# questions

tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2024 (27 Jan Shift 1)']
source_db: MarksBatch2_P1.db
title:
let-a-1-a-2-a-10-be-10-observations-such-that-k-1-10-a-k-50-and-k-j-a-k-a-j-1100-then-the-standard-deviation-of-a-1-a-2-a-10-is-equal-to

slug:
let-a-1-a-2-a-10-be-10-observations-such-that-k-1-10-a-k-50-and-k-j-a-k-a-j-1100-then-the-standard-deviation-of-a-1-a-2-a-10-is-equal-to-29188

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">a</mi><mn>1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">a</mi><mn>2</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi mathvariant="normal">a</mi><mn>10</mn></msub></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> observations such that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>â</mo><mrow><mi mathvariant="normal">k</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">k</mi></msub><mo>=</mo><mn>50</mn></math>Â and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mo>â</mo><mrow><mo>â</mo><mi mathvariant="normal">k</mi><mo>&lt;</mo><mi mathvariant="normal">j</mi></mrow></munder><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">k</mi></msub><mo>Â·</mo><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">j</mi></msub><mo>=</mo><mn>1100</mn></math>. Then the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">a</mi><mn>1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">a</mi><mn>2</mn></msub><mo>,</mo><mo>â¦</mo><mo>,</mo><msub><mi mathvariant="normal">a</mi><mn>10</mn></msub></math> is equal to :</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>5</mn></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"><msqrt><mn>5</mn></msqrt></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>10</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>115</mn></msqrt></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"><msqrt><mn>5</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p>Given:Â <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>â</mo><mrow><mi mathvariant="normal">k</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">k</mi></msub><mo>=</mo><mn>50</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><msub><mi mathvariant="normal">a</mi><mn>1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>2</mn></msub><mo>+</mo><mo>â¦</mo><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>10</mn></msub><mo>=</mo><mn>50</mn><mo>Â </mo><mo>Â </mo><mo>Â </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>i</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><msup><mfenced><mrow><msub><mi mathvariant="normal">a</mi><mn>1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>2</mn></msub><mo>+</mo><mo>â¦</mo><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>10</mn></msub></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2500</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><munderover><mo>â</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><munder><mo>â</mo><mrow><mi mathvariant="normal">k</mi><mo>&lt;</mo><mi mathvariant="normal">j</mi></mrow></munder><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">k</mi></msub><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">j</mi></msub><mo>=</mo><mn>2500</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><munderover><mo>â</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mo>Ã</mo><mn>1100</mn><mo>=</mo><mn>2500</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><munderover><mo>â</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>=</mo><mn>2500</mn><mo>-</mo><mn>2</mn><mfenced><mn>1100</mn></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><munderover><mo>â</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>=</mo><mn>300</mn></math></p><p>Â Standard deviation,Â <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ï</mi><mo>=</mo><msqrt><mfrac><mrow><mo>â</mo><msubsup><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup></mrow><mi mathvariant="normal">n</mi></mfrac><mo>-</mo><msup><mfenced><mfrac><mrow><mo>â</mo><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi></msub></mrow><mi mathvariant="normal">n</mi></mfrac></mfenced><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi mathvariant="normal">Ï</mi><mo>=</mo><msqrt><mfrac><mrow><mo>â</mo><msubsup><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mfrac><mrow><mo>â</mo><msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">i</mi></msub></mrow><mn>10</mn></mfrac></mfenced><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi mathvariant="normal">Ï</mi><mo>=</mo><msqrt><mfrac><mn>300</mn><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mfrac><mn>50</mn><mn>10</mn></mfrac></mfenced><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi mathvariant="normal">Ï</mi><mo>=</mo><msqrt><mn>30</mn><mo>-</mo><mn>25</mn></msqrt><mo>=</mo><msqrt><mn>5</mn></msqrt></math></p></div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (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:
let-a-2-j-3-k-b-j-3-k-and-c-3-i-3-j-k-let-n-be-a-unit-vector-such-that-a-n-b-n-0-what-is-the-value-of-c-n

slug:
let-a-2-j-3-k-b-j-3-k-and-c-3-i-3-j-k-let-n-be-a-unit-vector-such-that-a-n-b-n-0-what-is-the-value-of-c-n-39383

question:
<div class="question">$\quad$ Let $\overrightarrow{\mathbf{a}}=2 \overrightarrow{\mathbf{j}}-3 \overrightarrow{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=-3 \overrightarrow{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$. Let $\hat{\mathbf{n}}$ be<br/>a unit vector such that $\overrightarrow{\mathbf{a}} \hat{\mathbf{n}}=\overrightarrow{\mathbf{b}} \hat{\mathbf{n}}=0$. What is the value of $\overrightarrow{\mathbf{c}} . \hat{\mathbf{n}} ?$</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">$\sqrt{19}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">As given $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$<br/>and $\overrightarrow{\mathrm{c}}=-3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>Let $\overrightarrow{\mathrm{n}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$<br/>Since $\vec{a}$ and $\hat{n}$ are perpendicular to each other. $\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{n}}=0 \Rightarrow(2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}) \cdot(\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}})=0$<br/>$\Rightarrow 2 \mathrm{y}-3 \mathrm{z}=0$<br/>and $\overrightarrow{\mathrm{b}} \cdot \hat{\mathrm{n}}=0$<br/>$\Rightarrow(\hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \cdot(\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}})=0$<br/>$\Rightarrow \mathrm{y}+3 \mathrm{z}=0$<br/>$\ldots(2)$<br/>On solving Eqs. (1) and (2) $y=z=0$<br/>Since $\hat{n}$ is a unit vector, $\sqrt{x^{2}+y^{2}+z^{2}}=1 \Rightarrow x=1 \quad[$ since, $y=z=0]$<br/>hence, $\hat{n}=\hat{i}$<br/>This gives, $\overrightarrow{\text { c. }} \hat{\text { n }}=(-3 \hat{i}+3 \hat{j}+\hat{k}) \cdot(\hat{\mathrm{i}})=-3$</div>


title: let-a-5-2-6-1-1-5-let-there-exist-a-matrix-b-such-that-a-b-35-29-49-13-what-is-b-equal-to
slug: let-a-5-2-6-1-1-5-let-there-exist-a-matrix-b-such-that-a-b-35-29-49-13-what-is-b-equal-to-92487
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}=\left[\begin{array}{ccc}5 &amp; 6 &amp; 1 \\ 2 &amp; -1 &amp; 5\end{array}\right]$. Let there exist a matrix $B$ such that<br/>$A B=\left[\begin{array}{ll}35 &amp; 49 \\ 29 &amp; 13\end{array}\right] .$ What is $B$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{lll}5 &amp; 1 &amp; 4 \\ 2 &amp; 6 &amp; 3\end{array}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{lll}2 &amp; 6 &amp; 3 \\ 5 &amp; 1 &amp; 4\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{ll}5 &amp; 2 \\ 1 &amp; 6 \\ 4 &amp; 3\end{array}\right]$</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">$\left[\begin{array}{ll}2 &amp; 5 \\ 6 &amp; 1 \\ 3 &amp; 4\end{array}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{ll}5 &amp; 2 \\ 1 &amp; 6 \\ 4 &amp; 3\end{array}\right]$</span> </div>

answer:
<div class="solution">$A=\left[\begin{array}{ccc}5 &amp; 6 &amp; 1 \\ 2 &amp; -1 &amp; 5\end{array}\right]$ and let $B=\left[\begin{array}{ll}5 &amp; 2 \\ 1 &amp; 6 \\ 4 &amp; 3\end{array}\right]$<br/>$\therefore \quad A B=\left[\begin{array}{lrl}5 &amp; 6 &amp; 1 \\ 2 &amp; -1 &amp; 5\end{array}\right]\left[\begin{array}{ll}5 &amp; 2 \\ 1 &amp; 6 \\ 4 &amp; 3\end{array}\right]$<br/>$=\left[\begin{array}{ll}25+6+4 &amp; 10+36+3 \\ 10-1+20 &amp; 4-6+15\end{array}\right]$<br/>$=\left[\begin{array}{ll}35 &amp; 49 \\ 29 &amp; 13\end{array}\right]$<br/>Hence, option (c) is correct. <br/>$ALTERNATE SOLUTION$<br/>Given $A=\left[\begin{array}{lll}5 &amp; 6 &amp; 1 \\ 2 &amp; -1 &amp; 5\end{array}\right]_{2 \times 3}$ and $A B=\left[\begin{array}{cc}35 &amp; 49 \\ 29 &amp; 13\end{array}\right]_{2 \times 2}$<br/>Since, order of $\mathrm{A}=2 \times 3$ and order of $\mathrm{AB}=2 \times 2$ $\therefore$ order of $\mathrm{B}=3 \times 2$<br/>Let $B=\left[\begin{array}{ll}a &amp; b \\ c &amp; d \\ e &amp; f\end{array}\right]$<br/>$\therefore \quad A B=\left[\begin{array}{ccc}5 &amp; 6 &amp; 1 \\ 2 &amp; -1 &amp; 5\end{array}\right]\left[\begin{array}{ll}a &amp; b \\ c &amp; d \\ e &amp; f\end{array}\right]$<br/>$\left[\begin{array}{ll}35 &amp; 49 \\ 29 &amp; 13\end{array}\right]=\left[\begin{array}{cc}5 a+6 c+e &amp; 5 b+6 d+f \\ 2 a-c+5 e &amp; 2 b-d+5 f\end{array}\right]$<br/>$\Rightarrow 5 a+6 c+e=35$<br/>$5 b+6 d+f=49$<br/>$2 a-c+5 e=29$<br/>$2 b-d+5 f=13$<br/>On solving above four equations we get $\mathrm{a}=5, \mathrm{~b}=2, \mathrm{c}=1, \mathrm{~d}=6, \mathrm{e}=4$ and $\mathrm{f}=3$<br/>Hence $B=\left[\begin{array}{ll}5 &amp; 2 \\ 1 &amp; 6 \\ 4 &amp; 3\end{array}\right]$</div>


title: let-a-7-b-11-a-b-10-3-what-is-a-b-equal-to
slug: let-a-7-b-11-a-b-10-3-what-is-a-b-equal-to-62911
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $|\bar{a}|=7,|\bar{b}|=11,|\bar{a}+\bar{b}|=10 \sqrt{3}$<br/>What is $|\bar{a}-\bar{b}|$ equal to ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \sqrt{10}$</span> </div>

answer:
<div class="solution">Let angle between $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ be $\theta$.<br/>$|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=\sqrt{|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+2|\overrightarrow{\mathrm{a}}| \overrightarrow{\mathrm{b}}|\cos \theta|}$<br/>$\sqrt[10]{3}=\sqrt{49+121+2 \times 7 \times 11 \cos \theta}$<br/>$300=170+154 \cos \theta$<br/>$154 \cos \theta=130$<br/>$|\vec{a}-\vec{b}|=\sqrt{|\vec{a}|^{2}+|\vec{b}|^{2}-2|\vec{a} \| \vec{b}| \cos \theta}$<br/>$|\vec{a}-\vec{b}|=\sqrt{170-154 \cos \theta}$<br/>$|\vec{a}-\vec{b}|=\sqrt{170-130}=\sqrt{40}$ or $2 \sqrt{10}$</div>


title: let-a-7-b-11-a-b-10-3-what-is-the-angle-between-a-b-and-a-b
slug: let-a-7-b-11-a-b-10-3-what-is-the-angle-between-a-b-and-a-b-71886
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $|\bar{a}|=7,|\bar{b}|=11,|\bar{a}+\bar{b}|=10 \sqrt{3}$<br/>What is the angle between $(\bar{a}+\bar{b})$ and $(\bar{a}-\bar{b})$ ?</div>

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

answer:
<div class="solution">Let angle between $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$ be $\alpha$<br/>$\cos \alpha=\frac{(\vec{a}+\vec{b})(\vec{a}-\vec{b})}{|\vec{a}+\vec{b} \| \vec{a}-\vec{b}|}$<br/>$=\frac{(7)^{2}-(11)^{2}}{10 \sqrt{3} \times 2 \sqrt{10}}=\frac{(7+11)(7-11)}{20 \sqrt{3} \times \sqrt{10}}=\frac{-18}{5 \sqrt{30}}$<br/>$=\frac{-6 \times 3}{5 \sqrt{30}} \times \frac{\sqrt{30}}{\sqrt{30}}=-\frac{3 \sqrt{30}}{25}$<br/>$\alpha=\cos ^{-1}\left(\frac{-3}{5} \sqrt{\frac{6}{5}}\right)$</div>


title: let-a-a-b-c-d-and-b-x-y-z-what-is-the-number-of-element-in-a-b
slug: let-a-a-b-c-d-and-b-x-y-z-what-is-the-number-of-element-in-a-b-70583
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A=\{a, b, c, d\}$ and $B=\{x, y, z\} .$ What is the number of element in $A \times B$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">64</span> </li> </ul>

answer:
<div class="solution">$\mathrm{n}(\mathrm{A})=4$ and $\mathrm{n}(\mathrm{B})=3$<br/>Number of elements in $n(A \times B)=4 \times 3=12$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-a-ij-m-m-be-a-matrix-and-c-c-iij-m-m-be-another-matrix-where-c-ij-is-the-cofactor-of-a-ij-then-what-is-the-value-of-ac

slug:
let-a-a-ij-m-m-be-a-matrix-and-c-c-iij-m-m-be-another-matrix-where-c-ij-is-the-cofactor-of-a-ij-then-what-is-the-value-of-ac-93746

question:
<div class="question">Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{m}}$ be a matrix and $\mathrm{C}=\left[\mathrm{c}_{\mathrm{iij}}\right]_{\mathrm{m} \times \mathrm{m}}$ be another matrix where $\mathrm{c}_{\mathrm{ij}}$ is the cofactor of $\mathrm{a}_{\mathrm{ij}}$. Then, what is the value of $|\mathrm{AC}|$?</div>

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

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

answer:
<div class="solution">Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{i} j}\right]_{\mathrm{m} \times \mathrm{m}}$ be a matrix and $\mathrm{C}=\left[\mathrm{c}_{\mathrm{i} j}\right]_{\mathrm{m} \times \mathrm{m}}$ be<br/>another matrix where cij is the cofactor of $\mathrm{a}_{\mathrm{ij}}$ <br/>$\therefore \quad$ The value of $|\mathrm{AC}|=|\mathrm{A}|^{\mathrm{m}+1}$</div>


title: let-a-a-ij-n-n-and-adj-a-ij-if-a-1-4-2-2-5-3-3-4-1-what-is-the-value-of-23
slug: let-a-a-ij-n-n-and-adj-a-ij-if-a-1-4-2-2-5-3-3-4-1-what-is-the-value-of-23-22511
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{\mathrm{n} \times \mathrm{n}}$ and adj $\mathrm{A}=\left(\alpha_{\mathrm{ij}}\right)$<br/>If $\mathrm{A}=\left[\begin{array}{ccc}1 &amp; 2 &amp; 3 \\ 4 &amp; 5 &amp; 4 \\ 2 &amp; 3 &amp; -1\end{array}\right]$, what is the value of $\alpha_{23}$?</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">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">-8</span> </li> </ul>

answer:
<div class="solution">$\because \quad A=\left[\begin{array}{ccc}1 &amp; 2 &amp; 3 \\ 4 &amp; 5 &amp; 4 \\ 2 &amp; 3 &amp; -1\end{array}\right]$<br/>Since, adj $\mathrm{A}=\left(\alpha_{\mathrm{ij}}\right)$ so, $\alpha_{\mathrm{ij}}=\mathrm{a}_{\mathrm{ji}}$ of $\mathrm{A}$.<br/>$\therefore \alpha_{23}=\mathrm{a}_{32}=(-1)^{2+3}\left|\begin{array}{ll}1 &amp; 3 \\ 4 &amp; 4\end{array}\right|=-(4-12)=8$</div>


title: let-a-and-b-are-two-mutually-exclusive-events-with-p-a-3-1-p-b-4-1-what-is-the-value-of-p-a-b
slug: let-a-and-b-are-two-mutually-exclusive-events-with-p-a-3-1-p-b-4-1-what-is-the-value-of-p-a-b-51573
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A$ and $B$ are two mutually exclusive events with $\mathrm{P}(\mathrm{A})=$ $\frac{1}{3} \mathrm{P}(\mathrm{B})=\frac{1}{4} .$ What is the value of $\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{12}$</span> </div>

answer:
<div class="solution">$A \&amp; B$ are mutualy exclusive events. i.e., $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$<br/>We know, $\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=\mathrm{P}(\overline{\mathrm{A} \cup \mathrm{B}})$<br/>$=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})$<br/>$=1-[\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})]$<br/>$=1-\left[\frac{1}{3}+\frac{1}{4}-0\right]$<br/>$=1-\left[\frac{7}{12}\right]=\frac{5}{12} .$</div>


title: let-a-and-b-be-3-3-matrices-with-det-a-4-and-det-b-3-what-is-det-2-ab-equal-to
slug: let-a-and-b-be-3-3-matrices-with-det-a-4-and-det-b-3-what-is-det-2-ab-equal-to-92108
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">96</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A$ and $B$ be $(3 \times 3)$ matrices with $\operatorname{det} A=4$ and det $B=3$.<br/>What is det ( $2 \mathrm{AB}$ ) equal to ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">96</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">72</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">48</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">36</span> </li> </ul>

answer:
<div class="solution">A and B are $(3 \times 3)$ matrices det $\mathrm{A}=4$<br/>$\operatorname{det} B=3$<br/>$\therefore \operatorname{det}(2 \mathrm{AB})=(2)^{3}|\mathrm{~A}||\mathrm{B}|$<br/>$=2^{3}(4)(3)=8(4)(3)=96$</div>


title: let-a-and-b-be-3-3-matrices-with-det-a-4-and-det-b-3-what-is-det-3-ab-1-equal-to
slug: let-a-and-b-be-3-3-matrices-with-det-a-4-and-det-b-3-what-is-det-3-ab-1-equal-to-87181
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">36</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A$ and $B$ be $(3 \times 3)$ matrices with $\operatorname{det} A=4$ and det $B=3$.<br/>What is det $\left(3 \mathrm{AB}^{-1}\right)$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">18</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">36</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">48</span> </li> </ul>

answer:
<div class="solution">$\operatorname{det}\left(3 \mathrm{AB}^{-1}\right)=(3)^{3}|\mathrm{~A}|\left|\mathrm{B}^{-1}\right|$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/NrMU3GlF-Hzf2fkkM4fG3ZbuAQnf34UcmZajkgHDc_0.original.fullsize.png"/></div>


title: let-a-and-b-be-matrices-of-order-3-3-if-a-b-0-then-which-of-the-following-can-be-concluded
slug: let-a-and-b-be-matrices-of-order-3-3-if-a-b-0-then-which-of-the-following-can-be-concluded-70520
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A$ and $B$ be matrices of order $3 \times 3$. If $A B=0$, then which of the following can be concluded?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A=0$ and $B=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|A|=0$ and $|B|=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Either $|A|=0$ or $|B|=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Either $A=0$ or $B=0$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Either $A=0$ or $B=0$</span> </div>

answer:
<div class="solution">If AB = 0 , then it may be concluded that either $\mathrm{A}=0$ or $\mathrm{B}=0$<br/>But, it should be noticed that it is not necessary that either $\mathrm{A}=0$ or $\mathrm{B}=0$.</div>


title: let-a-and-b-be-obtuse-angles-such-that-sin-a-5-4-and-cos-b-13-12-what-is-the-value-of-sin-a-b
slug: let-a-and-b-be-obtuse-angles-such-that-sin-a-5-4-and-cos-b-13-12-what-is-the-value-of-sin-a-b-21883
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}$ and $\mathrm{B}$ be obtuse angles such that $\sin \mathrm{A}=\frac{4}{5}$ and<br/>$\cos B=-\frac{12}{13}$. What is the value of $\sin (A+B) ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{63}{65}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$-\frac{33}{65}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{33}{65}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{63}{65}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{63}{65}$</span> </div>

answer:
<div class="solution">$\sin \mathrm{A}=\frac{4}{5}$ and $\cos \mathrm{B}=-\frac{12}{13}$<br/>It is given that $\mathrm{A}$ and $\mathrm{B}$ are obtuse angle<br/>$\Rightarrow \cos \mathrm{A}-\pm \sqrt{1-\sin ^{2} \mathrm{~A}}-\pm \sqrt{1-\frac{16}{25}}--\frac{3}{5}$<br/>Negative sign is taken for cos A since A being obtuse lies in second quadrant. $\sin B=\pm \sqrt{1-\cos ^{2} B}=\pm \sqrt{1-\left(\frac{-12}{13}\right)^{2}}$<br/>$=\sqrt{\frac{169-144}{169}}=\frac{5}{13}$<br/>Positive sign is taken since, $\sin B$ is positive in second quadrant.<br/>$\Rightarrow \cos \mathrm{A}=\frac{-3}{5}$ and $\sin \mathrm{B}=\frac{3}{13}$<br/>$\therefore \sin (\mathrm{A}+\mathrm{B})=\sin \mathrm{A} \cos \mathrm{B}+\cos \mathrm{A} \sin \mathrm{B}$<br/>$=\frac{4}{5} \times\left(\frac{-12}{13}\right)+\frac{-3}{5} \times\left(\frac{5}{13}\right)=-\frac{48}{65}-\frac{15}{65}$<br/>$=\frac{-48-15}{65}=\frac{-63}{65}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-subsets-of-x-and-c-a-b-a-b-where-a-and-b-are-complements-of-a-and-b-respectively-in-x-what-is-c-equal-to

slug:
let-a-and-b-be-subsets-of-x-and-c-a-b-a-b-where-a-and-b-are-complements-of-a-and-b-respectively-in-x-what-is-c-equal-to-94367

question:
<div class="question">Let A and B be subsets of $\mathrm{X}$ and $\mathrm{C}=\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right) \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)$,<br/>where $\mathrm{A}^{\prime}$ and $\mathrm{B}^{\prime}$ are complements of A and $\mathrm{B}$ respectively in<br/>X. What is $C$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\mathrm{A} \cup \mathrm{B}^{\prime}\right)-\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\mathrm{A}^{\prime} \cup \mathrm{B}\right)-\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(\mathrm{A} \cup \mathrm{B})-(\mathrm{A} \cap \mathrm{B})$</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">D</span> <span class="option-data">$\left(\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}\right)-\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)$</span> </li> </ul>

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

answer:
<div class="solution">$C=\left(A \cap B^{\prime}\right) \cup\left(A^{\prime} \cap B\right)$<br/>Let usdraw venn diagram and compare it with options.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/qR0LDDuorNbVDdRa27Nw16QSwIXjKAz8slktGy7f1_g.original.fullsize.png"/><br/><br/>This also represents $(\mathrm{A} \cup \mathrm{B})-(\mathrm{A} \cap \mathrm{B})$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-the-position-vectors-of-a-and-b-respectively-if-c-is-the-point-3-a-2-b-then-which-one-of-the-following-is-correct

slug:
let-a-and-b-be-the-position-vectors-of-a-and-b-respectively-if-c-is-the-point-3-a-2-b-then-which-one-of-the-following-is-correct-19615

question:
<div class="question">Let $\vec{a}$ and $\vec{b}$ be the position vectors of $A$ and $B$ respectively.<br/>If $\mathrm{C}$ is the point $3 \overrightarrow{\mathrm{a}}-2 \overrightarrow{\mathrm{b}}$, 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">$\mathrm{C}$ is in between $\mathrm{A}$ and $\mathrm{B}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">A is in between $\mathrm{C}$ and $\mathrm{B}$</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">$\mathrm{B}$ is in between $\mathrm{A}$ and $\mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A, B, C are not collinear</span> </li> </ul>

