# questions

title: if-x-1-y-y-1-x-0-what-is-dx-dy-equal-to
slug: if-x-1-y-y-1-x-0-what-is-dx-dy-equal-to-74102
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x} \sqrt{1+\mathrm{y}}+\mathrm{y} \sqrt{1+\mathrm{x}}=0$, what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{1+\mathrm{x}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{(1+x)^{2}}$</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">$\frac{1}{(1+\mathrm{x})^{2}} \quad$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{\mathrm{x}}}{\sqrt{1+\mathrm{x}}}$</span> </li> </ul>

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

answer:
<div class="solution">Given equation<br/>$\mathrm{x} \sqrt{1+\mathrm{y}}+\mathrm{y} \sqrt{1+\mathrm{x}}=0$<br/>Can be written as:<br/>$\mathrm{x} \sqrt{1+\mathrm{y}}=-\mathrm{y} \sqrt{1+\mathrm{x}}$<br/>Squaring both sides, we get $\begin{aligned} &amp; x^{2}(1+y)=y^{2}(1+x) \\ \Rightarrow &amp; x^{2}+x^{2} y=y^{2}+y^{2} x \quad \Rightarrow \quad x^{2}-y^{2}=y^{2} x-x^{2} y \\ \Rightarrow &amp;(x-y)(x+y)=-x y(x-y) \\ \Rightarrow &amp; x+y=-x y \quad \Rightarrow y(1+x)=-x \end{aligned}$<br/>$\mathrm{y}=\frac{-\mathrm{x}}{1+\mathrm{x}}$ which is in explicit form.<br/>Differentiating w.r.t.x, we get<br/>$\frac{d y}{d x}=\frac{(1+x)(-1)+x(1)}{(1+x)^{2}}=\frac{-1}{(1+x)^{2}}$</div>


title: if-x-1-y-y-2-y-3-up-to-infinite-terms-where-y-1-then-which-one-of-the-following-is-correct
slug: if-x-1-y-y-2-y-3-up-to-infinite-terms-where-y-1-then-which-one-of-the-following-is-correct-94873
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}=1-\mathrm{y}+\mathrm{y}^{2}-\mathrm{y}^{3}+\ldots$ up to infinite terms, where $|\mathrm{y}|   &lt; 1$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}=\frac{1}{1+\mathrm{y}}$</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">$x=\frac{y}{1-y}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}=\frac{\mathrm{y}}{1+\mathrm{y}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=\frac{y}{1-y}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{x}=1-\mathrm{y}+\mathrm{y}^{2}-\mathrm{y}^{3}+\ldots$ up to infinite terms.<br/>We can see that the given series is geometric progression, with $\mathrm{a}=1$ and $\mathrm{r}=-\mathrm{y}$<br/>$\mathrm{S}_{\infty} \frac{\mathrm{a}}{1-\mathrm{r}}=\frac{1}{1-(-\mathrm{y})}=\frac{1}{1+\mathrm{y}}$<br/>$\therefore \mathrm{x}=\frac{1}{1+\mathrm{y}}$</div>


title: if-x-2-1-what-is-the-value-of-1-n-1-x-x-2-n-n-1-1-x-x-2
slug: if-x-2-1-what-is-the-value-of-1-n-1-x-x-2-n-n-1-1-x-x-2-33559
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\mathrm{x}|   &lt; \frac{1}{2}$, what is the value of<br/>$1+n\left[\frac{x}{1-x}\right]+\left[\frac{n(n+1)}{2 !}\right]\left[\frac{x}{1-x}\right]^{2}+\ldots \ldots \ldots \ldots \ldots \infty ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\frac{1-x}{1-2 x}\right]^{n}$</span> </div>

answer:
<div class="solution">Given that $1+\mathrm{n}\left[\frac{\mathrm{x}}{1-\mathrm{x}}\right]+\frac{\mathrm{n}(\mathrm{n}+1)}{2 !}\left[\frac{\mathrm{x}}{1-\mathrm{x}}\right]^{2}+\ldots \infty$ is<br/>expansion of $\left[1-\frac{x}{1-x}\right]^{-n}$.<br/>So, it is $=\left[1-\frac{x}{1-x}\right]^{-n}=\left[\frac{1-x-x}{1-x}\right]^{-n}=\left[\frac{1-x}{1-2 x}\right]^{n}$</div>


title: if-x-2-2-13-2-23-then-what-is-the-value-of-x-3-6-x-2-6-x
slug: if-x-2-2-13-2-23-then-what-is-the-value-of-x-3-6-x-2-6-x-34246
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}=2+2^{1 / 3}+2^{2 / 3}$, then what is the value of $x^{3}-6 x^{2}+6 x ?$</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">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Let $\begin{aligned} x=2+2^{1 / 3}+2^{2 / 3} \\ &amp;=2+2^{1 / 3}+\left(2^{1 / 3}\right)^{2}=2+2^{1 / 3}\left(1+2^{1 / 3}\right) \\ &amp; \Rightarrow x-2=2^{1 / 3}\left(1+2^{1 / 3}\right) \end{aligned}$<br/>On cubing both sides, we get<br/>$x^{3}-8-6 x^{2}+12 x=2\left(1+2+3.2^{1 / 3}+3.2^{2 / 3}\right)$<br/>$\Rightarrow x^{3}-6 x^{2}+6 x=14+6.2^{1 / 3}+6.2^{2 / 3}-6 x$<br/>$\quad=14+6.2^{1 / 3}+6.2^{2 / 3}-6\left(2+2^{1 / 3}+2^{2 / 3}\right)=2$</div>


title: if-x-2-3-x-4-0-then-which-one-of-the-following-is-correct
slug: if-x-2-3-x-4-0-then-which-one-of-the-following-is-correct-88982
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $-\mathrm{x}^{2}+3 \mathrm{x}+4&gt;0$, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x \in(-1,4)$</span> </div>

answer:
<div class="solution">$-x^{2}+3 x+4&gt;0$<br/>$\Rightarrow x^{2}-3 x-4   &lt; 0 \Rightarrow(x-4)(x+1) &lt; 0$<br/>Noting the sign of expression around $-1$ and 4 , we use way curve method.<img src="https://cdn-question-pool.getmarks.app/pyq/nda/DEZlDlz4vbK4KxW3DGDZRc6UHbiHLU2Aftc2C5leYW8.original.fullsize.png"/><br/><br/>$\Rightarrow \mathrm{x} \in(-1,4)$</div>


title: if-x-2-3-z-are-in-ap-x-3-z-are-in-gp-then-which-one-of-the-following-will-be-in-hp
slug: if-x-2-3-z-are-in-ap-x-3-z-are-in-gp-then-which-one-of-the-following-will-be-in-hp-75723
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}, \frac{3}{2}, \mathrm{z}$ are in $\mathrm{AP} ; \mathrm{x}, 3, \mathrm{z}$ are in GP; then which one of the following will be in HP?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}, 6, \mathrm{z}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{x}, 4, \mathrm{z}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}, 2, \mathrm{z}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x, 1, z$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{x}, 6, \mathrm{z}$</span> </div>

answer:
<div class="solution">$\mathrm{x}, \frac{3}{2}, \mathrm{z}$ are in A.P<br/>If $a, b, c$ are in $A \cdot P 2 b=a+c$<br/>$\therefore 2\left(\frac{3}{2}\right)=x+z$<br/>$\Rightarrow 3=x+z$<br/>$\ldots .(1)$<br/>$\mathrm{x}, 3, \mathrm{z}$ are in G.P. If $a, b, c$ are in G.P. $b^{2}=a c$ $\therefore 3^{2}=\mathrm{xz}$<br/>$\Rightarrow 9=\mathrm{xz}$...(2<br/>If $\mathrm{x}, 6, \mathrm{z}$ are in $\mathrm{H.P.} \frac{2}{6}=\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{z}}$<br/>$\quad\left(\because \frac{2}{b}=\frac{1}{a}+\frac{1}{c}\right.$, if $a, b, c$ are is H.P. $)$<br/>$\Rightarrow \frac{1}{3}=\frac{z+x}{x z}=\frac{3}{9}=\frac{1}{3} \quad($ from $(1)(2))$<br/>L.HS. $=$ R.H.S.</div>


title: if-x-2-13-2-13-then-what-is-the-value-of-2-x-3-6-x
slug: if-x-2-13-2-13-then-what-is-the-value-of-2-x-3-6-x-30649
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 1)']
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">If $\mathrm{x}=2^{1 / 3}-2^{-1 / 3}$, then what is the value of $2 \mathrm{x}^{3}+6 \mathrm{x}$ ?</div>

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

answer:
<div class="solution">Given equation is: $\mathrm{x}=2^{1 / 3}-2^{-1 / 3}$ On cubing both sides, we get<br/>$\begin{aligned} x^{3} &amp;=\left(2^{1 / 3}-2^{-1 / 3}\right)^{3} \\ &amp;=2-2^{-1}-3.2^{1 / 3} \cdot 2^{-1 / 3}\left(2^{1 / 3}-2^{-1 / 3}\right) \\ \Rightarrow &amp; x^{3}=-3 x+2-\frac{1}{2} \Rightarrow x^{3}=-3 x+\frac{3}{2} \\ \Rightarrow &amp; 2 x^{3}+6 x=3 \end{aligned}$</div>


title: if-x-2-ln-x-d-x-m-x-3-ln-x-n-x-3-c-then-what-are-the-values-of-m-and-n-respectively
slug: if-x-2-ln-x-d-x-m-x-3-ln-x-n-x-3-c-then-what-are-the-values-of-m-and-n-respectively-51697
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3,-9$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int x^{2} \ln x d x=\frac{x^{3}}{m} \ln x+\frac{x^{3}}{n}+c$, then what are the values of $m$ and $n$ respectively?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3,}-\frac{1}{9}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3,-9$</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">3,9</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3,3 <br/>where $c$ is a constant of integration.</span> </li> </ul>

answer:
<div class="solution">$\int x^{2} \ln x d x=\ln x \frac{x^{3}}{3}-\int \frac{1}{x} \cdot \frac{x^{3}}{3} d x$<br/>$=\frac{x^{3}}{3} \ln x-\int \frac{x^{2}}{3} d x=\frac{x^{3}}{3} \ln x-\frac{1}{3} \cdot \frac{x^{3}}{3}+c$ ...(1)<br/>$=\frac{x^{3}}{3} \ln x-\frac{x^{3}}{9}+c$<br/>But $\int x^{2} \ln x d x=\frac{x^{3}}{m} \ln x+\frac{x^{3}}{n}+c$<br/>On comparing with (1), we get $m=3$ and $n=-9$</div>


title: if-x-2-p-x-4-0-for-all-real-values-of-x-then-which-one-of-the-following-is-correct
slug: if-x-2-p-x-4-0-for-all-real-values-of-x-then-which-one-of-the-following-is-correct-47056
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x^{2}+p x+4&gt;0$ for all real values of $x$, 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">$|\mathrm{p}|   &lt; 4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$|\mathrm{p}| \leq 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$|p|&gt;4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\mathrm{p}| \geq 4$</span> </li> </ul>

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

answer:
<div class="solution">$x^{2}-p x+4&gt;0 \quad \forall$ real values of $x$. If $b^{2}-4 a c \leq 0$<br/>$\Rightarrow p^{2}-4(1)(4) \leq 0$<br/>$\Rightarrow p^{2} \leq 16 \Rightarrow|p| \leq 4$<br/>Sol. $(137-138)$ :<br/>$f(x)=\frac{27}{4}\left(x^{2 / 3}-x\right)$<br/>$=\frac{27}{4}\left(\left(x^{2}\right)^{1 / 3}-x\right.$</div>


title: if-x-2-x-2-3-x-3-are-the-first-three-terms-of-a-gp-then-what-is-its-fourth-term
slug: if-x-2-x-2-3-x-3-are-the-first-three-terms-of-a-gp-then-what-is-its-fourth-term-26999
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x, 2 x+2,3 x+3$ are the first three terms of a GP, then what is its fourth term?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-27 / 2$</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">$27 / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-33 / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$33 / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-27 / 2$</span> </div>

answer:
<div class="solution">Since, $\mathrm{x}, 2 \mathrm{x}+2,3 \mathrm{x}+3$ are the terms of G.P<br/>therefore $\frac{2 x+2}{x}=\frac{3 x+3}{2 x+2}$<br/>$\Rightarrow(2 x+2)^{2}=x(3 x+3) \Rightarrow 4 x^{2}+4+8 x=3 x^{2}+3 x$<br/>$\Rightarrow x^{2}+5 x+4=0$<br/>$\Rightarrow x^{2}+4 x+x+4=0$<br/>$\Rightarrow x(x+4)+1(x+4)=0$<br/>$\Rightarrow \mathrm{x}=-1,-4$<br/>Now, first term $\mathrm{a}=\mathrm{x}$<br/>Second term, ar $=2(\mathrm{x}+1) \Rightarrow \mathrm{r}=\frac{2(x+1)}{x}$<br/>Fourth term $=\mathrm{ar}^{3}=\mathrm{x}\left(\frac{2(x+1)}{x}\right)^{3}$<br/>Put $x=-4$, we get<br/>Fourth term $=-4\left(\frac{2(-4+1)}{-4}\right)^{3}=-4 \times\left(\frac{3}{2}\right)^{3}=-\frac{27}{2}$</div>


title: if-x-2-y-2-1-then-what-is-1-x-i-y-1-x-i-y-equal-to
slug: if-x-2-y-2-1-then-what-is-1-x-i-y-1-x-i-y-equal-to-51363
question: <div class="question">If $x^{2}+y^{2}=1$, then what is $\frac{1+x+i y}{1+x-i y}$ equal to?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x-i y$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x+i y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-2 i y$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\frac{1+x+\text { iy }}{1+x-i y}=\frac{(1+x+i y)(1+x+i y)}{(1+x-i y)(1+x+i y)}$<br/>(By Rationalizing)<br/>$=\frac{(1+x)^{2}+i y(1+x)+i y(1+x)-y^{2}}{1+x^{2}+2 x+y^{2}}\left(\because i^{2}=-1\right)$<br/>$=\frac{1+x^{2}+2 x-y^{2}+2 i y(1+x)}{2(1+x)}\left(\because x^{2}+y^{2}=1\right)$<br/>$=\frac{1-y^{2}+2 x+x^{2}+2 i y(1+x)}{2(1+x)}$<br/>$=\frac{2 x^{2}+2 x+2 i y(1+x)}{2(1+x)}=x+i y \quad\left(\because 1-y^{2}=x^{2}\right)$</div>


title: if-x-2-y-2-z-2-1-then-what-is-the-value-of-1-z-y-z-1-x-y-x-1
slug: if-x-2-y-2-z-2-1-then-what-is-the-value-of-1-z-y-z-1-x-y-x-1-27808
tags: ['Mathematics', 'Determinants', '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">If $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1$, then what is the value of<br/>$\left|\begin{array}{ccc}1 &amp; z &amp; -y \\ -z &amp; 1 &amp; x \\ y &amp; -x &amp; 1\end{array}\right|=?$</div>

