# questions

slug: 0-0-0-a-0-0-0-b-0-and-0-0-c-are-four-distinct-points-what-are-t-99128
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch0.db
title:
( 0 , 0 , 0 ) , ( a , 0 , 0 ) , ( 0 , b , 0 ) and ( 0 , 0 , c ) are four distinct points. What are the coordinates of the point which is equidistant from the four points?

question:
<div class="question">$(0,0,0),(\mathrm{a}, 0,0),(0, \mathrm{~b}, 0)$ and $(0,0, \mathrm{c})$ are four distinct points. What are the coordinates of the point which is equidistant from the four points?</div>

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

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

answer:
<div class="solution">$\mathrm{A}(0,0,0), \mathrm{B}(\mathrm{a}, 0,0), \mathrm{C}(0, \mathrm{~b}, 0), \mathrm{D}(0,0, \mathrm{c})$<br/>Let the equidistant point be $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$<br/>i.e, $\mathrm{AP}=\mathrm{BP}, \mathrm{AP}=\mathrm{CP}, \mathrm{AP}=\mathrm{DP}$<br/>$\Rightarrow \sqrt{(x-0)^{2}+(y-0)^{2}+(z-0)^{2}}$<br/>$=\sqrt{(x-a)^{2}+y^{2}+z^{2}}$<br/>$\Rightarrow x^{2}+y^{\prime}+z^{\prime}=(x-a)^{2}+y^{\prime}+z^{\prime}$<br/>$\Rightarrow x^{2}=x^{2}-2 a x+a^{2}$<br/>$\Rightarrow a^{2}-2 a x=0$<br/>$\Rightarrow a(a-2 x)=0$<br/>Since $a \neq 0, a=2 x \Rightarrow x=\frac{a}{2}$<br/>Similarly, we will get $\mathrm{y}=\frac{\mathrm{b}}{2}, \mathrm{z}=\frac{\mathrm{c}}{2}$</div>


title: â« 0 2 a â f ( x ) d x =
slug: 0-2-a-f-x-d-x-73311
question: <div class="question">$\int_0^{2 a} f(x) d x=$</div>
tags: ['Mathematics', 'Definite Integration', 'JEE Main']
answer: <div class="solution">It is a fundamental property.</div>
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \int_0^a f(x) d x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\int_0^a f(x) d x+\int_0^a f(2 a-x) d x$</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">D</span> <span class="option-data">$\int_0^a f(x) d x+\int_0^{2 a} f(2 a-x) d x$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\int_0^a f(x) d x+\int_0^a f(2 a-x) d x$</span> </div>


title: â« 0 Ï /2 â e s i n x cos x d x is equal to
slug: 0-2-e-s-i-n-x-cos-x-d-x-is-equal-to-19146
question: <div class="question">$\int_{0}^{\pi / 2} e^{\sin } x \cos x d x$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$e-1$</span> </div>
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$e+1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$e-1$</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">$e+2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$e$</span> </li> </ul>

answer:
<div class="solution">$\int_{0}^{\frac{\pi}{2}} e^{\sin x} \cdot \cos x d x$<br/>Let $\sin x=t \Rightarrow \cos x \cdot d x=d t$<br/>$\therefore \int_{0}^{\frac{\pi}{2}} e^{\sin x} \cdot \cos x d x=\int_{0}^{1} e^{t} . d t$<br/>$=\left(e^{t}\right)_{0}^{1}=e^{1}-e^{0}=e-1$</div>


title: â« 0 2 Ï â sin 5 ( 4 x â ) dx is equal to
slug: 0-2-sin-5-4-x-dx-is-equal-to-26986
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">$\int_{0}^{2 \pi} \sin ^{5}\left(\frac{\mathrm{x}}{4}\right) \mathrm{d} \mathrm{x}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{8}{15}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{16}{15}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{32}{15}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{32}{15}$</span> </div>

answer:
<div class="solution">$\int_{0}^{2 \pi} \sin ^{5}\left(\frac{x}{4}\right) d x=\int_{0}^{2 \pi}\left(1-\cos ^{2} \frac{x}{4}\right)\left(1-\cos ^{2} \frac{x}{4}\right) \sin \frac{x}{4} d x$<br/>Put $\cos \left(\frac{x}{4}\right)=t$<br/>$\Rightarrow-\sin \left(\frac{x}{4}\right) \cdot \frac{d x}{4}=d t$<br/>$\Rightarrow \sin \left(\frac{x}{4}\right) d x=-4 d t$<br/>$\Rightarrow \int_{0}^{2 \pi} \sin ^{5}\left(\frac{x}{4}\right) d x=-4 \int\left(1-t^{2}\right)\left(1-t^{2}\right) \mathrm{dt}$<br/>$=-4 \int\left(1+t^{4}-2 t^{2}\right) \mathrm{dt}$<br/>$=-4\left[t+\frac{t^{5}}{5}-\frac{2 t^{3}}{3}\right]$<br/>$=-4\left[\cos \left(\frac{x}{4}\right)+\frac{\cos ^{5}\left(\frac{x}{4}\right)}{5}-\frac{ \left.2 \cos ^{3}\left(\frac{x}{4}\right)\right]_{0}^{2 \pi}}{3}\right]$<br/>$=-4\left[(0+0-0)-\left(1+\frac{1}{5}-\frac{2}{3}\right)\right]=\frac{32}{15}$</div>


title: â« 0 Ï /2 â â£ sin x â cos x â£ d x is equal to.
slug: 0-2-sin-x-cos-x-d-x-is-equal-to-16778
question: <div class="question">$\int_{0}^{\pi / 2}|\sin x-\cos x| d x$ is equal to.</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch0.db
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">$2(\sqrt{2}-1)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \sqrt{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2(\sqrt{2}+1)$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp; \int_{0}^{\pi / 2}|\sin x-\cos x| d x \\ &amp;=\int_{0}^{\pi / 4}(\sin x-\cos x) \cdot d x+\int_{\pi / 4}^{\pi / 2}(\sin x-\cos x) d x \\ &amp;=\int_{0}^{\pi / 4}(\cos x-\sin x) \cdot d x+\int_{\pi / 4}^{\pi / 2}(\sin x-\cos x) \cdot d x \\ &amp;=(\sin x+\cos x)_{0}^{\pi / 4}+(-\cos x-\sin x)_{\pi / 4}^{\pi / 2} \\ &amp;=2 \sqrt{2}-2=2(\sqrt{2}-1) \end{aligned}$</div>


title: â« 0 a â f ( x ) d x =
slug: 0-a-f-x-d-x-91690
question: <div class="question">$\int_0^a f(x) d x=$</div>
tags: ['Mathematics', 'Definite Integration', 'JEE Main']
answer: <div class="solution">It is a fundamental property.</div>
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\int_0^a f(a+x) d x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\int_0^a f(2 a+x) d x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\int_0^a f(x-a) d x$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\int_0^a f(a-x) d x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\int_0^a f(a-x) d x$</span> </div>


title: â« â 1 1 â x â£ x â£ d x is equal to
slug: 1-1-x-x-d-x-is-equal-to-45768
question: <div class="question">$\int_{-1}^{1} x|x| d x$ is equal to</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch0.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">$\frac{2}{3}$</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">$-2$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \int_{-1}^{1} x|x| d x \\ &amp;=\int_{-1}^{0} x|x| d x+\int_{0}^{1} x|x| d x \\ &amp;=\int_{-1}^{0} x(-x) d x+\int_{0}^{1} x \cdot x d x \\ &amp;=-\int_{-1}^{0} x^{2} d x+\int_{0}^{1} x^{2} d x \\ &amp;=-\left[\frac{x^{3}}{3}\right]_{-1}^{0}+\left[\frac{x^{3}}{3}\right]_{0}^{1} \\ &amp;=-\left[0-\frac{(-1)^{3}}{3}\right]+\frac{1}{3}\left[(1)^{3}-(0)^{3}\right] \\ &amp;=-\frac{1}{3}+\frac{1}{3}=0 \end{aligned}$</div>


title: 1 4 - 5 4 Â· 8 + 5 Â· 9 4 Â· 8 Â· 12 - â¦ =
slug: 1-4-5-4-8-5-9-4-8-12-40267
tags: ['Mathematics', 'Sequences and Series', 'AP EAMCET', 'AP EAMCET 2018 (25 Apr Shift 1)']
source_db: MarksBatch0.db
question:
<div class="question"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>-</mo><mfrac><mn>5</mn><mrow><mn>4</mn><mo>·</mo><mn>8</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><mo>·</mo><mn>9</mn></mrow><mrow><mn>4</mn><mo>·</mo><mn>8</mn><mo>·</mo><mn>12</mn></mrow></mfrac><mo>-</mo><mo>…</mo><mo>=</mo></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>2</mn><msqrt><mn>5</mn></msqrt></mrow><mrow><mn>9</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>9</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></msup><mo>-</mo><mn>1</mn></mrow><msup><mn>2</mn><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></msup></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow><mrow><mn>9</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></msup><mo>-</mo><mn>1</mn></mrow><msup><mn>2</mn><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></msup></mfrac></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>-</mo><mfrac><mn>5</mn><mrow><mn>4</mn><mo>·</mo><mn>8</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mn>5</mn><mo>·</mo><mn>9</mn></mrow><mrow><mn>4</mn><mo>·</mo><mn>8</mn><mo>·</mo><mn>12</mn></mrow></mfrac><mo>-</mo><mo>…</mo><mo>…</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>·</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>3</mn></msup><mo>-</mo><mo>…</mo><mo>…</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="]" open="["><mrow><mn>1</mn><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mo>-</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>·</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>3</mn></msup><mo>-</mo><mo>…</mo><mo>…</mo></mrow></mfenced><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>i</mi></mfenced></math></p><p>Now, we know that</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mi>p</mi><mi>q</mi></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mi>p</mi><mfenced><mfrac><mi>x</mi><mi>q</mi></mfrac></mfenced><mo>+</mo><mfrac><mrow><mi>p</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mi>x</mi><mi>q</mi></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo></math></p><p>Therefore, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>i</mi></mfenced></math>, we have</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="]" open="["><mrow><mn>1</mn><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mo>-</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>·</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>3</mn></msup><mo>-</mo><mo>…</mo><mo>…</mo></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfenced close="]" open="["><mrow><mn>1</mn><mo>-</mo><mn>1</mn><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mo>+</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mfrac><mrow><mn>1</mn><mo>·</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>·</mo><mn>4</mn></mrow></mfenced></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced><mn>3</mn></msup><mo>+</mo><mo>…</mo><mo>…</mo></mrow></mfenced></math></p><p>Comparing with the above standard series, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mfrac><mi>x</mi><mi>q</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>4</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>1</mn></math></p><p>Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mi>p</mi><mi>q</mi></mfrac></mrow></msup><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></msup><mo>=</mo><msup><mn>2</mn><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></msup></math></p><p>Therefore, required sum is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mfrac><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></msup><mo>-</mo><mn>1</mn></mrow><msup><mn>2</mn><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></msup></mfrac></mfenced></math></p></div>


title: â« 1 + e â x dx â is equal to
slug: 1-e-x-dx-is-equal-to-42072
question: <div class="question">$\int \frac{\mathrm{dx}}{1+\mathrm{e}^{-\mathrm{x}}}$ is equal to</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+\mathrm{e}^{\mathrm{x}}+\mathrm{c}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\ln \left(1+\mathrm{e}^{-x}\right)+\mathrm{c}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\ln \left(1+\mathrm{e}^{x}\right)+\mathrm{c}$</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">D</span> <span class="option-data">$2 \ln \left(1+\mathrm{e}^{-x}\right)+\mathrm{c}$<br/>where $c$ is the constant of integration</span> </li> </ul>

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

answer:
<div class="solution">$\int \frac{\mathrm{dx}}{1+\mathrm{e}^{-\mathrm{x}}}$<br/>$\Rightarrow \int \frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{\mathrm{x}}+1} \mathrm{dx}$<br/>Let $\mathrm{e}^{\mathrm{x}}+1=\mathrm{t}$<br/>$\mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$<br/>$=\int \frac{\mathrm{dt}}{\mathrm{t}}$<br/>$\Rightarrow \log \mathrm{t}+\mathrm{c} \Rightarrow \log \left(\mathrm{e}^{\mathrm{x}}+1\right)+\mathrm{c}$</div>


title: 1 + sin A â = â ( sin 2 A â + cos 2 A â ) is true if
slug: 1-sin-a-sin-2-a-cos-2-a-is-true-if-63199
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">$\sqrt{1+\sin \mathrm{A}}=-\left(\sin \frac{\mathrm{A}}{2}+\cos \frac{\mathrm{A}}{2}\right)$ is true if</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3 \pi}{2}   &lt; \mathrm{A} &lt; \frac{7 \pi}{2}$</span> </div>

answer:
<div class="solution">$\sqrt{1+\sin A}=-\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)$<br/>We know, $1+\sin A=\left(\cos \frac{A}{2}+\sin \frac{A}{2}\right)^{2}$<br/>$\therefore \sqrt{1+\sin A}=\left|\cos \frac{A}{2}+\sin \frac{A}{2}\right|$<br/>We know, $|x|=\left\{\begin{array}{l}x \text { if } x \geq 0 \\ -x \text { if } x   &lt; 0\end{array}\right.$<br/>$\sin \frac{A}{2}+\cos \frac{A}{2} \mid=\left\{\begin{array}{l}\sin \frac{A}{2}+\cos \frac{A}{2}, \quad \text { if } 2 n \pi-\frac{\pi}{4} \leq \frac{A}{2} \leq 2 n \pi+\frac{3 \pi}{4} \\ -\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right),\end{array}\right.$<br/>$\quad \therefore \sqrt{1+\sin A}=-\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)$ when<br/>$\frac{3 \pi}{4} &lt; \frac{A}{2} &lt; \frac{7 \pi}{4}$<br/>$\Rightarrow \frac{3 \pi}{2} &lt; A &lt; \frac{7 \pi}{2}$</div>


title: 1-Propanol can be distinguished from 2-propanol by which test?
slug: 1-propanol-can-be-distinguished-from-2-propanol-by-which-test-96404
question: <div class="question">1-Propanol can be distinguished from 2-propanol by which test?</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'TS EAMCET', 'TS EAMCET 2023 (13 May Shift 1)']
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2, 4-DNP test</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Tollen's test</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Lucas test</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Fehling's test</span> </li> </ul>

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

answer:
<div class="solution">Primary and Secondary alcohols can be distinguished by the Lucas test. It involves heating the alcohol with anhydrous $\mathrm{ZnCl}_2$ in concentrated $\mathrm{HCl}$.<br/>Turbidity is observed for $2^{\circ}$ alcohols after some time while no turbidity is observed for a $1^{\circ}$ alcohol.</div>


title: (1-\sin A+\cos A)^{2}$ is equal to
slug: 1-sin-a-cos-a-2-is-equal-to-70294
question: <div class="question">(1-\sin A+\cos A)^{2}$ is equal to</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2(1-\cos A)(1+\sin A)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2(1-\sin A)(1+\cos A)$</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">C</span> <span class="option-data">$2(1-\cos A)(1-\sin A)$</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">$2(1-\sin A)(1+\cos A)$</span> </div>

answer:
<div class="solution">$(1-\sin A+\cos A)^{2}$<br/>$=1+\sin ^{2} A+\cos ^{2} A-2 \sin A$<br/>$\quad-2 \sin A \cdot \cos A+2 \cos A$<br/>$=2-2 \sin A-2 \sin A \cos A+2 \cos A$<br/>$=2(1-\sin A)+2 \cos A(1-\sin A)$<br/>$=2(1+\cos A)(1-\sin A)$<br/>$\therefore \quad$ Option (b) is correct.</div>


title: 2, 3-dimethyl hexane contains tertiary...... secondary and primary carbon atoms, respectively:
slug: 2-3-dimethyl-hexane-contains-tertiary-secondary-and-primary-carbon-atoms-respectively-53198
tags: ['Chemistry', 'General Organic Chemistry', 'AP EAMCET', 'AP EAMCET 2003']
source_db: MarksBatch0.db
question:
<div class="question">2, 3-dimethyl hexane contains tertiary...... secondary and primary carbon atoms, respectively:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2, 2, 4$</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">$2,4,3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4,3,2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3,2,4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2, 2, 4$</span> </div>

answer:
<div class="solution">The structure of 2, 3-dimethyl hexane is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/yNSH8g1J6SMNAZKIMu-hxZNojqGutSQPLUZo6F-xZvI.original.fullsize.png"/><br/><br/>So, the number of tertiary carbon atoms $=2$<br/>The number of secondary carbon atoms $=2$<br/>The number of primary carbon atoms $=4$</div>


title: 2, 3-dimethyl hexane contains tertiary...... secondary and primary carbon atoms, respectively:
slug: 2-3-dimethyl-hexane-contains-tertiary-secondary-and-primary-carbon-atoms-respectively-80238
tags: ['Chemistry', 'General Organic Chemistry', 'TS EAMCET', 'TS EAMCET 2003']
source_db: MarksBatch0.db
question:
<div class="question">2, 3-dimethyl hexane contains tertiary...... secondary and primary carbon atoms, respectively:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2, 2, 4$</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">$2,4,3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4,3,2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3,2,4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2, 2, 4$</span> </div>

answer:
<div class="solution">The structure of 2, 3-dimethyl hexane is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/yNSH8g1J6SMNAZKIMu-hxZNojqGutSQPLUZo6F-xZvI.original.fullsize.png"/><br/><br/>So, the number of tertiary carbon atoms $=2$<br/>The number of secondary carbon atoms $=2$<br/>The number of primary carbon atoms $=4$</div>


slug: 2-a-g-a-q-c-u-s-c-u-2-a-q-2-a-g-s-the-standard-potential-e-for-this-r-14450
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch0.db
title:
2 A g + ( a q ) + C u ( s ) â C u 2 + ( a q ) + 2 A g ( s ) The standard potential E Â° for this reaction is 0.46 V. Which change will increase the potential the most?

question:
<div class="question"><math><mn>2</mn><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">C</mi><mi mathvariant="normal">u</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>â</mo><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">A</mi><mi mathvariant="normal">g</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></math><br/>
The standard potential <math><mi mathvariant="normal">E</mi><mo>Â°</mo></math> for this reaction is 0.46 V. Which change will increase the potential the most?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Doubling the <math><mo>[</mo><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>]</mo></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">Halving the <math><mo>[</mo><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>]</mo></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Doubling the size of the Cu(s) electrode</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Decreasing the size of the Ag electrode by one-half</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Doubling the <math><mo>[</mo><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>]</mo></math></span> </div>

answer:
<div class="solution"><math><mi mathvariant="normal">E</mi><mo>=</mo><mi mathvariant="normal">E</mi><mo>Â°</mo><mo>-</mo><mfrac><mrow><mn>0.059</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mfrac><mrow><mo>[</mo><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>]</mo></mrow><mrow><mo>[</mo><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math><br/>The cell potential will increase the most by doubling the concentration of <math><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>+</mo></mrow></msup></math> ion.</div>


slug: 2-o-3-g-3-o-2-g-at-300-k-ozone-is-fifty-percent-dissociated-the-standard-free-energy-change-a-62151
tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Main', 'JEE Main 2022 (24 Jun Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">747</span> </div>
source_db: MarksBatch0.db
title:
2 O 3 g â 3 O 2 g At 300 K , ozone is fifty percent dissociated. The standard free energy change at this temperature and 1 atm pressure is - . . . . . . J mol - 1 . (Nearest integer) [Given: ln 1 . 35 = 0 . 3 and R = 8 . 3 J K - 1 mol - 1 ]

question:
<div class="question"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>⇌</mo><mn>3</mn><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced></math></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><mi mathvariant="normal">K</mi></math>, ozone is fifty percent dissociated. The standard free energy change at this temperature and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>atm</mi></math> pressure is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>-</mo></mfenced><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi>mol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. (Nearest integer)</p><p>[Given: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mn>1</mn><mo>.</mo><mn>35</mn><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>3</mn><mo> </mo><mi mathvariant="normal">J</mi><msup><mrow><mo> </mo><mi mathvariant="normal">K</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mo> </mo><mi>mol</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>]</p></div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>⇌</mo><mn>3</mn><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced></math></p><p>Initially <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> mole <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> mole <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>75</mn></math> mole</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">K</mi><mi mathvariant="normal">p</mi></msub><mo>=</mo><mfrac><msup><mfenced><msub><mi mathvariant="normal">P</mi><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub></msub></mfenced><mn>3</mn></msup><msup><mfenced><msub><mi mathvariant="normal">P</mi><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></msub></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><msup><mfenced><mfrac><mrow><mn>0</mn><mo>.</mo><mn>75</mn></mrow><mrow><mn>1</mn><mo>.</mo><mn>25</mn></mrow></mfrac></mfenced><mn>3</mn></msup><msup><mfenced><mfrac><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow><mrow><mn>1</mn><mo>.</mo><mn>25</mn></mrow></mfrac></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><msup><mfenced><mfrac><mn>3</mn><mn>5</mn></mfrac></mfenced><mn>3</mn></msup><msup><mfenced><mfrac><mn>2</mn><mn>5</mn></mfrac></mfenced><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>6</mn></mrow></mfenced><mn>3</mn></msup><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><mfenced><mrow><mn>0</mn><mo>.</mo><mn>216</mn></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></mfenced></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>35</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔG</mi><mo>°</mo><mo>=</mo><mo>-</mo><msub><mi>RTlnK</mi><mi mathvariant="normal">p</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>3</mn><mo>×</mo><mn>300</mn><mi>ln</mi><mn>1</mn><mo>.</mo><mn>35</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>3</mn><mo>×</mo><mn>300</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>747</mn><mi mathvariant="normal">J</mi><mo>/</mo><mi>mole</mi></math></p></div>


slug: 2-x-2-3-x-0-has-roots-2-and-while-the-equation-x-2-3-m-x-2-m-2-0-has-bot-41618
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch0.db
title:
2 x 2 + 3 x â Î± â 0 has roots â 2 and Î² while the equation x 2 â 3 m x + 2 m 2 = 0 has both roots positive, where Î± > 0 and Î² > 0 . If Î² , 2 , 2 m are in GP, then what is the value of Î² m â ?

question:
<div class="question">$2 x^{2}+3 x-\alpha-0$ has roots $-2$ and $\beta$ while the equation $x^{2}-3 m x+$ $2 \mathrm{~m}^{2}=0$ has both roots positive, where $\alpha&gt;0$ and $\beta&gt;0$.<br/>If $\beta, 2,2 \mathrm{~m}$ are in GP, then what is the value of $\beta \sqrt{\mathrm{m}}$ ?</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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">$\beta=\frac{1}{2}$<br/>$\Rightarrow \frac{2}{\beta}=\frac{2 m}{2}$<br/>$\Rightarrow \mathrm{m}=\frac{2}{\beta}=2 \times \frac{2}{1}$<br/>$\Rightarrow \mathrm{m}=4$<br/>$\Rightarrow \beta \sqrt{m}=\frac{1}{2} \times \sqrt{4}=1$</div>


slug: 2-x-2-3-x-0-has-roots-2-and-while-the-equation-x-2-3-m-x-2-m-2-0-has-bot-39087
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch0.db
title:
2 x 2 + 3 x â Î± â 0 has roots â 2 and Î² while the equation x 2 â 3 m x + 2 m 2 = 0 has both roots positive, where Î± > 0 and Î² > 0 . What is the value of Î± ?