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

answer:
<div class="solution">As given : $\vec{a}$ and $\vec{b}$ are position vectors of $A$ and $B$<br/>respectively and position vector of $\mathrm{C}$ is $3 \overrightarrow{\mathrm{a}}-2 \overrightarrow{\mathrm{b}}$<br/>$\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}}$, where $\mathrm{O}$ is the origin and<br/>$\overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{a}}-2 \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>$\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{3 \mathrm{a}}-2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}=2 \overrightarrow{\mathrm{a}}-2 \overrightarrow{\mathrm{b}}$<br/>$\Rightarrow \overrightarrow{\mathrm{AC}}=2(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})=-2(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}})=-2 \overrightarrow{\mathrm{AB}}$<br/>So, $\overrightarrow{\mathrm{AC}}$ is opposite to $\overrightarrow{\mathrm{AB}}$ so<br/>A is between $\mathrm{C}$ and $\mathrm{B}$ and position vector of $\mathrm{C}$ shows an external division by $\mathrm{C}$.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-two-distinct-roots-of-a-polynomial-equation-f-x-0-then-there-exists-at-least-one-root-lying-between-a-and-b-of-the-polynomial-equation

slug:
let-a-and-b-be-two-distinct-roots-of-a-polynomial-equation-f-x-0-then-there-exists-at-least-one-root-lying-between-a-and-b-of-the-polynomial-equation-57696

question:
<div class="question">Let $a$ and $b$ be two distinct roots of a polynomial equation $f(x)=0$. Then there exists at least one root lying between $a$ and $b$ of the polynomial equation.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f^{\prime}(x)=0$</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><li class=""> <span class="option-label">C</span> <span class="option-data">$f^{\prime \prime}(x)=0$</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">$f^{\prime}(x)=0$</span> </div>

answer:
<div class="solution">Rolle's theorem says between any two roots of a polynomial $f(x)$, there is always a root of its derivative $f^{\prime}(x)$<br/>Therefore between $a$ and $b$. There exist at least one root of the polynomial equation $f^{\prime}(x)=0$</div>


title: let-a-and-b-be-two-events-with-p-a-3-1-p-b-6-1-and-p-a-b-12-1-what-is-p-b-a-equal-to
slug: let-a-and-b-be-two-events-with-p-a-3-1-p-b-6-1-and-p-a-b-12-1-what-is-p-b-a-equal-to-78238
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}$ and $\mathrm{B}$ be two events with $\mathrm{P}(\mathrm{A})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{6}$ and<br/>$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{12} .$ What is $\mathrm{P}(\mathrm{B} \mid \overline{\mathrm{A}})$ equal to?</div>

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

answer:
<div class="solution">$\mathrm{P}(\mathrm{A})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{6}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{12}$<br/>$\mathrm{P}(\mathrm{B} \mid \overline{\mathrm{A}})=\frac{\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{A}})}{\mathrm{P}(\overline{\mathrm{A}})}$<br/>Now, $\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{A}})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{6}-\frac{1}{12}=\frac{1}{12}$<br/>$\mathrm{P}(\overline{\mathrm{A}})=1-\frac{1}{3}=\frac{2}{3}$<br/>$\therefore \mathrm{P}(\mathrm{B} \mid \overline{\mathrm{A}})=\frac{\frac{1}{12}}{\frac{2}{3}}=\frac{1}{12} \times \frac{3}{2}=\frac{1}{8}$</div>


title: let-a-and-b-be-two-invertible-square-matrices-each-of-order-n-what-is-adj-ab-equal-to
slug: let-a-and-b-be-two-invertible-square-matrices-each-of-order-n-what-is-adj-ab-equal-to-21773
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let A and B be two invertible square matrices each of order $\mathrm{n}$. What is adj $(\mathrm{AB})$ equal to?</div>

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

answer:
<div class="solution">Since, $\mathrm{A}$ and $\mathrm{B}$ be two invertible square matrices each of order $\mathrm{n}$, then $(\mathrm{AB})^{-1}=\left(\mathrm{B}^{-1}\right)\left(\mathrm{A}^{-1}\right)$<br/>$\Rightarrow \frac{\operatorname{adj}(\mathrm{AB})}{|\mathrm{AB}|}=\frac{\operatorname{adj} \mathrm{B}}{|\mathrm{B}|} \cdot \frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}$<br/>Since $|\mathrm{AB}|=|\mathrm{B}||\mathrm{A}|$<br/>$\operatorname{adj}(\mathrm{AB})=(\operatorname{adj} \mathrm{B})(\operatorname{adj} \mathrm{A})$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-two-matrices-such-that-a-b-a-and-b-a-b-which-of-the-following-statements-are-correct-2014ii-1-a-2-a-2-b-2-b-3-a-b-2-a-b-select-the-corr

slug:
let-a-and-b-be-two-matrices-such-that-a-b-a-and-b-a-b-which-of-the-following-statements-are-correct-2014ii-1-a-2-a-2-b-2-b-3-a-b-2-a-b-select-the-corr-80587

question:
<div class="question">Let $A$ and $B$ be two matrices such that $A B=A$ and $B A=B$. Which of the following statements are correct? [2014-II]<br/>$1.$ $A^{2}=A$<br/>$2.$ $\quad B^{2}=B$<br/>$3.$ $\quad(A B)^{2}=A B$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">We have, $\mathrm{AB}=\mathrm{A}$<br/>$\therefore \quad \mathrm{A}^{2}=(\mathrm{AB}) \cdot(\mathrm{AB})=\mathrm{A} .(\mathrm{BA}) \mathrm{B}$<br/>=ABB $\quad(\because \mathrm{BA}=\mathrm{B})$<br/>$=\mathrm{AB}=\mathrm{A}$ $\quad(\because \mathrm{AB}=\mathrm{A})$<br/>Also, $B^{2}=(B A) \cdot(B A)=B(A B) \cdot A$<br/>$=$ B.A.A $\quad(\because \mathrm{AB}=\mathrm{A})$<br/>$=$ B.A $=B$ $\quad(\because \mathrm{BA}=\mathrm{B})$<br/>Again, $(\mathrm{AB})^{2}=(\mathrm{AB}) \cdot(\mathrm{AB})=\mathrm{A}$. (BA) B<br/>$=$ A.B.B $\quad(\because \mathrm{BA}=\mathrm{B})$<br/>$=$ A. $\mathrm{B}=\mathrm{A}$ $\quad(\because \mathrm{AB}=\mathrm{A})$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-two-matrices-such-that-ab-is-defined-if-ab-0-then-which-one-of-the-following-can-be-definitely-concluded

slug:
let-a-and-b-be-two-matrices-such-that-ab-is-defined-if-ab-0-then-which-one-of-the-following-can-be-definitely-concluded-12753

question:
<div class="question">Let A and $\mathrm{B}$ be two matrices such that $\mathrm{AB}$ is defined. If $\mathrm{AB}=0$, then which one of the following can be definitely concluded?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A and B are non-zero square matrices</span> </div>

answer:
<div class="solution">Since, $\mathrm{AB}$ is defined, neither A nor $\mathrm{B}$ is singular i.e., they are non-zero matrix and if $\mathrm{AB}=0$ both $\mathrm{A}$ and $\mathrm{B}$ are square matrix. So, Aand B are non-zerosquare matrices.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-two-nonempty-subsets-of-a-set-x-if-a-b-b-a-a-b-then-which-one-of-the-following-is-correct

slug:
let-a-and-b-be-two-nonempty-subsets-of-a-set-x-if-a-b-b-a-a-b-then-which-one-of-the-following-is-correct-71837

question:
<div class="question">Let $A$ and $B$ be two non-empty subsets of a set $X$. If $(A-B) \cup(B-A)=A \cup B$, 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">$\mathrm{A} \subset \mathrm{B}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{A} \subset(\mathrm{X}-\mathrm{B})$</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">$\mathrm{A}=\mathrm{B}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{B} \subset \mathrm{A}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{A}$ and $\mathrm{B}$ are subsets of $\mathrm{X}$ and its, Venn diagram is shown.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/DDRkjqwFEaaitRcMMGONF32y1RN1EsZGB06ASCgZkcE.original.fullsize.png"/><br/><br/>$\mathrm{A}=\mathrm{B}$ indicates region a, $\mathrm{B}-\mathrm{A}$ indicates $\mathrm{c}, \mathrm{A} \cup \mathrm{B}$<br/>indicates $\mathrm{a}, \mathrm{b}, \mathrm{c}$. If $a, c=a, b, c$ indicates, the $b$ is zero. $A$ and $B$ are mutually exclusive. So, when $(\mathrm{A}-\mathrm{B}) \cup(\mathrm{B}-\mathrm{A})=\mathrm{A} \cup \mathrm{B}$<br/>$\mathrm{A} \subset(\mathrm{X}-\mathrm{B})$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-be-two-unit-vectors-and-be-the-angle-between-them-if-a-b-is-also-the-unit-vectors-then-what-is-the-value-of

slug:
let-a-and-b-be-two-unit-vectors-and-be-the-angle-between-them-if-a-b-is-also-the-unit-vectors-then-what-is-the-value-of-37576

question:
<div class="question">Let $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ be two unit vectors and $\alpha$ be the angle between<br/>them. If $(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})$ is also the unit vectors, then what is the<br/>value of $\alpha ?$</div>

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

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

answer:
<div class="solution">Let $\vec{a}$ and $\vec{b}$ be two unit vectors.<br/>$\therefore \quad|a|=1$ and $|b|=1$<br/>Since, $\alpha$ is the angle between $\vec{a}$ and $\vec{b}$<br/>$\therefore \quad \cos \alpha=\frac{\vec{a} \cdot \vec{b}}{|a \| b|}$<br/>$\cos \alpha=\frac{\vec{a} \cdot \vec{b}}{1}$<br/>$\cos \alpha=\vec{a} \cdot \vec{b}$<br/>Now, $|\vec{a}+\vec{b}|=1 \quad(\because \vec{a}+\vec{b}$ is unit vector $)$<br/>Squaring both sides<br/>$\Rightarrow|\vec{a}|^{2}+|\vec{b}|^{2}+2 \vec{a} \cdot \vec{b}=1$<br/>$\Rightarrow 1+1+2 \cos \alpha=1$<br/>$\Rightarrow 2 \cos \alpha=-1$<br/>$\Rightarrow \quad \cos \alpha=-\frac{1}{2}=\cos \frac{2 \pi}{3}$<br/>$\Rightarrow \alpha=\frac{2 \pi}{3}$</div>


tags: ['Mathematics', 'Hyperbola', 'JEE Main', 'JEE Main 2016 (09 Apr Online)']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-respectively-be-the-semitransverse-and-semiconjugate-axes-of-a-standard-hyperbola-whose-eccentricity-satisfies-the-equation-9-e-2-18-e-5-0

slug:
let-a-and-b-respectively-be-the-semitransverse-and-semiconjugate-axes-of-a-standard-hyperbola-whose-eccentricity-satisfies-the-equation-9-e-2-18-e-5-0-67584

question:
<div class="question">Let <math><mi>a</mi></math> and <math><mi>b</mi></math> respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation <math><mn>9</mn><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>18</mn><mi>e</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math>. If <math><mi>S</mi><mfenced separators="|"><mrow><mn>5</mn><mo>,</mo><mi> </mi><mn>0</mn></mrow></mfenced></math> is a focus and <math><mn>5</mn><mi>x</mi><mo>=</mo><mn>9</mn></math> is the corresponding directrix of this hyperbola, then <math><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></math> is equal to</div>

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

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

answer:
<div class="solution"><p>Given that focus of hyperbola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mfenced separators="|"><mrow><mn>5</mn><mn>, 0</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mi>e</mi><mo>=</mo><mn>5</mn></math>......<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced></math><br/><br/>Therefore, Directrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi>a</mi><mn>e</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi> </mi><mfrac><mi>a</mi><mi>e</mi></mfrac><mo>=</mo><mfrac><mn>9</mn><mn>5</mn></mfrac></math>  ......<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced></math><br/><br/>From <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced></math>, we get <math><mo>⇒</mo><mi> </mi><mi> </mi><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math>    </p><p>(It satisfies <math><mn>9</mn><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>18</mn><mi>e</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math> )</p><p>We also know that for an ellipse,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced separators="|"><mrow><msup><mi>e</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>16</mn></math><br/><br/><math><mo>⇒</mo></math> <math><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>9</mn><mo>-</mo><mn>16</mn><mo>=</mo><mi> </mi><mo>-</mo><mn>7</mn></math></p></div>


title: let-a-b-1-2-3-what-is-the-number-of-equations
slug: let-a-b-1-2-3-what-is-the-number-of-equations-24496
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{a}, \mathrm{b} \in\{1,2,3\}$. What is the number of equations</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=""> <span class="option-label">C</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">The given equation is $a x^{2}+b x+1=0$ This equation has real roots When discriminant $\geq 0$ $b^{2}-4 a \geq 0$<br/>$\Rightarrow b^{2} \geq 4 a$<br/>$\mathrm{a}, \mathrm{b}$ has to be selected from, three numbers, so total selections are possible when $(\mathrm{a}, \mathrm{b})$ are $(1,2),(1,3)$ and<br/>$(2,3) .$<br/>Thus, the number of equations of the form $a x^{2}$ $\mathrm{bx}+1=0$ having real root is 3</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-b-and-c-be-the-distinct-nonnegative-numbers-if-the-vectors-a-i-a-j-c-k-i-k-c-i-c-j-b-k-lie-on-a-plane-then-which-one-of-the-following-is-correct

slug:
let-a-b-and-c-be-the-distinct-nonnegative-numbers-if-the-vectors-a-i-a-j-c-k-i-k-c-i-c-j-b-k-lie-on-a-plane-then-which-one-of-the-following-is-correct-32943

question:
<div class="question">Let $a, b$ and $c$ be the distinct non-negative numbers. If the vectors $a \hat{\mathbf{i}}+a \hat{\mathbf{j}}+c \hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}, c \hat{\mathbf{i}}+c \hat{\mathbf{j}}+b \hat{\mathbf{k}}$ lie on a plane, then<br/>which one of the following is correct ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$c$ is the arithmetic mean of a and $b$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$c$ is the geometric mean of a and $b$</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">$c$ is the harmonic mean of a and $b$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$c$ is equal to zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$c$ is the geometric mean of a and $b$</span> </div>

answer:
<div class="solution">Given $a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$. and $c \hat{i}+c \hat{j}+b \hat{k}$ lie on a<br/>plane.<br/>$\Rightarrow$ vectors $a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $c \hat{i}+c \hat{j}+b \hat{k}$ are<br/>coplanar.<br/>$\therefore \quad\left|\begin{array}{lll}a &amp; a &amp; c \\ 1 &amp; 0 &amp; 1 \\ c &amp; c &amp; b\end{array}\right|=0$<br/>$\Rightarrow a(-c)-a(b-c)+c(c)=0$<br/>$\Rightarrow \quad-a c-a b+a c+c^{2}=0$<br/>$\Rightarrow \quad c^{2}=a b$<br/>$\Rightarrow c$ is the geometric mean of $a$ and $b$.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-b-and-c-be-three-mutually-exclusive-and-exhaustive-events-associated-with-a-random-experiment-if-p-b-15-p-a-and-p-c-05-p-b-then-p-a-0-is-equal-t

slug:
let-a-b-and-c-be-three-mutually-exclusive-and-exhaustive-events-associated-with-a-random-experiment-if-p-b-15-p-a-and-p-c-05-p-b-then-p-a-0-is-equal-t-90119

question:
<div class="question">Let $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ be three mutually exclusive and exhaustive events associated with a random experiment. If $\mathrm{P}(\mathrm{B})=1.5$ $\mathrm{P}(\mathrm{A})$ and $\mathrm{P}(\mathrm{C})=0.5 \mathrm{P}(\mathrm{B})$, then $\mathrm{P}(\mathrm{A} 0$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{4}{13}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{13}$</span> </div>

answer:
<div class="solution">As A, B and C are mutually exclusive and exhaustive event. $\therefore \quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{C})=1$<br/>$\Rightarrow \mathrm{P}(\mathrm{A})+\frac{3}{2} \mathrm{P}(\mathrm{A})+\frac{1}{2} \times \frac{3}{2} \mathrm{P}(\mathrm{A})=1$<br/>$\Rightarrow \quad \frac{13}{4} \mathrm{P}(\mathrm{A})=1$<br/>$\Rightarrow \quad \mathrm{P}(\mathrm{A})=\frac{4}{13}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-b-and-c-be-three-mutually-perpendicular-vectors-each-of-unit-magnitud-if-a-a-b-c-b-a-b-c-and-c-a-b-c-then-which-one-of-the-following-is-correct

slug:
let-a-b-and-c-be-three-mutually-perpendicular-vectors-each-of-unit-magnitud-if-a-a-b-c-b-a-b-c-and-c-a-b-c-then-which-one-of-the-following-is-correct-72771

question:
<div class="question">Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three mutually perpendicular vectors each<br/>of unit magnitud. If $\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}, \quad \overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$ and<br/>$\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}$, 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">$|\overrightarrow{\mathrm{A}}|&gt;|\overrightarrow{\mathrm{B}}|&gt;|\overrightarrow{\mathrm{C}}|$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|\overrightarrow{\mathrm{A}}|=|\overrightarrow{\mathrm{B}}| \neq|\overrightarrow{\mathrm{C}}|$</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">$|\overrightarrow{\mathrm{A}}| \neq|\overrightarrow{\mathrm{B}}| \neq|\overrightarrow{\mathrm{C}}|$</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">For simplicity let us take $\vec{a}, \vec{b}, \vec{c}$ as $\hat{i}, \hat{j}, \hat{k}$<br/>Now magnitude of $\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{C}}$ will be $\sqrt{3}$.</div>


title: let-a-b-and-c-be-three-vectors-such-that-a-b-c-0-and-a-10-b-6-and-c-14-what-is-a-b-b-c-c-a-equal-to
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-166$</span> </div>
source_db: MarksBatch2_P1.db
slug:
let-a-b-and-c-be-three-vectors-such-that-a-b-c-0-and-a-10-b-6-and-c-14-what-is-a-b-b-c-c-a-equal-to-86929

question:
<div class="question">Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}+\vec{b}+\vec{c}=0$ and $|\overrightarrow{\mathrm{a}}|=10,|\overrightarrow{\mathrm{b}}|=6$ and $|\overrightarrow{\mathrm{c}}|=14$<br/>What is a. $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}$. $\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}$. a. equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-332$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-166$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">166</span> </li> </ul>

answer:
<div class="solution">We have $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$<br/>So $|\vec{a}+\vec{b}+\vec{c}|=0$<br/>$\Rightarrow|\vec{a}+\vec{b}+\vec{c}|^{2}=|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})$<br/>$\Rightarrow 0=(10)^{2}+(6)^{2}+(14)^{2}+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})$<br/>$\Rightarrow 0=100+36+196+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})$<br/>$\Rightarrow-\frac{332}{2}=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$<br/>$\Rightarrow-166=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-b-and-c-be-three-vectors-such-that-a-b-c-0-and-a-10-b-6-and-c-14-what-is-the-angle-between-a-and-b

slug:
let-a-b-and-c-be-three-vectors-such-that-a-b-c-0-and-a-10-b-6-and-c-14-what-is-the-angle-between-a-and-b-96490

question:
<div class="question">Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}+\vec{b}+\vec{c}=0$ and $|\overrightarrow{\mathrm{a}}|=10,|\overrightarrow{\mathrm{b}}|=6$ and $|\overrightarrow{\mathrm{c}}|=14$<br/>What is the angle between $\vec{a}$ and $\vec{b}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">$75^{\circ}$</span> </li> </ul>

answer:
<div class="solution">Since $\vec{a}+\vec{b}+\vec{c}=0$<br/>$\Rightarrow \vec{a}+\vec{b}=-\vec{c}$<br/>$\Rightarrow|\vec{a}+\vec{b}|=|-\vec{c}|=|\vec{c}|$<br/>$\Rightarrow|\vec{a}+\vec{b}|^{2}=|\vec{c}|^{2}$<br/>$\Rightarrow|\vec{a}|^{2}+|\vec{b}|^{2}+2(\vec{a} \cdot \vec{b})=|\vec{c}|^{2}$<br/>$\Rightarrow(10)^{2}+(6)^{2}+2(\vec{a} \cdot \vec{b})=(14)^{2}$<br/>$\Rightarrow 2(\vec{a} \cdot \vec{b})=60$<br/>$\Rightarrow \vec{a} \cdot \vec{b}=30$<br/>$\Rightarrow \cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}=\frac{30}{10 \times 6}=\frac{1}{2}$<br/>$\Rightarrow \theta=\cos ^{-1}\left(\frac{1}{2}\right) \Rightarrow \theta=60^{\circ}$</div>


title: let-a-b-be-two-unit-vectors-and-be-the-angle-between-them-what-is-cos-2-equal-to
slug: let-a-b-be-two-unit-vectors-and-be-the-angle-between-them-what-is-cos-2-equal-to-54394
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\hat{a}, \hat{\mathrm{b}}$ be two unit vectors and $\theta$ be the angle between them.<br/>What is $\cos \left(\frac{\theta}{2}\right)$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{|\hat{a}+\hat{b}|}{2}$</span> </div>

answer:
<div class="solution">$\vec{a}$ and $\vec{b}$ are two unit vectors. . Hence.<br/>$\begin{aligned}|\vec{a}+\vec{b}|^{2} &amp;=(\hat{a}+\hat{b}) \cdot(\hat{a}+\hat{b}) \\ &amp;=\hat{a} \cdot \hat{a}+\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{a}+\hat{b} \cdot \hat{b} \\ &amp;=\hat{a} \cdot \hat{a}+\hat{b} \cdot \hat{b}+2 \hat{a} \cdot \hat{b} \\ &amp;=|\hat{a}|^{2}+2|\hat{a}||\hat{b}| \cos \theta+|\hat{b}||\hat{b}| \end{aligned}$<br/>$=1+2 \cos \theta+1 \quad[\because \hat{b}, \hat{a}$ are unit vector $]$<br/>$=2(1+\cos \theta)$<br/>$|\hat{a}+\hat{b}|^{2}=2 \cdot 2 \cos ^{2} \frac{\theta}{2}$<br/>$\cos \frac{\theta}{2}=\frac{|\hat{a}+\hat{b}|}{2}$</div>


title: let-a-b-be-two-unit-vectors-and-be-the-angle-between-them-what-is-sin-2-equal-to
slug: let-a-b-be-two-unit-vectors-and-be-the-angle-between-them-what-is-sin-2-equal-to-45380
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\hat{a}, \hat{\mathrm{b}}$ be two unit vectors and $\theta$ be the angle between them.<br/>What is $\sin \left(\frac{\theta}{2}\right)$ equal to?</div>