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

answer:
<div class="solution">Let $D=\left|\begin{array}{ccc}1 &amp; z &amp; -y \\ -z &amp; 1 &amp; x \\ y &amp; -x &amp; 1\end{array}\right|$<br/>Expanding along $\mathrm{R}_{1}$ $\Rightarrow \mathrm{D}=1\left|\begin{array}{cc}1 &amp; \mathrm{x} \\ -\mathrm{x} &amp; 1\end{array}\right|-\mathrm{z}\left|\begin{array}{cc}-\mathrm{z} &amp; \mathrm{x} \\ \mathrm{y} &amp; 1\end{array}\right|-\mathrm{y}\left|\begin{array}{cc}-\mathrm{z} &amp; 1 \\ \mathrm{y} &amp; -\mathrm{x}\end{array}\right|$<br/>$=\left(1+x^{2}\right)-z(-z-x y)-y(x z-y)$<br/>$=1+x^{2}+z^{2}+x y z-x y z+y^{2}$<br/>$=1+x^{2}+y^{2}+z^{2}=1+1=2$</div>


title: if-x-2-y-2-z-2-are-in-ap-then-y-z-z-x-x-y-are-in
slug: if-x-2-y-2-z-2-are-in-ap-then-y-z-z-x-x-y-are-in-60148
question: <div class="question">If $x^{2}, y^{2}, z^{2}$ are in AP, then $y+z, z+x, x+y$ are in</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">AP</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">AP</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{HP}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">GP</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{x}^{2}, \mathrm{y}^{2}, \mathrm{z}^{2}$ are in A.P $\Rightarrow \mathrm{y}^{2}-\mathrm{x}^{2}=\mathrm{z}^{2}-\mathrm{y}^{2}$<br/>$2 y^{2}=x^{2}+z^{2}$<br/>(a) Suppose $\mathrm{y}+\mathrm{z}, \mathrm{z}+\mathrm{x}$ and $\mathrm{x}+\mathrm{y}$ are in $\mathrm{A}-\mathrm{P}$<br/>$\therefore(\mathrm{z}+\mathrm{x})-(\mathrm{y}+\mathrm{z})=(\mathrm{x}+\mathrm{y})-(\mathrm{z}+\mathrm{x})$<br/>$2(z+x)=(y+z)+(x+y)$<br/>$\Rightarrow 2 z+2 x=2 y+z+x \Rightarrow z+x=2 y$<br/>$\Rightarrow \mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are in $\mathrm{AP}$. Which is true<br/>(b) Let $\mathrm{y}+\mathrm{z}, \mathrm{z}+\mathrm{x}, \mathrm{x}+\mathrm{y}$ are in $\mathrm{HP}$.<br/>$z+x=\frac{2(y+z)(x+y)}{y+z+x+y}$<br/>$\Rightarrow \quad z+x=\frac{2(y+z)(x+y)}{2 y+z+x}$<br/>$\Rightarrow \quad 2 y z+z^{2}+z x+2 x y+x z+x^{2}$<br/>$\quad=2 y x+2 y^{2}+2 z x+2 y z$<br/>$\Rightarrow z^{2}+x^{2}=2 y^{2}$<br/>$\Rightarrow \mathrm{x}^{2}, \mathrm{y}^{2}$ and $\mathrm{z}^{2}$ are in AP. Which is true. Hence, $\mathrm{y}+\mathrm{z}, \mathrm{z}+\mathrm{x}$ and $\mathrm{x}+$ y are in A.P.</div>


title: if-x-2-y-b-x-y-a-b-then-the-value-of-d-x-d-y-x-y-is-equal-to
slug: if-x-2-y-b-x-y-a-b-then-the-value-of-d-x-d-y-x-y-is-equal-to-69082
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x^{2} y^{b}=(x-y)^{a+b}$, then the value of $\frac{d y}{d x}-\frac{y}{x}$ is equal to</div>

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

answer:
<div class="solution">$x^{a} y^{b}=(x-y)^{a+b}$<br/>taking log both the sides.<br/>$\log \left(x^{a} y^{b}\right)=\log (x-y)^{(a+b)}$<br/>$a \log x+b \log y=(a+b) \log (x-y)$<br/>differentiating both sides w.r.t 'x'.<br/>$\frac{a}{x}+\frac{b}{y} \frac{d y}{d x}=\frac{(a+b)}{(x-y)}\left[1-\frac{d y}{d x}\right]$<br/>$\frac{d y}{d x}\left[\frac{b}{y}+\frac{a+b}{x-y}\right]=\frac{a+b}{x-y}-\frac{a}{x}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}\left[\frac{\mathrm{bx}-\mathrm{by}+\mathrm{ay}+\mathrm{by}}{\mathrm{y}(\mathrm{x}-\mathrm{y})}\right]=\frac{\mathrm{ax}+\mathrm{bx}-\mathrm{ax}+\mathrm{ay}}{\mathrm{x}(\mathrm{x}-\mathrm{y})}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}\left[\frac{\mathrm{bx}+\mathrm{ay}}{\mathrm{y}}\right]=\frac{\mathrm{bx}+\mathrm{ay}}{\mathrm{x}}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{y}}{\mathrm{x}}=0$</div>


title: if-x-3-1-4-1-b-5-2-2-1-and-a-p-r-q-s-satisfy-the-equation-a-x-b-then-the-matrix-a-is-equal-to
slug: if-x-3-1-4-1-b-5-2-2-1-and-a-p-r-q-s-satisfy-the-equation-a-x-b-then-the-matrix-a-is-equal-to-79453
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\quad X=\left[\begin{array}{ll}3 &amp; -4 \\ 1 &amp; -1\end{array}\right], B=\left[\begin{array}{cc}5 &amp; 2 \\ -2 &amp; 1\end{array}\right]$ and $A=\left[\begin{array}{cc}p &amp; q \\ r &amp; s\end{array}\right]$ satisfy<br/>the equation $A X=B$, then the matrix $A$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{cc}-7 &amp; 26 \\ 1 &amp; -5\end{array}\right]$</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">$\left[\begin{array}{ll}7 &amp; 26 \\ 4 &amp; 17\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{ll}-7 &amp; -4 \\ 26 &amp; 13\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\begin{array}{rr}-7 &amp; 26 \\ -6 &amp; 23\end{array}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{cc}-7 &amp; 26 \\ 1 &amp; -5\end{array}\right]$</span> </div>

answer:
<div class="solution">$\quad \mathrm{X}=\left[\begin{array}{cc}3 &amp; -4 \\ 1 &amp; -1\end{array}\right], \mathrm{B}=\left[\begin{array}{cc}5 &amp; 2 \\ -2 &amp; 1\end{array}\right]$ and $\mathrm{A}=\left[\begin{array}{ll}\mathrm{p} &amp; \mathrm{q} \\ \mathrm{r} &amp; \mathrm{s}\end{array}\right]$<br/>Now, $\mathrm{AX}=\mathrm{B}$<br/>$\left[\begin{array}{ll}\mathrm{p} &amp; \mathrm{q} \\ \mathrm{r} &amp; \mathrm{s}\end{array}\right]\left[\begin{array}{ll}3 &amp; -4 \\ 1 &amp; -1\end{array}\right]=\left[\begin{array}{cc}5 &amp; 2 \\ -2 &amp; 1\end{array}\right]$<br/>$\Rightarrow \quad\left[\begin{array}{ll}3 \mathrm{p}+\mathrm{q} &amp; -4 \mathrm{p}-\mathrm{q} \\ 3 \mathrm{r}+\mathrm{s} &amp; -4 \mathrm{r}-\mathrm{s}\end{array}\right]=\left[\begin{array}{cc}5 &amp; 2 \\ -2 &amp; 1\end{array}\right]$<br/>$3 p+q=5$ ...(i)<br/>$-4 p-q=2$ ...(ii)<br/>$3 r+s=-2$ ...(iii)<br/>$-4 r-s=1$ ...(iv)<br/>From equations (i) and (ii), we get $-p=7$<br/>$\therefore \mathrm{p}=-7$<br/>$\begin{aligned} \Rightarrow \quad \mathrm{q} &amp;=5-3(-7) \\ \mathrm{q} &amp;=26 \end{aligned}$<br/>From<br/>(iii) and (iv), <br/>$\begin{array}{ll} &amp; -r=-1 \\ \therefore &amp; \quad r=1 \\ \Rightarrow &amp; s=-2-3=-5 \\ \therefore &amp; s=-5\end{array}$<br/>Hence, $\mathrm{A}=\left[\begin{array}{ll}\mathrm{p} &amp; \mathrm{q} \\ \mathrm{r} &amp; \mathrm{s}\end{array}\right]=\left[\begin{array}{cc}-7 &amp; 26 \\ 1 &amp; -5\end{array}\right]$<br/>$\therefore$ Option (a) is correct.</div>


title: if-x-4-n-3-n-1-n-n-and-y-9-n-1-n-n-then-what-is-x-y-equal-to
slug: if-x-4-n-3-n-1-n-n-and-y-9-n-1-n-n-then-what-is-x-y-equal-to-19113
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $X=\left\{\left(4^{n}-3 n-1\right) \mid n \in N\right\}$ and $\mathrm{Y}=\{9(n-1) n \in N\}$, then<br/>what is $X \cup Y$ equal to?</div>

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

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

answer:
<div class="solution">Let $X=\left\{\left(4^{n}-3 n-1\right) \mid n \in N\right\}$<br/>and $Y=\{9(n-1) \mid n \in N\}$<br/>$\Rightarrow X=\{0,9,54, \ldots\}$ (By putting $n=1,2, \ldots .)$<br/>and $Y=\{0,9,18,27,36,54, \ldots$ ?<br/>(By putting $n=1,2, \ldots$ $\therefore X \cup Y=\{0,9,18,27,36,54, \ldots\}=Y$</div>


title: if-x-4-occurs-in-the-rth-term-in-the-expansion-of-x-4-x-3-1-15-then-what-is-the-value-of-r
slug: if-x-4-occurs-in-the-rth-term-in-the-expansion-of-x-4-x-3-1-15-then-what-is-the-value-of-r-16699
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}^{4}$ occurs in the rth term in the expansion of<br/>$\left(\mathrm{x}^{4}+\frac{1}{\mathrm{x}^{3}}\right)^{15}$, then what is the value of r?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">In the expansion of $\left(x^{4}+\frac{1}{x^{3}}\right)^{15}$, let $T_{r}$ is the $n_{\text {th }}$<br/>term<br/>$\mathrm{T}_{\mathrm{r}}=15_{\mathrm{C}_{\mathrm{r}-1}}\left(\mathrm{x}^{4}\right)^{15-\mathrm{r}+1}\left(\frac{1}{\mathrm{x}^{3}}\right)^{\mathrm{r}-1}$<br/>$=15_{\mathrm{C}_{\mathrm{r}-1}} \mathrm{x}^{64-4 \mathrm{r}-3 \mathrm{r}+3}=15_{\mathrm{C}_{\mathrm{r}-1}} \mathrm{x}^{67-7 \mathrm{r}}$<br/>$\mathrm{x}^{4}$ occurs in this term<br/>$\Rightarrow 4=67-7 \mathrm{r}$<br/>$\Rightarrow 7 \mathrm{r}=63$<br/>$\Rightarrow \quad \mathrm{r}=9$</div>


title: if-x-13-y-13-z-13-0-then-what-is-x-y-z-3-equal-to
slug: if-x-13-y-13-z-13-0-then-what-is-x-y-z-3-equal-to-35964
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">27</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x^{1 / 3}+y^{1 / 3}+z^{1 / 3}=0$ then what is $(x+y+z)^{3}$ 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">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 \mathrm{xy}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">27</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$x^{1 / 3}+y^{1 / 3}+z^{1 / 3}=0$<br/>Given formula, $(a+b)^{3}=a^{3}+b^{3}+3 a b(A+b)$<br/>so, $x^{1 / 3}=-\left(y^{1 / 3}+z^{1 / 3}\right)$<br/>Raising both the to the of cube $x=-\left\{y+z+3 y^{1 / 3} z^{1 / 3}\left(y^{1 / 3}+z^{1 / 3}\right)\right\}$<br/>$=-\left\{y+z+3 y^{1 / 3} z^{1 / 3}\left(-x^{1 / 3}\right)\right\}=-\left\{y+z-3 x^{1 / 3} y^{1 / 3} z^{1 / 3}\right\}$<br/>$=-y-z+3 x^{1 / 3} y^{1 / 3} z^{1 / 3} \Rightarrow x+y+z=3 x^{1 / 3} y^{1 / 3} z^{1 / 3}$<br/>or $x+y+z=3(x y z)^{1 / 3}$<br/>$\Rightarrow(x+y+z)^{3}=27 x y z$</div>