question:
<div class="question">$2 x^{2}+3 x-\alpha-0$ has roots $-2$ and $\beta$ while the equation $x^{2}-3 m x+$ $2 \mathrm{~m}^{2}=0$ has both roots positive, where $\alpha&gt;0$ and $\beta&gt;0$.<br/>What is the value of $\alpha ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 2$</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">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">4</span> </li> </ul>

answer:
<div class="solution">$2 x^{2}+3 x-\alpha=0$<br/>Its roots are: $-2 \&amp; \beta .$<br/>i.e., $\frac{-3}{2}=\beta-2 \Rightarrow \beta=2-\frac{3}{2}=\frac{1}{2} \Rightarrow \beta=\frac{1}{2}$<br/>$\frac{\alpha}{2}=2 \beta \Rightarrow \alpha=4 \times \frac{1}{2} \Rightarrow \alpha=2$</div>


tags: ['Chemistry', 'Coordination Compounds', 'JEE Main', 'JEE Main 2021 (27 Jul Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch0.db
title:
3 moles of metal complex with formula Co ( en ) 2 Cl 3 gives 3 moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of Co in the complex is (Round off to the nearest integer)

slug:
3-moles-of-metal-complex-with-formula-co-en-2-cl-3-gives-3-moles-of-silver-chloride-on-treatment-22099

question:
<div class="question"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> moles of metal complex with formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Co</mi><mo>(</mo><mi>en</mi><msub><mo>)</mo><mn>2</mn></msub><msub><mi>Cl</mi><mn>3</mn></msub></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Co</mi></math> in the complex is (Round off to the nearest integer)</div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced close="]" open="["><mrow><mi>Co</mi><mo>(</mo><mi>en</mi><msub><mo>)</mo><mn>2</mn></msub><msub><mi>Cl</mi><mn>2</mn></msub></mrow></mfenced><mi>Cl</mi><mo>+</mo><munder><msub><mi>AgNO</mi><mn>3</mn></msub><mtext> (excess)</mtext></munder><mo>→</mo><munder><mrow><mn>3</mn><mi>AgCl</mi></mrow><mrow><mtext> (white ppt. </mtext><mo>)</mo></mrow></munder></math></p><p>Secondary valency is equal to coordination number of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Co</mi><mo>=</mo><mn>6</mn></math> </p></div>


title: 3-Pentanol on reaction with aluminium tertiary butoxide in the presence of acetone gives
slug: 3-pentanol-on-reaction-with-aluminium-tertiary-butoxide-in-the-presence-of-acetone-gives-37014
tags: ['Chemistry', 'Aldehydes and Ketones', 'NEET']
source_db: MarksBatch0.db
question:
<div class="question">3-Pentanol on reaction with aluminium tertiary butoxide in the presence of acetone gives</div>

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

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

answer:
<div class="solution">Aluminium tertiary butoxide is an oxidising agent used for the oxidation of secondary alcohols into ketones.<br/><br/><img alt="" height="128" src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/EM0059861_Ans_1.jpg" width="546"/><br/> </div>


tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$m=n$</span> </div>
source_db: MarksBatch0.db
title:
5 books are to be chosen from a lot of 10 books. If m is the number of ways of choice when one specified book is always included and n is the number of ways of choice when a specified book is always excluded, then which one of the following is correct?

slug:
5-books-are-to-be-chosen-from-a-lot-of-10-books-if-m-is-the-number-of-ways-of-choice-when-one-speci-51293

question:
<div class="question">5 books are to be chosen from a lot of 10 books. If $m$ is the number of ways of choice when one specified book is always included and $n$ is the number of ways of choice when a specified book is always excluded, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$m&gt;n$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$m=n$</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">$m=n-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$m=n-2$</span> </li> </ul>

answer:
<div class="solution">Number of ways when one specified book is always included. So, 4 books can be chosen from remaining 9 ways:<br/>$=m={ }^{9} C_{4}$<br/>$\Rightarrow m=126$<br/>and number of ways when one specified book is always excluded $=n={ }^{9} C_{5}$<br/>$\Rightarrow n=126$<br/>$\Rightarrow m=n$</div>


title: 8 coins are tossed simultaneously. The probability of getting at least 6 heads is
slug: 8-coins-are-tossed-simultaneously-the-probability-of-getting-at-least-6-heads-is-46953
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">8 coins are tossed simultaneously. The probability of getting at least 6 heads is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{64}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{57}{64}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{37}{256}$</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">D</span> <span class="option-data">$\frac{229}{256}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{37}{256}$</span> </div>

answer:
<div class="solution">In tossing of coin getting $r$ head out of $n$ tossing<br/>$={ }^{n} C_{r} \cdot\left(\frac{1}{2}\right)^{n}$<br/>$\therefore \quad$ Required probability<br/>$=\left({ }^{8} \mathrm{C}_{6}+{ }^{8} \mathrm{C}_{7}+{ }^{8} \mathrm{C}_{8}\right)\left(\frac{1}{2}\right)^{8}$<br/>$=(28+8+1) \times \frac{1}{256}=\frac{37}{256}$</div>


tags: ['Chemistry', 'Electrochemistry', 'KCET', 'KCET 2012']
source_db: MarksBatch0.db
title:
100 mL of 0.1 M acetic acid is completely neutralized using a standard solution of NaOH . The volume of ethane obtained at STP after the complete electrolysis of the resulting solution is

slug:
100-ml-of-0-1-m-acetic-acid-is-completely-neutralized-using-a-standard-solution-of-naoh-the-volume-18724

question:
<div class="question">$100 \mathrm{~mL}$ of $0.1 \mathrm{M}$ acetic acid is completely neutralized using a standard solution of $\mathrm{NaOH}$. The volume of ethane obtained at STP after the complete electrolysis of the resulting solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$56 \mathrm{~mL}$.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$224 \mathrm{~mL}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$560 \mathrm{~mL}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$112 \mathrm{~mL}$.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$112 \mathrm{~mL}$.</span> </div>

answer:
<div class="solution">Acetic acid on electrolysis gives ethane as<br/>Given moles of acetic acid,<br/>$$<br/>\mathrm{CH}_{3} \mathrm{COOH}=\frac{0.1 \times 100}{1000}=0.01<br/>$$<br/>$\because 2 \mathrm{~mol} \mathrm{CH}_{3} \mathrm{COOH}$ gives $\mathrm{C}_{2} \mathrm{H}_{6}=22.4 \mathrm{~L}$<br/>$\therefore 0.01 \mathrm{~mol} \mathrm{CH} \mathrm{CHOH}_{3} \mathrm{COOH}$ give<br/>$$<br/>\begin{aligned}<br/>\mathrm{C}_{2} \mathrm{H}_{6} &amp;=\frac{22.4 \times 0.01}{2} \\<br/>&amp;=0.112 \mathrm{~L} \\<br/>&amp;=112 \mathrm{~mL}<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'AP EAMCET', 'AP EAMCET 2021 (23 Aug Shift 1)']
source_db: MarksBatch0.db
title:
100 mL of 0.2 M acetic acid is completely neutralised using a standard solution of NaOH . The volume of ethane obtained at STP after complete electrolysis of the resulting solution is

slug:
100-ml-of-0-2-m-acetic-acid-is-completely-neutralised-using-a-standard-solution-of-naoh-the-volume-20062

question:
<div class="question">$100 \mathrm{~mL}$ of $0.2 \mathrm{M}$ acetic acid is completely neutralised using a standard solution of $\mathrm{NaOH}$. The volume of ethane obtained at STP after complete electrolysis of the resulting solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$11.2 \mathrm{~L}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.24 \mathrm{~L}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.224 \mathrm{~L}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$22.4 \mathrm{~L}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.224 \mathrm{~L}$</span> </div>

answer:
<div class="solution">Given moles of acetic acid<br/>$$<br/>\begin{array}{r}<br/>=\frac{0.2 \times 100}{1000}=0.02 \text { moles } \\<br/>2 \mathrm{CH}_3 \mathrm{COOH} \longrightarrow \mathrm{C}_2 \mathrm{H}_6+2 \mathrm{CO}_2+\mathrm{H}_2<br/>\end{array}<br/>$$<br/>2 moles of $\mathrm{CH}_3 \mathrm{COOH}$ gives 1 mole of ethane $=22.4 \mathrm{~L}$<br/>0.02 moles of $\mathrm{CH}_3 \mathrm{COOH}$ will give ethane<br/>$$<br/>=\frac{22.4 \times 0.02}{2}=0.224 \mathrm{~L}<br/>$$<br/>$\therefore 0.224 \mathrm{~L}$ of ethane is obtained.</div>


tags: ['Chemistry', 'Thermodynamics (C)']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
18.0 g of water completely vaporizes at 10 0 â C and 1 bar pressure and the enthalpy change in the process is 40.79 kJ mol â 1 . What will be the enthalpy change for vaporizing two moles of water under the same conditions? What is the standard enthalpy of vaporization for water?

slug:
18-0-g-of-water-completely-vaporizes-at-10-0-c-and-1-bar-pressure-and-the-enthalpy-change-in-the-79603

question:
<div class="question">$18.0 \mathrm{~g}$ of water completely vaporizes at $100^{\circ} \mathrm{C}$ and 1 bar pressure and the enthalpy change in the process is $40.79 \mathrm{~kJ} \mathrm{~mol}^{-1}$. What will be the enthalpy change for vaporizing two moles of water under the same conditions? What is the standard enthalpy of vaporization for water?</div>

answer:
<div class="solution">$+81.58 \mathrm{~kJ}, \Delta_{\text {vap }} \mathrm{H}^{\circ}=+40.79 \mathrm{~kJ} \mathrm{~mol}^{-1}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch0.db
title:
18.0 g of water completely vapourises at 100Â°C and 1 bar pressure and the enthalpy change in the process is 40.79 kJ mol â1 . What will be the enthalpy change for vapourising two moles of water under the same conditions? What is the standard enthalphy of vapourisation for water?

slug:
18-0-g-of-water-completely-vapourises-at-100-c-and-1-bar-pressure-and-the-enthalpy-change-in-the-pr-15693

question:
<div class="question">18.0 g of water completely vapourises at 100Â°C and 1 bar pressure and the enthalpy change in the process is 40.79 kJ mol<sup>â1</sup>. What will be the enthalpy change for vapourising two moles of water under the same conditions? What is the standard enthalphy of vapourisation for water?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Enthalpy change for vapourising two moles of water under the same conditions is +69.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +45.79 kJ mol<sup>â1</sup></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Enthalpy change for vapourising two moles of water under the same conditions is +69.58 kJ and â<sub>vap</sub>H = +45.79 kJ mol<sup>â1</sup></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></span> </div>

answer:
<div class="solution">Correct Option is : (A)<br/>Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></div>


slug: 2-33-g-of-compound-x-empirical-formula-coh-12-n-4-cl-3-upon-treatment-with-excess-agn-17142
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'KVPY', 'KVPY 2016 (SB/SX)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3,6</span> </div>
source_db: MarksBatch0.db
title:
2.33 g of compound X (empirical formula CoH 12 â N 4 â Cl 3 â ) upon treatment with excess AgNO 3 â solution produces 1.435 g of a white precipitate. The primary and secondary valences of cobalt in compound X , respectively, are [Given : Atomic mass : Co = 59 , Cl = 35.5 , Ag = 108 ]

question:
<div class="question">$2.33 \mathrm{~g}$ of compound $\mathrm{X}$ (empirical formula $\left.\mathrm{CoH}_{12} \mathrm{~N}_{4} \mathrm{Cl}_{3}\right)$ upon treatment with excess $\mathrm{AgNO}_{3}$ solution produces $1.435 \mathrm{~g}$ of a white precipitate. The primary and secondary valences of cobalt in compound $\mathrm{X}$, respectively, are<br/>[Given : Atomic mass : $\mathrm{Co}=59, \mathrm{Cl}=35.5, \mathrm{Ag}=108$ ]</div>

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

answer:
<div class="solution">$$<br/>\begin{array}{l}<br/>\text { M.wt }=59+12+14 \times 4+35.5 \times 3=233.5 \\<br/>\mathrm{C}_{0} \mathrm{H}_{12} \mathrm{~N}_{4} \mathrm{Cl}_{3} \stackrel{\mathrm{AgNO}_{3}}{\longrightarrow} 1.435 \mathrm{~g} \mathrm{AgCl} \\<br/>\frac{2.33}{233.5} \mathrm{~g}=0.01 \text { mole } \quad(0.01 \mathrm{~mole})<br/>\end{array}<br/>$$<br/>(i) $0.01$ mole molecule produce $0.01$ mole $\mathrm{AgCl}$ $\therefore$ one replaceable $\mathrm{Cl}^{\underline{\Theta}}$ ion so formula of complex is<br/>$$<br/>\left[\mathrm{CO}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right] \mathrm{Cl}<br/>$$<br/>(ii) Oxidation no. of Co is $+3$ so primary valency is 3 .<br/>(iii) Coordination no. is 6 so sec. valency is 6 so ans. is 3,6</div>


slug: a-1-2-1-3-and-b-2-1-3-2-then-which-of-the-following-is-are-15463
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch0.db
title:
A = [ 1 2 â â 1 3 â ] and B = [ 2 â 1 â 3 â 2 â ] , then which of the following is/are correct? 1. AB ( A â 1 B â 1 ) is a unit matrix. 2. ( AB ) â 1 = A â 1 B â 1 Select the correct answer using the code given below.

question:
<div class="question">$\mathrm{A}=\left[\begin{array}{cc}1 &amp; -1 \\ 2 &amp; 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rr}2 &amp; 3 \\ -1 &amp; -2\end{array}\right]$, then which of the following is/are correct? <br/>$1.$ $\mathrm{AB}\left(\mathrm{A}^{-1} \mathrm{~B}^{-1}\right)$ is a unit matrix.<br/>$2.$ $(\mathrm{AB})^{-1}=\mathrm{A}^{-1} \mathrm{~B}^{-1}$<br/>Select the correct answer using the code given below.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 only 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 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">Neither 1 nor 2</span> </div>

answer:
<div class="solution">Here, $A=\left[\begin{array}{cc}1 &amp; -1 \\ 2 &amp; 3\end{array}\right]$ and $B=\left[\begin{array}{cc}2 &amp; 3 \\ -1 &amp; -2\end{array}\right]$<br/>$|A|=3-(-2)=5$ and $|B|=-4-(-3)=-1$<br/>$\Rightarrow A^{-1}=\frac{1}{5}\left[\begin{array}{cc}3 &amp; 1 \\ -2 &amp; 1\end{array}\right]$ and $B^{-1}=-1\left[\begin{array}{cc}-2 &amp; -3 \\ 1 &amp; 2\end{array}\right]$<br/>$A B=\left[\begin{array}{ll}3 &amp; 5 \\ 1 &amp; 0\end{array}\right]$ and $A^{-1} B^{-1}=\frac{1}{5}\left[\begin{array}{cc}5 &amp; 7 \\ -5 &amp; 8\end{array}\right]$<br/>$\Rightarrow A B\left(A^{-1} B^{-1}\right)=\frac{1}{5}\left[\begin{array}{cc}-10 &amp; -61 \\ 5 &amp; 7\end{array}\right] \neq 1 .$<br/>$|A B|=0-5=-5$<br/>$\therefore(A B)^{-1}=\frac{-1}{5}\left[\begin{array}{cc}0 &amp; -5 \\ -1 &amp; 3\end{array}\right] \neq A^{-1} B^{-1}$</div>


slug: a-2-a-4-b-2-c-3-r-6-s-3-t-x-2-y-z-a-b-c-r-s-t-x-y-21770
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-12$</span> </div>
source_db: MarksBatch0.db
title:
A = â 2 a 4 b â 2 c â 3 r 6 s â 3 t â x 2 y â z â â = Î» â a b c â r s t â x y z â â , then what is the value of Î» ?

question:
<div class="question">$A=\left|\begin{array}{ccc}2 a &amp; 3 r &amp; x \\ 4 b &amp; 6 s &amp; 2 y \\ -2 c &amp; -3 t &amp; -z\end{array}\right|=\lambda\left|\begin{array}{lll}a &amp; r &amp; x \\ b &amp; s &amp; y \\ c &amp; t &amp; z\end{array}\right|$, then what is the value of $\lambda ?$</div>

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

answer:
<div class="solution">Given, $\left|\begin{array}{ccc}2 a &amp; 3 r &amp; x \\ 4 b &amp; 6 s &amp; 2 y \\ -2 c &amp; -3 t &amp; -z\end{array}\right|=\lambda\left|\begin{array}{ccc}a &amp; r &amp; x \\ b &amp; s &amp; y \\ c &amp; t &amp; z\end{array}\right|$<br/>Taking 2 common from $C_{1}$ and 3 from $C_{2}$ in LHS<br/>$\therefore 2 \times 3\left|\begin{array}{ccc}a &amp; r &amp; x \\ 2 b &amp; 2 s &amp; 2 y \\ -c &amp; -t &amp; -z\end{array}\right|=\lambda\left|\begin{array}{lll}a &amp; r &amp; x \\ b &amp; s &amp; y \\ c &amp; t &amp; z\end{array}\right|$<br/>Taking 2 common from $R_{2}$ and $-1$ from $R_{3}$ in LHS<br/>$\therefore-12\left|\begin{array}{lll}a &amp; r &amp; x \\ b &amp; s &amp; y \\ c &amp; t &amp; z\end{array}\right|=\lambda\left|\begin{array}{ccc}a &amp; r &amp; x \\ b &amp; s &amp; y \\ c &amp; t &amp; z\end{array}\right|$<br/>$\Rightarrow \lambda=-12$</div>


slug: a-2-b-is-the-mid-point-of-the-line-segment-joining-the-points-10-6-and-k-4-if-62160
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch0.db
title:
(a, 2 b ) is the mid-point of the line segment joining the points ( 10 , â 6 ) and ( k , 4 ) . If a â 2 b = 7 , then what is the value of k ?

question:
<div class="question">(a, $2 \mathrm{~b}$ ) is the mid-point of the line segment joining the points $(10,-6)$ and $(\mathrm{k}, 4)$. If $\mathrm{a}-2 \mathrm{~b}=7$, then what is the value of $\mathrm{k}$ ?</div>

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

answer:
<div class="solution">$M \equiv$ mid point of line segment $P Q$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/4eS-Lm4AMjrATqEMDn8KKzNen8rxUuQN0fPOu15uCSU.original.fullsize.png"/><br/><br/>$\frac{K+10}{2}=a \Rightarrow K=(2 a-10)$<br/>$2 b=\frac{4-6}{2}=-1$<br/>given $a-2 b=7$...(1)<br/>Put the values of $a$ \&amp; $b$ in $e q(1)$, we get<br/>$\frac{K+10}{2}+1=7$<br/>$\frac{K+10}{2}=6 \Rightarrow K=12-10$<br/>$K=2$</div>


slug: a-3-0-1-4-and-b-1-0-1-2-then-which-of-the-following-is-are-correct-i-55555
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch0.db
title:
A = [ 3 0 â 1 4 â ] and B = [ 1 0 â 1 2 â ] , then which of the following is/are correct? I. A B is defined II. B A is defined III. A B = B A Select the correct answer using the codes given below.

question:
<div class="question">$A=\left[\begin{array}{ll}3 &amp; 1 \\ 0 &amp; 4\end{array}\right]$ and $B=\left[\begin{array}{ll}1 &amp; 1 \\ 0 &amp; 2\end{array}\right]$, then which of the following is/are correct? <br/>I. $A B$ is defined<br/>II. $B A$ is defined<br/>III. $A B=B A$<br/>Select the correct answer using the codes given below.</div>

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

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

answer:
<div class="solution">Let $\mathrm{A}=\left[\begin{array}{ll}3 &amp; 1 \\ 0 &amp; 4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 &amp; 1 \\ 0 &amp; 2\end{array}\right]$<br/>Now, $\mathrm{AB}=\left[\begin{array}{ll}3 &amp; 1 \\ 0 &amp; 4\end{array}\right]\left[\begin{array}{ll}1 &amp; 1 \\ 0 &amp; 2\end{array}\right]=\left[\begin{array}{ll}3 &amp; 5 \\ 0 &amp; 8\end{array}\right]$<br/>and $\mathrm{BA}=\left[\begin{array}{ll}1 &amp; 1 \\ 0 &amp; 2\end{array}\right]\left[\begin{array}{ll}3 &amp; 1 \\ 0 &amp; 4\end{array}\right]=\left[\begin{array}{ll}3 &amp; 5 \\ 0 &amp; 8\end{array}\right]$<br/>$\Rightarrow \quad \mathrm{AB}=\mathrm{BA}$<br/>Hence, all the three statements are correct.</div>


slug: a-3-4-and-b-5-2-are-two-points-and-p-is-a-point-such-that-pa-pb-if-the-area-of-t-10270
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch0.db
title:
A ( 3 , 4 ) and B ( 5 , â 2 ) are two points and P is a point such that PA = PB . If the area of triangle PAB is 10 square unit, what are the coordinates of P ?

question:
<div class="question">$\mathrm{A}(3,4)$ and $\mathrm{B}(5,-2)$ are two points and $\mathrm{P}$ is a point such that $\mathrm{PA}=\mathrm{PB}$. If the area of triangle $\mathrm{PAB}$ is 10 square unit, what are the coordinates of P ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(1,0)$ only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(7,2)$ only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(1,0)$ or $(7,2)$</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">Neither $(1,0)$ nor $(7,2)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(1,0)$ or $(7,2)$</span> </div>

answer:
<div class="solution">Given $\mathrm{A}(3,4)$ and $\mathrm{B}(5,-2)$ Let, $\mathrm{P}(\mathrm{x}, \mathrm{y})$<br/>Given that, $\mathrm{PA}=\mathrm{PB}$ $\Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2}$<br/>$\Rightarrow(x-3)^{2}+(y-4)^{2}=(x-5)^{2}+(y+2)^{2}$<br/>$\Rightarrow x^{2}-6 x+9+y^{2}-8 y+16$<br/>$=x^{2}-10 x+25+y^{2}+4 y+4$<br/>$\Rightarrow 4 x-12 y=4$<br/>$\Rightarrow x-3 y=1$ ...(i)<br/>$\therefore$ Area of $\Delta \mathrm{PAB}=10$<br/>$\frac{1}{2}\left|\begin{array}{lll}x &amp; y &amp; 1 \\ 3 &amp; 4 &amp; 1 \\ 5 &amp; -2 &amp; 1\end{array}\right|=\pm 10$<br/>$\Rightarrow x(4+2)-y(3-5)+1(-6-20)=\pm 20$<br/>$\Rightarrow 6 \mathrm{x}+2 \mathrm{y}-26=\pm 20$<br/>$\Rightarrow 6 \mathrm{x}+2 \mathrm{y}-26=20$<br/>or, $6 x+2 y-26=-20$<br/>$\Rightarrow 6 x+2 y=46$...(ii)<br/>or $6 x+2 y=6$...(iii)<br/>From eqs. (i) and (ii), we get $x=7, y=2$<br/>Similarly, from eqs. (i) and (iii), we get $\mathrm{x}=1, \mathrm{y}=0$<br/>Hence, coordinates of $\mathrm{P}$ are $(7,2)$ or $(1,0)$</div>


tags: ['Physics', 'Alternating Current', 'NEET', 'NEET 2023 (All India)']
source_db: MarksBatch0.db
title:
A 12 V , 60 W lamp is connected to the secondary of a step down transformer, whose primary is connected to ac mains of 220 V . Assuming the transformer to be ideal, what is the current in the primary winding?

