# questions

title: if-a-8-where-a-is-square-matrix-of-order-3-then-what-is-adj-a-equal-to
slug: if-a-8-where-a-is-square-matrix-of-order-3-then-what-is-adj-a-equal-to-64414
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">64</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|A|=8$, where $A$ is square matrix of order 3 , then what is $|\operatorname{adj} A|$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">24</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">64</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">512</span> </li> </ul>

answer:
<div class="solution">Let $|\mathrm{A}|=8$ and $\mathrm{A}$ is a square matrix of order $3 .$ We know that $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{n-1} . \mathrm{I}$ where<br/>$n^{\prime}$ is the order of the matrix $\mathrm{A}$.<br/>$\therefore|\operatorname{adj} \mathrm{A}|=8^{3-1}=8^{2}=64$</div>


title: if-a-10-b-2-and-a-b-12-then-what-is-the-value-of-a-b
slug: if-a-10-b-2-and-a-b-12-then-what-is-the-value-of-a-b-20815
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}|=10,|\overrightarrow{\mathrm{b}}|=2$ and $\overrightarrow{\mathrm{a}} \bullet \overrightarrow{\mathrm{b}}=12$, then what is the value of<br/>$|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}| ?$</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">16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">20</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">24</span> </li> </ul>

answer:
<div class="solution">We have<br/>$\cos \theta=\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}$ and $\sin \theta=\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}{|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|}$<br/>Given: $|\overrightarrow{\mathrm{a}}|=10,|\overrightarrow{\mathrm{b}}|=2, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=12$<br/>$\therefore \cos \theta=\frac{12}{20}$ and $\sin \theta=\frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|}{20}$<br/>Now, By squaring and adding, we get $\sin ^{2} \theta+\cos ^{2} \theta=1$<br/>$\Rightarrow \frac{|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}}{400}+\frac{144}{400}=1 \Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}=256$<br/>$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=16$</div>


title: if-a-12-41-then-what-is-the-value-of-3-t-a-n-a-t-a-n-3-a-1-3-t-a-n-2-a
slug: if-a-12-41-then-what-is-the-value-of-3-t-a-n-a-t-a-n-3-a-1-3-t-a-n-2-a-72771
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\frac{41 \pi}{12}$, then what is the value of $\frac{1-3 \tan ^{2} A}{3 \tan A-\tan ^{3} A}$ ?</div>

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

answer:
<div class="solution">Consider $\frac{1-3 \tan ^{2} A}{3 \tan A-\tan ^{3} A}=\frac{1}{\tan 3 A}$<br/>$\left(\because \tan 3 A=\frac{3 \tan A-\tan ^{3} A}{1-3 \tan ^{2} A}\right)$<br/>$=\frac{1}{\tan \frac{41 \pi}{4}}\left(\because A=\frac{4 \pi}{12}\right)$<br/>$=\frac{1}{\tan \left(10 \pi+\frac{\pi}{4}\right)}=\frac{1}{\tan \frac{\pi}{4}}=1$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-a-b-c-and-r-a-a-a-b-b-c-b-b-c-c-c-a-is-a-binary-relation-of-a-then-which-one-of-the-following-is-correct

slug:
if-a-a-b-c-and-r-a-a-a-b-b-c-b-b-c-c-c-a-is-a-binary-relation-of-a-then-which-one-of-the-following-is-correct-39091

question:
<div class="question">If $A=\{a, b, c\}$ and $R=\{(a, a),(a, b),(b, c),(b, b),(c, c),(c, a)\}$<br/>is a binary relation of $A$, then which one of the following is<br/>correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$R$ is reflexive, but neither symmetric nor transitive</span> </div>

answer:
<div class="solution">Let $\mathrm{A}=\{a, b, c\}$ and<br/>$\mathrm{R}=\{(a, a),(a, b),(b, c),(b, b),(c, c),(c, a)\}$<br/>Since, $(a, a),(b, b),(c, c) \in k$<br/>$R$ is reflexive relation But $(a, b) \in R$ and $(b, a) \notin R$ $R$ is not symmetric relation. Also, $(a, b),(b, c) \in \bar{k}$<br/>$\Rightarrow(c, a) \in R$ But $(a, c) \in R$<br/>$R$ isnot transitive relations.</div>


title: if-a-a-b-c-d-then-what-is-the-number-of-proper-subsets-of-a
slug: if-a-a-b-c-d-then-what-is-the-number-of-proper-subsets-of-a-83895
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\{a, b, c, d\}$, then what is the number of proper subsets<br/>of $A$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">16</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">C</span> <span class="option-data">14</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">12</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{A}=\{a, b, c, d\}$<br/>Let $n=$ no. of elementsin $\mathrm{A}=n(\mathrm{~A})=4$<br/>Now, number of subsets $=2^{n}=2^{4}=16$ As we know that no. of proper subsets $=2^{n}-1$ Number of proper subsets $=2^{4}-1=16-1=15$</div>


title: if-a-a-b-c-then-what-is-thenumber-of-proper-subsets-a
slug: if-a-a-b-c-then-what-is-thenumber-of-proper-subsets-a-15336
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$, then what is thenumber of proper subsets A?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Let $A=\{a, b, c\} \Rightarrow O(A)=3$<br/>Now, number of proper subsets of $\mathrm{A}=2^{O(A)}-1=2^{3}-1=7$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.6$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-any-two-events-such-that-p-a-04-p-b-03-p-a-b-09-then-what-is-the-value-of-p-a-b-equal-to

slug:
if-a-and-b-are-any-two-events-such-that-p-a-04-p-b-03-p-a-b-09-then-what-is-the-value-of-p-a-b-equal-to-11466

question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are any two events such that<br/>$\mathrm{P}(\overline{\mathrm{A}})=0.4, \mathrm{P}(\overline{\mathrm{B}})=0.3, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.9$, then what is the<br/>value of $\mathrm{P}(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.5$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.6$</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">$0.7$</span> </li> </ul>

answer:
<div class="solution">We have $\mathrm{P}(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})=\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})-\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})$<br/>$=\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})-\mathrm{P}(\overline{\mathrm{A} \cup \mathrm{B}})$<br/>By (De-Morgan's law)<br/>$=\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})-(1-\mathrm{P}(\mathrm{A} \cup \mathrm{B}))=.4+.3-(1-.9)=.6$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-any-two-events-such-that-p-a-b-4-3-a-b-4-1-and-p-a-3-2-where-a-stands-for-the-complementary-event-of-a-then-what-is-p-b

slug:
if-a-and-b-are-any-two-events-such-that-p-a-b-4-3-a-b-4-1-and-p-a-3-2-where-a-stands-for-the-complementary-event-of-a-then-what-is-p-b-31885

question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are any two events such that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}$,<br/>$(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, where $\bar{A}$ stands for the complementary event of $A$, then what is $\mathrm{P}(\mathrm{B})$ ?</div>

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

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

answer:
<div class="solution">As given, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}$<br/>and $\mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3}$<br/>$\mathrm{P}(\overline{\mathrm{A}})=1-\mathrm{P}(\mathrm{A})=\frac{2}{3} \Rightarrow \mathrm{P}(\mathrm{A})=\frac{1}{3}$<br/>We know that<br/>$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$\Rightarrow \frac{3}{4}=\frac{1}{3}+\mathrm{P}(\mathrm{B})-\frac{1}{4}$<br/>$\Rightarrow 1-\frac{1}{3}=\mathrm{P}(\mathrm{B}) \Rightarrow \frac{2}{3}=\mathrm{P}(\mathrm{B})$</div>


title: if-a-and-b-are-any-two-sets-then-what-is-a-a-b-equal-to
slug: if-a-and-b-are-any-two-sets-then-what-is-a-a-b-equal-to-99014
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are any two sets, then what is $\mathrm{A} \cap(\mathrm{A} \cup \mathrm{B})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Complement of $\mathrm{A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Complement of $\mathrm{B}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{B}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\bar{A}$</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> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/tI1y3nCbKFgFf8sMeCv4XyQa2A7ZL96z-a3dG3qwJ1I.original.fullsize.png"/><br/><br/>$\mathrm{A} \cap(\mathrm{A} \cup \mathrm{B})=(\mathrm{A} \cap \mathrm{A}) \cup(\mathrm{A} \cap \mathrm{B})=\mathrm{A} \cup(\mathrm{A} \cap \mathrm{B})$<br/>$=\mathrm{A}($ By diagram $)$<br/>Thus, $A \cap(A \cup B)=A$</div>


title: if-a-and-b-are-disjoint-sets-then-a-a-b-is-equal-to-which-one-of-the-following
slug: if-a-and-b-are-disjoint-sets-then-a-a-b-is-equal-to-which-one-of-the-following-86593
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\phi$</span> </div>
answer: <div class="solution">$A \cap\left(A^{\prime} \cup B\right)=A \cap A^{\prime}=\phi$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are disjoint sets, then $A \cap\left(A^{\prime} \cup B\right)$ is equal to which one of the following?</div>

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


title: if-a-and-b-are-events-such-that-p-a-b-05-p-b-08-and-p-a-b-04-then-what-is-p-a-b-equal-to
slug: if-a-and-b-are-events-such-that-p-a-b-05-p-b-08-and-p-a-b-04-then-what-is-p-a-b-equal-to-26346
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.08$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are events such that $P(A \cup B)=0.5, P(\bar{B})=0.8$<br/>and $P(A / B)=0.4$, then what is $P(A \cap B)$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.08$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.02</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.8$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.2$</span> </li> </ul>

answer:
<div class="solution">Let $P(A \cup B)=0.5, P(\bar{B})=0.8, P\left(\frac{A}{B}\right)=0.4$<br/>$\quad P(\bar{B})=1-P(B)$<br/>$\Rightarrow 0.8=1-P(B)$<br/>$\Rightarrow P(B)=1-0.8=0.2$<br/>Now, $P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}$<br/>$\Rightarrow P(B) \times P\left(\frac{A}{B}\right)=P(A \cap B)$<br/>$\Rightarrow 0.4 \times 0.2=P(A \cap B)$<br/>$\Rightarrow 0.08=P(A \cap B)$</div>


title: if-a-and-b-are-independent-events-such-that-p-a-5-1-p-a-b-10-7-then-what-is-p-b-equal-to
slug: if-a-and-b-are-independent-events-such-that-p-a-5-1-p-a-b-10-7-then-what-is-p-b-equal-to-64180
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are independent events such that<br/>$P(A)=\frac{1}{5}, P(A \cup B)=\frac{7}{10}$, then what is $P(\bar{B})$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{7}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{7}{9}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{8}$</span> </div>

answer:
<div class="solution">As A and B are independent event So, $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})$<br/>$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{5} \cdot \mathrm{P}(\mathrm{B})$<br/>Now, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$\frac{7}{10}=\frac{1}{5}+\mathrm{P}(\mathrm{B})-\frac{1}{5} \cdot \mathrm{P}(\mathrm{B})$<br/>$\mathrm{P}(\mathrm{B})\left(1-\frac{1}{5}\right)=\frac{7}{10}-\frac{1}{5}$<br/>$\frac{4}{5} \cdot \mathrm{P}(\mathrm{B})=\frac{1}{2} \Rightarrow \mathrm{P}(\mathrm{B})=\frac{5}{8}$<br/>Now, $\mathrm{P}(\overline{\mathrm{B}})=1-\mathrm{P}(\mathrm{B})=1-\frac{5}{8}=\frac{3}{8}$.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-position-vectors-of-the-points-a-and-b-respectively-then-what-is-the-position-vector-of-a-point-c-on-ab-produced-such-that-ac-2-ab

slug:
if-a-and-b-are-position-vectors-of-the-points-a-and-b-respectively-then-what-is-the-position-vector-of-a-point-c-on-ab-produced-such-that-ac-2-ab-41768

question:
<div class="question">If $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ are position vectors of the points $\mathrm{A}$ and $\mathrm{B}$ respectively, then what is the position vector of a point $\mathrm{C}$ on $\mathrm{AB}$ produced such that $\overrightarrow{\mathrm{AC}}=\overrightarrow{2 \mathrm{AB}}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$</span> </div>

answer:
<div class="solution">Let $\vec{c}$ be the position vector of point $C$ on AB produced. From the laws of vector addition,<br/>$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}$<br/>$\Rightarrow \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>and similarly in $\Delta \mathrm{AOC}, \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}$<br/>$\Rightarrow \overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}$<br/>As given, $\overrightarrow{\mathrm{AC}}=2 \overrightarrow{\mathrm{AB}}=2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/6t6v6v3Qq-PXiyZMWXVkMoPoBrM0HhxoAVS7aqayOcE.original.fullsize.png"/><br/><br/>$\Rightarrow 2 \mathrm{~b}-2 \mathrm{a}=\mathrm{c}-\mathrm{a}$<br/>$\Rightarrow \overrightarrow{\mathrm{c}}=2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{a}}=2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-rational-and-b-is-not-perfect-square-then-the-quadratic-equation-with-rational-coefficients-whose-one-root-is-3-a-b-is

slug:
if-a-and-b-are-rational-and-b-is-not-perfect-square-then-the-quadratic-equation-with-rational-coefficients-whose-one-root-is-3-a-b-is-18158

question:
<div class="question">If $a$ and $b$ are rational and $b$ is not perfect square, then the quadratic equation with rational coefficients whose one root is $3 a+\sqrt{b}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}-6 a x+9 a^{2}-b=0$</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">$3 a x^{2}+x-\sqrt{b}=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+3 a x+\sqrt{b}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{b} x^{2}+x-3 a=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}-6 a x+9 a^{2}-b=0$</span> </div>

answer:
<div class="solution">Sinceb is not a perfect square, therefore other root will be $3 a-\sqrt{b}$<br/>Required quadratic equation is $x^{2}-[(3 a+\sqrt{b})+(3 a-\sqrt{b})] x+(3 a+\sqrt{b})(3 a-\sqrt{b})=0$<br/>$\Rightarrow x^{2}-6 a x+9 a^{2}-b=0$</div>


title: if-a-and-b-are-square-matrices-of-second-order-such-that-a-1-b-3-then-what-is-3-ab-equal-to
slug: if-a-and-b-are-square-matrices-of-second-order-such-that-a-1-b-3-then-what-is-3-ab-equal-to-45103
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-27$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of second order such that $|\mathrm{A}|=-1,|\mathrm{~B}|=3$, then what is $|3 \mathrm{AB}|$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-9$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-27$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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>

answer:
<div class="solution">We know that,$|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|$, where $\mathrm{n}$ is order of matrix $\mathrm{A}$.<br/>$\therefore|3 \mathrm{AB}|=3^{2}|\mathrm{~A}||\mathrm{B}| \quad(\because|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|)$<br/>$=9(-1)(3)$<br/>$=-27 \quad(\because|\mathrm{A}|=-1,|\mathrm{~B}|=3)$</div>


title: if-a-and-b-are-subsets-of-a-set-x-then-what-is-a-x-b-b-equal-to
slug: if-a-and-b-are-subsets-of-a-set-x-then-what-is-a-x-b-b-equal-to-89033
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are subsets of a set $X$, then what is $\{A \cap(X-B)\} \cup B$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A \cup B$</span> </div>

answer:
<div class="solution">Since, $A$ and $B$ are subsets of set $X$ therefore $A \subseteq X$ and $B \subseteq X$<br/>Consider $\{(A \cap(X-B)\} \cup B$<br/>$=\left(A \cap B^{\prime}\right) \cup B$<br/>$=A \cup B$<br/>$\left(\because B^{\prime} \cap B=B\right)$</div>


title: if-a-and-b-are-the-unit-vectors-along-a-and-b-respectively-then-what-is-the-projection-of-b-on-a
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-and-b-are-the-unit-vectors-along-a-and-b-respectively-then-what-is-the-projection-of-b-on-a-62435

question:
<div class="question">If $\hat{a}$ and $\hat{b}$ are the unit vectors along $\vec{a}$ and $\vec{b}$ respectively, then what is the projection of $\vec{b}$ on $\vec{a}$ ?</div>

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

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

answer:
<div class="solution">The projection of $\vec{b}$ on $\vec{a}=\frac{\vec{a} \vec{b}}{|\vec{a}|}$<br/>Since, $\vec{a}$ is the unit vector<br/>$\therefore|\vec{a}|=1$<br/>Hence, projection of $\vec{b}$ on $\vec{a}=\frac{\vec{a} \cdot \vec{b}}{1}=\vec{a} \cdot \vec{b}$</div>


title: if-a-and-b-are-two-disjoint-sets-then-which-one-of-the-following-is-correct
slug: if-a-and-b-are-two-disjoint-sets-then-which-one-of-the-following-is-correct-51462
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are two disjoint sets, then which one of the <br/>following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A-B=A-(A \cap B)$</span> </div>

answer:
<div class="solution">Since, Aand $B$ are two disjoints therefore $A \cap B=\phi$<br/>$A \quad A-B=A-(A \cap B)$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-two-events-such-that-2-p-a-3-p-b-where-0-p-a-p-b-1-then-which-one-of-the-following-is-correct

slug:
if-a-and-b-are-two-events-such-that-2-p-a-3-p-b-where-0-p-a-p-b-1-then-which-one-of-the-following-is-correct-44007

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})   &lt; \mathrm{P}(\mathrm{B} \mid \mathrm{A}) &lt; \mathrm{P}(\mathrm{A} \mid \mathrm{B})$</span> </div>

answer:
<div class="solution">Given, $2 . \mathrm{P}(\mathrm{A})=3 . \mathrm{P}(\mathrm{B})$<br/>$\Rightarrow \frac{2 \mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}=\frac{3 \mathrm{P}(\mathrm{B})}{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}$<br/>$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{2 \cdot \mathrm{P}(\mathrm{A})}=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{3 \mathrm{P}(\mathrm{B})}$<br/>$\Rightarrow \frac{1}{2} \cdot \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{1}{3} \cdot \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)$<br/>$\Rightarrow \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)   &lt; \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) .$</div>


title: if-a-and-b-are-two-events-such-that-p-a-05-p-b-06-and-p-a-b-04-then-what-is-p-a-b-equal-to
slug: if-a-and-b-are-two-events-such-that-p-a-05-p-b-06-and-p-a-b-04-then-what-is-p-a-b-equal-to-89076
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.3$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are two events such that $P(A)=0.5, P(B)=0.6$ and $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.4$, then what is $P(\overline{A \cup B})$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.9$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.5$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp;=\frac{{ }^{4} C_{1}}{52} C_{1} \\ &amp; P(\mathrm{~A})=0.5, \mathrm{P}(\mathrm{B})=0.6, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.4 \\ &amp; P(\overline{A \cup B})=1-P(A \cup B) \\ &amp;=1-[\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})] \\ &amp;=1-[0.5+0.6-0.4] \\ &amp;=1-0.7=0.3 \end{aligned}$</div>


title: if-a-and-b-are-two-events-such-that-p-a-b-4-3-p-a-b-4-1-and-p-a-3-2-then-what-is-p-b-equal-to
slug: if-a-and-b-are-two-events-such-that-p-a-b-4-3-p-a-b-4-1-and-p-a-3-2-then-what-is-p-b-equal-to-38466
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are two events such that $P(A \cup B)=\frac{3}{4}$,<br/>$P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then what is $P(B)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{2}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{8}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{9}$</span> </li> </ul>

answer:
<div class="solution">$\begin{array}{ll} &amp; \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4} \\ &amp; \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4} \\ &amp; \mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3}, \mathrm{P}(\mathrm{A})=\frac{1}{3} \\ \therefore \quad &amp; \mathrm{P}(\mathrm{B})=\frac{3}{4}-\frac{1}{3}+\frac{1}{4}=\frac{9-4+3}{12}=\frac{\not 8}{12}=\frac{2}{3} \\ \therefore &amp; \text { Option (b) is correct. }\end{array}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-two-events-such-that-p-a-b-4-3-p-a-b-4-1-p-a-3-2-where-a-is-the-complement-of-a-then-what-is-p-b-equal-to

slug:
if-a-and-b-are-two-events-such-that-p-a-b-4-3-p-a-b-4-1-p-a-3-2-where-a-is-the-complement-of-a-then-what-is-p-b-equal-to-73432

question:
<div class="question">If $A$ and $B$ are two events such that $P(A \cup B)=\frac{3}{4}$, $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}, \mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3}$ where $\overline{\mathrm{A}}$ is the complement of<br/>A, then what is $\mathrm{P}(\mathrm{B})$ equal to ?</div>