title: if-x-1101-2-and-y-110-2-then-what-is-the-value-of-x-2-y-2
slug: if-x-1101-2-and-y-110-2-then-what-is-the-value-of-x-2-y-2-24986
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=(1101)_{2}$ and $y=(110)_{2}$, then what is the value of $x^{2}-y^{2}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\begin{array}{llll}10 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\end{array}\right)_{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\begin{array}{llllll}1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\end{array}\right)_{2}$</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">$\left(\begin{array}{llllll}1 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 0 &amp; 1\end{array}\right)_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\begin{array}{llllll}1 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1 &amp; 0 &amp; 1\end{array}\right)_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\begin{array}{llllll}1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\end{array}\right)_{2}$</span> </div>

answer:
<div class="solution">Given, $x=(1101)_{2}=1 \times 2^{3}+1 \times 2^{2}+0 \times 2^{1}+1$<br/>$=8+4+1=13$<br/>and $\begin{aligned} y=(110)_{2}=&amp; 1 \times 2^{2}+1 \times 2^{1}+0 \times 2^{0} \\=4+2=6 \end{aligned}$<br/>$\therefore x^{2}-y^{2}=(13)^{2}-(6)^{2}=169-36=133$<br/>Now, \begin{array}{l|l|l}<br/>2 &amp; 133 &amp; \\<br/>\hline 2 &amp; 66 &amp; 1 \\<br/>\hline 2 &amp; 33 &amp; 0 \\<br/>\hline 2 &amp; 16 &amp; 1 \\<br/>\hline 2 &amp; 8 &amp; 0 \\<br/>\hline 2 &amp; 4 &amp; 0 \\<br/>\hline 2 &amp; 2 &amp; 0 \\<br/>\hline &amp; 1 &amp; 0<br/>\end{array}<br/>$\therefore 133=(10000101)_{2}$</div>


title: if-x-a-13-a-13-then-what-is-x-3-3-x-equal-to
slug: if-x-a-13-a-13-then-what-is-x-3-3-x-equal-to-38555
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}=\mathrm{a}^{1 / 3}-\mathrm{a}^{-1 / 3}$, then what is $\mathrm{x}^{3}+3 \mathrm{x}$ equal to?</div>

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

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

answer:
<div class="solution">In the given equation, Given formula, $(a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}$<br/>$x=a^{1 / 3}-a^{-1 / 3}$<br/>Raising both the sides to the power of cube $\begin{aligned} x^{3} &amp;=a-3 a^{2 / 3} \cdot a^{-1 / 3}+3 a^{1 / 3} \times a^{-2 / 3}-a^{-1} \\ &amp;=a-3 a^{1 / 3}+3 a^{-1 / 3}-a^{-1}=a-3\left(a^{1 / 3}-a^{-1 / 3}\right)-a^{-1} \end{aligned}$<br/>$\Rightarrow x^{3}=a-3 x-a^{-1}$ or $x^{3}+3 x=a-\frac{1}{a}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-x-a-is-a-factor-of-both-the-quadratic-polynomials-x-2-p-x-q-and-x-2-lx-m-where-p-q-l-and-m-are-constants-then-which-one-of-the-following-is-correct

slug:
if-x-a-is-a-factor-of-both-the-quadratic-polynomials-x-2-p-x-q-and-x-2-lx-m-where-p-q-l-and-m-are-constants-then-which-one-of-the-following-is-correct-13765

question:
<div class="question">If $(x+a)$ is a factor of both the quadratic polynomials $x^{2}+p x+q$ and $x^{2}+L x+m$, where $p, q, l$ and $m$ are constants, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$a=(m-q) /(l-p)(l \neq p)$</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">$a=(m+q) /(l+p)(l \neq-p)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$l=(m-q) /(a-p)(a \neq p)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$p=(m-q) /(a-l)(a \neq l)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a=(m-q) /(l-p)(l \neq p)$</span> </div>

answer:
<div class="solution">Given $(x+a)$ is a factor of quadratic polynomials $x^{2}+p x+q$ and $x^{2}+b x+m$<br/>then $a^{2}-a p+q=0$<br/>and $a^{2}-l a+m=0$<br/>(i) $-($ ii $) \Rightarrow-a p+q+l a-m=0$<br/>$\Rightarrow(I-p) a=m-q$<br/>$\Rightarrow a=\frac{m-q}{l-p}(l \neq p)$</div>


title: if-x-a-sec-cos-y-bsec-sin-z-ctan-then-what-is-a-2-x-2-b-2-y-2-c-2-z-2-equal-to
slug: if-x-a-sec-cos-y-bsec-sin-z-ctan-then-what-is-a-2-x-2-b-2-y-2-c-2-z-2-equal-to-23497
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}=\mathrm{a} \sec \theta \cos \phi, \mathrm{y}=\mathrm{bsec} \theta \sin \phi, \mathrm{z}=\operatorname{ctan} \theta$, then what<br/>is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}$ equal to?</div>

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

answer:
<div class="solution">As given:<br/>$\mathrm{x}=\mathrm{a} \sec \theta \cos \phi, \mathrm{y}=\mathrm{b} \sec \theta \sin \phi, \mathrm{z}=\mathrm{c} \tan \theta$<br/>So, $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=\frac{a^{2} \sec ^{2} \theta \cos ^{2} \phi}{a^{2}}$<br/>$+\frac{b^{2} \sec ^{2} \theta \sin ^{2} \phi}{b^{2}}-\frac{c^{2} \tan ^{2} \theta}{c^{2}}$<br/>$=\sec ^{2} \theta\left(\cos ^{2} \phi+\sin ^{2} \phi\right)-\tan ^{2} \theta=\sec ^{2} \theta-\tan ^{2} \theta=1$</div>


title: if-x-and-y-are-any-two-nonempty-sets-then-what-is-x-y-equal-to
slug: if-x-and-y-are-any-two-nonempty-sets-then-what-is-x-y-equal-to-53004
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $X$ and $Y$ are any two non-empty sets, then what is $(X-Y)$ ' equal to?</div>

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

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

answer:
<div class="solution">We know <br/>$X-Y=X \cap Y^{\prime}$<br/>$(X-Y)^{\prime}=\left(X \cap Y^{\prime}\right)^{\prime}$<br/>$=X^{\prime} \cup\left(Y^{\prime}\right)^{\prime}=X^{\prime} \cup Y$</div>


title: if-x-and-y-are-positive-and-x-y-1-then-what-is-tan-1-x-tan-1-y-equal-to
slug: if-x-and-y-are-positive-and-x-y-1-then-what-is-tan-1-x-tan-1-y-equal-to-19926
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x$ and $y$ are positive and $x y&gt;1$, then what is $\tan ^{-1} x+$ $\tan ^{-1} y$ equal to ?</div>

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

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

answer:
<div class="solution">$\tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{y}=\tan ^{-1}\left[\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right]$, when $\mathrm{xy}   &lt; 1$.<br/>And if $x &lt; 0, y &lt; 0$ and $x y&gt;1$, then $\tan ^{-1} x+\tan ^{-1} y=\pi+\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-x-and-y-are-the-matrices-of-order-2-2-each-and-2-x-3-y-7-7-0-13-and-3-x-2-y-9-4-13-13-then-what-is-y-equal-to

slug:
if-x-and-y-are-the-matrices-of-order-2-2-each-and-2-x-3-y-7-7-0-13-and-3-x-2-y-9-4-13-13-then-what-is-y-equal-to-21785

question:
<div class="question">If $X$ and $Y$ are the matrices of order $2 \times 2$ each and<br/>$2 X-3 Y=\left[\begin{array}{cc}-7 &amp; 0 \\ 7 &amp; -13\end{array}\right]$ and $3 X+2 Y=\left[\begin{array}{ll}9 &amp; 13 \\ 4 &amp; 13\end{array}\right]$, then what is $Y$ equal to?</div>

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

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

answer:
<div class="solution">Let $X$ and $Y$ be two matrices of order $2 \times 2$ each.<br/>Given, $2 X-3 Y=\left[\begin{array}{cc}-7 &amp; 0 \\ 7 &amp; -13\end{array}\right]$ ...(i)<br/>and $3 X+2 Y=\left[\begin{array}{ll}9 &amp; 13 \\ 4 &amp; 13\end{array}\right]$ ...(ii)<br/>On multiplying Eq. (i) by 3 and Eq. (ii) by 2, we get<br/>$6 X-9 Y=\left[\begin{array}{cc}-21 &amp; 0 \\ 21 &amp; -39\end{array}\right]$ ...(iii)<br/>$6 X+4 Y=\left[\begin{array}{cc}18 &amp; 26 \\ 8 &amp; 26\end{array}\right]$ ...(iv)<br/>On subtracting Eqs. (iii) from (iv), we get<br/>$13 Y=\left[\begin{array}{cc}39 &amp; 26 \\ -13 &amp; 65\end{array}\right] \Rightarrow Y=\left[\begin{array}{cc}3 &amp; 2 \\ -1 &amp; 5\end{array}\right]$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 2)']
answer: <div class="solution">It is obvious that, $\bar{x} &lt; \bar{z} &lt; \bar{y}$.</div>
source_db: MarksBatch2_P1.db
title:
if-x-and-y-are-the-means-of-two-distrubutions-such-that-x-y-and-z-is-the-mean-of-the-combined-distrubution-then-which-one-of-the-following-statements-

slug:
if-x-and-y-are-the-means-of-two-distrubutions-such-that-x-y-and-z-is-the-mean-of-the-combined-distrubution-then-which-one-of-the-following-statements-44165

question:
<div class="question">If $\bar{x}$ and $\bar{y}$ are the means of two distrubutions such that<br/>$\bar{x}   &lt; \bar{y}$ and $\bar{z}$ is the mean of the combined distrubution, then which one of the following statements is correc</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\bar{x}   &lt; \bar{z} &lt; \bar{v}$</span> </div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-x-axis-is-tangent-to-the-circle-x-2-y-2-2-gx-2-f-y-k-0-then-which-one-of-the-following-is-correct

slug:
if-x-axis-is-tangent-to-the-circle-x-2-y-2-2-gx-2-f-y-k-0-then-which-one-of-the-following-is-correct-56141

question:
<div class="question">If $x$ -axis is tangent to the circle $x^{2}+y^{2}+2 g x+2 f y+k=0$, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">Length of intercept on the $x$ -axis made by the circle<br/>$$<br/>x^{2}+y^{2}+2 g x+2 f y+k=0 \text { is } 2 \sqrt{g^{2}-k}<br/>$$<br/>Since, circle touches the $\mathrm{x}$ -axis therefore intercept on $\mathrm{x}=$ axis $=0$<br/>$$<br/>\begin{array}{l}<br/>\therefore \quad \sqrt{g^{2}-k}=0 \\<br/>\Rightarrow \quad g^{2}=k<br/>\end{array}<br/>$$</div>


title: if-x-cos-2-t-and-y-sin-2-t-then-what-is-d-x-2-d-2-y-equal-to
slug: if-x-cos-2-t-and-y-sin-2-t-then-what-is-d-x-2-d-2-y-equal-to-29842
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=\cos (2 \mathrm{t})$ and $y=\sin ^{2} \mathrm{t}$, then what is $\frac{d^{2} y}{d x^{2}}$ equal to?</div>

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

answer:
<div class="solution">Let $x=\cos 2 t$ and $y=\sin ^{2} t$<br/>Differentiate both the functions w.r.t. ' $\mathrm{t}^{\prime}$<br/>$\frac{\mathrm{dx}}{\mathrm{dt}}=-2 \sin 2 \mathrm{t}$ and $\frac{\mathrm{dy}}{\mathrm{dt}}=2 \sin t \cos \mathrm{t}=\sin 2 \mathrm{t}$<br/>$\Rightarrow \quad \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{dx} / \mathrm{dt}}=\frac{\sin 2 \mathrm{t}}{-2 \sin 2 \mathrm{t}}=-\frac{1}{2}$<br/>$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=0$</div>


title: if-x-cos-t-y-sin-t-then-what-is-dx-2-d-2-y-equal-to
slug: if-x-cos-t-y-sin-t-then-what-is-dx-2-d-2-y-equal-to-26631
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}=\cos \mathrm{t}, \mathrm{y}=\sin \mathrm{t}$, then what is $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{y}^{-3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{y}^{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\mathrm{y}^{-3}$</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">$-y^{3}$</span> </li> </ul>

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

answer:
<div class="solution">Given that $x=\cos t, y=\sin t$<br/>$\Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=-\sin \mathrm{t}$ and $\frac{\mathrm{dy}}{\mathrm{dt}}=\cot \mathrm{t}$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=-\frac{\cos \mathrm{t}}{\sin \mathrm{t}} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=-\cot \mathrm{t}$<br/>$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=\operatorname{cosec}^{2} \mathrm{t} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}=\operatorname{cosec}^{2} \mathrm{t} \frac{1}{-\sin \mathrm{t}}=-\frac{1}{\sin ^{3} \mathrm{t}}$<br/>$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=-\mathrm{y}^{-3}$</div>


title: if-x-cos-y-sin-2-is-perpendicular-to-the-line-x-y-3-then-what-is-one-of-the-value-of
slug: if-x-cos-y-sin-2-is-perpendicular-to-the-line-x-y-3-then-what-is-one-of-the-value-of-76331
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x \cos \theta+y \sin \theta=2$ is perpendicular to the line $x-y=3$, then what is one of the value of $\theta ?$</div>