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

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

answer:
<div class="solution">$\begin{aligned}|\hat{a}-\hat{b}|^{2} &amp;=(\hat{a}-\hat{b}) \cdot(\hat{a}-\hat{b}) \\ &amp;=\hat{a} \cdot \hat{a}-\hat{b} \cdot \hat{a}-\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{b} \\ &amp;=|\hat{a}|^{2}-2|\hat{a}||\hat{b}| \cos \theta+|\hat{b}|^{2} \\ &amp;=2-2 \cos \theta \\ &amp;=2(1-\cos \theta) \\|\hat{a}-\hat{b}|^{2} &amp;=2 \cdot 2 \sin ^{2} \frac{\theta}{2} \\ \sin \frac{\theta}{2}=\frac{|\hat{a}-\hat{b}|}{2} \end{aligned}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-b-c-be-in-ap-consider-the-following-statements-1-ab-1-c-a-1-and-b-c-1-are-in-ap-2-b-c-1-c-a-1-and-a-b-1-are-in-ap-which-of-the-statements-given-

slug:
let-a-b-c-be-in-ap-consider-the-following-statements-1-ab-1-c-a-1-and-b-c-1-are-in-ap-2-b-c-1-c-a-1-and-a-b-1-are-in-ap-which-of-the-statements-given-34264

question:
<div class="question">Let $a, b, c$ be in AP. <br/>Consider the following statements:<br/>1) $\frac{1}{a b}, \frac{1}{c a}$ and $\frac{1}{b c}$ are in AP.<br/>2) $\frac{1}{\sqrt{b}+\sqrt{c}}, \frac{1}{\sqrt{c}+\sqrt{a}}$ and $\frac{1}{\sqrt{a}+\sqrt{b}}$ are in AP.<br/>Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Let $\frac{1}{a b}, \frac{1}{c a}, \frac{1}{b c}$ are in AP.<br/>$\Rightarrow \quad \frac{1}{c a}-\frac{1}{a b}=\frac{1}{b c}-\frac{1}{c a}$<br/>$\Rightarrow \frac{1}{a}\left(\frac{1}{c}-\frac{1}{b}\right)=\frac{1}{c}\left(\frac{1}{b}-\frac{1}{a}\right)$<br/>$\Rightarrow \frac{b-c}{a b c}=\frac{a-b}{a b c}$<br/>$\Rightarrow b-c=a-b \Rightarrow 2 b=a+c$<br/>$\Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$. Which is true<br/>Now, $\frac{1}{\sqrt{b}+\sqrt{c}}, \frac{1}{\sqrt{c}+\sqrt{a}}, \frac{1}{\sqrt{a}+\sqrt{b}}$ are in A.P.<br/>$\therefore \frac{2}{\sqrt{c}+\sqrt{a}}=\frac{1}{\sqrt{b}+\sqrt{c}}+\frac{1}{\sqrt{a}+\sqrt{b}}$<br/>$\Rightarrow 2(\sqrt{b}+\sqrt{c})(\sqrt{a}+\sqrt{b})$<br/>$=(\sqrt{c}+\sqrt{a})(\sqrt{a}+2 \sqrt{b}+\sqrt{c})$<br/>$\Rightarrow \quad 2(\sqrt{a b}+b+\sqrt{a c}+\sqrt{b c})=\sqrt{a c}+2 \sqrt{b c}+c+a$<br/>$+2 \sqrt{a b}+\sqrt{a c}$<br/>$\Rightarrow \quad 2 \sqrt{a b}+2 b+2 \sqrt{a c}+2 \sqrt{b c}$<br/>$=2 \sqrt{a c}+2 \sqrt{b c}+2 \sqrt{a b}+c+a$<br/>$\Rightarrow \quad 2 b=a+c$<br/>$\Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{A} . \mathrm{P}$. Which is true.<br/>Hence. both the statements are correct.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (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:
let-a-b-c-be-noncoplanar-vectors-and-p-abc-b-c-q-abc-c-a-r-abc-a-b-what-is-the-value-of-a-b-c-p-b-c-a-q-c-a-b-r

slug:
let-a-b-c-be-noncoplanar-vectors-and-p-abc-b-c-q-abc-c-a-r-abc-a-b-what-is-the-value-of-a-b-c-p-b-c-a-q-c-a-b-r-38615

question:
<div class="question">Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be non-coplanar vectors and $\overrightarrow{\mathrm{p}}=\frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}}{[\overrightarrow{\mathrm{abc}}]}$,<br/>$\overrightarrow{\mathrm{q}}=\frac{\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}}{[\overrightarrow{\mathrm{abc}}]}, \overrightarrow{\mathrm{r}}=\frac{\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}}{[\overrightarrow{\mathrm{abc}}]}$<br/>What is the value of<br/>$(\vec{a}-\vec{b}-\vec{c}) \cdot \vec{p}+(\vec{b}-\vec{c}-\vec{a}) \cdot \vec{q}+(\vec{c}-\vec{a}-\vec{b}) \cdot \vec{r} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">$-9$</span> </li> </ul>

answer:
<div class="solution">As given $\overrightarrow{\mathrm{p}}=\frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}}{[\overrightarrow{\mathrm{abc}}]}, \overrightarrow{\mathrm{q}}=\frac{\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}}{[\overrightarrow{\mathrm{abc}}]}$, and $\overrightarrow{\mathrm{r}}=\frac{\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}}{[\overrightarrow{\mathrm{abc}}]}$<br/>$\begin{aligned} \therefore &amp;(\vec{a}-\vec{b}-\vec{c}) \cdot \vec{p}+(\vec{b}-\vec{c}-\vec{a}) \cdot \vec{q}+(\vec{c}-\vec{a}-\vec{b}) \cdot \vec{r} \\ &amp;=\frac{a \cdot(\vec{b} \times \vec{c})}{[\overrightarrow{a b c}]}+\frac{\vec{b} \cdot(\vec{c} \times \vec{a})}{[a \vec{b} c]}+\frac{\vec{c} \cdot(\vec{a} \times \vec{b})}{[\overrightarrow{a b c}]} \end{aligned}$<br/>$[$ Since $\overrightarrow{\mathrm{b}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=0, \overrightarrow{\mathrm{c}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=0, \overrightarrow{\mathrm{c}} \cdot(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}})=0$<br/>$\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}})=0, \overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=0$ and $\overrightarrow{\mathrm{b}} \cdot(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=0]$<br/>$=\frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} c]}+\frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} c]}+\frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} c]}=3$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-b-c-be-the-position-vectors-of-points-a-b-c-respectively-under-which-one-of-the-following-conditions-are-the-points-a-b-c-collinear

slug:
let-a-b-c-be-the-position-vectors-of-points-a-b-c-respectively-under-which-one-of-the-following-conditions-are-the-points-a-b-c-collinear-98426

question:
<div class="question">Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of points $A, B, C$ respectively. Under which one of the following conditions are the points $A, B, C$ collinear?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \vec{a} \times \vec{b}=\overrightarrow{0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\vec{b} \times \vec{c}$ is parallel to $\vec{a} \times \vec{b}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\vec{a} \times \vec{b}$ is perpendicular to $\vec{b} \times \vec{c}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(\vec{a} \times \vec{b})+(\vec{b} \times \vec{c})+(\vec{c} \times \vec{a})=\overrightarrow{0}$</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">$(\vec{a} \times \vec{b})+(\vec{b} \times \vec{c})+(\vec{c} \times \vec{a})=\overrightarrow{0}$</span> </div>

answer:
<div class="solution">Points A, B and $\mathrm{C}$ are collinear, if<br/>$(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})=\overrightarrow{0}$</div>


tags: ['Mathematics', 'Matrices', 'AP EAMCET', 'AP EAMCET 2020 (21 Sep Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-b-in-mathbfr0-and-i2-be-the-identity-matrix-of-order-2-then-the-rank-of-the-block-matrix-leftbeginarraylla-i2-b2-a-i2-b-i2endarrayright-is

slug:
let-a-b-in-mathbfr0-and-i2-be-the-identity-matrix-of-order-2-then-the-rank-of-the-block-matrix-leftbeginarraylla-i2-b2-a-i2-b-i2endarrayright-is-73592

question:
<div class="question">Let \(a, b \in \mathbf{R}-\{0\}\), and \(I_2\) be the identity matrix of order 2. Then the rank of the (block) matrix \(\left[\begin{array}{ll}a I_2 &amp; b_2 \\ a I_2 &amp; b I_2\end{array}\right]\) is</div>

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

answer:
<div class="solution">Since \(a, b \in \mathbf{R}-\{0\}\) and \(I_2\) is a identify matrix of order 2 .<br/>Then the block matrix \(\left[\begin{array}{ll}a I_2 &amp; b I_2 \\ a I_2 &amp; b I_2\end{array}\right]\) is a non-singular matrix, so the rank of the given block matrix is 2.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-a-be-a-square-matrix-of-order-n-n-where-n-2-let-b-be-a-matrix-obtained-from-a-with-first-and-second-rows-interchanged-then-which-one-of-the-follow

slug:
let-a-be-a-square-matrix-of-order-n-n-where-n-2-let-b-be-a-matrix-obtained-from-a-with-first-and-second-rows-interchanged-then-which-one-of-the-follow-52877

question:
<div class="question">Let A be a square matrix of order $\mathrm{n} \times \mathrm{n}$ where $\mathrm{n} \geq 2 .$ Let $\mathrm{B}$ be a matrix obtained from A with first and second rows interchanged. 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">$\operatorname{det} \mathrm{A}=\operatorname{det} \mathrm{B}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\operatorname{det} \mathrm{A}=-\operatorname{det} \mathrm{B}$</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">$\mathrm{A}=\mathrm{B}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$A=-B$</span> </li> </ul>

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

answer:
<div class="solution">A be a square matrix of order $\mathrm{n} \times \mathrm{n}$ where $\mathrm{n} \geq 2 . \mathrm{B}$ be a matrix obtained from A with first and second rows interchanged. Then, det $A=-$ det $B$. Since interchanging any two rows makes the sign change with same value.</div>


title: let-a-be-an-m-n-matrix-under-which-one-of-the-following-conditions-does-a-1-exist
slug: let-a-be-an-m-n-matrix-under-which-one-of-the-following-conditions-does-a-1-exist-32597
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}$ be an $\mathrm{m} \times \mathrm{n}$ matrix. Under which one of the following conditions does $\mathrm{A}^{-1}$ exist?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{m}=\mathrm{n}$ only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{m}=\mathrm{n}$ and $\operatorname{det} \mathrm{A} \neq 0$</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">$\mathrm{m}=\mathrm{n}$ and $\mathrm{det} \mathrm{A}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$m \neq n$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{m}=\mathrm{n}$ and $\operatorname{det} \mathrm{A} \neq 0$</span> </div>

answer:
<div class="solution">Let a be an $\mathrm{m} \times \mathrm{n}$ matrix, then $\mathrm{A}^{-1}$ will exist if $\mathrm{m}=\mathrm{n}$ since only square matrix has determinant and det A $\neq 0$<br/>$\left[\right.$ Since $\left.A^{-1}=\frac{\text { adjA }}{|A|}\right]$</div>


title: let-a-be-an-n-n-matrix-if-det-a-s-det-a-what-is-the-value-of-s
slug: let-a-be-an-n-n-matrix-if-det-a-s-det-a-what-is-the-value-of-s-70058
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$n$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A$ be an $n \times n$ matrix. If det $(\lambda A)=\lambda^{s}$ det $(A)$, what is the value of $s ?$</div>

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

answer:
<div class="solution">We know, if $A$ is an $n \times n$ matrix, then<br/>$\operatorname{det}(\lambda A)=\lambda^{s} \operatorname{det}(A)$<br/>But given det $(\lambda A)=\lambda^{s} \operatorname{det} A$<br/>$\Rightarrow s=n$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-die-be-loaded-in-such-a-way-that-even-faces-are-twice-likely-to-occur-as-the-odd-faces-what-is-the-probability-that-a-prime-number-will-show-up-

slug:
let-a-die-be-loaded-in-such-a-way-that-even-faces-are-twice-likely-to-occur-as-the-odd-faces-what-is-the-probability-that-a-prime-number-will-show-up-30086

question:
<div class="question">Let a die be loaded in such a way that even faces are twice likely to occur as the odd faces. What is the probability that a prime number will show up when the die is tossed?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{9}$</span> </div>

answer:
<div class="solution">Possible prime numbers on the dice are 2,3 and 5 .<br/>Probability of getting prime number $=\frac{2}{3} \times \frac{1}{3}+\frac{1}{3} \times \frac{2}{3}$.<br/>$=\frac{2}{9}+\frac{2}{9}$<br/>$=\frac{4}{9} .$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-i-j-b-3-i-4-k-and-b-c-d-where-c-is-parallel-to-a-and-d-is-perpendicular-to-a-if-d-x-i-y-j-z-k-then-which-of-the-following-equations-isare-correc

slug:
let-a-i-j-b-3-i-4-k-and-b-c-d-where-c-is-parallel-to-a-and-d-is-perpendicular-to-a-if-d-x-i-y-j-z-k-then-which-of-the-following-equations-isare-correc-17604

question:
<div class="question">Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}$, where $\vec{c}$ is parallel to<br/>$\vec{a}$ and $\vec{d}$ is perpendicular to $\vec{a}$<br/>If $\overrightarrow{\mathrm{d}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$, then which of the following equations<br/>is/are correct?<br/>$1.$ $\quad \mathrm{y}-\mathrm{x}=4$<br/>$2.$ $\quad 2 z-3=0$<br/>Select the correct answer using the code given below:</div>

answer:
<div class="solution">Since $z=4$ and $x=\frac{3}{2}, y=-\frac{3}{2}$.<br/>So, neither 1 nor 2 is correct.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-i-j-b-3-i-4-k-and-b-c-d-where-c-is-parallel-to-a-and-d-is-perpendicular-to-a-what-is-c-equal-to

slug:
let-a-i-j-b-3-i-4-k-and-b-c-d-where-c-is-parallel-to-a-and-d-is-perpendicular-to-a-what-is-c-equal-to-59712

question:
<div class="question">Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}$, where $\vec{c}$ is parallel to<br/>$\vec{a}$ and $\vec{d}$ is perpendicular to $\vec{a}$<br/>What is $\vec{c}$ equal to?</div>

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

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

answer:
<div class="solution">Since $\vec{c}$ is parallel to $\vec{a}$<br/>$\vec{c}=\lambda \vec{a}$<br/>Now $\vec{b}=\vec{c}+\vec{d}=\lambda \vec{a}+\vec{d}$<br/>$=\lambda(\hat{i}+\hat{j})+x \hat{i}+y \hat{j}+z \hat{k}$<br/>$3 \hat{i}+4 \hat{k}=(\lambda+x) \hat{i}+(\lambda+y) \hat{j}+z \hat{k}$<br/>Comparing we get $z=4, \lambda+y=0, \lambda+x=3 \Rightarrow-y+x=3 \quad($ From $(1))$<br/>$\Rightarrow \lambda=-y \quad \ldots(1) \Rightarrow x-y=3$<br/>Now $\vec{d}$ is $\perp \mathrm{r}$ to $\vec{a}$<br/>So, $\cos \theta=0$<br/>$\Rightarrow x+y=0$<br/>Solving $(2)$ and $(3)$ we get<br/>$2 x=3$<br/>$\Rightarrow x=\frac{3}{2}, y=-\frac{3}{2}$<br/>$\Rightarrow \vec{c}=\lambda(\vec{a})=\frac{3}{2}(\hat{i}+\hat{j})$</div>


title: let-a-n-2-n-n-n-and-b-2-n-3-n-n-n-what-is-a-b-equal-to
slug: let-a-n-2-n-n-n-and-b-2-n-3-n-n-n-what-is-a-b-equal-to-40311
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (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">Let $\mathrm{A}=\{(\mathrm{n}, 2 \mathrm{n}): \mathrm{n} \in \mathrm{N}\}$ and $\mathrm{B}=\{(2 \mathrm{n}, 3 \mathrm{n}): \mathrm{n} \in \mathrm{N}\} .$ What is A $\cap$ B equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{(\mathrm{n}, 6 \mathrm{n}): \mathrm{n} \in \mathrm{N}\}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{(2 n, 6 n): n \in N\}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\{(\mathrm{n}, 3 \mathrm{n}): \mathrm{n} \in \mathrm{N}\}$</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">$\mathrm{A}=\{(\mathrm{n}, 2 \mathrm{n}): \mathrm{n} \in \mathrm{N}\}$ and $\mathrm{B}=\{(2 \mathrm{n}, 3 \mathrm{n})\}: \mathrm{n} \in \mathrm{N}$<br/>Listing few members of each set $\mathrm{A}=\{(1,2),(2,4),(3,6), \ldots\}$<br/>$\mathrm{B}=\{(2,3),(4,6),(6,9) \ldots \ldots\}$<br/>There is no member common to both these sets, hence $A \cap B=\phi$</div>


title: let-a-vector-r-make-angle-6-0-3-0-with-x-and-yaxes-respectively-what-angle-does-r-make-with-z-axis
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
let-a-vector-r-make-angle-6-0-3-0-with-x-and-yaxes-respectively-what-angle-does-r-make-with-z-axis-39838

question:
<div class="question">Let a vector $\bar{r}$ make angle $60^{\circ}, 30^{\circ}$ with $\mathrm{x}$ and y-axes respectively.<br/>What angle does $\bar{r}$ make with $z$ -axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$60^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$90^{\circ}$</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">D</span> <span class="option-data">$120^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$90^{\circ}$</span> </div>

answer:
<div class="solution">$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$<br/>$\cos ^{2} 60^{\circ}+\cos ^{2} 30^{\circ}+\cos ^{2} \gamma=1$<br/>$\frac{1}{4}+\frac{3}{4}+\cos ^{2} \gamma=1$<br/>$\cos ^{2} \gamma=0 \Rightarrow \gamma=90^{\circ}$</div>


title: let-a-vector-r-make-angle-6-0-3-0-with-x-and-yaxes-respectively-what-are-the-direction-cosines-of-r
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
let-a-vector-r-make-angle-6-0-3-0-with-x-and-yaxes-respectively-what-are-the-direction-cosines-of-r-34002

question:
<div class="question">Let a vector $\bar{r}$ make angle $60^{\circ}, 30^{\circ}$ with $\mathrm{x}$ and y-axes respectively.<br/>What are the direction cosines of $\bar{r}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left\langle\frac{1}{2}, \frac{\sqrt{3}}{2}, 0\right\rangle$</span> </div>

answer:
<div class="solution">$\left.\mathrm{r}=   &lt; 1, \mathrm{~m}, \mathrm{n}&gt;; \mathrm{r}= &lt; \cos 60^{\circ}, \cos 30^{\circ}, \cos 90^{\circ}\right\rangle$<br/>Direction cosines of $\overrightarrow{\mathrm{r}}= &lt; \frac{1}{2}, \frac{\sqrt{3}}{2}, 0&gt;$</div>


title: let-a-x-3-b-x-2-c-x-d-x-1-2-x-3-2-x-2-x-x-1-3-x-4-3-x-x-5-x-1-then-what-is-the-value-of-a-b-c-d
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">63</span> </div>
answer: <div class="solution">$a+b+c+d=63$</div>
source_db: MarksBatch2_P1.db
slug:
let-a-x-3-b-x-2-c-x-d-x-1-2-x-3-2-x-2-x-x-1-3-x-4-3-x-x-5-x-1-then-what-is-the-value-of-a-b-c-d-77228

question:
<div class="question">Let $a x^{3}+b x^{2}+c x+d=\left|\begin{array}{lcc}x+1 &amp; 2 x &amp; 3 x \\ 2 x+3 &amp; x+1 &amp; x \\ 2-x &amp; 3 x+4 &amp; 5 x-1\end{array}\right|$ then<br/>What is the value of $a+b+c+d$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">62</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">63</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">65</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">68</span> </li> </ul>


title: let-a-x-3-b-x-2-c-x-d-x-1-2-x-3-2-x-2-x-x-1-3-x-4-3-x-x-5-x-1-then-what-is-the-value-of-c
slug: let-a-x-3-b-x-2-c-x-d-x-1-2-x-3-2-x-2-x-x-1-3-x-4-3-x-x-5-x-1-then-what-is-the-value-of-c-81262
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">35</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $a x^{3}+b x^{2}+c x+d=\left|\begin{array}{lcc}x+1 &amp; 2 x &amp; 3 x \\ 2 x+3 &amp; x+1 &amp; x \\ 2-x &amp; 3 x+4 &amp; 5 x-1\end{array}\right|$ then<br/>What is the value of $\mathrm{c}$ ?</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">34</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">35</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">50</span> </li> </ul>

answer:
<div class="solution">$a x^{3}+b x^{2}+c x+d$<br/>$=(x+1)[(x+1)(5 x-1)-x(3 x+4)]-2 x[(2 x+3)(5 x-1)$<br/>$-x(2-x)]+3 x[(2 x+3)(3 x+4)-(2-x)(x+1)]$<br/>$\Rightarrow a x^{3}+b x^{2}+c x+d=x^{3}+28 x^{2}+35 x-1$<br/>$\Rightarrow c=35$</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-x-r-1-x-1-and-s-be-the-subset-of-a-b-defined-by-s-x-y-a-b-x-2-y-2-1-which-one-of-the-following-is-correct

slug:
let-a-x-r-1-x-1-and-s-be-the-subset-of-a-b-defined-by-s-x-y-a-b-x-2-y-2-1-which-one-of-the-following-is-correct-91339

question:
<div class="question">Let $\mathrm{A}=(\mathrm{x} \in \mathrm{R}:-1 \leq \mathrm{x} \leq 1)$ and $\mathrm{S}$ be the subset of $\mathrm{A} \times \mathrm{B}$<br/>defined by $\mathrm{S}=\left[(\mathrm{x}, \mathrm{y}) \in \mathrm{A} \times \mathrm{B}: \mathrm{x}^{2}+\mathrm{y}^{2}=1\right]$<br/>Which one of the following is correct?</div>

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

answer:
<div class="solution">$\mathrm{S}$ is not a function (By vertical line test)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/QXx8XbSqDtTyWWW93OiZUJUBjVy6TXzWK1R7MZpwKVo.original.fullsize.png"/></div>


title: let-a-x-r-9-x-4-b-x-r-13-x-5-and-c-x-r-7-x-8-then-which-one-of-the-following-is-correct
slug: let-a-x-r-9-x-4-b-x-r-13-x-5-and-c-x-r-7-x-8-then-which-one-of-the-following-is-correct-65122
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}=\{\mathrm{x} \in \mathrm{R} \mid-9 \leq \mathrm{x}   &lt; 4\} ; \mathrm{B}=\{\mathrm{x} \in \mathrm{R} \mid-13 &lt; \mathrm{x} \leq 5\}$ and<br/>$\mathrm{C}=\{\mathrm{x} \in \mathrm{R} \mid-7 \leq \mathrm{x} \leq 8\}$<br/>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">$-9 \in(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-7 \in(A \cap B \cap C)$</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">C</span> <span class="option-data">$4 \in(A \cap B \cap C)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 \in(A \cap B \cap C)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-7 \in(A \cap B \cap C)$</span> </div>

answer:
<div class="solution">Given sets in set builder form are:<br/>$A=\{x \in R:-9 \leq x   &lt; 4\}$<br/>$B=\{x \in R:-13 &lt; x 5\}$<br/>and $C=\{x \in R:-7 \leq x \leq 8\}$<br/>$\Rightarrow A \cap B \cap C=\{x \in R:-7 \leq x &lt; 4\}$<br/>Hence, $-7 \in(A \cap B \cap C)$</div>


title: let-a-x-r-x-0-a-function-f-a-a-is-defined-by-f-x-x-2-which-one-of-the-following-is-correct
slug: let-a-x-r-x-0-a-function-f-a-a-is-defined-by-f-x-x-2-which-one-of-the-following-is-correct-77685
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A=\{x \in R \mid x \geq 0\}$. <br/>A function $f: A \rightarrow A$ is defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$. <br/>Which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The function does not have inverse $\quad[2012-I I]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}$ is its own inverse</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The function has an inverse but $\mathrm{f}$ is not its own inverse</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">The function has an inverse but $\mathrm{f}$ is not its own inverse</span> </div>

answer:
<div class="solution">Let $f: A \rightarrow$ A defined as $f(x)=x^{2}$. Let $f(x)=y \quad \therefore x=f^{-1}(y)$<br/>Now $\Rightarrow x^{2}=y$<br/>$x=\sqrt{y}$<br/>Thus, $f^{-1}(y)=\sqrt{y}$<br/>Hence, the function has an inverse but $\mathrm{f}$ is not its own inverse.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-x-w-the-set-of-whole-numbers-and-x-3-b-x-n-the-set-of-natural-numbers-and-2-x-4-and-c-3-4-then-how-many-elements-will-a-b-c-contain

slug:
let-a-x-w-the-set-of-whole-numbers-and-x-3-b-x-n-the-set-of-natural-numbers-and-2-x-4-and-c-3-4-then-how-many-elements-will-a-b-c-contain-38286

question:
<div class="question">Let $A=\{x \in W$, the set of whole numbers and $x   &lt; 3\}$, $\mathrm{B}=\{\mathrm{x} \in \mathrm{N}$, the set of natural numbers and $2 \leq \mathrm{x} &lt; 4\}$ and<br/>$\mathrm{C}=\{3,4\}$, then how many elements will $(\mathrm{A} \cup \mathrm{B}) \times \mathrm{C}$<br/>contain?</div>

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

answer:
<div class="solution">We have $\mathrm{A}=\{0,1,2\}$<br/>$\mathrm{B}=\{2,3\}$<br/>$\mathrm{C}=\{3,4\}$<br/>$\mathrm{A} \cup \mathrm{B}=\{0,1,2,3\}$<br/>$(\mathrm{A} \cup \mathrm{B}) \times \mathrm{C}=\{(0,3),(0,4),(1,3),(1,4) ;(2,3)$<br/>$(2,4),(3,3) ;(3,4)\}$<br/>$\therefore \mathrm{n}[(\mathrm{A} \cup \mathrm{B}) \times \mathrm{C}]=8$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">30</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-x-x-9-x-n-let-b-a-b-c-be-the-subset-of-a-where-a-b-c-is-a-multiple-of-3-what-is-the-largest-possible-number-of-subsets-like-b

slug:
let-a-x-x-9-x-n-let-b-a-b-c-be-the-subset-of-a-where-a-b-c-is-a-multiple-of-3-what-is-the-largest-possible-number-of-subsets-like-b-29058

question:
<div class="question">Let $A=\{x \mid x \leq 9, x \in N\} .$ Let $B=\{a, b, c\}$ be the subset of $A$ where $(a+b+c)$ is a multiple of $3 .$ What is the largest possible number of subsets like B ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">21</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">27</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">30</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given<br/>$\mathrm{A}=\{\mathrm{x}: \mathrm{x} \leq 9, \mathrm{x} \in \mathrm{N}\}=\{1,2,3,4,5,6,7,8,9\}$<br/>Total possible multiple of 3 are<br/>$3,6,9,12,15,18,21,24,27$<br/>But 3 and 27 are not possible because 3 and 27 can not be express as such that $a+b+c$ is multiple of 3 $6 \rightarrow 1+2+3$<br/>$9 \rightarrow 2+3+4,5+3+1,6+2+1$<br/>$12 \rightarrow 9+2+1,8+3+1,7+1+4,7+2+3$<br/>$6+4+2,6+5+1,5+4+3$<br/>$15 \rightarrow 9+4+2,9+5+1,8+6+1,8+5+2$,<br/>$8+4+3,7+6+2,7+5+3,6+5+4$<br/>$18 \rightarrow 9+8+1,9+7+2,9+6+3$<br/>$9+5+4,8+7+3,8+6+4,7+6+5$<br/>$21 \rightarrow 9+8+4,9+7+5,8+7+6$<br/>$24 \rightarrow 9+8+7$<br/>Hence, total largest possible subsets are 30 .</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-x-x-is-a-square-of-anatural-number-and-x-is-less-than-100-and-b-is-a-set-of-even-natural-numbers-what-is-the-cardinality-of-a-b

slug:
let-a-x-x-is-a-square-of-anatural-number-and-x-is-less-than-100-and-b-is-a-set-of-even-natural-numbers-what-is-the-cardinality-of-a-b-15462

question:
<div class="question">Let $\mathrm{A}=\{\mathrm{x}: \mathrm{x}$ is a square of anatural number and $\mathrm{x}$ is less than $100\}$ and $B$ is a set of even natural numbers. What is the cardinality of $\mathrm{A} \cap \mathrm{B}$?</div>