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

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

answer:
<div class="solution">Given $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}$,<br/>$\mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3} \Rightarrow \mathrm{P}(\mathrm{A})=\frac{1}{3}$<br/>As we know<br/>$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$\therefore \quad \frac{3}{4}=\frac{1}{3}+\mathrm{P}(\mathrm{B})-\frac{1}{4}$<br/>$\Rightarrow \quad \frac{3}{4}+\frac{1}{4}-\frac{1}{3}=\mathrm{P}(\mathrm{B})$<br/>$\Rightarrow \mathrm{P}(\mathrm{B})=\frac{2}{3}$</div>


title: if-a-and-b-are-two-events-then-what-is-the-probability-of-occurrence-of-either-event-a-or-event-b
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-a-and-b-are-two-events-then-what-is-the-probability-of-occurrence-of-either-event-a-or-event-b-42680

question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are two events, then what is the probability of occurrence of either event A or event B?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}(\mathrm{A} \cup \mathrm{B})$</span> </div>

answer:
<div class="solution">Probability of occurence of either event A or event $\mathrm{B}=$ $\mathrm{P}(\mathrm{A} \cup \mathrm{B})$</div>


title: if-a-and-b-are-two-invertible-square-matrices-of-same-order-then-what-is-ab-1-equal-to
slug: if-a-and-b-are-two-invertible-square-matrices-of-same-order-then-what-is-ab-1-equal-to-73255
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
answer: <div class="solution">$(\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1}$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are two invertible square matrices of same order, then what is $(\mathrm{AB})^{-1}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{B}^{-1} \mathrm{~A}^{-1}$</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">$\mathrm{A}^{-1} \mathrm{~B}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{B}^{-1} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{A}^{-1} \mathrm{~B}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{B}^{-1} \mathrm{~A}^{-1}$</span> </div>


title: if-a-and-b-are-two-matrices-such-that-a-b-a-and-b-a-b-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-and-b-are-two-matrices-such-that-a-b-a-and-b-a-b-then-which-one-of-the-following-is-correct-48242

question:
<div class="question">If $A$ and $B$ are two matrices such that $A B=A$ and $B A=B$, then which one of the following is correct?</div>

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

answer:
<div class="solution">Let $A$ and $B$ be two matrices such that $A B=A$ and $B A$ $=B$<br/>Now, consider $A B=A$ Take Transpose on both side $(A B)^{T}=A^{T}$<br/>$\Rightarrow A^{T}=B^{T} . A^{T}$ ...(1)<br/>Now, $\mathrm{BA}=\mathrm{B}$<br/>Take, Transpose on both side $(B A)^{T}=\mathrm{B}^{\mathrm{T}}$<br/>$\Rightarrow \mathrm{B}^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}} \cdot \mathrm{B}^{\mathrm{T}}$ ...(2)<br/>Now, from equation (1) and (2). we have $A^{T}=\left(A^{T} \cdot B^{T}\right) A^{T}$<br/>$A^{T}=A^{T}\left(B^{T} A^{T}\right)$<br/>$=A^{T}(A B)^{T} \quad\left(\because(A B)^{T}=B^{T}=B^{T} A^{T}\right)$<br/>$=A^{T} \cdot A^{T}$<br/>Thus, $A^{T}=\left(A^{T}\right)^{2}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-two-mutually-exclusive-and-exhaustive-events-with-p-b-3-p-a-then-what-is-the-value-of-p-b

slug:
if-a-and-b-are-two-mutually-exclusive-and-exhaustive-events-with-p-b-3-p-a-then-what-is-the-value-of-p-b-15837

question:
<div class="question">If $A$ and $B$ are two mutually exclusive and exhaustive events<br/>with $P(B)=3 P(A)$, then what is the value of $P(\bar{B})$ ?</div>

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

answer:
<div class="solution">A and B are mutually exclusive and exhaustive events with $P(A \cap B)=0, P(A \cup B)=1$<br/>we know that $P(A \cup B)=P(A)+P(B)-P(A \cap B)$<br/>$\Rightarrow 1=P(A)+3 P(A)$<br/>$\Rightarrow \quad P(A)=\frac{1}{4}$<br/>$\therefore P(B)=\frac{3}{4}$<br/>Hence, $P(\bar{B})=1-P(B)=1-\frac{3}{4}=\frac{1}{4}$</div>


title: if-a-and-b-are-two-mutually-exclusive-events-then-what-is-pa-b-equal-to
slug: if-a-and-b-are-two-mutually-exclusive-events-then-what-is-pa-b-equal-to-22573
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are two mutually exclusive events, then what is P(A B) equal to?</div>

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

answer:
<div class="solution">Since, A and B are mutually exclusive events. $\therefore \quad \mathrm{P}(\mathrm{AB})=\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-two-nonempty-sets-having-n-elements-in-common-then-what-is-the-number-of-common-elements-in-the-sets-a-b-and-b-a

slug:
if-a-and-b-are-two-nonempty-sets-having-n-elements-in-common-then-what-is-the-number-of-common-elements-in-the-sets-a-b-and-b-a-66877

question:
<div class="question">If A and B are two non-empty sets having $n$ elements in common, then what is the number of common elements in<br/>the sets $\mathrm{A} \times \mathrm{B}$ and $\mathrm{B} \times \mathrm{A}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">n</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{n}^{2}$</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">$2 \mathrm{n}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zero</span> </li> </ul>

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

answer:
<div class="solution">Since A and B have $n$ elements in common so, $A \times B$ and $\mathrm{B} \times \mathrm{A}$ have $\mathrm{n}^{2}$ elements in common.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-two-nonsingular-square-matrices-such-that-ab-a-then-which-one-of-the-following-is-correct

slug:
if-a-and-b-are-two-nonsingular-square-matrices-such-that-ab-a-then-which-one-of-the-following-is-correct-55208

question:
<div class="question">If A and B are two non-singular square matrices such that $\mathrm{AB}=\mathrm{A}$, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{B}$ is an identity matrix</span> </div>

answer:
<div class="solution">Since, A and B are twonon-singular matrices therefore their determinant is non-zero. $\therefore \mathrm{A}^{-1}$ and $\mathrm{B}^{-1}$ defined.<br/>Consider $\mathrm{AB}=\mathrm{A} \Rightarrow \mathrm{A}^{-1} \mathrm{AB}=\mathrm{A}^{-1} \mathrm{~A} \Rightarrow \mathrm{B}=\mathrm{I}$</div>


title: if-a-and-b-are-two-subsets-of-a-set-x-then-what-is-a-a-b-equal-to
slug: if-a-and-b-are-two-subsets-of-a-set-x-then-what-is-a-a-b-equal-to-85670
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\phi$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ and $\mathrm{B}$ are two subsets of a set $\mathrm{X}$, then what is $A \cap(A \cup B)^{\prime}$ equal to?</div>

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

answer:
<div class="solution">On applying Demorgan's law, we get<br/>On applying associative law, we get,<br/>$=\left(\mathrm{A} \cap \mathrm{A}^{\prime}\right) \cap \mathrm{B}^{\prime}=\phi \cap \mathrm{B}^{\prime}=\phi$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-two-unit-vectors-inclined-at-an-angle-6-0-to-each-other-then-which-one-of-the-following-is-correct

slug:
if-a-and-b-are-two-unit-vectors-inclined-at-an-angle-6-0-to-each-other-then-which-one-of-the-following-is-correct-90761

question:
<div class="question">If $\vec{a}$ and $\vec{b}$ are two unit vectors inclined at an angle $60^{\circ}$ to each other, 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">$|\vec{a}+\vec{b}|   &lt; 1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$|\vec{a}+\vec{b}|&gt;1$</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">$|\vec{a}-\vec{b}|   &lt; 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\vec{a}-\vec{b}|&gt;1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|\vec{a}+\vec{b}|&gt;1$</span> </div>

answer:
<div class="solution">$\begin{aligned}|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}| &amp;=\sqrt{|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}} \\ &amp;=\sqrt{|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2|\overrightarrow{\mathbf{a}}| \cdot|\overrightarrow{\mathbf{b}}| \cos 60^{\circ}} \\ &amp;=\sqrt{1^{2}+1^{2}+2 \cdot 1 \cdot 1 \cdot \frac{1}{2}}=\sqrt{3} \\ \therefore|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|&gt;1 \end{aligned}$</div>


title: if-a-and-b-are-two-unit-vectors-then-the-vector-a-b-a-b-is-parallel-to
slug: if-a-and-b-are-two-unit-vectors-then-the-vector-a-b-a-b-is-parallel-to-78967
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\hat{a}$ and $\hat{b}$ are two unit vectors, then the vector<br/>$(\hat{a}+\hat{b}) \times(\hat{a} \times \hat{b})$ is parallel to</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp;  \\ \text {  } &amp;(\hat{a}+\hat{b}) \times(\hat{a} \times \hat{b}) \\ &amp;=\hat{a} \times(\hat{a} \times \hat{b})+\hat{b} \times(\hat{a} \times \hat{b}) \\ &amp;=(\hat{a} \cdot \hat{b}) \hat{a}-(\hat{a} \cdot \hat{a}) \hat{b}+(\hat{b} \cdot \hat{b}) \hat{a}-(\hat{b} \cdot \hat{a}) \hat{b} \\ &amp;=\lambda \hat{a}-\hat{b}+\hat{a}-\lambda \hat{b} \\ &amp;=\lambda(\hat{a}-\hat{b})+1(\hat{a}-\hat{b}) . \\ &amp;=(\lambda+1)(\hat{a}-\hat{b}) \end{aligned}$<br/>So, it is parallel to $\hat{a}-\hat{b}$.</div>


title: if-a-and-b-are-two-vectors-such-that-a-b-0-and-a-b-0-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-and-b-are-two-vectors-such-that-a-b-0-and-a-b-0-then-which-one-of-the-following-is-correct-75195

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

answer:
<div class="solution">Let $\vec{a}$ and $\vec{b}$ are two vectors such that $\vec{a} \cdot \vec{b}=0$ and<br/>$\vec{a} \times \vec{b}=0$ then either $\vec{a}$ or $\vec{b}$ is a null vector.</div>


title: if-a-and-b-are-twosets-satisfying-a-b-b-a-then-which-one-of-the-following-is-correct
slug: if-a-and-b-are-twosets-satisfying-a-b-b-a-then-which-one-of-the-following-is-correct-53942
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$A=B$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are twosets satisfying $A-B=B-A$, 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">$\quad A=\phi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$A \cap B=\phi$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$A=B$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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>

answer:
<div class="solution">We drawthe Venn diagram,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/c5s8KxJDwbuGip3V38IJBd5fnytIZF_YIUenZPv9BZE.original.fullsize.png"/><br/><br/>$A-B=A-(A \cap B)$ and $B-A=B-(A \cap B)$<br/>$\mathrm{A}-\mathrm{B}=\mathrm{B}-\mathrm{A}$<br/>$\Rightarrow \quad A-(A \cap B)=B-(A \cap B)$<br/>$\Rightarrow A=B$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-unit-vectors-inclined-at-an-angle-of-3-0-to-each-other-then-which-one-of-the-following-is-correct

slug:
if-a-and-b-are-unit-vectors-inclined-at-an-angle-of-3-0-to-each-other-then-which-one-of-the-following-is-correct-49408

question:
<div class="question">$\mathrm{If} \overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are unit vectors inclined at an angle of $30^{\circ}$ to each other, 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">$|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|&gt;1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\quad 1   &lt; |\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}| &lt; 2$</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">$|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|&gt;2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 1   &lt; |\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}| &lt; 2$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp;|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+2 \overrightarrow{\mathrm{a} .} \overrightarrow{\mathrm{b}} \cos \theta \\ \Rightarrow &amp;|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=1+1+2|\overrightarrow{\mathrm{a}} \| \overrightarrow{\mathrm{b}}| \cos 30^{\circ} \\ &amp;=1+1+2 \times \frac{\sqrt{3}}{2} \\ \Rightarrow &amp;|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=2+\sqrt{3} \\ \Rightarrow &amp;|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=\sqrt{2+\sqrt{3}} \\ &amp; 1   &lt; \sqrt{2+\sqrt{3}} &lt; 2 \\ \Rightarrow &amp; 1 &lt; |\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}| &lt; 2 \end{aligned}$</div>


title: if-a-and-b-are-unit-vectors-then-what-is-the-value-of-a-b-2-a-b-2
slug: if-a-and-b-are-unit-vectors-then-what-is-the-value-of-a-b-2-a-b-2-26445
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}$ and $\vec{b}$ are unit vectors, then what is the value of<br/>$|\vec{a} \times \vec{b}|^{2}+(\vec{a} \cdot \vec{b})^{2}$ ?</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">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 2$</span> </li> </ul>

answer:
<div class="solution">$\quad|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|^{2}+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})^{2}$<br/>$=(|\overrightarrow{\mathbf{a}}| \cdot|\overrightarrow{\mathbf{b}}| \cdot \sin \theta)^{2}+(|\overrightarrow{\mathbf{a}}| \cdot|\overrightarrow{\mathbf{b}}| \cdot \cos \theta)^{2}$<br/>$=(1.1 \cdot \sin \theta)^{2}+(1.1 \cdot \cos \theta)^{2}$<br/>$=\sin ^{2} \theta+\cos ^{2} \theta=1$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-vectors-such-that-a-2-b-7-and-a-b-3-i-2-j-6-k-then-what-is-the-acute-angle-between-a-and-b

slug:
if-a-and-b-are-vectors-such-that-a-2-b-7-and-a-b-3-i-2-j-6-k-then-what-is-the-acute-angle-between-a-and-b-60970

question:
<div class="question">If $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ are vectors such that $|\vec{a}|=2,|\overrightarrow{\mathrm{b}}|=7$ and<br/>$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$, then what is the acute angle between<br/>$\vec{a}$ and $\overrightarrow{\mathrm{b}}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</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">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$60^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$90^{\circ}$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp;|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=7 \\ &amp; \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}} \end{aligned}$<br/>We know, $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta \cdot \hat{\mathrm{n}}$ where $\hat{\mathrm{n}}$ is unit<br/>vector.<br/>$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \sin \theta$<br/>$\Rightarrow|3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}|=(2)(7) \sin \theta$<br/>$\Rightarrow \sqrt{9+4+36}=(2)(7) \sin \theta .$<br/>$\Rightarrow \pm 7=(2)(7) \sin \theta .$<br/>$\therefore \sin \theta=\pm \frac{1}{2} .$<br/>$\sin \theta$ is acute angle, $\theta=30^{\circ}$</div>


title: if-a-b-0-and-a-b-0-then-which-one-of-the-following-is-correct
slug: if-a-b-0-and-a-b-0-then-which-one-of-the-following-is-correct-72240
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0}$ then which one of the following is<br/>correct ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\overrightarrow{\mathrm{a}}=\overrightarrow{0}$ or $\overrightarrow{\mathrm{b}}=\overrightarrow{0}$</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0 \Rightarrow \overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}}$<br/>$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=0 \Rightarrow \overrightarrow{\mathrm{a}} \| \overrightarrow{\mathrm{b}}$<br/>(ii) it is clear that $\overrightarrow{\mathrm{a}}=0$ or $\overrightarrow{\mathrm{b}}=0$ From (i) and</div>


title: if-a-b-9-0-then-what-is-sin-a-sec-b-sin-a-cos-b-beginarraylltext-equal-to
slug: if-a-b-9-0-then-what-is-sin-a-sec-b-sin-a-cos-b-beginarraylltext-equal-to-37975
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}+\mathrm{B}=90^{\circ}$, then what is $\sqrt{\sin \mathrm{A} \sec \mathrm{B}-\sin \mathrm{A} \cos \mathrm{B}}$<br/>$\begin{array}{ll}\text { equal to ? }</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin \mathrm{A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\cos A$</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">$\tan \mathrm{A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{A}+\mathrm{B}=90^{\circ}$<br/>Consider $\sqrt{\sin A \sec B-\sin A \cos B}$<br/>$\begin{aligned} &amp;=\sqrt{\sin \mathrm{A} \sec \left(90^{\circ}-\mathrm{A}\right)-\sin \mathrm{A} \cos \left(90^{\circ}-\mathrm{A}\right)} \\ &amp;=\sqrt{\sin \mathrm{A} \operatorname{cosec} \mathrm{A}-\sin \mathrm{A} \sin \mathrm{A}}=\sqrt{1-\sin ^{2} \mathrm{~A}}=\cos \mathrm{A} . \\ \text { 140. (a) } &amp; \text { Consider } \tan ^{4} \mathrm{~A}-\sec ^{4} \mathrm{~A}+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A} \\ &amp;=\left(\tan ^{2} \mathrm{~A}\right)^{2}-\left(\sec ^{2} \mathrm{~A}\right)^{2}+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A} . \\ &amp;=\left(\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}\right)\left(\tan ^{2} \mathrm{~A}-\sec ^{2} \mathrm{~A}\right) \\ &amp;=\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A} \\ &amp;=-\left(\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}\right)+\tan ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}=0 . \end{aligned}$</div>


title: if-a-b-a-b-then-which-one-of-the-following-is-correct-1
slug: if-a-b-a-b-then-which-one-of-the-following-is-correct-1-58699
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}+\vec{b}|=\vec{a}-\vec{b} \mid$, 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">$\vec{a}=\lambda \vec{b}$ for some scalar $\lambda$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\vec{a}$ is parallel to $\vec{b}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\vec{a}$ is perpendicular to $\vec{b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\vec{a}=\vec{b}=\overrightarrow{0}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{a}$ is perpendicular to $\vec{b}$</span> </div>

answer:
<div class="solution">If $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$<br/>We know that when $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=0$, then $\vec{a}$ is<br/>perpendicular to $\overrightarrow{\mathrm{b}}$. $=\vec{a} \cdot \vec{a}-\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{a}-\vec{b} \cdot \vec{b}=0$<br/>$\vec{a}$ is perpendicular to $\overrightarrow{\mathrm{b}}$<br/>$\therefore \quad$ Option (c) is correct.</div>