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

answer:
<div class="solution">Consider a line<br/>$x \cos \theta+y \sin \theta=2$<br/>$\Rightarrow y \sin \theta=-x \cos \theta+2$<br/>$\Rightarrow y=-x \frac{\cos \theta}{\sin \theta}+\frac{2}{\sin \theta}$<br/>$\Rightarrow \mathrm{y}=-x \cot \theta+2 \operatorname{cosec} \theta$<br/>On comparing this equation with $y=m x+c$ we get slope of line $x \cos \theta+y \sin \theta=2$ is $-\cot \theta$<br/>Also, we have a line $x-y=3$ $\Rightarrow y=x-3$<br/>slope of line $x-y=3$ is 1 . Since, both the lines are perpendicular to each other Product of their slopes $=-1$ $\Rightarrow(-\cot \theta)(1)=-1$<br/>$\Rightarrow \cot \theta=1=\cot \frac{\pi}{4}$<br/>$\Rightarrow \theta=\frac{\pi}{4}$</div>


title: if-x-cos-y-sin-z-then-what-is-the-value-of-x-sin-y-cos-2
slug: if-x-cos-y-sin-z-then-what-is-the-value-of-x-sin-y-cos-2-12355
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x \cos \theta+y \sin \theta=z$, then what is the value of $(x \sin \theta-y \cos \theta)^{2} ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}-z^{2}$</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">$x^{2}-y^{2}-z^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}-y^{2}+z^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}+z^{2}$</span> </li> </ul>

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

answer:
<div class="solution">Here, $z=x \cos \theta+y \sin \theta$<br/>$z^{2}=x^{2} \cos ^{2} \theta+y^{2} \sin ^{2} \theta+2 x y \sin \theta \cos \theta$<br/>$\Rightarrow 2 x y \sin \theta \cos \theta=z^{2}-x^{2} \cos ^{2} \theta-y^{2} \sin ^{2} \theta$<br/>Let, $L=(x \sin \theta-y \cos \theta)^{2}$<br/>$\Rightarrow L=x^{2} \sin ^{2} \theta+y^{2} \cos ^{2} \theta-2 x y \sin \theta \cos \theta$<br/>$\Rightarrow L=x^{2} \sin ^{2} \theta+y^{2} \cos ^{2} \theta-\left[z^{2}-x^{2} \cos ^{2} \theta-y^{2} \sin ^{2} \theta\right]$<br/>$\Rightarrow L=x^{2}\left[\sin ^{2} \theta+\cos ^{2} \theta\right]+y^{2}\left[\sin ^{2} \theta+\cos ^{2} \theta\right]-z^{2}$<br/>$\Rightarrow L=x^{2}+y^{2}-z^{2}$</div>


title: if-x-d-y-y-d-x-y-d-y-y-1-1-and-y-x-0-then-what-is-y-3-equal-to
slug: if-x-d-y-y-d-x-y-d-y-y-1-1-and-y-x-0-then-what-is-y-3-equal-to-52978
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2017 (Phase 1)']
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">If $x d y=y(d x+y d y) ; y(1)=1$ and $y(x)&gt;0$, then what is $y(-$ 3) equal to? $\quad$</div>

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

answer:
<div class="solution">$x d y=y d x+y^{2} d y$<br/>$\Rightarrow x \cdot \frac{d y}{d x}=y+y^{2} \cdot \frac{d y}{d x} \Rightarrow \frac{y-x \cdot \frac{d y}{d x}}{y^{2}}=\frac{-d y}{d x}$<br/>Integrating both sides<br/>$\frac{x}{y}=-y+c$<br/>Given, $x=1, y=1$<br/>$\Rightarrow \frac{1}{1}=-1+c \quad \Rightarrow c=2$<br/>$\therefore \frac{\mathrm{x}}{\mathrm{y}}+\mathrm{y}=2$<br/>$\Rightarrow x+y^{2}=2 y$<br/>$\Rightarrow-3+y^{2}-2 y=0$<br/>$\Rightarrow y^{2}-2 y-3=0$<br/>$y=3,-1$<br/>$\therefore y=3$</div>


title: if-x-follows-a-binomial-distribution-with-parameters-n-100-and-p-13-then-p-x-r-is-maximum-when
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-x-follows-a-binomial-distribution-with-parameters-n-100-and-p-13-then-p-x-r-is-maximum-when-72950

question:
<div class="question">If $\mathrm{X}$ follows a binomial distribution with parameters $\mathrm{n}=100$ and $\mathrm{p}=1 / 3$, then $\mathrm{P}(\mathrm{X}=\mathrm{r})$ is maximum when</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad r=16$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad r=32$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad r=33$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad r=34$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad r=33$</span> </div>

answer:
<div class="solution">$\mathrm{P}(\mathrm{X}=r)$ will be maximum when $r$ is mode. There are 2 cases.<br/>(i) If $(n+1) \mathrm{P}$ is an integer, binomial distribution is bimodal and two modal values are $(n+1) \mathrm{P}$ and $(n+1) \mathrm{P}-1$<br/>(ii) If $(n+1) \mathrm{P}$ is not an integer, then modal value is integral part of $(n+1) \mathrm{P}$.<br/>Here, $(n+1) \mathrm{P}=(100+1) \frac{1}{3}=101 \times \frac{1}{3}=33.69$, which is<br/>not integer. $\therefore \mathrm{P}(\mathrm{X}=r)$ is maximum at integral part of $33.69$ i.e., 33 .</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-x-i-0-y-i-0-i-1-2-3-n-are-the-values-of-two-variable-x-and-y-with-geometric-mean-pand-q-respectively-then-the-geometric-mean-of-y-x-is

slug:
if-x-i-0-y-i-0-i-1-2-3-n-are-the-values-of-two-variable-x-and-y-with-geometric-mean-pand-q-respectively-then-the-geometric-mean-of-y-x-is-16573

question:
<div class="question">If $\mathrm{x}_{\mathrm{i}}&gt;0, \mathrm{y}_{\mathrm{i}}&gt;0(\mathrm{i}=1,2,3, \ldots \ldots \mathrm{n})$ are the values of two variable $X$ and Y with geometric mean Pand Q respectively, then the geometric mean of $\frac{\mathrm{X}}{\mathrm{Y}}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">antilog $\left(\frac{\mathrm{P}}{\mathrm{Q}}\right)$</span> </div>

answer:
<div class="solution">$\left(x_{1} x_{2} \ldots \ldots \ldots x_{n}\right)^{1 / n}=\mathrm{P}$<br/>$\left(y_{1} y_{2} \ldots \ldots . y_{n}\right)^{1 / n}=\mathrm{Q}$<br/>$\left(\frac{x_{1} \cdot x_{2} \ldots \ldots \ldots \ldots x_{n}}{y_{1} \cdot y_{2} \ldots \ldots \ldots . y_{n}}\right)^{1 / n}=\frac{\left(x_{1} \cdot x_{2} \ldots \ldots \ldots \ldots x_{n}\right)^{1 / n}}{\left(y_{1} y_{2} \ldots \ldots \ldots . y_{n}\right)^{1 / n}}=\frac{\mathrm{P}}{\mathrm{Q}}$</div>


title: if-x-i-2-110-n-x-i-5-20-then-what-is-the-mean
slug: if-x-i-2-110-n-x-i-5-20-then-what-is-the-mean-34567
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sum\left(x_{i}-2\right)=110, \quad \sum^{n}\left(x_{i}-5\right)=20$, then what is the<br/>mean?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$11 / 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 / 11$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$17 / 3$</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">$17 h$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$17 / 3$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \sum_{t=1}^{n}\left(x_{i}-2\right)=110 \\ &amp; \therefore x_{1}+x_{2}+\ldots+x_{n}-2 n=110 \\ &amp; \Rightarrow x_{1}+x_{2}+\ldots+x_{n}=2 n+110 \\ &amp; \text { and } \sum_{i=1}^{n}\left(x_{i}-5\right)=20 \\ &amp; \Rightarrow x_{1}+x_{2}+\ldots+x_{n}-5 n=20 \\ &amp; \Rightarrow x_{1}+x_{2}+\ldots+x_{n}=5 n+20 \end{aligned}$<br/>From equations (i) and (ii), we get $5 n+20=2 n+110$<br/>$\Rightarrow 3 n=90 \Rightarrow n=30$<br/>Now, mean $=\frac{x_{1}+x_{2}+\ldots x_{n}}{n}$<br/>$=\frac{5 \times 30+20}{30}=\frac{170}{30}=\frac{17}{3}$</div>


title: if-x-i-y-6-i-4-20-3-i-3-i-3-1-1-i-then-what-is-x-i-y-equal-to
slug: if-x-i-y-6-i-4-20-3-i-3-i-3-1-1-i-then-what-is-x-i-y-equal-to-75837
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x+i y=\left|\begin{array}{ccc}6 i &amp; -3 i &amp; 1 \\ 4 &amp; 3 i &amp; -1 \\ 20 &amp; 3 &amp; i\end{array}\right|$, then what is $x-i y$ equal to?</div>

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

answer:
<div class="solution">Let $x+i y=\left|\begin{array}{ccc}6 i &amp; -3 i &amp; 1 \\ 4 &amp; 3 i &amp; -1 \\ 20 &amp; 3 &amp; i\end{array}\right|$<br/>$\Rightarrow \quad x+i y=6 i\left(3 i^{2}+3\right)+3 i(4 i+20)+1(12-60 i)$<br/>$\quad=-12+60 i+12-60 i=0\left(\because i^{2}=-1\right)$<br/>$\Rightarrow \quad x+i y=0$<br/>$\Rightarrow \quad x=0$ and $y=0$<br/>Hence, $x-i y=0-i(0)=0$</div>


title: if-x-i-y-j-z-k-is-a-unit-vector-and-x-y-z-3-2-3-then-what-is-the-value-of-z
slug: if-x-i-y-j-z-k-is-a-unit-vector-and-x-y-z-3-2-3-then-what-is-the-value-of-z-21659
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x} \hat{\mathbf{i}}+\mathrm{y} \hat{\mathbf{j}}+\mathrm{z} \hat{\mathbf{k}}$ is a unit vector and $\mathrm{x}: \mathrm{y}: \mathrm{z}=\sqrt{3}: 2: 3$, then what is the value of $z ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{16}$</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">$\frac{3}{4}$</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">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Let $x \hat{i}+y \hat{j}+z \hat{k}$ is a unit vector.<br/>$\therefore \quad x^{2}+y^{2}+z^{2}=1$<br/>Given $x: y: z=\sqrt{3}: 2: 3$<br/>$\Rightarrow x=\sqrt{3} k, y=2 k$ and $z=3 k$<br/>$\therefore \quad(\sqrt{3} k)^{2}+(2 k)^{2}+(3 k)^{2}=1$<br/>$\Rightarrow 3 k^{2}+4 k^{2}+9 k^{2}=1$<br/>$\Rightarrow k^{2}=\frac{1}{16} \Rightarrow k=\frac{1}{4}$<br/>Hence, $z=3 k=3 \times \frac{1}{4}=\frac{3}{4}$</div>


title: if-x-is-a-poisson-variate-such-that-2-p-x-1-5-p-x-5-2-p-x-3-then-the-standard-deviation-of-x-is
tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2019 (21 Apr Shift 2)']
source_db: MarksBatch2_P1.db
slug:
if-x-is-a-poisson-variate-such-that-2-p-x-1-5-p-x-5-2-p-x-3-then-the-standard-deviation-of-x-is-19621

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> is a Poisson variate such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>5</mn><mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mn>5</mn><mo>)</mo><mo>+</mo><mn>2</mn><mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo>)</mo><mo>,</mo></math> then the standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</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> </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>2</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p>It is given that,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>5</mn><mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mn>5</mn><mo>)</mo><mo>+</mo><mn>2</mn><mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mn>3</mn><mo>)</mo></math>,</p><p>The formula for Poisson <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo>=</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo>-</mo><mi>λ</mi></mrow></msup><msup><mi>λ</mi><mi>r</mi></msup></mrow><mrow><mi>r</mi><mo>!</mo></mrow></mfrac></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>2</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>λ</mi></mrow></msup><mi>λ</mi></mrow><mrow><mn>1</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>λ</mi></mrow></msup><msup><mi>λ</mi><mn>5</mn></msup></mrow><mrow><mn>5</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>λ</mi></mrow></msup><msup><mi>λ</mi><mn>3</mn></msup></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mo>=</mo><mfrac><msup><mi>λ</mi><mn>4</mn></msup><mn>24</mn></mfrac><mo>+</mo><mfrac><msup><mi>λ</mi><mn>2</mn></msup><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>λ</mi><mn>4</mn></msup><mo>+</mo><mn>24</mn><msup><mi>λ</mi><mn>2</mn></msup></mrow><mn>72</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>144</mn><mo>=</mo><mn>3</mn><msup><mi>λ</mi><mn>4</mn></msup><mo>+</mo><mn>24</mn><msup><mi>λ</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>λ</mi><mn>4</mn></msup><mo>+</mo><mn>8</mn><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>48</mn><mo>=</mo><mn>0</mn></math></p><p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn mathvariant="italic">2</mn></msup><mo mathvariant="italic">=</mo><mi>x</mi></math>,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>-</mo><mn>48</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>(</mo><mi>x</mi><mo>+</mo><mn>12</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>4</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mo>-</mo><mn>12</mn></math></p><p>The standard deviation is,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>λ</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>λ</mi></msqrt><mo>=</mo><msqrt><mn>2</mn></msqrt></math></p></div>


title: if-x-is-an-integer-and-satisfies-9-4-x-1-19-then-x-is-an-element-of-which-one-of-the-following-sets
tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-x-is-an-integer-and-satisfies-9-4-x-1-19-then-x-is-an-element-of-which-one-of-the-following-sets-99116

question:
<div class="question">If $x$ is an integer and satisfies $9   &lt; 4 x-1 \leq 19$, then $x$ is an element of which one of the following sets?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{3,4\}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{2,3,4\}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\{3,4,5\}$</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">D</span> <span class="option-data">$\{2,3,4,5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\{3,4,5\}$</span> </div>

answer:
<div class="solution">Given, $9   &lt; 4 x-1 \leq 19$<br/>$\Rightarrow 9 &lt; 4 x-1$ and $4 x-1 \leq 19$<br/>$\Rightarrow 9+1 &lt; 4 x$ and $4 x \leq 19+1$<br/>$\Rightarrow x&gt;\frac{5}{2}$ and $x \leq 5$<br/>$\therefore x \in\{3,4,5\}$</div>


title: if-x-is-anyreal-number-then-1-x-4-x-2-belongs-to-which-one-of-the-following-intervals
slug: if-x-is-anyreal-number-then-1-x-4-x-2-belongs-to-which-one-of-the-following-intervals-51216
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If x is anyreal number, then $\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}$ belongs to which one of the following intervals?</div>