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

answer:
<div class="solution">Here $\mathrm{A}=\{1,4,9,16,25,36,49,64,81\}$<br/>$\mathrm{B}=$ \{even natural numbers $\mathrm{A} \cap \mathrm{B}=\{4,16,36,64\}$<br/>So, cardinal ity of $\mathrm{A} \cap \mathrm{B}=4$</div>


title: let-a-x-y-2-x-y-x-y-b-2-1-and-c-3-2-if-a-b-c-then-what-is-a-2-equal-to
slug: let-a-x-y-2-x-y-x-y-b-2-1-and-c-3-2-if-a-b-c-then-what-is-a-2-equal-to-10377
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let<br/>$A=\left[\begin{array}{cc}x+y &amp; y \\ 2 x &amp; x-y\end{array}\right], B=\left[\begin{array}{c}2 \\ -1\end{array}\right]$ and $C=\left[\begin{array}{l}3 \\ 2\end{array}\right]$<br/>If $A B=C$, then what is $A^{2}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{cc}6 &amp; -10 \\ 4 &amp; 26\end{array}\right]$</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">$\left[\begin{array}{cc}-10 &amp; 5 \\ 4 &amp; 24\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{cc}-5 &amp; -6 \\ -4 &amp; -20\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\begin{array}{ll}-5 &amp; -7 \\ -5 &amp; 20\end{array}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{cc}6 &amp; -10 \\ 4 &amp; 26\end{array}\right]$</span> </div>

answer:
<div class="solution">$A=\left[\begin{array}{cc}x+y &amp; y \\ 2 x &amp; x-y\end{array}\right]$<br/>$B=\left[\begin{array}{c}2 \\ -1\end{array}\right]$ and $C=\left[\begin{array}{l}3 \\ 2\end{array}\right]$<br/>Here $\mathrm{AB}=\mathrm{C}$<br/>$\therefore \quad\left[\begin{array}{cc}\mathrm{x}+\mathrm{y} &amp; \mathrm{y} \\ 2 \mathrm{x} &amp; \mathrm{x}-\mathrm{y}\end{array}\right]\left[\begin{array}{c}2 \\ -1\end{array}\right]=\left[\begin{array}{l}3 \\ 2\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{cc}2(\mathrm{x}+\mathrm{y}) &amp; -\mathrm{y} \\ 4 \mathrm{x} &amp; -\mathrm{x}+\mathrm{y}\end{array}\right]=\left[\begin{array}{l}3 \\ 2\end{array}\right]$<br/>$2 x+y=3$ ...(i)<br/>$3 x+y=2$ ...(ii)<br/>From equations (i) and (ii), we get $x=-1$ and $y=5$<br/>$\therefore  A=\left[\begin{array}{cc}4 &amp; 5 \\ -2 &amp; -6\end{array}\right]$<br/>Now, $A^{2}=\left[\begin{array}{cc}4 &amp; 5 \\ -2 &amp; -6\end{array}\right]\left[\begin{array}{cc}4 &amp; 5 \\ -2 &amp; -6\end{array}\right]$<br/>$=\left[\begin{array}{cc}16-10 &amp; 20-30 \\ -8+12 &amp; -10+36\end{array}\right]=\left[\begin{array}{cc}6 &amp; -10 \\ 4 &amp; 26\end{array}\right]$<br/>$\therefore$ Option (a) is correct.</div>


title: let-a-x-y-z-and-b-p-q-r-s-what-is-the-number-of-distinct-relations-from-b-to-a
slug: let-a-x-y-z-and-b-p-q-r-s-what-is-the-number-of-distinct-relations-from-b-to-a-89809
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4096</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A=\{x, y, z\}$ and $B=\{p, q, r, s\} .$ What is the number of distinct relations from B to A?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">4096</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">4094</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">128</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">126</span> </li> </ul>

answer:
<div class="solution">$\mathrm{A}=\{\mathrm{x}, \mathrm{y}, \mathrm{z}\}$<br/>$\mathrm{B}=(\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s})$<br/>$\mathrm{n}(\mathrm{A})=3$<br/>$\mathrm{n}(\mathrm{B})=4$<br/>Number of distinct relations $=2^{n(A) \times n(B)}=2^{3 \times 4}=2^{12}=4096$<br/>$\therefore$ option (a) is correct.</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-x-y-z-b-be-in-ap-where-x-y-z-15-let-a-p-q-r-b-be-in-hp-where-p-1-q-1-r-1-3-5-what-is-the-value-of-ab

slug:
let-a-x-y-z-b-be-in-ap-where-x-y-z-15-let-a-p-q-r-b-be-in-hp-where-p-1-q-1-r-1-3-5-what-is-the-value-of-ab-49329

question:
<div class="question">Let $\mathrm{a}, \mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{b}$ be in $\mathrm{AP}$, where $\mathrm{x}+\mathrm{y}+\mathrm{z}=15$. Let $\mathrm{a}, \mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{b}$ be<br/>in HP, where $\mathrm{p}^{-1}+\mathrm{q}^{-1}+\mathrm{r}^{-1}=\frac{5}{3}$<br/>What is the value of ab?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">$\mathrm{S}_{\mathrm{n}}=\frac{n}{2}(a+l)$<br/>$\Rightarrow \mathrm{a}+\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{b}=\frac{5}{2}(a+b)$<br/>$a+b+15=\frac{5}{2}(a+b)$<br/>$\Rightarrow \mathrm{a}+\mathrm{b}=10$ ...(1)<br/>$\&amp; \frac{1}{a}+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{b}=\frac{5}{2}\left(\frac{1}{a}+\frac{1}{b}\right)$<br/>$\Rightarrow \quad \frac{1}{a}+\frac{1}{b}+\frac{5}{3}=\frac{5}{2}\left(\frac{1}{a}+\frac{1}{b}\right)$<br/>$\Rightarrow \frac{3(a+b)}{a b}=\frac{10}{3}$ ...(2)<br/>$\Rightarrow \quad \frac{3 \times 10}{a b}=\frac{10}{3}$<br/>$\Rightarrow \quad a b=9$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-a-x-y-z-b-be-in-ap-where-x-y-z-15-let-a-p-q-r-b-be-in-hp-where-p-1-q-1-r-1-3-5-what-is-the-value-of-pqr

slug:
let-a-x-y-z-b-be-in-ap-where-x-y-z-15-let-a-p-q-r-b-be-in-hp-where-p-1-q-1-r-1-3-5-what-is-the-value-of-pqr-58477

question:
<div class="question">Let $\mathrm{a}, \mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{b}$ be in $\mathrm{AP}$, where $\mathrm{x}+\mathrm{y}+\mathrm{z}=15$. Let $\mathrm{a}, \mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{b}$ be<br/>in HP, where $\mathrm{p}^{-1}+\mathrm{q}^{-1}+\mathrm{r}^{-1}=\frac{5}{3}$<br/>What is the value of pqr?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$35 / 243$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$81 / 35$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$243 / 35$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Cannot be determined</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$243 / 35$</span> </div>

answer:
<div class="solution">since a, $\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{b}$ or $1, \mathrm{p}, \mathrm{q}, \mathrm{r}, 9$ are in H.P.<br/>$\begin{aligned} \Rightarrow &amp; \frac{1}{1+4 \mathrm{~d}}=9 \Rightarrow d=-\frac{2}{9} . \\ &amp; \frac{1}{\mathrm{p}}=1-\frac{2}{9}=\frac{7}{9} \Rightarrow p=\frac{9}{7} \\ &amp; \frac{1}{\mathrm{q}}=\frac{7}{9}-\frac{2}{9}=\frac{5}{9} \Rightarrow q=\frac{9}{5} \\ &amp; \&amp; \frac{1}{\mathrm{r}}=\frac{5}{9}-\frac{2}{9}=\frac{3}{9} \Rightarrow r=\frac{9}{3} \\ \Rightarrow &amp; \mathrm{p} \times \mathrm{q} \times \mathrm{r}=\frac{243}{35} \end{aligned}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">105</span> </div>
source_db: MarksBatch2_P1.db
title:
let-a-x-y-z-b-be-in-ap-where-x-y-z-15-let-a-p-q-r-b-be-in-hp-where-p-1-q-1-r-1-3-5-what-is-the-value-of-x-yz

slug:
let-a-x-y-z-b-be-in-ap-where-x-y-z-15-let-a-p-q-r-b-be-in-hp-where-p-1-q-1-r-1-3-5-what-is-the-value-of-x-yz-62609

question:
<div class="question">Let $\mathrm{a}, \mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{b}$ be in $\mathrm{AP}$, where $\mathrm{x}+\mathrm{y}+\mathrm{z}=15$. Let $\mathrm{a}, \mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{b}$ be<br/>in HP, where $\mathrm{p}^{-1}+\mathrm{q}^{-1}+\mathrm{r}^{-1}=\frac{5}{3}$<br/>What is the value of $x y z$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">120</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">105</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">90</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cannot be determined</span> </li> </ul>

answer:
<div class="solution">On solving eq (1) \&amp; (2), we get<br/>(i) $a=1 \&amp; b=9 \Rightarrow a+4 d=9 \Rightarrow d=2$<br/>(ii) $a=9 \&amp; b=1 \Rightarrow a+4 d=1 \Rightarrow d=-2$.<br/>For $a=1 \&amp; d=2$, $x=3, y=5 \&amp; z=7$<br/>For $a=9 \&amp; d=-2$<br/>$x=7, y=5 \&amp; z=3$<br/>$\Rightarrow \quad x y z=7 \times 5 \times 3=105$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-aand-b-be-two-matrices-of-order-n-n-let-a-be-nonsingular-and-b-be-singular-consider-the-following-1-ab-is-singular-2-ab-is-nonsingular-3-a-1-b-is-

slug:
let-aand-b-be-two-matrices-of-order-n-n-let-a-be-nonsingular-and-b-be-singular-consider-the-following-1-ab-is-singular-2-ab-is-nonsingular-3-a-1-b-is-16146

question:
<div class="question">Let Aand $\mathrm{B}$ be two matrices of order $\mathrm{n} \times \mathrm{n}$. Let $\mathrm{A}$ be nonsingular and $\mathrm{B}$ be singular. Consider the following:<br/>$1. \quad \mathrm{AB}$ is singular<br/>$2. \quad \mathrm{AB}$ is non-singular<br/>$3. \quad \mathrm{A}^{-1} \mathrm{~B}$ is singular<br/>$4. \quad \mathrm{A}^{-1} \mathrm{~B}$ is non singular <br/>Which of the above is/ are correct?</div>

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

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

answer:
<div class="solution">If A is non-singular and B is singular, then $\mathrm{AB}$ and $\mathrm{A}^{-1} \mathrm{~B}$ are non-singular. Statements (2) and (4) are correct.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-abcd-be-a-parallelogram-whose-diagonals-intersect-at-p-and-let-o-be-the-origin-then-what-is-oa-ob-oc-od-equal-to

slug:
let-abcd-be-a-parallelogram-whose-diagonals-intersect-at-p-and-let-o-be-the-origin-then-what-is-oa-ob-oc-od-equal-to-40853

question:
<div class="question">Let $\mathrm{ABCD}$ be a parallelogram whose diagonals intersect at $\mathrm{P}$ and let $\mathrm{O}$ be the origin, then what is<br/>$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}$ equal to $?$</div>

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

answer:
<div class="solution">Diagonals of a parallelogram bisect each other. Therefore, $\mathrm{P}$ is the mid point of $\mathrm{AC}$ and $\mathrm{BD}$ both.<br/>So, in $\Delta \mathrm{OAC}, \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}}=2 \overrightarrow{\mathrm{OP}}$<br/>and in $\Delta \mathrm{ODB}, \overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OD}}=2 \overrightarrow{\mathrm{OP}}$<br/>$\Rightarrow \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}}=2 \overrightarrow{\mathrm{OP}}$ and $\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OD}}=2 \overrightarrow{\mathrm{OP}}$<br/>$\Rightarrow \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}=4 \overrightarrow{\mathrm{OP}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/rPFueBK91MxZYI-GDV2a19j7-pFYuDH4qoe8W-k-gg4.original.fullsize.png"/></div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-abcd-be-a-parallelogram-whose-diagonals-intersect-at-p-and-let-o-be-the-origin-what-is-oa-ob-oc-od-equal-to

slug:
let-abcd-be-a-parallelogram-whose-diagonals-intersect-at-p-and-let-o-be-the-origin-what-is-oa-ob-oc-od-equal-to-18686

question:
<div class="question">Let $\mathrm{ABCD}$ be a parallelogram whose diagonals intersect at $\mathrm{P}$ and let $\mathrm{O}$ be the origin. What is $\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}$<br/>equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \overrightarrow{\mathrm{OP}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4 \overrightarrow{\mathrm{OP}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$6 \overrightarrow{\mathrm{OP}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$8 \overrightarrow{\mathrm{OP}}$</span> </li> </ul>

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

answer:
<div class="solution">From the figure, observe that<br/>$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{OP}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/ORp-wP3XiGEkhAfSyiTB_md5NU0VChCEK4PUGj_8bAQ.original.fullsize.png"/><br/><br/>Similarly, $\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{BP}}=\overrightarrow{\mathrm{OP}}$<br/>$\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{CP}}=\overrightarrow{\mathrm{OP}}$<br/>$\overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{DP}}=\overrightarrow{\mathrm{OP}}$<br/>$\Rightarrow \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{AP}}$<br/>$+\overrightarrow{\mathrm{BP}}+\overrightarrow{\mathrm{CP}}+\overrightarrow{\mathrm{DP}}=4 \overrightarrow{\mathrm{OP}} \quad \rightarrow$ (i)<br/>In a parallelogram, diagonals bisect each other.<br/>$\therefore \overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{PC}} \quad(\therefore$ Since $\mathrm{AP}=\mathrm{CP})$<br/>$\Rightarrow \overrightarrow{\mathrm{AP}}=-\overrightarrow{\mathrm{PC}}$<br/>$\Rightarrow \overrightarrow{\mathrm{AP}}+\overrightarrow{\mathrm{PC}}=\mathrm{O}$<br/>Similarly $\overrightarrow{\mathrm{BP}}+\overrightarrow{\mathrm{DP}}=\mathrm{O}$<br/>$\therefore(\mathrm{i}) \Rightarrow \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}+\overrightarrow{\mathrm{OD}}=4 \overrightarrow{\mathrm{OP}}$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-abcd-be-a-square-and-let-p-be-a-point-on-ab-such-that-ap-pb-1-2-if-apd-then-what-is-the-value-of-cos

slug:
let-abcd-be-a-square-and-let-p-be-a-point-on-ab-such-that-ap-pb-1-2-if-apd-then-what-is-the-value-of-cos-57768

question:
<div class="question">Let $\mathrm{ABCD}$ be a square and let $\mathrm{P}$ be a point on $\mathrm{AB}$ such that $\mathrm{AP}: \mathrm{PB}=1: 2 .$ If $\angle \mathrm{APD}=\theta$, then what is the value of<br/>$\cos \theta ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\sqrt{10}}$</span> </div>

answer:
<div class="solution">Let, ABCD be a square having length of side $=\mathrm{x}$ $\therefore \quad$ So, $A D=A B=x$<br/>As given : $\frac{\mathrm{AP}}{\mathrm{PB}}=\frac{1}{2}$<br/>$\Rightarrow \frac{\mathrm{AP}}{\mathrm{PB}+\mathrm{AP}}=\frac{1}{2+1}$<br/>$\Rightarrow \mathrm{AP}=\frac{1}{1+2} \mathrm{x}=\frac{\mathrm{x}}{3}$<br/>$\cos \theta=\frac{\mathrm{AP}}{\mathrm{PD}}$<br/>Now, $\mathrm{PD}^{2}=\mathrm{AP}^{2}+\mathrm{AD}^{2}=\left(\frac{x}{3}\right)^{2}+x^{2}$<br/>$=\frac{x^{2}}{9}+\mathrm{x}^{2}=\frac{10 \mathrm{x}^{2}}{9}$<br/>$\Rightarrow \mathrm{PD}=\frac{\sqrt{10} \mathrm{x}}{3}$<br/>$\cos \theta=\frac{\mathrm{x} / 3}{\sqrt{10} \mathrm{x} / 3}=\frac{1}{\sqrt{10}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/oQDEaNnS5m4iykFOF34U4j7ui6PKNy7cov52OPDbzw8.original.fullsize.png"/></div>


title: let-aleftbeginarrayccc2-0-3-4-7-11-5-4-8endarrayright-then
slug: let-aleftbeginarrayccc2-0-3-4-7-11-5-4-8endarrayright-then-88281
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let \(A=\left(\begin{array}{ccc}2 &amp; 0 &amp; 3 \\ 4 &amp; 7 &amp; 11 \\ 5 &amp; 4 &amp; 8\end{array}\right)\). Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\operatorname{det} A\) is divisible by 11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">\(\operatorname{det} A\) is not divisible by 11</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\operatorname{det} A=0\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A is orthogonal matrix</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\operatorname{det} A\) is divisible by 11</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint: } \operatorname{det}(A)=2(56-44)+0+3(16-35) \\<br/>&amp; =24-57 \\<br/>&amp; =-33 \text { which is divisible by } 11<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2023']
source_db: MarksBatch2_P1.db
title:
let-aleftbeginarraylll0-0-1-1-0-0-0-0-0endarrayright-bleftbeginarraylll0-1-0-0-0-1-0-0-0endarrayright-and-pleftbeginarraylll0-1-0-x-0-0-0-0-yendarrayr

slug:
let-aleftbeginarraylll0-0-1-1-0-0-0-0-0endarrayright-bleftbeginarraylll0-1-0-0-0-1-0-0-0endarrayright-and-pleftbeginarraylll0-1-0-x-0-0-0-0-yendarrayr-53048

question:
<div class="question">Let \(A=\left(\begin{array}{lll}0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0\end{array}\right), B=\left(\begin{array}{lll}0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 0\end{array}\right)\) and \(P=\left(\begin{array}{lll}0 &amp; 1 &amp; 0 \\ x &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; y\end{array}\right)\) be an orthogonal matrix such that \(B=PAP^{-1}\) holds. Then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x=1=y\)</span> </div>

answer:
<div class="solution">Hint : \(B=P A P^{-1}\)<br/>\(P^{-1} B P=P^{-1} P A P^{-1} P\)<br/>\(\begin{aligned}<br/>&amp; {\left[\begin{array}{ccc}<br/>0 &amp; 0 &amp; x y \\<br/>x &amp; 0 &amp; 0 \\<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right]=A=\left[\begin{array}{lll}<br/>0 &amp; 0 &amp; 1 \\<br/>1 &amp; 0 &amp; 0 \\<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right]} \\<br/>&amp; \therefore x=1=y<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Matrices', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
let-alpha-be-a-root-of-the-equation-x2x10-and-the-matrix-afrac1sqrt3leftbeginarrayccc1-1-1-1-alpha-alpha2-1-alpha2-alpha4endarrayright-then-the-matrix

slug:
let-alpha-be-a-root-of-the-equation-x2x10-and-the-matrix-afrac1sqrt3leftbeginarrayccc1-1-1-1-alpha-alpha2-1-alpha2-alpha4endarrayright-then-the-matrix-64308

question:
<div class="question">Let \( \alpha \) be a root of the equation \( x^{2}+x+1=0 \) and the matrix \( A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; \alpha &amp; \alpha^{2} \\ 1 &amp; \alpha^{2} &amp; \alpha^{4}\end{array}\right] \), then the matrix \( A^{31} \) is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\( A^{3} \)</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">\( I_{3} \)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\( A^{2} \)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( A \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( A^{3} \)</span> </div>

answer:
<div class="solution"><p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>ω</mi><mo>,</mo><msup><mi>ω</mi><mn>2</mn></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> is complex cube root of unity.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced close="]" open="[" separators="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>ω</mi></mtd><mtd><msup><mi>ω</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><msup><mi>ω</mi><mn>2</mn></msup></mtd><mtd><mi>ω</mi></mtd></mtr></mtable></mfenced><mfenced close="]" open="[" separators="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>ω</mi></mtd><mtd><msup><mi>ω</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><msup><mi>ω</mi><mn>2</mn></msup></mtd><mtd><mi>ω</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>A</mi><mn>4</mn></msup><mo>=</mo><mfenced close="]" open="[" separators="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mfenced close="]" open="[" separators="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced close="]" open="[" separators="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>A</mi><mn>31</mn></msup><mo>=</mo><msup><mi>A</mi><mn>28</mn></msup><mo>×</mo><msup><mi>A</mi><mn>3</mn></msup><mo>=</mo><mi>I</mi><mo>×</mo><msup><mi>A</mi><mn>3</mn></msup><mo>=</mo><msup><mi>A</mi><mn>3</mn></msup></math></p></div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: MarksBatch2_P1.db
title:
let-alpha-beta-be-the-roots-of-the-equation-a-x2b-xc0-a-b-c-real-and-snalphanbetan-and-leftbeginarrayccc3-1s1-1s2-1s1-1s2-1s3-1s2-1s3-1s4endarrayright

slug:
let-alpha-beta-be-the-roots-of-the-equation-a-x2b-xc0-a-b-c-real-and-snalphanbetan-and-leftbeginarrayccc3-1s1-1s2-1s1-1s2-1s3-1s2-1s3-1s4endarrayright-53034

question:
<div class="question">Let \(\alpha, \beta\) be the roots of the equation \(a x^2+b x+c=0, a, b, c\) real and \(s_n=\alpha^n+\beta^n\) and \(\left|\begin{array}{ccc}3 &amp; 1+s_1 &amp; 1+s_2 \\ 1+s_1 &amp; 1+s_2 &amp; 1+s_3 \\ 1+s_2 &amp; 1+s_3 &amp; 1+s_4\end{array}\right|=k \frac{(a+b+c)^2}{a^4}\) then \(k=\)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(b^2-4 a c\)</span> </div>

answer:
<div class="solution">Hint: \(\left|\begin{array}{ccc}3 &amp; 1+s_1 &amp; 1+s_2 \\ 1+s_1 &amp; 1+s_2 &amp; 1+s_3 \\ 1+s_2 &amp; 1+s_3 &amp; 1+s_4\end{array}\right|\)<br/>\(=\left|\begin{array}{ccc}<br/>1+1+1 &amp; 1+\alpha+\beta &amp; 1+\alpha^2+\beta^2 \\<br/>1+\alpha+\beta &amp; 1+\alpha^2+\beta^2 &amp; 1+\alpha^3+\beta^3 \\<br/>1+\alpha^2+\beta^2 &amp; 1+\alpha^3+\beta^3 &amp; 1+\alpha^4+\beta^4<br/>\end{array}\right|\)<br/>\(\begin{aligned} &amp; =\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; \alpha &amp; \beta \\ 1 &amp; \alpha^2 &amp; \beta^2\end{array}\right| \times\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; \alpha &amp; \alpha^2 \\ 1 &amp; \beta &amp; \beta^2\end{array}\right| \\ &amp; =\{(1-\alpha)(\alpha-\beta)(1-\beta)\}^2 \\ &amp; =(1-(\alpha+\beta)+\alpha \beta)^2(\alpha-\beta)^2 \\ &amp; =\left(1+\frac{b}{a}+\frac{c}{a}\right)^2\left(\frac{b^2}{a^2}-\frac{4 c}{a}\right) \\ &amp; =\frac{(a+b+c)^2}{a^2} \times \frac{b^2-4 a c}{a^2}=\left(b^2-4 a c\right) \frac{(a+b+c)^2}{a^4} \\ &amp; \therefore k=b^2-4 a c\end{aligned}\)</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-and-be-real-numbers-and-z-be-a-complex-number-if-z-2-z-0-has-two-distinct-nonreal-roots-with-re-z-1-then-it-is-necessary-that

slug:
let-and-be-real-numbers-and-z-be-a-complex-number-if-z-2-z-0-has-two-distinct-nonreal-roots-with-re-z-1-then-it-is-necessary-that-69860

question:
<div class="question">Let $\alpha$ and $\beta$ be real numbers and $z$ be a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct non-real roots with $\operatorname{Re}(\mathrm{z})=1$, then it is necessary that</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\beta \in(1, \infty)$</span> </div>

answer:
<div class="solution">Let $z=x+$ iy $\therefore z^{2}+\alpha z+\beta=(x+i y)^{2}+\alpha(x+i y)+\beta$<br/>$=\mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{ixy}+\alpha \mathrm{x}+\mathrm{i} \alpha \mathrm{y}+\beta$<br/>Given, $z^{2}+\alpha z+\beta=0$<br/>$\therefore x^{2}-y^{2}+2 i x y+\alpha x+i \alpha y+\beta=0$<br/>$\Rightarrow x^{2}-y^{2}+\alpha x+\beta+i(2 x y+\alpha y)=0+i .0$<br/>Comparing real and imaginary parts, we get $\mathrm{x}^{2}-\mathrm{y}^{2}+\alpha \mathrm{x}+\beta=0$<br/>$\mathrm{y}(2 \mathrm{x}+\alpha)=0$<br/>$(2) \Rightarrow 2 \mathrm{x}+\alpha=0 \quad(\because \mathrm{y} \neq 0)$<br/>$\Rightarrow 2(1)+\alpha=0 \quad(\because$ Given $\operatorname{Re}(z)=1)$<br/>$\Rightarrow \alpha=-2$<br/>Now, (1) $\Rightarrow \mathrm{x}^{2}-\mathrm{y}^{2}+\alpha \mathrm{x}+\beta=0$<br/>$\Rightarrow(1)^{2}-\mathrm{y}^{2}+(-2)(1)+\beta=0$<br/>$\Rightarrow 1-\mathrm{y}^{2}-2+\beta=0$<br/>$\Rightarrow-1-\mathrm{y}^{2}+\beta=0$<br/>$\Rightarrow \beta=1+\mathrm{y}^{2}$<br/>Since, $\mathrm{y} \in \mathrm{R}$ and $\mathrm{y} \neq 0$,<br/>$\beta$ is always greater than $1 .$ So, $\beta \in(1, \infty)$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-and-be-the-roots-of-the-equation-x-2-1-2-a-2-x-1-2-a-2-0-under-what-condition-does-the-above-equation-have-real-roots

slug:
let-and-be-the-roots-of-the-equation-x-2-1-2-a-2-x-1-2-a-2-0-under-what-condition-does-the-above-equation-have-real-roots-77379

question:
<div class="question">Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-\left(1-2 a^{2}\right) x+\left(1-2 a^{2}\right)=0$<br/>Under what condition does the above equation have real roots?</div>