title: if-a-b-a-b-then-which-one-of-the-following-is-correct-2
slug: if-a-b-a-b-then-which-one-of-the-following-is-correct-2-87205
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$,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">$|\vec{a}|=|\vec{b}|$.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\vec{a}$ is parrallel to $\vec{b}$.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\vec{a}$ is perpendicular to $\vec{b}$.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\vec{a}$ is a unit vector .</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{a}$ is perpendicular to $\vec{b}$.</span> </div>

answer:
<div class="solution">Since, $|a+b|=|a-b|$ $\Rightarrow[|a+b|]^{2}=[a-b]^{2}$<br/>$\Rightarrow a \cdot a+b \cdot b+a \cdot b+b \cdot a=a \cdot a+b \cdot b-a \cdot b-b \cdot a$<br/>$\Rightarrow 4 a \cdot b=0 \quad(\because a \cdot b=b . a)$<br/>$\Rightarrow a \cdot b=0$<br/>Hence, a is perpendicular to $\mathrm{b}$.</div>


title: if-a-b-a-b-then-which-one-of-the-following-is-correct
slug: if-a-b-a-b-then-which-one-of-the-following-is-correct-18950
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, 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">$\vec{a}$ is parallel to $\vec{b}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\vec{a}$ is perpendicular to $\vec{b}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\vec{a}=\vec{b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Both $\vec{a}$ and $\vec{b}$ are unit vectors</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{a}$ is perpendicular to $\vec{b}$</span> </div>

answer:
<div class="solution">Given, $|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|$<br/>$\Rightarrow|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|^{2}$<br/>$\Rightarrow|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2|\overrightarrow{\mathbf{a}}| \cdot|\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}-2|\overrightarrow{\mathbf{a}}| \cdot|\overrightarrow{\mathbf{b}}|$<br/>$\Rightarrow 4|\overrightarrow{\mathbf{a}}| \cdot|\overrightarrow{\mathbf{b}}|=0$<br/>$\Rightarrow \overrightarrow{\mathbf{a}}$ is perpendicular to $\overrightarrow{\mathbf{b}}$</div>


title: if-a-b-and-c-are-angles-of-a-triangle-such-that-tan-a-1-tan-b-2-then-what-is-the-value-of-tan-c
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-a-b-and-c-are-angles-of-a-triangle-such-that-tan-a-1-tan-b-2-then-what-is-the-value-of-tan-c-27285

question:
<div class="question">If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are angles of a triangle such that $\tan \mathrm{A}=1$, tan $B=2$, then what is the value of $\tan C$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">3</span> </li> </ul>

answer:
<div class="solution">In any triangle $\mathrm{ABC}, \mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$<br/>or, $\mathrm{A}+\mathrm{B}=\pi-\mathrm{C}$<br/>so, $\tan (A+B)=\tan (\pi-C)$<br/>or $-\tan C=\tan (A+B)$<br/>so, $\tan C=-\tan (A+B)$<br/>$=-\frac{\tan A+\tan B}{1-\tan A \cdot \tan B}$<br/>As given, $\tan A=1$, and $\tan B=2$<br/>putting these values,<br/>$\tan C=-\frac{1+3}{1-1 \times 3}=\frac{4}{2}=2$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-b-and-c-are-any-three-arbitraryevents-then-which-one-of-the-following-expressions-shows-that-both-a-and-b-occur-but-not-c

slug:
if-a-b-and-c-are-any-three-arbitraryevents-then-which-one-of-the-following-expressions-shows-that-both-a-and-b-occur-but-not-c-31060

question:
<div class="question">If A. B and $\mathrm{C}$ are any three arbitraryevents then which one of the following expressions shows that both $\mathrm{A}$ and $\mathrm{B}$ occur but not C ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}}$</span> </div>

answer:
<div class="solution">If A, B and C are any three arbitrary events occurrence of both A and B is given by A $\cap$ Band non-occurrence<br/>of $\mathrm{C} a s \overline{\mathrm{C}}$ then the event both $\mathrm{A}$ and $\mathrm{B}$ occur but not $\mathrm{C}$<br/>is represented by $\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}}$.</div>


title: if-a-b-and-c-are-in-ap-and-b-c-3-2-then-what-is-the-value-of-sin-c
slug: if-a-b-and-c-are-in-ap-and-b-c-3-2-then-what-is-the-value-of-sin-c-67188
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A, B$ and $C$ are in AP and $b: c=\sqrt{3}: \sqrt{2}$, then what is the value of sin C?</div>

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

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

answer:
<div class="solution">Let $\mathrm{a}-\mathrm{d}$, a and $\mathrm{a}+\mathrm{d}$ be three numbers which are in<br/>A.P. since $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are in $\mathrm{A} . \mathrm{P}$. $\therefore \quad A=a-d, B=a, C=a+d$<br/>$\Rightarrow \quad a-d+a+a+d=180^{\circ}$<br/>$(\because \mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are angles of a triangle $)$ $\Rightarrow \mathrm{a}=60^{\circ}$<br/>$\Rightarrow A=60^{\circ}-d, B=60^{\circ}, C=60^{\circ}+d$<br/>Now by sine rule,<br/>$\frac{b}{c}=\frac{\sin B}{\sin C} \Rightarrow \frac{\sqrt{3}}{\sqrt{2}}=\frac{\sin 60^{\circ}}{\sin C}$<br/>$\Rightarrow \quad \sin C=\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{\sqrt{3}}=\frac{1}{\sqrt{2}}$</div>


tags: ['Mathematics', 'Mathematical Reasoning', 'KCET', 'KCET 2014']
source_db: MarksBatch2_P1.db
title:
if-a-b-and-c-are-in-ap-then-the-value-of-leftbeginarraylllx2-x3-xa-x4-x5-xb-x6-x7-xcendarrayright-is

slug:
if-a-b-and-c-are-in-ap-then-the-value-of-leftbeginarraylllx2-x3-xa-x4-x5-xb-x6-x7-xcendarrayright-is-99211

question:
<div class="question">If \( a, b \) and \( c \) are in A.P., then the value of \( \left|\begin{array}{lll}x+2 &amp; x+3 &amp; x+a \\ x+4 &amp; x+5 &amp; x+b \\ x+6 &amp; x+7 &amp; x+c\end{array}\right| \) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( x-(a+b+c) \)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\( 9 x^{2}+a+b+c \)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\( 00 \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\( a+b+c \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( 00 \)</span> </div>

answer:
<div class="solution">Given \( \left|\begin{array}{ccc}x+2 &amp; x+3 &amp; x+a \\ x+4 &amp; x+5 &amp; x+b \\ x+6 &amp; x+7 &amp; x+6\end{array}\right| \) \( R_{1} \rightarrow R_{1}-R_{2} ; R_{2} \rightarrow R_{2}-R_{3} \) \( \Delta=\left|\begin{array}{ccc}-2 &amp; -2 &amp; a-b \\ -2 &amp; -2 &amp; b-c \\ x+6 &amp; x+7 &amp; x+c\end{array}\right| \) Since, a, b, c are in A.P. then \( a-b=b-c \) So, \( R_{1} \equiv R_{2} \) Therefore, \( \Delta=0 \)</div>


title: if-a-b-and-c-are-nonempty-sets-such-that-a-c-then-what-is-a-b-c-b-equal-to
slug: if-a-b-and-c-are-nonempty-sets-such-that-a-c-then-what-is-a-b-c-b-equal-to-74916
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\phi$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A, B$ and $C$ are non-empty sets such that<br/>$A \cap C=\phi$, then what is $(A \times B) \cap(C \times B)$ equal to?</div>

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

answer:
<div class="solution">Let $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are non-empty sets such that $\mathrm{A} \cap \mathrm{C}=\phi$<br/>Consider, $(A \times B) \cap(C \times B)=(A \cap C) \times(B \cap B)$<br/>$=(\mathrm{A} \cap \mathrm{C}) \times \mathrm{B}=\phi \times \mathrm{B}=\phi$</div>


title: if-a-b-and-c-are-real-numbers-then-the-roots-of-the-equation-x-a-x-b-x-b-x-c-x-c-x-a-0-are-always
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">real</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-a-b-and-c-are-real-numbers-then-the-roots-of-the-equation-x-a-x-b-x-b-x-c-x-c-x-a-0-are-always-23742

question:
<div class="question">If $a, b$ and $c$ are real numbers then the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are always</div>

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

answer:
<div class="solution">Given equation is $(x-a)(x-b)+(x-b)(x-c)+(x-c$ $(x-a)=0$<br/>$\Rightarrow 3 x^{2}-2(b+a+c) x+a b+b c+c a=0$<br/>Now, here $A=3, B=-2(a+b+c)$<br/>$C=a b+b c+c a$<br/>$\therefore D=\sqrt{B^{2}-4 A C}$<br/>$=\sqrt{(-2(a+b+c))^{2}-4(3)(a b+b c+c a)}$<br/>$=\sqrt{4(a+b+c)^{2}-12(a b+b c+c a)}$<br/>$=2 \sqrt{a^{2}+b^{2}+c^{2}-a b-b c-c a}$<br/>$=2 \sqrt{\frac{1}{2}\left\{(a-b)^{2}-(b-c)^{2}+(c-a)^{2}\right\}}$</div>


title: if-a-b-and-c-are-subsets-of-a-universal-set-then-which-one-of-the-following-is-not-correct
slug: if-a-b-and-c-are-subsets-of-a-universal-set-then-which-one-of-the-following-is-not-correct-32671
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 2)']
answer: <div class="solution">Checkingthrough option 'c' is incorrect</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If A, B and C are subsets of a Universal set, then which one of the following is not correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{A} \cup(\mathrm{B} \cup \mathrm{C})=\left(\mathrm{C}^{\prime} \cap \mathrm{B}\right)^{\prime} \cap \mathrm{A}$</span> </div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-b-and-c-are-the-angles-of-a-triangle-and-1-1-sin-a-sin-a-sin-2-a-1-1-sin-b-sin-b-sin-2-b-1-1-sin-c-sin-c-sin-2-c-0-then-which-one-of-the-followin

slug:
if-a-b-and-c-are-the-angles-of-a-triangle-and-1-1-sin-a-sin-a-sin-2-a-1-1-sin-b-sin-b-sin-2-b-1-1-sin-c-sin-c-sin-2-c-0-then-which-one-of-the-followin-18268

question:
<div class="question">If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are the angles of a triangle and<br/>$\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1+\sin A &amp; 1+\sin B &amp; 1+\sin C \\ \sin A+\sin ^{2} A &amp; \sin B+\sin ^{2} B &amp; \sin C+\sin ^{2} C\end{array}\right|=0$<br/>then which one of the following is correct?</div>

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

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

answer:
<div class="solution">$\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1+\sin A &amp; 1+\sin B &amp; 1+\sin C \\ \sin A+\sin ^{2} A &amp; \sin B+\sin ^{2} B &amp; \sin C+\sin ^{2} C\end{array}\right|$<br/>$\begin{array}{l}<br/>\left|\begin{array}{ccc}<br/>1 &amp; 0 &amp; 0 \\<br/>1+\sin A &amp; \sin A-\sin B &amp; \sin A-\sin C \\<br/>\sin A+\sin ^{2} A &amp; \sin ^{2} A-\sin ^{2} B &amp; \sin ^{2} A-\sin ^{2} C<br/>\end{array}\right|+0=0 \\<br/>\left\{\text { Applying } C_{2} \rightarrow C_{1}-C_{2} \text { and } C_{3} \rightarrow C_{1}-C_{3}\right\} \\<br/>=(\sin A-\sin B) \times(\sin A-\sin C)<br/>\end{array}$<br/>$\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 1+\sin \mathrm{A} &amp; 1 &amp; 1 \\ \sin \mathrm{A}+\sin ^{2} \mathrm{~A} &amp; (\sin \mathrm{A}+\sin \mathrm{B}) &amp; (\sin \mathrm{A}+\sin \mathrm{C})\end{array}\right|=0$<br/>$(\sin A-\sin B)(\sin A-\sin C)(\sin B-\sin C)=0$<br/>$\therefore \sin \mathrm{A}=\sin \mathrm{B}$ or $\sin \mathrm{A}=\sin \mathrm{C}$ or $\sin \mathrm{B}=\sin \mathrm{C}$ either $\mathrm{A}=\mathrm{B}$<br/>or $\mathrm{B}=\mathrm{C}$ or $\mathrm{A}=\mathrm{C}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-b-and-c-are-the-position-vectors-of-the-vertices-of-an-equilateral-triangle-whose-orthocentre-is-at-the-origin-then-which-one-of-the-following-is

slug:
if-a-b-and-c-are-the-position-vectors-of-the-vertices-of-an-equilateral-triangle-whose-orthocentre-is-at-the-origin-then-which-one-of-the-following-is-78722

question:
<div class="question">If $\vec{a}, \vec{b}$ and $\vec{c}$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$</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">$\vec{a}+\vec{b}+\vec{c}=$ unit vector</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \vec{a}+\vec{b}=\vec{c}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$</span> </div>

answer:
<div class="solution">Position vectors of vertices $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $\vec{a}, \vec{b}$ and $\vec{c}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/_NI94zGg33F-ypHO_SI8XhdKz8BbU5_XbRhQ5bQhSBY.original.fullsize.png"/><br/><br/>$\because$ triangle is equilateral.<br/>$\therefore$ Centroid and orthocenter will coincide. Centroid $\equiv$ orthocenter position vector<br/>$=\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})$<br/>$\because$ given in question orthocenter is at origin.<br/>Hence $\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})=0$<br/>$\vec{a}+\vec{b}+\vec{c}=0$</div>


title: if-a-b-and-c-are-three-finite-sets-then-what-is-a-b-c-equal-to
slug: if-a-b-and-c-are-three-finite-sets-then-what-is-a-b-c-equal-to-76189
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A, B$ and $C$ are three finite sets, then what is $[(A \cup B) \cap C]$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A^{\prime} \cap B^{\prime} \cup C^{\prime}$</span> </div>

answer:
<div class="solution">We know that <br/>$[(A \cup B) \cap C]^{\prime}=A^{\prime} \cap B^{\prime} \cup C$</div>


title: if-a-b-and-c-are-three-positive-numbers-in-an-arithmetic-progression-then
slug: if-a-b-and-c-are-three-positive-numbers-in-an-arithmetic-progression-then-97800
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are three positive numbers in an arithmetic progression, then:</div>

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

answer:
<div class="solution">Since, $\mathrm{a}, \mathrm{b}, \mathrm{c}$, are in $\mathrm{AP}$<br/>$\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}}$ are in $\mathrm{HP}$.<br/>Since, $\mathrm{AM} \geq \mathrm{HM}$<br/>$\Rightarrow \quad \mathrm{b} \geq \frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}}$<br/>[since $\mathrm{AM}=\mathrm{b}$ and $\left.\mathrm{HM}=\frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}}\right]$<br/>$\Rightarrow \quad \mathrm{ab}+\mathrm{bc} \geq 2 \mathrm{ac}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">300</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-b-and-c-are-three-sets-and-u-is-the-universal-set-such-thatn-u-700-n-a-200-n-b-300-and-n-a-b-100-then-what-is-the-value-of-a-b

slug:
if-a-b-and-c-are-three-sets-and-u-is-the-universal-set-such-thatn-u-700-n-a-200-n-b-300-and-n-a-b-100-then-what-is-the-value-of-a-b-66864

question:
<div class="question">If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are three sets and $\mathrm{U}$ is the universal set such thatn $(\mathrm{U})=700, \mathrm{n}(\mathrm{A})=200, \mathrm{n}(\mathrm{B})=300$ and $\mathrm{n}(\mathrm{A} \cap \mathrm{B})=100$,<br/>then what is the value of $\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">100</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">200</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">300</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">400</span> </li> </ul>

answer:
<div class="solution">From the given data $\mathrm{n}(\mathrm{U})=700, \mathrm{n}(\mathrm{A})=200, \mathrm{n}(\mathrm{B})=300$ and<br/>$n(A \cap B)=100$<br/>We know that, $n(A \cup B)=n(A)+n(B)-n(A \cap B)$<br/>$=200+300-100=400$<br/>Now, $\mathrm{n}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=\sim(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{n}(\mathrm{U})-\mathrm{n}(\mathrm{A} \cup \mathrm{B})$<br/>$=700-400=300$</div>


title: if-a-b-b-a-a-for-subsets-a-and-b-of-the-universal-set-u-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-b-b-a-a-for-subsets-a-and-b-of-the-universal-set-u-then-which-one-of-the-following-is-correct-66200

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

answer:
<div class="solution">For subsets $\mathrm{A}$ and $\mathrm{B}$ of $\mathrm{U}$. If $(A-B) \cup(B-A)=A$<br/>$\Rightarrow \mathrm{B}=\phi$</div>


title: if-a-b-c-0-then-one-of-the-solution-of-a-x-c-b-c-b-x-a-b-a-c-x-0-is
slug: if-a-b-c-0-then-one-of-the-solution-of-a-x-c-b-c-b-x-a-b-a-c-x-0-is-78708
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a+b+c=0$, then one of the solution of <br/>$\left|\begin{array}{ccc}\mathrm{a}-\mathrm{x} &amp; \mathrm{c} &amp; \mathrm{b} \\ \mathrm{c} &amp; \mathrm{b}-\mathrm{x} &amp; \mathrm{a} \\ \mathrm{b} &amp; \mathrm{a} &amp; \mathrm{c}-\mathrm{x}\end{array}\right|=0$ is</div>

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

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

answer:
<div class="solution">$\left|\begin{array}{ccc}a-x &amp; c &amp; b \\ c &amp; b-x &amp; a \\ b &amp; a &amp; c-x\end{array}\right|=0$<br/>$-\left|\begin{array}{ccc}x-(a+b+c) &amp; c &amp; b \\ x-(a+b+c) &amp; b-x &amp; a \\ x-(a+b+c) &amp; a &amp; c-x\end{array}\right|=0$<br/>$\Rightarrow\left|\begin{array}{ccc}x &amp; c &amp; b \\ x &amp; b-x &amp; a \\ x &amp; a &amp; c-x\end{array}\right|=0$<br/>$\left\{\right.$ Applying $\left.\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\right\}$<br/>$\left|\begin{array}{ccc}x &amp; c &amp; b \\ 0 &amp; c+x-b &amp; b-a \\ 0 &amp; c-a &amp; b+x-c\end{array}\right|=0$<br/>$\left\{\right.$ Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ and $\left.\mathrm{R}_{3} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{3}\right\}$<br/>$x\{(c+x-b)(b+x-c)-(b-a)(c-a)\}=0$<br/>$x\left\{\left(x^{2}-(b-c)^{2}-b c+a c+a b-a^{2}\right\}=0\right.$<br/>$x\left(x^{2}-a^{2}-b^{2}-c^{2}+a b+b c+c a\right)=0$<br/>$x\left\{x^{2}-(a-b)^{2}-(b-c)^{2}-(c-a)^{2}\right\}=0$<br/>$\therefore  x=0$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-b-c-0-then-which-of-the-following-isare-correct-1-a-b-c-are-coplanar-2-a-b-b-c-c-a-select-the-correct-answer-using-the-code-given-below

slug:
if-a-b-c-0-then-which-of-the-following-isare-correct-1-a-b-c-are-coplanar-2-a-b-b-c-c-a-select-the-correct-answer-using-the-code-given-below-46102

question:
<div class="question">If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, then which of the following is/are correct?<br/>$1.$ $\vec{a}, \vec{b}, \vec{c}$ are coplanar.<br/>$2.$ $\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}$<br/>Select the correct answer using the code given below.</div>

answer:
<div class="solution">According to statements (1) and (2), when $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$, then both statements are correct. If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}=0$, then $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ are<br/>collinear. Therefore, option (c) is correct.</div>


title: if-a-b-c-2-then-what-is-the-value-of-tan-a-tan-btan-b-tan-ctan-c-tan-a
slug: if-a-b-c-2-then-what-is-the-value-of-tan-a-tan-btan-b-tan-ctan-c-tan-a-69244
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A+B+C=\pi / 2$, then what is the value of $\tan A \tan B+\tan B \tan C+\tan C \tan A ?</div>