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

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

answer:
<div class="solution">$\mathrm{y}=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}} \Rightarrow \mathrm{y} \geq 0 .$<br/>Also, $\mathrm{y}=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}=\frac{1}{\mathrm{x}^{2}+\frac{1}{\mathrm{x}^{2}}}$<br/>$\Rightarrow \mathrm{y} \leq \frac{1}{2}$<br/>$\therefore \frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}$ belongs to $\left(0, \frac{1}{2}\right]$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-x-is-changed-to-hu-and-y-to-b-kv-then-which-one-of-the-following-is-the-correct-relation-between-the-regression-coefficients-b-xy-and-b-uv

slug:
if-x-is-changed-to-hu-and-y-to-b-kv-then-which-one-of-the-following-is-the-correct-relation-between-the-regression-coefficients-b-xy-and-b-uv-82357

question:
<div class="question">If $\mathrm{X}$ is changed to $+\mathrm{hU}$ and $\mathrm{Y}$ to $\mathrm{b}+\mathrm{kV}$, then which one of the following is the correct relation between the regression coefficients $\mathrm{b}_{\mathrm{XY}}$ and $\mathrm{b}_{\mathrm{UV}}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{hb}_{\mathrm{XY}}=\mathrm{k} \mathrm{b}_{\mathrm{UV}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{k} \mathrm{b}_{\mathrm{XY}}=\mathrm{h} \mathrm{b}$</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">$b_{X Y}=b_{U V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{k}^{2} \mathrm{~b}_{\mathrm{XY}}=\mathrm{h}^{2} \mathrm{~b}_{\mathrm{UV}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{k} \mathrm{b}_{\mathrm{XY}}=\mathrm{h} \mathrm{b}$</span> </div>

answer:
<div class="solution">If $\mathrm{X}$ is changed to $\mathrm{a}+\mathrm{hU}$ and $\mathrm{Y}$ to $\mathrm{b}+\mathrm{k} \mathrm{V}$, then<br/>$\mathbf{b}_{\mathrm{XY}}=\left|\frac{\mathrm{h}}{\mathrm{k}}\right| \mathrm{b}_{\mathrm{UV}}$<br/>$\Rightarrow \mathrm{kb}_{\mathrm{XY}}=\mathrm{h} \cdot \mathrm{b}_{\mathrm{UV}}$</div>


title: if-x-is-real-and-x-2-3-x-2-0-x-2-3-x-4-0-then-which-on-of-the-following-is-correct
slug: if-x-is-real-and-x-2-3-x-2-0-x-2-3-x-4-0-then-which-on-of-the-following-is-correct-93617
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}$ is real and $\mathrm{x}^{2}-3 \mathrm{x}+2&gt;0, \mathrm{x}^{2}-3 \mathrm{x}-4 \leq 0$, then which on<br/>of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad-1 \leq x \leq 4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad 2 \leq x \leq 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-1   &lt; x \leq 1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-1 \leq x   &lt; 1$ or $2 &lt; x \leq 4$</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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-1 \leq x   &lt; 1$ or $2 &lt; x \leq 4$</span> </div>

answer:
<div class="solution">Consider first: $\mathrm{x}^{2}-3 \mathrm{x}+2&gt;0$ $\Rightarrow \quad(x-1)(x-2)&gt;0$...(1)<br/>$\Rightarrow \quad x   &lt; 1$ or $x&gt;2$<br/>and $x^{2}-3 x-4 \leq 0$<br/>$\Rightarrow \quad(x-4)(x+1) \leq 0$<br/>$\Rightarrow-1 \leq x \leq 4$...(2)<br/>Combining $(1)$ and $(2)$ $-1 \leq x &lt; 1$ or $2 &lt; x \leq 4$<br/>Drawing on number line:<img src="https://cdn-question-pool.getmarks.app/pyq/nda/_82ZpE0jpkNTPhLguMQI_zWrYtastoG8hX_H8486LA4.original.fullsize.png"/></div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2022 (18 Jul Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-x-is-the-mean-of-n-observations-x-1-x-2-x-n-then-the-mean-of-the-absolute-deviations-of-these-observations-from-x-is

slug:
if-x-is-the-mean-of-n-observations-x-1-x-2-x-n-then-the-mean-of-the-absolute-deviations-of-these-observations-from-x-is-82558

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover></math> is the mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></math> then the mean of the absolute deviations of these observations from <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover></math> is</div>

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

answer:
<div class="solution"><blockquote><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover></math> is the mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></math> then the mean of the absolute deviations of these observations from <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover></math> is the mean deviation of the data.</p><p>Here,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><mi>n</mi></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math></p><p>Mean deviation</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced close="|" open="|"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mo>+</mo><mfenced close="|" open="|"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><mfenced close="|" open="|"><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced></mrow><mi>n</mi></mfrac></math></p></blockquote></div>


title: if-x-is-the-number-of-tails-in-three-tosses-of-a-coin-then-determine-the-standard-deviation-of-x
tags: ['Mathematics', 'Probability']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
slug:
if-x-is-the-number-of-tails-in-three-tosses-of-a-coin-then-determine-the-standard-deviation-of-x-78110

question:
<div class="question">If $X$ is the number of tails in three tosses of a coin, then determine the standard deviation of $\mathrm{X}$.</div>

answer:
<div class="solution">Let, random variable $\mathrm{X}$ is the number of tails in three tosses of the coin.<br/>$$<br/>\begin{aligned}<br/>&amp;\therefore \quad \mathrm{X}=0,1,2,3 \\<br/>&amp;\Rightarrow \quad \mathrm{P}(\mathrm{X}=\mathrm{x})={ }^{\mathrm{n}} C_{\mathrm{x}}(\mathrm{p})^{\mathrm{x}} \mathrm{q}^{\mathrm{n}-\mathrm{x}} \\<br/>&amp;\text { where } \\<br/>&amp;\quad \mathrm{n}=3, \mathrm{p}=1 / 2, \mathrm{q}=1 / 2 \text { and } \mathrm{x}=0,1,2,3 .<br/>\end{aligned}<br/>$$<br/>\begin{array}{|l|l|l|l|l|}<br/>\hline \mathbf{X} &amp; \mathbf{0} &amp; \mathbf{1} &amp; \mathbf{2} &amp; \mathbf{3} \\<br/>\hline \mathrm{P}(\mathrm{X}) &amp; \frac{1}{8} &amp; \frac{3}{8} &amp; \frac{3}{8} &amp; \frac{1}{8} \\<br/>\mathrm{XP}(\mathrm{X}) &amp; 0 &amp; \frac{3}{8} &amp; \frac{3}{4} &amp; \frac{3}{8} \\<br/>\mathrm{X}^2 \mathrm{P}(\mathrm{X}) &amp; 0 &amp; \frac{3}{8} &amp; \frac{3}{2} &amp; \frac{9}{8} \\<br/>\hline<br/>\end{array}<br/>We know that, $\operatorname{Var}(X)=E\left(X^2\right)-[E(X)]^2$<br/>Also $\mathrm{E}\left(\mathrm{X}^2\right)$<br/>$$<br/>\begin{aligned}<br/>&amp;=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{x}_{\mathrm{i}}^2 \mathrm{P}\left(\mathrm{X}_{\mathrm{i}}\right)=0+\frac{3}{8}+\frac{3}{2}+\frac{9}{8}=\frac{24}{8}=3 \\<br/>&amp;\text { and }[\mathrm{E}(\mathrm{X})]^2 \\<br/>&amp;=\left[\sum_{\mathrm{i}=0}^{\mathrm{n}} \mathrm{x}_{\mathrm{i}}^2 \mathrm{P}\left(\mathrm{x}_{\mathrm{i}}\right)\right]^2=\left[0+\frac{3}{8}+\frac{3}{4}+\frac{3}{8}\right]^2=\left[\frac{12}{8}\right]^2=\frac{9}{4} \\<br/>&amp;\left.\therefore \operatorname{Var}(\mathrm{X})=3-\frac{9}{4}=\frac{3}{4} \quad \text { [using Eq. (i) }\right]<br/>\end{aligned}<br/>$$<br/>and S.D of $X=\sqrt{\operatorname{Var}(X)}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$</div>


title: if-x-k-sin-and-y-k-1-cos-then-what-is-the-derivative-of-y-with-respect-to-x-at-2
slug: if-x-k-sin-and-y-k-1-cos-then-what-is-the-derivative-of-y-with-respect-to-x-at-2-71214
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=k(\theta+\sin \theta)$ and $y=k(1+\cos \theta)$, then what is the derivative of $y$ with respect to $x$ at $\theta=\pi / 2 ?$</div>

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

answer:
<div class="solution">Let $x=k(\theta+\sin \theta)$ and $y=k(1+\cos \theta)$<br/>Differentiate both the functions w.r.t. ' $\theta$ '<br/>$\Rightarrow \quad \frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{k}(1+\cos \theta)$<br/>and $\frac{\mathrm{dy}}{\mathrm{d} \theta}=-\mathrm{k} \sin \theta$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{d} \mathrm{x}}=\frac{\mathrm{d} \mathrm{y} / \mathrm{d} \theta}{\mathrm{d} \mathrm{x} / \mathrm{d} \theta}$<br/>$=\frac{-\mathrm{k} \sin \theta}{\mathrm{k}(1+\cos \theta)}=\frac{-2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}}=-\tan \frac{\theta}{2}$<br/>$\left(\because \sin 2 \theta=2 \sin \theta \cos \theta\right.$ and $\left.\cos 2 \theta=2 \cos ^{2} \theta-1\right)$<br/>$\Rightarrow\left(\frac{d y}{d x}\right)_{\theta=\frac{\pi}{2}}=-\tan \frac{\pi}{4}=-1$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-x-l-n-z-y-y-l-n-x-z-2-z-l-n-y-x-y-4-n-y-for-any-x-1-y-1-and-z-1-then-which-one-of-the-following-is-correct

slug:
if-x-l-n-z-y-y-l-n-x-z-2-z-l-n-y-x-y-4-n-y-for-any-x-1-y-1-and-z-1-then-which-one-of-the-following-is-correct-86304

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ell \mathrm{n} \mathrm{y}$ is the $\mathrm{AM}$ of $\ell \mathrm{n} \mathrm{x}, \ell \mathrm{n} \mathrm{x}, \ln \mathrm{x}$ and $\ell \mathrm{n} \mathrm{z}$</span> </div>

answer:
<div class="solution">$x^{\ln \left(\frac{y}{z}\right)} \cdot y^{\ln (x z)^{2}} \cdot z^{\ln \left(\frac{x}{y}\right)}=y^{4 \ln y}$<br/>$\Rightarrow \ln \left[x^{ \left.\ln \left(\frac{y}{z}\right)\right]}+\ln \left[y^{\ln (x z)^{2}}\right]+\ln \left[z^{ \left.\ln \left(\frac{x}{y}\right)\right]=\ln \left[y^{4 \ln y}\right]}\right.\right.$<br/>$\Rightarrow\left[\ln \left(\frac{y}{z}\right) \ln x\right]+[2 \ln (x z) \ln y]+\left[\ln \left(\frac{x}{y}\right) \ln z\right]=4[\ln y]^{2}$<br/>$\Rightarrow \ln x[\ln y-\ln z]+2 \ln y[\ln x+\ln z]$<br/>$\quad+\ln z[\ln x-\ln y]=4[\ln y]^{2}$<br/>$\Rightarrow 3 \ln x+\ln z=4 \ln y$<br/>$\Rightarrow \frac{\ln x+\ln x+\ln x+\ln z}{4}=\ln y$<br/>$\therefore \quad$ lny is the AM of $\ln x, \ln x, \ln x, \ln x \&amp; \ln z .$</div>


title: if-x-lo-g-10-1-2-x-x-lo-g-10-5-lo-g-10-6-then-x-is-equal-to
slug: if-x-lo-g-10-1-2-x-x-lo-g-10-5-lo-g-10-6-then-x-is-equal-to-88194
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6$ then $x$ is equal to</div>

