>

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


title: what-is-the-standard-temperature-at-which-data-are-usually-taken-for-standard-enthalpy-changes
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">298 K</span> </div>
answer: <div class="solution">Data for standard enthalpy changes are usually taken at 298 K (25Â°</div>
source_db: MarksBatch2_P4.db
slug:
what-is-the-standard-temperature-at-which-data-are-usually-taken-for-standard-enthalpy-changes-41494

question:
<div class="question">What is the standard temperature at which data are usually taken for standard enthalpy changes?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0 K</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">25 K</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">273 K</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">298 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>


title: what-is-the-structure-of-the-vascular-bundles-in-a-dicot-stem
slug: what-is-the-structure-of-the-vascular-bundles-in-a-dicot-stem-36921
question: <div class="question">What is the structure of the vascular bundles in a dicot stem?</div>
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Concentric</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Radial</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Conjoint</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Closed</span> </li> </ul>

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

answer:
<div class="solution">A large number of vascular bundles are arranged in a ring in a dicot stem and they are conjoint, open and with endarch protoxylem.</div>


title: what-is-the-sum-of-3-3-1-3-3-1
slug: what-is-the-sum-of-3-3-1-3-3-1-44831
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of $\sqrt{3}+\frac{1}{\sqrt{3}}+\frac{1}{3 \sqrt{3}}+\ldots . . ?$</div>

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

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

answer:
<div class="solution">Given series is<br/>$$<br/>\sqrt{3}+\frac{1}{\sqrt{3}}+\frac{1}{3 \sqrt{3}}+\ldots \ldots \ldots<br/>$$<br/>Since, $\left(\frac{1}{\sqrt{3}}\right)^{2}=\sqrt{3} \times \frac{1}{3 \sqrt{3}}$<br/>$\Rightarrow$ Given series is a G.P.<br/>$\therefore \quad$ Sum upto $\infty=\frac{a}{1-r}$<br/>where $\mathrm{a}=\sqrt{3}=$ first term and $\mathrm{r}=\frac{1}{3}=$ common ratio<br/>$$<br/>\therefore \quad S_{\infty}=\frac{\sqrt{3}}{1-\frac{1}{3}}=\frac{3 \sqrt{3}}{2}<br/>$$</div>


title: what-is-the-sum-of-all-natural-numbers-between-200-and-400-which-are-divisible-by-7
slug: what-is-the-sum-of-all-natural-numbers-between-200-and-400-which-are-divisible-by-7-48253
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8729</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of all natural numbers between 200 and 400 which are divisible by $7 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6729</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8712</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">8729</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">9276</span> </li> </ul>

answer:
<div class="solution">The numbers between 200 and 400 which are divisible by 7, are $203,210,217, \ldots \ldots, 399$<br/>This is an A.P with first term $=\mathrm{a}=203$ and common difference $=d=7$ Now, let number of terms be n. Therefore from the $\mathrm{n}^{\text {th }}$ term of $\mathrm{A} . \mathrm{P}=\mathrm{a}+(\mathrm{n}-1) \mathrm{d}$ we have $399=203+(\mathrm{n}-1) 7$<br/>$\Rightarrow \frac{196}{7}=(n-1) \Rightarrow n=29$<br/>Required sum $=\frac{n}{2}[a+\ell]$ where $\ell=$ last term<br/>Thus, required sum $=\frac{29}{2}[203+399]$<br/>$=\frac{29 \times 602}{2}=8729$</div>


title: what-is-the-sum-of-all-the-coefficients-in-the-expansion-of-1-x-n
slug: what-is-the-sum-of-all-the-coefficients-in-the-expansion-of-1-x-n-23742
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of all the coefficients in the expansion of $(1+x)^{n} ?$</div>

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

answer:
<div class="solution">Given expansion is $(1+x)^{\mathrm{n}}$. Put $x=1$, we get Required sum $=(1+1)^{n}=2^{n}$</div>


tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2664</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-sum-of-all-threedigit-numbers-that-can-be-formed-using-all-the-digits-34-and-5-when-repetition-of-digits-is-not-allowed

slug:
what-is-the-sum-of-all-threedigit-numbers-that-can-be-formed-using-all-the-digits-34-and-5-when-repetition-of-digits-is-not-allowed-68024

question:
<div class="question">What is the sum of all three-digit numbers that can be formed using all the digits 3,4 and 5 , when repetition of digits is not allowed?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">2664</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">3882</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4044</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4444</span> </li> </ul>

answer:
<div class="solution">Number of 3 digit number made from digit 3,4 or 5<br/>and having all distinct digit $=3 !=6$<br/>and sum of such numbers are<br/>$543+534+345+354+435+453=2664$</div>


title: what-is-the-sum-of-all-twodigit-numbers-which-when-divided-by-3-leave-2-as-the-remainder
slug: what-is-the-sum-of-all-twodigit-numbers-which-when-divided-by-3-leave-2-as-the-remainder-28328
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1635</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of all two-digit numbers which when divided by 3 leave 2 as the remainder?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1565</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1585</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1635</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1655</span> </li> </ul>

answer:
<div class="solution">The numbers which divided by 3, leaving remainder 2 will be of the form $3 \mathrm{x}+2$ Given, $3 \mathrm{x}+2$ is 2-digit number So, $x$ can be from 3 to 32<br/>Sum of numbers $=\sum_{\mathrm{x}=3}^{32}(3 \mathrm{x}+2)$<br/>$=3(3+4+5+\ldots .+32)+(2+2+\ldots .)$<br/>$3,4,5 \ldots .32$ is an $\mathrm{A}$. $\mathrm{P}$ with $\mathrm{a}=3, \mathrm{~d}=1, \mathrm{~T}_{\mathrm{n}}=32$<br/>$\therefore \mathrm{T}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1) 1$<br/>$32=3+(\mathrm{n}-1) 1$<br/>$\Rightarrow \mathrm{n}-1=29 \Rightarrow \mathrm{n}=30$ terms<br/>$\therefore(1) \Rightarrow 3(3+4+5+\ldots .+32)+(2+2+\ldots .30$ times $)$<br/>$=3\left(\frac{30}{2}(3+32)\right)+(2 \times 30)\left(\because \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}(\mathrm{a}+\ell)\right)$<br/>$=\frac{90}{2}(35)+60$<br/>$=(45 \times 35)+60$<br/>$=1575+60=1635$</div>


title: what-is-the-sum-of-first-eight-terms-of-the-series-1-2-1-4-1-8-1
slug: what-is-the-sum-of-first-eight-terms-of-the-series-1-2-1-4-1-8-1-15751
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of first eight terms of the series<br/>$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots . . ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{89}{128}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{57}{384}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{85}{128}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{85}{128}$</span> </div>

answer:
<div class="solution">$a=1$ $\mathrm{r}=-\frac{1}{2}(   &lt; 1)$<br/>$\therefore \quad$ Sum of Ist 8 terms is :-<br/>$\mathrm{S}_{8}=\frac{1\left[1-\left(\frac{-1}{2}\right)^{8}\right]}{1+\frac{1}{2}} \quad\left[\right.$ For $\left.\mathrm{G} . \mathrm{P} . \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{a}\left(1-\mathrm{r}^{\mathrm{n}}\right)}{1-\mathrm{r}}\right]$<br/>$=\frac{\frac{1}{1}-\frac{1}{256}}{\frac{3}{2}}=\frac{85}{128}$</div>


title: what-is-the-sum-of-focal-radii-of-any-point-on-an-ellipse-equal-to
slug: what-is-the-sum-of-focal-radii-of-any-point-on-an-ellipse-equal-to-97397
question: <div class="question">What is the sum of focal radii of any point on an ellipse equal to?</div>
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Length oflatusrectum</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Length of major-axis</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Length of minor-axis</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Length of semi-latusrectum</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Length of major-axis</span> </div>

answer:
<div class="solution">We know sum of the focal distances of a point on the ellipse is constant and is equal to the length of the major axis. Thus, we know that, the sum of focal radii of any point on ellipse is equal to length of major axis</div>


title: what-is-the-sum-of-n-terms-of-the-series-2-8-18-32
slug: what-is-the-sum-of-n-terms-of-the-series-2-8-18-32-95012
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of $\mathrm{n}$ terms of the series<br/>$\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\ldots ?$</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\ldots . . \\ &amp;=\sqrt{2}+2 \sqrt{2}+3 \sqrt{2}+4 \sqrt{2}+\ldots . . \\ &amp; \therefore \mathrm{S}_{\mathrm{n}}=\sqrt{2}[1+2+3+4+\ldots . \mathrm{n} \text { terms }] \\ &amp;=\sqrt{2} \frac{\mathrm{n}(\mathrm{n}+1)}{2}=\frac{\mathrm{n}(\mathrm{n}+1)}{\sqrt{2}} \end{aligned}$</div>


title: what-is-the-sum-of-the-coefficients-in-the-expansion-of-1-x-n
slug: what-is-the-sum-of-the-coefficients-in-the-expansion-of-1-x-n-75485
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the coefficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}} ?$</div>

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

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

answer:
<div class="solution">Given expansion is $(1+\mathrm{x})^{\mathrm{n}}$. <br/>Put $\mathrm{x}=1$, we get Sum of coefficient $=2^{\mathrm{n}}$.</div>


title: what-is-the-sum-of-the-coefficients-of-all-the-terms-in-the-expansion-of-45-x-49-4
slug: what-is-the-sum-of-the-coefficients-of-all-the-terms-in-the-expansion-of-45-x-49-4-53179
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">256</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the coefficients of all the terms in the expansion of $(45 x-49)^{4} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-256</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-100</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">100</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">256</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Given expansion is $(45 x-49)^{4}$. To find the sum of the coefficients of all the terms in the expansion, we have to put $x=1$ in the expansion. Thus, required sum of coefficients $=(45-49)^{4}$ $=(-4)^{4}=256$</div>


title: what-is-the-sum-of-the-first-50-terms-of-the-series-1-3-3-5-5-7
slug: what-is-the-sum-of-the-first-50-terms-of-the-series-1-3-3-5-5-7-77164
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the first 50 terms of the series $(1 \times 3)+(3 \times 5)+(5 \times 7)+\ldots . ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1,71,650$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">26600</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">26650</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">26900</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1,71,650$</span> </div>

answer:
<div class="solution">The given series is $(1 \times 3)+(3 \times 5)+(5 \times 7)+\ldots \ldots \ldots . .$<br/>Its general term is given by<br/>$\mathrm{T}_{\mathrm{n}}=(2 \mathrm{n}-1)(2 \mathrm{n}+1)=4 \mathrm{n}^{2}-1$<br/>Sum of series $=4 \Sigma \mathrm{n}^{2}-\Sigma 1$<br/>$\mathrm{S}_{\mathrm{n}}=\frac{4 \mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}-\mathrm{n}$<br/>$\mathrm{S}_{\mathrm{n}}=\mathrm{n}\left[\frac{2\left(2 \mathrm{n}^{2}+3 \mathrm{n}+1\right)}{3}-1\right]$<br/>$\mathrm{S}_{\mathrm{n}}=\mathrm{n}\left[\frac{4 \mathrm{n}^{2}+6 \mathrm{n}+2-3}{3}\right]$<br/>$\mathrm{S}_{\mathrm{n}}=\left[\frac{\mathrm{n}\left(4 \mathrm{n}^{2}+6 \mathrm{n}-1\right)}{3}\right]$<br/>For sum of first 50 terms of the series, $\mathrm{n}=50$, $\mathrm{S}_{50}=\frac{50\left[4(50)^{2}+6(50)-1\right]}{3}$<br/>$=\frac{50 \times(10000+300-1)}{3}=\frac{50 \times 10299}{3}=171650$</div>


title: what-is-the-sum-of-the-focal-distances-of-a-point-of-an-ellipse-a-2-x-2-b-2-y-2-1
slug: what-is-the-sum-of-the-focal-distances-of-a-point-of-an-ellipse-a-2-x-2-b-2-y-2-1-18893
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 a$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the focal distances of a point of an<br/>ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 ?$</div>