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

answer:
<div class="solution">Given $A+B+C=\frac{\pi}{2}$ Take tan on both sides,<br/>$\Rightarrow \tan (A+B+C)=\tan \left(\frac{\pi}{2}\right)$<br/>$\Rightarrow \frac{\tan A+\tan B+\tan C-\tan A \tan B \tan C}{1-\tan A \tan B-\tan B \tan C-\tan C \tan A}=\frac{1}{0}$<br/>$\left(\because \frac{1}{0}=\infty\right)$<br/>$\Rightarrow \tan A \tan B+\tan B \tan C+\tan C \tan A=1$</div>


title: if-a-b-c-18-0-then-what-is-sin-2-a-sin-2-b-sin-2-c-equal-to
slug: if-a-b-c-18-0-then-what-is-sin-2-a-sin-2-b-sin-2-c-equal-to-58829
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}+\mathrm{B}+\mathrm{C}=180^{\circ}$, then what is $\sin 2 \mathrm{~A}-\sin 2 \mathrm{~B}-\sin 2 \mathrm{C}$ equal<br/>to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-4 \sin A \cos B \cos C$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \sin 2 \mathrm{~A}-\sin 2 \mathrm{~B}-\sin 2 \mathrm{C} \\ &amp;=2 \cos (\mathrm{A}+\mathrm{B}) \cdot \sin (\mathrm{A}-\mathrm{B})+\sin (2 \mathrm{~A}+2 \mathrm{~B}) \\ &amp;=2 \cos (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})+2 \sin (\mathrm{A}+\mathrm{B}) \cos (\mathrm{A}+\mathrm{B}) \\ &amp;=2 \cos (\mathrm{A}+\mathrm{B})[\sin (\mathrm{A}-\mathrm{B})+\sin (\mathrm{A}+\mathrm{B})] \\ &amp;=-2 \cos \mathrm{C}[2 \sin \mathrm{A} \cdot \cos \mathrm{B}] \\ &amp;=-4 \sin \mathrm{A} \cos \mathrm{B} \cos \mathrm{C} \end{aligned}$</div>


title: if-a-b-c-and-b-c-a-then-which-one-of-the-following-is-correct
slug: if-a-b-c-and-b-c-a-then-which-one-of-the-following-is-correct-63335
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$, then which one of the following<br/>is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\vec{a}, \vec{b}, \vec{c}$ are orthogonal in pairs and $|\vec{a}|=|\vec{c}|$ and<br/>$|\overrightarrow{\mathrm{b}}|=1$</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">$\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are non-orthogonal to each other</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are orthogonal in pairs but $|\overrightarrow{\mathrm{a}}| \neq|\overrightarrow{\mathrm{c}}|$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are orthogonal is pairs but $|\overrightarrow{\mathrm{b}}| \neq 1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{a}, \vec{b}, \vec{c}$ are orthogonal in pairs and $|\vec{a}|=|\vec{c}|$ and<br/>$|\overrightarrow{\mathrm{b}}|=1$</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}, \quad \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$<br/>$\Rightarrow \vec{c} \perp \vec{a} \&amp; \vec{c} \perp \vec{b}$<br/>$\Rightarrow \vec{a} \perp \vec{b} \perp \vec{c}$<br/>$\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are orthogonal.<br/>Now, $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{c}}|$<br/>$\Rightarrow|\vec{a}||\vec{b}| \sin 90^{\circ}=|\vec{c}|$<br/>$\Rightarrow|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{c}}|$<br/>$\Rightarrow|\vec{b}||\vec{\phi}| \vec{b}|=| \vec{\not} \mid$<br/>$\Rightarrow|\overrightarrow{\mathrm{b}}|^{2}=1$<br/>$\Rightarrow|\vec{b}|=1$...(i)<br/>$\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$<br/>$\Rightarrow \overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}} \perp \overrightarrow{\mathrm{c}}$<br/>Also, $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}|$<br/>$\Rightarrow|\vec{b}||\vec{c}| \sin 90^{\circ}=|\vec{a}|$<br/>$\Rightarrow|\overrightarrow{\mathrm{b}}| \cdot|\overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}|$<br/>$\Rightarrow|\vec{c}|=|\vec{a}| \quad$ (from $\quad$ (i)</div>


title: if-a-b-c-are-all-positive-then-the-value-of-the-determinant-a-b-c-b-c-a-c-a-b-is
slug: if-a-b-c-are-all-positive-then-the-value-of-the-determinant-a-b-c-b-c-a-c-a-b-is-36219
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a \neq b \neq c$ are all positive, then the value of the determinant<br/>$\left|\begin{array}{lll}a &amp; b &amp; c \\ b &amp; c &amp; a \\ c &amp; a &amp; b\end{array}\right|$ is</div>

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

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

answer:
<div class="solution">$\left|\begin{array}{lll}a &amp; b &amp; c \\ b &amp; c &amp; a \\ c &amp; a &amp; b\end{array}\right|=\left|\begin{array}{lll}a+b+c &amp; b &amp; c \\ a+b+c &amp; c &amp; a \\ a+b+c &amp; a &amp; b\end{array}\right|$<br/>$\left(\right.$ Applying $\left.\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\right)$<br/>$=(a+b+c)\left|\begin{array}{lll}1 &amp; b &amp; c \\ 1 &amp; c &amp; a \\ 1 &amp; a &amp; b\end{array}\right|$<br/>[on taking $(\mathrm{a}+\mathrm{b}+\mathrm{c})$ common from $\left.\mathrm{C}_{1}\right]$<br/>$=(a+b+c)\left[1\left(b c-a^{2}\right)-b(b-a)+c(a-c)\right]$<br/>$=(a+b+c)\left[b c-a^{2}-b^{2}+a b+a c-c^{2}\right]$<br/>$=(a+b+c)\left[-\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)\right]$<br/>$=-\frac{1}{2}(a+b+c)\left[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}\right]$<br/>Hence, the determinant value is Negative</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-any-three-arbitrary-events-then-which-one-of-the-following-expressions-shows-that-both-a-and-b-occur-but-not-c

slug:
if-a-b-c-are-any-three-arbitrary-events-then-which-one-of-the-following-expressions-shows-that-both-a-and-b-occur-but-not-c-14628

question:
<div class="question">If $A, B, C$ are any three arbitrary events, then which one of the following expressions shows that both $A$ and $B$ occur but not $C$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A \cap B \cap \bar{C}$</span> </div>

answer:
<div class="solution">If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are any three arbitary events then only expression $A \cap B \cap \bar{C}$ will show that both $A$ and $B$ occur but not $\mathrm{C}$.</div>


title: if-a-b-c-are-in-apor-gphp-then-b-c-a-b-is-equal-to
slug: if-a-b-c-are-in-apor-gphp-then-b-c-a-b-is-equal-to-58869
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If a, b, c are in APor GPHP, then $\frac{\mathrm{a}-\mathrm{b}}{\mathrm{b}-\mathrm{c}}$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 or $\frac{\mathrm{a}}{\mathrm{b}}$ or $\frac{\mathrm{a}}{\mathrm{c}}$</span> </div>

answer:
<div class="solution">From question,<br/>$\begin{aligned} &amp; a, b, c \in \text { A.P. } \Rightarrow a-b=b-c \Rightarrow \frac{a-b}{b-c}=1 \\ &amp; a, b, c \in \text { G.P. } \Rightarrow \frac{a}{b}=\frac{b}{c}=\frac{a-b}{b-c} \\ &amp; a, b, c \in \text { H.P. } \Rightarrow b=\frac{2 a c}{a+c} \Rightarrow a b+b c=2 a c \\ \Rightarrow &amp; a b-a c=a c-b c \\ \Rightarrow &amp; \frac{a-b}{b-c}=\frac{a}{c} \end{aligned}$</div>


title: if-a-b-c-are-in-gp-then-what-is-the-value-of-a-b-a-b-b-c-b-c-a-b-b-c-0
slug: if-a-b-c-are-in-gp-then-what-is-the-value-of-a-b-a-b-b-c-b-c-a-b-b-c-0-24817
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then what is the value of<br/>$\left|\begin{array}{lll}a &amp; b &amp; a+b \\ b &amp; c &amp; b+c \\ a+b &amp; b+c &amp; 0\end{array}\right|$</div>

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

answer:
<div class="solution">Since, $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$. <br/>$\Rightarrow \mathrm{b}^{2}=\mathrm{ac}$<br/>Explanding the determinant we get,<br/>$\left|\begin{array}{ccc}a &amp; b &amp; a+b \\ b &amp; c &amp; b+c \\ a+b &amp; b+c &amp; 0\end{array}\right|$<br/>$=a\left|\begin{array}{cc}c &amp; b+c \\ b+c &amp; 0\end{array}\right|-b\left|\begin{array}{cc}b &amp; b+c \\ a+b &amp; 0\end{array}\right|+(a+b)\left|\begin{array}{cc}b &amp; c \\ a+b &amp; b+c\end{array}\right|$<br/>$=-a(b+c)^{2}+b(a+b)(b+c)+(a+b)\left(b^{2}+b c-a c-b c\right)$<br/>$=-a\left(b^{2}+c^{2}+2 b c\right)+b\left(a b+a c+b^{2}+b c\right)$<br/>$=-a b^{2}-a c^{2}-2 a b c+a b^{2}+2 a b c+b^{2} c \quad\left(\because b^{2}=a c\right)$<br/>$=-a c^{2}+b^{2} c=-a c^{2}+a c \cdot c .$<br/>$=-a c^{2}+a c^{2}=0$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-nonzero-real-numbers-and-1-a-1-1-1-1-b-1-1-1-1-c-0-then-what-is-the-value-of-a-1-b-1-c-1

slug:
if-a-b-c-are-nonzero-real-numbers-and-1-a-1-1-1-1-b-1-1-1-1-c-0-then-what-is-the-value-of-a-1-b-1-c-1-30473

question:
<div class="question">If $\mathrm{a}, \mathrm{b}, \mathrm{c}$, are non-zero real numbers and <br/>$\left[\begin{array}{ccc}1+a &amp; 1 &amp; 1 \\ 1 &amp; 1+b &amp; 1 \\ 1 &amp; 1 &amp; 1+c\end{array}\right]=0$<br/>then what is the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ ?</div>

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

answer:
<div class="solution">Given $\left|\begin{array}{ccc}1+a &amp; 1 &amp; 1 \\ 1 &amp; 1+b &amp; 1 \\ 1 &amp; 1 &amp; 1+c\end{array}\right|=0$<br/>Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1}$<br/>$\Rightarrow\left|\begin{array}{ccc}1+a &amp; -a &amp; -a \\ 1 &amp; b &amp; 0 \\ 1 &amp; 0 &amp; c\end{array}\right|=0$<br/>$\Rightarrow$ Expanding along $R_{3}, 1(a b)+c(b+a b+a)=0$<br/>$\Rightarrow a b+b c+c a+a b c=0$<br/>$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=-1$</div>


title: if-a-b-c-are-nonzero-real-numbers-then-the-inverse-of-the-matrix-a-a-0-0-0-b-0-0-0-c-is-equal-to
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-b-c-are-nonzero-real-numbers-then-the-inverse-of-the-matrix-a-a-0-0-0-b-0-0-0-c-is-equal-to-93642

question:
<div class="question">If $a, b, c$ are non-zero real numbers, then the inverse of the<br/>matrix $\mathrm{A}=\left[\begin{array}{lll}\mathrm{a} &amp; 0 &amp; 0 \\ 0 &amp; \mathrm{~b} &amp; 0 \\ 0 &amp; 0 &amp; \mathrm{c}\end{array}\right]$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{lll}a^{-1} &amp; 0 &amp; 0 \\ 0 &amp; b^{-1} &amp; 0 \\ 0 &amp; 0 &amp; c^{-1}\end{array}\right]$</span> </div>

answer:
<div class="solution">$\mathrm{A}=\left[\begin{array}{lll}\mathrm{a} &amp; 0 &amp; 0 \\ 0 &amp; \mathrm{~b} &amp; 0 \\ 0 &amp; 0 &amp; \mathrm{c}\end{array}\right]$<br/>$\mathrm{A}^{-1}=\left[\begin{array}{ccc}\frac{1}{\mathrm{a}} &amp; 0 &amp; 0 \\ 0 &amp; \frac{1}{\mathrm{~b}} &amp; 0 \\ 0 &amp; 0 &amp; \frac{1}{\mathrm{c}}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{a}^{-1} &amp; 0 &amp; 0 \\ 0 &amp; \mathrm{~b}^{-1} &amp; 0 \\ 0 &amp; 0 &amp; \mathrm{c}^{-1}\end{array}\right]$</div>


title: if-a-b-c-are-nonzero-vectors-and-a-b-c-a-b-c-then-which-one-of-the-following-is-correct
slug: if-a-b-c-are-nonzero-vectors-and-a-b-c-a-b-c-then-which-one-of-the-following-is-correct-82220
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are non-zero vectors and $|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|$<br/>$|\overrightarrow{\mathrm{c}}|$, 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">$\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=\vec{c} \cdot \vec{a} \neq 0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$ only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=0$ only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=0$</span> </div>

answer:
<div class="solution">Refer to the figure. $|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{c}}|$ is triple dot product and is volume $(\mathrm{V})$ of<br/>the parallelepiped whose adjacent edges are a, b, and c.<br/>i.e., $V=|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})| \cdot \overrightarrow{\mathrm{OC}}$<br/>$\Rightarrow \quad \mathrm{V}=|\overrightarrow{\mathrm{a}} \| \overrightarrow{\mathrm{b}}|(\sin \theta)(\cos \phi)|\overrightarrow{\mathrm{c}}|$<br/>where $0 \leq \theta \leq \pi$ is the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$.<br/>As given $|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}\|\overrightarrow{\mathrm{b}}\| \overrightarrow{\mathrm{c}}|$<br/>$\Rightarrow \quad \sin \theta \cos \phi \mid .=1$<br/>$\Rightarrow \quad \sin \theta=1, \cos \phi=1$<br/>$\Rightarrow \quad \theta=\frac{\pi}{2}, \phi=0$<br/>$\Rightarrow \quad \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/mFHT9NQuz5JBRqj_aR1G0VFEDYK5PV1cJfOwH86D1pU.original.fullsize.png"/></div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-real-numbers-then-the-value-of-the-determinant-1-a-1-b-1-c-a-b-c-b-c-a-c-a-b-b-c-c-a-a-b-is

slug:
if-a-b-c-are-real-numbers-then-the-value-of-the-determinant-1-a-1-b-1-c-a-b-c-b-c-a-c-a-b-b-c-c-a-a-b-is-91599

question:
<div class="question">If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real numbers, then the value of the determinant $\left|\begin{array}{ccc}1-a &amp; a-b-c &amp; b+c \\ 1-b &amp; b-c-a &amp; c+a \\ 1-c &amp; c-a-b &amp; a+b\end{array}\right|$ is</div>

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

answer:
<div class="solution">$\left|\begin{array}{lll}1-a &amp; a-b-c &amp; b+c \\ 1-b &amp; b-c-a &amp; c+a \\ 1-c &amp; c-a-b &amp; a+b\end{array}\right|$<br/>apply $C_{2} \rightarrow C_{2}+C_{3}$<br/>$\left|\begin{array}{lll}1-a &amp; a &amp; b+c \\ 1-b &amp; b &amp; c+a \\ 1-c &amp; c &amp; a+b\end{array}\right|$<br/>apply $C_{1} \rightarrow C_{1}+C_{2}$<br/>$\left|\begin{array}{lll}1 &amp; a &amp; b+c \\ 1 &amp; b &amp; c+a \\ 1 &amp; c &amp; a+b\end{array}\right|$<br/>apply $C_{3} \rightarrow C_{2}+C_{3}$<br/>$(a+b+c)\left|\begin{array}{lll}1 &amp; a &amp; 1 \\ 1 &amp; b &amp; 1 \\ 1 &amp; c &amp; 1\end{array}\right|=0$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-the-position-vectors-of-corners-a-b-c-of-a-parallelogram-abcd-then-what-is-the-position-vector-of-the-corner-d

slug:
if-a-b-c-are-the-position-vectors-of-corners-a-b-c-of-a-parallelogram-abcd-then-what-is-the-position-vector-of-the-corner-d-19707

question:
<div class="question">If $\vec{a}, \vec{b}, \vec{c}$ are the position vectors of corners $A, B, C$ of a parallelogram $\mathrm{ABCD}$, then what is the position vector of the corner $\mathrm{D}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$</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">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$</span> </li> </ul>

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

answer:
<div class="solution">Let O be the origin and $\mathrm{ABCD}$ be the parallelogram. $\operatorname{In} \Delta \mathrm{ODC}$,<br/>$\overrightarrow{\mathrm{OD}}=\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{CD}}$<br/>$\overrightarrow{\mathrm{CD}}=-\overrightarrow{\mathrm{AB}}$<br/>and $\operatorname{In} \Delta \mathrm{AOB}, \quad \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>Thus, $\overrightarrow{\mathrm{CD}}=-\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$<br/>So, $\overrightarrow{\mathrm{OD}}=\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}[$ since, $\overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{CD}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}]$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/3YIr15mAVRWzNThpYHdENpfpI2agFjsjfOoFQrQ4gvQ.original.fullsize.png"/></div>


title: if-a-b-c-are-the-sides-of-a-triangle-a-bc-then-a-p-1-b-p-1-c-p-1-where-p-1-is
slug: if-a-b-c-are-the-sides-of-a-triangle-a-bc-then-a-p-1-b-p-1-c-p-1-where-p-1-is-28125
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a, b, c$ are the sides of a triangle $A B C$, then $a^{\frac{1}{p}}+b^{\frac{1}{p}}-c^{\frac{1}{p}}$ where $\mathrm{p}&gt;1$, is</div>