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

answer:
<div class="solution">$\begin{aligned} &amp; x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6 \\ &amp; \Rightarrow x-x \log _{10} 5=\log _{10} 6-\log _{10}\left(1+2^{x}\right) \\ &amp; \Rightarrow x\left(1-\log _{10} 5\right)=\log _{10} 6-\log _{10}\left(1+2^{x}\right) \\ &amp; \Rightarrow x\left(\log _{10} 10-\log _{10} 5\right)=\log _{10}\left(\frac{6}{1+2^{x}}\right) \\ &amp; \Rightarrow x\left(\log _{10}\left(\frac{10}{5}\right)\right)=\log _{10}\left(\frac{6}{1+2^{x}}\right) \\ \Rightarrow &amp; x \log _{10} 2=\log _{10}\left(\frac{6}{1+2^{x}}\right) \end{aligned}$</div>


title: if-x-lo-g-15-1-3-x-x-lo-g-15-5-lo-g-15-12-where-x-is-an-integer-then-what-is-x-equal-to
slug: if-x-lo-g-15-1-3-x-x-lo-g-15-5-lo-g-15-12-where-x-is-an-integer-then-what-is-x-equal-to-65339
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x+\log _{15}\left(1+3^{x}\right)=x \log _{15} 5+\log _{15} 12$, where $x$ is an integer, then what is $x$ equal to?</div>

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

answer:
<div class="solution">$x+\log _{15}\left(1+3^{x}\right)=x \log _{15} 5+\log _{15} 12$<br/>$\Rightarrow \quad x \cdot \log _{15} 15+\log _{15}\left(1+3^{x}\right)=x \log _{15} 5+\log _{15} 12$<br/>$\left(\because \log _{15} 15=1\right)$<br/>$\Rightarrow \log _{15} 15^{x}+\log _{15}\left(1+3^{x}\right)=\log _{15} 5^{x}+\log _{15} 12$<br/>$\Rightarrow \log _{15} 15^{x}\left(1+3^{x}\right)=\log _{15}\left(5^{x} \times 12\right)$<br/>$(\because \log \mathrm{a}+\log \mathrm{b}=\log \mathrm{ab})$<br/>$\Rightarrow \quad 15^{x}\left(1+3^{x}\right)=5^{x} \times 12$<br/>$3^{x}\left(1+3^{x}\right)=12$<br/>$3^{x}+3^{2 x}=12$<br/>$\mathrm{x}=1$ satisfies the above equation.</div>


title: if-x-lo-g-t-and-y-t-2-1-then-what-is-d-x-2-d-2-y-at-t-1-equal-to
slug: if-x-lo-g-t-and-y-t-2-1-then-what-is-d-x-2-d-2-y-at-t-1-equal-to-32077
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=\log t$ and $y=t^{2}-1$, then what is $\frac{d^{2} y}{d x^{2}}$ at $t=1$ 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">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $x=\log t$ and $y=t^{2}-1$<br/>$x=\log t$<br/>$\Rightarrow 2 x=2 \log t$<br/>$\Rightarrow 2 x=\log t^{2}$<br/>$\Rightarrow 2 x=\log (y+1) \Rightarrow e^{2 x}=y+1$<br/>On differentiating w.r.t. $x$, twice, we get<br/>$e^{2 x} 2=\frac{d y}{d x} \Rightarrow 4 e^{2 x}=\frac{d^{2} y}{d x^{2}}$<br/>At $t=1, x=0$<br/>$\frac{d^{2} y}{d x^{2}}=4 e^{2(0)}=4 \quad\left(\because e^{0}=1\right)$</div>


title: if-x-m-y-m-1-such-that-d-x-d-y-y-x-then-what-should-be-the-value-of-m
slug: if-x-m-y-m-1-such-that-d-x-d-y-y-x-then-what-should-be-the-value-of-m-13671
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x^{m}+y^{m}=1$ such that $\frac{d y}{d x}=-\frac{x}{y}$, then what should be the value of $\mathrm{m} ?$</div>

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

answer:
<div class="solution">Let $x^{\mathrm{m}}+\mathrm{y}^{\mathrm{m}}=1$<br/>Differentiate both the sides w.r.t $^{\prime} x^{\prime}$<br/>$\mathrm{m} \cdot \mathrm{x}^{\mathrm{m}-1}+\mathrm{m} \cdot \mathrm{y}^{\mathrm{m}-1} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=0$<br/>$\Rightarrow \quad \mathrm{m} \mathrm{x}^{\mathrm{m}-1}=-\mathrm{m} \mathrm{y}^{\mathrm{m}-1} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$<br/>$\Rightarrow \frac{\mathrm{x}^{\mathrm{m}-1}}{\mathrm{y}^{\mathrm{m}-1}}=-\frac{\mathrm{dy}}{\mathrm{dx}}$<br/>$\Rightarrow \left(\frac{x}{y}\right)^{m-1}=\frac{x}{y} \quad\left(\because \frac{d y}{d x}=-\frac{x}{y}\right)$<br/>$\Rightarrow \quad m-1=1 \Rightarrow m=2$</div>


title: if-x-r-sin-cos-y-r-sin-sin-and-z-r-cos-then-x-2-y-2-z-2-is-independent-of-which-of-the-following
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-x-r-sin-cos-y-r-sin-sin-and-z-r-cos-then-x-2-y-2-z-2-is-independent-of-which-of-the-following-68590

question:
<div class="question">If $\mathrm{x}=\mathrm{r} \sin \theta \cos \phi, \mathrm{y}=\mathrm{r} \sin \theta \sin \phi$ and $\mathrm{z}=\mathrm{r} \cos \theta$, then<br/>$\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}$ is independent of which of the following?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\theta, \phi$</span> </div>

answer:
<div class="solution">As given, $x=r \sin \theta \cos \phi, y=r \sin \theta \sin \phi$.<br/>and $z=r \cos \theta$<br/>Now, $x^{2}+y^{2}+z^{2}=r^{2} \sin ^{2} \theta \cos ^{2} \phi+r^{2} \sin ^{2} \phi \sin ^{2} \theta$<br/>$+r^{2} \cos ^{2} \theta$<br/>$=r^{2} \sin ^{2} \theta\left(\sin ^{2} \phi+\cos ^{2} \phi\right)+r^{2} \cos ^{2} \theta$<br/>$=r^{2} \sin ^{2} \theta+r^{2} \cos ^{2} \theta$<br/>$=r^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)$<br/>$=r^{2}$<br/>Thus, $x^{2}+y^{2}+z^{2}$ is independent of $\theta$ and $\phi$.</div>


title: if-x-sin-7-0-sin-5-0-and-y-cos-6-0-cos-8-0-then-what-is-xy-equal-to
slug: if-x-sin-7-0-sin-5-0-and-y-cos-6-0-cos-8-0-then-what-is-xy-equal-to-22285
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{x}=\sin 70^{\circ} \sin 50^{\circ}$ and $\mathrm{y}=\cos 60^{\circ} \cos 80^{\circ}$, then what is $\mathrm{xy}$<br/>equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 16$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 8$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 16$</span> </div>

answer:
<div class="solution">$x=\sin 70^{\circ} \cdot \sin 50^{\circ}$ and $y=\cos 60^{\circ} \cdot \cos 80^{\circ}$<br/>$\begin{aligned} \Rightarrow x y=&amp; \cos 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 50^{\circ} \cdot \cos 80^{\circ} \\ x y=\frac{1}{2} \cdot &amp; \sin (90-20) \cdot \sin (90-40) \cdot \cos 80 \\ \Rightarrow x y=&amp; \frac{1}{2} \cdot \cos 20 \cdot \cos 40 \cdot \cos 80 \\(\because \sin (90-x)=\cos x) \\ \Rightarrow x y=&amp; \frac{1}{2} \cdot \cos 20^{\circ} \cdot \cos (60-20)^{\circ} \cdot \cos (60+20)^{\circ} \\ \Rightarrow x y=&amp; \frac{1}{2}\left[\frac{1}{4} \cos 3\left(20^{\circ}\right)\right]=\frac{1}{2} \times \frac{1}{4} \times \cos 60^{\circ}=\frac{1}{16} . \\ &amp;\left[\because \cos \theta \cdot \cos (60-\theta) \cdot \cos (60+\theta)=\frac{1}{4} \cos 3 \theta\right] \\</div>


title: if-x-sin-cos-and-y-sin-cos-then-what-is-the-value-of-x-4-4-x-2-y-2-x-2-4-y-2-4-y-1
slug: if-x-sin-cos-and-y-sin-cos-then-what-is-the-value-of-x-4-4-x-2-y-2-x-2-4-y-2-4-y-1-26612
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=\sin \theta+\cos \theta$ and $y=\sin \theta \cdot \cos \theta$, then what is the value<br/>of $x^{4}-4 x^{2} y-2 x^{2}+4 y^{2}+4 y+1 ?$</div>

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

answer:
<div class="solution">Let $x=\sin \theta+\cos \theta$ and $y=\sin \theta \cdot \cos \theta$<br/>Now, $x^{4}-4 x^{2} y-2 x^{2}+4 y^{2}+4 y+1$<br/>$=(\sin \theta+\cos \theta)^{4}-4(\sin \theta+\cos \theta)^{2} y-$<br/>$\quad 2(\sin \theta+\cos \theta)^{2}+4 y^{2}+4 y+1$<br/>$=\left(\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta\right)^{2}-$<br/>$\quad 4\left(\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta\right) y$<br/>$\quad-2\left(\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta\right)+4 y^{2}+4 y+1$<br/>$=(1+2 y)^{2}-4(1+2 y) y-2(1+2 y)+4 y^{2}+4 y+1$<br/>$=1+4 y^{2}+4 y-4 y-8 y^{2}-2-4 y+4 y^{2}+4 y+1=0$</div>


title: if-x-sin-t-t-cos-t-and-y-t-sin-t-cos-t-then-what-is-d-x-d-y-at-point-t-2
slug: if-x-sin-t-t-cos-t-and-y-t-sin-t-cos-t-then-what-is-d-x-d-y-at-point-t-2-17225
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=\sin t-t \cos t$ and $y=t \sin t+\cos t$, then what is $\frac{d y}{d x}$ at point $\mathrm{t}=\frac{\pi}{2} ?$</div>

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

answer:
<div class="solution">As given :<br/>$\mathrm{x}=\sin \mathrm{t}-\mathrm{t} \cos \mathrm{t}$ and $\mathrm{y}=\mathrm{t} \sin \mathrm{t}+\cos \mathrm{t}$<br/>On differentiating w.r.t. t, we get<br/>$\frac{\mathrm{dx}}{\mathrm{dt}}=\cos \mathrm{t}-\{\cos \mathrm{t}+\mathrm{t}(-\sin \mathrm{t})\}$<br/>$\Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\cos \mathrm{t}-\cos \mathrm{t}+\mathrm{t} \sin \mathrm{t}=\mathrm{t} \sin \mathrm{t}$<br/>and, $\frac{d y}{d t}=t \cos t+\sin t-\sin t=t \cos t$<br/>Hence, $\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{t \cos t}{t \sin t}=\cot t$<br/>$\Rightarrow\left(\frac{d y}{d x}\right)_{t=\frac{\pi}{2}}=\cot \frac{\pi}{2}=0$</div>


title: if-x-t-2-y-t-3-then-what-is-d-x-2-d-2-y-equal-to
slug: if-x-t-2-y-t-3-then-what-is-d-x-2-d-2-y-equal-to-71114
question: <div class="question">If $x=t^{2}, y=t^{3}$, then what is $\frac{d^{2} y}{d x^{2}}$ equal to?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{2 t}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3}{4 t}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Let $x=t^{2}$ and $y=t^{3}$<br/>$\Rightarrow \frac{d x}{d t}=2 t$ and $\frac{d y}{d t}=3 t^{2}$<br/>$\therefore \quad \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{3 t^{2}}{2 t}=\frac{3}{2} t$<br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{3}{2} \cdot \frac{d t}{d x}=\frac{3}{2} \cdot \frac{1}{2 t}\left(\because \frac{d x}{d t}=2 t\right)$<br/>$=\frac{3}{4 t}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-x-x-x-0-x-2-0-and-y-flower-churchill-moon-kargil-then-which-one-of-the-following-is-a-correct-statement

slug:
if-x-x-x-0-x-2-0-and-y-flower-churchill-moon-kargil-then-which-one-of-the-following-is-a-correct-statement-35944

question:
<div class="question">If $\mathrm{X}=\left\{\mathrm{x}: \mathrm{x}&gt;0, \mathrm{x}^{2}   &lt; 0\right\}$, and $\mathrm{Y}=$ flower, Churchill, moon, Kargil), then which one of the following is a correct statement?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{X}$ is well defined but $\mathrm{Y}$ is not a well defined set</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Y}$ is well defined but $\mathrm{X}$ is not a well defined set</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both $\mathrm{X}$ and $\mathrm{Y}$ are well defined sets</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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 X nor Y isa well defined set</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both $\mathrm{X}$ and $\mathrm{Y}$ are well defined sets</span> </div>

answer:
<div class="solution">$X=\left\{x: x&gt;0, x^{2}   &lt; 0\right\}$<br/>We know that the square of each number greater thar zero is always greater than zero. So, X contains member and so, $\mathrm{X}$ is null set but a well defined set Also, $\mathrm{Y}=\{$ flower, Churchill, Moon, Kargil\} is wel<br/>defined. So, Y is also a well defined set</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-x-x-y-and-x-y-are-the-angles-of-a-triangle-not-an-equilateral-triangle-such-that-tan-x-y-tan-x-and-tan-x-y-are-in-gp-then-what-is-x-equal-to

slug:
if-x-x-y-and-x-y-are-the-angles-of-a-triangle-not-an-equilateral-triangle-such-that-tan-x-y-tan-x-and-tan-x-y-are-in-gp-then-what-is-x-equal-to-19620

question:
<div class="question">If $\mathrm{x}, \mathrm{x}-\mathrm{y}$ and $\mathrm{x}+\mathrm{y}$ are the angles of a triangle (not an equilateral triangle) such that $\tan (\mathrm{x}-\mathrm{y}), \tan \mathrm{x}$ and $\tan (\mathrm{x}+\mathrm{y})$<br/>are in GP, then what is $x$ equal to?</div>