slug:
a-12-v-60-w-lamp-is-connected-to-the-secondary-of-a-step-down-transformer-whose-primary-is-connec-77369

question:
<div class="question">A <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>,</mo><mo> </mo><mn>60</mn><mo> </mo><mi mathvariant="normal">W</mi></math> lamp is connected to the secondary of a step down transformer, whose primary is connected to ac mains of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>220</mn><mo> </mo><mi mathvariant="normal">V</mi></math>. Assuming the transformer to be ideal, what is the current in the primary winding?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>37</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>27</mn><mo> </mo><mi mathvariant="normal">A</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">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>7</mn><mo> </mo><mi mathvariant="normal">A</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"><mn>3</mn><mo>.</mo><mn>7</mn><mo> </mo><mi mathvariant="normal">A</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"><mn>0</mn><mo>.</mo><mn>27</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution"><blockquote><p>For an ideal transformer,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mrow><mi>P</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow></mfenced><mrow><mi>i</mi><mi>n</mi><mi>p</mi><mi>u</mi><mi>t</mi></mrow></msub><mo>=</mo><msub><mfenced><mrow><mi>P</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow></mfenced><mrow><mi>o</mi><mi>u</mi><mi>t</mi><mi>p</mi><mi>u</mi><mi>t</mi></mrow></msub><mspace linebreak="newline"></mspace><mo>⇒</mo><msub><mi>V</mi><mi>P</mi></msub><msub><mi>I</mi><mi>P</mi></msub><mo>=</mo><msub><mfenced><mrow><mi>P</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow></mfenced><mrow><mi>o</mi><mi>u</mi><mi>t</mi><mi>p</mi><mi>u</mi><mi>t</mi></mrow></msub><mspace linebreak="newline"></mspace><mo>⇒</mo><msub><mi>I</mi><mi>P</mi></msub><mo>=</mo><mfrac><mn>60</mn><mn>220</mn></mfrac><mo>≈</mo><mn>0</mn><mo>.</mo><mn>27</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p></blockquote></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">59.29</span> </div>
source_db: MarksBatch0.db
title:
A 25 cm long string is fixed at both ends. It has a mass of 2 . 5 g and vibrates in its first overtone under tension. Also given is a closed pipe which is 30 cm long which vibrates in its fundamental frequency. When both of these are sounded simultaneously, 8 beats per second are heard. It is also observed that decreasing the tension in the string also decreases the beat frequency. What is the tension (in N ) in the string? [Given, speed of sound in air is 360 ms - 1 ]

slug:
a-25-cm-long-string-is-fixed-at-both-ends-it-has-a-mass-of-2-5-g-and-vibrates-in-its-first-overto-79372

question:
<div class="question">A <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mi>cm</mi></math> long string is fixed at both ends. It has a mass of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">g</mi></math> and vibrates in its first overtone under tension. Also given is a closed pipe which is  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><mi>cm</mi></math> long which vibrates in its fundamental frequency. When both of these are sounded simultaneously, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> beats per second are heard. It is also observed that decreasing the tension in the string also decreases the beat frequency. What is the tension (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math>) in the string? [Given, speed of sound in air is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo> </mo><msup><mi>ms</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>]</div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mo>.</mo><mn>5</mn></mrow><mn>25</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mi mathvariant="normal">g</mi><mo>/</mo><mi>cm</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn></mrow></msup><mi>kg</mi><mo>/</mo><mi mathvariant="normal">m</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>1</mn><mtext>st </mtext></msup></math> overtone,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi mathvariant="normal">s</mi></msub><mo>=</mo><mn>25</mn><mo> </mo><mi>cm</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn><mo> </mo><mi mathvariant="normal">m</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi></msub><mo>=</mo><mfrac><mn>1</mn><msub><mi>λ</mi><mi mathvariant="normal">s</mi></msub></mfrac><msqrt><mfrac><mi mathvariant="normal">T</mi><mi>μ</mi></mfrac></msqrt></math></p><p><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/abs/bd3625fe-696c-4fdf-868b-f043979c5d38-10.JPG" style="width: 147px; height: 94px;"/><br/>Pipe in fundamental frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>λ</mi><mi>p</mi></msub><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>p</mi></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>2</mn><mi mathvariant="normal">m</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi mathvariant="normal">f</mi><mi mathvariant="normal">p</mi></msub><mo>=</mo><mfrac><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">s</mi></msub><msub><mi>λ</mi><mi mathvariant="normal">p</mi></msub></mfrac><mo>=</mo><mfrac><mn>360</mn><mrow><mn>1</mn><mo>.</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>300</mn><mo> </mo><mi>Hz</mi></math></p><p><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/abs/1e111a34-03a4-4668-b7b5-c5ba39013803-11.JPG" style="width: 112px; height: 181px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> By decreasing the tension, beat<br/>frequency is decreased.<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo>  </mo><msub><mi>f</mi><mi>s</mi></msub><mo>&gt;</mo><msub><mi>f</mi><mi>p</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi></msub><mo>-</mo><msub><mi mathvariant="normal">f</mi><mi mathvariant="normal">p</mi></msub><mo>=</mo><mn>8</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>1</mn><mrow><mn>0</mn><mo>.</mo><mn>25</mn></mrow></mfrac><msqrt><mfrac><mi>T</mi><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfrac></msqrt><mo>-</mo><mfrac><mn>360</mn><mrow><mn>1</mn><mo>.</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>8</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>10</mn><mo>×</mo><mn>100</mn></mrow><mn>25</mn></mfrac><msqrt><mi mathvariant="normal">T</mi></msqrt><mo>-</mo><mn>300</mn><mo>=</mo><mn>8</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msqrt><mi mathvariant="normal">T</mi></msqrt><mo>=</mo><mfrac><mrow><mn>308</mn><mo>×</mo><mn>25</mn></mrow><mn>1000</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi mathvariant="normal">T</mi><mo>=</mo><mn>59</mn><mo>.</mo><mn>29</mn><mo> </mo><mi mathvariant="normal">N</mi></math></p></div>


tags: ['Chemistry', 'Alternating Current', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.27 A</span> </div>
answer: <div class="solution">Correct Option is : (B)<br/>0.27 A</div>
source_db: MarksBatch0.db
title:
A 60 W load is connected to the secondary of a transformer whose primary draws line voltage. If a current of 0.54 A flows in the load, what is the current in the primary coil?

slug:
a-60-w-load-is-connected-to-the-secondary-of-a-transformer-whose-primary-draws-line-voltage-if-a-cu-22181

question:
<div class="question">A 60 W load is connected to the secondary of a transformer whose primary draws line voltage. If a current of 0.54 A flows in the load, what is the current in the primary coil?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.54 A</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0.27 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">C</span> <span class="option-data">1.08 A</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2.16 A</span> </li> </ul>


tags: ['Physics', 'Alternating Current']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A 60 W load is connected to the secondary of a transformer whose primary draws line voltage. If a current of 0.54 A flows in the load, what is the current in the primary coil? Comment on the type of tansformer being used.

slug:
a-60-w-load-is-connected-to-the-secondary-of-a-transformer-whose-primary-draws-line-voltage-if-a-cu-65402

question:
<div class="question">A $60 \mathrm{~W}$ load is connected to the secondary of a transformer whose primary draws line voltage. If a current of $0.54 \mathrm{~A}$ flows in the load, what is the current in the primary coil? Comment on the type of tansformer being used.</div>

answer:
<div class="solution">As given that,<br/>$$<br/>\begin{aligned}<br/>&amp;\mathrm{P}_{\mathrm{S}}=60 \mathrm{~W} \\<br/>&amp;\mathrm{I}_{\mathrm{S}}=0.54 \mathrm{~A}<br/>\end{aligned}<br/>$$<br/>Primary current $\mathrm{I}_{\mathrm{p}}=$ ?<br/>Taking line voltage as $220 \mathrm{~V}\left(\mathrm{~V}_{\mathrm{P}}=220 \mathrm{~V}\right)$<br/>$$<br/>\begin{aligned}<br/>&amp;\mathrm{P}_{\mathrm{S}}=\mathrm{V}_{\mathrm{S}} \mathrm{I}_{\mathrm{S}} \\<br/>&amp;\mathrm{P}_{\mathrm{S}}=60 \mathrm{~W}, \mathrm{I}_{\mathrm{S}}=0.54 \mathrm{~A} \\<br/>&amp;\mathrm{~V}_{\mathrm{S}}=\frac{60}{0.54}=110 \mathrm{Volt} .<br/>\end{aligned}<br/>$$<br/>Voltage in the secondary $\left(\mathrm{V}_{\mathrm{S}}\right)$ is less than voltage in the primary $\left(\mathrm{V}_{\mathrm{p}}\right)$.<br/>So, the transformer is step down transformer.<br/>since, the transformation ratio factor is<br/>$$<br/>\begin{aligned}<br/>&amp;r=\frac{\text { Secondary voltage }}{\text { Primary voltage }} \\<br/>&amp;r=\frac{V_S}{V_P}<br/>\end{aligned}<br/>$$<br/>$\therefore$ For transformer $\frac{V_S}{V_{\mathrm{p}}}=\frac{I_P}{I_s}$<br/>Substituting the values,<br/>$$<br/>\begin{aligned}<br/>\frac{110 \mathrm{~V}}{220 \mathrm{~V}} &amp;=\frac{\mathrm{I}_{\mathrm{P}}}{0.54 \mathrm{~A}} \\<br/>\Rightarrow \quad \quad \mathrm{I}_{\mathrm{P}} &amp;=0.27 \text { Ampere }<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'TS EAMCET', 'TS EAMCET 2021 (05 Aug Shift 2)']
source_db: MarksBatch0.db
title:
A 90 g of ethyl amine on reaction with methyl chloride produced a tertiary amine as an exclusive product. The amount of methyl chloride required is [Given mass in amu : H = 1 , C = 12 , N = 14 , Cl = 35 . 5 ]

slug:
a-90-g-of-ethyl-amine-on-reaction-with-methyl-chloride-produced-a-tertiary-amine-as-an-exclusive-pro-33395

question:
<div class="question">A <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo> </mo><mi mathvariant="normal">g</mi></math> of ethyl amine on reaction with methyl chloride produced a tertiary amine as an exclusive product. The amount of methyl chloride required is<br/>[Given mass in amu : <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi mathvariant="normal">C</mi><mo>=</mo><mn>12</mn><mo>,</mo><mo> </mo><mi mathvariant="normal">N</mi><mo>=</mo><mn>14</mn><mo>,</mo><mo> </mo><mi>Cl</mi><mo>=</mo><mn>35</mn><mo>.</mo><mn>5</mn></math>]</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">g</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>101</mn><mo> </mo><mi mathvariant="normal">g</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>202</mn><mo> </mo><mi mathvariant="normal">g</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">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>303</mn><mo> </mo><mi mathvariant="normal">g</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"><mn>202</mn><mo> </mo><mi mathvariant="normal">g</mi></math></span> </div>

answer:
<div class="solution"><p>The Standard Reaction is:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NH</mi><mn>2</mn></msub><mo> </mo><mo>+</mo><mo> </mo><mn>2</mn><msub><mi>CH</mi><mn>3</mn></msub><mi>Cl</mi><mo> </mo><mo>→</mo><mo> </mo><msub><mfenced><msub><mi>CH</mi><mn>3</mn></msub></mfenced><mn>3</mn></msub><mi mathvariant="normal">N</mi><mo> </mo><mo>+</mo><mo> </mo><mi>HCl</mi></math></p><p>Molar mass of ethyl amine <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mn>45</mn><mo> </mo><mi mathvariant="normal">g</mi></math></p><p>The molar mass of methyl chloride <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mn>50</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">g</mi></math></p><p>Given Moles ethyl amine<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mfrac><mn>90</mn><mn>45</mn></mfrac><mo>=</mo><mn>2</mn></math> </p><p>As per the Standard reaction,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> mole of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NH</mi><mn>2</mn></msub></math> react with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> mole of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mi>Cl</mi></math>, </p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> moles of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NH</mi><mn>2</mn></msub></math> react with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> moles of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mi>Cl</mi></math></p><p>Hence, the amount methyl chloride obtained is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mn>50</mn><mo>.</mo><mn>5</mn><mo> </mo><mo>=</mo><mo> </mo><mn>202</mn><mo> </mo><mi mathvariant="normal">g</mi></math></p></div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch0.db
title:
A 200 Î© resistance has a certain colour code as per standard colour coding. If one replaces the red colour by green in the code, the new resistance will be

slug:
a-200-resistance-has-a-certain-colour-code-as-per-standard-colour-coding-if-one-replaces-the-red-25402

question:
<div class="question">A <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200 </mn><mi mathvariant="normal">Ω</mi></math> resistance has a certain colour code as per standard colour coding. If one replaces the red colour by green in the code, the new resistance will be</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"><mn>500</mn><mo> </mo><mi mathvariant="normal">Ω</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"><mn>400</mn><mo> </mo><mi mathvariant="normal">Ω</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"><mn>300</mn><mo> </mo><mi mathvariant="normal">Ω</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"><mn>100</mn><mo> </mo><mi mathvariant="normal">Ω</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"><mn>500</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> </div>

answer:
<div class="solution">When red is replaced by green, then the first significant figure in the value of resistance changes from 2 to 5. So the new resistance is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></div>


slug: a-and-b-are-two-events-such-that-a-and-b-are-mutually-exclusive-if-p-a-0-5-and-p-b-25850
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch0.db
title:
A and B are two events such that A Ë and B Ë are mutually exclusive. If P ( A ) = 0.5 and P ( B ) = 0.6 , then what is the value of P ( A â£ B ) ?

question:
<div class="question">A and B are two events such that $\bar{A}$ and $\bar{B}$ are mutually exclusive. If $\mathrm{P}(\mathrm{A})=0.5$ and $\mathrm{P}(\mathrm{B})=0.6$, then what is the value of $\mathrm{P}(\mathrm{A} \mid \mathrm{B})$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{6}$</span> </div>

answer:
<div class="solution">$\bar{A}$ and $\bar{B}$ are mutually exclusive $\therefore P(\bar{A} \cap \bar{B})=0$<br/>Given, $\mathrm{P}(\mathrm{A})=0.5 \Rightarrow P(\bar{A})=1-0.5=0.5$<br/>$\mathrm{P}(\mathrm{B})=0.6 \Rightarrow P(\bar{B})=1-0.6=0.4$<br/>$P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{1-P(\bar{A} \cup \bar{B})}{P(B)}$<br/>$=\frac{1-(P(\bar{A})+P(\bar{B}))}{P(B)}=\frac{1-(0.5+0.4)}{0.6}$<br/>$=\frac{1-0.9}{0.6}=\frac{0.1}{0.6}=\frac{1}{6}$</div>


slug: a-and-b-are-two-metals-the-standard-reduction-potential-of-a-aq-a-s-and-b-aq-b-12298
tags: ['Chemistry', 'Electrochemistry', 'TS EAMCET', 'TS EAMCET 2023 (13 May Shift 2)']
source_db: MarksBatch0.db
title:
A and B are two metals. The standard reduction potential of A + ( aq ) / A ( s ) and B + ( aq ) / B ( s ) are â 0.5 V and + 0.5 V respectively. What is the lo g K C â value for the following reaction at 298 K ? $ A ( s ) + B + ( aq ) â A + ( aq ) + B ( s ) \left(\right. G i v e n : \left.\frac{2.303 R T}{F}=0.06 \mathrm{~V}\right)$

question:
<div class="question">A and B are two metals. The standard reduction potential of $\mathrm{A}^{+}(\mathrm{aq}) / \mathrm{A}(\mathrm{s})$ and $\mathrm{B}^{+}(\mathrm{aq}) / \mathrm{B}(\mathrm{s})$ are $-0.5 \mathrm{~V}$ and $+0.5 \mathrm{~V}$ respectively. What is the $\log K_C$ value for the following reaction at $298 \mathrm{~K}$ ?<br/>$$<br/>\mathrm{A}(\mathrm{s})+\mathrm{B}^{+}(\mathrm{aq}) \rightleftharpoons \mathrm{A}^{+}(\mathrm{aq})+\mathrm{B}(\mathrm{s})<br/>$$<br/>$\left(\right.$ Given : $\left.\frac{2.303 R T}{F}=0.06 \mathrm{~V}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{6}{100}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{100}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{}{200}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{200}{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{100}{6}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \text { } \Delta \mathrm{G}^{\circ}=\mathrm{nFE}_{\text {cell }}^{\circ} \\ &amp; \Delta \mathrm{G}^{\circ}=-\mathrm{RT} \ln \mathrm{K}_{\mathrm{c}} \\ &amp; \therefore \quad \mathrm{nFE}_{\text {cell }}^{\circ}=-\mathrm{RT} \ln \mathrm{K}_{\mathrm{c}}=\frac{-2.303 \mathrm{RT}}{\mathrm{nF}} \log \mathrm{K}_{\mathrm{c}} \\ &amp; \mathrm{n}=\text { no. of electrons involved }=1 \\ &amp; \mathrm{E}_{\text {cell }}^{\circ}=\mathrm{E}_{\text {cathode }}^{\circ}-\mathrm{E}_{\text {Anode }}^{\circ}=0.5-(-0.5)=1 \text {. } \\ &amp; \therefore \quad \mathrm{E}_{\text {cell }}^{\circ}=\frac{-2.303 \mathrm{RT} \log \mathrm{K}_{\mathrm{c}}}{\mathrm{nF}} \\ &amp; 1=0.06 \log \mathrm{K}_{\mathrm{c}} \text {. } \\ &amp; \therefore \log \mathrm{K}_{\mathrm{c}}=\frac{100}{6} \\ &amp; \end{aligned}$</div>


slug: a-and-b-are-two-sets-having-3-elements-in-common-lfn-a-5-n-b-4-then-what-is-n-a-57991
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20</span> </div>
source_db: MarksBatch0.db
title:
A and B are two sets having 3 elements in common lfn ( A ) = 5 , n ( B ) = 4 , then what is n ( A Ã B ) equal to

question:
<div class="question">A and $\mathrm{B}$ are two sets having 3 elements in common $\operatorname{lfn}(\mathrm{A})=5, \mathrm{n}(\mathrm{B})=4$, then what is $\mathrm{n}(\mathrm{A} \times \mathrm{B})$ equal to</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">9</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">15</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">20</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Here, $\mathrm{n}(\mathrm{A})=5$ and $\mathrm{n}(\mathrm{B})=$<br/>$\cdot n(A \times B)=5 \times 4=20$<br/>$[\because \mathrm{n}(\mathrm{A})=\mathrm{m}, \mathrm{n}(\mathrm{B})=\mathrm{n} \Rightarrow \mathrm{n}(\mathrm{A} \times \mathrm{B})=\mathrm{mn}]$</div>


slug: a-b-are-two-events-and-a-denotes-the-complements-of-a-consider-the-following-statements-1-p-a-89400
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch0.db
title:
A, B are two events and A Ë denotes the complements of A. Consider the following statements 1 P ( A âª B ) â¤ P ( B ) + P ( A ) 2 P ( A ) + P ( A âª B ) â¤ 1 + P ( B ) Which of the above statements is/are correct?

question:
<div class="question">A, B are two events and $\bar{A}$ denotes the complements of A. Consider the following statements <br/>$1 \quad P(A \cup B) \leq P(B)+P(A)$<br/>$2 \quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\overline{\mathrm{A}} \cup \mathrm{B}) \leq 1+\mathrm{P}(\mathrm{B})$<br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and 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">Neither 1 nor 2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both 1 and 2</span> </div>

answer:
<div class="solution">(1) $\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$ is correct<br/>and (2) $\mathrm{P}(\mathrm{A})+\mathrm{P}(\overline{\mathrm{A}} \cup \mathrm{B}) \leq 1+\mathrm{P}(\mathrm{B})$ is also correct<br/>$\Rightarrow$ Both the given statement are correct.</div>


slug: a-b-c-0-such-that-a-3-b-5-and-c-7-what-is-a-b-b-c-48409
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch0.db
title:
a + b + c = 0 such that â£ a â£ = 3 , â£ b â£ = 5 and â£ c â£ = 7 What is a â b + b â c + c a equal to ?

question:
<div class="question">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ such that $|\vec{a}|=3,|\overrightarrow{\mathrm{b}}|=5$ and $|\overrightarrow{\mathrm{c}}|=7$<br/>What is $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c}$ a equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-83$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-83 / 2$</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">75</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-75 / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-83 / 2$</span> </div>

answer:
<div class="solution">We have, $\overline{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overline{\mathrm{c}}=0$ ...(i)<br/>On squaring both sides request.<br/>$\bar{a}^{2}+\vec{b}^{2}+\bar{c}^{2}+2 \bar{a} \vec{b}+2 \vec{b} \bar{c}+2 \vec{c} \bar{a}=0$<br/>$\quad(\because \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}, \vec{b} \cdot \vec{c}=\bar{c} \vec{b}$ and $\bar{c} \cdot \vec{a}=\bar{a} \cdot \vec{c})$<br/>$\Rightarrow|a|^{2}+|b|^{2}+|c|^{2}=-2[a \cdot b+b . c+c . a]$<br/>$\Rightarrow(3)^{2}+(5)^{2}+(7)^{2}=-2[a \cdot b+b \cdot c+c . a]$<br/>$\Rightarrow a \cdot b+b . c+c . a=\frac{9+25+49}{-2}=-\frac{83}{2}$<br/>Now $a+b+c=0 \quad$ [using eq. (i)]<br/>$\Rightarrow \mathrm{a}+\mathrm{b}=-\mathrm{c}$<br/>On squaring both sides, we get $\Rightarrow \mathrm{a}^{2}+\mathrm{b}^{2}+2 \mathrm{a} \cdot \mathrm{b}=\mathrm{c}^{2}$<br/>$\Rightarrow(3)^{2}+(5)^{2}+2 \mathrm{ab}=(7)^{2}$<br/>$\Rightarrow \overline{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\frac{15}{2}$<br/>$\Rightarrow|\overline{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta=\frac{15}{2} \Rightarrow 3.5 \cos \theta=\frac{15}{2}$<br/>$\Rightarrow \cos \theta=\frac{1}{2}=\cos \frac{\pi}{3}$<br/>$\theta=\frac{\pi}{3}$<br/>From eq. (i), $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=-\overline{\mathrm{a}}$<br/>$\Rightarrow b^{2}+c^{2}+2 b \cdot c=a^{2}$<br/>$\Rightarrow 2 \mathrm{~b} \cdot \mathrm{c}=\mathrm{a}^{2}-\mathrm{b}^{2}-\mathrm{c}^{2}=9-25-49=-65$<br/>$\Rightarrow \mathrm{b.c}=-\frac{65}{2} \Rightarrow|\mathrm{b}||\mathrm{c}| \cos \theta=-\frac{65}{2}$<br/>$\Rightarrow \cos \theta=\frac{65}{2} \times \frac{1}{5} \times \frac{1}{7}=-\frac{13}{14}$<br/>Also, $|\vec{a}+\vec{b}|=|-\vec{c}|=|\vec{c}|=7$</div>


slug: a-b-c-0-such-that-a-3-b-5-and-c-7-what-is-a-b-equal-38277
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch0.db
title:
a + b + c = 0 such that â£ a â£ = 3 , â£ b â£ = 5 and â£ c â£ = 7 What is â£ a + b â£ equal to ?

question:
<div class="question">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ such that $|\vec{a}|=3,|\overrightarrow{\mathrm{b}}|=5$ and $|\overrightarrow{\mathrm{c}}|=7$<br/>What is $|\vec{a}+\vec{b}|$ equal to ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">10</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">11</span> </li> </ul>

answer:
<div class="solution">We have, $\overline{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overline{\mathrm{c}}=0$ ...(i)<br/>On squaring both sides request.<br/>$\bar{a}^{2}+\vec{b}^{2}+\bar{c}^{2}+2 \bar{a} \vec{b}+2 \vec{b} \bar{c}+2 \vec{c} \bar{a}=0$<br/>$\quad(\because \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}, \vec{b} \cdot \vec{c}=\bar{c} \vec{b}$ and $\bar{c} \cdot \vec{a}=\bar{a} \cdot \vec{c})$<br/>$\Rightarrow|a|^{2}+|b|^{2}+|c|^{2}=-2[a \cdot b+b . c+c . a]$<br/>$\Rightarrow(3)^{2}+(5)^{2}+(7)^{2}=-2[a \cdot b+b \cdot c+c . a]$<br/>$\Rightarrow a \cdot b+b . c+c . a=\frac{9+25+49}{-2}=-\frac{83}{2}$<br/>Now $a+b+c=0 \quad$ [using eq. (i)]<br/>$\Rightarrow \mathrm{a}+\mathrm{b}=-\mathrm{c}$<br/>On squaring both sides, we get $\Rightarrow \mathrm{a}^{2}+\mathrm{b}^{2}+2 \mathrm{a} \cdot \mathrm{b}=\mathrm{c}^{2}$<br/>$\Rightarrow(3)^{2}+(5)^{2}+2 \mathrm{ab}=(7)^{2}$<br/>$\Rightarrow \overline{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\frac{15}{2}$<br/>$\Rightarrow|\overline{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta=\frac{15}{2} \Rightarrow 3.5 \cos \theta=\frac{15}{2}$<br/>$\Rightarrow \cos \theta=\frac{1}{2}=\cos \frac{\pi}{3}$<br/>$\theta=\frac{\pi}{3}$<br/>From eq. (i), $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=-\overline{\mathrm{a}}$<br/>$\Rightarrow b^{2}+c^{2}+2 b \cdot c=a^{2}$<br/>$\Rightarrow 2 \mathrm{~b} \cdot \mathrm{c}=\mathrm{a}^{2}-\mathrm{b}^{2}-\mathrm{c}^{2}=9-25-49=-65$<br/>$\Rightarrow \mathrm{b.c}=-\frac{65}{2} \Rightarrow|\mathrm{b}||\mathrm{c}| \cos \theta=-\frac{65}{2}$<br/>$\Rightarrow \cos \theta=\frac{65}{2} \times \frac{1}{5} \times \frac{1}{7}=-\frac{13}{14}$<br/>Also, $|\vec{a}+\vec{b}|=|-\vec{c}|=|\vec{c}|=7$</div>


slug: a-b-c-0-such-that-a-3-b-5-and-c-7-what-is-cosine-of-the-angle-78568
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch0.db
title:
a + b + c = 0 such that â£ a â£ = 3 , â£ b â£ = 5 and â£ c â£ = 7 What is cosine of the angle between b and c ?