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

answer:
<div class="solution">Bydefinition of ellipse, <br/>we have $\mathrm{PS}+\mathrm{PS}^{\prime}=2 a$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/VyVu5UzqrA-ECTkHht3zSd4GI8zogUupw2xjyEFetL4.original.fullsize.png"/></div>


tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P4.db
title:
what-is-the-sum-of-the-major-and-minor-axes-of-the-ellipse-whose-eccentricity-is-45-and-length-of-latus-rectum-is-144-unit

slug:
what-is-the-sum-of-the-major-and-minor-axes-of-the-ellipse-whose-eccentricity-is-45-and-length-of-latus-rectum-is-144-unit-56326

question:
<div class="question">What is the sum of the major and minor axes of the ellipse whose eccentricity is $4 / 5$ and length of latus rectum is $14.4$ unit</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">32 units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48 units</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">64 units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">64 units</span> </div>

answer:
<div class="solution">Let $2 \mathrm{a}$ and $2 \mathrm{~b}$ be the length of major and minor axis respectively.<br/>$\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}=\frac{9}{25}$<br/>Also, $\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=14.4$<br/>$\frac{\mathrm{b}^{2}}{\mathrm{a}}=7.2, \mathrm{~b}^{2}=7.2 \mathrm{a}$<br/>Putting value of $\frac{\mathrm{b}^{2}}{\mathrm{a}}$ in equation (i)<br/>$\frac{7.2}{\mathrm{a}}=\frac{9}{25} \Rightarrow \mathrm{a}=20$<br/>$b^{2}=7.2 \times 20=144$<br/>$b=12$<br/>the sum of the major and minor axes $=2 a+2 b$<br/>$=2(\mathrm{a}+\mathrm{b})=2(20+12)=64$ units</div>


title: what-is-the-sum-of-the-roots-of-the-equation-2-3-x-2-7-4-3-x-2-3-0
slug: what-is-the-sum-of-the-roots-of-the-equation-2-3-x-2-7-4-3-x-2-3-0-30891
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the roots of the equation<br/>$(2-\sqrt{3}) x^{2}-(7-4 \sqrt{3}) x+(2+\sqrt{3})=0 ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2-\sqrt{3}$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$2+\sqrt{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$7-4 \sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

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

answer:
<div class="solution">The given equation is $(2-\sqrt{3}) x^{2}-(7-4 \sqrt{3}) x+(2+\sqrt{3})=0$<br/>Sum of roots $=\frac{(7-4 \sqrt{3})}{2-\sqrt{3}}$<br/>$=\frac{(7-4 \sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=\frac{14+7 \sqrt{3}-8 \sqrt{3}-12}{4-3}$<br/>$=2-\sqrt{3}$</div>


title: what-is-the-sum-of-the-series-1-2-1-4-1-8-1-equal-to
slug: what-is-the-sum-of-the-series-1-2-1-4-1-8-1-equal-to-42540
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots .$ equal to?</div>

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

answer:
<div class="solution">Given series is a GP. with $a=1, r=-\frac{1}{2}$ $\therefore \mathrm{S}_{\infty}=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{1}{1-\left(\frac{-1}{2}\right)}=\frac{2}{3}$</div>


title: what-is-the-sum-of-the-series-1-2-1-4-1-8-1
slug: what-is-the-sum-of-the-series-1-2-1-4-1-8-1-74241
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots . . ?$</div>

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

answer:
<div class="solution">$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots$ can be written as<br/>$1+\left(-\frac{1}{2}\right)+\left(-\frac{1}{2}\right)^{2}+\left(-\frac{1}{2}\right)^{3}+\ldots$<br/>$[\because$ This is a GP with first term $=1$ and common ratio<br/>$\left.=-\frac{1}{2}\right]$<br/>So, sum of the series<br/>$=\frac{1}{1-\left(-\frac{1}{2}\right)}=\frac{1}{1+\frac{1}{2}}=\frac{2}{3}$</div>


title: what-is-the-sum-of-the-series-1-8-1-816-13-81624-135
slug: what-is-the-sum-of-the-series-1-8-1-816-13-81624-135-88500
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the series<br/>$1+\frac{1}{8}+\frac{1.3}{8.16}+\frac{1.3 .5}{8.16 .24}+\ldots . \infty ?$</div>

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

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

answer:
<div class="solution">As given the series is<br/>$\mathrm{S}=1+\frac{1}{8}+\frac{1.3}{8.16}+\frac{1.3 .5}{8.16 .24}+\ldots . \infty$<br/>On comparing this series with<br/>$\mathrm{S}=(1+\mathrm{x})^{\mathrm{n}}=1+\mathrm{nx}+\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}+\ldots \infty$, we get<br/>$\mathrm{nx}=\frac{1}{8}$ ...(i)<br/>and $\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}=\frac{1.3}{8.16}$ ...(ii)<br/>From Eqs. (1) and (2), we get $\frac{\frac{n(n-1)}{2 !} x^{2}}{n^{2} x^{2}}=\frac{\frac{1.3}{8.16}}{\frac{1}{8} \cdot \frac{1}{8}}$<br/>$\Rightarrow \quad \frac{n-1}{2 n}=\frac{3}{2}$<br/>$\Rightarrow \quad n-1=3 n$<br/>$\Rightarrow \quad n=-\frac{1}{2}$<br/>On putting this value in Eq. (i) $\Rightarrow\left(-\frac{1}{2}\right) \mathrm{x}=\frac{1}{8}$<br/>$\begin{aligned} \Rightarrow \quad x &amp;=-\frac{1}{4} \\ \text { But } \quad &amp; S=(1+x)^{n}=\left(1-\frac{1}{4}\right)^{-1 / 2} \\ &amp;=\left(\frac{3}{4}\right)^{-1 / 2}=\frac{2}{\sqrt{3}} \end{aligned}$</div>


title: what-is-the-sum-of-the-series-03-033-0333-n-terms
slug: what-is-the-sum-of-the-series-03-033-0333-n-terms-97754
question: <div class="question">What is the sum of the series $0.3+0.33+0.333+\ldots . n$ terms?</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}\left[\mathrm{n}-\frac{2}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{3}\left[\mathrm{n}-\frac{1}{3}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1+\frac{1}{10^{\mathrm{n}}}\right)\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; 0.3+0.33+0.333+\ldots \ldots \mathrm{n} \text { terms } \\ &amp;=\frac{3}{10}+\frac{33}{100}+\frac{333}{1000}+\ldots \ldots . \\ &amp;=3\left(\frac{1}{10}+\frac{11}{100}+\frac{111}{1000}+\ldots \ldots\right) \\ &amp;=\frac{3}{9}\left(\frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+\ldots . .\right) \\ &amp;=\frac{1}{3}\left[\left(1-\frac{1}{10}\right)+\left(1-\frac{1}{100}\right)+\left(1-\frac{1}{1000}\right)+\ldots \ldots .\right] \\ &amp;=\frac{1}{3}\left[\mathrm{n}-\left(\frac{1}{10}+\frac{1}{10^{2}}+\frac{1}{10^{3}}+\ldots \ldots \frac{1}{10^{\mathrm{n}}}\right)\right] \end{aligned}$<br/>$=\frac{1}{3}\left[\mathrm{n}-\frac{\frac{1}{10}\left(1-\frac{1}{10^{\mathrm{n}}}\right)}{1-\frac{1}{10}}\right]$<br/>$\left(\right.$ Since, Sum of $n$ terms of G.P $\left.=\frac{a\left(1-r^{n}\right)}{1-r}\right]$<br/>$=\frac{1}{3}\left[n-\frac{10^{\mathrm{n}}-1}{9\left(10^{\mathrm{n}}\right)}\right]=\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$</div>


title: what-is-the-sum-of-the-series-05-055-0555-to-n-terms
slug: what-is-the-sum-of-the-series-05-055-0555-to-n-terms-74534
question: <div class="question">What is the sum of the series $0.5+0.55+0.555+\ldots$ to $n$ terms?</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{9}\left[n-\frac{2}{9}\left(1-\frac{1}{10^{n}}\right)\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{9}\left[5-\frac{2}{9}\left(1-\frac{1}{10^{n}}\right)\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{9}\left[n-\frac{5}{9}\left(1-\frac{1}{10^{n}}\right)\right]$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{5}{9}\left[n-\frac{1}{9}\left(1-\frac{1}{10^{n}}\right)\right]$</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">$\frac{5}{9}\left[n-\frac{1}{9}\left(1-\frac{1}{10^{n}}\right)\right]$</span> </div>

answer:
<div class="solution">Given $0.5+0.55+0.555+\ldots .$ to n<br/>$=5[0.1+0.11+0.111+\ldots . .$ to n terms $]$<br/>$=\frac{5}{9}[0.9+0.99+0.999+\ldots . .$ to $\mathrm{n}$ terms $]$<br/>$=\frac{5}{9}\left[\frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+\ldots . .\right.$ to $\mathrm{n}$ terms $]$<br/>$=\frac{5}{9}\left[\left(1-\frac{1}{10}\right)+\left(1-\frac{1}{100}\right)+\left(1-\frac{1}{1000}\right)+\ldots . .\right]$<br/>$=\frac{5}{9}\left[\left(1-\frac{1}{10}\right)+\left(1-\frac{1}{10^{2}}\right)+\left(1-\frac{1}{10^{3}}\right)\right.$<br/>$\left.+\ldots . .+\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$<br/>$=\frac{5}{9}\left[\mathrm{n}-\left(\frac{1}{10}+\frac{1}{10^{2}}+\ldots \cdot \frac{1}{10^{\mathrm{n}}}\right)\right]$<br/>$=\frac{5}{9}\left[\mathrm{n}-\frac{1}{10} \frac{\left\{1-\left(\frac{1}{10}\right)^{\mathrm{n}}\right\}}{\left(1-\frac{1}{10}\right)}\right]$<br/>$=\frac{5}{9}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$</div>


title: what-is-the-sum-of-the-squares-of-direction-cosines-of-the-line-joining-the-points-1-2-3-and-2-3-1
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
slug:
what-is-the-sum-of-the-squares-of-direction-cosines-of-the-line-joining-the-points-1-2-3-and-2-3-1-46652

question:
<div class="question">What is the sum of the squares of direction cosines of the line joining the points $(1,2,-3)$ and $(-2,3,1) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{\sqrt{26}}$</span> </li> </ul>

answer:
<div class="solution">Let the direction cosines be $\ell, \mathrm{m}, \mathrm{n}$. Let $\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and $\mathrm{Q}\left(\mathrm{x}_{2}, \mathrm{y}_{2}, \mathrm{z}_{2}\right)$ be two points which<br/>joins a line.<br/>$\therefore \mathrm{x}_{1}=1, \mathrm{y}_{1}=2, \mathrm{z}_{1}=-3$<br/>$\mathrm{x}_{2}=-2, \mathrm{y}_{2}=3, \mathrm{z}_{2}=1$<br/>Now, $\ell=\frac{\mathrm{x}_{2}-\mathrm{x}_{1}}{\sqrt{\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}+\left(\mathrm{z}_{2}-\mathrm{z}_{1}\right)^{2}}}$<br/>$=\frac{-3}{\sqrt{26}}$<br/>$\mathrm{m}=\frac{\mathrm{y}_{2}-\mathrm{y}_{1}}{\sqrt{\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}+\left(\mathrm{z}_{2}-\mathrm{z}_{1}\right)^{2}}}=\frac{1}{\sqrt{26}}$<br/>$\mathrm{n}=\frac{4}{\sqrt{26}}$<br/>$\therefore \ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}=\frac{9}{26}+\frac{1}{26}+\frac{16}{26}=\frac{26}{26}=1$<br/>Always equal to 1</div>


title: what-is-the-sum-of-the-squares-of-direction-cosines-of-x-axis
slug: what-is-the-sum-of-the-squares-of-direction-cosines-of-x-axis-89165
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
answer: <div class="solution">Sum of squares of direction cosines $=(1)^{2}+(0)^{2}+(0)^{2}=1$</div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the squares of direction cosines of $\begin{array}{ll}x \text { -axis ? } &amp; {}\end{array}$</div>