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

answer:
<div class="solution">$\mathrm{x}, \mathrm{x}-\mathrm{y}, \mathrm{x}+\mathrm{y}$ are angles of a triangle. $\tan (\mathrm{x}-\mathrm{y}), \tan \mathrm{x}, \tan (\mathrm{x}+\mathrm{y})$ are in G.P.<br/>Now, $\mathrm{x}+\mathrm{x}-\mathrm{y}+\mathrm{x}+\mathrm{y}=\pi$ (Sum of angles in triangle $\left.=180^{\circ}=\pi\right)$<br/>$\Rightarrow 3 x=\pi$<br/>$\Rightarrow x=\frac{\pi}{3}$</div>


title: if-x-y-0-3-i-1-2-i-1-i-i-6-11-i-then-what-are-the-values-of-x-and-y-respectively
slug: if-x-y-0-3-i-1-2-i-1-i-i-6-11-i-then-what-are-the-values-of-x-and-y-respectively-11228
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-3,4$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left[\begin{array}{ccc}x &amp; -3 i &amp; 1 \\ y &amp; 1 &amp; i \\ 0 &amp; 2 i &amp; -i\end{array}\right]=6+11$ i, then what are the values of $\mathrm{x}$ and<br/>y respectively ?</div>

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

answer:
<div class="solution">$\begin{array}{l}{\left[\begin{array}{lcc}x &amp; -3 i &amp; 1 \\ y &amp; 1 &amp; i \\ 0 &amp; 2 i &amp; -i\end{array}\right]=6+11 i} \\ &amp; \Rightarrow \mathrm{x}[-\mathrm{i}-(2 \mathrm{i})(\mathrm{i})]-\mathrm{y}[(-3 \mathrm{i})(-\mathrm{i})-(2 \mathrm{i})(1)]+0=6+11 \mathrm{i} \\ &amp; \Rightarrow \mathrm{x}(-\mathrm{i}+2)-\mathrm{y}(-3-2 \mathrm{i})=6+11 \mathrm{i} \\ &amp; \Rightarrow-\mathrm{x} \mathrm{i}+2 \mathrm{x}+3 \mathrm{y}+2 \mathrm{yi}=6+11 \mathrm{i} \\ &amp; \Rightarrow(2 \mathrm{x}+3 \mathrm{y})+(-\mathrm{x}+2 \mathrm{y}) \mathrm{i}=6+11 \mathrm{i} \\ &amp; \therefore 2 \mathrm{x}+3 \mathrm{y}=6 \text { and }-\mathrm{x}+2 \mathrm{y}=11\end{array}$<br/>Solving the equations,<br/>$2 \not x+3 y=6$<br/>$\frac{-2 \not x+4 y=22}{7 y=28} \Rightarrow y=4$<br/>$2 x+3 y=6 \Rightarrow 2 x+12=6$<br/>$\Rightarrow 2 x=-6 \Rightarrow x=-3$<br/>$\therefore \mathrm{x}=-3, \mathrm{y}=4$</div>


title: if-x-y-2-then-what-is-d-x-d-y-at-y-1-and-x-1-equal-to
slug: if-x-y-2-then-what-is-d-x-d-y-at-y-1-and-x-1-equal-to-91696
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sqrt{x}+\sqrt{y}=2$, then what is $\frac{d y}{d x}$ at $y=1$ and $\mathrm{x}=1$ equal to?</div>

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

answer:
<div class="solution">Let $\sqrt{x}+\sqrt{y}=2$ <br/>Differentiate w.r.t.x, we get<br/>$\frac{1}{2 \sqrt{x}}+\frac{1}{2 \sqrt{y}} \frac{d y}{d x}=0$ ...(1)<br/>Put $\mathrm{y}=1, \mathrm{x}=1$ in equation (1)<br/>$\frac{1}{2}+\frac{1}{2} \frac{d y}{d x}=0$<br/>$\Rightarrow \quad \frac{d y}{d x}=-1$</div>


title: if-x-y-4-then-the-how-many-nonzero-positive-integer-ordered-pair-x-y
slug: if-x-y-4-then-the-how-many-nonzero-positive-integer-ordered-pair-x-y-60345
tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x+y \leq 4$, then the how many non-zero positive integer ordered pair $(x, y) ?$</div>

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

answer:
<div class="solution">Since $x$ and $y$ are non-zero positive integers therefore $x=1,2,3, \ldots \ldots \ldots \ldots$ and $y=1,2,3, \ldots$<br/>Now, given $x+y \leq 4$ $(x, y)$ can be $(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)$<br/>Required number of ordered pairs $=6$</div>


title: if-x-y-12-what-is-the-maximum-value-of-xy
slug: if-x-y-12-what-is-the-maximum-value-of-xy-50603
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (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">If $\mathrm{x}+\mathrm{y}=12$, what is the maximum value of $\mathrm{xy}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">25</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">49</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">64</span> </li> </ul>

answer:
<div class="solution">Given $x+y=12$ $\mathrm{y}=12-\mathrm{x}$<br/>so, $x y=x(12-x)=12 x-x^{2}$<br/>Let $f(x)=12 x-x^{2}$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=12-2 \mathrm{x}$<br/>To get maximum or minimum value $\mathrm{f}^{\prime}(\mathrm{x})=0$ and $\mathrm{f}^{\prime \prime}(\mathrm{x})   &lt; 0$ it is maximum<br/>$\mathrm{f}^{\prime \prime}(\mathrm{x})=-2 &lt; 0$<br/>so, $\mathrm{f}^{\prime}(\mathrm{x})=0$ will give maximum value. so, $12-2 x=0 \Rightarrow x=6$ and $x+y=12 \Rightarrow y=6$<br/>Hence, $\mathrm{y}=6$ and $\mathrm{f}(\mathrm{x})=12 \mathrm{x}-\mathrm{x}^{2}=12 \times 6-36=36$</div>


title: if-x-y-cos-3-2-z-cos-3-4-then-what-is-x-y-yz-z-x-equal-to
slug: if-x-y-cos-3-2-z-cos-3-4-then-what-is-x-y-yz-z-x-equal-to-55944
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=y \cos \left(\frac{2 \pi}{3}\right)=z \cos \left(\frac{4 \pi}{3}\right)$, then what is<br/>$x y+y z+z x$ equal to?</div>

answer:
<div class="solution">Let $x=y \cos \left(\frac{2 \pi}{3}\right)=z \cos \left(\frac{4 \pi}{3}\right)$<br/>$\Rightarrow x=y \cos \left(\pi-\frac{\pi}{3}\right)=-y \cos \frac{\pi}{3}=\frac{-y}{2}$<br/>and $x=z \cos \left(\pi+\frac{\pi}{3}\right)=-z \cos \frac{\pi}{3}=\frac{-z}{2}$<br/>from (1) and (2)<br/>$\frac{-y}{2}=\frac{-z}{2} \Rightarrow y=z$<br/>$\begin{aligned} \text { Thus, } x y+y z+z x &amp;=z x+z^{2}+x z=2 x z+z^{2} \\ &amp;=-y \cdot(y)+y^{2}=-y^{2}+y^{2}=0</div>


title: if-x-y-e-x-y-then-d-y-d-x-is-equal-to-which-one-of-the-following
slug: if-x-y-e-x-y-then-d-y-d-x-is-equal-to-which-one-of-the-following-93127
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x^{y}=e^{x-y}$ then $d y / d x$ is equal to which one of the following?</div>

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

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

answer:
<div class="solution">$x^{y}=e^{x-y}$<br/>Taking log both sides, we get $\Rightarrow y \cdot \log x=x-y$<br/>$\Rightarrow y=\frac{x}{1+\log x}$<br/>$\Rightarrow \frac{d y}{d x}=\frac{(1+\log x)-x\left(0+\frac{1}{x}\right)}{(1+\log x)^{2}}$<br/>$=\frac{(1+\log x)-1}{(1+\log x)^{2}}=\frac{\log x}{(1+\log x)^{2}}$</div>


title: if-x-y-t-t-1-x-2-y-2-t-2-t-2-1-what-is-d-x-d-y-equal-to
slug: if-x-y-t-t-1-x-2-y-2-t-2-t-2-1-what-is-d-x-d-y-equal-to-22901
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x+y=t-\frac{1}{t}, x^{2}+y^{2}=t^{2}+\frac{1}{t^{2}}$, what is $\frac{d y}{d x}$ equal to?</div>

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

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

answer:
<div class="solution">Given that $x+y=t-\frac{1}{t}$ and $x^{2}+y^{2}=t^{2}+\frac{1}{t^{2}}$<br/>$\therefore (\mathrm{x}+\mathrm{y})^{2}=\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{xy}$<br/>$\Rightarrow\left(\mathrm{t}-\frac{1}{\mathrm{t}}\right)^{2}=\left(\mathrm{t}^{2}+\frac{1}{\mathrm{t}^{2}}\right)+2 \mathrm{xy}$<br/>$-2=2 x y \Rightarrow x y=-1$<br/>$(x-y)^{2}=(x+y)^{2}-4 x y$<br/>$=\left(t-\frac{1}{t}\right)^{2}-4 \times-1=t^{2}+\frac{1}{t^{2}}-2+4=\left(t+\frac{1}{t}\right)^{2}$<br/>$\mathrm{x}-\mathrm{y}=\mathrm{t}+\frac{1}{\mathrm{t}}$<br/>$\Rightarrow \mathrm{x}=\mathrm{t}, \mathrm{y}=-\frac{1}{\mathrm{t}}$<br/>$\mathrm{xy}=-1$<br/>$\Rightarrow x \frac{d y}{d x}+y=0$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{y}}{\mathrm{x}}=\frac{1}{\mathrm{t}^{2}}=\frac{1}{\mathrm{x}^{2}}$</div>


tags: ['Mathematics', 'Determinants', 'KCET', 'KCET 2016']
source_db: MarksBatch2_P1.db
title:
if-x-y-z-are-all-different-and-not-equal-to-zero-and-leftbeginarraycccc1x-1-1-1-1y-1-1-1-1zendarrayright0-then-the-value-of-x1y1z1-is-equal-to

slug:
if-x-y-z-are-all-different-and-not-equal-to-zero-and-leftbeginarraycccc1x-1-1-1-1y-1-1-1-1zendarrayright0-then-the-value-of-x1y1z1-is-equal-to-27097

question:
<div class="question">If \( x y z \) are all different and not equal to zero and \( \left|\begin{array}{cccc}1+x &amp; 1 &amp; 1 \\ 1 &amp; 1+y &amp; 1 \\ 1 &amp; 1 &amp; 1+z\end{array}\right|=0 \)<br/>then the value of \( x^{-1}+y^{-1}+z^{-1} \) is equal to</div>

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

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

answer:
<div class="solution">Given that,<br/>\[<br/>\begin{array}{l}<br/>\left|\begin{array}{ccc}<br/>1+x &amp; 1 &amp; 1 \\<br/>1 &amp; 1+y &amp; 1 \\<br/>1 &amp; 1 &amp; 1+z<br/>\end{array}\right|=0 \\<br/>R_{1} \rightarrow R_{1}-R_{2} \text { and } R_{2} \rightarrow R_{2}-R_{3} \\<br/>\left|\begin{array}{ccc}<br/>x &amp; -y &amp; 0 \\<br/>0 &amp; y &amp; -z \\<br/>1 &amp; 1 &amp; 1+z<br/>\end{array}\right|=0 \\<br/>\Rightarrow x[y(1+z)+z]+y(z)=0 \\<br/>\Rightarrow x y+y z+z x+x y z=0<br/>\end{array}<br/>\]<br/>Divide by xyz both side, we get<br/>\[<br/>\begin{array}{l}<br/>\Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+1=0 \\<br/>\Rightarrow x^{-1}+y^{-1}+z^{-1}=-1<br/>\end{array}<br/>\]</div>


tags: ['Mathematics', 'Determinants', 'KCET', 'KCET 2016']
source_db: MarksBatch2_P1.db
title:
if-x-y-z-are-not-equal-and-neq-0-neq-1-the-value-of-leftbeginarrayccclog-x-log-y-log-z-log-2-x-log-2-y-log-2-z-log-3-x-log-3-y-log-3-zendarrayright-is

slug:
if-x-y-z-are-not-equal-and-neq-0-neq-1-the-value-of-leftbeginarrayccclog-x-log-y-log-z-log-2-x-log-2-y-log-2-z-log-3-x-log-3-y-log-3-zendarrayright-is-55876

question:
<div class="question">If \( x y z \) are not equal and \( \neq 0, \neq 1 \) the value of \( \left|\begin{array}{ccc}\log x &amp; \log y &amp; \log z \\ \log 2 x &amp; \log 2 y &amp; \log 2 z \\ \log 3 x &amp; \log 3 y &amp; \log 3 z\end{array}\right| \) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( \log (x y z) \)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\( \log (6 \times y z) \)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\( 00 \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( \log (x+y+z) \)</span> </li> </ul>