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

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

answer:
<div class="solution">Using $a x^{2}+b x+c=0$<br/>$a=1, b=-\left(1-2 a^{2}\right) \&amp; c=\left(1-2 a^{2}\right)$<br/>For roots to be real,<br/>$b^{2}-4 a c&gt;0$<br/>$\Rightarrow\left[-\left(1-2 a^{2}\right)\right]^{2}-4(1)\left(1-2 a^{2}\right) \geq 0$<br/>$\Rightarrow 4 a^{4}+4 a^{2}-3 \geq 0$<br/>$\Rightarrow\left(2 a^{2}-1\right)\left(2 a^{2}+3\right) \geq 0$<br/>$\Rightarrow a^{2} \geq \frac{1}{2}$ or $a^{2} \leq-\frac{3}{2}$</div>


title: let-and-be-the-roots-of-the-equation-x-2-1-2-a-2-x-1-2-a-2-0-under-what-condition-is-2-1-2-1-1
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
let-and-be-the-roots-of-the-equation-x-2-1-2-a-2-x-1-2-a-2-0-under-what-condition-is-2-1-2-1-1-90432

question:
<div class="question">Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-\left(1-2 a^{2}\right) x+\left(1-2 a^{2}\right)=0$<br/>Under what condition is $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}   &lt; 1 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{a}^{2}   &lt; \frac{1}{2}$</span> </div>

answer:
<div class="solution">$\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}   &lt; 1 \Rightarrow \frac{\alpha^{2}+\beta^{2}}{(\alpha \beta)^{2}} &lt; 1$<br/>$\Rightarrow \frac{(\alpha+\beta)^{2}-2 \alpha \beta}{(\alpha \beta)^{2}} &lt; 1$<br/>$\alpha+\beta=\frac{-b}{a}=\left(1-2 a^{2}\right) \&amp; \alpha \beta=\frac{c}{a}=\left(1-2 a^{2}\right)$<br/>On solving: $\frac{4 a^{4}-1}{4 a^{4}-4 a^{2}+1} &lt; 1$<br/>$\Rightarrow 4 a^{4}-1 &lt; 4 a^{4}-4 a^{2}+1$<br/>$4 a^{2} &lt; 2 \Rightarrow a^{2} &lt; \frac{1}{2}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-and-be-the-roots-of-the-equation-x-2-b-x-c-0-where-b-0-and-c-0-consider-the-following-1-2-which-of-the-above-isare-correct

slug:
let-and-be-the-roots-of-the-equation-x-2-b-x-c-0-where-b-0-and-c-0-consider-the-following-1-2-which-of-the-above-isare-correct-33204

question:
<div class="question">Let $\alpha$ and $\beta(\alpha   &lt; \beta)$ be the roots of the equation $x^{2}+b x+c=0$, where $\mathrm{b}&gt;0$ and $\mathrm{c} &lt; 0 .$<br/>Consider the following:<br/>1.$\beta &lt; -\alpha$<br/>2.$\beta &lt; \alpha \mid$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">Given quadratic equation, Hence roots of given quadratic equation are<br/>$\beta=\frac{-b+\sqrt{b^{2}-4 c}}{2}$<br/>$\alpha=\frac{-b-\sqrt{b^{2}-4 c}}{2} \quad(\because \alpha   &lt; \beta)$<br/>$\Rightarrow-\alpha=\frac{b+\sqrt{b^{2}-4 c}}{2}$ and $\mid \alpha \models \frac{b+\sqrt{b^{2}-4 c}}{2}$</div>


title: let-and-be-the-roots-of-the-equation-x-2-x-1-0-the-equation-whose-roots-are-19-and-7-is
slug: let-and-be-the-roots-of-the-equation-x-2-x-1-0-the-equation-whose-roots-are-19-and-7-is-21153
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+x+1=0 .$ The equation whose roots are $\alpha^{19}$ and $\beta^{7}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}-x+1=0$</span> </div>

answer:
<div class="solution">Given Equation is $x^{2}+x+1=0$<br/>$\Rightarrow x=\frac{-1 \pm \sqrt{3} i}{2}$<br/>Thus, roots are $\omega$ and $\omega^{2}$ $\alpha=\omega$ and $\beta=\omega^{2}$<br/>$\mathrm{S} o, \alpha^{19}=(\omega)^{19}=\left(\omega^{3}\right)^{6}, \omega=\omega \quad\left(\because \omega^{3}=1\right)$<br/>and $\beta^{7}=\left(\omega^{2}\right)^{7}=\omega^{14}=\left(\omega^{3}\right)^{4} \cdot \omega^{2}=\omega^{2}$<br/>Now, $\alpha^{19}+\beta^{7}=\omega+\omega^{2}=-1\left(\because 1+\omega+\omega^{2}=0\right)$<br/>and $\left(\alpha^{19}\right)\left(\beta^{7}\right)=(\omega)\left(\omega^{2}\right)=1$<br/>$\therefore$ Required Quadratic equation whose roots are $\alpha^{19}$<br/>and $\beta^{7}$ is<br/>$x^{2}+\left(\alpha^{19}+\beta^{7}\right) x+\left(\alpha^{19}\right)\left(\beta^{7}\right)=0$<br/>$\Rightarrow x^{2}+(-1) x+1=0$<br/>$\Rightarrow x^{2}-x+1=0$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-be-a-positive-angle-if-the-number-of-degrees-in-is-divided-by-the-number-of-radians-in-then-an-irrational-number-180-results-if-the-number-of-degr

slug:
let-be-a-positive-angle-if-the-number-of-degrees-in-is-divided-by-the-number-of-radians-in-then-an-irrational-number-180-results-if-the-number-of-degr-99974

question:
<div class="question">Let $\theta$ be a positive angle, If the number of degrees in $\theta$ is divided by the number of radians in $\theta$, then an irrational<br/>number $\frac{180}{\pi}$ results. If the number of degrees in $\theta$ is<br/>multiplied by the number of radians in $\theta$, then an irrational<br/>number $\frac{125 \pi}{9}$ results. The angle $\theta$ must be equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$50^{\circ}$</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">D</span> <span class="option-data">$60^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$50^{\circ}$</span> </div>

answer:
<div class="solution">From going by the options, option (a), $\theta=30^{\circ}$, as we know that $180^{\circ}=\pi \mathrm{radian}$<br/>$\therefore \quad 30^{\circ}=\frac{30 \pi}{180}$ radian<br/>Now according to question,<br/>$\frac{30^{\circ} \times 180^{\circ}}{30^{\circ} \pi}=\frac{180}{\pi}$<br/>Now number of degree in $\theta$ is multiplied by number of radians in $\theta$.<br/>$\therefore \quad 30^{\circ} \times \frac{30 \pi}{180}=\frac{900 \pi}{180}=\frac{10 \pi}{2}=5 \pi \neq \frac{125 \pi}{9}$<br/>From option (b), $\theta=45^{\circ}$<br/>$\therefore \quad 45^{\circ}=\frac{45^{\circ} \pi}{180}$ radian<br/>Now according to question,<br/>$\frac{45^{\circ} \times 180}{45^{\circ} \pi}=\frac{180}{\pi}$<br/>Now number of degree in $\theta$ is multiplied by number of $\operatorname{radian}$ in $\theta$.<br/>$\therefore \quad 45^{\circ} \times \frac{45^{\circ} \pi}{180^{\circ}}=\frac{45 \pi}{4} \neq \frac{125 \pi}{9}$<br/>From option (c), $\theta=50^{\circ}$<br/>As we know that $180^{\circ}=\pi$ radian<br/>$\therefore \quad 50^{\circ}=\frac{50 \pi}{180}$ radian<br/>Now according to question<br/>$\frac{50^{\circ} \times 180^{\circ}}{50^{\circ} \pi}=\frac{180}{\pi}$<br/>Now number of degree in ' $\theta$ ' is multiplied bynumber of $\operatorname{radian} \operatorname{in} \theta .$<br/>$\therefore \quad 50^{\circ} \times \frac{50 \pi}{180}=\frac{2500 \pi}{180}=\frac{125 \pi}{9}$<br/>$\therefore \quad$ Option (c) is correct.</div>


title: let-be-an-angle-in-the-standard-position-such-that-the-point-5-12-lies-on-its-terminal-side-then
source_db: MarksBatch2_P1.db
slug:
let-be-an-angle-in-the-standard-position-such-that-the-point-5-12-lies-on-its-terminal-side-then-41646

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> be an angle in the standard position such that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>12</mn><mo>)</mo></math> lies on its terminal side, then</div>

tags:
['Mathematics', 'Trigonometric Ratios & Identities', 'AP EAMCET', 'AP EAMCET 2021 (20 Aug Shift 2)']

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

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

answer:
<div class="solution"><p>Clearly, the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mn>12</mn></mrow></mfenced></math> lies in second quadrant. Since, it's <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo></math>coordinate is negative and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo></math>coordinate is positive.</p><p>We know that, in second quadrant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>θ</mi><mo> </mo><mo>&amp;</mo><mo> </mo><mi>cos</mi><mi>e</mi><mi>c</mi><mi>θ</mi></math> are positive and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>θ</mi><mo>,</mo><mo> </mo><mi>s</mi><mi>e</mi><mi>c</mi><mi>θ</mi><mo>,</mo><mo> </mo><mi>tan</mi><mi>θ</mi><mo> </mo><mo>&amp;</mo><mo> </mo><mi>c</mi><mi>o</mi><mi>t</mi><mi>θ</mi></math> are negative.</p><p>So, by using definition of modulus function, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>sin</mi><mi>θ</mi><mo>|</mo><mo>=</mo><mi>sin</mi><mi>θ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>cos</mi><mi>θ</mi><mo>|</mo><mo>=</mo><mo>-</mo><mi>cos</mi><mi>θ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>tan</mi><mi>θ</mi><mo>|</mo><mo>=</mo><mo>-</mo><mi>tan</mi><mi>θ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>cos</mi><mi>e</mi><mi>c</mi><mi>θ</mi><mo>|</mo><mo>=</mo><mi>cos</mi><mi>e</mi><mi>c</mi><mi>θ</mi></math>.</p></div>


title: let-be-distinct-real-numbers-the-points-with-position-vectors-i-j-k-i-j-k-and-i-j-k
slug: let-be-distinct-real-numbers-the-points-with-position-vectors-i-j-k-i-j-k-and-i-j-k-43096
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\alpha, \beta, \gamma$ be distinct real numbers. The points with position<br/>vectors $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}, \hat{\beta} \hat{i}+\gamma \hat{j}+\alpha \hat{k}$ and $\gamma \hat{i}+\alpha \hat{j}+\beta \hat{k}$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">form an equilateral triangle</span> </div>

answer:
<div class="solution">$\alpha, \beta$ and $\gamma$ be distinct real numbers $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{b}}=\beta \hat{\mathrm{i}}+\gamma \hat{\mathrm{j}}+\alpha \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{c}}=\gamma \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}$<br/>Let $\alpha=1, \beta=2$ and $\gamma=3$<br/>then, $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{c}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$<br/>Now, $|\overrightarrow{a b}|=\sqrt{(2-1)^{2}}+(3-2)^{2}+(1-3)^{2}=\sqrt{6}$<br/>Similarly $|\overrightarrow{\mathrm{bc}}|=|\overrightarrow{\mathrm{ac}}|=\sqrt{6}$<br/>Hence, point for equilateral triangle.</div>


title: let-be-the-angle-which-the-vector-v-2-i-j-2-k-makes-with-the-zaxis-then-what-is-the-value-of-sin
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
let-be-the-angle-which-the-vector-v-2-i-j-2-k-makes-with-the-zaxis-then-what-is-the-value-of-sin-68379

question:
<div class="question">Let $\alpha$ be the angle which the vector $\overrightarrow{\mathrm{V}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ makes<br/>with the z-axis, Then, what is the value of $\sin \alpha$ ?</div>

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

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

answer:
<div class="solution">The given vector is $\overrightarrow{\mathrm{V}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$<br/>and for z-axis $\mathrm{x}=0$ and $\mathrm{y}=0$, so the vector equation is $\overrightarrow{\mathrm{A}}=0 \hat{\mathrm{i}}+0 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\cos \alpha=\frac{\overrightarrow{\mathrm{V}} \cdot \overrightarrow{\mathrm{A}}}{|\overrightarrow{\mathrm{V}}| \cdot|\overrightarrow{\mathrm{A}}|}$<br/>$\cos \alpha=\frac{2.0+(-1) \cdot(0)+2 \cdot(1)}{\sqrt{4+1+4} \sqrt{0+0+1}}=\frac{2}{3}$<br/>Hence, $\sin \alpha=\sqrt{1-\cos ^{2} \alpha}$<br/>$=\sqrt{1-\frac{4}{9}}=\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{3}$</div>


title: let-be-the-root-of-the-equation-25-cos-2-5-cos-12-0-where-2-what-is-sin-2-equal-to
slug: let-be-the-root-of-the-equation-25-cos-2-5-cos-12-0-where-2-what-is-sin-2-equal-to-21401
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\alpha$ be the root of the equation $25 \cos ^{2} \theta+5 \cos \theta-12=0$,<br/>where $\frac{\pi}{2}   &lt; \alpha &lt; \pi$<br/>What is $\sin 2 \alpha$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{24}{25}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{-24}{25}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{-5}{12}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{-21}{25}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-24}{25}$</span> </div>

answer:
<div class="solution">$\sin 2 \alpha=2 \sin \alpha \cdot \cos \alpha$<br/>$=2\left(\frac{3}{5}\right)\left(\frac{-4}{5}\right)$<br/>$=\frac{6}{5} \times \frac{-4}{5}=\frac{-24}{25}$<br/>$\therefore \quad$ Option (b) is correct.</div>


title: let-be-the-root-of-the-equation-25-cos-2-5-cos-12-0-where-2-what-is-tan-equal-to
slug: let-be-the-root-of-the-equation-25-cos-2-5-cos-12-0-where-2-what-is-tan-equal-to-19360
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\alpha$ be the root of the equation $25 \cos ^{2} \theta+5 \cos \theta-12=0$,<br/>where $\frac{\pi}{2}   &lt; \alpha &lt; \pi$<br/>What is tan $\alpha$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{-3}{4}$</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">$\frac{3}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-4}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{-4}{5}$</span> </li> </ul>

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

answer:
<div class="solution">Here $\alpha$ is the root of equation $25 \cos ^{2} \theta+5 \cos \theta-12=0$<br/>$\Rightarrow \quad 25 \cos ^{2} \alpha+5 \cos \alpha-12=0$<br/>$\Rightarrow \quad 25 \cos ^{2} \alpha+20 \cos \alpha-15 \cos \alpha-12=0$<br/>$\Rightarrow \quad 5 \cos \alpha(5 \cos \alpha+4)-3(5 \cos \alpha+4)=0$<br/>$(5 \cos \alpha-3)(5 \cos \alpha+4)=0$<br/>$\cos \alpha=\frac{3}{5}$ or $\cos \alpha=\frac{-4}{5}$<br/>Here, $\frac{\pi}{2}   &lt; \alpha &lt; \pi$<br/>$\therefore \quad \cos \alpha=\frac{-4}{5}$<br/>$\left(\because\right.$ In $2^{\text {nd }}$ quadrant, $\cos \alpha$ value is negative)<br/>Now, $\sin \alpha=\sqrt{1-\cos ^{2} \alpha}=\sqrt{1-\frac{16}{25}}$<br/>$\therefore \quad \sin \alpha=\frac{3}{5}$<br/>$\therefore \quad \tan \alpha=\frac{\sin \alpha}{\cos \alpha}=\frac{3}{5} \times \frac{-5}{4}=\frac{-3}{4}$<br/>$\therefore \quad$ Option (a) is correct.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-3,8$</span> </div>
source_db: MarksBatch2_P1.db
title:
let-be-the-roots-of-a-x-2-4-x-1-0-and-be-the-rootsof-b-x-2-6-x-1-0-if-are-in-hp-then-what-are-the-values-of-a-and-b-respectively

slug:
let-be-the-roots-of-a-x-2-4-x-1-0-and-be-the-rootsof-b-x-2-6-x-1-0-if-are-in-hp-then-what-are-the-values-of-a-and-b-respectively-61075

question:
<div class="question">Let $\alpha, \gamma$ be the roots of $A x^{2}-4 x+1=0$ and $\beta, \delta$ be the<br/>rootsof $B x^{2}-6 x+1=0 .$ If $\alpha, \gamma, \beta, \delta$ are in $\mathrm{HP}$, then what are the values of $A$ and $B$ respectively?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3,8</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-3,-8$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 .$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-3,8$</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> </ul>

answer:
<div class="solution">Sum and product of roots $=0$ will be $\alpha+\gamma=\frac{4}{A}$ and $\alpha \gamma=\frac{1}{A}$ respectively<br/>Sum and product of roots of $\mathrm{Bx}^{2}-6 \mathrm{x}+1=0$ will be<br/>$\beta+\delta=\frac{6}{B}$ and $\beta \delta=\frac{1}{B}$ respectively<br/>Since, $\alpha, \beta, \gamma$ and $\delta$ are in HP.<br/>Then, $\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}$ and $\frac{1}{\delta}$ will be in AP.<br/>$\Rightarrow \frac{1}{\beta}-\frac{1}{\alpha}=\frac{1}{\delta}-\frac{1}{\gamma} \Rightarrow \frac{1}{\beta}-\frac{1}{\delta}=\frac{1}{\alpha}-\frac{1}{\delta}$<br/>$\Rightarrow \frac{\delta-\beta}{\beta \delta}=\frac{\gamma-\alpha}{\alpha \gamma}$<br/>$\Rightarrow \frac{\sqrt{(\delta+\beta)^{2}-4 \beta \delta}}{\beta \delta}=\frac{\sqrt{(\gamma+\alpha)^{2}-4 \alpha \gamma}}{\alpha \gamma}$<br/>$\Rightarrow 36-4 B=16-4 A$<br/>$\Rightarrow 4 A+4 B=20$<br/>$\Rightarrow A+B=5$<br/>It is possible only, if\<br/>$A=-3$ and $B=8$</div>


title: let-be-the-roots-of-the-equation-x-a-x-b-c-c-0-then-the-roots-of-the-equation-x-x-c-0-are
slug: let-be-the-roots-of-the-equation-x-a-x-b-c-c-0-then-the-roots-of-the-equation-x-x-c-0-are-79915
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (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">Let $\alpha, \beta$ be the roots of the equation $(x-a)(x-b)=c, c \neq$ $0 .$ Then the roots of the equation $(x-\alpha)(x-\beta)+\mathrm{c}=0$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a, c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$b,c$</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">$a+b, a+c$</span> </li> </ul>

answer:
<div class="solution">Given equation is $(x-a)(x-b)=c, c \neq 0$<br/>$\Rightarrow x^{2}-(a+b) x+a b-c=0$<br/>Let $\alpha, \beta$ be the roots of this equation. $\alpha+\beta=a+b, \alpha \beta=a b-c$<br/>Consider $(x-\alpha)(x-\beta)+c=0$<br/>$\Rightarrow x^{2}-(\alpha+\beta) x+\alpha \beta+c=0$<br/>Roots of this equation is<br/>$x=\frac{(\alpha+\beta) \pm \sqrt{(\alpha+\beta)^{2}-4(\alpha \beta+c)}}{2}$<br/>$=\frac{(a+b) \pm \sqrt{(a+b)^{2}-4(a b-c+c)}}{2}$<br/>$=\frac{(a+b) \pm \sqrt{a^{2}+b^{2}-2 a b}}{2}$<br/>$=\frac{(a+b) \pm \sqrt{(a-b)^{2}}}{2}=\frac{(a+b) \pm(a-b)}{2}$<br/>$=\frac{a+b+a-b}{2}, \frac{a+b-a+b}{2}=a, b$.<br/>Hence, roots of the equation $(x-\alpha)(x-\beta)+\mathrm{c}=0$ are $a$ and $b$.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-be-the-set-of-all-numbers-which-are-perfect-squares-b-be-the-set-of-all-numbers-which-are-multiples-of-5-and-c-be-the-set-of-all-numbers-which-are

slug:
let-be-the-set-of-all-numbers-which-are-perfect-squares-b-be-the-set-of-all-numbers-which-are-multiples-of-5-and-c-be-the-set-of-all-numbers-which-are-57511

question:
<div class="question">Let be the set of all numbers, which are perfect squares, $\mathrm{B}$ be the set of all numbers which are multiples of 5 and $C$ be the set of all numbers, which are divisible by 2 and 3 . Consider<br/>the following statements : <br/>I. $A, B, C$ are mutually exdusive<br/>II. $A, B, C$ are mutually exhaustive. III. The number of elements in the complement set of $\mathrm{A} \cup \mathrm{B}$ is 12<br/>Which of the statements given above are the correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I and II only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">I and III 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">II and III only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">I, II and III</span> </li> </ul>