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


title: what-is-the-sum-of-the-squares-of-the-roots-of-the-equation-x-2-2-x-143-0
slug: what-is-the-sum-of-the-squares-of-the-roots-of-the-equation-x-2-2-x-143-0-51172
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">290</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the sum of the squares of the roots of the equation $\mathrm{x}^{2}+2 \mathrm{x}-143=0$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">170</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">180</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">190</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">290</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Let $\alpha, \beta$ be the roots of given equation<br/>and $\alpha \beta=-$ Consider $\beta^{2}=(\alpha+\beta)^{2}-2 \alpha \beta$<br/>$=(-2)^{2}-2(-143)=4+286=290$</div>


title: what-is-the-sum-of-the-two-numbers-11110-2-and-1010
slug: what-is-the-sum-of-the-two-numbers-11110-2-and-1010-43792
question: <div class="question">What is the sum of the two numbers $(11110)_{2}$ and $(1010)$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(101000)$,</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$(110000)_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(100100)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(101100)_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(101000)$,</span> </div>

answer:
<div class="solution">Therefore relation is equivalence.<br/>$(11110)_{2}=2^{4} \times 1+2^{3} \times 1+2^{2} \times 1+2^{1} \times 1+2^{0} \times 0$<br/>$=16+8+4+2+0=30$<br/>$(1010)_{2}=\left(2^{3} \times 1+2^{2} \times 0+2^{1} \times 1+2^{0} \times 0=8+0+2\right.$<br/>$+0)=10$<br/>Sum $=30+10=40$<br/>$=(101000)_{3}$<br/>\begin{array}{l|l|l}<br/>2 &amp; 40 &amp; \\<br/>\hline 2 &amp; 20 &amp; 0 \\<br/>\hline 2 &amp; 10 &amp; 0 \\<br/>\hline 2 &amp; 5 &amp; 0 \\<br/>\hline 2 &amp; 2 &amp; 1 \\<br/>\hline &amp; 1 &amp; 0<br/>\end{array}</div>


tags: ['Chemistry', 'States of Matter', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-temperature-at-which-the-clear-boundary-between-the-liquid-and-vapor-phases-disappears-called

slug:
what-is-the-temperature-at-which-the-clear-boundary-between-the-liquid-and-vapor-phases-disappears-called-95495

question:
<div class="question">What is the temperature at which the clear boundary between the liquid and vapor phases disappears called?</div>

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

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

answer:
<div class="solution">The temperature at which the clear boundary between the liquid and vapor phases disappears is called the critical temperature.</div>


tags: ['Chemistry', 'Waves and Sound', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">248 N</span> </div>
answer: <div class="solution">Correct Option is : (B)<br/>248 N</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-tension-in-the-wire-stretched-between-two-rigid-supports-if-it-vibrates-in-its-fundamental-mode-with-a-frequency-of-45-hz-given-that-the-m

slug:
what-is-the-tension-in-the-wire-stretched-between-two-rigid-supports-if-it-vibrates-in-its-fundamental-mode-with-a-frequency-of-45-hz-given-that-the-m-81625

question:
<div class="question">What is the tension in the wire stretched between two rigid supports if it vibrates in its fundamental mode with a frequency of 45 Hz, given that the mass of the wire is 3.5 Ã${{10}^{-2}}$ kg and its linear mass density is 4.0 Ã ${{10}^{-2}}$kg ${{m}^{-1}}$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">157 N</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">248 N</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">394 N</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">197 N</span> </li> </ul>


title: what-is-the-term-for-a-ligand-that-can-bind-through-two-donor-atoms
slug: what-is-the-term-for-a-ligand-that-can-bind-through-two-donor-atoms-79482
question: <div class="question">What is the term for a ligand that can bind through two donor atoms?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Didendate</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Hexadentate</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Polydentate</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Unidentate</span> </li> </ul>

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

answer:
<div class="solution">Ligands that can bind through two donor atoms are described as didendate ligands.</div>


tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-term-for-a-ligand-that-uses-two-or-more-donor-atoms-simultaneously-to-bind-a-single-metal-ion

slug:
what-is-the-term-for-a-ligand-that-uses-two-or-more-donor-atoms-simultaneously-to-bind-a-single-metal-ion-73126

question:
<div class="question">What is the term for a ligand that uses two or more donor atoms simultaneously to bind a single metal ion?</div>

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

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

answer:
<div class="solution">Chelate ligands use two or more donor atoms simultaneously to bind a single metal ion, forming chelate complexes.</div>


title: what-is-the-term-for-the-boiling-temperature-at-1-atm-pressure
slug: what-is-the-term-for-the-boiling-temperature-at-1-atm-pressure-41031
question: <div class="question">What is the term for the boiling temperature at 1 atm pressure?</div>
tags: ['Chemistry', 'States of Matter', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Standard boiling point</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Evaporation point</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Normal boiling point</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Solidification point</span> </li> </ul>

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

answer:
<div class="solution">The boiling temperature at 1 atm pressure is called the normal boiling point.</div>


title: what-is-the-term-for-the-potential-difference-between-the-two-electrodes-of-a-galvanic-cell
slug: what-is-the-term-for-the-potential-difference-between-the-two-electrodes-of-a-galvanic-cell-95229
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the term for the potential difference between the two electrodes of a galvanic cell?</div>

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

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

answer:
<div class="solution">The potential difference between the two electrodes of a galvanic cell is called the cell potential.</div>


title: what-is-the-term-for-the-primary-structure-of-proteins
slug: what-is-the-term-for-the-primary-structure-of-proteins-61739
question: <div class="question">What is the term for the primary structure of proteins?</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Primary conformation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Tertiary structure</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Secondary structure</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Quaternary structure</span> </li> </ul>

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

answer:
<div class="solution">The primary structure of proteins refers to the specific sequence of amino acids in a polypeptide chain, also known as primary conformation.</div>


title: what-is-the-term-for-the-shape-in-which-a-long-polypeptide-chain-can-exist-in-proteins
slug: what-is-the-term-for-the-shape-in-which-a-long-polypeptide-chain-can-exist-in-proteins-73670
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the term for the shape in which a long polypeptide chain can exist in proteins?</div>

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

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

answer:
<div class="solution">The secondary structure of proteins refers to the shape in which a long polypeptide chain can exist, such as alpha-helix or beta-pleated sheet.</div>


title: what-is-the-term-for-the-specific-sequence-of-amino-acids-in-a-protein
slug: what-is-the-term-for-the-specific-sequence-of-amino-acids-in-a-protein-11264
question: <div class="question">What is the term for the specific sequence of amino acids in a protein?</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Tertiary structure</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Quaternary structure</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Primary structure</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Secondary structure</span> </li> </ul>

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

answer:
<div class="solution">The primary structure of a protein is defined by the specific sequence of amino acids in its polypeptide chain.</div>


title: what-is-the-term-independent-of-x-in-the-expansion-of-1-x-2-x-3-2-3-x-2-3-x-1-9
slug: what-is-the-term-independent-of-x-in-the-expansion-of-1-x-2-x-3-2-3-x-2-3-x-1-9-32329
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the term independent of $x$ in the expansion of $\left(1+x+2 x^{3}\right)\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{9}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$17 / 54$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$1 / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">No such term exists in the expansion</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$17 / 54$</span> </div>

answer:
<div class="solution">Given expansion is<br/>$\left(1+x+2 x^{3}\right)\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{9}$<br/>$=\left(1+x+2 x^{3}\right)\left[\left(\frac{3}{2} x^{2}\right)^{9}-{ }^{9} C_{1}\left(\frac{3}{2} x^{2}\right)^{8} \cdot \frac{1}{3 x}\right.$<br/>$\left.+{ }^{9} C_{6}\left(\frac{3}{2} x^{2}\right)^{3}\left(\frac{1}{3 x}\right)^{6}-{ }^{9} C_{7}\left(\frac{3}{2} x^{2}\right)^{2}\left(\frac{1}{3 x}\right)^{7} \ldots \ldots \ldots .\right]$<br/>In the second bracket we have to search out terms of $x^{\circ}$<br/>and $\frac{1}{x^{3}}$ which when multiplied with the terms 1 and<br/>$2 x^{3}$ in the first bracket will give a term independent of<br/>$x$. The term containing $\frac{1}{x}$ will not occur in the 2 nd<br/>bracket.<br/>- Term independent of $\mathrm{x}$<br/>$=1-\left[{ }^{9} C_{6} \frac{3^{3}}{2^{3}} \cdot \frac{1}{3^{6}}\right]-2 x^{3}\left[{ }^{9} C_{7} \frac{3^{2}}{2^{2}} \cdot \frac{1}{3^{7}} \cdot \frac{1}{x^{3}}\right]$<br/>$=\left[\frac{9.8 .7}{1.2 .3} \cdot \frac{1}{8.27}\right]-2\left[\frac{9.8}{1.2} \cdot \frac{1}{4.243}\right]$<br/>$=\frac{7}{18}-\frac{2}{27}=\frac{17}{54}$</div>


tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-term-used-for-the-atoms-within-the-square-brackets-in-the-compound-formulation-shown-in-table-51

slug:
what-is-the-term-used-for-the-atoms-within-the-square-brackets-in-the-compound-formulation-shown-in-table-51-51880

question:
<div class="question">What is the term used for the atoms within the square brackets in the compound formulation shown in Table 5.1?</div>

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

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

answer:
<div class="solution">The atoms within the square brackets in the compound formulation are referred to as "primary valence."</div>


tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-term-used-for-the-compounds-behavior-under-the-reaction-conditions-as-described-in-the-paragraph

slug:
what-is-the-term-used-for-the-compounds-behavior-under-the-reaction-conditions-as-described-in-the-paragraph-30717

question:
<div class="question">What is the term used for the compounds' behavior under the reaction conditions as described in the paragraph?</div>

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

answer:
<div class="solution">The term "quaternary valence" is not mentioned in the paragraph. It is not used to describe the compounds' behavior under the reaction conditions.</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-term-used-for-the-electrode-potential-when-the-concentrations-of-all-species-in-a-halfcell-are-unity

slug:
what-is-the-term-used-for-the-electrode-potential-when-the-concentrations-of-all-species-in-a-halfcell-are-unity-42379

question:
<div class="question">What is the term used for the electrode potential when the concentrations of all species in a half-cell are unity?</div>