question:
<div class="question">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ such that $|\vec{a}|=3,|\overrightarrow{\mathrm{b}}|=5$ and $|\overrightarrow{\mathrm{c}}|=7$<br/>What is cosine of the angle between $\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$11 / 12$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$13 / 14$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-11 / 12$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-13 / 14$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$-13 / 14$</span> </div>

answer:
<div class="solution">We have, $\overline{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overline{\mathrm{c}}=0$ ...(i)<br/>On squaring both sides request.<br/>$\bar{a}^{2}+\vec{b}^{2}+\bar{c}^{2}+2 \bar{a} \vec{b}+2 \vec{b} \bar{c}+2 \vec{c} \bar{a}=0$<br/>$\quad(\because \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}, \vec{b} \cdot \vec{c}=\bar{c} \vec{b}$ and $\bar{c} \cdot \vec{a}=\bar{a} \cdot \vec{c})$<br/>$\Rightarrow|a|^{2}+|b|^{2}+|c|^{2}=-2[a \cdot b+b . c+c . a]$<br/>$\Rightarrow(3)^{2}+(5)^{2}+(7)^{2}=-2[a \cdot b+b \cdot c+c . a]$<br/>$\Rightarrow a \cdot b+b . c+c . a=\frac{9+25+49}{-2}=-\frac{83}{2}$<br/>Now $a+b+c=0 \quad$ [using eq. (i)]<br/>$\Rightarrow \mathrm{a}+\mathrm{b}=-\mathrm{c}$<br/>On squaring both sides, we get $\Rightarrow \mathrm{a}^{2}+\mathrm{b}^{2}+2 \mathrm{a} \cdot \mathrm{b}=\mathrm{c}^{2}$<br/>$\Rightarrow(3)^{2}+(5)^{2}+2 \mathrm{ab}=(7)^{2}$<br/>$\Rightarrow \overline{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\frac{15}{2}$<br/>$\Rightarrow|\overline{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta=\frac{15}{2} \Rightarrow 3.5 \cos \theta=\frac{15}{2}$<br/>$\Rightarrow \cos \theta=\frac{1}{2}=\cos \frac{\pi}{3}$<br/>$\theta=\frac{\pi}{3}$<br/>From eq. (i), $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=-\overline{\mathrm{a}}$<br/>$\Rightarrow b^{2}+c^{2}+2 b \cdot c=a^{2}$<br/>$\Rightarrow 2 \mathrm{~b} \cdot \mathrm{c}=\mathrm{a}^{2}-\mathrm{b}^{2}-\mathrm{c}^{2}=9-25-49=-65$<br/>$\Rightarrow \mathrm{b.c}=-\frac{65}{2} \Rightarrow|\mathrm{b}||\mathrm{c}| \cos \theta=-\frac{65}{2}$<br/>$\Rightarrow \cos \theta=\frac{65}{2} \times \frac{1}{5} \times \frac{1}{7}=-\frac{13}{14}$<br/>Also, $|\vec{a}+\vec{b}|=|-\vec{c}|=|\vec{c}|=7$</div>


slug: a-b-c-0-such-that-a-3-b-5-and-c-7-what-is-the-angle-between-71994
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch0.db
title:
a + b + c = 0 such that â£ a â£ = 3 , â£ b â£ = 5 and â£ c â£ = 7 What is the angle between â£ a â£ and â£ b â£ ?

question:
<div class="question">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ such that $|\vec{a}|=3,|\overrightarrow{\mathrm{b}}|=5$ and $|\overrightarrow{\mathrm{c}}|=7$<br/>What is the angle between $|\vec{a}|$ and $|\vec{b}|$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi / 6$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pi / 4$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\pi / 3$</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">$\pi / 2$</span> </li> </ul>

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


slug: a-b-c-and-d-are-four-sets-such-that-a-b-c-d-consider-the-following-1-a-c-26803
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch0.db
title:
A , B , C and D are four sets such that A â© B = C â© D = Ï . Consider the following: 1. A âª C and B âª D are always disjoint. 2. A â© C and B â© D are always disjoint Which of the above statements is/are correct?

question:
<div class="question">$\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are four sets such that $\mathrm{A} \cap \mathrm{B}=\mathrm{C} \cap \mathrm{D}=\phi$.<br/>Consider the following: <br/>1.$\mathrm{A} \cup \mathrm{C}$ and $\mathrm{B} \cup \mathrm{D}$ are always disjoint.<br/>2.$\mathrm{A} \cap \mathrm{C}$ and $\mathrm{B} \cap \mathrm{D}$ are always disjoint Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Both 1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 2</span> </li> </ul>

answer:
<div class="solution">$\quad$ Let $\mathrm{A}=\{1,2\}$<br/>$\mathrm{B}=\{3,4,0\}$<br/>$\mathrm{C}=\{5,6,0\}$<br/>$\mathrm{D}=\{7,8\}$<br/>Such that $(\mathrm{A} \cap \mathrm{B})=(\mathrm{C} \cap \mathrm{D})=\phi$<br/>$\Rightarrow(\mathrm{A} \cup \mathrm{C})=\{1,2,5,6,0\}$<br/>$\Rightarrow(\mathrm{B} \cup \mathrm{D})=\{3,4,7,8,0\}$<br/>$\Rightarrow(\mathrm{A} \cup \mathrm{C}) \cap(\mathrm{B} \cup \mathrm{D})=\{0\}$<br/>So $(\mathrm{A} \cup \mathrm{C})$ and $(\mathrm{B} \cup \mathrm{D})$ are not always dispoint $\Rightarrow(\mathrm{A} \cap \mathrm{C})=\phi$ and $(\mathrm{B} \cap \mathrm{D})=\phi$<br/>So $(\mathrm{A} \cap \mathrm{C})$ and $(\mathrm{B} \cap \mathrm{D})$ are always disjoint.</div>


tags: ['Biology', 'The Living World', 'NEET']
source_db: MarksBatch0.db
title:
A, B, C and D are respectively :- Common Name Man Housefly C Wheat Biological Name A Musca domestica Mangifera indica Triticum aestivum Genus Homo Musca Mangifera Triticum Family Hominidae B Anacardiaceae D

slug:
a-b-c-and-d-are-respectively-common-name-man-housefly-c-wheat-biological-name-a-musca-domestica-28837

question:
<div class="question">A, B, C and D are respectively :-<br/>
 
<table border="1"> <tbody> <tr> <td> Common Name</td> <td> Man</td> <td> Housefly</td> <td> C</td> <td> Wheat</td> </tr> <tr> <td> Biological Name</td> <td> A</td> <td> <em>Musca domestica</em></td> <td> <em>Mangifera indica</em></td> <td> <em>Triticum aestivum</em></td> </tr> <tr> <td> Genus</td> <td> Homo</td> <td> <em>Musca</em></td> <td> <em>Mangifera</em></td> <td> <em>Triticum</em></td> </tr> <tr> <td> Family</td> <td> Hominidae</td> <td> B</td> <td> Anacardiaceae</td> <td> D</td> </tr> </tbody> </table> </div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A - Homo sapiens,<br/>B - Muscidae,<br/>C - Mango,<br/>D - Poaceae</span> </div>

answer:
<div class="solution"><table border="1" cellpadding="1" cellspacing="1" style="width: 100%;"> <tbody> <tr> <td> CommonName</td> <td> Man</td> <td> Housefl<strong>y</strong></td> <td><strong> Mango</strong></td> <td> Wheat</td> </tr> <tr> <td> BiologicalName</td> <td> <strong><em>Homo  sapien</em></strong></td> <td> <em>Musca domestica</em></td> <td> <em>Mangifera indica </em></td> <td> <em>Triticum aestivum</em></td> </tr> <tr> <td> Genus</td> <td> Homo</td> <td> Musca</td> <td> Mangifera</td> <td> Triticum</td> </tr> <tr> <td> Family</td> <td> Hominidae</td> <td> <strong>Muscidae</strong></td> <td><strong> </strong>Anacardiaceae</td> <td><strong> Poaceae</strong></td> </tr> </tbody></table></div>


slug: a-b-c-are-in-g-p-with-1-a-b-n-and-n-1-is-an-integer-lo-g-an-n-lo-g-b-n-lo-g-c-55819
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch0.db
title:
a , b , c are in G.P. with 1 a b n , and n > 1 is an integer. lo g an â n , lo g b â n , lo g c â n form a sequence. This sequence is which one of the following ?

question:
<div class="question">$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in G.P. with $1   &lt; \mathrm{a} &lt; \mathrm{b} &lt; \mathrm{n}$, and $\mathrm{n}&gt;1$ is an integer. $\log _{\text {an }}$ $\mathrm{n}, \log _{\mathrm{b}} \mathrm{n}, \log _{\mathrm{c}} \mathrm{n}$ form a sequence. This sequence is which<br/>one of the following ?</div>

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

answer:
<div class="solution">If a, b, c are in G.P. then, $\mathrm{b}^{2}=\mathrm{ac} \Rightarrow \mathrm{b}=(\mathrm{ac})^{1 / 2} \quad \ldots .(1)$<br/>Taking $\log _{\mathrm{n}}$ on both the sides of eq. (1).<br/>$\log _{\mathrm{n}} \mathrm{b}=\frac{1}{2}\left[\left(\log _{\mathrm{n}}(\mathrm{ac})\right]=\frac{\log _{\mathrm{n}} \mathrm{a}+\log _{\mathrm{n}} \mathrm{c}}{2}\right.$<br/>or, $\frac{\log _{n} a+\log _{n} c}{2}=\log _{n} b$<br/>So, $\log _{\mathrm{n}} \mathrm{a}, \log _{\mathrm{n}} \mathrm{b}$ and $\log _{\mathrm{n}} \mathrm{c}$ are in $\mathrm{AP}$.<br/>Hence, $\frac{1}{\log _{n} a}, \frac{1}{\log _{n} b}, \frac{1}{\log _{n} c}$ are in H.P.<br/>$\log _{\mathrm{a}} \mathrm{n}=\frac{1}{\log _{\mathrm{n}} \mathrm{a}}$<br/>$\log _{\mathrm{b}} \mathrm{n}=\frac{1}{\log _{\mathrm{n}} \mathrm{b}}$<br/>$\log _{\mathrm{c}} \mathrm{n}=\frac{1}{\log _{\mathrm{n}} \mathrm{c}}$<br/>i.e. $\log _{\mathrm{a}} \mathrm{n}, \log _{\mathrm{b}} \mathrm{n}$, and $\log _{\mathrm{c}} \mathrm{n}$ are in $\mathrm{HP}$.</div>


slug: a-b-c-d-and-e-are-coplanar-points-and-three-of-them-lie-in-a-straight-line-what-is-the-maximu-28388
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch0.db
title:
A , B , C , D and E are coplanar points and three of them lie in a straight line. What is the maximum number of triangles that can be drawn with these points as their vertices?

question:
<div class="question">$A, B, C, D$ and $E$ are coplanar points and three of them lie in a straight line. What is the maximum number of triangles that can be drawn with these points as their vertices?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">12</span> </li> </ul>

answer:
<div class="solution">Number of triangles using 5 points out of three are on a straight line $={ }^{5} C_{3}-{ }^{3} C_{3}=\frac{5 !}{3 ! 2 !}-1=\frac{5 \times 4}{2}-1$<br/>$=10-1=9$</div>


title: a Ã ( b Ã c ) is equal to
slug: a-b-c-is-equal-to-16745
question: <div class="question">$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ is equal to</div>
tags: ['Mathematics', 'Vector Algebra', 'JEE Main']
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{a}) \mathbf{b}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(\mathbf{a} \cdot \mathbf{c}) \mathbf{a}-(\mathbf{b} \cdot \mathbf{c}) \mathbf{a}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$</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">$(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}-(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$</span> </div>

answer:
<div class="solution">It is a fundamental concept.<br/>(a.c)b-(a.b)c.<br/>This is a fundamental concept of vector calculus. The vector triple product is a binary operation that takes three vectors and produces a new vector. It is also known as the cross product of a vector and a cross product.<br/><br/>The formula for the vector triple product is as follows:<br/>$a \times(b \times c)=(a . c) b-(a . b) c$</div>


tags: ['Physics', 'Current Electricity']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A B is a potentiometer wire (figure). If the value of R is increased, in which direction will the balance point J shift?

slug:
a-b-is-a-potentiometer-wire-figure-if-the-value-of-r-is-increased-in-which-direction-will-the-ba-41934

question:
<div class="question">$\quad A B$ is a potentiometer wire (figure). If the value of $R$ is increased, in which direction will the balance point $\mathrm{J}$ shift?<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/ZUra6zegeUhWhzbHbkYM0yWKdXiFoREUcgTfaVVudX4.original.fullsize.png"/><br/></div>

answer:
<div class="solution">If $\mathrm{R}$ increases then, the current in main circuit decreases which in turn, decreases the potential difference across $\mathrm{AB}$ and hence potential gradient $(\mathrm{K})$ across $\mathrm{AB}$ decreases. Since, at neutral point, for given emf of cell, I increases as potential gradient $(\mathrm{K})$ across $\mathrm{AB}$ has decreased because.<br/>$$<br/>\mathrm{E}=\mathrm{kI} \quad\left[\because \mathrm{K}=\frac{\mathrm{V}}{(\mathrm{AB})}\right]<br/>$$<br/>So balance potential accross $\mathrm{AB}$ equal to potential of secondary circuit the length $(\mathrm{AJ})$ must be longer then earlier.<br/>Thus, with increase of 1 , the balance point neutral point will shift towards B.</div>


title: A baby has been born with a small tail. It is a case exhibiting.
slug: a-baby-has-been-born-with-a-small-tail-it-is-a-case-exhibiting-69700
question: <div class="question">A baby has been born with a small tail. It is a case exhibiting.</div>
tags: ['Biology', 'Evolution', 'NEET']
source_db: MarksBatch0.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Retrogresive evolution</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Mutation</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Reversion</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Metamorphosis</span> </li> </ul>

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

answer:
<div class="solution">Atavism or reversion is the sudden reappearance of some ancestral features. Atavisms is observed in humans, as with the infants that are born with the vestigial tails called a "coccygeal process" or "coccygeal projection", or "caudal appendage".</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch0.db
title:
A bag contains 4 white and 2 black balls and another bag contains 3 white and 5 black balls. If one ball is drawn from each bag, then the probability that one ball is white and one ball is black is

slug:
a-bag-contains-4-white-and-2-black-balls-and-another-bag-contains-3-white-and-5-black-balls-if-one-27461

question:
<div class="question">A bag contains 4 white and 2 black balls and another bag contains 3 white and 5 black balls. If one ball is drawn from each bag, then the probability that one ball is white and one ball is black is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{24}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{13}{24}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{13}{24}$</span> </div>

answer:
<div class="solution">$\mathrm{P}$ (one white ball \&amp; one black ball) $=\mathrm{P}\{[$ black from 1 st bag $\&amp;$ white from $2 \mathrm{nd}]$ or [white from 1 st $\&amp;$ black from $2 \mathrm{nd}]\}$ $=\mathrm{P}\left(\left[\mathrm{B}_{1} \cap \mathrm{W}_{2}\right) \cup\left(\mathrm{W}_{1} \cap \mathrm{B}_{2}\right)\right]$<br/>$=\mathrm{P}\left(\mathrm{B}_{1} \cap \mathrm{W}_{2}\right)+\mathrm{P}\left(\mathrm{W}_{1} \cap \mathrm{B}_{2}\right)$<br/>(By addition theorem for mutually<br/>exclusive events) $=\mathrm{P}\left(\mathrm{B}_{1}\right) \mathrm{P}\left(\mathrm{W}_{2}\right)+\mathrm{P}\left(\mathrm{W}_{1}\right) \mathrm{P}\left(\mathrm{B}_{2}\right)$<br/>$\left(\therefore \mathrm{B}_{2} \&amp; \mathrm{~W}_{2} ; \mathrm{B}_{2} \&amp; \mathrm{~W}_{1}\right.$ are pairs of<br/>independent events) $=\left[\frac{2}{6} \times \frac{3}{8}\right]+\left[\frac{4}{6} \times \frac{5}{8}\right]$<br/>$=\frac{13}{24}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch0.db
title:
A bag contains 5 black and 3 white balls. Two balls are drawn at random one after the other without replacement. What is the probability that both are white?

slug:
a-bag-contains-5-black-and-3-white-balls-two-balls-are-drawn-at-random-one-after-the-other-without-14039

question:
<div class="question">A bag contains 5 black and 3 white balls. Two balls are drawn at random one after the other without replacement. What is the probability that both are white?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{28}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{14}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3}{28}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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 above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{28}$</span> </div>

answer:
<div class="solution">Total number of balls $=5+3=8$. Prob (both ball are white)<br/>$=\frac{3}{8} \times \frac{2}{7}(\because$ No. of white ball $=3)$<br/>$=\frac{3}{28}$</div>


tags: ['Mathematics', 'Probability', 'JEE Main', 'JEE Main 2019 (11 Jan Shift 2)']
source_db: MarksBatch0.db
title:
A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then ( standard deviation of X mean of X â ) is equal to:

slug:
a-bag-contains-30-white-balls-and-10-red-balls-16-balls-are-drawn-one-by-one-randomly-from-the-bag-81967

question:
<div class="question">A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If $X$ be the number of white balls drawn, then
<br/> <br/>$\left(\frac{\text { mean of } \mathrm{X}}{\text { standard deviation of } \mathrm{X}}\right)$ is equal to:</div>

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

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

answer:
<div class="solution">$P($ white ball $)=\frac{30}{40}=\frac{3}{4}, Q$ (red ball) $=\frac{10}{40}=\frac{1}{4}, n=16$
<br/> <br/>$\frac{\text { Mean of } X}{\text { standard deviation of } X}=\frac{n P}{\sqrt{n P Q}}=\frac{\sqrt{n P}}{\sqrt{Q}}$ $=\sqrt{\frac{16 \times \frac{3}{4}}{\frac{1}{4}}}=\sqrt{48}=4 \sqrt{3}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch0.db
title:
A bag contains balls of two colours, 3 black and 3 white. What is the smallest number of balls which must be drawn from the bag, without looking, so that among these three are two of the same colour?

slug:
a-bag-contains-balls-of-two-colours-3-black-and-3-white-what-is-the-smallest-number-of-balls-which-16135

question:
<div class="question">A bag contains balls of two colours, 3 black and 3 white. What is the smallest number of balls which must be drawn from the bag, without looking, so that among these three are two of the same colour?</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">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">5</span> </li> </ul>

answer:
<div class="solution">Minimum 4 balls have to be drawn so that among these there are two of the same colour</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch0.db
title:
A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is 4 8 â . If the height of the balloon above the bridge is 122 m , then what is the length of the bridge?

slug:
a-balloon-is-directly-above-one-end-of-a-bridge-the-angle-of-depression-of-the-other-end-of-the-bri-24785

question:
<div class="question">A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is $48^{\circ} .$ If the height of the balloon above the bridge is $122 \mathrm{~m}$, then what is the length of the bridge?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$122 \sin 48^{\circ} \mathrm{m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$122 \tan 42^{\circ} \mathrm{m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$122 \cos 48^{\circ} \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$122 \tan 48^{\circ} \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$122 \sin 48^{\circ} \mathrm{m}$</span> </div>

answer:
<div class="solution">Let the bridge is $\mathrm{BC}$ and height above the bridge $\mathrm{AB}=122 \mathrm{~m}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/kzw3_7M5y8SN4f4OnyPx-2Elr5e2vUcBQTCP4Qr0zro.original.fullsize.png"/><br/><br/>From DABC,<br/>$\begin{aligned} \cot 48^{\circ}=\frac{\mathrm{BC}}{\mathrm{AB}}=\frac{\mathrm{BC}}{122} \\ \mathrm{BC}=122 \cdot \cot \left(48^{\circ}\right) \\=122 \cdot \cot \left(90^{\circ}-42^{\circ}\right) \\=&amp; 122 \cdot \tan 42^{\circ} \end{aligned}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch0.db
title:
A balloon is pumped at the rate of 4 cm 3 per second. What is the rate at which its surface area increases and radius is 4 cm ?

slug:
a-balloon-is-pumped-at-the-rate-of-4-cm-3-per-second-what-is-the-rate-at-which-its-surface-area-inc-88217

question:
<div class="question">A balloon is pumped at the rate of $4 \mathrm{~cm}^{3}$ per second. What is the rate at which its surface area increases and radius is $4 \mathrm{~cm} ?$</div>

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

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

answer:
<div class="solution">Let $r$ be the radius of balloon.<br/>Balloon is like a sphere and volume of sphere $=\frac{4}{3} \pi r^{3}$<br/>$\therefore \quad V=\frac{4}{3} \pi r^{3}$<br/>Differentiate both side w.r.t ' $\mathrm{t}^{\prime}$<br/>$\Rightarrow \frac{d V}{d t}=\frac{4}{3} \pi \cdot 3 r^{2} \frac{d r}{d t}$<br/>$\Rightarrow 4=\frac{4}{3} \pi .3(4)^{2} \frac{d r}{d t} \quad\left(\because \frac{d V}{d t}=4 \mathrm{~cm}^{3} / \mathrm{s}\right)$<br/>$\Rightarrow \frac{d r}{d t}=\frac{1}{16 \pi}$<br/>Now, surface area of balloon $=S=4 \pi r^{2}$<br/>$$<br/>\begin{aligned}<br/>\frac{d S}{d t} &amp;=4 \pi \cdot 2 r \frac{d r}{d t} \\<br/>&amp;=4 \pi \cdot 2 \times 4 \frac{1}{16 \pi}(\text { from }(1))=2 \mathrm{~cm}^{2} / \mathrm{s}<br/>\end{aligned}<br/>$$</div>


slug: a-balloon-of-mass-m-descends-with-an-acceleration-a-g-what-mass-need-to-be-removed-from-49709
tags: ['Physics', 'Laws of Motion', 'AP EAMCET', 'AP EAMCET 2020 (18 Sep Shift 2)']
source_db: MarksBatch0.db
title:
A balloon of mass \(M\) descends with an acceleration \(a( < g)\). What mass need to be removed from the balloon, so that it starts ascending with acceleration, \(a\) ?