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

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

answer:
<div class="solution">Consider any equilateral triangle:<br/>$c=b=a=1$ unit<br/>Take value of $p$ between $1 \&amp; 2$ i.e., $\frac{3}{2}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/zXzVGhlHjzWKpz2fDmmTp6ozGkQuZMOo2VSi5ETQqkw.original.fullsize.png"/><br/><br/>$=1+1-1=1&gt;0$<br/>Take value of $\mathrm{p}$ greater than 2 i.e; $3 .$<img src="https://cdn-question-pool.getmarks.app/pyq/nda/axwGQMxVo20-581kURYHwpQJ5pZvVFcv0GGj3ytgFSA.original.fullsize.png"/><br/><br/>$\therefore$ By considering all the options carefully; we came to<br/>a conclusion that opton (b) is correct.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-three-events-then-what-is-the-probability-that-at-least-two-of-these-events-occur-together

slug:
if-a-b-c-are-three-events-then-what-is-the-probability-that-at-least-two-of-these-events-occur-together-82352

question:
<div class="question">If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are three events, then what is the probability that at least two of these events occur together?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{C} \cap \mathrm{A})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{C} \cap \mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{C} \cap \mathrm{A})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$</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">$P(A \cap B)+P(B \cap C)+P(C \cap A)-3 P(A \cap B \cap C)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{C} \cap \mathrm{A})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$</span> </div>

answer:
<div class="solution">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{A} \cap \mathrm{C})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/sqA8V8kABYrcMJlpZBVoQALvOW58ZQ7DZ6fQjUTeGkI.original.fullsize.png"/></div>


title: if-a-b-c-are-three-sets-such-that-a-b-a-c-and-a-b-a-c-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-b-c-are-three-sets-such-that-a-b-a-c-and-a-b-a-c-then-which-one-of-the-following-is-correct-87891

question:
<div class="question">If $A, B, C$ are three sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$, 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">$A=\underline{B}$ only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$B=C$ 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">$A=C$ only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$A=B=C$</span> </li> </ul>

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

answer:
<div class="solution">Let $A, B, C$ be three sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$<br/>Let $x \in A \cup B \Rightarrow x \in A$ or $x \in B$<br/>(1) Since, $A \cup B=A \cup C$ there fore<br/>$x \in A$ or $x \in C$<br/>Also, Given $A \cap B=A \cap C$ $x \in A \cap B \Rightarrow x \in A$ and $x \in B$<br/>and $x \in A$ and $x \in C(\because A \cap B=A \cap C)$<br/>Thus, from $(1),(2),(3)$, and $(4)$, we have $B=C$ only</div>


title: if-a-b-c-are-three-sets-then-what-is-a-b-c-equal-to
slug: if-a-b-c-are-three-sets-then-what-is-a-b-c-equal-to-96365
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are three sets, then what is $\mathrm{A}-(\mathrm{B}-\mathrm{C})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A-(B \cap C)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(A-B) \cup C$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(A-B) \cup(A \cap C)$</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">$\quad(A-B) \cup(A-C)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(A-B) \cup(A \cap C)$</span> </div>

answer:
<div class="solution">Following venn diagram shows the relation $\mathrm{A}-(\mathrm{B}-\mathrm{C})$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/qoaInInFbNUMih1gPE3D3YvmQvs6GlbPOGuC0BnIqpM.original.fullsize.png"/><br/><br/>In the above venn diagram, horizontal lines shows<br/>(A-B) and vertical lines show $(\mathrm{A} \cap \mathrm{C})$<br/>$(A-B) \cup(A \cap C)=A-(B-C)$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-unit-vectors-such-that-a-is-perpendicular-to-the-plane-of-b-c-and-the-angle-between-b-and-c-is-3-then-what-is-a-b-c

slug:
if-a-b-c-are-unit-vectors-such-that-a-is-perpendicular-to-the-plane-of-b-c-and-the-angle-between-b-and-c-is-3-then-what-is-a-b-c-64062

question:
<div class="question">If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to<br/>the plane of $\vec{b}, \vec{c}$; and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$<br/>Then, what is $|\vec{a}+\vec{b}+\vec{c}| ?$</div>

answer:
<div class="solution">As given: a is perpendicular to $\overrightarrow{\mathrm{b}}$ and $\mathrm{c}$ $\Rightarrow \vec{a} \cdot \vec{b}=0 \&amp; \overrightarrow{a \cdot c}=0$<br/>and angle between $\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}=\frac{\pi}{3}$<br/>$\therefore \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}| \cos \frac{\pi}{3}=1.1 \cdot \frac{1}{2}$<br/>$=\frac{1}{2} \quad(\because \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ are unit vectors)<br/>Now, $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+|\vec{c}|^{2}$<br/>$+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \vec{a})$<br/>$=1+1+1+2 \cdot\left(0+\frac{1}{2}+0\right)$<br/>$=1+1+1+1=4$<br/>$\Rightarrow|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}|=2$</div>


title: if-a-b-c-d-and-a-c-b-d-are-collinear-then-which-one-of-the-following-is-correct
slug: if-a-b-c-d-and-a-c-b-d-are-collinear-then-which-one-of-the-following-is-correct-62508
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $(a, b),(c, d)$ and $(a-c, b-d)$ are collinear, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{bc}-\mathrm{ad}=0$</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">$a b-c d=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{bc}+\mathrm{ad}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a b+c d=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{bc}-\mathrm{ad}=0$</span> </div>

answer:
<div class="solution">Let $A, B$ and $C$ having co-ordinates $(a, b),(c, d)$ and $\{(a-c),(b-d)\}$ respectively be the points If these poins are collinear then<br/>$\left|\begin{array}{ccc}a &amp; b &amp; 1 \\ c &amp; d &amp; 1 \\ a-c &amp; b-d &amp; 1\end{array}\right|=0$<br/>$\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ gives<br/>$\left|\begin{array}{ccc}a &amp; b &amp; 1 \\ c-a &amp; d-b &amp; 0 \\ a-c &amp; b-d &amp; 1\end{array}\right|=0$<br/>$\mathrm{R}_{3} \rightarrow \mathrm{R}_{2}+\mathrm{R}_{3}$ gives<br/>$\left|\begin{array}{ccc}a &amp; b &amp; 1 \\ c-a &amp; d-b &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right|=0$<br/>$\Rightarrow 1 .\{\mathrm{a}(\mathrm{d}-\mathrm{b})-\mathrm{b}(\mathrm{c}-\mathrm{a})\}=0$<br/>$\Rightarrow \mathrm{ad}-\mathrm{ab}-\mathrm{bc}+\mathrm{ab}=0$<br/>$\Rightarrow b c-a d=0$</div>


title: if-a-b-c-d-are-in-harmonical-progression-such-that-a-d-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-b-c-d-are-in-harmonical-progression-such-that-a-d-then-which-one-of-the-following-is-correct-28183

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a+c&gt;b+d$</span> </div>

answer:
<div class="solution">Since, $\mathrm{a}&gt;\mathrm{d}$ and $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in $\mathrm{HP}$.<br/>$\Rightarrow \mathrm{a}&gt;\mathrm{b}&gt;\mathrm{c}&gt;\mathrm{d}$.<br/>$\mathrm{b}=\frac{2 \mathrm{a} \mathrm{c}}{\mathrm{a}+\mathrm{c}} \Rightarrow \mathrm{b}^{2} \mathrm{~d}=\frac{2 \mathrm{a} \mathrm{b} \mathrm{cd}}{\mathrm{a}+\mathrm{c}}$<br/>and $c=\frac{2 b d}{b+d} \Rightarrow c^{2} a=\frac{2 a b c d}{b+d}$<br/>$\frac{c^{2} a}{b^{2} d}=\frac{a+c}{b+d}$<br/>$\Rightarrow \quad \frac{a+c}{b+d}=\left(\frac{a}{b}\right) \cdot\left(\frac{c}{d}\right) \cdot\left(\frac{c}{b}\right)&gt;1$<br/>$\Rightarrow a+c&gt;b+d$</div>


title: if-a-b-c-d-e-f-are-in-ap-then-e-c-is-equal-to-which-one-of-the-following
slug: if-a-b-c-d-e-f-are-in-ap-then-e-c-is-equal-to-which-one-of-the-following-19888
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a, b, c, d, e, f$ are in AP, then $(e-c)$ is equal to which one of the following?</div>

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

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

answer:
<div class="solution">Given, $a, b, c, d, e, f$ are in $A . P$<br/>$\therefore \quad 2 \mathrm{~d}=\mathrm{e}+\mathrm{c}$...(1)<br/>Consider $\mathrm{e}-\mathrm{c}=2 \mathrm{~d}-\mathrm{c}-\mathrm{c} \quad$ (from 1$)$<br/>$=2 \mathrm{~d}-2 \mathrm{c}=2(\mathrm{~d}-\mathrm{c})$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-b-c-then-one-value-of-x-which-satisfies-the-equation-0-x-a-x-b-x-a-0-x-c-x-b-x-c-0-0-is-given-by

slug:
if-a-b-c-then-one-value-of-x-which-satisfies-the-equation-0-x-a-x-b-x-a-0-x-c-x-b-x-c-0-0-is-given-by-73235

question:
<div class="question">If $a \neq b \neq c$, then one value of $x$ which satisfies the equation $\left|\begin{array}{ccc}0 &amp; \mathrm{x}-\mathrm{a} &amp; \mathrm{x}-\mathrm{b} \\ \mathrm{x}+\mathrm{a} &amp; 0 &amp; \mathrm{x}-\mathrm{c} \\ \mathrm{x}+\mathrm{b} &amp; \mathrm{x}+\mathrm{c} &amp; 0\end{array}\right|=0$ is given by</div>

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

answer:
<div class="solution">$\left|\begin{array}{ccc}0 &amp; x-a &amp; x-b \\ x+a &amp; 0 &amp; x-c \\ x+b &amp; x+c &amp; 0\end{array}\right|=0$<br/>We know, the value of symmetric matrix's determinant is $0\left|\begin{array}{ccc}0 &amp; -a &amp; -b \\ a &amp; 0 &amp; -c \\ b &amp; c &amp; 0\end{array}\right|=0$<br/>In the given matrix, determinant is 0, if $x=0$</div>


title: if-a-b-c-then-what-is-cos-a-b-cos-c-equal-to
slug: if-a-b-c-then-what-is-cos-a-b-cos-c-equal-to-13133
question: <div class="question">If $A+B+C=\pi$, then what is $\cos (A+B)+\cos C$ equal to ?</div>
tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.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">$2 \cos C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\cos C-\sin C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sin C$</span> </li> </ul>

answer:
<div class="solution">$A+B+C=\pi$<br/>$\quad A+B=\pi-C$<br/>$\quad \cos (A+B)=\cos (\pi-C)$<br/>$\quad \cos (A+B)=-\cos C$<br/>$\quad$ or $\cos (A+B)+\cos C=0$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-b-care-in-geometric-progression-and-a-2-b-3-c-are-in-arithmetic-progression-then-what-is-the-common-ratio-r-such-that-0-r-1

slug:
if-a-b-care-in-geometric-progression-and-a-2-b-3-c-are-in-arithmetic-progression-then-what-is-the-common-ratio-r-such-that-0-r-1-78434

question:
<div class="question">If a, b, care in geometric progression and a, $2 \mathrm{~b}, 3 \mathrm{c}$ are in arithmetic progression, then what is the common ratio $\mathrm{r}$ such that $0   &lt; \mathrm{r} &lt; 1 ?$</div>

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

answer:
<div class="solution">Given that a, b, c, are in GP. Let r be common ratio of GP. So, $a=a, b=a r$ and $c=a r^{2}$ ...(i)<br/>Also, given that $a, 2 b, 3 c$ are in AP.<br/>$\Rightarrow 2 b=\frac{a+3 c}{2}$<br/>$\Rightarrow \quad 4 \mathrm{~b}=\mathrm{a}+3 \mathrm{c}$<br/>From Eq. (1) $4 a r=a+3 a r^{2}$<br/>$\Rightarrow \quad 3 \mathrm{ar}^{2}-4 \mathrm{ar}+\mathrm{a}=0$<br/>$\Rightarrow \quad 3 r^{2}-4 r+1=0$<br/>$\Rightarrow \quad 3 r^{2}-3 r-r+1=0$<br/>$\Rightarrow 3 \mathrm{r}(\mathrm{r}-1)-1(\mathrm{r}-1)=0$<br/>$\Rightarrow(\mathrm{r}-1)(3 \mathrm{r}-1)=0$<br/>$\Rightarrow \mathrm{r}=1$ or $\mathrm{r}=\frac{1}{3}, \mathrm{r}=\frac{1}{3}$ is in the choice.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-b-is-at-unit-distance-from-the-line-8-x-6-y-1-0-then-which-of-the-following-conditions-are-correct-1-3-a-4-b-4-0-2-8-a-6-b-11-0-3-8-a-6-b-9-0-sel

slug:
if-a-b-is-at-unit-distance-from-the-line-8-x-6-y-1-0-then-which-of-the-following-conditions-are-correct-1-3-a-4-b-4-0-2-8-a-6-b-11-0-3-8-a-6-b-9-0-sel-52119

question:
<div class="question">If $(\mathrm{a}, \mathrm{b})$ is at unit distance from the line $8 \mathrm{x}+6 \mathrm{y}+1=0$, then which of the following conditions are correct? <br/>1) $\quad 3 \mathrm{a}-4 \mathrm{~b}-4=0$<br/>2) $8 \mathrm{a}+6 \mathrm{~b}+11=0$<br/>3) $\quad 8 a+6 b-9=0$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">Here $\frac{|8 a+6 b+1|}{\sqrt{8^{2}+6^{2}}}=1 \Rightarrow|8 a+6 b+1|=10$<br/>$\Rightarrow 8 a+6 b+1=\pm 10$<br/>$\Rightarrow 8 \mathrm{a}+6 \mathrm{~b}+1=10 \&amp; 8 \mathrm{a}+6 \mathrm{~b}+1=-10$<br/>$\Rightarrow 8 a+6 b-9=0 \&amp; 8 a+6 b+11=0$</div>


title: if-a-b-then-which-one-of-the-following-is-not-correct
slug: if-a-b-then-which-one-of-the-following-is-not-correct-40893
question: <div class="question">If $A \subseteq B$, then which one of the following is not correct?</div>
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad P(A \cap \bar{B})=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad P(A \mid B)=\frac{P(A)}{P(B)}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad P(B \mid A)=\frac{P(B)}{P(A)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$P(A \mid(A \cup B))=\frac{P(A)}{P(B)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad P(B \mid A)=\frac{P(B)}{P(A)}$</span> </div>


title: if-a-b-x-3-dx-0-and-a-b-x-2-dx-3-2-then-what-are-the-values-of-a-and-b-respectively
slug: if-a-b-x-3-dx-0-and-a-b-x-2-dx-3-2-then-what-are-the-values-of-a-and-b-respectively-69629
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1,1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x}^{3} \mathrm{dx}=0$ and $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x}^{2} \mathrm{dx}=\frac{2}{3}$, then what are the values of a and b respectively?</div>

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

answer:
<div class="solution">$\begin{aligned} &amp; \int_{a}^{b} x^{3} \cdot d x=0 \\ \Rightarrow &amp;\left(\frac{x^{4}}{4}\right)_{a}^{b}=0 \\ \Rightarrow &amp; b^{4}-a^{4}=0 \\ \Rightarrow &amp;\left(b^{2}-a^{2}\right)\left(b^{2}+a^{2}\right)=0 \\ \Rightarrow &amp; b^{2}-a^{2}=0(\text { or }) b^{2}+a^{2}=0 \\ \Rightarrow &amp;(b+a)(b-a)=0(\text { or }) b^{2}+a^{2}=0 \\ \Rightarrow &amp; b=-a(\text { or }) a=b(\text { or }) b^{2}+a^{2}=0 \end{aligned}$<br/>$\int_{a}^{b} x^{2} \cdot d x=\frac{2}{3}$<br/>$\left(\frac{x^{3}}{3}\right)_{a}^{b}=\frac{2}{3}$<br/>$\Rightarrow b^{3}-a^{3}=2$<br/>Since, $b=-a$<br/>$\Rightarrow b^{3}-\left(-b^{3}\right)=2$<br/>$\Rightarrow 2 b^{3}=2$<br/>$\Rightarrow b^{3}=1$<br/>$\Rightarrow b=1, a=-1$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-be-a-real-skew-symmetric-matrix-of-order-n-such-that-a-2-i-0-i-being-the-identity-matrix-of-the-same-order-as-that-of-a-then-what-is-the-order-of

slug:
if-a-be-a-real-skew-symmetric-matrix-of-order-n-such-that-a-2-i-0-i-being-the-identity-matrix-of-the-same-order-as-that-of-a-then-what-is-the-order-of-95815

question:
<div class="question">If $A$ be a real skew- symmetric matrix of order $\mathrm{n}$ such that $A^{2}+I=0$, I being the identity matrix of the same order as that of $A$, then what is the order of $A$?</div>

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

answer:
<div class="solution">We know if $A$ is a real skew-symmetric matrix of order $n$ such that $A^{2}+I=0$, then order of $A$ is 3.</div>


title: if-a-circle-of-radius-b-units-with-centre-at-0-b-touches-the-line-y-x-2-then-what-is-the-value-of-b
tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-a-circle-of-radius-b-units-with-centre-at-0-b-touches-the-line-y-x-2-then-what-is-the-value-of-b-54232

question:
<div class="question">If a circle of radius b units with centre at $(0, \mathrm{~b})$ touches the<br/>line $\mathrm{y}=\mathrm{x}-\sqrt{2}$, then what is the value of b?</div>

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

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

answer:
<div class="solution">Distance from the centre to the point of line which touches circle is $\mathrm{OM}=$ radius<br/>$$<br/>y=x-\sqrt{2} \text { or } x-y-\sqrt{2}=0<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/6MeBeZBRqmhfCQak-ldmYVfR3x01Mo2roYYVCzO-92U.original.fullsize.png"/><br/><br/>$r=\frac{\left|A x_{0}+B y_{0}+C\right|}{\sqrt{A^{2}+B^{2}}}$<br/>$\because x_{0}=0$<br/>$\&amp; Y_{0}=b$<br/>$b=\frac{|1(0)+(-1)(b)-\sqrt{2}|}{\sqrt{(1)^{2}+(-1)^{2}}}$<br/>$b=\frac{b+\sqrt{2}}{\sqrt{2}}$<br/>$(\sqrt{2}-1) b=\sqrt{2}$<br/>$\Rightarrow b=2+\sqrt{2}$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-cos-1-2-cos-3-6-sin-9-6-sin-2-4-and-b-sin-6-0-sin-1-2-cos-4-8-cos-7-2-then-what-is-b-a-equal-to

slug:
if-a-cos-1-2-cos-3-6-sin-9-6-sin-2-4-and-b-sin-6-0-sin-1-2-cos-4-8-cos-7-2-then-what-is-b-a-equal-to-78117

question:
<div class="question">If $\mathrm{A}=\left(\cos 12^{\circ}-\cos 36^{\circ}\right)\left(\sin 96^{\circ}+\sin 24^{\circ}\right)$<br/>and $\mathrm{B}=\left(\sin 60^{\circ}-\sin 12^{\circ}\right)\left(\cos 48^{\circ}-\cos 72^{\circ}\right)$, then what is<br/>$\frac{\mathrm{A}}{\mathrm{B}}$ equal to?</div>