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

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

answer:
<div class="solution">When the concentrations of all species in a half-cell are unity, the electrode potential is known as standard electrode potential.</div>


title: what-is-the-term-used-for-the-lateral-roots-produced-by-the-primary-root-in-dicotyledonous-plants
tags: ['Biology', 'Morphology of Flowering Plants', 'NEET']
source_db: MarksBatch2_P4.db
slug:
what-is-the-term-used-for-the-lateral-roots-produced-by-the-primary-root-in-dicotyledonous-plants-92149

question:
<div class="question">What is the term used for the lateral roots produced by the primary root in dicotyledonous plants?</div>

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

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

answer:
<div class="solution">The lateral roots produced by the primary root in dicotyledonous plants are referred to as secondary roots. They continue to grow and develop, forming more branches and helping the plant to absorb water and nutrients from a wider area of the soil.</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-term-used-for-the-potential-difference-between-the-electrode-and-the-electrolyte-in-a-halfcell

slug:
what-is-the-term-used-for-the-potential-difference-between-the-electrode-and-the-electrolyte-in-a-halfcell-25116

question:
<div class="question">What is the term used for the potential difference between the electrode and the electrolyte in a half-cell?</div>

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

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

answer:
<div class="solution">A potential difference develops between the electrode and the electrolyte which is called electrode potential.</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-term-used-to-describe-the-additional-potential-required-for-slow-kinetically-feasible-electrochemical-processes

slug:
what-is-the-term-used-to-describe-the-additional-potential-required-for-slow-kinetically-feasible-electrochemical-processes-59925

question:
<div class="question">What is the term used to describe the additional potential required for slow kinetically feasible electrochemical processes?</div>

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

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

answer:
<div class="solution">Slow kinetically feasible processes, an extra potential called "overpotential" has to be applied.</div>


title: what-is-the-total-number-of-combination-of-n-different-things-taken-1-2-3-n-at-a-time
slug: what-is-the-total-number-of-combination-of-n-different-things-taken-1-2-3-n-at-a-time-84612
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the total number of combination of $n$ different things taken $1,2,3, \ldots,, \mathrm{n}$ at a time?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{n+1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2^{2 n+1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{n-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2^{n}-1$</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">$2^{n}-1$</span> </div>

answer:
<div class="solution">Since, combinations of taking $1,2,3, \ldots \ldots \ldots, n$ things at a times are ${ }^{n} C_{1},{ }^{n} C_{2}, \ldots \ldots . .,{ }^{n} C_{n}$<br/>$\therefore \quad$ Total number of combinations $={ }^{n} C_{1}+{ }^{n} C_{2}+\ldots \ldots . .+{ }^{n} C_{n}$<br/>$=1+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots \ldots . .+{ }^{n} C_{n}-1$<br/>$=2^{n}-1$</div>


title: what-is-the-trend-in-the-relative-stability-of-the-1-oxidation-state-as-you-move-down-the-group
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'NEET']
source_db: MarksBatch2_P4.db
slug:
what-is-the-trend-in-the-relative-stability-of-the-1-oxidation-state-as-you-move-down-the-group-52231

question:
<div class="question">What is the trend in the relative stability of the +1 oxidation state as you move down the group?</div>

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

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

answer:
<div class="solution">The relative stability of +1 oxidation state progressively increases for heavier elements: Al<ga> <!-- /Solution --> <!-- Call to Action --> <section class="call-to-action" data-pagefind-ignore=""> <div class="call-to-action-header"> <svg fill="none" height="24" viewbox="0 0 25 24" width="25" xmlns="http://www.w3.org/2000/svg"> <path d="M19.9999 13.5C20.0018 13.8058 19.9089 14.1047 19.734 14.3555C19.559 14.6063 19.3107 14.7968 19.023 14.9006L14.1874 16.6875L12.4062 21.5269C12.3007 21.8134 12.1098 22.0608 11.8594 22.2355C11.6089 22.4101 11.3109 22.5038 11.0055 22.5038C10.7002 22.5038 10.4022 22.4101 10.1517 22.2355C9.90124 22.0608 9.71039 21.8134 9.60491 21.5269L7.8124 16.6875L2.97303 14.9062C2.68646 14.8008 2.43915 14.6099 2.26445 14.3595C2.08976 14.109 1.99609 13.811 1.99609 13.5056C1.99609 13.2003 2.08976 12.9022 2.26445 12.6518C2.43915 12.4013 2.68646 12.2105 2.97303 12.105L7.8124 10.3125L9.59366 5.47313C9.69914 5.18656 9.88998 4.93924 10.1404 4.76455C10.3909 4.58985 10.6889 4.49619 10.9943 4.49619C11.2996 4.49619 11.5977 4.58985 11.8481 4.76455C12.0986 4.93924 12.2894 5.18656 12.3949 5.47313L14.1874 10.3125L19.0268 12.0938C19.3146 12.1986 19.5628 12.3901 19.7371 12.642C19.9114 12.8939 20.0033 13.1937 19.9999 13.5ZM14.7499 4.5H16.2499V6C16.2499 6.19891 16.3289 6.38968 16.4696 6.53033C16.6102 6.67098 16.801 6.75 16.9999 6.75C17.1988 6.75 17.3896 6.67098 17.5302 6.53033C17.6709 6.38968 17.7499 6.19891 17.7499 6V4.5H19.2499C19.4488 4.5 19.6396 4.42098 19.7802 4.28033C19.9209 4.13968 19.9999 3.94891 19.9999 3.75C19.9999 3.55109 19.9209 3.36032 19.7802 3.21967C19.6396 3.07902 19.4488 3 19.2499 3H17.7499V1.5C17.7499 1.30109 17.6709 1.11032 17.5302 0.96967C17.3896 0.829018 17.1988 0.75 16.9999 0.75C16.801 0.75 16.6102 0.829018 16.4696 0.96967C16.3289 1.11032 16.2499 1.30109 16.2499 1.5V3H14.7499C14.551 3 14.3602 3.07902 14.2196 3.21967C14.0789 3.36032 13.9999 3.55109 13.9999 3.75C13.9999 3.94891 14.0789 4.13968 14.2196 4.28033C14.3602 4.42098 14.551 4.5 14.7499 4.5ZM22.9999 7.5H22.2499V6.75C22.2499 6.55109 22.1709 6.36032 22.0302 6.21967C21.8896 6.07902 21.6988 6 21.4999 6C21.301 6 21.1102 6.07902 20.9696 6.21967C20.8289 6.36032 20.7499 6.55109 20.7499 6.75V7.5H19.9999C19.801 7.5 19.6102 7.57902 19.4696 7.71967C19.3289 7.86032 19.2499 8.05109 19.2499 8.25C19.2499 8.44891 19.3289 8.63968 19.4696 8.78033C19.6102 8.92098 19.801 9 19.9999 9H20.7499V9.75C20.7499 9.94891 20.8289 10.1397 20.9696 10.2803C21.1102 10.421 21.301 10.5 21.4999 10.5C21.6988 10.5 21.8896 10.421 22.0302 10.2803C22.1709 10.1397 22.2499 9.94891 22.2499 9.75V9H22.9999C23.1988 9 23.3896 8.92098 23.5302 8.78033C23.6709 8.63968 23.7499 8.44891 23.7499 8.25C23.7499 8.05109 23.6709 7.86032 23.5302 7.71967C23.3896 7.57902 23.1988 7.5 22.9999 7.5Z" fill="#1589EE"></path> </svg> <h3>Looking for more such questions to practice?</h3> </div> <p>
Download the MARKS App - The ultimate prep app for IIT JEE &amp;
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title: what-is-the-unique-behaviour-of-cu-with-regard-to-its-standard-electrode-potential-eo
slug: what-is-the-unique-behaviour-of-cu-with-regard-to-its-standard-electrode-potential-eo-70515
tags: ['Chemistry', 'd and f Block Elements', 'NEET']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the unique behaviour of Cu with regard to its standard electrode potential (Eo)?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Cu has a positive Eo.</span> </div>

answer:
<div class="solution">The unique behaviour of Cu, having a positive E_, accounts for its inability to liberate H<sub>2</sub> from acids.</div>


title: what-is-the-value-of-0-1-x-1-e-x-d-x
slug: what-is-the-value-of-0-1-x-1-e-x-d-x-60534
question: <div class="question">What is the value of $\int_{0}^{1}(x-1) e^{-x} d x$ ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">e</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\mathrm{e}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{-1}{\mathrm{e}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{-1}{\mathrm{e}}$</span> </div>

answer:
<div class="solution">Given integral is $\mathrm{I}=\int_{0}^{1}(\mathrm{x}-1) \mathrm{e}^{-\mathrm{x}} \mathrm{dx}$<br/>Integrating by parts taking $(\mathrm{x}-1)$ as first function<br/>We get, $\mathrm{I}=\left[(\mathrm{x}-1)\left\{-\mathrm{e}^{-\mathrm{x}}\right\}\right]_{0}^{1}-\int_{0}^{1} 1 .\left(-\mathrm{e}^{-\mathrm{x}}\right) \mathrm{dx}$<br/>$=-(1-1) \frac{1}{\mathrm{e}}+(-1) \mathrm{e}^{0}+\left[-\mathrm{e}^{-\mathrm{x}}\right]_{0}^{1}=-1-\frac{1}{\mathrm{e}}+1=-\frac{1}{\mathrm{e}}$</div>


title: what-is-the-value-of-0-1-x-e-x-2-d-x
slug: what-is-the-value-of-0-1-x-e-x-2-d-x-23946
question: <div class="question">What is the value of $\int_{0}^{1} x e^{x^{2}} d x$</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{(\mathrm{e}-1)}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{e}^{2}-1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2(\mathrm{e}-1)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{e}-1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{(\mathrm{e}-1)}{2}$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int_{0}^{1} \mathrm{xe}^{\mathrm{x}^{2}} \cdot \mathrm{dx}$<br/>Let $x^{2}=t$<br/>$\Rightarrow 2 x d x=d t$<br/>$\Rightarrow x d x=\frac{d t}{2}$<br/>when $\mathrm{x}=0, \mathrm{t}=0$ then $\mathrm{x}=1, \mathrm{t}=1$<br/>$\Rightarrow \mathrm{I}=\frac{1}{2} \int_{0}^{1} \mathrm{e}^{\mathrm{t}} \mathrm{dt}=\frac{1}{2}\left[\mathrm{e}^{\mathrm{t}}\right]_{0}^{1}$<br/>$=\frac{1}{2}\left[\mathrm{e}^{\mathrm{x}^{2}}\right]_{0}^{1}=\frac{1}{2}\left[\mathrm{e}-\mathrm{e}^{0}\right]=\frac{\mathrm{e}-1}{2}$</div>


title: what-is-the-value-of-0-2-0-23
slug: what-is-the-value-of-0-2-0-23-73393
question: <div class="question">What is the value of $0 . \overline{2}+0 . \overline{23}$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0 . \overline{43}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0 . \overline{45}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$0 . \overline{223}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.2 \overline{23}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0 . \overline{45}$</span> </div>

answer:
<div class="solution">$0 . \overline{2}+0 . \overline{23}$<br/>$=\frac{2}{9}+\frac{23}{99}=\frac{22+23}{99}=\frac{45}{99}=0 . \overline{45}$</div>