question:
<div class="question">A balloon of mass \(M\) descends with an acceleration \(a( &lt; g)\). What mass need to be removed from the balloon, so that it starts ascending with acceleration, \(a\) ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{2 M a}{(a+g)}\)</span> </div>

answer:
<div class="solution">The given situation is shown in the figure<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/_pkzq6rjZmg5oF6-jLY5QcdNDASBnZaxwTaREMNmL1Y.original.fullsize.png"/><br/><br/>In above figures, \(M\) is the mass of balloon and \(M^{\prime}\) is the mass remove from the ballon.<br/>According to Newton's law of motion in first case,<br/>\(M g-F=M a \Rightarrow F=M g-M a\) ...(i)<br/>According to Newton's second law of motion, for second case,<br/>\(\begin{aligned}<br/>&amp; F-\left(M-M^{\prime}\right) g=\left(M-M^{\prime}\right) a \\<br/>&amp; \Rightarrow F-M g+M^{\prime} g=M a-M^{\prime} a \\<br/>&amp; \Rightarrow M g-M a-M g+M^{\prime} g=M a-M^{\prime} a \text { [from Eq. (i)] } \\<br/>&amp; \Rightarrow M^{\prime} g+M^{\prime} a=M a+M a \\<br/>&amp; \Rightarrow M^{\prime}(g+a)=2 M a \\<br/>&amp; \Rightarrow M^{\prime}=\frac{2 M a}{g+a} \\<br/>\end{aligned}\)</div>


tags: ['Physics', 'Current Electricity', 'AP EAMCET', 'AP EAMCET 2018 (22 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">52 cm</span> </div>
source_db: MarksBatch0.db
title:
A battery of emf 10 V is connected to a uniform wire A B of 1 m length and having a resistance of 10Î© in series with a 10Î© resistor as shown in the figure. Two cells of emf 2 V and 3 V having internal resistance 2Î© and 3Î© , respectively are connected as shown in the figure. If the galvanometer shows null deflection at point J on the wire, then the distance of point J from the point B is.

slug:
a-battery-of-emf-10-v-is-connected-to-a-uniform-wire-a-b-of-1-m-length-and-having-a-resistance-of-10-48484

question:
<div class="question">A battery of emf $10 \mathrm{~V}$ is connected to a uniform wire $A B$ of $1 \mathrm{~m}$ length and having a resistance of $10 \Omega$ in series with a $10 \Omega$ resistor as shown in the figure. Two cells of emf $2 \mathrm{~V}$ and $3 \mathrm{~V}$ having internal resistance $2 \Omega$ and $3 \Omega$, respectively are connected as shown in the figure. If the galvanometer shows null deflection at point $J$ on the wire, then the distance of point $J$ from the point $B$ is.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/B57eDYWstbt5s9VoSvoYHiRHUj7HdqmDeTlxZEgxTQg.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">48 cm</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">50 cm</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">52 cm</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">54 cm</span> </li> </ul>

answer:
<div class="solution">Given $r=10 \Omega, R=10 \Omega, E=10 \mathrm{~V}$ and $L=100 \mathrm{~cm}$.<br/>Now, voltage drop on the wire, $E^{\prime}=\frac{E}{(r+R)} \times R$<br/>$$<br/>\Rightarrow \quad E^{\prime}=\frac{10}{20} \times 10=5 R<br/>$$<br/>So, potential gradient on wire,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/zdMB-s7PaFiW8uExjzllj1nBXTTQHpW5yDCBlCXX4hU.original.fullsize.png"/><br/><br/>Effective emf of combination in secondary circuit,<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; &amp; \frac{V}{r_{\text {eff }}} &amp; =\frac{\varepsilon_1}{r_1}+\frac{\varepsilon_2}{r_2} \\<br/>\Rightarrow \text { Here, } \quad &amp; \frac{1}{r_{\text {eff }}} &amp; =\frac{1}{2}+\frac{1}{3}=\frac{5}{6} \\<br/>\Rightarrow \quad &amp; V &amp; =\frac{6}{5}(1+1)=\frac{12}{5}<br/>\end{array}<br/>$$<br/>Now, at balancing point (from point $A$ ) is<br/>$$<br/>\begin{aligned}<br/>&amp; l=V / x=\frac{12}{5} \times \frac{100}{5} \\<br/>&amp; l=48 \mathrm{~cm}<br/>\end{aligned}<br/>$$<br/>So, length from point $B$ is<br/>$$<br/>100-48=52 \mathrm{~cm}<br/>$$</div>


slug: a-bc-is-a-triangle-in-which-a-b-6-cm-bc-8-cm-and-c-a-10-cm-what-is-the-value-of-cot-a-4-92551
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch0.db
title:
A BC is a triangle in which A B = 6 cm , BC = 8 cm and C A = 10 cm . What is the value of cot ( A /4 )?

question:
<div class="question">$A B C$ is a triangle in which $A B=6 \mathrm{~cm}, B C=8 \mathrm{~cm}$ and $C A=10 \mathrm{~cm} .$ What is the value of $\cot (A / 4) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{5}-2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{5}+2$</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">$\sqrt{3}-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{3}+1$</span> </li> </ul>

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

answer:
<div class="solution">Here, $\mathrm{AB}=6 \mathrm{~cm}, \mathrm{BC}=8 \mathrm{~cm}$ and $\mathrm{CA}=10 \mathrm{~cm}$<br/>So, $c=6 \mathrm{~cm}, \mathrm{a}=8 \mathrm{~cm}, \mathrm{~b}=10 \mathrm{~cm}$<br/>$\mathrm{S}=\frac{a+b+c}{2}=\frac{24}{2}=12$<br/>$\therefore \tan \frac{A}{2}=\sqrt{\frac{(12-10)(12-6)}{12(12-8)}}$<br/>$\left(\because \tan \frac{\mathrm{A}}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\right)$<br/>$=\sqrt{\frac{1}{4}}=\frac{1}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/r0Y6vJgH3IGr9GmbCfSUvwPSWPsVqjR3lvxaOZcV70w.original.fullsize.png"/><br/><br/>$\therefore \cot \frac{A}{2}=2$<br/>Now, $\cot \left(\frac{A}{4}+\frac{A}{4}\right)=\frac{\cot ^{2} \frac{A}{4}-1}{2 \cot \frac{A}{4}}$<br/>$\cot \left(\frac{A}{2}\right)=\frac{\cot ^{2} \frac{A}{4}-1}{2 \cot \frac{A}{4}}$<br/>Let $\cot \left(\frac{A}{4}\right)=x$<br/>$\therefore \quad 2=\frac{x^{2}-1}{2 x}$<br/>$\Rightarrow x^{2}-4 x-1=0$<br/>$\Rightarrow x=\frac{4 \pm \sqrt{16+4}}{2}$<br/>$\Rightarrow x=\frac{4 \pm 2 \sqrt{5}}{2}=2 \pm \sqrt{5}$<br/>So, $\cot \left(\frac{A}{4}\right)=\sqrt{5}+2$ or $2-\sqrt{5}$</div>


slug: a-bc-is-a-triangle-in-which-bc-10-cm-c-a-6-cm-and-a-b-8-cm-which-one-of-the-following-is-c-89887
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch0.db
title:
A BC is a triangle in which BC = 10 cm , C A = 6 cm and A B = 8 cm . Which one of the following is correct?

question:
<div class="question">$A B C$ is a triangle in which $B C=10 \mathrm{~cm}, C A=6 \mathrm{~cm}$ and $A B=8 \mathrm{~cm} .$ Which one of the following is correct?</div>

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

answer:
<div class="solution">Given a $\triangle \mathrm{ABC}$ in which $\mathrm{BC}=10 \mathrm{~cm}$, $\mathrm{CA}=6 \mathrm{~cm}$ and $\mathrm{AB}=8 \mathrm{~cm} .$<br/>Since, $\mathrm{CA}^{2}+\mathrm{AB}^{2}=36+64=100=\mathrm{BC}^{2}$<br/>$\therefore \Delta \mathrm{ABC}$ is a right angled triangle.</div>


slug: a-bc-is-a-triangle-right-angled-at-b-the-hypotenuse-a-c-is-four-times-the-perpendicular-b-d-62909
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch0.db
title:
A BC is a triangle right-angled at B . The hypotenuse ( A C ) is four times the perpendicular ( B D ) drawn to it from the opposite vertex and A D D C . What is one of the acute angle of the triangle?

question:
<div class="question">$A B C$ is a triangle right-angled at $B$. The hypotenuse $(A C)$ is four times the perpendicular $(B D)$ drawn to it from the opposite vertex and $A D   &lt; D C$.<br/>What is one of the acute angle of the triangle?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$15^{\circ}$</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">$30^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$45^{\circ}$</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">$15^{\circ}$</span> </div>

answer:
<div class="solution">Let $\mathrm{BD}=\mathrm{P}$ and $\mathrm{DE}=x$ <br/>$\Rightarrow <br/>\quad A C=4 P$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Zzx-LY9gIT819g0yc0XqPk9noHDWI5ll19PNrz0-NjA.original.fullsize.png"/><br/><br/>Let $\mathrm{E}$ be mid-point of $\mathrm{AC}$. Then, $\mathrm{AE}=\mathrm{EC}=\mathrm{BE}=2 \mathrm{P}$<br/>In $\Delta \mathrm{BDE},(\mathrm{BE})^{2}=(\mathrm{BD})^{2}+(\mathrm{ED})^{2}$<br/>$\Rightarrow(2 P)^{2}=(P)^{2}+x^{2}$<br/>$\Rightarrow \quad 4 \mathrm{P}^{2}=\mathrm{P}^{2}+\mathrm{x}^{2}$<br/>$\Rightarrow \quad 3 \mathrm{P}^{2}=\mathrm{x}^{2} \Rightarrow x=\sqrt{3} \mathrm{P}$<br/>Now, $\mathrm{AD}=2 \mathrm{P}-x=2 \mathrm{P}-\sqrt{3} \mathrm{P}=\mathrm{P}(2-\sqrt{3})$<br/>$\mathrm{DC}=2 \mathrm{P}+x=2 \mathrm{P}+\sqrt{3} \mathrm{P}=\mathrm{P}(2+\sqrt{3})$<br/>In $\Delta \mathrm{BAD}, \tan \mathrm{A}=\frac{\mathrm{BD}}{\mathrm{AD}}=\frac{\mathrm{P}}{\mathrm{P}(2-\sqrt{3})}$<br/>$=\frac{1}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2-\sqrt{3}}=2+\sqrt{3}=\tan 75^{\circ}$<br/>$\tan \alpha=\frac{\mathrm{AD}}{\mathrm{BD}}=\frac{\mathrm{P}(2-\sqrt{3})}{\mathrm{P}}=2-\sqrt{3}=\tan 15^{\circ}$<br/>$\Rightarrow \quad \alpha=15^{\circ}$<br/>As, $\Delta \mathrm{ABC}$ is right angled at $\mathrm{B}$, from figure $\alpha+\beta=90^{\circ}$ $\Rightarrow \quad 15+\beta=90^{\circ} \Rightarrow \beta=75^{\circ}$<br/>In $\triangle \mathrm{ABC}, \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$<br/>$\Rightarrow \quad 75^{\circ}+90^{\circ}+\angle \mathrm{C}=180^{\circ}$<br/>$\Rightarrow \quad \angle \mathrm{C}=180^{\circ}-165^{\circ}=15^{\circ}$<br/>One of the a cute angle is $15^{\circ}$</div>


slug: a-bc-is-a-triangle-right-angled-at-b-the-hypotenuse-a-c-is-four-times-the-perpendicular-b-d-34268
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2011 (Phase 1)']
answer: <div class="solution">$\angle \mathrm{ABD}=\alpha=15^{\circ}$</div>
source_db: MarksBatch0.db
title:
A BC is a triangle right-angled at B . The hypotenuse ( A C ) is four times the perpendicular ( B D ) drawn to it from the opposite vertex and AD DC . What is â  A B D ? â

question:
<div class="question">$A B C$ is a triangle right-angled at $B$. The hypotenuse $(A C)$ is four times the perpendicular $(B D)$ drawn to it from the opposite vertex and $\mathrm{AD}   &lt; \mathrm{DC}$.<br/>What is $\angle A B D_{?}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$15^{\circ}$</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">$30^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$45^{\circ}$</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">$15^{\circ}$</span> </div>


slug: a-bc-is-a-triangle-right-angled-at-b-the-hypotenuse-a-c-is-four-times-the-perpendicular-b-d-57093
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch0.db
title:
A BC is a triangle right-angled at B . The hypotenuse ( A C ) is four times the perpendicular ( B D ) drawn to it from the opposite vertex and AD DC . What is A D : D C equal to?

question:
<div class="question">$A B C$ is a triangle right-angled at $B$. The hypotenuse $(A C)$ is four times the perpendicular $(B D)$ drawn to it from the opposite vertex and $\mathrm{AD}   &lt; \mathrm{DC}$.<br/>What is $A D: D C$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(7-2 \sqrt{3}): 1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(7-4 \sqrt{3}): 1$</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">$1: 2$</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">$(7-4 \sqrt{3}): 1$</span> </div>

answer:
<div class="solution">$\mathrm{AD}: \mathrm{DC}=\frac{\mathrm{AD}}{\mathrm{DC}}=\frac{\mathrm{P}(2-\sqrt{3})}{\mathrm{P}(2+\sqrt{3})}=\frac{2-\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}$<br/>$=\frac{4+3-4 \sqrt{3}}{1}=\frac{7-4 \sqrt{3}}{1}$<br/>$\mathrm{AD}: \mathrm{DC}=7-4 \sqrt{3}: 1$</div>


slug: a-bc-is-a-triangle-right-angled-at-b-the-hypotenuse-a-c-is-four-times-the-perpendicular-b-d-10763
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch0.db
title:
A BC is a triangle right-angled at B . The hypotenuse ( A C ) is four times the perpendicular ( B D ) drawn to it from the opposite vertex and AD DC . What is tan ( A â C ) equal to?

question:
<div class="question">$A B C$ is a triangle right-angled at $B$. The hypotenuse $(A C)$ is four times the perpendicular $(B D)$ drawn to it from the opposite vertex and $\mathrm{AD}   &lt; \mathrm{DC}$.<br/>What is tan $(A-C)$ equal to?</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=""> <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">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">None of these</span> </div>

answer:
<div class="solution">$\tan (A-C)=\tan \left(75^{\circ}-15^{\circ}\right)=\tan 60^{\circ}=\sqrt{3}$</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: MarksBatch0.db
title:
A BC is an equilateral triangle inscribed in a circle of centre O and radius 5 cm . Let the diameter through C meet the circle again at D . Assertion (A): A D â B D OB â OC Reason (R): 2 ( A D 2 + B D 2 ) = C D 2 = 100 sq cm

slug:
a-bc-is-an-equilateral-triangle-inscribed-in-a-circle-of-centre-o-and-radius-5-cm-let-the-diameter-77544

question:
<div class="question">$A B C$ is an equilateral triangle inscribed in a circle of centre $O$ and radius $5 \mathrm{~cm}$. Let the diameter through $C$ meet the circle again at $\mathrm{D}$. <br/>Assertion (A): $A D \cdot B D   &lt; O B \cdot O C$<br/>Reason (R): $2\left(A D^{2}+B D^{2}\right)=C D^{2}=100$ sq cm</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A is false but $\mathrm{R}$ is true</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/JaogXx5q3BI9d6hOZ1NgOpS6oT4gJ7w1tzoVkYUI77g.original.fullsize.png"/><br/><br/>(A) : Consider $A D \cdot B D &lt; O B \cdot O C$<br/>Now, $\mathrm{OA}=5 \mathrm{~cm}$ (Radius), $\mathrm{OB}=5 \mathrm{~cm}$ (Radius) and $\mathrm{OC}=5 \mathrm{~cm}($ Radius $)$<br/>Since, $\Delta O A D$ and $\triangle O B D$<br/>are congruent by SAS therefore<br/>$$<br/>A D=O A=5 \mathrm{~cm}<br/>$$<br/>and $B D=O B=5 \mathrm{~cm}$<br/>Thus, $A D \cdot B D=5 \times 5=25$<br/>and $\quad O B \cdot O C=5 \times 5=25$<br/>Thus, we have<br/>$A D \cdot B D=25=O B \cdot O C$<br/>$$<br/>\begin{aligned}<br/>\operatorname{Now}(\mathrm{R}): &amp; 2\left(A D^{2}+B D^{2}\right) \\<br/>&amp;=2[25+25]=100 \\<br/>\text { and } \quad &amp; C D^{2}=(10)^{2}=100<br/>\end{aligned}<br/>$$<br/>Thus, $2\left(A D^{2}+B D^{2}\right)=C D^{2}=100$ sq. $\mathrm{cm}$.<br/>Hence, (A) is false and (R) is true.</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch0.db
title:
A BC is an equilateral triangle inscribed in a circle of centre O and radius 5 cm . Let the diameter through C meet the circle again at D . Assertion (A): A D â B D OB â OC Reason (R): 2 ( A D 2 + B D 2 ) = C D 2 = 100 sq cm

slug:
a-bc-is-an-equilateral-triangle-inscribed-in-a-circle-of-centre-o-and-radius-5-cm-let-the-diameter-28473

question:
<div class="question">$A B C$ is an equilateral triangle inscribed in a circle of centre $O$ and radius $5 \mathrm{~cm}$. Let the diameter through $C$ meet the circle again at $\mathrm{D}$. <br/>Assertion (A): $A D \cdot B D   &lt; O B \cdot O C$<br/>Reason (R): $2\left(A D^{2}+B D^{2}\right)=C D^{2}=100$ sq cm</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A is false but $\mathrm{R}$ is true</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/JaogXx5q3BI9d6hOZ1NgOpS6oT4gJ7w1tzoVkYUI77g.original.fullsize.png"/><br/><br/>(A) : Consider $A D \cdot B D &lt; O B \cdot O C$<br/>Now, $\mathrm{OA}=5 \mathrm{~cm}$ (Radius), $\mathrm{OB}=5 \mathrm{~cm}$ (Radius) and $\mathrm{OC}=5 \mathrm{~cm}($ Radius $)$<br/>Since, $\Delta O A D$ and $\triangle O B D$<br/>are congruent by SAS therefore<br/>$$<br/>A D=O A=5 \mathrm{~cm}<br/>$$<br/>and $B D=O B=5 \mathrm{~cm}$<br/>Thus, $A D \cdot B D=5 \times 5=25$<br/>and $\quad O B \cdot O C=5 \times 5=25$<br/>Thus, we have<br/>$A D \cdot B D=25=O B \cdot O C$<br/>$$<br/>\begin{aligned}<br/>\operatorname{Now}(\mathrm{R}): &amp; 2\left(A D^{2}+B D^{2}\right) \\<br/>&amp;=2[25+25]=100 \\<br/>\text { and } \quad &amp; C D^{2}=(10)^{2}=100<br/>\end{aligned}<br/>$$<br/>Thus, $2\left(A D^{2}+B D^{2}\right)=C D^{2}=100$ sq. $\mathrm{cm}$.<br/>Hence, (A) is false and (R) is true.</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2022 (04 Jul Shift 1)']
source_db: MarksBatch0.db
title:
A body is suspended from a string of length 1 m and mass 2 g . The mass of the body to produce a fundamental mode of 100 Hz frequency in the string is (Acceleration due to gravity = 10 m s - 2 )

slug:
a-body-is-suspended-from-a-string-of-length-1-m-and-mass-2-g-the-mass-of-the-body-to-produce-a-fun-94455

question:
<div class="question">A body is suspended from a string of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> and mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">g</mi></math>. The mass of the body to produce a fundamental mode of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mi>Hz</mi></math> frequency in the string is<br/>(Acceleration due to gravity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo> </mo><mi mathvariant="normal">g</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>kg</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"><mn>400</mn><mo> </mo><mi mathvariant="normal">g</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>kg</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>kg</mi></math></span> </div>

answer:
<div class="solution"><blockquote><p>Mass per unit length of the given string is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mi>m</mi><mi>l</mi></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow><mn>1</mn></mfrac><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Speed of transverse waves on a string is given by,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msqrt><mfrac><mi>T</mi><mi>μ</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mrow><mi>m</mi><mi>g</mi></mrow><mi>μ</mi></mfrac></msqrt></math></p><p>Fundamental frequency on a string fixed at both ends is given by,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><msqrt><mfrac><mrow><mi>m</mi><mi>g</mi></mrow><mi>μ</mi></mfrac></msqrt><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>m</mi><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>l</mi><mn>2</mn></msup><msup><msub><mi>f</mi><mn>0</mn></msub><mn>2</mn></msup><mi>μ</mi></mrow><mi>g</mi></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>×</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo>×</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow><mn>10</mn></mfrac><mo>=</mo><mn>8</mn><mo> </mo><mi>kg</mi></math></p></blockquote></div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'NEET']
source_db: MarksBatch0.db
title:
A body of density d 1 is counterpoised by M g of weights of density d 2 in the air of density d . Then the true mass of the body is

slug:
a-body-of-density-d-1-is-counterpoised-by-m-g-of-weights-of-density-d-2-in-the-air-of-density-d-th-18740

question:
<div class="question">A body of density<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msub><mi>d</mi><mn>1</mn></msub></math><sub> </sub>is counterpoised by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>g</mi></math> of weights of density <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mn>2</mn></msub></math> in the air of density <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>. Then the true mass of the body is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math><div id="gtx-trans" style="position: absolute; left: 86px; top: -5.77778px;"><div class="gtx-trans-icon"> </div></div><div id="gtx-trans" style="position: absolute; left: 88px; top: -5.77778px;"><div class="gtx-trans-icon"> </div></div></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>d</mi><msub><mi>d</mi><mn>2</mn></msub></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>d</mi><msub><mi>d</mi><mn>1</mn></msub></mfrac></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>d</mi><mo>/</mo><msub><mi>d</mi><mn>2</mn></msub></mrow></mfenced></mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>d</mi><mo>/</mo><msub><mi>d</mi><mn>1</mn></msub></mrow></mfenced></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>d</mi><mo>/</mo><msub><mi>d</mi><mn>2</mn></msub></mrow></mfenced></mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>d</mi><mo>/</mo><msub><mi>d</mi><mn>1</mn></msub></mrow></mfenced></mfrac></math></span> </div>

answer:
<div class="solution">Let M<sub>0</sub> = mass of body in vacuum.<br/>Apparent weight of the body in air=Apparent weight of standard weights in air <br/>â Actual weight - upthrust due to displaced air = Actual weight - upthrust due to displaced air<br/><br/><math><mrow><mo> â</mo> <mspace width="0.5em"></mspace> <msub> <mtext>M</mtext> <mn>0</mn> </msub> <mtext>g</mtext> <mo>-</mo> <mfenced close=")" open="(" separators=","> <mrow> <mfrac> <mrow> <msub> <mtext>M</mtext> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mtext>d</mtext> <mn>1</mn> </msub> </mrow> </mfrac> </mrow> </mfenced> <mtext>d</mtext> <mtext>g</mtext> <mo>=</mo> <mtext>M</mtext> <mtext>g</mtext> <mo>-</mo> <mfenced close=")" open="(" separators=","> <mrow> <mfrac> <mrow> <mtext>M</mtext> </mrow> <mrow> <msub> <mtext>d</mtext> <mn>2</mn> </msub> </mrow> </mfrac> </mrow> </mfenced> <mtext>dg</mtext> <mspace width="0.5em"></mspace> <mo>â</mo> <msub> <mtext>M</mtext> <mn>0</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mtext>M</mtext> <mfenced close="]" open="[" separators=","> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mrow> <mtext>d</mtext> </mrow> <mrow> <msub> <mtext>d</mtext> <mn>2</mn> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mfenced close="]" open="[" separators=","> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mrow> <mtext>d</mtext> </mrow> <mrow> <msub> <mtext>d</mtext> <mn>1</mn> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mfrac> </mrow> </math></div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'NEET']
source_db: MarksBatch0.db
title:
A body of density d , is counterpoised by M g of weight of density d 2 in air of density d . Then, the true mass of the body is

slug:
a-body-of-density-d-is-counterpoised-by-m-g-of-weight-of-density-d-2-in-air-of-density-d-then-t-72757

question:
<div class="question">A body of density <math><mi>d</mi></math>, is counterpoised by <math><mi>M</mi><mi>g</mi></math> of weight of density <math><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></math> in air of density <math><mi>d</mi></math> . Then, the true mass of the body is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mi>M</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mi>M</mi><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mi>M</mi><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mi>M</mi><mfenced separators="|"><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfrac></mrow></mfenced></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> </ul>