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

answer:
<div class="solution">Let, $\mathrm{U}=\{1,2,3, \ldots, 20\}$<br/>$\mathrm{A}=$ Set of all numbers which are perfect square $=\{1,4,9,16\}$<br/>$\mathrm{B}=$ Set of all numbers which are multiples of $5=\{5,10,15,20\}$<br/>$\mathrm{C}=$ Set of all numbers which are divisible by 2 and 3 $=\{6,12,18\}$<br/>$\mathrm{A} \cup \mathrm{B}=\{1,4,9,16,5,10,15,20\}$<br/>$\Rightarrow \mathrm{n}(\mathrm{A} \cup \mathrm{B})=8$<br/>$\Rightarrow \mathrm{n}(\mathrm{A} \cup \mathrm{B})^{\prime}=20-8=12$<br/>Also, $\mathrm{n}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})+\mathrm{n}(\mathrm{C})$<br/>$\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\{1,4,9,16,5,10,15,20,6,12,18\}$<br/>$\mathrm{n}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=11$<br/>and $\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})+\mathrm{n}(\mathrm{C})=4+4+3=11$</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2008 (Phase 1)']
answer: <div class="solution">Both statements are not correct.</div>
source_db: MarksBatch2_P1.db
title:
let-c-be-the-set-of-complex-number-and-z-1-z-2-are-in-c-1-agr-z-1-agr-z-2-z-1-z-2-2-z-1-z-2-z-1-z-2-which-of-the-statements-given-above-isare-correct

slug:
let-c-be-the-set-of-complex-number-and-z-1-z-2-are-in-c-1-agr-z-1-agr-z-2-z-1-z-2-2-z-1-z-2-z-1-z-2-which-of-the-statements-given-above-isare-correct-14803

question:
<div class="question">Let $C$ be the set of complex number and $z_{1}, z_{2}$ are in $C$<br/>1- agr $\left(z_{1}\right)=\operatorname{agr}\left(z_{2}\right) \Rightarrow z_{1}=z_{2}$<br/>2- $\left|z_{1}\right|=\left|z_{2}\right| \Rightarrow z_{1}=z_{2}$<br/>Which of the statements given above is/are correct?</div>


tags: ['Mathematics', 'Indefinite Integration', 'KCET', 'KCET 2017']
source_db: MarksBatch2_P1.db
title:
let-deltaleftbeginarrayccca-x-x2-1-b-y-y2-1-c-z-z2-1endarrayright-and-delta1leftbeginarraycccca-b-c-x-y-z-z-y-z-x-x-y-endarrayright-then

slug:
let-deltaleftbeginarrayccca-x-x2-1-b-y-y2-1-c-z-z2-1endarrayright-and-delta1leftbeginarraycccca-b-c-x-y-z-z-y-z-x-x-y-endarrayright-then-33787

question:
<div class="question">Let \( \Delta=\left|\begin{array}{ccc}A x &amp; x^{2} &amp; 1 \\ B y &amp; y^{2} &amp; 1 \\ C z &amp; z^{2} &amp; 1\end{array}\right| \) and<br/>\( \Delta_{1}=\left|\begin{array}{cccc}A &amp; B &amp; C &amp; \\ x &amp; y &amp; z &amp; \\ z y &amp; z x &amp; x y &amp; \end{array}\right| \) then</div>

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

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

answer:
<div class="solution">Given that, \( \Delta=\left|\begin{array}{ccc}A x &amp; x^{2} &amp; 1 \\ B y &amp; y^{2} &amp; 1 \\ C z &amp; z^{2} &amp; 1\end{array}\right| \rightarrow(1) \)<br/>\[<br/>\text { and } \Delta_{1}=\left|\begin{array}{ccc}<br/>A &amp; B &amp; C \\<br/>x &amp; y &amp; z \\<br/>z y &amp; z x &amp; x y<br/>\end{array}\right| \rightarrow(2)<br/>\]<br/>Take out common \( x, y \) and \( z \) in Eq. (1) from \( R_{1}, R_{2} \) and \( R_{3} \) respectively we get<br/>\( \Delta=x y z\left|\begin{array}{ccc}A &amp; x &amp; \frac{1}{x} \\ B &amp; y &amp; \frac{1}{y} \\ C &amp; z &amp; \frac{1}{z}\end{array}\right| \)<br/>\( C_{3} \rightarrow x y z C_{3} \), we get<br/>\( \Delta=\left|\begin{array}{ccc}A &amp; x &amp; y z \\ B &amp; y &amp; z x \\ C &amp; z &amp; x y\end{array}\right| \)<br/>\( =\left|\begin{array}{ccc}A &amp; B &amp; C \\ x &amp; y &amp; z \\ y z &amp; z x &amp; x y\end{array}\right|=\Delta_{1} \)<br/>Therefore, \( \Delta=\Delta_{1} \)</div>


tags: ['Mathematics', 'Ellipse', 'JEE Main', 'JEE Main 2021 (27 Jul Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
let-e-be-an-ellipse-whose-axes-are-parallel-to-the-coordinates-axes-having-its-centre-at-3-4-one-focus-at-4-4-and-one-vertex-at-5-4-if-m-x-y-4-m-0-is-

slug:
let-e-be-an-ellipse-whose-axes-are-parallel-to-the-coordinates-axes-having-its-centre-at-3-4-one-focus-at-4-4-and-one-vertex-at-5-4-if-m-x-y-4-m-0-is-67788

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>,</mo></math> one focus at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced></math> and one vertex at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>m</mi><mo>&gt;</mo><mn>0</mn></math> is a tangent to the ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>,</mo></math> then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msup><mi>m</mi><mn>2</mn></msup></math> is equal to _____ .</div>

answer:
<div class="solution"><p>We know that, If the standard ellipse is shifted to have center <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced></math>, then it's equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p>The center is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mfenced><mrow><mi>h</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced></math><br/>The vertices on the major axis are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mi>h</mi><mo>+</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mo>'</mo></msup><mfenced><mrow><mi>h</mi><mo>-</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced></math><br/>The foci are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mfenced><mrow><mi>h</mi><mo>+</mo><mi>a</mi><mi>e</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mo>'</mo></msup><mfenced><mrow><mi>h</mi><mo>-</mo><mi>a</mi><mi>e</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced></math></p><p>The eccentricity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><msqrt><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><msup><mi>a</mi><mn>2</mn></msup></mfrac></msqrt></math></p><p>Given <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mfenced><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>,</mo><mi>S</mi><mfenced><mrow><mn>4</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mn>5</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced></math></p><p>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>+</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>+</mo><mi>a</mi><mi>e</mi><mo>,</mo><mo> </mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/0323cc0c-5292-493b-894c-5a34c1360e5e-image.png" style="width: 282px; height: 164px;"/></p><p>Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn><mo> </mo><mo>&amp;</mo><mi>a</mi><mi>e</mi><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>e</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p>We know that, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>e</mi><mn>2</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>e</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup><mo>-</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn></math></p><p>So, equation of ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mn>4</mn></mfrac><mo>+</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mn>3</mn></mfrac><mo>=</mo><mn>1</mn></math></p><p>Intersecting with given tangent</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mi>m</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>3</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mfenced><mo>+</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>27</mn><mo>+</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced><mrow><mn>3</mn><mo>+</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>15</mn><mo>=</mo><mn>0</mn></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mn>0</mn></math> (as it is tangent)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mn>18</mn></mfenced><mn>2</mn></msup><mo>=</mo><mn>4</mn><mfenced><mrow><mn>3</mn><mo>+</mo><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced><mo>×</mo><mn>15</mn></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msup><mi>m</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><mo>.</mo></math></p></div>


tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-e-be-the-ellipe-9-x-2-4-y-2-1-and-c-be-the-circle-x-2-y-2-9-let-p-1-2-and-q-2-1-which-one-of-the-following

slug:
let-e-be-the-ellipe-9-x-2-4-y-2-1-and-c-be-the-circle-x-2-y-2-9-let-p-1-2-and-q-2-1-which-one-of-the-following-97986

question:
<div class="question">Let E be the ellipe $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and $C$ be the circle $x^{2}+y^{2}=9$. Let $P=(1,2)$ and $Q=(2,1) .$ Which one of the following</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$P$ lies inside $C$ but outside $E$.</span> </div>

answer:
<div class="solution">Given equation of ellipse $\mathrm{E}$ is $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$<br/>$\Rightarrow \frac{4 x^{2}+9 y^{2}}{36}=1 \Rightarrow 4 x^{2}+9 y^{2}=36$<br/>$\Rightarrow 4 x^{2}+9 y^{2}-36=0$ $\ldots(1)$<br/>and $C:$ Eqn of circle is $x^{2}+y^{2}=9$ Which can be rewritten as<br/>$x^{2}+y^{2}-9=0$ $\ldots(2)$<br/>For a point $P(1,2)$ we have<br/>$4(1)^{2}+9(2)^{2}-36=40-36&gt;0 \quad[$ from $(1)]$<br/>and $1^{2}+2^{2}-9=5-9   &lt; 0 \quad[$ from $(2)]$<br/>Point $P$ lies outside of $E$ and inside of $C$.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-6-6-r-be-defined-by-f-x-x-2-3-consider-the-following-2017-i-1-f-f-f-1-f-f-f-1-2-f-f-f-1-4-f-f-f-1-f-f-0-which-of-the-above-isare-correct

slug:
let-f-6-6-r-be-defined-by-f-x-x-2-3-consider-the-following-2017-i-1-f-f-f-1-f-f-f-1-2-f-f-f-1-4-f-f-f-1-f-f-0-which-of-the-above-isare-correct-85822

question:
<div class="question">Let $f:[-6,6] \rightarrow \mathrm{R}$ be defined by $f(\mathrm{x})=\mathrm{x}^{2}-3$. Consider the $\begin{array}{ll}\text { following : } &amp; {[2017-I]}\end{array}$<br/>1- $\left(f^{\circ} f^{\circ} f\right)(-1)=\left(f^{\circ} f^{\circ} f\right)(1)$<br/>2- $\left(f^{\circ} f^{\circ} f\right)(-1)-4\left(f^{\circ} f^{\circ} f\right)(1)=\left(f^{\circ} f\right)(0)$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-3$<br/>$\operatorname{fof}(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x}))=\mathrm{f}\left(\mathrm{x}^{2}-3\right)=\left(2^{2}-3\right)^{2}-3$<br/>$=2^{4}-6 \mathrm{x}^{2}+9-3=\mathrm{x}^{4}-6 \mathrm{x}^{2}+6$<br/>$\mathrm{fof}(0)=0-0+6=6$<br/>$\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))=\mathrm{f}\left(\mathrm{x}^{4}-6 \mathrm{x}^{2}+6\right)=\left(\mathrm{x}^{4}-6 \mathrm{x}^{2}+6\right)^{2}-3$<br/>$\operatorname{fofof}(1)=\left((-1)^{4}-6(-1)^{2}+6\right)^{2}-3$<br/>$=(1-6+6)-3=-2$<br/>$\quad \mathrm{fofof}(1)=\left(1^{4}-6(1)^{2}+6\right)^{2}-3$<br/>$=(1-6+6)-3=-2$</div>


title: let-f-100-100-1-1-be-defined-by-f-sin-then-what-is-the-number-of-values-of-100-1000-for-which-f-0
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1101</span> </div>
source_db: MarksBatch2_P1.db
slug:
let-f-100-100-1-1-be-defined-by-f-sin-then-what-is-the-number-of-values-of-100-1000-for-which-f-0-47625

question:
<div class="question">Let f $[-100 \pi, 100 \pi] \rightarrow[-1,1]$ be defined by $f(\theta)=\sin \theta$.<br/>Then what is the number of values of $\theta \in[-100 \pi, 1000 \pi]$ for which $\mathrm{f}(\theta)=0$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1000</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1101</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1100</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1110</span> </li> </ul>

answer:
<div class="solution">$f(\theta)=0$, if $\theta$ is an integral multiple of $\pi$. From $-100$ m $\operatorname{to} 0 \pi$ there are 101 values of $\theta$ for which $f(\theta)=0$. From $\pi$ to $1000 \pi$, there are 1000 values for which $\mathrm{f}(\theta)$ $=0$<br/>so, total values number is $101+1000=1101$</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-a-a-1-a-1-2017-i-consider-the-following-1-f-2-a-f-a-1-2-f-a-1-f-a-which-of-the-above-isare-correct

slug:
let-f-a-a-1-a-1-2017-i-consider-the-following-1-f-2-a-f-a-1-2-f-a-1-f-a-which-of-the-above-isare-correct-43373

question:
<div class="question">Let $f(\mathrm{a})=\frac{\mathrm{a}-1}{\mathrm{a}+1} . \quad[2017-\mathrm{I}]$<br/>Consider the following :<br/>1- $ f(2 \mathrm{a})=f(\mathrm{a})+1$<br/>2- $f\left(\frac{1}{\mathrm{a}}\right)=-f(\mathrm{a})$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{a})=\frac{\mathrm{a}-1}{\mathrm{a}+1}$<br/>$\mathrm{f}(2 \mathrm{a})=\frac{2 \mathrm{a}-1}{2 \mathrm{a}+1}$<br/>$\mathrm{f}(\mathrm{a})+1=\frac{\mathrm{a}-1}{\mathrm{a}+1}+1=\frac{2 \mathrm{a}}{\mathrm{a}+1}$<br/>So, $\mathrm{f}(2 \mathrm{a}) \neq \mathrm{f}(\mathrm{a})+1$<br/>Now, $f\left(\frac{1}{a}\right)=\frac{\frac{1}{a}-1}{\frac{l}{a}+1}=\frac{1-a}{1+a}=-\left(\frac{a-1}{a+1}\right)=-f(a)$</div>


title: let-f-a-r-where-a-r-0-is-such-that-f-x-x-x-x-on-which-one-of-the-following-sets-is-f-x-continuous
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">A</span> </div>
source_db: MarksBatch2_P1.db
slug:
let-f-a-r-where-a-r-0-is-such-that-f-x-x-x-x-on-which-one-of-the-following-sets-is-f-x-continuous-92506

question:
<div class="question">Let $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{R}$, where $\mathrm{A}=\mathrm{R} \backslash\{0\}$ is such that $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}+|\mathrm{x}|}{\mathrm{x}}$.<br/>On which one of the following sets is $\mathrm{f}(\mathrm{x})$ continuous?</div>

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

answer:
<div class="solution">For $x \geq 0$<br/>$f(x)=\frac{x+x}{x}=2$<br/>For $x   &lt; 0$<br/>$f(x)=\frac{x-x}{x}=0$<br/>$\lim _{x \rightarrow 0^{+}} f(x)=2$<br/>$\lim _{x \rightarrow 0^{-}} f(x)=0$<br/>$f(0)=2$<br/>$\Rightarrow$ It is discontinuous at $x=0$. Option (a) is correct.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-be-a-function-from-the-set-of-natural-numbers-to-the-set-of-even-natural-numbers-given-by-f-x-2-x-then-f-is

slug:
let-f-be-a-function-from-the-set-of-natural-numbers-to-the-set-of-even-natural-numbers-given-by-f-x-2-x-then-f-is-55581

question:
<div class="question">Let $f$ be a function from the set of natural numbers to the set of even natural numbers given by $f(x)=2 x$. Then $f$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">both one-one and onto</span> </div>

answer:
<div class="solution">$1 ..... 2$<br/>$2 ..... 4$<br/>$3 ..... 6$<br/>$4 ..... 8$<br/>Domain (x) $\quad$ Co-domain $(2 \mathrm{x})$ For each element in the domain there is only one element in the therefore $\mathrm{f}(\mathrm{x})$ is one-one. No image in the co-domain which has no pre-image in the domain, therefore $\mathrm{f}(\mathrm{x})$ is onto Hence $\mathrm{f}(\mathrm{x})$ both is one-one and onto.</div>


tags: ['Mathematics', 'Functions', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-i-x-s-i-n-2-p-i-x-for-i-12-3-p-i-n-it-is-given-that-the-fundamental-periods-of-f-1-x-f-2-x-f-3-x-f-1-x-f-2-x-and-f-1-x-f-3-x-are-2-and-3-respect

slug:
let-f-i-x-s-i-n-2-p-i-x-for-i-12-3-p-i-n-it-is-given-that-the-fundamental-periods-of-f-1-x-f-2-x-f-3-x-f-1-x-f-2-x-and-f-1-x-f-3-x-are-2-and-3-respect-29877

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>i</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mn>2</mn><msub><mi>p</mi><mi>i</mi></msub><mi>x</mi></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo> </mo><mo>&amp;</mo><mo> </mo><msub><mi>p</mi><mi>i</mi></msub><mo>â</mo><mi>N</mi></math>. It is given that the fundamental periods of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mfenced separators="|"><mi>x</mi></mfenced><mo>+</mo><msub><mi>f</mi><mn>2</mn></msub><mfenced separators="|"><mi>x</mi></mfenced><mo>+</mo><msub><mi>f</mi><mn>3</mn></msub><mfenced separators="|"><mi>x</mi></mfenced><mo>,</mo><mo> </mo><msub><mi>f</mi><mn>1</mn></msub><mfenced separators="|"><mi>x</mi></mfenced><mo>+</mo><msub><mi>f</mi><mn>2</mn></msub><mfenced separators="|"><mi>x</mi></mfenced></math> and <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></math> are <math><mi>Ï</mi><mo>,</mo><mfrac><mrow><mi>Ï</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math> and <math><mfrac><mrow><mi>Ï</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math> respectively, then the minimum value of <math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub></math> is</div>

answer:
<div class="solution"><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mn>2</mn></mrow></mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi></math><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mn>2</mn></mrow></mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>x</mi></math><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mn>2</mn></mrow></mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mi>x</mi></math><br/>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï</mi><mo>=</mo><mi>L</mi><mi>C</mi><mi>M</mi><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>2</mn></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>3</mn></msub></mrow></mfrac></mrow></mfenced></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi>H</mi><mi>C</mi><mi>F</mi><mfenced separators="|"><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><msub><mi>p</mi><mn>3</mn></msub></mrow></mfenced><mo>=</mo><mn>1</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>Ï</mi><mn>2</mn></mfrac><mo>=</mo><mi>L</mi><mi>C</mi><mi>M</mi><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>2</mn></msub></mrow></mfrac></mrow></mfenced></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi>H</mi><mi>C</mi><mi>F</mi><mfenced separators="|"><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mn>2</mn><mo>â</mo><mfenced close="" open="[" separators="|"><mtable columnalign="left"><mtr><mtd><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi></mtd></mtr><mtr><mtd><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>b</mi></mtd></mtr></mtable></mfenced></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>Ï</mi><mn>3</mn></mfrac><mo>=</mo><mi>L</mi><mi>C</mi><mi>M</mi><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>1</mn></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>2</mn><mi>Ï</mi></mrow><mrow><mn>2</mn><msub><mi>p</mi><mn>3</mn></msub></mrow></mfrac></mrow></mfenced></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi>H</mi><mi>C</mi><mi>F</mi><mfenced separators="|"><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>3</mn></msub></mrow></mfenced><mo>=</mo><mn>3</mn><mo>â</mo><mtable columnalign="left"><mtr><mtd><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mi>x</mi></mtd></mtr><mtr><mtd><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mn>3</mn><mi>y</mi></mtd></mtr></mtable></math><br/>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mn>6</mn><mo>,</mo><mo> </mo><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mn>3</mn></math> (for minimum value)</div>


title: let-f-n-4-1-1000-n-where-x-denote-the-integral-part-of-x-then-the-value-of-f-n-is
slug: let-f-n-4-1-1000-n-where-x-denote-the-integral-part-of-x-then-the-value-of-f-n-is-25385
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">251</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(\mathrm{n})=\left[\frac{1}{4}+\frac{\mathrm{n}}{1000}\right]$, where $[\mathrm{x}]$ denote the integral part<br/>of $\mathrm{x}$. Then the value of $\sum f(\mathrm{n})$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">251</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">250</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">$\mathrm{f}(\mathrm{n})=\left[\frac{1}{4}+\frac{\mathrm{n}}{1000}\right]$<br/>$\sum_{\mathrm{n}=1}^{1000} \mathrm{f}(\mathrm{n})=\left[\frac{1}{4}+\frac{1}{1000}\right]+\left[\frac{1}{4}+\frac{2}{1000}\right]+\ldots .+\left[\frac{1}{4}+\frac{1000}{1000}\right]$<br/>$=[0.25+0.001]+[0.25+0.002]+\ldots .+[0.25+1]$<br/>We get ' 0 ' for all values of $\mathrm{n}$ from 1 to 750 . From $\mathrm{n}=750$, we get all the values as 1 . So,<br/>$\begin{array}{rl}\sum_{n=1}^{1000} &amp; f(n)=0+0+0+\ldots .+\left[\frac{1}{4}+\frac{750}{1000}\right]+\left[\frac{1}{4}+\frac{751}{1000}\right]+\ldots .[1.25] \\ &amp; =1+1+1+\ldots .(251 \text { times }) \\ &amp; =251 .\end{array}$</div>


tags: ['Mathematics', 'Differential Equations', 'KCET', 'KCET 2014']
source_db: MarksBatch2_P1.db
title:
let-f-n-rightarrow-n-defined-by-fnleftbeginarrayclfracn12-text-if-n-text-is-odd-fracn2-text-if-n-text-is-even-endarrayright-then-mathrmf-is

slug:
let-f-n-rightarrow-n-defined-by-fnleftbeginarrayclfracn12-text-if-n-text-is-odd-fracn2-text-if-n-text-is-even-endarrayright-then-mathrmf-is-83425

question:
<div class="question">Let \( f: N \rightarrow N \) defined by \( f(n)=\left\{\begin{array}{cl}\frac{n+1}{2} ; &amp; \text { if } n \text { is odd } \\ \frac{n}{2} ; &amp; \text { if } n \text { is even }\end{array}\right. \)<br/>then \( \mathrm{f} \) is</div>

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

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

answer:
<div class="solution">Given that, \( f: N \rightarrow N \) defined by<br/>\( f(n)=\left\{\begin{array}{cc}\frac{n+1}{2} &amp; n \text { is odd } \\ \frac{n}{2} &amp; n \text { is even }\end{array}\right. \)<br/>For \( n=1 \), we have \( f(1)=\frac{1+1}{2}=1 \)<br/>and, if \( n=2 \), we have \( f(2)=\frac{2}{2}=1 \)<br/>So, \( f(1)=f(2) \) but \( 1 \neq 2 \). Therefore, \( f(x) \) is not one-one.<br/>Now, \( f(x)=\frac{n+1}{2} \) if \( \mathrm{n} \) is odd<br/>if \( y=\frac{n+1}{2} \), then \( n=2 y-1, \forall y \)<br/>Also, \( f(x)=\frac{n}{2} \) if \( n \) is even. That is,<br/>\( y=\frac{n}{2} \) or \( n=2 y, \forall y \)<br/>Therefore, \( f(x) \) is onto.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-r-r-be-a-function-defined-as-f-x-x-x-for-each-x-r-r-being-the-set-of-real-numbers-which-one-of-the-following-is-correct

slug:
let-f-r-r-be-a-function-defined-as-f-x-x-x-for-each-x-r-r-being-the-set-of-real-numbers-which-one-of-the-following-is-correct-39505

question:
<div class="question">Let $f: R \rightarrow R$ be a function defined as $f(x)=x|x|$; for each $x \in R, R$ being the set of real numbers. Which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ is both one-one and onto</span> </div>

answer:
<div class="solution">Given, $f(x)=x|x|$<br/>If $\quad f\left(x_{1}\right)=f\left(x_{2}\right)$<br/>$\Rightarrow x_{1}\left|x_{1}\right|=x_{2}\left|x_{2}\right|$<br/>$\Rightarrow x_{1}=x_{2}$<br/>$\therefore f(x)$ is one-one.<br/>Also, range of $f(x)=$ co-domain of $f(x)$ <br/>$\therefore f(x)$ is onto.<br/>Hence, $f(x)$ is both one-one and onto.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-consider-the-following-1-f-2-f-1-f-0-2-f-2-2-f-1-12-which-of-the-above-isare-correc

slug:
let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-consider-the-following-1-f-2-f-1-f-0-2-f-2-2-f-1-12-which-of-the-above-isare-correc-21926

question:
<div class="question">Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that<br/>$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2} \mathrm{f}^{\prime}(1)+\mathrm{xf}^{\prime}(2)+\mathrm{f}^{\prime \prime}(3)$<br/>for $\mathrm{x} \in \mathbf{R}$<br/>Consider the following:<br/>$1 \quad f(2)=f(1)-f(0)$<br/>$2 \quad \mathrm{f}^{\prime \prime}(2)-2 \mathrm{f}^{\prime}(1)=12$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">1 $f(1)-f(0)=4-6$<br/>$=-2$<br/>$f(2)=8-20+4+6=-2$<br/>$\quad$ Hence $f(2)=f(1)-f(0)$<br/>$\therefore$ Statement $(1)$ is correct.<br/>2 $f^{\prime \prime}(2)-2 f^{\prime}(1)=2-2(-5)$<br/>$f^{\prime \prime}(2)-2 \mathrm{f}^{\prime}(1)=12$<br/>$\therefore$ Statement $(2)$ is correct.</div>


title: let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-what-is-f-1-equal-to-1
slug: let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-what-is-f-1-equal-to-1-49426
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that<br/>$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2} \mathrm{f}^{\prime}(1)+\mathrm{xf}^{\prime}(2)+\mathrm{f}^{\prime \prime}(3)$<br/>for $\mathrm{x} \in \mathbf{R}$ <br/>What is $\mathrm{f}(1)$ equal to?</div>