title: what-is-the-value-of-0-2-cos-8-x-d-x
slug: what-is-the-value-of-0-2-cos-8-x-d-x-55572
question: <div class="question">What is the value of $\int_{0}^{\pi / 2} \cos ^{8} x d x$ ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{35 \pi}{256}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{70}{256}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{16}{35}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{8 \pi}{35}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{35 \pi}{256}$</span> </div>

answer:
<div class="solution">Let, $I=\int_{0}^{\pi / 2} \cos ^{8} x d x$<br/>Given integral can be also be written as:<br/>$\mathrm{I}=\int_{0}^{\pi / 2} \sin ^{\circ} \mathrm{x} \cos ^{8} \mathrm{x} \mathrm{dx}$, which is known as Gamma<br/>function.<br/>$$<br/>\begin{array}{l}<br/>\text { Solution is : } \int_{0}^{\pi / 2} \sin ^{\mathrm{m}} x \cos ^{\mathrm{n}} x \mathrm{dx} \\<br/>=\frac{[(\mathrm{m}-1)(\mathrm{m}-3) \ldots 2 \operatorname{or} 1][(\mathrm{n}-1)(\mathrm{n}-3) \ldots 2 \mathrm{or} 1]}{[(\mathrm{m}+\mathrm{n})(\mathrm{m}+\mathrm{n}-2) \ldots .2 \mathrm{or} 1]} \cdot \theta<br/>\end{array}<br/>$$<br/>If $m$ and $n$ both are even then RHS should be multiplied<br/>by $\frac{\pi}{2}$ here, $\mathrm{m}=0, \mathrm{n}=8$<br/>$$<br/>\begin{aligned}<br/>\Rightarrow \quad \mathrm{I} &amp;=\frac{(8-1)(8-3)(8-5) \ldots(8-7)}{8 .(8-2)(8-4)(8-6)} \frac{\pi}{2} \\<br/>&amp;=\frac{7.5 .3 .1}{8.6 .4 .2} \frac{\pi}{2}=\frac{35 \pi}{256}<br/>\end{aligned}<br/>$$</div>


title: what-is-the-value-of-0-2-lo-g-tan-x-d-x
slug: what-is-the-value-of-0-2-lo-g-tan-x-d-x-18633
question: <div class="question">What is the value of $\int_{0}^{\pi / 2} \log (\tan x) d x ? \quad$</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi / 4$</span> </li> </ul>

answer:
<div class="solution">Let $I=\int_{0}^{\pi / 2} \log (\tan x) d x$<br/>and $I=\int_{0}^{\pi / 2} \log \left\{\tan \left(\frac{\pi}{2}-x\right)\right\} d x$<br/>[By the property of definite integral which says<br/>$\left.\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right]$<br/>$\Rightarrow I=\int_{0}^{\pi / 2} \log (\cot x) d x$<br/>By adding equation (i) and (ii), we get $2 \mathrm{I}=\int_{0}^{\pi / 2} \log (\tan x) d x+\int_{0}^{\pi / 2} \log (\cot x) d x$<br/>$\Rightarrow 2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \log (\tan x \cot x) d x$<br/>$[\because \log m+\log n=\log (m n)]$<br/>$=\int_{0}^{\frac{\pi}{2}} \log \left(\tan x \cdot \frac{1}{\tan x}\right) d x=\int_{0}^{\frac{\pi}{2}} \log 1 d x=0$<br/>$\Rightarrow \mathrm{I}=0$</div>


title: what-is-the-value-of-1-1-x-x-dx
slug: what-is-the-value-of-1-1-x-x-dx-18590
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\int_{-1}^{1} \mathrm{x}|\mathrm{x}| \mathrm{dx}$ ?</div>

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

answer:
<div class="solution">Let $\mathrm{f}(\mathrm{x})=\mathrm{x}|\mathrm{x}|$<br/>$\mathrm{f}(-\mathrm{x})=-\mathrm{x}|-\mathrm{x}|=-\mathrm{x}|\mathrm{x}|=-\mathrm{f}(\mathrm{x})$<br/>$\Rightarrow \mathrm{x}|\mathrm{x}|$ is an odd function<br/>Hence, $\int_{-1}^{1} x|x| d x=0$</div>


title: what-is-the-value-of-1-2-3-2-2-3-3-2-2-4-3-6-4-where-is-the-cube-root-of-unity
slug: what-is-the-value-of-1-2-3-2-2-3-3-2-2-4-3-6-4-where-is-the-cube-root-of-unity-56378
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\left|\begin{array}{ccc}1 &amp; \omega &amp; 2 \omega^{2} \\ 2 &amp; 2 \omega^{2} &amp; 4 \omega^{3} \\ 3 &amp; 3 \omega^{3} &amp; 6 \omega^{4}\end{array}\right|$<br/>where $\omega$ is the cube root of unity?</div>

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

answer:
<div class="solution">Consider $\left|\begin{array}{ccc}1 &amp; \omega &amp; 2 \omega^{2} \\ 2 &amp; 2 \omega^{2} &amp; 4 \omega^{3} \\ 3 &amp; 3 \omega^{3} &amp; 6 \omega^{4}\end{array}\right|=\left|\begin{array}{ccc}1 &amp; \omega &amp; 2 \omega^{2} \\ 2 &amp; 2 \omega^{2} &amp; 4 \\ 3 &amp; 3 &amp; 6 \omega\end{array}\right|$<br/>$\left(\because \omega^{3}=1\right.$ and $\left.\omega^{4}=\omega\right)$<br/>$=2 \times 3\left|\begin{array}{ccc}1 &amp; \omega &amp; 2 \omega^{2} \\ 1 &amp; \omega^{2} &amp; 2 \\ 1 &amp; 1 &amp; 2 \omega\end{array}\right|$<br/>$=6\left[1\left(2 \omega^{3}-2\right)-\omega(2 \omega-2)+2 \omega^{2}\left(1-\omega^{2}\right)\right]$<br/>$=6\left[0-2 \omega^{2}-2 \omega+2 \omega^{2}-2 \omega\right]=0$</div>


title: what-is-the-value-of-1-2-e-x-x-1-x-2-1-d-x
slug: what-is-the-value-of-1-2-e-x-x-1-x-2-1-d-x-10617
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\int_{1}^{2} e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right) d x^{\prime} ? \quad$</div>

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

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

answer:
<div class="solution">Let $\mathrm{I}=\int_{1}^{2} e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right) d x$<br/>$=\int_{1}^{2} e^{x}\left(f(x)+f^{\prime}(x)\right) d x$ where $f(x)=\frac{1}{x}$<br/>$=\left.e^{x} f(x)\right|_{1} ^{2}$<br/>$\therefore \quad \mathrm{I}=\left.\frac{e^{x}}{x}\right|_{1} ^{2}=\frac{e^{2}}{2}-e=e\left(\frac{e}{2}-1\right)$</div>


title: what-is-the-value-of-1-4-n-3-i-41-i-257-9-where-n-n
slug: what-is-the-value-of-1-4-n-3-i-41-i-257-9-where-n-n-10556
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">i.</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $(-\sqrt{-1})^{4 \mathrm{n}+3}+\left(\mathrm{i}^{41}+i^{-257}\right)^{9}$,<br/>where $n \in N ?$</div>

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

answer:
<div class="solution">Consider $(-\sqrt{-1})^{4 n+3}+\left(i^{41}+i^{-257}\right)^{9}$<br/>$=(-i)^{4 n+3}+\left[\left(i^{4}\right)^{10} . i^{1}+\left(i^{3}\right)^{-85} \cdot i^{-2}\right]^{9}$<br/>$=(-i)^{4 n+3}+\left[i+\frac{1}{\left(i^{3}\right)^{85}} \cdot \frac{1}{i^{2}}\right]^{9}=(-i)^{4 n+3}+\left(i+\frac{1}{i}\right)^{9}$<br/>$=-(-1)^{4 n+3}(i)^{4 n}(i)^{3}+(i-i)^{9}=-(1)(-i)+0=i$</div>


title: what-is-the-value-of-1-cos-8-1-cos-8-3-1-cos-8-5-1-cos-8-7
slug: what-is-the-value-of-1-cos-8-1-cos-8-3-1-cos-8-5-1-cos-8-7-33564
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right) ?$</div>

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


title: what-is-the-value-of-1-i-2-i-1-2-i-2-2-2-i-i-where-is-the-cube-root-of-unity
slug: what-is-the-value-of-1-i-2-i-1-2-i-2-2-2-i-i-where-is-the-cube-root-of-unity-23494
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left|\begin{array}{ccc}1-i &amp; \omega^{2} &amp; -\omega \\ \omega^{2}+i &amp; \omega &amp; -i \\ 1-2 i-\omega^{2} &amp; \omega^{2}-\omega &amp; i-\omega\end{array}\right|$, where $\omega$ is the cube root of unity?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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>

answer:
<div class="solution">Let $\Delta=\left|\begin{array}{ccc}1-i &amp; \omega^{2} &amp; -\omega \\ \omega^{2}+i &amp; \omega &amp; -i \\ 1-2 i-\omega^{2} &amp; \omega^{2}-\omega &amp; i-\omega\end{array}\right|$<br/>Applying $R_{3} \rightarrow R_{1}-R_{2}-R_{3}$<br/>$=\left|\begin{array}{ccc}1-i &amp; \omega^{2} &amp; -\omega \\ \omega^{2}+i &amp; \omega &amp; -i \\ 0 &amp; 0 &amp; 0\end{array}\right|=0$<br/>$(\because$ one row of determinant is zero)</div>


title: what-is-the-value-of-1-i-2-i-4-i-6-i-100-w-h-ere-isqrt1-2010i-i
slug: what-is-the-value-of-1-i-2-i-4-i-6-i-100-w-h-ere-isqrt1-2010i-i-63380
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$$<br/>1+i^{2}+i^{4}+i^{6}+\ldots . .+i^{100}<br/>$$<br/>where $i=\sqrt{-1}$ ? $[2010-I I]$</div>

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

answer:
<div class="solution">Consider $1+\mathrm{i}^{2}+\mathrm{i}^{4}+\mathrm{i}^{6}+\ldots+\mathrm{i}^{100}$<br/>$=1+\left[i^{2}+i^{4+} i^{6}+\ldots+i^{100}\right]$<br/>$=1+[(-1)+(1)-1+\ldots+1]$<br/>$=1-1+1-1+\ldots+1=1$</div>


title: what-is-the-value-of-1-i-3-2-10-1-i-3-2-10
slug: what-is-the-value-of-1-i-3-2-10-1-i-3-2-10-25193
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\left.[-1+\mathrm{i} \sqrt{3}) / 2]^{10}+[-1-\mathrm{i} \sqrt{3}) / 2\right]^{10}$</div>