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

answer:
<div class="solution">Given A=\left(\cos 12^{\circ}-\cos 36^{\circ}\right)\left(\sin 96^{\circ}+\sin 24^{\circ}\right) \\<br/>B=\left(\sin 60^{\circ}-\sin 12^{\circ}\right)\left(\cos 48^{\circ}-\cos 72^{\circ}\right) \\<br/>&amp; \frac{A}{B}=\frac{\left[2 \sin 24^{\circ} \sin 12^{\circ}\right]\left[2 \sin 60^{\circ} \cos 36^{\circ}\right]}{\left[2 \cos 36^{\circ} \sin 24^{\circ}\right]\left[2 \sin 60^{\circ} \sin 12^{\circ}\right]} \\<br/>&amp; \Rightarrow \frac{A}{B}=1 \\</div>


title: if-a-cos-sin-sin-cos-then-what-is-a-3-equal-to
slug: if-a-cos-sin-sin-cos-then-what-is-a-3-equal-to-95812
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}\cos \theta &amp; \sin \theta \\ -\sin \theta &amp; \cos \theta\end{array}\right]$, then what is $\mathbf{A}^{3}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{ll}\cos 3 \theta &amp; \sin 3 \theta \\ -\sin 3 \theta &amp; \cos 3 \theta\end{array}\right]$</span> </div>

answer:
<div class="solution">$A=\left[\begin{array}{cc}\cos \theta &amp; \sin \theta \\ -\sin \theta &amp; \cos \theta\end{array}\right]$<br/>We know, $A^{n}=\left[\begin{array}{cc}\cos n \theta &amp; \sin n \theta \\ -\sin n \theta &amp; \cos n \theta\end{array}\right]$<br/>$\therefore A^{3}=\left[\begin{array}{cc}\cos 3 \theta &amp; \sin 3 \theta \\ -\sin 3 \theta &amp; \cos 3 \theta\end{array}\right]$</div>


title: if-a-cos-sin-sin-cos-then-what-is-aa-t-equal-to-where-a-t-is-the-transpose-of-a
slug: if-a-cos-sin-sin-cos-then-what-is-aa-t-equal-to-where-a-t-is-the-transpose-of-a-64004
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right]$ then what is $\mathrm{AA}^{\mathrm{T}}$ equal to (where $\begin{array}{ll} \left.\mathrm{A}^{\mathrm{T}} \text { is the transpose of } \mathrm{A}\right) ?\end{array}$</div>

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

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

answer:
<div class="solution">$A=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right]$<br/>$A A^{T}=\left[\begin{array}{cc}\cos \alpha &amp; \sin \alpha \\ -\sin \alpha &amp; \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha &amp; -\sin \alpha \\ \sin \alpha &amp; \cos \alpha\end{array}\right]$<br/>$=\left[\begin{array}{cc}\cos ^{2} \alpha+\sin ^{2} \alpha &amp; -\cos \alpha \sin \alpha+\sin \alpha \cos \alpha \\ -\sin \alpha \cos \alpha+\sin \alpha \cos \alpha &amp; \sin ^{2} \alpha+\cos ^{2} \alpha\end{array}\right]$<br/>$=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$</div>


tags: ['Physics', 'Electromagnetic Induction', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-a-current-of-30-amperes-flowing-in-the-primary-coil-is-reduced-to-zero-in-0001-second-then-the-induced-emf-in-the-secondary-coil-is-15000-volts-the

slug:
if-a-current-of-30-amperes-flowing-in-the-primary-coil-is-reduced-to-zero-in-0001-second-then-the-induced-emf-in-the-secondary-coil-is-15000-volts-the-56622

question:
<div class="question">If a current of 3.0 amperes flowing in the primary coil is reduced to zero in 0.001 second, then the induced e.m.f. in the secondary coil is 15000 volts. The mutual inductance between the two coils is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.5$ henry</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5$ henry</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$1.5$ henry</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10$ henry</span> </li> </ul>

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

answer:
<div class="solution">$e=M \frac{d i}{d t} \Rightarrow$ $M=\frac{15000}{3} \times 0.001=5 \mathrm{H}$</div>


tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2017 (Phase 2)']
answer: <div class="solution">All the given three options are correct.</div>
source_db: MarksBatch2_P1.db
title:
if-a-denotes-the-absolutevalue-of-an-integer-then-which-of-the-following-are-correct-1-ab-a-b-2-a-b-a-b-3-a-b-a-b-select-the-correct-answer-using-the-

slug:
if-a-denotes-the-absolutevalue-of-an-integer-then-which-of-the-following-are-correct-1-ab-a-b-2-a-b-a-b-3-a-b-a-b-select-the-correct-answer-using-the-86965

question:
<div class="question">If | a | denotes the absolutevalue of an integer, then which of the following are correct? <br/>1.$|\mathrm{ab}|=|\mathrm{a}||\mathrm{b}|$<br/>$2.|\mathrm{a}+\mathrm{b}| \leq|\mathrm{a}|+|\mathrm{b}|$<br/>$3.\quad|\mathrm{a}-\mathrm{b}| \geq|| \mathrm{a}|-| \mathrm{b} \mid$<br/>Select the correct answer using the code given below.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-differentiable-function-f-defined-for-x-0-satisfies-the-relation-f-x-2-x-3-x-0-then-what-is-the-value-of-f-4

slug:
if-a-differentiable-function-f-defined-for-x-0-satisfies-the-relation-f-x-2-x-3-x-0-then-what-is-the-value-of-f-4-80267

question:
<div class="question">If a differentiable function $\mathrm{f}$ defined for $\mathrm{x}&gt;0$ satisfies the relation $\mathrm{f}\left(\mathrm{x}^{2}\right)=\mathrm{x}^{3}, \mathrm{x}&gt;0$, then what is the value of $\mathrm{f}^{\prime}(4) ?$</div>

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

answer:
<div class="solution">According to given relation. <br/>$\because \mathrm{f}\left(\mathrm{x}^{2}\right)=\mathrm{x}^{3}$<br/>Putting $\mathrm{x}=\sqrt{\mathrm{x}}$<br/>$\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{x}^{3 / 2}$<br/>Differentiating both the sides,<br/>$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\frac{3}{2} \mathrm{x}^{1 / 2}$<br/>$\Rightarrow \quad f^{\prime}(4)=\frac{3}{2} \cdot 4^{1 / 2}=\frac{3}{2}(2)=3$</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-flagstaff-of-6-m-height-placed-on-the-top-of-a-tower-throws-a-shadow-of-2-3-m-along-the-ground-then-what-is-the-angle-that-the-sun-makes-with-the

slug:
if-a-flagstaff-of-6-m-height-placed-on-the-top-of-a-tower-throws-a-shadow-of-2-3-m-along-the-ground-then-what-is-the-angle-that-the-sun-makes-with-the-40542

question:
<div class="question">If a flag-staff of $6 \mathrm{~m}$ height placed on the top of a tower<br/>throws a shadow of $2 \sqrt{3} \mathrm{~m}$ along the ground, then what is the angle that the sun makes with the ground?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$60^{\circ}$</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">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$15^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$60^{\circ}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/CNlyUqJFdUBtDD8FTDTUrZ_2M52dmfpcIPdQ-M9LSY4.original.fullsize.png"/><br/><br/>$\mathrm{AB}$ is the tower. $\mathrm{BC}$ is flag staff. The angles made by the shadows of tower and flag staff are same.<br/>In $\Delta \mathrm{ABE}, \tan \theta=\frac{\mathrm{h}}{\mathrm{x}}$<br/>In $\Delta \mathrm{ACD}, \tan \theta=\frac{\mathrm{h}+6}{\mathrm{x}+2 \sqrt{3}}$<br/>(2), $\frac{\mathrm{h}}{\mathrm{x}}=\frac{\mathrm{h}+6}{\mathrm{x}+2 \sqrt{3}} \Rightarrow \mathrm{hx}+2 \sqrt{3} \mathrm{~h}=\mathrm{hx}+6 \mathrm{x}$<br/>from(1),<br/>$\Rightarrow \frac{\mathrm{h}}{\mathrm{x}}=\frac{6}{2 \sqrt{3}}=\frac{3}{\sqrt{3}}=\frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}}=\sqrt{3}$<br/>from(1), $\tan \theta=\frac{\mathrm{h}}{\mathrm{x}}=\sqrt{3} \Rightarrow \theta=60^{\circ}$.</div>


tags: ['Physics', 'Semiconductors', 'NEET', 'NEET 2003']
source_db: MarksBatch2_P1.db
title:
if-a-full-wave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be-1

slug:
if-a-full-wave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be-1-80015

question:
<div class="question">If a full wave rectifier circuit is operating from $50 \mathrm{~Hz}$ mains, the fundamental frequency in the ripple will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$25 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$50 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$70.7 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$100 \mathrm{~Hz}$</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> </ul>

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

answer:
<div class="solution">In full wave rectifier the fundamental frequency in ripple is twice of input frequency.</div>


tags: ['Physics', 'Semiconductors', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-a-full-wave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be-2

slug:
if-a-full-wave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be-2-50734

question:
<div class="question">If a full wave rectifier circuit is operating from $50 \mathrm{~Hz}$ mains, the fundamental frequency in the ripple will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$25 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$50 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$70.7 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$100 \mathrm{~Hz}$</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> </ul>

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

answer:
<div class="solution">In full wave rectifier the fundamental frequency in ripple is twice of input frequency.</div>


tags: ['Physics', 'Magnetic Effects of Current', 'MHT CET', 'MHT CET 2007']
source_db: MarksBatch2_P1.db
title:
if-a-full-wave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be

slug:
if-a-full-wave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be-27350

question:
<div class="question">If a full wave rectifier circuit is operating from $50 \mathrm{~Hz}$ mains, the fundamental frequency in the ripple will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$70.7 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$100 \mathrm{~Hz}$</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">C</span> <span class="option-data">$25 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$59 \mathrm{~Hz}$</span> </li> </ul>

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

answer:
<div class="solution">For full wave rectifier, ripple frequency<br/>$=2 \times$ input frequency<br/>$$<br/>\begin{array}{l}<br/>=2 \times 50 \\<br/>=100 \mathrm{~Hz}<br/>\end{array}<br/>$$<br/>Note A full wave rectifier consists of two junction diodes, so, its efficiency is twice that of half wave rectifier.</div>


tags: ['Physics', 'Semiconductors', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-a-fullwave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be

slug:
if-a-fullwave-rectifier-circuit-is-operating-from-50-hz-mains-the-fundamental-frequency-in-the-ripple-will-be-10469

question:
<div class="question">If a full-wave rectifier circuit is operating from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mi>Hz</mi></math> mains, the fundamental frequency in the ripple will be</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>70</mn><mo>.</mo><mn>7</mn><mo> </mo><mi>Hz</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>100</mn><mo> </mo><mi>Hz</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>25</mn><mo> </mo><mi>Hz</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>59</mn><mo> </mo><mi>Hz</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>100</mn><mo> </mo><mi>Hz</mi></math></span> </div>

answer:
<div class="solution">For full-wave rectifier, ripple frequency<br/><br/>= <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo></math> input frequency<br/><br/>= <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>50</mn><mo>=</mo><mn>100</mn><mo> </mo><mi>Hz</mi></math></div>


tags: ['Mathematics', 'Continuity and Differentiability', 'AP EAMCET', 'AP EAMCET 2019 (23 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-a-function-f-is-defined-by-beginaligned-fx-0-text-when-x1-x31-text-when-1-x-infty-endaligned-x1-when-infty-x-1-then-at-x1-f-is

slug:
if-a-function-f-is-defined-by-beginaligned-fx-0-text-when-x1-x31-text-when-1-x-infty-endaligned-x1-when-infty-x-1-then-at-x1-f-is-35160

question:
<div class="question">If a function \(f\) is defined by :<br/>\(\begin{aligned}<br/>f(x) &amp; =0, \text { when } x=1, \\<br/>&amp; =x^3-1, \text { when } 1 &lt; x &lt; \infty,<br/>\end{aligned}\)<br/>\(=x-1\), when \(-\infty &lt; x &lt; 1\), then at \(x=1, f\) is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">continuous but not differentiable</span> </div>

answer:
<div class="solution">We have,<br/>\(f(x)=\left\{\begin{array}{rc}<br/>x-1, &amp; -\infty  &lt; x &lt; 1 \\<br/>0, &amp; x=1 \\<br/>x^3-1, &amp; 1 &lt; x &lt; \infty<br/>\end{array}\right.\)<br/>Now, (LHL at \(x=1\) )<br/>\(\begin{aligned}<br/>&amp; =\lim _{x \rightarrow 1}(x-1) \\<br/>&amp; =\mathbf{l}-\mathbf{l}=\mathbf{0}<br/>\end{aligned}\)<br/>(RHL at \(x=1\) ) \(=\lim _{x \rightarrow 1} x^3-1\)<br/>\(=(\mathrm{l})^3-\mathbf{1}=0 \text { and } f(\mathrm{l})=0\)<br/>\(\therefore \quad \text { LHL }=\text { RHL }=f(\mathrm{l})\)<br/>So, \(f(x)\) is continuous at \(x=1\)<br/>Now, (LHD at \(x=1\) )<br/>\(=\lim _{x \rightarrow 1} \frac{(x-1)-0}{x-1}=1\)<br/>and (RHD at \(x=1\) )<br/>\(\begin{aligned}<br/>&amp; =\lim _{x \rightarrow 1} \frac{\left(x^3-1\right)-0}{x-1} \\<br/>&amp; =\lim _{x \rightarrow 1}\left(x^2+x+1\right)=3<br/>\end{aligned}\)<br/>\(\therefore\) LHD \(\neq\) RHD<br/>So, \(f(x)\) is not differentiable at \(x=1\)</div>


tags: ['Biology', 'Molecular Basis of Inheritance', 'NEET', 'NEET 2022 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-geneticist-uses-the-blind-approach-for-sequencing-the-whole-genome-of-an-organism-followed-by-assignment-of-function-to-different-segments-the-me

slug:
if-a-geneticist-uses-the-blind-approach-for-sequencing-the-whole-genome-of-an-organism-followed-by-assignment-of-function-to-different-segments-the-me-56347

question:
<div class="question">If a geneticist uses the blind approach for sequencing the whole genome of an organism, followed by assignment of function to different segments, the methodology adopted by him is called as:</div>

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

answer:
<div class="solution"><p style="text-align: justify;">In sequence annotation, the whole set of genome containing all coding and non-coding sequences is sequenced and the functions are assigned to different segments.</p><p style="text-align: justify;">It could be used for marking specific features in a DNA, RNA or protein sequence with descriptive information about structure or function.</p><p style="text-align: justify;">It helps in describing regions or sites of interest in the protein sequence, such as enzyme active sites, secondary structure or other characteristics reported in the cited references.</p></div>


title: if-a-i-0-0-i-b-0-1-1-0-c-0-i-i-0-where-i-1-then-which-one-of-the-following-is-correct
slug: if-a-i-0-0-i-b-0-1-1-0-c-0-i-i-0-where-i-1-then-which-one-of-the-following-is-correct-48777
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left(\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right), B=\left(\begin{array}{cc}0 &amp; -1 \\ 1 &amp; 0\end{array}\right), C=\left(\begin{array}{ll}0 &amp; i \\ i &amp; 0\end{array}\right)$<br/>where $i=\sqrt{-1}$, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">$\mathrm{AB}=\left(\begin{array}{cc}\mathrm{i} &amp; 0 \\ 0 &amp; -1\end{array}\right)\left(\begin{array}{cc}0 &amp; -1 \\ 1 &amp; 0\end{array}\right)=\left(\begin{array}{cc}0 &amp; -1 \\ -\mathrm{i} &amp; 0\end{array}\right)$<br/>$\mathrm{AB}=-\left(\begin{array}{ll}0 &amp; \mathrm{i} \\ \mathrm{i} &amp; 0\end{array}\right)=-\mathrm{C}$</div>


title: if-a-i-2-j-3-k-and-b-3-i-j-k-and-a-b-is-perpendicular-to-a-b-then-what-is-the-value-of
slug: if-a-i-2-j-3-k-and-b-3-i-j-k-and-a-b-is-perpendicular-to-a-b-then-what-is-the-value-of-83343
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \overrightarrow{\mathrm{i}}-\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \quad$ and $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$ is<br/>perpendicular to $\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$, then what is the value of $\lambda ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2$ only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\pm 2$</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">3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pm 3$</span> </li> </ul>

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

answer:
<div class="solution">As given:<br/>$\begin{array}{ll} &amp; \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}} \\ &amp; \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}+3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}} \\ &amp; =4 \hat{\mathrm{i}}+\hat{\mathrm{j}}+(\lambda-3) \hat{\mathrm{k}} \\ \text { and } &amp; \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}-3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}} \\ &amp; =-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-(3+\lambda) \hat{\mathrm{k}}\end{array}$<br/>$(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$<br/>$\Rightarrow \quad(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \cdot(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})=0$<br/>$\Rightarrow \quad\{\hat{\mathrm{i}}+\hat{\mathrm{j}}+(\lambda-3) \hat{\mathrm{k}}\}\{-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-(3-\lambda) \hat{\mathrm{k}}\}=0$<br/>$\Rightarrow-8+3+\left(3^{2}-\lambda^{2}\right)=0$<br/>$\Rightarrow 4-\lambda^{2}=0$<br/>$\Rightarrow \lambda=\pm 2$</div>


title: if-a-i-2-j-5-k-and-b-2-i-j-3-k-then-what-is-b-a-3-a-b-equal-to
slug: if-a-i-2-j-5-k-and-b-2-i-j-3-k-then-what-is-b-a-3-a-b-equal-to-44486
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-106$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}=\hat{i}-2 \hat{j}+5 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-3 \hat{k}$ then what is<br/>$(\vec{b}-\vec{a}) \cdot(3 \vec{a}+\vec{b})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">106</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-106$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">53</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-53$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} \vec{a} &amp;=\hat{i}-2 \hat{j}+5 \hat{k} \\ \vec{b} &amp;=2 \hat{i}+\hat{j}-3 \hat{k} \\ &amp; \vec{b}-\vec{a}=2 \hat{i}+\hat{j}-3 \hat{k}-(\hat{i}-2 \hat{j}+5 \hat{k}) \\ &amp;=\hat{i}(2-1)+\hat{j}(1+2)+\hat{k}(-3-5)=\hat{i}+3 \hat{j}-8 \hat{k} \\ &amp; 3 \vec{a}+\vec{b}=3(\hat{i}-2 \hat{j}+5 \hat{k})+(2 \hat{i}+\hat{j}-3 \hat{k}) \\ &amp;=3 \hat{i}-6 \hat{j}+15 \hat{k}+2 \hat{i}+\hat{j}-3 \hat{k} \\ &amp;=\hat{i}(3+2)+\hat{j}(-6+1)+\hat{k}(15-3) \\ &amp;=5 \hat{i}-5 \hat{j}+12 \hat{k} \\ &amp;(\vec{b}-\vec{a}) \cdot(3 \vec{a}+\vec{b})=(\hat{i}+3 \hat{j}-8 \hat{k}) \cdot(5 \hat{i}-5 \hat{j}+12 \hat{k}) \\ &amp;=(1)(5)+(3)(-5)+(-8)(12) \\ &amp;=5-15-96=-106 \end{aligned}$</div>


title: if-a-i-2-j-5-k-b-2-i-j-3-k-then-what-is-b-a-3-a-b-equal-to
slug: if-a-i-2-j-5-k-b-2-i-j-3-k-then-what-is-b-a-3-a-b-equal-to-54254
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-106$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}=\hat{i}-2 \hat{j}+5 \hat{k}, \quad \vec{b}=2 \hat{i}+\hat{j}-3 \hat{k}$, then what is<br/>$(\vec{b}-\vec{a}) \cdot(3 \vec{a}+\vec{b})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">106</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-106$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">53</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-53$</span> </li> </ul>

answer:
<div class="solution">Given, $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$<br/>and $\overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$<br/>$\therefore \quad \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}$<br/>and $(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})=(3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+15 \hat{\mathbf{k}})+(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})$<br/>$\begin{aligned}=5 \hat{i}-5 \hat{j}+12 \hat{k} &amp; \\ \text { Hence, }(\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}) \cdot(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) &amp;=(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}) \cdot(5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+12 \hat{\mathbf{k}}) \\ &amp;=5-15-96 \\ &amp;=-106 \end{aligned}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-i-j-k-b-2-i-3-j-2-k-and-c-i-m-j-n-k-are-three-coplanar-vectors-and-c-6-then-which-one-of-the-following-is-correct

slug:
if-a-i-j-k-b-2-i-3-j-2-k-and-c-i-m-j-n-k-are-three-coplanar-vectors-and-c-6-then-which-one-of-the-following-is-correct-16062