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

answer:
<div class="solution">$f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$ ...(1)<br/>$f^{\prime}(x)=3 x^{2}+2 x f^{\prime}(1)+f^{\prime \prime}(2)$ ...(2)<br/>$f^{\prime \prime}(x)=6 x+2 f^{\prime}(1)$ ...(3)<br/>$f^{\prime \prime \prime}(x)=6$ ...(4)<br/>$f^{\prime}(1)=3+2 f^{\prime}(1)+f^{\prime \prime}(2)$ ...(5)<br/>$f^{\prime}(2)=12+2 f^{\prime}(1)$ ...(6)<br/>Using $(6)$ in $(5)$, we get <br/>$f^{\prime}(1)=3+2 f^{\prime}(1)+12+2 f^{\prime \prime}(1)$<br/>$-3 f^{\prime}(1)=15$<br/>$f^{\prime}(1)=-5$<br/>Using this value in eqn (6) we get <br/>$f^{\prime}(2)=12+2 \times(-5)$<br/>$f^{\prime \prime}(2)=2$<br/>Using $x=3$ in eqn (4), <br/>$f^{\prime \prime \prime}(3)=6$<br/>Putting value of $f^{\prime}(1)+f^{\prime \prime}(2)$ and $\mathrm{f}^{\prime \prime \prime}(3)$ in eqn (1) <br/>We get <br/>$\begin{aligned} f(x) &amp;=x^{3}+x^{2}(-5)+x(2)+6 \\ &amp;=x^{3}-5 x^{2}+2 x+6 \end{aligned}$<br/>Putting $\mathrm{x}=1$ $f(1)=(1)^{3}-5(1)^{2}+2(1)+6$<br/>$f(1)=4$</div>


title: let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-what-is-f-1-equal-to
slug: let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-what-is-f-1-equal-to-43675
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-5$</span> </div>
answer: <div class="solution">$\mathrm{f}^{\prime}(1)=-5$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that<br/>$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2} \mathrm{f}^{\prime}(1)+\mathrm{xf}^{\prime}(2)+\mathrm{f}^{\prime \prime}(3)$<br/>for $\mathrm{x} \in \mathbf{R}$<br/>What is $\mathrm{f}^{\prime}(1)$ equal to?</div>

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


title: let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-what-is-f-10-equal-to
slug: let-f-r-r-be-a-function-such-that-f-x-x-3-x-2-f-1-xf-2-f-3-for-x-r-what-is-f-10-equal-to-26032
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
answer: <div class="solution">$f^{\prime \prime \prime}(10)=6$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that<br/>$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2} \mathrm{f}^{\prime}(1)+\mathrm{xf}^{\prime}(2)+\mathrm{f}^{\prime \prime}(3)$<br/>for $\mathrm{x} \in \mathbf{R}$<br/>What is $\mathrm{f}^{\prime \prime}(10)$ 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">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">8</span> </li> </ul>


title: let-f-r-r-be-a-function-whose-inverse-is-3-x-5-what-is-f-x-equal-to-2012-ii
slug: let-f-r-r-be-a-function-whose-inverse-is-3-x-5-what-is-f-x-equal-to-2012-ii-74792
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function whose inverse is $\frac{\mathrm{x}+5}{3}$. What is<br/>$\begin{array}{ll}\mathrm{f}(\mathrm{x}) \text { equal to } ? &amp; {[2012-\mathrm{II}]}\end{array}$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}(\mathrm{x})=3 \mathrm{x}-5$</span> </div>

answer:
<div class="solution">Let $\mathrm{y}=\mathrm{f}^{-1}(\mathrm{x})=\frac{\mathrm{x}+5}{3}$<br/>$\Rightarrow 3 y-5=x$<br/>Now, $y=f^{-1}(x)$<br/>$\Rightarrow x=f(y)$<br/>$\Rightarrow x=3 y-5$<br/>Hence, $f(x)=3 x-5$</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-r-r-be-defined-as-f-x-ax-2-bx-c-a-b-c-being-fixed-nonzero-real-numbers-which-one-of-the-following-statements-is-correct-in-general

slug:
let-f-r-r-be-defined-as-f-x-ax-2-bx-c-a-b-c-being-fixed-nonzero-real-numbers-which-one-of-the-following-statements-is-correct-in-general-42717

question:
<div class="question">Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}, \mathrm{a}, \mathrm{b}, \mathrm{c}$ being fixed<br/>non-zero real numbers. Which one of the following statements is correct, in general?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">If $b^{2}-4 a c&gt;0$, then $f^{-1}(0)$ does not contain 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">If b $^{2}-4$ ac $   &lt; 0$, then $\mathrm{f}^{-1}(0)$ must contain 0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">If $b^{2}-4 a c&gt;0$, then $f^{-1}(0)$ may contain 0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">If b $^{2}-4 a c   &lt; 0$, then $\mathrm{f}^{-1}(0)$ may contain 0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">If $b^{2}-4 a c&gt;0$, then $f^{-1}(0)$ does not contain 0</span> </div>

answer:
<div class="solution">As given $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$, Let $\mathrm{y}=\mathrm{f}(\mathrm{x})$. To get $\mathrm{f}^{-1}(\mathrm{x})$<br/>we express $x$ is terms of y in equation $y=a x^{2}+b x+c$.<br/>$\frac{y}{a}=x^{2}+\frac{b}{a} x+\frac{c}{a}=\left(x+\frac{b}{2 a}\right)^{2}-\left(\frac{b}{2 a}\right)^{2}+\frac{c}{a}$<br/>$\Rightarrow x+\frac{b}{2 a}=\sqrt{\frac{y}{a}+\frac{b^{2}}{4 a^{2}}-\frac{c}{a}}=\frac{\pm \sqrt{4 a y+b^{2}-4 a c}}{2 a}$<br/>$x=\frac{-b \pm \sqrt{4 a y+b^{2}-4 a c}}{2 a}$<br/>or, $f^{1}(x)=\frac{-b \pm \sqrt{4 a x+b^{2}-4 a c}}{2 a}$<br/>Putting $\mathrm{x}=0$<br/>$\mathrm{f}^{-1}(0)=\frac{-\mathrm{b} \pm \sqrt{0+\mathrm{b}^{2}-4 \mathrm{ac}}}{2 \mathrm{a}} \neq 0 \quad\left[\right.$ if $\left.\mathrm{b}^{2}-4 \mathrm{ac}&gt;0\right]$<br/>so, $\mathrm{f}^{-1}(0) \neq 0$. i.e. if $b^{2}-4 a c&gt;0$<br/>$\mathrm{f}^{-1}(0)$ does not contain 0, if $b^{2}-4 a c&gt;0 .$</div>


title: let-f-r-r-be-defined-as-f-x-sin-x-which-one-of-the-following-is-correct
slug: let-f-r-r-be-defined-as-f-x-sin-x-which-one-of-the-following-is-correct-54290
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $\mathrm{f}(\mathrm{x})=\sin (|\mathrm{x}|)$ <br/>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">fis not differentiable only at 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">fis differentiable at 0 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">fis differentiable everywhere</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">fis non-differentiable at many points</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">fis not differentiable only at 0</span> </div>

answer:
<div class="solution">Given function is $: \mathrm{f}(\mathrm{x})=\sin |\mathrm{x}|$<br/>$=\left\{\begin{array}{ll}\sin (\mathrm{x}), &amp; \mathrm{x} \geq 0 \\ \sin (-\mathrm{x}), &amp; \mathrm{x}   &lt; 0\end{array}\right.$<br/>$=\left\{\begin{array}{cl}\sin \mathrm{x}, &amp; \mathrm{x} \geq 0 \\ -\sin \mathrm{x}, &amp; \mathrm{x} &lt; 0\end{array}\right.$<br/>$\mathrm{LHD}$ at $\mathrm{x}=0=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0-\mathrm{h})-\mathrm{f}(0)}{0-\mathrm{h}-0}$<br/>$=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0} \frac{-\sin (-h)-0}{-h}=-1$<br/>$\mathrm{RHD}$ at $\mathrm{x}=0=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{0+\mathrm{h}-0}$<br/>$=\lim _{h \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{\sin (\mathrm{h}-0)}{\mathrm{h}}=1$<br/>$\mathrm{LHD} \neq \mathrm{RHD}$<br/>$\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=0$.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main', 'JEE Main 2021 (22 Jul Shift 1)']
source_db: MarksBatch2_P1.db
title:
let-f-r-r-be-defined-as-f-x-x-3-1-cos-2-x-2-log-e-1-2-x-e-2-x-1-x-e-x-2-x-0-x-0-if-f-is-continuous-at-x-0-then-is-equal-to

slug:
let-f-r-r-be-defined-as-f-x-x-3-1-cos-2-x-2-log-e-1-2-x-e-2-x-1-x-e-x-2-x-0-x-0-if-f-is-continuous-at-x-0-then-is-equal-to-84115

question:
<div class="question"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math> be defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{"><mtable columnalign="center left"><mtr><mtd><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mn>2</mn><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac><msub><mi>log</mi><mi>e</mi></msub><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mfrac></mfenced></mtd><mtd><mo>,</mo><mo>  </mo><mi>x</mi><mo>≠</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>α</mi></mtd><mtd><mo>,</mo><mo> </mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> is equal to:</p></div>

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

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

answer:
<div class="solution"><p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi><mo>,</mo></math> if <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>a</mi></mfenced><mo>.</mo></math></p><p>Thus, for continuity of the given function, we have</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mn>2</mn><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac><msub><mi>log</mi><mi>e</mi></msub><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mfrac></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math></p><p>Using, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi><mo>,</mo><mo> </mo><msub><mi>log</mi><mi>e</mi></msub><mfenced><mfrac><mi>m</mi><mi>n</mi></mfrac></mfenced><mo>=</mo><msub><mi>log</mi><mi>e</mi></msub><mi>m</mi><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><mi>n</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&amp;</mo><mo> </mo><msub><mi>log</mi><mi>e</mi></msub><msup><mi>m</mi><mi>n</mi></msup><mo>=</mo><mi>n</mi><msub><mi>log</mi><mi>e</mi></msub><mi>m</mi><mo>,</mo></math></p><p>we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><msup><mi>x</mi><mn>3</mn></msup><msup><mfenced><mrow><mn>2</mn><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mfenced><mrow><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow></mfenced><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>4</mn><msup><mi>sin</mi><mn>4</mn></msup><mi>x</mi></mrow></mfrac><mfenced><mrow><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mi>x</mi><msup><mi>sin</mi><mn>4</mn></msup><mi>x</mi></mrow></mfrac><mfenced><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup><mo>·</mo><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>·</mo><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math></p><p>Now, using the standard limits <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>sin</mi><mi>x</mi></mrow><mi>x</mi></mfrac></mfenced><mo>=</mo><mn>1</mn><mo> </mo><mo>&amp;</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow><mi>x</mi></mfrac></mfenced><mo>=</mo><mn>1</mn><mo>,</mo></math> we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi>x</mi></mrow></mfrac><mfenced><mrow><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>4</mn><mi>x</mi></mrow></mfrac><mfenced><mrow><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mn>2</mn></mfenced><mo>=</mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>α</mi><mo>=</mo><mn>1</mn><mo>.</mo></math></p></div>


title: let-f-r-r-be-defined-by-f-x-x-x-x-0-f-0-2-what-is-the-range-of-f
slug: let-f-r-r-be-defined-by-f-x-x-x-x-0-f-0-2-what-is-the-range-of-f-86247
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be defined by $f(x)=|x| / x, x \neq 0, f(0)=2$ What is the range of $f$ ?</div>

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

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

answer:
<div class="solution">Let $f: R \rightarrow R$ be defined as $f(x)=\frac{|x|}{x}, x \neq 0$<br/>Also, $f(0)=2$ ie. value of function at $x=0$ is 2 .<br/>Consider, $f(x)=\left\{\begin{array}{l}\frac{x}{x}=1 \text { if } x&gt;0 \\ \frac{-x}{x}=-1 \text { if } x   &lt; 0\end{array}\right.$<br/>because we have $\mid x \vDash\left\{\begin{array}{ll}x, &amp; x \geq 0 \\ -x, &amp; x &lt; 0\end{array}\right.$<br/>Thus, Range of $f(x)=\{1,-1,2\}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'AP EAMCET', 'AP EAMCET 2019 (20 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
let-f-r-rightarrow-r-be-the-function-defined-by-fxleftbeginarraycl5-text-if-x-leq-1-ab-x-text-if-1-x-3-b5-x-text-if-3-leq-x-5-30-text-if-x-geq-5endarr

slug:
let-f-r-rightarrow-r-be-the-function-defined-by-fxleftbeginarraycl5-text-if-x-leq-1-ab-x-text-if-1-x-3-b5-x-text-if-3-leq-x-5-30-text-if-x-geq-5endarr-19148

question:
<div class="question">Let \(f: R \rightarrow R\) be the function defined by \(f(x)=\left\{\begin{array}{cl}5, &amp; \text { if } x \leq 1 \\ a+b x, &amp; \text { if } 1  &lt; x &lt; 3 \\ b+5 x, &amp; \text { if } 3 \leq x &lt; 5 \\ 30, &amp; \text { if } x \geq 5\end{array}\right.\) then \(f\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">continuous if \(a=5\) and \(b=5\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">continuous if \(a=0\) and \(b=5\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">continuous if \(a=-5\) and \(b=10\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">not continuous for any values of \(a\) and \(b\)</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">not continuous for any values of \(a\) and \(b\)</span> </div>

answer:
<div class="solution">Given function \(f: R \rightarrow R\), such that<br/>\(f(x)=\left[\begin{array}{cl}<br/>5, &amp; \text { if } x \leq 1 \\<br/>a+b x, &amp; \text { if } 1  &lt; x &lt; 3 \\<br/>b+5 x, &amp; \text { if } 3 \leq x &lt; 5 \\<br/>30, &amp; \text { if } x \geq 5<br/>\end{array}\right.\)<br/>If \(f\) is continuous at \(x=1\), then<br/>\(a+b=5\) ...(i)<br/>If \(f\) is continuous at \(x=3\), then<br/>\(\Rightarrow \quad \begin{gathered}<br/>a+3 b=b+15 \\<br/>a+2 b=15 \quad \ldots (ii)<br/>\end{gathered}\)<br/>and if \(f\) is continuous at \(x=5\), then<br/>\(b+25=30 \Rightarrow b=5\) ...(iii)<br/>From Eqs. (ii) and (iii), we get<br/>\(a=5\) ...(iv)<br/>but \(a=5\) and \(b=5\) doesn't satisfy the Eq. (i). so, \(f: R \rightarrow R\) is not continuous for any values of \(a\) and \(b\).<br/>Hence, option (4) is correct.</div>


title: let-f-x-1-1-x-1-then-what-is-lim-x-0-f-x-equal-to
slug: let-f-x-1-1-x-1-then-what-is-lim-x-0-f-x-equal-to-97154
tags: ['Mathematics', 'Limits', '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">Let $f(x)=\frac{1}{1-|1-x|} .$ Then, what is $\lim _{x \rightarrow 0} f(x)$ equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">of</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</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \text { LHL }=\lim _{h \rightarrow 0} \frac{1}{1-|1-(1-h)|} \\ &amp;=\lim _{h \rightarrow 0} \frac{1}{1-|h|}=\lim _{h \rightarrow 0} \frac{1}{1-h}=1 \\ &amp; \text { RHL }=\lim _{h \rightarrow 0} \frac{1}{1-|1-(1+h)|} \\ &amp;=\lim _{h \rightarrow 0} \frac{1}{1-|-h|}=\lim _{h \rightarrow 0} \frac{1}{1-h}=1 \\ &amp; \therefore \lim _{x \rightarrow 0} f(x)=1 \end{aligned}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (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:
let-f-x-2-x-2-3-x-0-0-x-3-and-g-x-f-x-f-x-what-is-the-value-of-the-differential-coefficient-of-g-x-at-x-2

slug:
let-f-x-2-x-2-3-x-0-0-x-3-and-g-x-f-x-f-x-what-is-the-value-of-the-differential-coefficient-of-g-x-at-x-2-63337

question:
<div class="question">Let $f(x)=\left\{\begin{array}{ll}-2, &amp; -3 \leq x \leq 0 \\ x-2, &amp; 0   &lt; x \leq 3\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$<br/>What is the value of the differential coefficient of $g(x)$ at $x=-2$ ?</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">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">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">For $x=-2$ $g(x)=-2+|-2|=-2+2$<br/>$\Rightarrow g(x)=0$<br/>$\Rightarrow$ differential coefficient at $x=-2$ is given as :<br/>$g^{\prime}(x)=\lim _{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}=\lim _{h \rightarrow 0} \frac{0-0}{4}=0 .$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-2-x-2-3-x-0-0-x-3-and-g-x-f-x-f-x-which-of-the-following-statements-isare-correct-1-g-x-is-differentiable-at-x-0-2-g-x-is-differentiable-at-x-

slug:
let-f-x-2-x-2-3-x-0-0-x-3-and-g-x-f-x-f-x-which-of-the-following-statements-isare-correct-1-g-x-is-differentiable-at-x-0-2-g-x-is-differentiable-at-x-31793

question:
<div class="question">Let $f(x)=\left\{\begin{array}{ll}-2, &amp; -3 \leq x \leq 0 \\ x-2, &amp; 0   &lt; x \leq 3\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$ <br/>Which of the following statements is/are correct?<br/>1- $\mathrm{g}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$.<br/>2- $\mathrm{g}(\mathrm{x})$ is differentiable at $\mathrm{x}=2$. Select the correct answer using the code given below:</div>

answer:
<div class="solution">$f(x)=\left\{\begin{array}{ll}-2, &amp; -3 \leq x \leq 0 \\ x-2, &amp; 0   &lt; x \leq 3 \text { and }\end{array}\right.$<br/>$g(x)=f(|x|)+|f(x)|$<br/>At $x=0$<br/>For LHD $: g(x)=-2+|-2|=-2+2=0 \Rightarrow g(x)=0$<br/>$\begin{array}{l}\text { LHD }=\lim _{x \rightarrow 0^{-}} \frac{g(x)-g(0)}{x-0}=\lim _{h \rightarrow 0} \frac{g(-h)-g(0)}{-h} \\ \quad= &amp; \lim _{h \rightarrow 0} \frac{0-0}{-h}=\lim _{h \rightarrow 0} 0 \\ \text { LHD }=0\end{array}$<br/>For RHD : $g(x)=|x|-2+|x-2|$<br/>$g(x)=x-2-(x-2) \quad x&gt;0$ (and just greater<br/>than zero)<br/>$g(x)=x-2-x+2=0$<br/>Now $g(x)$ is not continuous at $x=0$, hence $g(x)$ is not differentiable at $\mathrm{x}=0$ At $x=2$ For LHD :<br/>$g(x)=|x|-2+|x-2|=x-2-(x-2)$<br/>$\quad=x-2-x+2=0$<br/>$\therefore \mathrm{LHD}=\lim _{x \rightarrow 2^{-}} \frac{g(x)-g(2)}{x-2}=\lim _{x \rightarrow 2} \frac{0}{x-2}=0$<br/>For RHD:<br/>$g(x)=|x|-2+|x-2|=x-2+x-2=2 x-4$<br/>$\begin{aligned} \Rightarrow \mathrm{RHD} &amp;=\lim _{x \rightarrow 2^{+}} \frac{g(x)-g(2)}{x-2}=\lim _{x \rightarrow 2} \frac{2 x-4-2(2)+4}{x-2} \\ &amp;=\lim _{x \rightarrow 2} \frac{2(x-2)}{x-2}=2 \\ \Rightarrow \mathrm{LHD} &amp; \neq \mathrm{RHD} \end{aligned}$<br/>Thus $g(x)$ is not differentiable at $x=2$.</div>


title: let-f-x-3-x-4-2-x-0-x-2-2-x-9-if-f-is-continuous-at-x-2-then-what-is-the-value-of
slug: let-f-x-3-x-4-2-x-0-x-2-2-x-9-if-f-is-continuous-at-x-2-then-what-is-the-value-of-61826
tags: ['Mathematics', 'Continuity and Differentiability', '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
question:
<div class="question">Let $f(x)=\left\{\begin{array}{ll}3 x-4, &amp; 0 \leq x \leq 2 \\ 2 x+\ell, &amp; 2   &lt; x \leq 9\end{array}\right.$<br/>If $\mathrm{f}$ is continuous at $\mathrm{x}=2$, then what is the value of $\ell$ ?</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">-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">-1</span> </li> </ul>

answer:
<div class="solution">Given function is :<br/>$f(x)=\left\{\begin{array}{ll}3 x-4, &amp; 0 \leq x \leq 2 \\ 2 x+\ell, &amp; 2   &lt; x \leq 9\end{array}\right.$<br/>and also given that $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$ For a function to be continuous at a point $\mathrm{LHL}=\mathrm{RHL}$ $=\mathrm{V} . \mathrm{F}$. at that point. $\mathrm{f}(2)=2=\mathrm{V}$.F.<br/>$\Rightarrow \mathrm{RHL}: \lim _{\mathrm{x} \rightarrow 2}(2 \mathrm{x}+\ell)=3(2)-4$<br/>$\Rightarrow \lim _{\mathrm{h} \rightarrow 0}\{2(2+\mathrm{h})+\ell\}=6-4$<br/>$\Rightarrow 4+\ell=2$<br/>$\Rightarrow \ell=-2$</div>


title: let-f-x-18-x-2-1-what-is-the-value-of-lim-x-3-x-3-f-x-f-3
slug: let-f-x-18-x-2-1-what-is-the-value-of-lim-x-3-x-3-f-x-f-3-56966
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">$\quad \operatorname{Let} f(x)=\frac{1}{\sqrt{18-x^{2}}}$.<br/>What is the value of $\lim _{x \rightarrow 3} \frac{f(x)-f(3)}{x-3}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{9}$</span> </div>

answer:
<div class="solution">The given functions is $\mathrm{f}(\mathrm{x})=\frac{1}{\sqrt{18-\mathrm{x}^{2}}}$<br/>So, $\mathrm{f}(3)=\frac{1}{\sqrt{18-9}}=\frac{1}{3}$<br/>$\Rightarrow \lim _{x \rightarrow 3} \frac{f(x)-f(3)}{x-3}=\lim _{x \rightarrow 3} \frac{\frac{1}{\sqrt{18-x^{2}}}-\frac{1}{3}}{x-3}$<br/>Putting $\mathrm{x}=3$ makes the that of $\frac{0}{0}$ form<br/>$\lim _{x \rightarrow 3}-\frac{1}{2}\left(18-x^{2}\right)^{-3 / 2}(-2 x)$<br/>(Applying L' Hospital's Rule)<br/>$=-\frac{1}{2}(9)^{-3 / 2}(-2 \times 3)=\frac{1}{27} \times 3=\frac{1}{9}$</div>


title: let-f-x-a-x-2-b-x-c-such-that-f-1-f-1-and-a-b-c-are-in-arithmetic-progression-f-a-f-b-f-c-are-1
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">A.P.</span> </div>
source_db: MarksBatch2_P1.db
slug:
let-f-x-a-x-2-b-x-c-such-that-f-1-f-1-and-a-b-c-are-in-arithmetic-progression-f-a-f-b-f-c-are-1-23034

question:
<div class="question">Let $f(x)=a x^{2}+b x+c$ such that $f(1)=f(-1)$ and $a, b, c$ are in Arithmetic Progression.<br/>$\mathrm{f}^{\prime}(\mathrm{a}), \mathrm{f}^{\prime}(\mathrm{b}), \mathrm{f}^{\prime}(\mathrm{c})$ are</div>