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

answer:
<div class="solution">We know that $\frac{-1+\mathrm{i} \sqrt{3}}{2}=\omega$,<br/>so, $\frac{-1-i \sqrt{3}}{2}=\omega^{2}$<br/>$\therefore\left[\frac{-1+\mathrm{i} \sqrt{3}}{2}\right]^{10}+\left[\frac{-1-\mathrm{i} \sqrt{3}}{2}\right]^{10}$<br/>$=\omega^{10}+\omega^{20}=\omega^{3 \times 3+1}+\omega^{3 \times 6+2}$<br/>$=\omega^{3} \omega^{3} \omega+\omega^{3} \omega^{6} \omega^{2}=\omega+\omega^{2} \quad$ (since $\left.\omega^{3}=1\right)$<br/>and $\omega+\omega^{2}=-1 \quad\left(1+\omega+\omega^{2}=0\right)$</div>


title: what-is-the-value-of-1-i-3-48
slug: what-is-the-value-of-1-i-3-48-67996
question: <div class="question">What is the value of $(-1+i \sqrt{3})^{48}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{24}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2^{48}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$2^{48}$</span> </div>

answer:
<div class="solution">We know, $\omega=\frac{1+i \sqrt{3}}{2}$ and<br/>$\omega^{2}=\frac{1-i \sqrt{3}}{2} \Rightarrow 1-i \sqrt{3}=2 \omega^{2}$<br/>Consider $(-1+i \sqrt{3})^{48}=[-(1-i \sqrt{3})]^{48}$<br/>$=(1-i \sqrt{3})^{48}$<br/>$\Rightarrow \quad\left(2 \omega^{2}\right)^{48}=2^{48} \cdot \omega^{96}=2^{48}\left(\omega^{3}\right)^{32}=2^{48}$</div>


title: what-is-the-value-of-1-i-5-1-i-5-where-i-1
slug: what-is-the-value-of-1-i-5-1-i-5-where-i-1-14013
question: <div class="question">What is the value of $(1+i)^{5}+(1-i)^{5}$ where $i=\sqrt{-1}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-8</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">8</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8i</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-8i</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp;(1+i)^{5}={ }^{5} \mathrm{C}_{0}(1)^{5-0} i^{0}+{ }^{5} \mathrm{C}_{1}(1)^{5-1}(i)^{1}+{ }^{5} \mathrm{C}_{2}(1)^{5-2}(i)^{2} \\ &amp;+{ }^{5} \mathrm{C}_{3}(1)^{5-3}(i)^{3}+{ }^{5} \mathrm{C}_{4}(1)^{5-4}(i)^{4}+{ }^{5} \mathrm{C}_{5}(1)^{5-5}(i)^{5} \\ &amp;=1+5 i+10 i^{2}+10 i^{3}+5 i^{4}+i^{5} \\ &amp;(1-i)^{5}={ }^{5} \mathrm{C}_{0} i^{0}-{ }^{5} \mathrm{C}_{1} i+{ }^{5} \mathrm{C}_{2} i^{2}-{ }^{5} \mathrm{C}_{3} i^{3} \\ &amp; \quad+{ }^{5} \mathrm{C}_{4} i^{4}-{ }^{5} \mathrm{C}_{5} i^{5} \\ &amp;=1-5 i+10 i^{2}-10 i^{3}+5 i^{4}-i^{5} \ldots \ldots(2) \end{aligned}$<br/>By adding $(1)$ and $(2)$, we get $(1+i)^{5}+(1-i)^{5}=1+5 i+10 i^{2}+10 i^{3}+5 i^{4}+i^{5}$<br/>$\quad+1-5 i+10 i^{2}-10 i^{3}+5 i^{4}-\mathrm{i}^{5}$<br/>$=2+20 i^{2}+10 i^{4}=2-20+10=-8$</div>


title: what-is-the-value-of-1-t-a-n-1-5-1-t-a-n-1-5
slug: what-is-the-value-of-1-t-a-n-1-5-1-t-a-n-1-5-65179
question: <div class="question">What is the value of $\frac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\sqrt{3}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\sqrt{3}$</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>

answer:
<div class="solution">Consider $\frac{1+\tan 15^{\circ}}{1-\tan 15^{\circ}}=\frac{\tan 45^{\circ}+\tan 15^{\circ}}{1-\tan 45^{\circ} \tan 15^{\circ}}\left(\because \tan 45^{\circ}=1\right)$<br/>$\begin{aligned}=&amp; \tan \left(45^{\circ}+15^{\circ}\right) \quad\left(\because \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}\right) \\ &amp;=\tan 60^{\circ}=\sqrt{3} \end{aligned}$</div>


title: what-is-the-value-of-1sin-10circ-sin-50circ-sin-70circ
slug: what-is-the-value-of-1sin-10circ-sin-50circ-sin-70circ-49196
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $1-\sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} ?</div>

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

answer:
<div class="solution">$\begin{aligned} 1 &amp;-\sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} \\=&amp; 1-\frac{1}{2}\left[2 \sin 70^{\circ} \sin 10^{\circ} \sin 50^{\circ}\right] \\=&amp; 1-\frac{1}{2}\left[\left(\cos 60^{\circ}-\cos 80^{\circ}\right) \sin 50^{\circ}\right] \\[\because 2 \sin \mathrm{A} \sin \mathrm{B}=\cos (\mathrm{A}-\mathrm{B})-\cos (\mathrm{A}+\mathrm{B})] \\=&amp; 1-\frac{1}{2}\left[\frac{1}{2} \sin 50^{\circ}-\frac{1}{2} 2 \cos 80^{\circ} \sin 50^{\circ}\right] \\=&amp; 1-\frac{1}{4}\left[\sin 50^{\circ}-\sin 130^{\circ}+\sin 30^{\circ}\right] \\(\because 2 \cos \mathrm{A} \sin \mathrm{B}=\sin (\mathrm{A}+\mathrm{B})-\sin (\mathrm{A}-\mathrm{B})) \\=&amp; 1-\frac{1}{8}=\frac{7}{8} \quad\left(\because \sin 130^{\circ}=\sin \left(180^{\circ}-50^{\circ}=\sin 50^{\circ}\right)\right.\end{aligned}$</div>


title: what-is-the-value-of-2-1-i-3-3-n-2-1-i-3-3-n-where-i-1
slug: what-is-the-value-of-2-1-i-3-3-n-2-1-i-3-3-n-where-i-1-88819
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left(\frac{-1+i \sqrt{3}}{2}\right)^{3 n}+\left(\frac{-1+i \sqrt{3}}{2}\right)^{3 n}$ where<br/>$\mathrm{i}=\sqrt{-1}$ ?</div>

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

answer:
<div class="solution">From cube root of unity<br/>$\omega=\frac{-1+\sqrt{3} i}{2}$ and<br/>$$<br/>\begin{aligned} \omega^{2} &amp;=\frac{-1-\sqrt{3} i}{2} \\ \omega^{3} &amp;=\left(\frac{-1+\sqrt{3} i}{2}\right)\left(\frac{-1-\sqrt{3} i}{2}\right) \\ &amp;=\frac{(-1)^{2}-(\sqrt{3} i)^{2}}{4}=\frac{4}{4}=1 \\ \text { Now, }(\omega)^{3 n}+\left(\omega^{2}\right)^{3 n} \\=1+1=2 \end{aligned}<br/>$$</div>


title: what-is-the-value-of-2-lo-g-8-2-2-10-g-3-7
slug: what-is-the-value-of-2-lo-g-8-2-2-10-g-3-7-80694
question: <div class="question">What is the value of $2 \log _{8} 2-\frac{10 \mathrm{~g}_{3} 7}{2}$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 / 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$16 / 3$</span> </li> </ul>

answer:
<div class="solution">$2 \log _{8} 2-\frac{\log _{3} 9}{3}=2 \log _{2} 32-\frac{\log _{3} 3^{2}}{3}$<br/>$=\frac{2}{3} \log _{2} 2-2 \frac{\log _{3} 3}{3}=\frac{2}{3}-\frac{2}{3}=0 \quad\left(\because \log _{a} a=1\right)$</div>


title: what-is-the-value-of-2-logg-2-3-1-lo-g-3
slug: what-is-the-value-of-2-logg-2-3-1-lo-g-3-50740
question: <div class="question">What is the value of 2 logg $2-\frac{1}{3} \log _{3}$</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 3$</span> </li> </ul>

answer:
<div class="solution">Consider $2 \log _{8} 2-\frac{1}{3} \log _{3} 9$<br/>$=2 \cdot \frac{\log 2}{\log 8}-\frac{1}{3} \cdot \frac{\log 9}{\log 3}=2 \cdot \frac{\log 2}{\log 2^{3}}-\frac{1}{3} \cdot \frac{\log 3^{2}}{\log 3}$<br/>$=2 \cdot \frac{\log 2}{3 \log 2}-\frac{1}{3} 2 \cdot \frac{\log 3}{\log 3}=\frac{2}{3}-\frac{2}{3}=0$</div>


title: what-is-the-value-of-3-cosec-2-0-sec-2-0-2009-ii
slug: what-is-the-value-of-3-cosec-2-0-sec-2-0-2009-ii-15256
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\sqrt{3}$ cosec $20^{\circ}-\sec 20^{\circ} ? \quad[2009-I I]$</div>

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

answer:
<div class="solution">Consider $\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}=\frac{\sqrt{3}}{\sin 20^{\circ}}-\frac{1}{\cos 20^{\circ}}$<br/>$=\frac{\sqrt{3} \cos 20^{\circ}-\sin 20^{\circ}}{\sin 20^{\circ} \cos 20^{\circ}}$<br/>Multiply and divide by 2 in $N^{r}$.<br/>$=2\left(\frac{\frac{\sqrt{3}}{2} \cos 20^{\circ}-\frac{1}{2} \sin 20^{\circ}}{\sin 20^{\circ} \cos 20^{\circ}}\right)$<br/>$=2\left(\frac{\sin 60^{\circ} \cos 20^{\circ}-\cos 60^{\circ} \sin 20^{\circ}}{\sin 20^{\circ} \cos 20^{\circ}}\right)$<br/>$\left(\because \sin 60^{\circ}=\frac{\sqrt{3}}{2}\right.$ and $\left.\cos 60^{\circ}=\frac{1}{2}\right)$<br/>$=\frac{2 \times 2\left[\sin \left(60^{\circ}-20^{\circ}\right)\right]}{2 \sin 20^{\circ} \cos 20^{\circ}}$<br/>$(\because \sin A \cos B-\cos A \sin B=\sin (A-B)$ and $\sin 2 \theta=$<br/>$2 \sin \theta \cos \theta)$<br/>$=\frac{4 \sin 40^{\circ}}{\sin 40^{\circ}}=4$</div>


title: what-is-the-value-of-3-cosec-2-0-sec-2-0
slug: what-is-the-value-of-3-cosec-2-0-sec-2-0-52919
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ} ?$</div>