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \mathrm{m}=\pm 2$ and $\mathrm{n}=1$</span> </div>

answer:
<div class="solution">$\quad \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$<br/>and $\vec{c}=\hat{i}+m \hat{j}+n \hat{k} ;|\vec{c}|=\sqrt{6}$<br/>Given, $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are coplanar.<br/>So, $\left|\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; 3 &amp; 2 \\ 1 &amp; \mathrm{~m} &amp; \mathrm{n}\end{array}\right|=0$<br/>$\mathrm{c}_{1} \rightarrow \mathrm{c}_{1}+\mathrm{c}_{2} \quad \mathrm{c}_{3} \rightarrow \mathrm{c}_{3}+\mathrm{c}_{2}$<br/>$\Rightarrow\left|\begin{array}{ccc}0 &amp; 1 &amp; 0 \\ 5 &amp; 3 &amp; 5 \\ 1+\mathrm{m} &amp; \mathrm{m} &amp; \mathrm{m}+\mathrm{n}\end{array}\right|=0$<br/>$\Rightarrow 0+1(5 m+5 n)-(5+5 m)=0$<br/>$\Rightarrow 5 \mathrm{~m}+5 \mathrm{n}-5-5 \mathrm{~m}=0 \Rightarrow 5 \mathrm{n}=5 \Rightarrow \mathrm{n}=1$<br/>$|\vec{c}|=6 \Rightarrow \sqrt{1+m^{2}+n^{2}}=\sqrt{6}$<br/>$\Rightarrow 1+\mathrm{m}^{2}+\mathrm{n}^{2}=6 \Rightarrow 2+\mathrm{m}^{2}=6 \Rightarrow \mathrm{m}^{2}=4 \Rightarrow \mathrm{m}=\pm 2$</div>


title: if-a-i-j-k-b-i-j-k-and-c-i-j-k-then-what-is-a-b-c-b-c-a-c-a-b-equal-to
slug: if-a-i-j-k-b-i-j-k-and-c-i-j-k-then-what-is-a-b-c-b-c-a-c-a-b-equal-to-80042
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}+\hat{j}-\hat{k}$, then what is<br/>$\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times(\vec{c}+\vec{a})+\vec{c} \times(\vec{a}+\vec{b})$ equal to?</div>

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

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

answer:
<div class="solution">Since, $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$<br/>$\Rightarrow \overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})+\overrightarrow{\mathbf{b}} \times(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}})+\overrightarrow{\mathbf{c}} \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})$<br/>$=(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{b}})$<br/>$=(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})-(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})-(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=0$</div>


title: if-a-i-j-k-i-b-k-and-i-j-c-k-are-coplanar-vectors-then-what-is-the-value-of-a-b-c-ab-c
slug: if-a-i-j-k-i-b-k-and-i-j-c-k-are-coplanar-vectors-then-what-is-the-value-of-a-b-c-ab-c-28776
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If a $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{b}}+\hat{\mathrm{k}}$, and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{c} \hat{\mathrm{k}}$ are coplanar vectors,<br/>then what is the value of $a+b+c-a b c ?$</div>

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

answer:
<div class="solution">Vectors a $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{c} \hat{\mathrm{k}}$ are<br/>coplanar vectors. $\Rightarrow\left|\begin{array}{lll}a &amp; 1 &amp; 1 \\ 1 &amp; b &amp; 1 \\ 1 &amp; 1 &amp; c\end{array}\right|=0$<br/>$\Rightarrow$ a $(b c-1)-1(c-1)+1(1-b)=0$<br/>$\Rightarrow \quad a b c-a-c+1+1-b=0$<br/>$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}-\mathrm{abc}=2$</div>


title: if-a-i-k-b-x-i-j-1-x-k-c-y-i-x-j-1-x-y-k-then-a-b-c-depends-on
slug: if-a-i-k-b-x-i-j-1-x-k-c-y-i-x-j-1-x-y-k-then-a-b-c-depends-on-82768
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathbf{a}}=\hat{\mathrm{i}}-\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=x \hat{\mathbf{i}}+\hat{\mathbf{j}}+(1-x) \hat{\mathbf{k}}$<br/>$\overrightarrow{\mathbf{c}}=y \hat{\mathbf{i}}+\mathrm{x} \hat{\mathbf{j}}+(1+x-y) \hat{\mathbf{k}}$<br/>then $\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})$ depends on</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x$ only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y$ only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both $x$ and $y$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither x nor $y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Neither x nor $y$</span> </div>

answer:
<div class="solution">Let $\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$<br/>and $\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$<br/>$\begin{array}{l}\text { Now, }(\vec{b} \times \vec{c})=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ x &amp; 1 &amp; (1-x) \\ y &amp; x &amp; (1+x-y)\end{array}\right| \\ \begin{aligned}=&amp; i\left(1+x-y-x+x^{2}\right)-\hat{j}\left(x+x^{2}-x y-y\right) \\+&amp; k\left(x^{2}-y\right) \end{aligned} \\ =\hat{i}\left(1-y+x^{2}\right)-\hat{j}\left(x+x^{2}-x y-y\right)+\hat{k}\left(x^{2}-y\right) \\ \text { Now, } \vec{a} .(\vec{b} \times \vec{c})=1\left(1-y+x^{2}\right)+0\left(x+x^{2}-x y-y\right) \\ -1\left(x^{2}-y\right) \\ =1-y+x^{2}-x^{2}+y &amp; \end{array}$<br/>$=1$ which shows that $\vec{a} .(\vec{b} \times \vec{c})$ does not depend<br/>on $x$ and $y$.</div>


title: if-a-ib-1-2-i-4-2-i-where-i-1-then-what-is-the-value-of-a
slug: if-a-ib-1-2-i-4-2-i-where-i-1-then-what-is-the-value-of-a-97691
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}+\mathrm{iB}=\frac{4+2 \mathrm{i}}{1-2 \mathrm{i}}$ where $\mathrm{i}=\sqrt{-1}$ then what is the value of $\mathrm{A}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-8</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">$\mathrm{A}+\mathrm{iB}=\frac{4+2 \mathrm{i}}{1-2 \mathrm{i}} \times \frac{1+2 \mathrm{i}}{1+2 \mathrm{i}}(\mathrm{ByRationalizing})$<br/>$=\frac{4+2 \mathrm{i}+8 \mathrm{i}-4}{(1)^{2}-(2 \mathrm{i})^{2}}=\frac{0+10 \mathrm{i}}{5}=2 \mathrm{i}=0+2 \mathrm{i}$<br/>$\Rightarrow \mathrm{A}=0, \mathrm{~B}=2$</div>


title: if-a-is-a-2-3-matrix-and-ab-is-a-2-5-matrix-then-b-must-be-a
slug: if-a-is-a-2-3-matrix-and-ab-is-a-2-5-matrix-then-b-must-be-a-76678
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If A is a $2 \times 3$ matrix and $\mathrm{AB}$ is a $2 \times 5$ matrix, then B must be a</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \times 5$ matrix</span> </div>

answer:
<div class="solution">A is $2 \times 3$ matrix <br/>$\mathrm{AB}$ is $2 \times 5$ matrix<br/>Let ' $\mathrm{B}$ ' bem $\times \mathrm{n}$ matrix<br/>$[\mathrm{A}]_{2 \times 3}[\mathrm{~B}]_{\mathrm{m} \times \mathrm{n}}=[\mathrm{AB}]_{2 \times 5}$<br/>number of columns of $\mathrm{A}=$ number of rows of $\mathrm{B}$. $\therefore m=3$<br/>we can observe that $\mathrm{n}=5$ from the product. So, $\mathrm{B}$ is $3 \times 5$ matrix.</div>


title: if-a-is-a-3-3-matrix-such-that-a-4-then-what-is-a-adj-a-equal-to
slug: if-a-is-a-3-3-matrix-such-that-a-4-then-what-is-a-adj-a-equal-to-21836
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ is a $3 \times 3$ matrix such that $|\mathrm{A}|=4$, then what is $\mathrm{A}(\operatorname{adj} \mathrm{A})$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{lll}4 &amp; 0 &amp; 0 \\ 0 &amp; 4 &amp; 0 \\ 0 &amp; 0 &amp; 4\end{array}\right]$</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">C</span> <span class="option-data">$\left[\begin{array}{ccc}16 &amp; 0 &amp; 0 \\ 0 &amp; 16 &amp; 0 \\ 0 &amp; 0 &amp; 16\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cannot be determined, as data is insufficient</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{lll}4 &amp; 0 &amp; 0 \\ 0 &amp; 4 &amp; 0 \\ 0 &amp; 0 &amp; 4\end{array}\right]$</span> </div>

answer:
<div class="solution">We know that $\mathrm{A}^{-1}=\frac{(\mathrm{adjA})}{|\mathrm{A}|}$<br/>or, $\mathrm{AA}^{-1}=\frac{\mathrm{A} \cdot \mathrm{Ad}_{\mathrm{j}} \mathrm{A}}{|\mathrm{A}|}$<br/>or, $\mathrm{I}_{\mathrm{n}}=\frac{\mathrm{A} \cdot \mathrm{Ad}_{\mathrm{j}} \mathrm{A}}{|\mathrm{A}|}$<br/>$\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}| \mathrm{I}_{\mathrm{n}}=|\mathrm{A}|\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]=\left[\begin{array}{lll}4 &amp; 0 &amp; 0 \\ 0 &amp; 4 &amp; 0 \\ 0 &amp; 0 &amp; 4\end{array}\right]$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-is-a-matrix-of-order-p-q-and-b-is-a-matrix-of-order-s-t-under-which-one-of-the-following-conditions-does-ab-exist

slug:
if-a-is-a-matrix-of-order-p-q-and-b-is-a-matrix-of-order-s-t-under-which-one-of-the-following-conditions-does-ab-exist-89035

question:
<div class="question">If A is a matrix of order $\mathrm{p} \times \mathrm{q}$ and $\mathrm{B}$ is a matrix of order $\mathrm{s} \times \mathrm{t}$, under which one of the following conditions does AB exist?</div>

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

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

answer:
<div class="solution">If A is a matrix of order $\mathrm{p} \times \mathrm{q}$ and $\mathrm{B}$ is a matrix of order s $\times \mathrm{t}$, then $\mathrm{AB}$ will exist when the number of column in $\mathrm{A}$ is equal to the number of rows in $\mathrm{B}$ $\Rightarrow \mathrm{q}=\mathrm{s}$</div>


title: if-a-is-a-n-n-matrix-then-adj-adj-a
slug: if-a-is-a-n-n-matrix-then-adj-adj-a-99356
tags: ['Mathematics', 'Matrices', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ is a $n \times n$ matrix, then $\operatorname{adj}(\operatorname{adj} A)=$</div>

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

answer:
<div class="solution">We know by the fundamental concept that $\operatorname{adj}(\operatorname{adj} A)=\left|A\right|^{n-2} A$.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-is-a-nonnull-row-matrix-with-5-columns-and-b-is-a-nonnull-column-matrix-with-5-rows-how-manyrows-are-there-in-a-b

slug:
if-a-is-a-nonnull-row-matrix-with-5-columns-and-b-is-a-nonnull-column-matrix-with-5-rows-how-manyrows-are-there-in-a-b-58222

question:
<div class="question">If A is a non-null row matrix with 5 columns and B is a nonnull column matrix with 5 rows, how manyrows are there in $\mathrm{A} \times \vec{B}$ ?</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">5</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">25</span> </li> </ul>

answer:
<div class="solution">If there is matrix of $m$ rows and $n$ columns and another with $n$ rows and k columns, their product will be a matrix of $\mathrm{m}$ rows and $\mathrm{k}$ column. A is a non-null matrix with one row and 5 columns and $\mathrm{B}$ is a non-null matrix with 5 rows and one column. Therefore number of row in $\mathrm{A} \times \mathrm{B}$ is 1.</div>


title: if-a-is-a-nonzero-vector-of-modulus-a-and-is-a-nonzero-scalar-and-a-is-a-unit-vector-then
slug: if-a-is-a-nonzero-vector-of-modulus-a-and-is-a-nonzero-scalar-and-a-is-a-unit-vector-then-68194
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}$ is a non-zero vector of modulus a and $\lambda$ is a non-zero<br/>scalar and $\lambda \vec{a}$ is a unit vector then</div>

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

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

answer:
<div class="solution">As given, $\lambda \overrightarrow{\mathrm{a}}$ is a unit vector.<br/>$\Rightarrow|\lambda \overrightarrow{\mathrm{a}}|=1$<br/>$\Rightarrow|\lambda| \overrightarrow{\mathrm{a}} \mid=1$<br/>$\Rightarrow \mathrm{a}=\frac{1}{|\lambda|} \quad[\because|\overrightarrow{\mathrm{a}}|=\mathrm{a}]$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-is-a-position-vector-of-a-point-1-3-and-a-is-another-point-1-5-then-what-are-the-coordinates-of-the-point-b-such-that-ab-a

slug:
if-a-is-a-position-vector-of-a-point-1-3-and-a-is-another-point-1-5-then-what-are-the-coordinates-of-the-point-b-such-that-ab-a-87625

question:
<div class="question">If $\vec{a}$ is a position vector of a point $(1,-3)$ and $A$ is another point $(-1,5)$, then what are the coordinates of the point $B$ such that $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{a}}$ ?</div>

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

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

answer:
<div class="solution">Let the coordinates of $\mathrm{B}$ be $(\mathrm{x}, \mathrm{y})$.<br/>$\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-3 \hat{\mathrm{j}}$<br/>P.V. of $\mathrm{A}$ is $(-1,5) \mathrm{so}, \overrightarrow{\mathrm{OA}}=\hat{\mathrm{i}}+5 \hat{\mathrm{j}}, \overrightarrow{\mathrm{OB}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}$<br/>$\therefore \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}$<br/>$\Rightarrow(\mathrm{x}+1) \hat{\mathrm{i}}+(\mathrm{y}-5) \overline{\mathrm{j}}=\overline{\mathrm{i}}-3 \overline{\mathrm{j}}$<br/>$\Rightarrow \mathrm{x}+1=1$ and $\mathrm{y}-5=-3$<br/>$\Rightarrow \mathrm{x}=0$ and $\mathrm{y}=2$<br/>$\therefore \quad$ Coordinates of $\mathrm{B}$ are $(0,2)$.</div>


title: if-a-is-a-relation-on-a-set-r-then-which-one-of-the-following-is-correct
slug: if-a-is-a-relation-on-a-set-r-then-which-one-of-the-following-is-correct-85055
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ is a relation on a set $\mathrm{R}$, 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">$\quad R \subseteq A$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{A} \subseteq \mathrm{R}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$A \subseteq(R \times R)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{R} \subseteq(\mathrm{A} \times \mathrm{A})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A \subseteq(R \times R)$</span> </div>

answer:
<div class="solution">Since A is a relation on a set $R$<br/>$\therefore \mathrm{A} \subseteq(\mathrm{R} \times \mathrm{R})$</div>


title: if-a-is-a-square-matrix-of-order-3-with-a-0-then-which-one-of-the-following-is-correct
slug: if-a-is-a-square-matrix-of-order-3-with-a-0-then-which-one-of-the-following-is-correct-36365
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ is a square matrix of order 3 with $|A| \neq 0$, 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">$|a d j A|=|A|$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$|a d j A|=|A|^{2}$</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">$|a d j A|=|A|^{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|a d j A|^{2}=|A|$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|a d j A|=|A|^{2}$</span> </div>

answer:
<div class="solution">$|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1} \quad$ \{n is order of square matrix $\}$<br/>If A is square matrix of order 3 , then $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{2}$</div>


title: if-a-is-a-square-matrix-of-order-n-1-then-which-one-of-the-following-is-correct
slug: if-a-is-a-square-matrix-of-order-n-1-then-which-one-of-the-following-is-correct-87156
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ is a square matrix of order $\mathrm{n}&gt;1$, then which one of the $\begin{array}{ll}\text { following is correct? } \end{array}$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\operatorname{det}(-\mathrm{A})=(-1)^{\mathrm{n}} \operatorname{det} \mathrm{A}$</span> </div>

answer:
<div class="solution">$\mathrm{A}$ is square matrix of order $\mathrm{n}&gt;1$. $\operatorname{det}(-\mathrm{A})=(-1)^{\mathrm{n}} \operatorname{det} \mathrm{A}$</div>


title: if-a-is-a-square-matrix-such-that-a-2-i-where-i-is-the-identity-matrix-then-what-is-a-1-equal-to
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-a-is-a-square-matrix-such-that-a-2-i-where-i-is-the-identity-matrix-then-what-is-a-1-equal-to-27765

question:
<div class="question">If A is a square matrix such that $\mathrm{A}^{2}=\mathrm{I}$ where $\mathrm{I}$ is the identity matrix, then what is $\mathrm{A}^{-1}$ equal to?</div>