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

answer:
<div class="solution">Let the <math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub></math>  is the mass of the body. Apparent weight of the body in air<br/><br/> <math><mo>=</mo></math>  Apparent weight of standard weight in air <br/><br/> <math><mo>â´</mo></math>  Actual weight - Upthrust due to displaced air <br/><br/> <math><mo>â</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>g</mi><mo>-</mo><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi>d</mi><mi>g</mi><mo>=</mo><mi>M</mi><mi>g</mi><mo>-</mo><mfrac><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mi>d</mi><mi>g</mi></math> <br/><br/> <math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>g</mi><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>=</mo><mi>M</mi><mi>g</mi><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></math> <br/><br/> <math><mo>â´</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>M</mi><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfrac></math></div>


tags: ['Physics', 'Oscillations', 'TS EAMCET', 'TS EAMCET 2019 (04 May Shift 1)']
source_db: MarksBatch0.db
title:
A body of mass 0.3 kg hangs by a spring with a force constant of 50 N / m . The amplitude of oscillations is damped and reaches e 1 â of its original value in about 100 oscillations. If Ï and Ï â² are the angular frequencies of undamped and damped oscillations respectively, then percentage of ( Ï Ï â Ï â ) is

slug:
a-body-of-mass-0-3-kg-hangs-by-a-spring-with-a-force-constant-of-50-n-m-the-amplitude-of-oscilla-38396

question:
<div class="question">A body of mass $0.3 \mathrm{~kg}$ hangs by a spring with a force constant of $50 \mathrm{~N} / \mathrm{m}$. The amplitude of oscillations is damped and reaches $\frac{1}{e}$ of its original value in about 100 oscillations. If $\omega$ and $\omega^{\prime}$ are the angular frequencies of undamped and damped oscillations respectively, then percentage of $\left(\frac{\omega-\omega}{\omega}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\frac{1}{800 \pi}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{\pi^2}{600}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{800 \pi^2}\right)$</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">D</span> <span class="option-data">$\left(\frac{\pi}{400}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{1}{800 \pi^2}\right)$</span> </div>

answer:
<div class="solution">Hint The displacement of a spring mass oscillator, $X=A e^{-b t / 2 m} \cos \left(\omega^{\prime} t+\phi\right)$ where, $\omega^{\prime}=$ angular frequency of damped oscillation and $\omega^{\prime}=\omega_0 \sqrt{1-\frac{b^2}{4 m^2 \omega_0^2}}$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'NEET']
source_db: MarksBatch0.db
title:
A body of mass M hits normally a rigid wall with velocity V and bounce s back with the same velocity. The impulse experienced by the body is :

slug:
a-body-of-mass-m-hits-normally-a-rigid-wall-with-velocity-v-and-bounce-s-back-with-the-same-velocity-76095

question:
<div class="question"><div> <p align="left" style="text-align:left;"><span>A body of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> hits normally a rigid wall with velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> and bounce</span><span>s</span><span> back with the same velocity. The impulse experienced by the body is :</span></p> </div> </div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><div><p align="left" style="text-align:left;"><i><span><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>V</mi></math></span></i></p></div></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><div><p align="left" style="text-align:left;"><span><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>M</mi><mi>V</mi></math></span></p></div><div id="gtx-trans" style="position: absolute; left: 89px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><div><p align="left" style="text-align:left;"><span><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>M</mi><mi>V</mi></math></span></p></div><div id="gtx-trans" style="position: absolute; left: 51px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><div><p align="left" style="text-align:left;"><span><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>M</mi><mi>V</mi></math></span></p></div><div id="gtx-trans" style="position: absolute; left: 51px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> </div>

answer:
<div class="solution"><div><p align="left" style="text-align:left;"><span>Impulse,  <math> <mrow> <mover accent="true"> <mi>J</mi> <mo>→</mo> </mover> <mo>=</mo><mstyle displaystyle="true"> <mrow> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </msubsup> <mrow> <mover accent="true"> <mi>F</mi> <mo>→</mo> </mover> <mo>.</mo><mi>d</mi><mi>t</mi> </mrow> </mrow> </mstyle> </mrow> </math> </span></p><p align="left" style="text-align:left;"><span>or            <math> <mrow> <mi>J</mi><mo>=</mo><msub> <mi>F</mi> <mrow> <mi>avg</mi> </mrow> </msub> <mo>.</mo><mtext> </mtext><mi>Δ</mi><mi>t</mi> </mrow> </math> </span></p><p align="left" style="text-align:left;"><span>Now, according to Newton’s Second Law</span></p><p align="left" style="text-align:left;"><span style="position:relative;top:0.0pt"> <math> <mrow> <mover accent="true"> <mi>F</mi> <mo>→</mo> </mover> <mo>=</mo><mfrac> <mrow> <mi>d</mi><mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>d</mi><mi>t</mi> </mrow> </mfrac> <mtext> </mtext><mtext> </mtext><mi>or</mi><mtext> </mtext><mtext> </mtext><mstyle displaystyle="true"> <mrow> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </msubsup> <mrow> <mover accent="true"> <mi>F</mi> <mo>→</mo> </mover> <mo>.</mo><mtext> </mtext><mi>d</mi><mi>t</mi> </mrow> </mrow> </mstyle><mo>=</mo><mstyle displaystyle="true"> <mrow> <msubsup> <mo>∫</mo> <mrow> <msub> <mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> <mn>2</mn> </msub> </mrow> </msubsup> <mrow> <mi>d</mi><mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> </math> </span></p><p align="left" style="text-align:left;"><span style="position:relative;top:0.0pt"> <math> <mrow> <mo>∴</mo><mtext> </mtext><mtext> </mtext><mover accent="true"> <mi>J</mi> <mo>→</mo> </mover> <mo>=</mo><msub> <mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> <mn>2</mn> </msub> <mo>−</mo><msub> <mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> <mn>1</mn> </msub> <mo>=</mo><mi>Δ</mi><mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> </mrow> </math> </span><span>  </span></p><p align="left" style="text-align:left;"><span>[<i>i.e.</i>, the action of impulse is to change  the momentum of a body and the impulse of force is equal to the change in momentum. This is also known as impulse-momentum theorem]</span></p><p align="left" style="text-align:left;"><span>In the given case,</span></p><p align="left" style="text-align:left;"><span><img alt="" height="133" src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/IL0001908_AnsExp3_1.png" width="132"/></span></p><p align="left" style="text-align:left;"><span>Impulse  <math> <mrow> <mo>=</mo><msub> <mover accent="true"> <mi>p</mi> <mo>→</mo> </mover> <mn>2</mn> </msub> <mo>−</mo><msub> <mover accent="true"> <mi>p</mi> <mo>→</mo> </mover> <mn>1</mn> </msub> </mrow> </math>  <br/> (<i>i.e.</i>, impulse  <math> <mo>=</mo> </math> change in momentum)</span></p><p align="left" style="text-align:left;"><span>or Impulse,  <math> <mrow> <mi>J</mi><mo>=</mo><mi>M</mi><mtext> </mtext><msub> <mover accent="true"> <mi>v</mi> <mo>→</mo> </mover> <mn>2</mn> </msub> <mo>−</mo><mi>M</mi><mtext> </mtext><msub> <mover accent="true"> <mi>v</mi> <mo>→</mo> </mover> <mn>1</mn> </msub> </mrow> </math> </span></p><p align="left" style="text-align:left;"><span style="position:relative;top:0.0pt"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mo>⇒</mo><mtext> </mtext><mtext> </mtext><mi>J</mi><mo>=</mo><mi>M</mi><mrow><mo>[</mo><mrow><mo>(</mo><mi>v</mi><mo>)</mo><mo>−</mo><mo>(</mo><mo>−</mo><mi>v</mi><mo>)</mo></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>⇒</mo><mtext> </mtext><mtext> </mtext><mi>J</mi><mo>=</mo><mn>2</mn><mi>M</mi><mi>v</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi mathvariant="normal">N</mi><mo> </mo><mi mathvariant="normal">s</mi></mtd></mtr></mtable></math> </span></p></div> <!-- /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>
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slug: a-body-of-mass-m-is-situated-in-a-potential-field-u-x-u-0-1-cos-x-when-u-0-92265
tags: ['Physics', 'Oscillations']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A body of mass m is situated in a potential field U ( x ) = U 0 â ( 1 â cos Î±x ) when, U 0 â and Î± are constants. Find the time period of small oscillations.

question:
<div class="question">A body of mass $m$ is situated in a potential field $U(x)=U_0(1-\cos \alpha x)$ when, $U_0$ and $\alpha$ are constants. Find the time period of small oscillations.</div>

answer:
<div class="solution">As given that, the potential energy (associated with the field) for a body of mass $m$.<br/>$$<br/>U(x)=U_0(1-\cos \alpha x)<br/>$$<br/>$\because$ for conservative force $f$, we can write $f=\frac{-d u}{d x}$<br/>We have assumed the field to be conservative<br/>$$<br/>\begin{aligned}<br/>F &amp;=-\frac{d}{d x}\left(U_0-U_0 \cos \alpha x\right) \\<br/>&amp;=-U_0 \alpha \sin \alpha x<br/>\end{aligned}<br/>$$<br/>For SHM, $\alpha x$ is small so $\sin \alpha x$ i.e. $\sin \alpha x=\alpha x$ so,<br/>$$<br/>\begin{aligned}<br/>&amp;F=-U_0 \alpha^2 x \\<br/>&amp;\therefore F \propto(-x)<br/>\end{aligned}<br/>$$<br/>As, $U_0, \alpha$ being constant.<br/>$\therefore$ So motion is SHM for small oscillations.<br/>Standard equation for SHM,<br/>$$<br/>F=-m \omega^2 x<br/>$$<br/>Comparing Eqs. (ii) and (iii), then<br/>$$<br/>\begin{aligned}<br/>&amp;-m \omega^2 x=-U_0 \alpha^2 x \\<br/>&amp;m \omega^2=U_0 \alpha^2 \\<br/>&amp;\omega^2=\frac{U_0 \alpha^2}{m} \text { or } \omega=\sqrt{\frac{U_0 \alpha^2}{m}} \\<br/>&amp;\therefore \text { Time period } T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{U_0 \alpha^2}}<br/>\end{aligned}<br/>$$<br/>From (i) this period is valid for small angle $\alpha x$.<br/>$$<br/>T=\frac{2 \pi}{\alpha} \sqrt{\frac{m}{U_0}}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch0.db
title:
A box contains 3 white and 2 black balls. Two balls are drawn at random one after the other. If the balls are not replaced, what is the probability that both the balls are black?

slug:
a-box-contains-3-white-and-2-black-balls-two-balls-are-drawn-at-random-one-after-the-other-if-the-35327

question:
<div class="question">A box contains 3 white and 2 black balls. Two balls are drawn at random one after the other. If the balls are not replaced, what is the probability that both the balls are black?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 / 5$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 / 5$</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">C</span> <span class="option-data">$1 / 10$</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">$1 / 5$</span> </div>

answer:
<div class="solution">Total number of balls $=5$ Number of black balls $=2$ Required probability $=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{{ }^{2} \mathrm{C}_{1} \times{ }^{1} \mathrm{C}_{1}}{{ }^{5} \mathrm{C}_{2}}=\frac{2}{5} \times \frac{1}{2}=\frac{1}{5}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch0.db
title:
A box contains 6 distinct dolls. From this box, 3 dolls are randomly selected one by one with replacement. What is the probability of selecting 3 distinct dolls?

slug:
a-box-contains-6-distinct-dolls-from-this-box-3-dolls-are-randomly-selected-one-by-one-with-replac-26390

question:
<div class="question">A box contains 6 distinct dolls. From this box, 3 dolls are randomly selected one by one with replacement. What is the probability of selecting 3 distinct dolls?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 / 54$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$12 / 25$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 20$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$5 / 9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 / 9$</span> </div>

answer:
<div class="solution">Required probability $=\frac{{ }^{6} \mathrm{C}_{1} \times{ }^{5} \mathrm{C}_{1} \times{ }^{4} \mathrm{C}_{1}}{{ }^{6} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{1}}$<br/>$=\frac{6 \times 5 \times 4}{6 \times 6 \times 6}=\frac{5}{9}$</div>


tags: ['Mathematics', 'Probability', 'JEE Main']
source_db: MarksBatch0.db
title:
A box contains 6 pens, 2 of which are defective. Two pens are taken randomly from the box. If random variable x : Number of defective pens obtained, then standard deviation of x =

slug:
a-box-contains-6-pens-2-of-which-are-defective-two-pens-are-taken-randomly-from-the-box-if-random-72713

question:
<div class="question">A box contains <math> <mn>6</mn> </math> pens, <math> <mn>2</mn> </math> of which are defective. Two pens are taken randomly from the box. If random variable <math><mi>x</mi><mo>:</mo></math> Number of defective pens obtained, then standard deviation of <math><mi>x</mi><mo>=</mo></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mo>Â±</mo><mi>Â </mi><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><mo>â</mo><mn>5</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mn>16</mn></mrow><mrow><mn>45</mn></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><math><mi>x</mi></math> : no. of defective pens<br/>Two pens are taken from box<br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi>x</mi></math> can taken values 0, 1, 2<br/><math><mi>p</mi><mfenced separators="|"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>4</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>6</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>Ã</mo><mn>3</mn></mrow><mrow><mn>6</mn><mo>Ã</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mi>p</mi><mi> </mi><mfenced separators="|"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>Ã</mo><msup><mrow><mi> </mi></mrow><mrow><mn>4</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>6</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>Ã</mo><mn>4</mn><mo>Ã</mo><mn>2</mn><mo>Ã</mo><mn>1</mn></mrow><mrow><mn>6</mn><mo>Ã</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mi>p</mi><mfenced separators="|"><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>6</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>Ã</mo><mn>2</mn><mo>Ã</mo><mn>1</mn></mrow><mrow><mn>6</mn><mo>Ã</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><table border="1"> <tbody> <tr> <td>x</td> <td>P</td> <td><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></td> <td><math><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></td> </tr> <tr> <td>0</td> <td><math><mfrac><mrow><mn>6</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td>0</td> <td>0</td> </tr> <tr> <td>1</td> <td><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> </tr> <tr> <td>2</td> <td><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> </tr> </tbody></table><math><mi>E</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math><br/><math><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>12</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math><br/>Standard deviation <math><mo>=</mo><msqrt><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>-</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/>Standard deviation <math><mo>=</mo><msqrt><mfenced separators="|"><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>9</mn></mrow></mfrac></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn><mo>Ã</mo><mn>4</mn></mrow><mrow><mn>45</mn></mrow></mfrac></msqrt></math><br/><math><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><mo>â</mo><mn>5</mn></mrow></mfrac></math></div>


tags: ['Mathematics', 'Probability', 'MHT CET', 'MHT CET 2017']
source_db: MarksBatch0.db
title:
A box contains 6 pens, 2 of which are defective. Two pens are taken randomly from the box. If random variable x : Number of defective pens obtained, then standard deviation of x =

slug:
a-box-contains-6-pens-2-of-which-are-defective-two-pens-are-taken-randomly-from-the-box-if-random-55998

question:
<div class="question">A box contains <math> <mn>6</mn> </math> pens, <math> <mn>2</mn> </math> of which are defective. Two pens are taken randomly from the box. If random variable <math><mi>x</mi><mo>:</mo></math> Number of defective pens obtained, then standard deviation of <math><mi>x</mi><mo>=</mo></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mo>Â±</mo><mi>Â </mi><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><mo>â</mo><mn>5</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mn>16</mn></mrow><mrow><mn>45</mn></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><math><mi>x</mi></math> : no. of defective pens<br/>Two pens are taken from box<br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi>x</mi></math> can taken values 0, 1, 2<br/><math><mi>p</mi><mfenced separators="|"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>4</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>6</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>Ã</mo><mn>3</mn></mrow><mrow><mn>6</mn><mo>Ã</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mi>p</mi><mi> </mi><mfenced separators="|"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>Ã</mo><msup><mrow><mi> </mi></mrow><mrow><mn>4</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>6</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>Ã</mo><mn>4</mn><mo>Ã</mo><mn>2</mn><mo>Ã</mo><mn>1</mn></mrow><mrow><mn>6</mn><mo>Ã</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mi>p</mi><mfenced separators="|"><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mrow><mi> </mi></mrow><mrow><mn>6</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>Ã</mo><mn>2</mn><mo>Ã</mo><mn>1</mn></mrow><mrow><mn>6</mn><mo>Ã</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><table border="1"> <tbody> <tr> <td>x</td> <td>P</td> <td><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></td> <td><math><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></td> </tr> <tr> <td>0</td> <td><math><mfrac><mrow><mn>6</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td>0</td> <td>0</td> </tr> <tr> <td>1</td> <td><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>8</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> </tr> <tr> <td>2</td> <td><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> <td><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math></td> </tr> </tbody></table><math><mi>E</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math><br/><math><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>12</mn></mrow><mrow><mn>15</mn></mrow></mfrac></math><br/><math><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math><br/>Standard deviation <math><mo>=</mo><msqrt><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>-</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/>Standard deviation <math><mo>=</mo><msqrt><mfenced separators="|"><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>9</mn></mrow></mfrac></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn><mo>Ã</mo><mn>4</mn></mrow><mrow><mn>45</mn></mrow></mfrac></msqrt></math><br/><math><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><mo>â</mo><mn>5</mn></mrow></mfrac></math></div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch0.db
title:
A box contains 10 identical electronic components of which 4 are defective. If 3 components are selected at random from the box in succession, without replacing the units already drawn, what is the probability that two of the selected components are defective?

slug:
a-box-contains-10-identical-electronic-components-of-which-4-are-defective-if-3-components-are-sele-58823

question:
<div class="question">A box contains 10 identical electronic components of which 4 are defective. If 3 components are selected at random from the box in succession, without replacing the units already drawn, what is the probability that two of the selected components are defective?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 / 24$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3 / 10$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$1 / 40$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 / 10$</span> </div>

answer:
<div class="solution">Total number of selecting 3 components out of $10=$ ${ }^{10} \mathrm{C}_{3}$. Out of 3 selected components two defective pieces can be selected in ${ }^{4} \mathrm{C}_{2}$ ways and one non-defective piece will be selected in ${ }^{6} \mathrm{C}_{1}$ ways, hence.<br/>Required probability $=\frac{{ }^{6} \mathrm{C}_{1} \times{ }^{4} \mathrm{C}_{2}}{{ }^{10} \mathrm{C}_{3}}=\frac{6 \times 6 \times 6}{10 \times 9 \times 8}=\frac{3}{10}$</div>


slug: a-box-has-ten-chits-numbered-0-1-2-3-9-first-one-chit-is-drawn-at-random-and-kept-as-52377
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch0.db
title:
A box has ten chits numbered 0 , 1 , 2 , 3 , â¦ , 9. First, one chit is drawn at random and kept aside. From the remaining, a second chit is drawn at random. What is the probability that the second chit drawn is " 9 "?

question:
<div class="question">A box has ten chits numbered $0,1,2,3, \ldots$, 9. First, one chit is drawn at random and kept aside. From the remaining, a second chit is drawn at random. What is the probability that the second chit drawn is " 9 "?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{10}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{9}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{90}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{1}{90}$</span> </div>

answer:
<div class="solution">First chit can be drawn in 10 ways. Second chit can be drawn in 9 ways. Total number of events, $\mathrm{n}(\mathrm{S})=10 \times 9=90$. Number of events of drawing chit numbered $9=\mathrm{n}(\mathrm{E})=1$<br/>Probability $=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{1}{90}$</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: MarksBatch0.db
title:
A boy throws a cricket ball from the boundary to the wicket keeper. If the frictional force due to air f a cannot be ignored, the forces acting on the ball at the position X are represented by

slug:
a-boy-throws-a-cricket-ball-from-the-boundary-to-the-wicket-keeper-if-the-frictional-force-due-to-a-87345

question:
<div class="question"><p>A boy throws a cricket ball from the boundary to the wicket keeper. If the frictional force due to air <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>f</mi><mi>a</mi></msub></mfenced></math> cannot be ignored, the forces acting on the ball at the position <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> are represented by</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/7ab689cd-874c-4388-a542-b56846632b47-image.png" style="width: 400px; height: 174px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/c66823f5-adf8-4f74-99a8-556f3aaa63b4-image.png" style="width: 109px; height: 192px;"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/064c67ce-578d-4bd5-a4fa-89946e7b72a5-image.png" style="width: 89px; height: 194px;"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/48c6e526-6611-412e-bf3d-f3e9b7b5f704-image.png" style="width: 171px; height: 148px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/3cc3c37f-e23f-493f-abdf-ca24ee379ccf-image.png" style="width: 145px; height: 175px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/48c6e526-6611-412e-bf3d-f3e9b7b5f704-image.png" style="width: 171px; height: 148px;"/></span> </div>

answer:
<div class="solution"><p>The fiction force due to air on the ball <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>a</mi></msub><mo> </mo></math>will always act in the opposite direction of the velocity of the ball, and the velocity of the ball at any point on the path is always in the tangential direction of the path of the projectile.</p><p>The weight of the body always acts downward perpendicular to the surface of the earth and along the centre of the earth.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/f9524d4c-db9e-4fbf-bf30-3a3e75472261-image.png" style="width: 240px; height: 96px;"/></p></div>


tags: ['Physics', 'Laws of Motion', 'AP EAMCET', 'AP EAMCET 2021 (20 Aug Shift 1)']
source_db: MarksBatch0.db
title:
A boy throws a cricket ball from the boundary to the wicket keeper. If the frictional force due to air f a cannot be ignored, the forces acting on the ball at the position X are represented by

slug:
a-boy-throws-a-cricket-ball-from-the-boundary-to-the-wicket-keeper-if-the-frictional-force-due-to-a-71911

question:
<div class="question"><p>A boy throws a cricket ball from the boundary to the wicket keeper. If the frictional force due to air <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>f</mi><mi>a</mi></msub></mfenced></math> cannot be ignored, the forces acting on the ball at the position <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> are represented by</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/7ab689cd-874c-4388-a542-b56846632b47-image.png" style="width: 400px; height: 174px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/c66823f5-adf8-4f74-99a8-556f3aaa63b4-image.png" style="width: 109px; height: 192px;"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/064c67ce-578d-4bd5-a4fa-89946e7b72a5-image.png" style="width: 89px; height: 194px;"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/48c6e526-6611-412e-bf3d-f3e9b7b5f704-image.png" style="width: 171px; height: 148px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/3cc3c37f-e23f-493f-abdf-ca24ee379ccf-image.png" style="width: 145px; height: 175px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/48c6e526-6611-412e-bf3d-f3e9b7b5f704-image.png" style="width: 171px; height: 148px;"/></span> </div>

answer:
<div class="solution"><p>The fiction force due to air on the ball <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>a</mi></msub><mo> </mo></math>will always act in the opposite direction of the velocity of the ball, and the velocity of the ball at any point on the path is always in the tangential direction of the path of the projectile.</p><p>The weight of the body always acts downward perpendicular to the surface of the earth and along the centre of the earth.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/f9524d4c-db9e-4fbf-bf30-3a3e75472261-image.png" style="width: 240px; height: 96px;"/></p></div>


tags: ['Physics', 'Laws of Motion', 'KCET', 'KCET 2010']
source_db: MarksBatch0.db
title:
A boy throws a cricket ball from the boundary to the wicket-keeper. If the frictional force due to air cannot be ignored, the forces acting on the ball at the position X are respected by

slug:
a-boy-throws-a-cricket-ball-from-the-boundary-to-the-wicket-keeper-if-the-frictional-force-due-to-a-43505

question:
<div class="question">A boy throws a cricket ball from the boundary to the wicket-keeper. If the frictional force due to air cannot be ignored, the forces acting on the ball at the position $X$ are respected by<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/iEKJYAsnSzU5bLNZ3GA4xGDR5vyBYQdXH-hGZdGYc_M.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/SEWsQrPmKAeJfxBtTj6VUNYqv3qjA6CnVD6iZ0LQBcs.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/rFQUgdCSqy5XzNGgGiv7_lFSY7kjhNhOb-XeK2tM53s.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/K0vBgrKXAHfhKxa-HcmWnejBZJongLFXWPwpeqf3yaE.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/bkluIkBSR7ZwsFY-hHRX5a6ZLZhG_rmV17GgIurQO8Q.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/K0vBgrKXAHfhKxa-HcmWnejBZJongLFXWPwpeqf3yaE.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution">The forces acting on the ball will be (i) in the direction opposite to its motion ie, frictional force and (ii) weight mg.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch0.db
title:
A can hit a target 4 times in 5 shots; B can hit a target 3 times in 4 shots: C can hit a target 2 times in 3 shots All the three fire a shot each. What is the probability that two shots are at least hit?