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

answer:
<div class="solution">$\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}$<br/>$=\frac{\sqrt{3}}{\sin 20^{\circ}}-\frac{1}{\cos 20^{\circ}}=\frac{\sqrt{3} \cos 20^{\circ}-\sin 20^{\circ}}{\sin 20^{\circ} \cos 20^{\circ}}$<br/>$=\frac{4 \cdot \frac{1}{2}\left(\sqrt{3} \cos 20^{\circ}-\sin 20^{\circ}\right)}{2 \sin 20^{\circ} \cos 20^{\circ}}$<br/>$=\left(\frac{\sqrt{3}}{2} \cos 20^{\circ}-\frac{1}{2} \sin 20^{\circ}\right)\left(\frac{4}{2 \sin 20^{\circ} \cos 20^{\circ}}\right)$<br/>$=\left(\sin 60^{\circ} \cos 20^{\circ}-\cos 60^{\circ} \sin 20^{\circ}\right)\left(\frac{4}{\sin 40^{\circ}}\right)$<br/>$=\sin 40^{\circ} \frac{4}{\sin 40^{\circ}}=4$<br/>$(\because \sin (A-B)=\sin A \cos B-\cos A \sin B)$</div>


title: what-is-the-value-of-3-cosec-20circsec-20circ
slug: what-is-the-value-of-3-cosec-20circsec-20circ-25810
question: <div class="question">What is the value of $\sqrt{3}$ cosec $20^{\circ}-\sec 20^{\circ} ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Consider $\sqrt{3}$ cosec $20^{\circ}-\sec 20^{\circ}$<br/>$=\frac{\sqrt{3}}{\sin 20^{\circ}}-\frac{1}{\cos 20^{\circ}}=\frac{\sqrt{3} \cos 20^{\circ}-\sin 20^{\circ}}{\sin 20^{\circ} \cos 20^{\circ}}$<br/>Multiply and divide by 4 $=\frac{4}{2 \sin 20^{\circ} \cos 20^{\circ}}\left(\frac{\sqrt{3}}{2} \cos 20^{\circ}-\frac{1}{2} \sin 20^{\circ}\right)$<br/>$=\frac{4}{\sin 40^{\circ}}\left(\frac{\sqrt{3}}{2} \cos 20^{\circ}-\frac{1}{2} \sin 20^{\circ}\right)$<br/>$(\because \sin 2 \theta=2 \sin \theta \cos \theta)$<br/>$=\frac{4}{\sin 40^{\circ}}\left(\sin 60^{\circ} \cos 20^{\circ}-\cos 60^{\circ} \sin 20^{\circ}\right)$<br/>$=\frac{4}{\sin 40^{\circ}} \sin \left(60^{\circ}-20^{\circ}\right)$<br/>$=4$<br/>$(\because \sin (\mathrm{A}-\mathrm{B})=\sin A \cos B-\cos A \sin B)$</div>


title: what-is-the-value-of-4-4-sin-tan-x-dx
slug: what-is-the-value-of-4-4-sin-tan-x-dx-20801
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}(\sin \times \tan x) \mathrm{dx} ? \quad$</div>

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

answer:
<div class="solution">$\begin{aligned} \text { (c) } &amp; \int_{-\pi}^{\frac{\pi}{4}}(\sin x-\tan x) d x . \\ &amp; \text { Let } f(x)=\sin x-\tan x \\ &amp; f(-x)=\sin (-x)-\tan (-x) \\ &amp;=-\sin x+\tan x=-(\sin x-\tan x) \\ &amp;=-f(x) . \end{aligned}$<br/>So, $f(x)$ is an odd function.<br/>We know $\int_{-a}^{a} f(x) d x=0$, if $f(x)$ is odd function.</div>


title: what-is-the-value-of-5-5-5
slug: what-is-the-value-of-5-5-5-67503
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{\ldots \infty}}}} ?$</div>

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

answer:
<div class="solution">Let $\mathrm{x}=\sqrt{5 \sqrt{5 \sqrt{5 \ldots \ldots \infty}}}$<br/>$\Rightarrow x=\sqrt{5 x} \Rightarrow x^{2}=5 x$<br/>$x^{2}-5 x=0 \Rightarrow x(x-5)=0$<br/>$\Rightarrow x=0$ or 5<br/>5 is given in the option.</div>


title: what-is-the-value-of-6-4-s-i-n-x-c-o-s-x-d-x
slug: what-is-the-value-of-6-4-s-i-n-x-c-o-s-x-d-x-15987
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\int_{\pi / 6}^{\pi / 4} \frac{d x}{\sin x \cos x} ? \quad\mathrm{}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \ln 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \ln \sqrt{3}$</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><li class=""> <span class="option-label">C</span> <span class="option-data">$\ln \sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \ln 3$</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{I}=\int_{\pi / 6}^{\pi / 4} \frac{d x}{\sin x \cos x}$<br/>multiply and divide by 2<br/>$=2 \int_{\pi / 6}^{\pi / 4} \frac{d x}{2 \sin x \cos x}=2 \int_{\pi / 6}^{\pi / 4} \frac{d x}{\sin 2 x}$<br/>$=2 \int_{\pi / 6}^{\pi / 4} \cos e c 2 x d x=2[\log \tan x]_{\pi / 6}^{\pi / 4} \cdot \frac{1}{2}$<br/>$=[\log \tan \pi / 4-\log \tan \pi / 6]$<br/>$=\log 1-\log \frac{1}{\sqrt{3}}=0-\log \frac{1}{\sqrt{3}}$<br/>$=\log \sqrt{3}(\because \log 1=0)$</div>


title: what-is-the-value-of-8-2-8-2-8-2-8
slug: what-is-the-value-of-8-2-8-2-8-2-8-38980
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\sqrt{8+2 \sqrt{8+2 \sqrt{8+2 \sqrt{8+\ldots .}}}}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $y=\sqrt{8+2 y}$<br/>$\Rightarrow y^{2}=8+2 y \Rightarrow y^{2}-2 y-8=0$<br/>$\Rightarrow(y+2)(y-4)=0$<br/>$\Rightarrow y=4,-2$<br/>Hence, required value of given expression is $4 .$</div>


title: what-is-the-value-of-8-c-0-8-c-1-8-c-2-8-c-3-8-c-4-8-c-5-8-c-6-8-c-7-8-c-8
slug: what-is-the-value-of-8-c-0-8-c-1-8-c-2-8-c-3-8-c-4-8-c-5-8-c-6-8-c-7-8-c-8-66764
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>${ }^{8} C_{0}-{ }^{8} C_{1}+{ }^{8} C_{2}-{ }^{8} C_{3}+{ }^{8} C_{4}-{ }^{8} C_{5}+{ }^{8} C_{6}-{ }^{8} C_{7}+{ }^{8} C_{8}$</div>

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

answer:
<div class="solution">$\begin{array}{ll}(1-x)^{\mathrm{n}}={ }^{n} \mathrm{C}_{0}-{ }^{n} \mathrm{C}_{1}(x)+{ }^{n} \mathrm{C}_{2} x^{2}-{ }^{n} \mathrm{C}_{3} x^{3} &amp; +\ldots \\ &amp; +(-1)^{n n} \mathrm{C}_{n} \\ &amp; \text { Put } x=1 \text { and } n=8 \\ &amp; \therefore(1-1)^{8}={ }^{8} \mathrm{C}_{0}-{ }^{8} \mathrm{C}_{1}+{ }^{8} \mathrm{C}_{2}-{ }^{8} \mathrm{C}_{3}+\ldots+{ }^{8} \mathrm{C}_{8} \\ \Rightarrow &amp; \left({ }^{8} \mathrm{C}_{0}-{ }^{8} \mathrm{C}_{1}+{ }^{8} \mathrm{C}_{2}-{ }^{8} \mathrm{C}_{3}+\ldots+{ }^{8} \mathrm{C}_{8}\right)=0\end{array}$</div>


title: what-is-the-value-of-9-13-9-119-9-127
slug: what-is-the-value-of-9-13-9-119-9-127-14781
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $9^{1 / 3} .9^{1 / 19} \cdot 9^{1 / 27} \ldots \ldots \infty$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9^{1 / 3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">The given expression $9^{1 / 3}, 9^{1 / 9} 9^{1 / 27} \ldots \infty$ Can be written as:<br/>$9^{\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots \infty}=9^{\frac{1}{3}}{1-\frac{1}{3}}=9^{\frac{1}{3} / \frac{2}{3}}=9^{1 / 2}=3$</div>


title: what-is-the-value-of-0101-2-10-2-0101-2-01-2-0011-2-01-2-0011-2-10-2-0101-2-11-2-0011-2-11-2
slug: what-is-the-value-of-0101-2-10-2-0101-2-01-2-0011-2-01-2-0011-2-10-2-0101-2-11-2-0011-2-11-2-70349
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\frac{(0.101)_{2}^{(11)_{2}}+(0.011)_{2}^{(11)_{2}}}{(0.101)_{2}^{(10)_{2}}-(0.101)_{2}^{(01)_{2}}(0.011)_{2}^{(01)_{2}}+(0.011)_{2}^{(10)_{2}}}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(0.001)_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(0.01)_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(0.1)_{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(1)_{2}$</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">$(1)_{2}$</span> </div>

answer:
<div class="solution">$(0.101)_{2}=2^{-1} \times 1+2^{-2} \times 0+2^{-3} \times 1$<br/>$=\frac{1}{2}+0+\frac{1}{8}=\frac{5}{8}$<br/>and $(0.011)_{2}=0 \times 2^{-1}+1 \times 2^{-2}+1 \times 2^{-3}$<br/>$=0+\frac{1}{4}+\frac{1}{8}=\frac{3}{8}$<br/>Also, $(11)_{2}=1 \times 2^{1}+1 \times 2^{0}=3$<br/>and $(01)_{2}=0 \times 2^{1} \times 1 \times 2^{0}=1$<br/>$\frac{(0.101)_{2}^{(11)}+(0.011)_{2}^{(11)_{2}}}{(0.101)_{2}^{(10)_{2}}-(0.101)_{2}^{(0)]_{2}}(0.011)_{2}^{(01)}+(0.011)_{2}^{(10)_{2}}}$</div>


title: what-is-the-value-of-1001-2-10-2-1001-2-0-2-101-2-01-2-101-2-10-2-1001-2-11-2-101-2-11-2
slug: what-is-the-value-of-1001-2-10-2-1001-2-0-2-101-2-01-2-101-2-10-2-1001-2-11-2-101-2-11-2-66892
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of<br/>$\frac{(1001)_{2}^{(11)_{2}}-(101)_{2}^{(11)_{2}}}{(1001)_{2}^{(10)_{2}}+(1001)_{2}^{(0) / 2}(101)_{2}^{(01) / 2}+(101)_{2}^{(10) / 2}} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(1001)_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(101)_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(110)_{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(100)_{2}$</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">$(100)_{2}$</span> </div>

answer:
<div class="solution">Converting from binary to decimal $(1001)_{2}=1 \times 2^{3}+2^{0}=8+1=9$<br/>$\begin{array}{l}(11)_{2}=2^{1}+2^{0}=2+1=3 \\ (101)_{2} \\ (10)_{2}=2^{1}=2\end{array} \quad=2^{2}+2^{0}=4+1=5$<br/>and $(01)_{2}=1$<br/>$\frac{(1001)_{2}^{(11)_{2}}-(101)_{2}^{(11)_{2}}}{(1001)_{2}^{(10)_{2}}+(1001)_{2}^{(01)_{2}}(101)_{2}^{(01)_{2}}+(101)_{2}^{(10)} 2}$<br/>$=\frac{9^{3}-5^{3}}{9^{2}+9 \times 5+5^{2}}$<br/>$=\frac{(9-5)\left(9^{2}+9 \times 5+5^{2}\right)}{\left(9^{2}+9 \times 5+5^{2}\right)}=\frac{4 \times(81+45+25)}{(81+45+25)}$<br/>$=4=(100)_{2}$ [Converting from decimal to binary]</div>


title: what-is-the-value-of-a-b-c-b-c-a-c-a-b-if-a-3-b-3-c-3-0
slug: what-is-the-value-of-a-b-c-b-c-a-c-a-b-if-a-3-b-3-c-3-0-30489
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\left|\begin{array}{lll}a &amp; b &amp; c \\ b &amp; c &amp; a \\ c &amp; a &amp; b\end{array}\right|$ if $a^{3}+b^{3}+c^{3}=0 ?$</div>