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

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

answer:
<div class="solution">Let $\mathrm{A}^{2}=\mathrm{I}$<br/>$\Rightarrow \mathrm{A}^{2} \mathrm{~A}^{-1}=\mathrm{IA}^{-1}$<br/>$\Rightarrow \mathrm{A}=\mathrm{A}^{-1}$</div>


title: if-a-is-a-square-matrix-such-that-a-a-t-0-then-which-one-of-the-following-is-correct
slug: if-a-is-a-square-matrix-such-that-a-a-t-0-then-which-one-of-the-following-is-correct-38121
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If A is a square matrix such that $\mathrm{A}-\mathrm{A}^{\mathrm{T}}=0$, 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">A must be a null matrix</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A must be a unit matrix</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A must be a scalar matrix</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of the above</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">None of the above</span> </div>

answer:
<div class="solution">Since, A is a square matrix and $\mathrm{A}-\mathrm{A}^{\mathrm{T}}=0 \Rightarrow \mathrm{A}=\mathrm{A}^{\mathrm{T}}$<br/>A is a symmetric matrix. Considering following two points.<br/>$1.$ No two rows or two columns should be identical <br/>and<br/>$2.$ There should be two l's and one 0 , in every row or column. Such determinant can be found.</div>


title: if-a-is-a-square-matrix-then-the-value-of-adj-a-t-adj-a-t-is-equal-to
slug: if-a-is-a-square-matrix-then-the-value-of-adj-a-t-adj-a-t-is-equal-to-82335
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ is a square matrix, then the value of $\operatorname{adj} \mathrm{A}^{\mathrm{T}}-(\operatorname{adj} \mathrm{A})^{\mathrm{T}}$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">null matrix whose order is same as that of A</span> </div>

answer:
<div class="solution">We know, Adj $\mathrm{A}^{\mathrm{T}}=(\operatorname{adj} \mathrm{A})^{\mathrm{T}}$<br/>$\therefore \mathrm{adj} \mathrm{A}^{\mathrm{T}}-(\mathrm{adj} \mathrm{A})^{\mathrm{T}}=\operatorname{adj} \mathrm{A}^{\mathrm{T}}-\mathrm{adj} \mathrm{A}^{\mathrm{T}}=0$</div>


title: if-a-is-a-square-matrix-then-what-is-adj-a-1-adj-a-1-equal-to
slug: if-a-is-a-square-matrix-then-what-is-adj-a-1-adj-a-1-equal-to-67584
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ is a square matrix, then what is $\operatorname{adj}\left(\mathrm{A}^{-1}\right)-(\operatorname{adj} \mathrm{A})^{-1}$ equal to?</div>

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

answer:
<div class="solution">Let $[A]=\left[\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right]$<br/>$A^{-1}=\frac{1}{|A|}($ adj. $A)$<br/>$\quad=\frac{1}{(a d-b c)}\left[\begin{array}{cc}d &amp; -c \\ -b &amp; a\end{array}\right]^{T}=\frac{1}{(a d-b c)}\left[\begin{array}{cc}d &amp; -b \\ -c &amp; a\end{array}\right]$<br/>$\operatorname{adj}\left(A^{-1}\right)=\frac{1}{(a d-b c)}\left[\begin{array}{ll}a &amp; c \\ b &amp; d\end{array}\right]^{T}=\frac{1}{(a d-b c)}\left[\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right]$ ...(1)<br/>$$<br/>\begin{array}{l}a d j A=\left[\begin{array}{cc}d &amp; -b \\ -c &amp; a\end{array}\right] \\ |\operatorname{adj} A|=(a d-b c)\end{array}<br/>$$<br/>$(\operatorname{adj} A)^{-1}=\frac{1}{|a d j A|}\left[\begin{array}{ll}a &amp; c \\ b &amp; d\end{array}\right]^{T}=\frac{1}{a d-b c}\left[\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right]$ ...(2)<br/>Subtracting eqn. (1) and (2),<br/>$\operatorname{adj}\left(A^{-1}\right)-(\operatorname{adj} A)^{-1}=\left[\begin{array}{ll}0 &amp; 0 \\ 0 &amp; 0\end{array}\right]$<br/>$=$ null matrix.</div>


title: if-a-is-a-square-matrix-then-what-is-adj-a-t-a-d-j-a-t-equal-to
slug: if-a-is-a-square-matrix-then-what-is-adj-a-t-a-d-j-a-t-equal-to-15581
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ is a square matrix, then what is adj $A^{T}-(a d j A)^{\mathrm{T}}$ equal to?</div>

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

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

answer:
<div class="solution">We know $\left(\operatorname{adj} \mathrm{A}^{\mathrm{T}}\right)=(\operatorname{adj} \mathrm{A})^{\mathrm{T}}$<br/>$\Rightarrow\left(\operatorname{adj} \mathrm{A}^{\mathrm{T}}\right)-(\operatorname{adj} \mathrm{A})^{\mathrm{T}}=$ Null matrix</div>


title: if-a-is-a-subset-of-b-then-which-one-of-the-following-is-correct
slug: if-a-is-a-subset-of-b-then-which-one-of-the-following-is-correct-47395
question: <div class="question">If $A$ is a subset of $B$, then which one of the following is correct?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A^{c} \subset B^{c}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$B^{c} \subset A^{c}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$A^{c}=B^{t}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$A \subseteq A \cap B$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$B^{c} \subset A^{c}$</span> </div>

answer:
<div class="solution">Suppose $U=\{a, b, c, d, e, f, g, h\}$ $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$<br/>$B=\{a, b, c, d, e\}$ Given $A \subseteq B$<br/>$\mathrm{A}^{\mathrm{c}}=\{\mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\}, \mathrm{B}^{\mathrm{C}}=\{\mathrm{f}, \mathrm{g}, \mathrm{h}\}$<br/>Hence, $\mathrm{B}^{\mathrm{C}} \subseteq \mathrm{A}^{\circ}$</div>


title: if-a-is-an-identity-matrix-of-order-3-then-its-inverse-a-1
slug: if-a-is-an-identity-matrix-of-order-3-then-its-inverse-a-1-60700
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}$ is an identity matrix of order 3 , then its inverse $\left(\mathrm{A}^{-1}\right)$</div>

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

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

answer:
<div class="solution">Given, A is an identity matrix. $\therefore \mathrm{A}=\mathrm{I}$<br/>We know, $\mathrm{I}^{-1}=\mathrm{I}$<br/>$\therefore \mathrm{A}^{-1}=\mathrm{A}$</div>


title: if-a-is-an-invertible-matrix-then-what-is-det-a-1-equal-to
slug: if-a-is-an-invertible-matrix-then-what-is-det-a-1-equal-to-79920
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ is an invertible matrix, then what is det $\left(A^{-1}\right)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\operatorname{det} A$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\operatorname{det} A}$</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</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">$\frac{1}{\operatorname{det} A}$</span> </div>

answer:
<div class="solution">$\operatorname{det}\left(\mathrm{A}^{-1}\right)=\frac{1}{\operatorname{det} \mathrm{A}}$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-is-an-orthogonal-matrix-of-order-3-and-b-1-3-2-2-0-5-3-2-0-then-which-of-the-following-isare-correct-1-ab-47-2-ab-ba-select-the-correct-answer-us

slug:
if-a-is-an-orthogonal-matrix-of-order-3-and-b-1-3-2-2-0-5-3-2-0-then-which-of-the-following-isare-correct-1-ab-47-2-ab-ba-select-the-correct-answer-us-15155

question:
<div class="question">If A is an orthogonal matrix of order 3 and $\mathrm{B}=\left[\begin{array}{ccc}1 &amp; 2 &amp; 3 \\ -3 &amp; 0 &amp; 2 \\ 2 &amp; 5 &amp; 0\end{array}\right]$, then which of the following is/are correct? <br/>$1.$ $|\mathrm{AB}|=\pm 47$<br/>$2. \mathrm{AB}=\mathrm{BA}$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">The determinent of a orthogonal matrix is always $\pm 1$ $|\mathrm{A}|=\pm 1$<br/>$\mathrm{B}=\left[\begin{array}{ccc}1 &amp; 2 &amp; 3 \\ -3 &amp; 0 &amp; 2 \\ 2 &amp; 5 &amp; 0\end{array}\right]$<br/>$|B|=-10-2(-4)+3(-15)$<br/>$=-47$<br/>$|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|$<br/>$=(\pm 1)(-47)$<br/>$=\pm 47$</div>


title: if-a-is-any-2-2-matrix-such-that-1-0-2-3-a-1-6-0-3-then-what-is-a-equal-to
slug: if-a-is-any-2-2-matrix-such-that-1-0-2-3-a-1-6-0-3-then-what-is-a-equal-to-83479
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If A is any $2 \times 2$ matrix such that $\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right] \mathrm{A}=\left[\begin{array}{cc}-1 &amp; 0 \\ 6 &amp; 3\end{array}\right]$<br/>then what is A equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{ll}-5 &amp; 1 \\ -2 &amp; 2\end{array}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{cc}-5 &amp; -2 \\ 1 &amp; 2\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{cc}-5 &amp; -2 \\ 2 &amp; 1\end{array}\right]$</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">$\left[\begin{array}{cc}5 &amp; 2 \\ -2 &amp; -1\end{array}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{cc}-5 &amp; -2 \\ 2 &amp; 1\end{array}\right]$</span> </div>

answer:
<div class="solution">Let $\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]=\mathrm{B}$<br/>Then $\mathrm{BA}=\left[\begin{array}{cc}-1 &amp; 0 \\ 6 &amp; 3\end{array}\right]$<br/>$\Rightarrow A=B^{-1}\left[\begin{array}{ll}-1 &amp; 0 \\ -6 &amp; 3\end{array}\right]$<br/>$|\mathrm{B}|=3$<br/>$\operatorname{adj} \mathrm{B}=\left[\begin{array}{cc}3 &amp; -2 \\ 0 &amp; 1\end{array}\right]$<br/>$\mathrm{B}^{-1}=\frac{1}{3}\left[\begin{array}{rr}3 &amp; -2 \\ 0 &amp; 1\end{array}\right]$<br/>$\Rightarrow \mathrm{A}=\frac{1}{3}\left[\begin{array}{rr}3 &amp; -2 \\ 0 &amp; 1\end{array}\right]\left[\begin{array}{rr}-1 &amp; 0 \\ 6 &amp; 3\end{array}\right]=\frac{1}{3}\left[\begin{array}{cc}-3-12 &amp; -6 \\ 6 &amp; 3\end{array}\right]$<br/>$=\left[\begin{array}{rr}-5 &amp; -2 \\ 2 &amp; 1\end{array}\right]$<br/>Aliter:<br/>$\operatorname{Let} A=\left[\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right]$<br/>then $\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]\left[\begin{array}{ll}\mathrm{a} &amp; \mathrm{b} \\ \mathrm{c} &amp; \mathrm{d}\end{array}\right]=\left[\begin{array}{ll}-1 &amp; 0 \\ 6 &amp; 3\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{ll}\mathrm{a}+2 \mathrm{c} &amp; \mathrm{b}+2 \mathrm{~d} \\ 0+3 \mathrm{c} &amp; 0+3 \mathrm{~d}\end{array}\right]=\left[\begin{array}{ll}-1 &amp; 0 \\ 6 &amp; 3\end{array}\right]$<br/>$\Rightarrow 3 \mathrm{c}=6$ or $\mathrm{c}=2$<br/>$3 \mathrm{~d}=3$ or $\mathrm{d}=1$<br/>$\mathrm{a}+2 \times 2=-1$ or $\mathrm{a}=-5$<br/>$\mathrm{~b}+2 \times 1=0, \mathrm{~b}=-2$<br/>So, $\mathrm{A}=\left[\begin{array}{cc}-5 &amp; -2 \\ 2 &amp; 1\end{array}\right]$</div>


title: if-a-is-any-matrix-then-the-product-aa-is-defined-only-when-a-is-a-matrix-of-order-m-n-where
slug: if-a-is-any-matrix-then-the-product-aa-is-defined-only-when-a-is-a-matrix-of-order-m-n-where-72838
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$m=n$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ is any matrix, then the product $A A$ is defined only when $A$ is a matrix of order $m \times n$ where:</div>

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

answer:
<div class="solution">A A is defined only when $A$ is a matrix of order $m \times n$ where $m=n$ $\mathrm{A} \times \mathrm{A}=(\mathrm{m} \times \mathrm{n})(\mathrm{m} \times \mathrm{n})=(\mathrm{m} \times \mathrm{n})(\mathrm{n} \times \mathrm{n})$ if $\mathrm{m}=\mathrm{n}$<br/>$=\mathrm{m} \times \mathrm{n}=\mathrm{n} \times \mathrm{n}$ or $\mathrm{m} \times \mathrm{m}$<br/>$=\mathrm{A}$ is a square matrix.</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-is-matrix-of-order-3-2-and-b-is-matrix-of-order-2-3-then-what-is-kab-equal-to-where-k-is-any-scalar-quantity

slug:
if-a-is-matrix-of-order-3-2-and-b-is-matrix-of-order-2-3-then-what-is-kab-equal-to-where-k-is-any-scalar-quantity-34483

question:
<div class="question">If A is matrix of order $3 \times 2$ and $\mathrm{B}$ is matrix of order $2 \times 3$, then what is $|\mathrm{kAB}|$ equal to (where $\mathrm{k}$ is any scalar quantity)?</div>

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

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

answer:
<div class="solution">As given of $\mathrm{A}$ and $\mathrm{B}$ are $3 \times 2$ and $2 \times 3$ respectively. $\Rightarrow$ order of $\mathrm{AB}$ is $3 \times 3$ $\Rightarrow|\mathrm{kAB}|=\mathrm{k}^{3} \mid \mathrm{AB}$</div>


title: if-a-is-semy-square-matrix-of-order-3-and-det-a-5-then-what-is-det-2-a-1-equal-to
slug: if-a-is-semy-square-matrix-of-order-3-and-det-a-5-then-what-is-det-2-a-1-equal-to-73913
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If A is semy square matrix of order 3 and det $\mathrm{A}=5$, then what is $\operatorname{det}\left[(2 \mathrm{~A})^{-1}\right]$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 10$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 / 5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 / 5$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 40$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 40$</span> </div>

answer:
<div class="solution">$|A|=5 \Rightarrow|2 A|=2^{3} \times 5=40$<br/>$\Rightarrow\left|A^{-1}\right|=\frac{1}{|A|} \Rightarrow\left|(2 A)^{-1}\right|=\frac{1}{|2 A|}=\frac{1}{40}$</div>


title: if-a-l-p-b-m-q-c-n-r-2-then-what-is-the-value-of-the-determinant-6-a-2-l-2-p-3-b-m-q-15-c-5-n-5-r
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">60</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-a-l-p-b-m-q-c-n-r-2-then-what-is-the-value-of-the-determinant-6-a-2-l-2-p-3-b-m-q-15-c-5-n-5-r-69456

question:
<div class="question">If $\left|\begin{array}{ccc}a &amp; b &amp; c \\ l &amp; m &amp; n \\ p &amp; q &amp; r\end{array}\right|=2$, then what is the value of the determinant $\left|\begin{array}{ccc}6 a &amp; 3 b &amp; 15 c \\ 2 l &amp; m &amp; 5 n \\ 2 p &amp; q &amp; 5 r\end{array}\right| ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">40</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">60</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $\left|\begin{array}{ccc}a &amp; b &amp; c \\ l &amp; m &amp; n \\ p &amp; q &amp; r\end{array}\right|=2$<br/>Consider $\left|\begin{array}{ccc}6 a &amp; 3 b &amp; 15 c \\ 2 l &amp; m &amp; 5 n \\ 2 p &amp; q &amp; 5 r\end{array}\right|=2 \times 5\left|\begin{array}{ccc}3 a &amp; 3 b &amp; 3 c \\ l &amp; m &amp; n \\ p &amp; q &amp; r\end{array}\right|$<br/>$=10 \times 3\left|\begin{array}{lll}a &amp; b &amp; c \\ l &amp; m &amp; n \\ p &amp; q &amp; r\end{array}\right|$<br/>$=30\left|\begin{array}{lll}a &amp; b &amp; c \\ l &amp; m &amp; n \\ p &amp; q &amp; r\end{array}\right|=30 \times 2=60$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-line-is-perpendicular-to-the-line-5-x-y-0-and-forms-a-triangle-of-area-5-square-units-with-coordinate-axes-then-its-equation-is

slug:
if-a-line-is-perpendicular-to-the-line-5-x-y-0-and-forms-a-triangle-of-area-5-square-units-with-coordinate-axes-then-its-equation-is-59631

question:
<div class="question">If a line is perpendicular to the line $5 \mathrm{x}-\mathrm{y}=0$ and forms a triangle of area 5 square units with co-ordinate axes, then its equation is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad x+5 y \pm 5 \sqrt{2}=0$</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">$x-5 y \pm 5 \sqrt{2}=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 x+y \pm 5 \sqrt{2}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 x-y \pm 5 \sqrt{2}=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad x+5 y \pm 5 \sqrt{2}=0$</span> </div>

answer:
<div class="solution">$\quad 5 \mathrm{x}-\mathrm{y}=0$<br/>$\mathrm{y}=5 \mathrm{x}$...(1)<br/>Slope $=5$<br/>Slope of perpendicular line will be $-\frac{1}{5}$<br/>Let equation of line is $y=\left(-\frac{1}{5}\right) x+c$...(2)<br/>Putting $\mathrm{y}=0$ $x=5 c$<br/>$O B=5 c$<br/>Intersecting point $\mathrm{A}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/-0mJFnImBHir5BURwVllW-3fzry04qTzja2xOCoKRCg.original.fullsize.png"/><br/><br/>Putting $\mathrm{x}=0$<br/>$\mathrm{y}=-\frac{1}{5} \times 0+\mathrm{c}$<br/>$\mathrm{y}=\mathrm{c}$<br/>area $\Delta \mathrm{AOB}=\frac{1}{2} \times \mathrm{c} \times 5 \mathrm{c}$<br/>$5=\frac{1}{2} \times 5 \mathrm{c}^{2}$<br/>$c=\pm \sqrt{2}$<br/>$\mathrm{y}=\left(-\frac{1}{5}\right) \mathrm{x} \pm \sqrt{2}$<br/>$5 y+x \pm 5 \sqrt{2}=0$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-a-line-makes-3-0-with-the-positive-direction-of-x-axis-angle-with-the-positive-direction-of-y-axis-and-angle-with-the-positive-direction-of-z-axis-

slug:
if-a-line-makes-3-0-with-the-positive-direction-of-x-axis-angle-with-the-positive-direction-of-y-axis-and-angle-with-the-positive-direction-of-z-axis-80126

question:
<div class="question">If a line makes $30^{\circ}$ with the positive direction of $\mathrm{x}$ -axis, angle $\beta$ with the positive direction of $y$ -axis and angle $\gamma$ with the positive direction of $\mathrm{z}$ -axis, then what is $\cos ^{2} \beta+\cos ^{2} \gamma$ equal to ?</div>

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

answer:
<div class="solution">Direction cosines are $\cos 30^{\circ}, \cos \beta$ and $\cos \gamma$<br/>Since we know $\cos ^{2} 30+\cos ^{2} \beta+\cos ^{2} \gamma=1$<br/>$\Rightarrow \cos ^{2} \beta+\cos ^{2} \gamma=\frac{1}{4} \quad\left(\because \cos 30^{\circ}=\frac{\sqrt{3}}{2}\right)$</div>