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

answer:
<div class="solution">We have $\mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{ax}$<br/>$\therefore \quad \mathrm{f}^{\prime}(\mathrm{a})=2 \mathrm{a}^{2}, \mathrm{f}^{\prime}(\mathrm{b})=2 \mathrm{ab}=0$<br/>and $\mathrm{f}^{\prime}(\mathrm{c})=2 \mathrm{ac} \quad(\because \mathrm{b}=0)$<br/>$\therefore \quad \mathrm{f}^{\prime}(\mathrm{a})=2 \mathrm{a}^{2}$<br/>$\mathrm{f}^{\prime}(\mathrm{b})=0$<br/>and $\mathrm{f}^{\prime}(\mathrm{c})=-2 \mathrm{a}^{2} \quad(\because 2 \mathrm{~b}=\mathrm{a}+\mathrm{c} \Rightarrow \mathrm{c}=-\mathrm{a})$<br/>Hence $\mathrm{f}^{\prime}(\mathrm{a}), \mathrm{f}^{\prime}(\mathrm{b})$ and $\mathrm{f}^{\prime}(\mathrm{c})$ are in $\mathrm{AP}$.</div>


title: let-f-x-a-x-2-b-x-c-such-that-f-1-f-1-and-a-b-c-are-in-arithmetic-progression-f-a-f-b-f-c-are
slug: let-f-x-a-x-2-b-x-c-such-that-f-1-f-1-and-a-b-c-are-in-arithmetic-progression-f-a-f-b-f-c-are-63460
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=a x^{2}+b x+c$ such that $f(1)=f(-1)$ and $a, b, c$ are in Arithmetic Progression.<br/>$\mathrm{f}^{\prime \prime}(\mathrm{a}), \mathrm{f}^{\prime \prime}(\mathrm{b}), \mathrm{f}^{\prime \prime}(\mathrm{c})$ are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">in both A.P. and GP.</span> </div>

answer:
<div class="solution">$\mathrm{f}^{\prime \prime}(\mathrm{x})=2 \mathrm{a}$<br/>$\therefore \mathrm{f}^{\prime \prime}(\mathrm{a})=\mathrm{f}^{\prime \prime}(\mathrm{b})=\mathrm{f}^{\prime \prime}(\mathrm{c})$<br/>Hence, $\mathrm{f}^{\prime \prime}(\mathrm{a}), \mathrm{f}^{\prime \prime}(\mathrm{b})$ and $\mathrm{f}^{\prime \prime}(\mathrm{c})$ are in both $\mathrm{AP}$ and $\mathrm{GP}$.</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2014 (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:
let-f-x-a-x-2-b-x-c-such-that-f-1-f-1-and-a-b-c-are-in-arithmetic-progression-what-is-the-value-of-b

slug:
let-f-x-a-x-2-b-x-c-such-that-f-1-f-1-and-a-b-c-are-in-arithmetic-progression-what-is-the-value-of-b-43926

question:
<div class="question">Let $f(x)=a x^{2}+b x+c$ such that $f(1)=f(-1)$ and $a, b, c$ are in Arithmetic Progression.<br/>What is the value of b?</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">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">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cannot be determined deu to insufficient data</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c} \\ &amp; \therefore \mathrm{f}(1)=\mathrm{a}+\mathrm{b}+\mathrm{c} \\ &amp; \text { and } \mathrm{f}(-1)=\mathrm{a}-\mathrm{b}+\mathrm{c} \\ &amp; \therefore \mathrm{f}(1)=\mathrm{f}(-1) \\ &amp; \Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{a}-\mathrm{b}+\mathrm{c} \Rightarrow \mathrm{b}=0 \end{aligned}$</div>


title: let-f-x-be-a-function-such-that-f-x-1-x-3-f-x-0-what-is-1-1-f-x-dx-equal-to
slug: let-f-x-be-a-function-such-that-f-x-1-x-3-f-x-0-what-is-1-1-f-x-dx-equal-to-99639
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2016 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{f}(\mathrm{x})$ be a function such that $\mathrm{f}^{\prime}\left(\frac{1}{\mathrm{x}}\right)+\mathrm{x}^{3} \mathrm{f}^{\prime}(\mathrm{x})=0$, What is<br/>$\int_{-1}^{1} \mathrm{f}(\mathrm{x}) \mathrm{dx}$ equal to? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{f}(1)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{f}(-1)$</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">$4 \mathrm{f}(1)$</span> </li> </ul>

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


tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-x-be-an-indefinite-integral-of-sin-2-x-consider-the-following-statements-statement-1-the-function-f-x-satisfies-f-x-f-x-for-all-real-x-statement

slug:
let-f-x-be-an-indefinite-integral-of-sin-2-x-consider-the-following-statements-statement-1-the-function-f-x-satisfies-f-x-f-x-for-all-real-x-statement-37083

question:
<div class="question">Let $f(x)$ be an indefinite integral of $\sin ^{2} x .$ Consider the following statements:<br/>Statement $1:$ The function $f(x)$ satisfies $f(x+\pi)$ $=f(\mathrm{x})$ for all real $\mathrm{x}$. Statement $2: \sin ^{2}(x+\pi)=\sin ^{2} x$ for all real $x$. Which one of the following is correct in respect of the above statements?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Both the statements are true and Statement 2 is the correct explanation of Statement 1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Both the statements are true but $\mathrm{Statement} 2$ is not the correct explanation of Statement 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">Statement 1 is true but Statement 2 is false</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Statement 1 is false but Statement 2 is true</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both the statements are true but $\mathrm{Statement} 2$ is not the correct explanation of Statement 1</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{f}(\mathrm{x})=\int \sin ^{2} \mathrm{x} \cdot \mathrm{dx}=\int \frac{1-\cos 2 \mathrm{x}}{2} \mathrm{~d} \mathrm{x} \\ &amp;=\frac{1}{2} \mathrm{x}-\frac{\sin 2 \mathrm{x}}{2} \cdot \frac{1}{2}+\mathrm{c}=\frac{1}{2} \mathrm{x}-\frac{1}{4} \sin 2 \mathrm{x}+\mathrm{c} \\ \text { (i) } &amp; \mathrm{f}(\pi+\mathrm{x})=\frac{1}{2}(\pi+\mathrm{x})-\frac{1}{4} \sin 2(\pi+\mathrm{x}) \\ &amp;=\frac{1}{2} \pi+\frac{1}{2} \mathrm{x}-\frac{1}{2} \sin (2 \pi+2 \mathrm{x})+\mathrm{c} \\ &amp;=\frac{\mathrm{x}}{2}-\frac{1}{2} \sin 2 \mathrm{x}+\mathrm{c}=\mathrm{f}(\mathrm{x}) \end{aligned}$<br/>So, statement 1 is true.<br/>(ii) $\sin ^{2}(\pi+x)=\sin ^{2} x$<br/>$(-\sin x)^{2}=\sin ^{2} x$<br/>$\Rightarrow \sin ^{2} x=\sin ^{2} x$<br/>So, statement 2 is true</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-x-be-defined-as-follows-f-x-2-x-1-x-1-x-2-3-x-2-2-x-0-0-x-1-which-one-of-the-following-statements-is-correct-in-respect-of-the-above-function

slug:
let-f-x-be-defined-as-follows-f-x-2-x-1-x-1-x-2-3-x-2-2-x-0-0-x-1-which-one-of-the-following-statements-is-correct-in-respect-of-the-above-function-48811

question:
<div class="question">Let $f(\mathrm{x})$ be defined as follows :<br/>$f(x)=\left\{\begin{array}{ll}2 x+1, &amp; -3   &lt; x &lt; -2 \\ x-1, &amp; -2 \leq x &lt; 0 \\ x+2, &amp; 0 \leq x &lt; 1\end{array}\right.$<br/>Which one of the following statements is correct in respect of the above function?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It is discontinuous at $\mathrm{x}=0$ but continuous at every other point.</span> </div>

answer:
<div class="solution">$f(x)=\left\{\begin{array}{cc}2 x+1, &amp; -3   &lt; x &lt; -2 \\ x-1, &amp; -2 \leq x &lt; 0 \\ x+2, &amp; 0 \leq x &lt; 1\end{array}\right.$<br/>Here, $\mathrm{f}\left(0^{-}\right)=-1$ and $\mathrm{f}\left(0^{+}\right)=2$<br/>So, $\mathrm{f}(\mathrm{x})$ is discontinuous at $\mathrm{x}=0$ and continuous at other points.</div>


tags: ['Mathematics', 'Functions', '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:
let-f-x-be-the-greatest-integer-function-and-g-x-be-the-modulus-function-what-is-f-f-5-9-g-g-2-equal-to

slug:
let-f-x-be-the-greatest-integer-function-and-g-x-be-the-modulus-function-what-is-f-f-5-9-g-g-2-equal-to-28285

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})$ be the greatest integer function and $\mathrm{g}(\mathrm{x})$ be the modulus function.<br/>What is $(\mathrm{f} \circ \mathrm{f})\left(-\frac{9}{5}\right)+(\mathrm{g} \circ \mathrm{g})(-2)$ equal to?</div>

answer:
<div class="solution">$f(x) \rightarrow$ greatest integer function<br/>$f(x)=[x]$<br/>$g(x) \rightarrow$ modulus function $g(x)=|x|$<br/>$f o f(x)=f(f(x))=f([x])$<br/>$\begin{array}{l}\text { fof }\left(-\frac{9}{5}\right)=f(-2) \quad \because\left[-\frac{9}{5}\right]=-2 \\ =[-2]=-2 &amp; \\ \operatorname{gog}(x)=g(|x|) &amp; \\ =|| x|| &amp; |x| \\ \quad=|x| &amp; \\ \operatorname{gog}(-2)=|-2|=2 &amp; \\ (\text { fof })\left(-\frac{9}{5}\right)+\operatorname{gog}(-2)=-2+2=0\end{array}$</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-x-be-the-greatest-integer-function-and-g-x-be-the-modulus-function-what-is-g-f-3-5-f-g-3-5-equal-to

slug:
let-f-x-be-the-greatest-integer-function-and-g-x-be-the-modulus-function-what-is-g-f-3-5-f-g-3-5-equal-to-74539

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})$ be the greatest integer function and $\mathrm{g}(\mathrm{x})$ be the modulus function.<br/>What is $(\mathrm{g} \circ \mathrm{f})\left(-\frac{5}{3}\right)-(\mathrm{f} \circ \mathrm{g})\left(-\frac{5}{3}\right)$ 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">0</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">2</span> </li> </ul>

answer:
<div class="solution">$f(x) \rightarrow$ greatest integer function<br/>$f(x)=[x]$<br/>$g(x) \rightarrow$ modulus function $g(x)=|x|$<br/>$\begin{aligned} \operatorname{gof}(x)=g(f(x)) &amp; \quad \operatorname{fog}(x)=f(g(x)) \\=g([x]) &amp;=f(|x|) \\=|[x]| &amp;=[|x|] \\ \operatorname{gof}\left(-\frac{5}{3}\right)=\left|\left[-\frac{5}{3}\right]\right| ; f o g\left(-\frac{5}{3}\right)=\left[\left|-\frac{5}{3}\right|\right] \\=|-2| ;=\left[\frac{5}{3}\right] \\=2 ;=1 \\ \operatorname{gof}\left(-\frac{5}{3}\right)-\operatorname{fog}\left(-\frac{5}{3}\right)=2-1=1 \end{aligned}$</div>


title: let-f-x-p-x-q-and-g-x-m-x-n-then-f-g-x-g-f-x-is-equivalent-to
slug: let-f-x-p-x-q-and-g-x-m-x-n-then-f-g-x-g-f-x-is-equivalent-to-71773
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=p x+q$ and $g(x)=m x+n$. Then $f(g(x))=g(f(x))$ is equivalent to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(\mathrm{n})=\mathrm{g}(\mathrm{q})$</span> </div>

answer:
<div class="solution">$\begin{array}{l}\mathrm{f}(\mathrm{x})=\mathrm{px}+\mathrm{q}, \mathrm{g}(\mathrm{x})=\mathrm{mx}+\mathrm{n} \\ \mathrm{f}(\mathrm{g}(\mathrm{x}))=\mathrm{g}(\mathrm{f}(\mathrm{x})) \\ \Rightarrow \mathrm{f}(\mathrm{mx}+\mathrm{n})=\mathrm{g}(\mathrm{px}+\mathrm{q}) \\ &amp; \Rightarrow \mathrm{p}(\mathrm{mx}+\mathrm{n})+\mathrm{q}=\mathrm{m}(\mathrm{px}+\mathrm{q})+\mathrm{n} \\ &amp; \Rightarrow \mathrm{pmx}+\mathrm{pn}+\mathrm{q}=\mathrm{pmx}+\mathrm{mq}+\mathrm{n} \\ &amp; \Rightarrow \mathrm{pn}+\mathrm{q}=\mathrm{mq}+\mathrm{n} \\ &amp; \Rightarrow \mathrm{f}(\mathrm{n})=\mathrm{g}(\mathrm{q})\end{array}$</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-0-x-is-rational-x-is-irrational-and-g-x-0-x-x-is-rational-x-is-irrational-if-f-r-r-and-g-r-r-then-f-g-is

slug:
let-f-x-x-0-x-is-rational-x-is-irrational-and-g-x-0-x-x-is-rational-x-is-irrational-if-f-r-r-and-g-r-r-then-f-g-is-78348

question:
<div class="question">Let $f(\mathrm{x}):\left\{\begin{array}{ll}\mathrm{x}, &amp; \mathrm{x} \text { is rational } \\ 0, &amp; \mathrm{x} \text { is irrational }\end{array}\right.$<br/>and $g(x):\left\{\begin{array}{ll}0, &amp; x \text { is rational } \\ x, &amp; x \text { is irrational }\end{array}\right.$<br/>If $f: \mathrm{R} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$, then $(f-\mathrm{g})$ is</div>

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

answer:
<div class="solution">$\quad(\mathrm{f}-\mathrm{g})(\mathrm{x})=\left\{\begin{array}{c}\mathrm{x}-0=\mathrm{x}, \mathrm{x} \text { is rational } \\ 0-\mathrm{x}=-\mathrm{x}, \mathrm{x} \text { is irrational }\end{array}\right.$<br/>Clearly, $\mathrm{f}-\mathrm{g}$ is one $-$ one and onto.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-x-x-e-x-1-0-x-0-x-0-be-a-real-valued-function-which-of-the-following-statements-isare-correct-1-f-x-is-right-continuous-at-x-0-2-f-x-is-disconti

slug:
let-f-x-x-e-x-1-0-x-0-x-0-be-a-real-valued-function-which-of-the-following-statements-isare-correct-1-f-x-is-right-continuous-at-x-0-2-f-x-is-disconti-15729

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\frac{\mathrm{e}^{\mathrm{x}}-1}{\mathrm{x}}, &amp; \mathrm{x}&gt;0 \\ 0, &amp; \mathrm{x}=0\end{array}\right.$ be a real valued function.<br/>Which of the following statements is/are correct?<br/>1- $\mathrm{f}(\mathrm{x})$ is right continuous at $\mathrm{x}=0$.<br/>2- $\mathrm{f}(\mathrm{x})$ is discontinuous at $\mathrm{x}=1$. Select the correct answer using the code given below:</div>

answer:
<div class="solution">For right hand continuity at $x=0$ $\begin{aligned} \mathrm{RHL} &amp;=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h) \\ &amp;=\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}=\lim _{h \rightarrow 0} \frac{\left(1+h+\frac{h^{2}}{2 !}+\ldots\right)-1}{h} \\=&amp; \lim _{h \rightarrow 0} \frac{h+\frac{h^{2}}{2 !}+\frac{h^{3}}{3 !}+\ldots .}{h} \\=&amp; \lim _{h \rightarrow 0} 1+\frac{h}{2 !}+\frac{h^{2}}{3 !}+\ldots . .=1 \end{aligned}$<br/>$f(0)=0$<br/>$\Rightarrow f(x)$ is not right continuous at $x=0$. For discontinuity at $x=1$<br/>$\begin{aligned} \mathrm{RHL} &amp;=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{h \rightarrow 0} f(1+h) \\ &amp;=\lim _{h \rightarrow 0} \frac{e^{1+h}-1}{1+h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\left(1+(1+h)+\frac{(1+h)^{2}}{2}+\ldots\right)-1}{1+h} \\=&amp; \lim _{h \rightarrow 0} \frac{(1+h)+\frac{(1+h)^{2}}{2 !}+\frac{(1+h)^{3}}{3 !}+\ldots .}{(1+h)} \\=&amp; \lim _{h \rightarrow 0} 1+\frac{(1+h)}{2 !}+\frac{(1+h)^{2}}{3 !}+\ldots . \end{aligned}$<br/>$\begin{aligned}=&amp; 1+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots \\ \mathrm{LHL} &amp;=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h) \\ &amp;=\lim _{h \rightarrow 0} \frac{e^{(1-h)}-1}{1-h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\left(1+(1-h)+\frac{(1-h)^{2}}{2 !}+\ldots .\right)-1}{(1-h)} \\ &amp;=\lim _{h \rightarrow 0} 1+\frac{(1-h)}{2 !}+\frac{(1-h)^{2}}{3 !}+\ldots . \\ &amp;=1+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots . . \end{aligned}$<br/>$f(1)=\frac{e^{1}-1}{1}=\left(1+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots .\right)-1$<br/>$\quad=\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots$<br/>$\Rightarrow \mathrm{RHL} \neq f(1), \mathrm{LHL} \neq f(1)$<br/>So $f$ is discontinuous.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-e-x-1-0-x-0-x-0-be-a-real-valued-function-which-one-of-the-following-statements-is-correct

slug:
let-f-x-x-e-x-1-0-x-0-x-0-be-a-real-valued-function-which-one-of-the-following-statements-is-correct-40346

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\frac{\mathrm{e}^{\mathrm{x}}-1}{\mathrm{x}}, &amp; \mathrm{x}&gt;0 \\ 0, &amp; \mathrm{x}=0\end{array}\right.$ be a real valued function.<br/>Which one of the following statements is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}(\mathrm{x})$ is a strictly increasing function in $(0, \mathrm{x})$,</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\frac{\mathrm{e}^{\mathrm{x}}-1}{\mathrm{x}}&gt;0$<br/>$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\frac{\mathrm{xe}^{\mathrm{x}}-\left(\mathrm{e}^{\mathrm{x}}-1\right)}{\mathrm{x}^{2}}=\frac{\mathrm{e}^{\mathrm{x}}(\mathrm{x}-1)+1}{\mathrm{x}^{2}}$<br/>$=\left(\frac{\mathrm{e}^{\mathrm{x}}(\mathrm{x}-1)}{\mathrm{x}^{2}}+\frac{1}{\mathrm{x}^{2}}\right)$, which is a strictly increasing function.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-of-the-following-statements-are-correct-1-fof

slug:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-of-the-following-statements-are-correct-1-fof-59357

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$, where $[.]$ is the greatest integer function and $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}$ be two real valued functions over R. <br/>Which of the following statements are correct?<br/>1- (fof)(x)=f(x)<br/>2- (gog)(x)=g(x) only when $x=0$.<br/>3- $(\mathrm{go}(\mathrm{fog}))(\mathrm{x})$ can take only three values. Select the correct answer using the code given below:</div>

answer:
<div class="solution">$($ fof $)(x)=[[x]]$ and $f(x)=[x]$<br/>Suppose $x=0.2$<br/>$\Rightarrow(f o f)(x)=[[0.2]]=[0]=0$<br/>$f(x)=[0.2]=0$<br/>$\Rightarrow f o f(x)=f(x)$<br/>Now, $(\operatorname{gog})(x)=\sin \sin x$ and $g(x)=\sin x$<br/>At $x=0$<br/>$(\operatorname{gog})(x)=\sin \sin 0=\sin 0=0$<br/>$g(x)=\sin 0=0$<br/>$\Rightarrow(\operatorname{gog})(x)=g(x)$ at $x=0$<br/>and this is true for $\mathrm{x}=\mathrm{n} \pi$, where $\mathrm{n}=0,1,2,3,4 \ldots \ldots$<br/>$\because(f o g)(x)=[\sin x]$<br/>$(g o(f o g))(x)=\sin [\sin x]$<br/>$\because \sin x$ has value from $-1$ to 1<br/>If $-1 \leq \sin x   &lt; 0$<br/>$(g o(f o g))(x)=\sin (-1)-\sin (-1)$<br/>If $0 \leq \sin x &lt; 1$<br/>$\therefore(g o(f o g))(x)=\sin (0)=\sin 0$ and<br/>If $\sin x=1$<br/>$\therefore(g o(f o g))(x)=\sin (1)=\sin 1$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-of-the-following-statements-is-correct

slug:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-of-the-following-statements-is-correct-39230

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$, where $[.]$ is the greatest integer function and $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}$ be two real valued functions over R. <br/>Which of the following statements is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\operatorname{Both} \mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are continuous at $\mathrm{x}=0$.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$, but $\mathrm{g}(\mathrm{x})$ is not continuous at $\mathrm{x}=0 .$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{g}(\mathrm{x})$ is continuous at $\mathrm{x}=0$, but $\mathrm{f}(\mathrm{x})$ is not continuous at $\mathrm{x}=0 .$</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><li class=""> <span class="option-label">D</span> <span class="option-data">Both $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are discontinuous at $\mathrm{x}=0$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{g}(\mathrm{x})$ is continuous at $\mathrm{x}=0$, but $\mathrm{f}(\mathrm{x})$ is not continuous at $\mathrm{x}=0 .$</span> </div>

answer:
<div class="solution">$f(x)=[x]$ and $g(x)=\sin x$ $\lim _{x \rightarrow 0^{+}} f(x)=[0+h]=0$<br/>$\lim _{x \rightarrow 0^{-}} f(x)=[0-h]=-1$<br/>$f(0)=0$<br/>$\Rightarrow f(x)$ is not continuous at $x=0$ and also $g(x)$ is<br/>continuous at $x=0$. (every trignometric function is continuous).</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-one-of-the-following-statements-is-correct-1

slug:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-one-of-the-following-statements-is-correct-1-65776

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$, where $[.]$ is the greatest integer function and $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}$ be two real valued functions over R. <br/>Which one of the following statements is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\lim _{x \rightarrow 0+}($ fog $)(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 0^{+}}(\mathrm{gof})(\mathrm{x})$</span> </div>

answer:
<div class="solution">At $x=0$<br/>For LHL: $g(x)=-2+|-2|=0$<br/>For RHL: $g(x)=|x|-2+|x-2|$<br/>$\begin{aligned} g(x) &amp;=x-2-(x-2)=0 \\ \mathrm{~g}(\mathrm{x}) &amp;=0 \\ \text { For }(x=0): g(x) &amp;=-2+|-2|=0 \end{aligned}$<br/>LHL $=\lim _{x \rightarrow 0^{-}} g(x)=0$<br/>$\mathrm{RHL}=\lim _{x \rightarrow 0^{+}} g(x)=0$<br/>$g(0)=0$<br/>$\Rightarrow g(x)$ is continuous at $x=0$ At $x=2$<br/>For LHL: $g(x)=|x|-2+|x-2|$<br/>$g(x)=x-2-(x-2)$<br/>$g(x)=0$<br/>For RHL: $g(x)=|x|-2+|x-2|$<br/>$g(x)=x-2+x-2$<br/>$2(x)=2 x-4$<br/>For $(x=2): g(x)=|x|-2+|x-2|$<br/>$\mathrm{LHL}=\lim _{x \rightarrow 2^{-}} g(x)=0$<br/>$\mathrm{RHL}=\lim _{x \rightarrow 2^{+}} g(x)=\lim _{x \rightarrow 2} 2 x-4=2(2)-4=0$<br/>$g(2)=|2|-2+|2-2|$<br/>$g(2)=2-2+2-2=0$<br/>$\Rightarrow g(x)$ is continuous at $x=2$. At $x=-1$<br/>For LHL: $g(x)=-2+|-2|=0$<br/>For RHL: $g(x)=-2+|-2|=0$<br/>For $(x=-1): g(x)=-2+|-2|=0$<br/>$\therefore \mathrm{LHL}=\lim _{x \rightarrow-1^{-}} g(x)=0$<br/>$\mathrm{RHL}=\lim _{x \rightarrow-1^{+}} g(x)=0$<br/>$g(-1)=0$<br/>$\Rightarrow g(\mathrm{x})$ is differentiable at $x=-1$.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-one-of-the-following-statements-is-correct

slug:
let-f-x-x-where-is-the-greatest-integer-function-and-g-x-sin-x-be-two-real-valued-functions-over-r-which-one-of-the-following-statements-is-correct-57022

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$, where $[.]$ is the greatest integer function and $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}$ be two real valued functions over R. <br/>Which one of the following statements is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\lim _{x \rightarrow 0+}($ fog $)(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 0^{+}}(\mathrm{gof})(\mathrm{x})$</span> </div>

answer:
<div class="solution">$(f \circ g)(x)=[\sin x]$<br/>$\lim _{x \rightarrow 0^{+}}(f o g)(x)=\lim _{x \rightarrow 0^{+}}[\sin x]=[h]$ where $h&gt;0$<br/>$\Rightarrow \lim _{x \rightarrow 0^{+}}(f \circ g)(x)=0$<br/>$\lim _{x \rightarrow 0^{-}}(f o g)(x)=\lim _{x \rightarrow 0^{-}}[\sin x]=[h]$ where $h   &lt; 0$<br/>$\Rightarrow \lim _{x \rightarrow 0^{-}}(f \circ g)(x)=-1$<br/>$\Rightarrow \lim _{x \rightarrow 0}(f \circ g)(x)$ does not exist.<br/>Now, $(g o f)(x)=\sin [x]$<br/>$\lim _{x \rightarrow 0^{+}}(g o f)(x)=\lim _{x \rightarrow 0^{+}} \sin [x]=\sin 0=0$<br/>$\lim _{x \rightarrow 0^{-}}($ gof $)(x)=\lim _{x \rightarrow 0^{-}} \sin [x]=\sin (-1)=-0.01745$<br/>$\Rightarrow \lim _{x \rightarrow 0}($ gof $)(x)$ does not exist.<br/>$\Rightarrow \lim _{x \rightarrow 0^{+}}(f o g)(x)=\lim _{x \rightarrow 0^{+}}(g o f)(x)$</div>