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

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

answer:
<div class="solution">$\left|\begin{array}{lll}\mathrm{a} &amp; \mathrm{b} &amp; \mathrm{c} \\ \mathrm{b} &amp; \mathrm{c} &amp; \mathrm{a} \\ \mathrm{c} &amp; \mathrm{a} &amp; \mathrm{b}\end{array}\right|$<br/>$=\mathrm{a}\left(\mathrm{bc}-\mathrm{a}^{2}\right)-\mathrm{b}\left(\mathrm{b}^{2}-\mathrm{ac}\right)+\mathrm{c}\left(\mathrm{ab}-\mathrm{c}^{2}\right)$<br/>$=\mathrm{abc}-\mathrm{a}^{3}-\mathrm{b}^{3}+\mathrm{abc}+\mathrm{abc}-\mathrm{c}^{3}=3 \mathrm{abc}-\left(\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}\right)$<br/>Given that $\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}=0$<br/>$\Rightarrow\left|\begin{array}{lll}\mathrm{a} &amp; \mathrm{b} &amp; \mathrm{c} \\ \mathrm{b} &amp; \mathrm{c} &amp; \mathrm{a} \\ \mathrm{c} &amp; \mathrm{a} &amp; \mathrm{b}\end{array}\right|=3 \mathrm{abc} .$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-a-given-a-function-f-x-1-a-x-b-1-if-if-if-x-0-0-x-1-x-1-where-a-b-are-constants-the-function-is-continuous-everywhere

slug:
what-is-the-value-of-a-given-a-function-f-x-1-a-x-b-1-if-if-if-x-0-0-x-1-x-1-where-a-b-are-constants-the-function-is-continuous-everywhere-59359

question:
<div class="question">What is the value of a?<br/>Given a function<br/>$f(x)=\left\{\begin{array}{ccc}-1 &amp; \text { If } &amp; x \leq 0 \\ a x+b &amp; \text { If } &amp; 0   &lt; x &lt; 1 \\ 1 &amp; \text { If } &amp; x \geq 1\end{array}\right.$<br/>where a, b are constants. The function is continuous everywhere.</div>

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

answer:
<div class="solution">$F(x)=\left\{\begin{array}{ccc}-1 &amp; , &amp; x \leq 0 \\ a x+b &amp; , &amp; 0   &lt; x &lt; 1 \\ 1 &amp; , &amp; x \geq 1\end{array}\right.$<br/>At $\mathrm{x}=0$,<br/>L.H.L. $=$ R.H.L.<br/>$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)$<br/>$-1=\lim _{x \rightarrow 0}(a x+b)$<br/>$-1=\mathrm{b}$<br/>Since the given function is also continuous at $\mathrm{x}=1$<br/>L.H.L. $=$ R.H.L.<br/>$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$<br/>$\mathrm{a}+\mathrm{b}=1$<br/>$\mathrm{a}-1=1$<br/>$\mathrm{a}=1+1=2 .$<br/>Function is continuous at $\mathrm{x}=0$<br/>$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)$<br/>$-1=a(0)+b \Rightarrow b=-1$<br/>Now, $\mathrm{f}(\mathrm{x})$ is also continuous at $\mathrm{x}=1$<br/>$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$<br/>$a(1)+b=1$<br/>$a=1-b=1+1$<br/>$\mathrm{a}=2 .$</div>


title: what-is-the-value-of-a-given-that-lim-x-1-x-2-x-2-a-x-b-3
slug: what-is-the-value-of-a-given-that-lim-x-1-x-2-x-2-a-x-b-3-17462
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of A?<br/>Given that $\lim _{x \rightarrow \infty}\left(\frac{2+x^{2}}{1+x}-A x-B\right)=3$.</div>

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

answer:
<div class="solution">$\lim _{x \rightarrow \infty}\left(\frac{2+x^{2}}{1+x}-A x-B\right)=3$<br/>$\Rightarrow \quad \lim _{x \rightarrow \infty}\left(\frac{2+x^{2}-A x-B-A x^{2}-B x}{1+x}\right)=3$<br/>$\Rightarrow \quad \lim _{x \rightarrow \infty}\left[\frac{(1-A) x^{2}-(A+B) x+2-B}{1+x}\right]=3$<br/>Applying L'Hospital rule, $2 x(1-A)-(A+B)=3$<br/>Comparing coefficients $\begin{array}{ll} &amp; 2(1-A)=0 \\ \therefore &amp; A=1 \text { and } \\ &amp; -(A+B)=3 \\ \therefore &amp; B=-3-1=-4 \\ &amp; A+B=-3 \\ &amp; A=1, B=-4\end{array}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-b-given-a-function-f-x-1-a-x-b-1-if-if-if-x-0-0-x-1-x-1-where-a-b-are-constants-the-function-is-continuous-everywhere

slug:
what-is-the-value-of-b-given-a-function-f-x-1-a-x-b-1-if-if-if-x-0-0-x-1-x-1-where-a-b-are-constants-the-function-is-continuous-everywhere-60947

question:
<div class="question">What is the value of b?<br/>Given a function<br/>$f(x)=\left\{\begin{array}{ccc}-1 &amp; \text { If } &amp; x \leq 0 \\ a x+b &amp; \text { If } &amp; 0   &lt; x &lt; 1 \\ 1 &amp; \text { If } &amp; x \geq 1\end{array}\right.$<br/>where a, b are constants. The function is continuous everywhere.</div>

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

answer:
<div class="solution">$F(x)=\left\{\begin{array}{ccc}-1 &amp; , &amp; x \leq 0 \\ a x+b &amp; , &amp; 0   &lt; x &lt; 1 \\ 1 &amp; , &amp; x \geq 1\end{array}\right.$<br/>At $\mathrm{x}=0$,<br/>L.H.L. $=$ R.H.L.<br/>$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)$<br/>$-1=\lim _{x \rightarrow 0}(a x+b)$<br/>$-1=\mathrm{b}$<br/>Since the given function is also continuous at $\mathrm{x}=1$<br/>L.H.L. $=$ R.H.L.<br/>$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$<br/>$\mathrm{a}+\mathrm{b}=1$<br/>$\mathrm{a}-1=1$<br/>$\mathrm{a}=1+1=2$.<br/>$\mathrm{b}=-1$</div>


title: what-is-the-value-of-b-given-that-lim-x-1-x-2-x-2-a-x-b-3
slug: what-is-the-value-of-b-given-that-lim-x-1-x-2-x-2-a-x-b-3-90949
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\mathrm{B}$ ?<br/>Given that $\lim _{x \rightarrow \infty}\left(\frac{2+x^{2}}{1+x}-A x-B\right)=3$.</div>

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

answer:
<div class="solution">$\lim _{x \rightarrow \infty}\left(\frac{2+x^{2}}{1+x}-A x-B\right)=3$<br/>$\Rightarrow \quad \lim _{x \rightarrow \infty}\left(\frac{2+x^{2}-A x-B-A x^{2}-B x}{1+x}\right)=3$<br/>$\Rightarrow \quad \lim _{x \rightarrow \infty}\left[\frac{(1-A) x^{2}-(A+B) x+2-B}{1+x}\right]=3$<br/>Applying L'Hospital rule, $2 x(1-A)-(A+B)=3$<br/>Comparing coefficients $\begin{array}{ll} &amp; 2(1-A)=0 \\ \therefore &amp; A=1 \text { and } \\ &amp; -(A+B)=3 \\ \therefore &amp; B=-3-1=-4 \\ &amp; A+B=-3 \\ &amp; A=1, B=-4\end{array}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-value-of-b-such-that-the-scalar-product-of-the-vector-i-j-k-with-the-unit-vector-parallel-to-the-sum-of-the-vectors-2-i-4-j-5-k-and-b-i-2-

slug:
what-is-the-value-of-b-such-that-the-scalar-product-of-the-vector-i-j-k-with-the-unit-vector-parallel-to-the-sum-of-the-vectors-2-i-4-j-5-k-and-b-i-2-10931

question:
<div class="question">What is the value of $\mathrm{b}$ such that the scalar product of the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ with the unit vector parallel to the sum of<br/>the vectors $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ and $\mathrm{b} \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is unity?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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>

answer:
<div class="solution">Let $\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{B}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$<br/>and $\overrightarrow{\mathrm{C}}=\mathrm{b} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$<br/>Now, $\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}+\mathrm{b} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$<br/>$=(2+\mathrm{b}) \hat{\mathrm{i}}+6 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$<br/>The unit vector parallel to $\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}$ is<br/>$\hat{n}=\frac{(2+b) \hat{i}+6 \hat{j}-2 \hat{k}}{\sqrt{(2+b)^{2}+6^{2}+(-2)^{2}}}=\frac{(2+b) \hat{i}+6 \hat{j}-2 \hat{k}}{\sqrt{b^{2}+4 b+44}}$<br/>Now, $(\hat{i}+\hat{j}+\hat{k}) \cdot \hat{n}=1 \Rightarrow \frac{2+b+6-2}{\sqrt{b^{2}+4 b+44}}=1$<br/>$\Rightarrow 2+b+6-2=\sqrt{b^{2}+4 b+44}$<br/>$\Rightarrow \quad 8 \mathrm{~b}=8$<br/>$\Rightarrow \quad b=1$</div>


title: what-is-the-value-of-c-o-s-1-0-s-i-n-1-0-c-o-s-1-0-s-i-n-1-0
slug: what-is-the-value-of-c-o-s-1-0-s-i-n-1-0-c-o-s-1-0-s-i-n-1-0-87496
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\frac{\cos 10^{\circ}-\sin 10^{\circ}}{\cos 10^{\circ}+\sin 10^{\circ}}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\tan 35^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\tan 10^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan 35^{\circ}$</span> </div>

answer:
<div class="solution">Given expression,<br/>$\frac{\cos 10^{\circ}-\sin 10^{\circ}}{\cos 10^{\circ}+\sin 10^{\circ}}=\frac{1-\tan 10^{\circ}}{1+\tan 10^{\circ}}=\mathrm{x} \quad$ (let)<br/>$x=\frac{\tan 45-\tan 10^{\circ}}{1+\tan 45 \cdot \tan 10^{\circ}}=\tan (45-10)=\tan 35^{\circ} .$</div>


title: what-is-the-value-of-c-o-s-2-0-s-i-n-1-0-c-o-s-1-0-s-i-n-2-0
slug: what-is-the-value-of-c-o-s-2-0-s-i-n-1-0-c-o-s-1-0-s-i-n-2-0-82964
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the value of $\frac{\left(\cos 10^{\circ}+\sin 20^{\circ}\right)}{\left(\cos 20^{\circ}-\sin 10^{\circ}\right)} ?$</div>

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

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

answer:
<div class="solution">The given expression is, $\frac{\cos 10^{\circ}+\sin 20^{\circ}}{\cos 20^{\circ}-\sin 10^{\circ}}$<br/>$=\frac{\cos \left(90^{\circ}-80^{\circ}\right)+\sin 20^{\circ}}{\operatorname{cso}\left(90^{\circ}-70^{\circ}\right)-\sin 10^{\circ}}$<br/>$=\frac{\sin 80^{\circ}+\sin 20^{\circ}}{\sin 70^{\circ}-\sin 10^{\circ}}=\frac{2 \sin \frac{80+20}{2} \cdot \cos \frac{80-20}{2}}{2 \cos \frac{70+10}{2} \cdot \sin \frac{70-10}{2}}$<br/>$=\frac{2 \sin 50^{\circ} \cos 30^{\circ}}{2 \cos 40^{\circ} \sin 30^{\circ}}=\frac{\sin \left(90^{\circ}-40^{\circ}\right) \cot 30^{\circ}}{\cos 40^{\circ}}$<br/>$=\frac{\cos 40^{\circ} \cot 30^{\circ}}{400}=\cot 30^{\circ}=\sqrt{3}$</div>