slug:
a-can-hit-a-target-4-times-in-5-shots-b-can-hit-a-target-3-times-in-4-shots-c-can-hit-a-target-2-t-45241

question:
<div class="question">A can hit a target 4 times in 5 shots; $\mathrm{B}$ can hit a target 3 times in 4 shots:<br/>$\mathrm{C}$ can hit a target 2 times in 3 shots All the three fire a shot each. What is the probability that two shots are at least hit?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 / 6$</span> </div>

answer:
<div class="solution">Probability of no one hitting the target<br/>$=\frac{1}{5 \times 4 \times 3}=\frac{1}{60}$<br/>Probability of one hitting the target<br/>$=\frac{4+3+2}{60}=\frac{9}{60}$<br/>$\therefore$ Probability of maximum one hit<br/>$=\frac{1}{60}+\frac{9}{60}=\frac{10}{60}=\frac{1}{6}$<br/>Probability that two shots are hit at least is the required<br/>probability $=1-\frac{1}{6}=\frac{5}{6}$</div>


tags: ['Mathematics', 'Probability', 'KCET', 'KCET 2021']
source_db: MarksBatch0.db
title:
A car manufacturing factory has two plants X and Y . Plant X manufactures 70% of cars and plant Y manufactures 30% of cars. 80% of cars at plant X and 90% of cars at plant Y are rated as standard quality. A car is chosen at random and is found to be standard quality. The probability that it has come from plant X is

slug:
a-car-manufacturing-factory-has-two-plants-x-and-y-plant-x-manufactures-70-of-cars-and-plant-y-ma-52721

question:
<div class="question">A car manufacturing factory has two plants $X$ and $Y$. Plant $X$ manufactures $70 \%$ of cars and plant $Y$ manufactures $30 \%$ of cars. $80 \%$ of cars at plant $X$ and $90 \%$ of cars at plant $Y$ are rated as standard quality. A car is chosen at random and is found to be standard quality. The probability that it has come from plant $X$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{56}{73}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{56}{84}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{56}{83}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{56}{79}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{56}{83}$</span> </div>

answer:
<div class="solution">Let $E=$ the event that the car is of standard quáº£lity.<br/>Let $A_{1}=$ the event that the car is manufactured in plant $X$.<br/>and $A_{2}=$ the event that the car is manufactured in plant $Y$.<br/>Now, $P\left(A_{1}\right)=\frac{70}{100}=\frac{7}{10}, P\left(A_{2}\right)=\frac{30}{100}=\frac{3}{10}$ $P\left(\frac{E}{A_{1}}\right)=$ Probability that a standard quality car is manufactured in plant $X=\frac{80}{100}=\frac{8}{10}$ $P\left(\frac{E}{A_{2}}\right)=$ Probability that a standard quality car is manu factured in blant $Y=\frac{90}{100}=\frac{9}{10}$ $P\left(\frac{A_{1}}{E}\right)=$ Probability that a standard quality car has come from plant $X$<br/>$\begin{aligned}<br/>=&amp; \frac{P\left(A_{1}\right) \times P\left(\frac{E}{A_{1}}\right)}{P\left(A_{1}\right) \cdot P\left(\frac{E}{A_{1}}\right)+P\left(A_{2}\right) \cdot P\left(\frac{E}{A_{2}}\right)} \\<br/>=&amp; \frac{\frac{7}{10} \times \frac{8}{10}}{\frac{7}{10} \times \frac{8}{10}+\frac{3}{10} \times \frac{9}{10}} \\<br/>=&amp; \frac{7 \times 8}{7 \times 8+3 \times 9} \\<br/>=&amp; \frac{56}{56+27}=\frac{56}{83}<br/>\end{aligned}$<br/>Hence, the required probability is $\frac{56}{83}$.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9 to 4</span> </div>
source_db: MarksBatch0.db
title:
A card is drawn from a pack of 52 cards and a gambler bets that it is a spade or an ace. Which one of the following are the odds against his winning this bet?

slug:
a-card-is-drawn-from-a-pack-of-52-cards-and-a-gambler-bets-that-it-is-a-spade-or-an-ace-which-one-o-63321

question:
<div class="question">A card is drawn from a pack of 52 cards and a gambler bets that it is a spade or an ace. Which one of the following are the odds against his winning this bet?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">13 to 4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4 to 13</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">9 to 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">4 to 9</span> </li> </ul>

answer:
<div class="solution">Probability of getting a spade $=\frac{13}{52}$<br/>Probability of an ace $=\frac{4}{52}$<br/>and probability of getting a spade ace $=\frac{1}{52}$<br/>$\therefore \quad$ Required probability $=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$<br/>Odds against his winning $=\frac{1-\frac{4}{13}}{\frac{4}{13}}=\frac{\frac{9}{13}}{\frac{4}{13}}=\frac{9}{4}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch0.db
title:
A card is drawn from a well-shuffled ordinary deck of 52 cards. What is the probability that it is an ace?

slug:
a-card-is-drawn-from-a-well-shuffled-ordinary-deck-of-52-cards-what-is-the-probability-that-it-is-a-33377

question:
<div class="question">A card is drawn from a well-shuffled ordinary deck of 52 cards. What is the probability that it is an ace?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{13}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{2}{13}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{13}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{52}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{13}$</span> </div>

answer:
<div class="solution">$\quad$ Number of ace cards $=4$<br/>So, probability $=\frac{4}{52}=\frac{1}{13}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch0.db
title:
A card is drawn from a wel-shuffled deck of 52 cards. What is the probability that it is queen of spade?

slug:
a-card-is-drawn-from-a-wel-shuffled-deck-of-52-cards-what-is-the-probability-that-it-is-queen-of-sp-54345

question:
<div class="question">A card is drawn from a wel-shuffled deck of 52 cards. What is the probability that it is queen of spade?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{52}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{1}{13}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{52}$</span> </div>

answer:
<div class="solution">Prob. of getting queen of spade $=\frac{{ }^{1} C_{1}}{{ }^{52} C_{1}}=\frac{1}{52}$</div>


tags: ['Physics', 'Ray Optics']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A Cassegrain telescope uses two mirrors as shown in figure. Such a telescope is built with the mirrors 20 mm apart. If the radius of curvature of the large mirror is 220 mm and the small mirror is 140 mm , where will the final image of an object at infinity be?

slug:
a-cassegrain-telescope-uses-two-mirrors-as-shown-in-figure-such-a-telescope-is-built-with-the-mirro-62279

question:
<div class="question">A Cassegrain telescope uses two mirrors as shown in figure. Such a telescope is built with the mirrors 20 $\mathrm{mm}$ apart. If the radius of curvature of the large mirror is $220 \mathrm{~mm}$ and the small mirror is $140 \mathrm{~mm}$, where will the final image of an object at infinity be?<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/slX6sPSbwIG_CWFT-sX9_XI5Hrrz3X67yHVlCu19ISY.original.fullsize.png"/><br/></div>

answer:
<div class="solution">For objective mirror (concave), $u=-\infty$<br/>Radius of curvature $R=-220 \mathrm{~mm}=-22 \mathrm{~cm}$<br/>Focal length $f=\frac{R}{2}=-11 \mathrm{~cm}$<br/>Image will be obtained at focus.<br/>$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$ or $\frac{1}{v}+\frac{1}{-\infty}=\frac{1}{-11}$ or $v=-11 \mathrm{~cm}$<br/>Image formed by objective mirror act as a virtual object<br/>for secondary mirror.<br/>Distance of virtual object for secondary mirror.<br/>$u_0=(11-2) \mathrm{cm}=+9 \mathrm{~cm}$<br/>Focal length of convex mirror used $=14 \mathrm{~cm}$<br/>So, $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$ or $\frac{1}{v}=\frac{1}{14}-\frac{1}{9}$ or $v=-25.5 \mathrm{~cm}$</div>


tags: ['Physics', 'Current Electricity', 'TS EAMCET', 'TS EAMCET 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3.5 m</span> </div>
source_db: MarksBatch0.db
title:
A cell in secondary circuit gives null deflection for 2.5 m length of potentiometer having 10 m length of wire. If the length of the potentiometer wire is increased by 1 m without changing the cell in the primary, the position of the null point now is

slug:
a-cell-in-secondary-circuit-gives-null-deflection-for-2-5-m-length-of-potentiometer-having-10-m-leng-39257

question:
<div class="question">A cell in secondary circuit gives null deflection for $2.5 \mathrm{~m}$ length of potentiometer having $10 \mathrm{~m}$ length of wire. If the length of the potentiometer wire is increased by $1 \mathrm{~m}$ without changing the cell in the primary, the position of the null point now is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3.5 m</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3 m</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2.75 m</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2.0 m</span> </li> </ul>

answer:
<div class="solution">Resistance of potentiometer wire<br/>$R=\rho \times \frac{l}{A}$<br/>or $\quad R=\left(\rho \times \frac{10}{A}\right)$<br/>The value of $2.5 \mathrm{~m}$ length wire<br/>$R^{\prime}=\frac{\rho \times 10}{A \times 10} \times 2.5$<br/>or $\quad R^{\prime}=\left(\frac{2.5 \rho}{A \times 10}\right)$<br/>Potential<br/>$\begin{aligned}<br/>V^{\prime} &amp; =I \times R^{\prime} \\<br/>&amp; =I\left(\frac{2.5 \rho}{A \times 10}\right)<br/>\end{aligned}$<br/>Now, again the length of potentiometer wire is increased by $1 \mathrm{~m}$, then resistance of null position wire.<br/>$\begin{aligned}<br/>R^{\prime \prime} &amp; =\left(\frac{\rho \times l}{11 \times A}\right) \\<br/>V^{\prime \prime} &amp; =I R^{\prime \prime} \\<br/>and \quad V &amp; =V^{\prime}<br/>\end{aligned}$<br/>$\begin{aligned} \frac{I \times 2.5 \rho}{A \times 10} &amp; =\frac{\rho \times l}{11 \times A} \times I \\ \text { or } \quad \frac{2.5 \times 11}{10} &amp; =l=2.75 \mathrm{~m}\end{aligned}$</div>


tags: ['Physics', 'Current Electricity', 'AP EAMCET', 'AP EAMCET 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3.5 m</span> </div>
source_db: MarksBatch0.db
title:
A cell in secondary circuit gives null deflection for 2.5 m length of potentiometer having 10 m length of wire. If the length of the potentiometer wire is increased by 1 m without changing the cell in the primary, the position of the null point now is

slug:
a-cell-in-secondary-circuit-gives-null-deflection-for-2-5-m-length-of-potentiometer-having-10-m-leng-50829

question:
<div class="question">A cell in secondary circuit gives null deflection for $2.5 \mathrm{~m}$ length of potentiometer having $10 \mathrm{~m}$ length of wire. If the length of the potentiometer wire is increased by $1 \mathrm{~m}$ without changing the cell in the primary, the position of the null point now is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3.5 m</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3 m</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2.75 m</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2.0 m</span> </li> </ul>

answer:
<div class="solution">Resistance of potentiometer wire<br/>$R=\rho \times \frac{l}{A}$<br/>or $\quad R=\left(\rho \times \frac{10}{A}\right)$<br/>The value of $2.5 \mathrm{~m}$ length wire<br/>$R^{\prime}=\frac{\rho \times 10}{A \times 10} \times 2.5$<br/>or $\quad R^{\prime}=\left(\frac{2.5 \rho}{A \times 10}\right)$<br/>Potential<br/>$\begin{aligned}<br/>V^{\prime} &amp; =I \times R^{\prime} \\<br/>&amp; =I\left(\frac{2.5 \rho}{A \times 10}\right)<br/>\end{aligned}$<br/>Now, again the length of potentiometer wire is increased by $1 \mathrm{~m}$, then resistance of null position wire.<br/>$\begin{aligned}<br/>R^{\prime \prime} &amp; =\left(\frac{\rho \times l}{11 \times A}\right) \\<br/>V^{\prime \prime} &amp; =I R^{\prime \prime} \\<br/>and \quad V &amp; =V^{\prime}<br/>\end{aligned}$<br/>$\begin{aligned} \frac{I \times 2.5 \rho}{A \times 10} &amp; =\frac{\rho \times l}{11 \times A} \times I \\ \text { or } \quad \frac{2.5 \times 11}{10} &amp; =l=2.75 \mathrm{~m}\end{aligned}$</div>


tags: ['Physics', 'Current Electricity', 'VITEEE', 'VITEEE 2009']
source_db: MarksBatch0.db
title:
A cell in secondary circuit gives null deflection for 2.5 m length of potentiometer having 10 m length of wire. If the length of the potentiometer wire is increased by 1 m without changing the cell in the primary, the position of the null point now is

slug:
a-cell-in-secondary-circuit-gives-null-deflection-for-2-5-m-length-of-potentiometer-having-10-m-leng-80172

question:
<div class="question">A cell in secondary circuit gives null deflection for $2.5 \mathrm{~m}$ length of potentiometer having $10 \mathrm{~m}$ length of wire. If the length of the potentiometer wire is increased by $1 \mathrm{~m}$ without changing the cell in the primary, the position of the null point now is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3.5 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2.75 \mathrm{~m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.0 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.75 \mathrm{~m}$</span> </div>

answer:
<div class="solution">Here length of potentiometer wire, $l=10 \mathrm{~m}$ Resistance of potentiometer wire<br/>$R=\rho \times \frac{l}{A}$ or $R=\left(\rho \times \frac{10}{A}\right)$<br/>The value of $2.5 \mathrm{~m}$ length wire<br/>$$<br/>R^{\prime}=\frac{\rho \times 10}{A \times 10} \times 2.5 \Rightarrow R^{\prime}=\left(\frac{2.5 \rho}{A \times 10}\right)<br/>$$<br/>Potential, $V^{\prime}=I \times R^{\prime}=I\left(\frac{2.5 \rho}{A \times 10}\right)$<br/>Again the length of potentiometer wire is increased by $1 \mathrm{~m}$.<br/>Resistance of null position<br/>$$<br/>R^{n}=\left(\frac{\rho \times l}{11 \times A}\right)<br/>$$<br/>$\therefore V^{\prime \prime}=I R^{\prime \prime}$ and $V=V^{\prime}$<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \frac{I \times 2.5 \rho}{A \times 10}=\frac{\rho \times I}{11 \times A} \times I \\<br/>\text { or } \frac{2.5 \times 11}{10}=I=2.75 \mathrm{~m}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch0.db
title:
A certain type of missile hits the target with probability p = 0.3 . What is the least number of missiles should be fired so that there is at least an 80% probability that the target is hit?

slug:
a-certain-type-of-missile-hits-the-target-with-probability-p-0-3-what-is-the-least-number-of-mis-95273

question:
<div class="question">A certain type of missile hits the target with probability $\mathrm{p}=0.3$. What is the least number of missiles should be fired so that there is at least an $80 \%$ probability that the target is hit?</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">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

answer:
<div class="solution">Probability of hitting the target $=0.3$ If'n' is the no. of times that the Missile is fired. $\therefore$ Probability ofhitting at least once $=1-[1-0.3]^{\mathrm{n}}=0.8$ $0.7^{\mathrm{n}}=0.2$<br/>$n \log 0.7=\log 0.2$<br/>$\Rightarrow \mathrm{n}=4.512$<br/>for $\mathrm{n}=4 ; \mathrm{p}   &lt; 0.8$<br/>take $\mathrm{n}=5$<br/>$n=5$<br/>Hence 5 missiles should be fired so that there is at least $80 \%$ prob. that the target is hit.</div>


tags: ['Biology', 'Evolution', 'NEET']
source_db: MarksBatch0.db
title:
A change in the relative abundance of an allele (the allelic frequency) within a population, over a succession of generations, is called

slug:
a-change-in-the-relative-abundance-of-an-allele-the-allelic-frequency-within-a-population-over-a-62679

question:
<div class="question">A change in the relative abundance of an allele (the allelic frequency) within a population, over a succession of generations, is called</div>

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

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

answer:
<div class="solution">Microevolution involves changes in allelic frequency within a gene pool for a given species in a relatively short period of geologic time by the alterations of successfully reproducing individuals within a population. The macroevolution involves large scale changes among groups of species. Co-evolution occurs when two or more species reciprocally affect each other's evolution through the process of natural selection. Phylogenetic evolution represents evolutionary relationships among organisms and reflects how species or other groups evolved from a series of common ancestors.</div>


tags: ['Chemistry', 'Biomolecules', 'AIIMS', 'AIIMS 2007']
source_db: MarksBatch0.db
title:
A child took sugar cane and sucked its juice. Regarding this which of the following match is correct?

slug:
a-child-took-sugar-cane-and-sucked-its-juice-regarding-this-which-of-the-following-match-is-correct-35928

question:
<div class="question">A child took sugar cane and sucked its juice. Regarding this which of the following match is correct?</div>

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

answer:
<div class="solution">Sucrose (cane sugar; beet sugar; saccharose) is a sugar comprising one molecule of glucose linked to a fructose molecule. It occurs widely in plants and is particularly abundant in sugar cane and sugar beet \((15-20 \%)\), from which it is extracted and refined for table sugar. If heated to \(200^{\circ} \mathrm{C}\), sucrose becomes caramel. Sucrose is broken down into the monosaccharides glucose and fructose with the help of carbohydrate digesting enzyme, invertase, produced in the small intestine (duodenum).</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch0.db
title:
A circle is drawn on the chord of a circle x 2 + y 2 = a 2 as diameter. The chord lies on the line x + y = a . What is the equation of the circle?

slug:
a-circle-is-drawn-on-the-chord-of-a-circle-x-2-y-2-a-2-as-diameter-the-chord-lies-on-the-line-x-50282

question:
<div class="question">A circle is drawn on the chord of a circle $x^{2}+y^{2}=a^{2}$ as diameter. The chord lies on the line $x+y=a$. What is the equation of the circle?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}-a x-a y=0$</span> </div>

answer:
<div class="solution">Given, equation of circle $\Rightarrow x^{2}+y^{2}=a^{2}$ Equation of chord $\Rightarrow x+y=a$ (1) $\Rightarrow x^{2}+(a-x)^{2}=a^{2}$<br/>$\Rightarrow x^{2}+a^{2}+x^{2}-2 a x=a^{2}$<br/>$\Rightarrow 2 x^{2}=2 a x$<br/>$\Rightarrow \mathrm{x}=0, \mathrm{a}$<br/>When, $x=0, y=a$ and when $x=a, y=0$. $\therefore$ Points of intersection are $(0, \mathrm{a})$ and $(\mathrm{a}, 0)$ $\therefore$ Equation of circle with chord as diameter is $(x-0)(x-a)+(y-a)(y-0)=0$<br/>$\Rightarrow x(x-a)+y(y-a)=0$<br/>$\Rightarrow x^{2}-a x+y^{2}-a y=0$<br/>$\Rightarrow x^{2}+y^{2}-a x-a y=0$</div>


slug: a-circle-is-drawn-with-the-two-foci-of-an-ellipse-a-2-x-2-b-2-y-2-1-at-the-end-of-the-di-30306
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch0.db
title:
A circle is drawn with the two foci of an ellipse a 2 x 2 â + b 2 y 2 â = 1 at the end of the diameter. What is the equation of the circle?

question:
<div class="question">A circle is drawn with the two foci of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$<br/>at the end of the diameter. What is the equation of the circle?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}=a^{2}+b^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}=a^{2}-b^{2}$ (20) 10-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">$x^{2}+y^{2}=2\left(a^{2}+b^{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}=2\left(a^{2}-b^{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}=a^{2}-b^{2}$ (20) 10-I]</span> </div>

answer:
<div class="solution">Foci of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ are given as $(a e, 0)$ and $(-a, 0)$.<br/>Since, two foci are at the end of the diameter $\therefore$ Equation of circle, is<br/>$(x-a e)(x+a e)+(y-0)(y-0)=0$<br/>$\Rightarrow x^{2}-a^{2} e^{2}+y^{2}=0$<br/>$\Rightarrow x^{2}+y^{2}-a^{2}\left(1-\frac{b^{2}}{a^{2}}\right)=0\left(\because e=\sqrt{1-\frac{b^{2}}{a^{2}}}\right)$<br/>$\Rightarrow x^{2}+y^{2}-a^{2}+b^{2}=0$<br/>$\Rightarrow x^{2}+y^{2}=a^{2}-b^{2}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch0.db
title:
A class consists of 3 sections A , B and C with 35,35 and 30 students respectively. The arithmetic means of the marks secured by students of sections A and B , who appeared fol a test of 100 marks are 74 and 70 respectively. The arithmetic mean of the marks secured by students of section C , who appeared for a test in the same subject which carried marks is 51 . What is the averagc perocntage of marks socuros by all the 100 student of the three sections?

slug:
a-class-consists-of-3-sections-a-b-and-c-with-35-35-and-30-students-respectively-the-arithmetic-m-31154

question:
<div class="question">A class consists of 3 sections $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ with 35,35 and 30 students respectively. The arithmetic means of the marks secured by students of sections $\mathrm{A}$ and $\mathrm{B}$, who appeared fol a test of 100 marks are 74 and 70 respectively. The arithmetic mean of the marks secured by students of section $\mathrm{C}$, who appeared for a test in the same subject which carried marks is 51 . What is the averagc perocntage of marks socuros by all the 100 student of the three sections? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 70.0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$70 . \mathrm{E}$</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><li class=""> <span class="option-label">C</span> <span class="option-data">65</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">67</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$70 . \mathrm{E}$</span> </div>

answer:
<div class="solution">Total no. of students in section A $=74 \times 35=2590$<br/>Total no. of students in section $\mathrm{B}$ $=70 \times 35=2450$<br/>Now, total no. of students in section $\mathrm{C}$<br/>$=\frac{51}{75} \times 100 \times 30=2040$<br/>Thus, total students in all $=2590+2450+2040$ $\begin{aligned} &amp;=7080 \\ \text { Thus, Required percentage } &amp;=7080 \div 100 \\ &amp;=70.8 \end{aligned}$</div>


slug: a-class-has-15-students-whose-ages-are-14-17-15-14-21-17-19-20-16-18-20-17-1-81600
tags: ['Mathematics', 'Probability']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A class has 15 students whose ages are 14 , 17 , 15 , 14 , 21 , 17 , 19 , 20 , 16 , 18 , 20 , 17 , 16 , 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded.What is the probability distribution of the random variable X ? Find mean, variance and standard deviation of X?

question:
<div class="question">A class has 15 students whose ages are $14,17,15,14,21$,<br/>$17,19,20,16,18,20,17,16,19$ and 20 years. One student<br/>is selected in such a manner that each has the same chance<br/>of being chosen and the age $\mathrm{X}$ of the selected student is<br/>recorded.What is the probability distribution of the random<br/>variable $\mathrm{X}$ ? Find mean, variance and standard deviation<br/>of X?</div>

answer:
<div class="solution">There are 15 students in a class. Each has the same chance<br/>of being choosen.<br/>The probability of each student to be selected $=\frac{1}{15}$<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/KRQ8ZRJsGBalGq7MRuKcibEHDTZdiXMtNSLDYt5NkPQ.original.fullsize.png"/><br/><br/>First and third columns from the probability distribution Mean $=\mathrm{E}(\mathrm{X})=\sum \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}=\frac{293}{15}=17.53333$<br/>Varience $=E\left(X^2\right)-[E(X)]^2=\sum p_i x_i{ }^2-\left(\sum p_i x_i\right)^2$<br/>$$<br/>\begin{aligned}<br/>&amp;=\frac{4683}{15}-(17.53333)^2 \\<br/>&amp;=312.20-307.41766=4.78<br/>\end{aligned}<br/>$$<br/>Standard deviation<br/>$$<br/>=\sqrt{\text { Variance }}=\sqrt{4 \cdot 78}=2 \cdot 19<br/>$$<br/>probability distribution is<img src="https://cdn-question-pool.getmarks.app/modules/cbse/h2DiXn2FndvK3yZPmOz10FiKPZwH9TVQpnC4r6iCXj4.original.fullsize.png"/><br/><br/>Mean $=17 \cdot 53$, variance $=4 \cdot 78$, S.D. $(X)=2 \cdot 19$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch0.db
title:
A closed and an open organ pipe have the same length. When they are vibrating simultaneously in their first overtone, they produce three beats. The length of the open pipe is now made one third the original length and one of its ends is closed. On the other hand, the length of the closed pipe is made three times the original length. The number of beats produced when they vibrate with fundamental frequencies will be

slug:
a-closed-and-an-open-organ-pipe-have-the-same-length-when-they-are-vibrating-simultaneously-in-thei-24957

question:
<div class="question">A closed and an open organ pipe have the same length. When they are vibrating simultaneously in their first overtone, they produce three beats. The length of the open pipe is now made one third the original length and one of its ends is closed. On the other hand, the length of the closed pipe is made three times the original length. The number of beats produced when they vibrate with fundamental frequencies will be</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"><mn>8</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></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"><mn>8</mn></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">V</mi><mo>=</mo></math> speed of sound and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">L</mi><mo>=</mo></math> lenght of pipes<br/><br/>For open pipe first overtone <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">f</mi><mn>1</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi></mfrac></math><br/><br/>For closed pipe first overtone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><msub><mo>'</mo><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">V</mi></mrow><mrow><mn>4</mn><mi mathvariant="normal">L</mi></mrow></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â´</mo><mtext>âââ</mtext><msub><mi mathvariant="normal">f</mi><mn>1</mn></msub><mo>â</mo><mi mathvariant="normal">f</mi><msub><mo>'</mo><mn>1</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi></mfrac><mo>â</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">V</mi></mrow><mrow><mn>4</mn><mi mathvariant="normal">L</mi></mrow></mfrac><mo>=</mo><mn>3</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â´</mo><mo>  </mo><mi mathvariant="normal"> </mi><mfrac><mi mathvariant="normal">V</mi><mrow><mn>4</mn><mi mathvariant="normal">L</mi></mrow></mfrac><mo>=</mo><mn>3</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â´</mo><mo>  </mo><mo> </mo><mfrac><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi></mfrac><mo>=</mo><mn>12</mn><mi mathvariant="normal"> </mi></math><br/><br/>When we close one end of the open pipe and make its length one-third of the original length, then its fundamental frequency is<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mo>=</mo><mfrac><mi mathvariant="normal">V</mi><mrow><mn>4</mn><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">L</mi><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">V</mi></mrow><mrow><mn>4</mn><mi mathvariant="normal">L</mi></mrow></mfrac></math><br/><br/>When the length of closed pipe is made <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> times, the fundamental frequency is<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mo>'</mo><mo>=</mo><mfrac><mi mathvariant="normal">V</mi><mrow><mn>4</mn><mrow><mo>(</mo><mn>3</mn><mi mathvariant="normal">L</mi><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">V</mi><mrow><mn>12</mn><mi mathvariant="normal">L</mi></mrow></mfrac></math><br/><br/>Beats produced <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">f</mi><mo>-</mo><mi mathvariant="normal">f</mi><mo>'</mo></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">V</mi></mrow><mrow><mn>4</mn><mi mathvariant="normal">L</mi></mrow></mfrac><mo>-</mo><mfrac><mi mathvariant="normal">V</mi><mrow><mn>12</mn><mi mathvariant="normal">L</mi></mrow></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>8</mn><mn>12</mn></mfrac><mo>.</mo><mfrac><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi></mfrac><mo>=</mo><mfrac><mn>8</mn><mn>12</mn></mfrac><mo>Ã</mo><mn>12</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2024 (27 Jan Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">294</span> </div>
source_db: MarksBatch0.db
title:
A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm , both vibrating in fundamental mode. The velocity of sound is ______ m s - 1 .

slug:
a-closed-organ-pipe-150-cm-long-gives-7-beats-per-second-with-an-open-organ-pipe-of-length-350-cm-97988

question:
<div class="question"><p>A closed organ pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>150</mn><mo>Â </mo><mi>cm</mi></math> long gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> beats per second with an open organ pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>350</mn><mo>Â </mo><mi>cm</mi></math>, both vibrating in fundamental mode. The velocity of sound is ______Â <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi><mo>Â </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p></div>

answer:
<div class="solution"><p>The formula to calculate the fundamental frequency in a closed pipe is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>c</mi></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>4</mn><msub><mi>l</mi><mn>1</mn></msub></mrow></mfrac><mo>Â </mo><mo>Â </mo><mo>Â </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math></p><p>And, that in an open pipe is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>o</mi></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><msub><mi>l</mi><mn>2</mn></msub></mrow></mfrac><mo>Â </mo><mo>Â </mo><mo>Â </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>2</mn></mfenced></math></p><p>Given that</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mrow><msub><mi>f</mi><mi>c</mi></msub><mo>-</mo><msub><mi>f</mi><mi>o</mi></msub></mrow></mfenced><mo>=</mo><mn>7</mn><mo>Â </mo><mo>Â </mo><mo>Â </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>3</mn></mfenced></math></p><p>From equations (1), (2) and (3), it follows that</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>v</mi><mrow><mn>4</mn><mo>Ã</mo><mn>150</mn></mrow></mfrac><mo>-</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mo>Ã</mo><mn>350</mn></mrow></mfrac><mo>=</mo><mn>7</mn><mspace linebreak="newline"></mspace><mo>â</mo><mfrac><mi>v</mi><mrow><mn>600</mn><mo>Â </mo><mi>cm</mi></mrow></mfrac><mo>-</mo><mfrac><mi>v</mi><mrow><mn>700</mn><mo>Â </mo><mi>cm</mi></mrow></mfrac><mo>=</mo><mn>7</mn><mspace linebreak="newline"></mspace><mo>â</mo><mfrac><mi>v</mi><mrow><mn>6</mn><mi>m</mi></mrow></mfrac><mo>-</mo><mfrac><mi>v</mi><mrow><mn>7</mn><mi>m</mi></mrow></mfrac><mo>=</mo><mn>7</mn><mspace linebreak="newline"></mspace><mo>â</mo><mi>v</mi><mfenced><mfrac><mn>1</mn><mn>42</mn></mfrac></mfenced><mo>=</mo><mn>7</mn><mspace linebreak="newline"></mspace><mo>â</mo><mi>v</mi><mo>=</mo><mn>42</mn><mo>Ã</mo><mn>7</mn><mspace linebreak="newline"></mspace><mo>=</mo><mn>294</mn><mo>Â </mo><mi mathvariant="normal">m</mi><mo>Â </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p></div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch0.db
title:
A closed organ pipe and an open organ pipe are tuned to the same fundamental frequency. The ratio of their lengths is

slug:
a-closed-organ-pipe-and-an-open-organ-pipe-are-tuned-to-the-same-fundamental-frequency-the-ratio-of-24154

question:
<div class="question">A closed organ pipe and an open organ pipe are tuned to the same fundamental frequency. The ratio of their lengths is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>:</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>4</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution">Fundamental frequency of close pipe,<br/><br/><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>4</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math><br/><br/>Fundamental frequency of open pipe,<br/><br/><math><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math><br/><br/>But <math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mo>â´</mo><mfrac><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2021 (24 Sep Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch0.db
title:
A closed organ pipe and an open organ pipe of same length produce 2 beats per second when they are set into vibrations together in fundamental mode. The length of open pipe is now halved and that of closed pipe is doubled. The number of beats produced per second will be

slug:
a-closed-organ-pipe-and-an-open-organ-pipe-of-same-length-produce-2-beats-per-second-when-they-are-s-65696

question:
<div class="question">A closed organ pipe and an open organ pipe of same length produce 2 beats per second when they are set into vibrations together in fundamental mode. The length of open pipe is now halved and that of closed pipe is doubled. The number of beats produced per second will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </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">8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Fundamental frequency of closed pipe, $\mathrm{n}_{\mathrm{c}}=\frac{\mathrm{V}}{4 \mathrm{~L}}$ Fundamental frequency of open pipe, $\mathrm{n}_0=\frac{\mathrm{v}}{2 \mathrm{~L}}$ They produce 2 beats per second<br/>$\begin{aligned} &amp; \therefore \mathrm{n}_{\mathrm{o}}-\mathrm{n}_{\mathrm{c}}=2, \\ &amp; \frac{\mathrm{V}}{2 \mathrm{~L}}-\frac{\mathrm{V}}{4 \mathrm{~L}}=2 \\ &amp; \frac{\mathrm{V}}{4 \mathrm{~L}}=2 \text { or } \frac{\mathrm{v}}{\mathrm{L}}=8\end{aligned}$<br/>When length of open pipe is halved<br/>$\mathrm{n}_{\mathrm{o}}^{\prime}=\frac{\mathrm{v}}{2\left(\frac{\mathrm{L}}{2}\right)}=\frac{\mathrm{v}}{\mathrm{L}}$<br/>When length of closed pipe is doubled<br/>$\mathrm{n}_{\mathrm{c}}^{\prime}=\frac{\mathrm{v}}{4 \times 2 \mathrm{~L}}=\frac{\mathrm{v}}{8 \mathrm{~L}}$<br/>New beat frequency $=\mathrm{n}_{\mathrm{o}}^{\prime}-\mathrm{n}_{\mathrm{c}}^{\prime}=\frac{\mathrm{v}}{\mathrm{L}}-\frac{\mathrm{v}}{8 \mathrm{~L}}=\frac{7 \mathrm{v}}{8 \mathrm{~L}}=\frac{7}{8} \times 8=7$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2020 (23 Sep Shift 1)']
source_db: MarksBatch0.db
title:
A closed organ pipe and an open organ pipe of same length produce 2 beats sec - 1 when they are set into vibrations together in fundamental mode. The length of open pipe is now halved and that of closed pipe is doubled. The number of beats produced will be

slug:
a-closed-organ-pipe-and-an-open-organ-pipe-of-same-length-produce-2-beats-sec-1-when-they-are-set-43963

question:
<div class="question">A closed organ pipe and an open organ pipe of same length produce <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>beats</mi><mo> </mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> when they are set into vibrations together in fundamental mode. The length of open pipe is now halved and that of closed pipe is doubled. The number of beats produced will be</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"><mn>7</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></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"><mn>7</mn></math></span> </div>

answer:
<div class="solution"><p>Fundamental frequency of closed organ pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>c</mi></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>4</mn><mi>l</mi></mrow></mfrac></math> </p><p>Fundamental frequency of open organ pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>o</mi></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math></p><p>Initial beats frequency </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>o</mi></msub><mo>-</mo><msub><mi>f</mi><mi>c</mi></msub><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mo>-</mo><mfrac><mi>v</mi><mrow><mn>4</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mi>v</mi><mrow><mn>4</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mi>v</mi><mi>l</mi></mfrac><mo>=</mo><mn>8</mn></math></p><p>After changing of length </p><p> Fundamental frequency of closed organ pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>f</mi><mi>c</mi><mo>'</mo></msubsup><mo>=</mo><mfrac><mi>v</mi><mrow><mn>4</mn><mfenced><mstyle displaystyle="true"><mn>2</mn><mi>l</mi></mstyle></mfenced></mrow></mfrac><mo>=</mo><mfrac><mi>v</mi><mrow><mn>8</mn><mi>l</mi></mrow></mfrac></math></p><p>Fundamental frequency of open organ pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>f</mi><mi>o</mi><mo>'</mo></msubsup><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mfenced><mstyle displaystyle="true"><mfrac><mstyle displaystyle="true"><mi>l</mi></mstyle><mn>2</mn></mfrac></mstyle></mfenced></mrow></mfrac><mo>=</mo><mfrac><mi>v</mi><mi>l</mi></mfrac></math></p><p>New beats frequency </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>f</mi><mi>o</mi><mo>'</mo></msubsup><mo>-</mo><msubsup><mi>f</mi><mi>c</mi><mo>'</mo></msubsup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>v</mi><mi>l</mi></mfrac><mo>-</mo><mfrac><mi>v</mi><mrow><mn>8</mn><mi>l</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>7</mn><mi>v</mi></mrow><mrow><mn>8</mn><mi>l</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>7</mn><mo>×</mo><mn>8</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mn>7</mn></math></p></div>


tags: ['Physics', 'Waves and Sound', 'KCET', 'KCET 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch0.db
title:
A closed organ pipe and an open organ pipe of same length produce 2 beats/second while vibrating in their fundamental modes. The length of the open organ pipe is halved and that of closed pipe is doubled. Then the number of beats produced per second while vibrating in the fundamental mode is

slug:
a-closed-organ-pipe-and-an-open-organ-pipe-of-same-length-produce-2-beats-second-while-vibrating-in-56077

question:
<div class="question">A closed organ pipe and an open organ pipe of same length produce 2 beats/second while vibrating in their fundamental modes. The length of the open organ pipe is halved and that of closed pipe is doubled. Then the number of beats produced per second while vibrating in the fundamental mode is</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">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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, $\quad \mathrm{f}_{\mathrm{o}}-\mathrm{f}_{\mathrm{c}}=2 \quad \text{...(i)}$ <br/>Frequency of fundamental mode for a closed Frequency of fundar organ pipe, $f_{c}=\frac{v}{4 L_{c}}$<br/>Similarly frequency of fundamental mode for an open organ pipe, $\mathrm{f}_{\mathrm{o}}=\frac{\mathrm{V}}{2 \mathrm{~L}_{\mathrm{o}}}$<br/>Given $\quad \mathrm{L}_{\mathrm{c}}=\mathrm{L}_{\mathrm{o}}$ <br/>$\Rightarrow$ $f_{0}=2 f_{c} \quad \text{...(ii)}$<br/>From Eqs. (i) and (ii), we get <br/>$\mathrm{f}_{\mathrm{o}}=2 \mathrm{f}_{\mathrm{c}}$ $\mathrm{f}_{\mathrm{o}}=4 \mathrm{~Hz}$ and $\mathrm{f}_{\mathrm{c}}=2 \mathrm{~Hz}$ <br/>When the length of the open pipe is halved, frequency of fundamental mode is $$ \mathrm{f}_{\mathrm{o}}^{\prime}=\frac{\mathrm{v}}{2\left[\frac{\mathrm{L}_{\mathrm{o}}}{2}\right]}=2 \mathrm{f}_{\mathrm{o}}=2 \times 4 \mathrm{~Hz}=8 \mathrm{~Hz} $$<br/>When the length of the closed pipe is doubled, its frequency of fundamental mode is<br/>$$<br/>\mathrm{f}_{\mathrm{o}}^{\prime}=\frac{\mathrm{v}}{4\left(2 \mathrm{~L}_{\mathrm{c}}\right)}=\frac{1}{2} \mathrm{f}_{\mathrm{c}}=\frac{1}{2} \times 2=1 \mathrm{~Hz}<br/>$$<br/>Hence, number of beats produced per second is<br/>$$<br/>\mathrm{f}_{\mathrm{o}}^{\prime}-\mathrm{f}_{\mathrm{c}}^{\prime}=8-1=7<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch0.db
title:
A closed organ pipe and an open pipe of same length produce 2 beats s - 1 when they are set into vibrations together in fundamental mode. The length of open pipe is now halved and that of closed pipe is doubled. The number of beats produced will be

slug:
a-closed-organ-pipe-and-an-open-pipe-of-same-length-produce-2-beats-s-1-when-they-are-set-into-vib-74483

question:
<div class="question">A closed organ pipe and an open pipe of same length produce 2 beats <math><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> when they are set into vibrations together in fundamental mode. The length of open pipe is now halved and that of closed pipe is doubled. The number of beats produced will be</div>

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

answer:
<div class="solution"><math><msub><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>4</mn><mi>l</mi></mrow></mfrac><mo>,</mo><mi> </mi><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math><br/><br/>As <math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mn>2</mn></math><br/><br/><math><mo>â´</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>4</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi> </mi><mi>o</mi><mi>r </mi><mfrac><mrow><mi>v</mi></mrow><mrow><mi>l</mi></mrow></mfrac><mo>=</mo><mn>8</mn></math><br/><br/>Now<br/><br/>Number of beats <math><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><msubsup><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>â²</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi>â²</mi></mrow></msubsup></math><br/><br/><math><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mi>l</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>8</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>7</mn><mi>v</mi></mrow><mrow><mn>8</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>Ã</mo><mn>8</mn><mo>=</mo><mn>7</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2019 (10 Jan Shift 2)']
source_db: MarksBatch0.db
title:
A closed organ pipe has a fundamental frequency of 1.5 kHz . The number of overtones that can be distinctly heard by a person with this organ pipe will be ( Assume that the highest frequency a person can hear is 20,000 Hz ).

slug:
a-closed-organ-pipe-has-a-fundamental-frequency-of-1-5-khz-the-number-of-overtones-that-can-be-dis-24411

question:
<div class="question">A closed organ pipe has a fundamental frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.5</mn><mtext> </mtext><mi>kHz</mi></math>. The number of overtones that can be distinctly heard by a person with this organ pipe will be <math><mo>(</mo></math>Assume that the highest frequency a person can hear is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20,000</mn><mtext> </mtext><mi>Hz</mi></math>).</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>4</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>5</mn></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"><mn>6</mn></math></span> </div>

answer:
<div class="solution"><blockquote><p>For a closed organ pipe, the resonate frequency(harmonics) is an odd multiple of the fundamental frequency.</p><p>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><msub><mi>f</mi><mn>0</mn></msub><mo>≤</mo><mn>20000</mn><mo> </mo><mi>Hz</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>.</mo><mo>.</mo><mo>.</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>kHz</mi><mo>⇒</mo></math> fundamental frequency</p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>7</mn></math></p><p>The total number of overtones that can be heard is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>. (<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>nd</mi></msup></math> harmonic to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>7</mn><mi>th</mi></msup></math> harmonic. )</p></blockquote></div>


slug: a-closed-organ-pipe-of-length-l-c-and-an-open-organ-pipe-of-length-l-0-contain-diffe-23161
tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2021 (21 Sep Shift 1)']
source_db: MarksBatch0.db
title:
A closed organ pipe of length ' L c â ' and an open organ pipe of length ' L 0 â ' contain different gases of densities ' Ï 1 â ' and ' Ï 2 â ' respectively. The compressibility of the gases is the same in both the pipes. The gases are vibrating in their first overtone with the same frequency. What is the length of open organ pipe?

question:
<div class="question">A closed organ pipe of length ' $L_c$ ' and an open organ pipe of length ' $L_0$ ' contain different gases of densities ' $\rho_1$ ' and ' $\rho_2$ ' respectively. The compressibility of the gases is the same in both the pipes. The gases are vibrating in their first overtone with the same frequency. What is the length of open organ pipe?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{4 L_c}{3} \sqrt{\frac{\rho_1}{\rho_2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3 L_c}{4} \sqrt{\frac{\rho_2}{\rho_1}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4 L_c}{3} \sqrt{\frac{\rho_2}{\rho_1}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2 L_c}{3} \sqrt{\frac{\rho_2}{\rho_1}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4 L_c}{3} \sqrt{\frac{\rho_1}{\rho_2}}$</span> </div>

answer:
<div class="solution">For open organ pipe:<br/>Fundamental frequency $\mathrm{n}=\frac{\mathrm{V}}{2 \mathrm{~L}_{\text {o }}}$<br/>First overtone $\mathrm{n}_1=2 \mathrm{n}=\frac{\mathrm{V}}{\mathrm{L}_{\text {ã }}}$<br/>For closed organ pipe<br/>Fundamental frequency $n^{\prime}=\frac{V^{\prime}}{4 L_c}$<br/>First overtone $n_1{ }^{\prime}=3 n^{\prime}=\frac{3 \mathrm{~V}^{\prime}}{4 \mathrm{~L}_{\mathrm{c}}}$<br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; \because \mathrm{n}_1=\mathrm{n}_1^{\prime} \\<br/>&amp; \therefore \frac{\mathrm{V}}{\mathrm{L}_{\mathrm{o}}}=\frac{3 \mathrm{~V}^{\prime}}{4 \mathrm{~L}_{\mathrm{c}}} \\<br/>&amp; \therefore \frac{\mathrm{L}_{\mathrm{o}}}{\mathrm{L}_{\mathrm{c}}}=\frac{4}{3} \frac{\mathrm{V}}{\mathrm{V}^{\prime}} \\<br/>&amp; \mathrm{V}=\sqrt{\frac{\mathrm{k}}{\rho_2}}, \mathrm{~V}^{\prime}=\sqrt{\frac{\mathrm{k}}{\rho_1}}<br/>\end{aligned}<br/>$$<br/>where $\mathrm{k}$ is the adiabatic bulk modulus, which is reciprocal of compressibility.<br/>$$<br/>\frac{L_0}{L_c}=\frac{4}{3} \sqrt{\frac{\rho_1}{\rho_2}}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'JIPMER', 'JIPMER 2005']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 2$</span> </div>
source_db: MarksBatch0.db
title:
A closed orgon pipe and an open orgon pipe are tuned to the same fundamental frequency. What is the ratio of their lengths?

slug:
a-closed-orgon-pipe-and-an-open-orgon-pipe-are-tuned-to-the-same-fundamental-frequency-what-is-the-24884

question:
<div class="question">A closed orgon pipe and an open orgon pipe are tuned to the same fundamental frequency. What is the ratio of their lengths?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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><li class=""> <span class="option-label">B</span> <span class="option-data">$2: 1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2: 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4: 3$</span> </li> </ul>

answer:
<div class="solution">Avoiding end correction, the length of the closed organ pipe is<br/>$l_1=\frac{\lambda_1}{4}$ or $\lambda_1=4 l_1$<br/>The length of open organ pipe is<br/>$l_2=\frac{\lambda_2}{2}$ or $\lambda_2=2 l_2$<br/>Here, $n_1=n_2$<br/>So, $\frac{v}{\lambda_1}=\frac{v}{\lambda_2}$ or $\frac{v}{4 l_1}=\frac{v}{2 l_2}$<br/>Therefore, $l_1: l_2=1: 2$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2014']
source_db: MarksBatch0.db
title:
A closed pipe is suddenly opened and changed to an open pipe of same length. The fundamental frequency of the resulting open pipe is less than that of 3rd harmonic of the earlier closed pipe by 55 Hz . Then, the value of fundamental frequency of the closed pipe is

slug:
a-closed-pipe-is-suddenly-opened-and-changed-to-an-open-pipe-of-same-length-the-fundamental-frequen-74263

question:
<div class="question">A closed pipe is suddenly opened and changed to an open pipe of same length. The fundamental frequency of the resulting open pipe is less than that of 3rd harmonic of the earlier closed pipe by $55 \mathrm{~Hz}$. Then, the value of fundamental frequency of the closed pipe is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$165 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$110 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$55 \mathrm{~Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$220 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$55 \mathrm{~Hz}$</span> </div>

answer:
<div class="solution">Frequency of first mode of vibration<br/>$$<br/>n_1=\frac{3 v}{4 /}<br/>$$<br/>(closed pipe) and frequency of first mode of vibration<br/>$$<br/>n_2=\frac{v}{2 l}<br/>$$<br/>(open pipe)<br/>According to the question,<br/>$$<br/>n_1-n_2=\frac{3 v}{4 l}-\frac{v}{2 l}=55<br/>$$<br/>So $\quad \frac{3 v-2 v}{4 !}=55$<br/>$$<br/>\Rightarrow \quad \frac{v}{4 l}=55 \mathrm{~Hz}<br/>$$</div>