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

answer:
<div class="solution">Given that, $x, y, z \neq 0 \neq 1$<br/>As we know, $\log 2 x-\log x=\log 2$<br/>and $\log 3 x-\log x=\log 3$<br/>Given that $\left|\begin{array}{ccc}\log x &amp; \log y &amp; \log z \\ \log 2 x &amp; \log 2 y &amp; \log 2 z \\ \log 3 x &amp; \log 3 y &amp; \log 3 z\end{array}\right|$<br/>$R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$<br/>$\mid \begin{array}{ll}\log x &amp; \log y &amp; \log z \\ \log 2 &amp; \log 2 &amp; \log 2 \\ \log 3 &amp; \log 3 &amp; \log 3 &amp; \\ = &amp; \log 2 \log 3 \\ \text { When the two rows of determinants are identical the value of determinant is equal to } 0 &amp; \begin{array}{ccc}\log x &amp; \log y &amp; \log z \\ 1 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 1\end{array} \mid\end{array}$</div>


tags: ['Mathematics', 'Determinants', 'AP EAMCET', 'AP EAMCET 2019 (23 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
title:
if-xalpha-ybeta-zgamma-is-the-solution-for-the-system-of-equations-beginaligned-2-xy8-z-13-3-x4-y5-z-18-5-x2-y7-z-20-endaligned-then-alpha-betabeta-ga

slug:
if-xalpha-ybeta-zgamma-is-the-solution-for-the-system-of-equations-beginaligned-2-xy8-z-13-3-x4-y5-z-18-5-x2-y7-z-20-endaligned-then-alpha-betabeta-ga-19527

question:
<div class="question">If \(x=\alpha, y=\beta, z=\gamma\) is the solution, for the system of equations<br/>\(\begin{aligned}<br/>2 x-y+8 z &amp; =13 \\<br/>3 x+4 y+5 z &amp; =18 \\<br/>5 x-2 y+7 z &amp; =20<br/>\end{aligned}\)<br/>then \(\alpha \beta+\beta \gamma+\gamma \alpha=\)</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">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-3</span> </li> </ul>

answer:
<div class="solution">According to Cramer's rule,<br/>\(\begin{aligned}<br/>&amp; x=\alpha=\frac{\Delta_1}{\Delta}, y=\beta=\frac{\Delta_2}{\Delta} \text { and } z=\gamma=\frac{\Delta_3}{\Delta} \\<br/>&amp; \text {where } \begin{aligned}<br/>\Delta &amp; =\left|\begin{array}{ccc}<br/>2 &amp; -1 &amp; 8 \\<br/>3 &amp; 4 &amp; 5 \\<br/>5 &amp; -2 &amp; 7<br/>\end{array}\right| \\<br/>&amp; =2(28+10)+1(21-25)+8(-6-20) \\<br/>&amp; =76-4-208=76-212=-136<br/>\end{aligned}<br/>\end{aligned}\)<br/>\(\begin{aligned}<br/>\Delta_1 &amp; =\left|\begin{array}{ccc}<br/>13 &amp; -1 &amp; 8 \\<br/>18 &amp; 4 &amp; 5 \\<br/>20 &amp; -2 &amp; 7<br/>\end{array}\right| \\<br/>&amp; =13(28+10)+1(126-100)+8(-36-80) \\<br/>&amp; =(13 \times 38)+26-(8 \times 116) \\<br/>&amp; =494+26-928=-408 \\<br/>\Delta_2 &amp; =\left|\begin{array}{lll}<br/>2 &amp; 13 &amp; 8 \\<br/>3 &amp; 18 &amp; 5 \\<br/>5 &amp; 20 &amp; 7<br/>\end{array}\right| \\<br/>&amp; =2(126-100)-13(21-25)+8(60-90) \\<br/>&amp; =(2 \times 26)+(13 \times 4)-(8 \times 30) \\<br/>&amp; =52+52-240=-136 \\<br/>\Delta_3 &amp; =\left|\begin{array}{ccc}<br/>2 &amp; -1 &amp; 13 \\<br/>3 &amp; 4 &amp; 18 \\<br/>5 &amp; -2 &amp; 20<br/>\end{array}\right| \\<br/>&amp; =2(80+36)+1(60-90)+13(-6-20) \\<br/>&amp; =(2 \times 116)-30-(13 \times 26) \\<br/>&amp; =232-30-338=-136<br/>\end{aligned}\)<br/>So, \(\alpha=3, \beta=1, \gamma=1\)<br/>\(\therefore \quad \alpha \beta+\beta \gamma+\gamma \alpha=3+1+3=7\)<br/>Hence, option (c) is correct.</div>


title: if-xdy-y-dx-y-2-dy-y-0-and-y-1-1-then-what-is-y-3-equal-to
slug: if-xdy-y-dx-y-2-dy-y-0-and-y-1-1-then-what-is-y-3-equal-to-72035
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3 only</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{xdy}=\mathrm{y} \mathrm{dx}+\mathrm{y}^{2} \mathrm{dy}, \mathrm{y}&gt;0$ and $\mathrm{y}(1)=1$, then what is<br/>$\mathrm{y}(-3)$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3 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">B</span> <span class="option-data">$-1$ only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both $-1$ and 3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither $-1$ nor 3</span> </li> </ul>

answer:
<div class="solution">Given, $x d y=y d x+y^{2} d y$<br/>$\Rightarrow 1=\frac{y}{x} \frac{d x}{d y}+\frac{y^{2}}{x}$<br/>$\Rightarrow \frac{d x}{d y}+y=\frac{x}{y}$<br/>$\Rightarrow \frac{d x}{d y}-\frac{x}{y}=-y$<br/>$P=-\frac{1}{y}, Q=-y$<br/>$\mathrm{IF}=e^{\int P d y}=e^{\int-\frac{1}{y} d y}=e^{-\log y}=\frac{1}{y}$<br/>Now, Solution of d.E. $\mathrm{x}(\mathrm{I.F})=\int(\mathrm{Q} . \mathrm{I.F}) \mathrm{dy}$<br/>$\frac{x}{y}=\int \frac{1}{y}(-y) d y+C$<br/>$\Rightarrow \frac{x}{y}=\int-1 d y+C$<br/>$\Rightarrow \frac{x}{y}=-y+C$<br/>$y(1)=1$<br/>$\frac{1}{1}=-1+C \Rightarrow C=2$<br/>$\Rightarrow \frac{x}{y}=-y+2 \Rightarrow x=-y^{2}+2 y$<br/>$\Rightarrow y(-3) \Rightarrow-3=-y^{2}+2 y$<br/>$\Rightarrow y^{2}-2 y-3=0$<br/>$\Rightarrow y=\frac{+2 \pm \sqrt{4+12}}{2}=\frac{2 \pm 4}{2}$<br/>$\Rightarrow y=3,-1$<br/>Since $y&gt;0$ so $y=3$.</div>


title: if-y-1-x-4-1-1-x-2-1-1-x-4-1-then-what-is-d-x-d-y-equal-to
slug: if-y-1-x-4-1-1-x-2-1-1-x-4-1-then-what-is-d-x-d-y-equal-to-53921
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\left(1+x^{\frac{1}{4}}\right)\left(1+x^{\frac{1}{2}}\right)\left(1-x^{\frac{1}{4}}\right)$, then what is $\frac{d y}{d x}$ equal to?</div>

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

answer:
<div class="solution">Let $y=\left(1+x^{1 / 4}\right)\left(1+x^{1 / 2}\right)\left(1-x^{1 / 4}\right)$<br/>$=\left(1+x^{1 / 4}\right)\left(1-x^{1 / 4}\right)\left(1+x^{1 / 2}\right)$<br/>$=\left(1-x^{1 / 2}\right)\left(1+x^{1 / 2}\right)$<br/>$\left(\because(a+b)(a-b)=a^{2}-b^{2}\right)$<br/>$\begin{array}{l}<br/>=(1-x)\left(\because(a+b)(a-b)=a^{2}-b^{2}\right) \\<br/>\Rightarrow y=1-x<br/>\end{array}$<br/>Differentiate both side w.r.t ' $x$, $\frac{d y}{d x}=-1$</div>


title: if-y-2-p-x-is-a-polynomial-of-degree-3-then-what-is-2-dx-d-y-3-dx-2-d-2-y-equal-to
slug: if-y-2-p-x-is-a-polynomial-of-degree-3-then-what-is-2-dx-d-y-3-dx-2-d-2-y-equal-to-65900
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}^{2}=\mathrm{p}(\mathrm{x})$ is a polynomial of degree 3, then what is<br/>$2 \frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{y}^{3} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right]$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{p}^{\prime}(\mathrm{x}) \mathrm{p}^{\prime \prime}(\mathrm{x})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}^{\prime \prime}(\mathrm{x}) \mathrm{p}^{\prime \prime}(\mathrm{x})$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{p}(\mathrm{x}) \mathrm{p}^{\prime \prime \prime}(\mathrm{x})$</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">D</span> <span class="option-data">$\begin{array}{ll}\text {  A constant } &amp; {}\end{array}$</span> </li> </ul>

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

answer:
<div class="solution">Given that $y^{2}=p(x)$<br/>Differentiating<br/>$\Rightarrow 2 \mathrm{yy}_{1}=\mathrm{p}^{\prime}(\mathrm{x})$<br/>$\left[\right.$ here $\left.\mathrm{y}_{1}=\frac{\mathrm{dy}}{\mathrm{dx}}\right]$<br/>$\Rightarrow \quad 2 \mathrm{y}_{1}=\frac{\mathrm{p}^{\prime}(\mathrm{x})}{\mathrm{y}}$<br/>Differentiating again, $\Rightarrow \quad 2 \mathrm{y}_{2}=\frac{\mathrm{yp}^{\prime \prime}(\mathrm{x})-\mathrm{p}^{\prime}(\mathrm{x}) \mathrm{y}_{1}}{\mathrm{y}^{2}},\left[\mathrm{y}_{2}=\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}\right]$<br/>$\Rightarrow 2 \mathrm{y}_{2}=\frac{\mathrm{yp}^{\prime \prime}(\mathrm{x})-\frac{\mathrm{p}^{\prime}(\mathrm{x}) \cdot \mathrm{p}^{\prime}(\mathrm{x})}{2 \mathrm{y}}}{\mathrm{y}^{2}}$<br/>$$<br/>\begin{array}{l}<br/>\quad=\frac{\left.2 \mathrm{y}^{2} \mathrm{p}^{\prime \prime}(\mathrm{x})-\mathrm{p}^{\prime}(\mathrm{x})\right)^{2}}{2 \mathrm{y}^{3}} \\<br/>\Rightarrow \quad 2 \mathrm{y}^{3} \mathrm{y}_{2}=\frac{1}{2}\left[2 \mathrm{y}^{2} \mathrm{p}^{\prime \prime}(\mathrm{x})-\left(\mathrm{p}^{\prime}(\mathrm{x})\right)^{2}\right] \\<br/>\Rightarrow \quad 2 \mathrm{y}^{3} \mathrm{y}_{2}=\frac{1}{2}\left[2 \mathrm{p}(\mathrm{x}) \mathrm{p}^{\prime \prime}(\mathrm{x})-\left(\mathrm{p}^{\prime}(\mathrm{x})\right)^{2}\right] \\<br/>\Rightarrow \quad 2 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{y}^{3} \mathrm{y}_{2}\right) \\<br/>=\frac{1}{2}\left[2 \mathrm{p}^{\prime}(\mathrm{x}) \mathrm{p}^{\prime \prime}(\mathrm{x})+2 \mathrm{p}(\mathrm{x}) \mathrm{p}^{\prime \prime}(\mathrm{x})-2 \mathrm{p}^{\prime}(\mathrm{x}) \mathrm{p}^{\prime \prime}(\mathrm{x})\right] \\<br/>=\mathrm{p}(\mathrm{x}) \mathrm{p}^{\prime \prime \prime}(\mathrm{x})<br/>\end{array}<br/>$$</div>


title: if-y-a-cos-2-x-b-sin-2-x-then
slug: if-y-a-cos-2-x-b-sin-2-x-then-70633
question: <div class="question">If $y=a \cos 2 x+b \sin 2 x$, then</div>
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}+y=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}+2 y=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}-4 y=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}+4 y=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{d^{2} y}{d x^{2}}+4 y=0$</span> </div>

answer:
<div class="solution">The equation of circle touching $\mathrm{y}$ -axis at origin is $(\mathrm{x}-\alpha)^{2}+(\mathrm{y}-0)^{2}=\alpha^{2}$<br/>$\Rightarrow \mathrm{x}^{2}+\alpha^{2}-2 \alpha \mathrm{x}+\mathrm{y}^{2}=\alpha^{2}$<br/>$\Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}-2 \alpha \mathrm{x}=0$<br/>$\Rightarrow x+\frac{y^{2}}{x}-2 \alpha=0$<br/>Differentiating, $1+\frac{x \cdot 2 y \cdot \frac{d y}{d x}-y^{2}}{x^{2}}=0$<br/>$\Rightarrow x^{2}+2 x y \cdot \frac{d y}{d x}-y^{2}=0$</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-y-axis-touches-the-circle-x-2-y-2-gx-f-y-4-c-0-then-the-normal-at-this-point-intersects-the-circle-at-the-point

slug:
if-y-axis-touches-the-circle-x-2-y-2-gx-f-y-4-c-0-then-the-normal-at-this-point-intersects-the-circle-at-the-point-47143

question:
<div class="question">If $\mathrm{y}$ -axis touches the circle<br/>$x^{2}+y^{2}+g x+f y+\frac{c}{4}=0$, then the normal at this point intersects the circle at the point</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Cbcn_Lg8_cl-4EvuMrv4ektJWw_eJJTUqS5jCFl7uws.original.fullsize.png"/></div>


title: if-y-cos-1-1-x-2-2-x-then-dx-dy-is-equal-to
slug: if-y-cos-1-1-x-2-2-x-then-dx-dy-is-equal-to-14562
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\cos ^{-1}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^{2}}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to</div>

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

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

answer:
<div class="solution">$\mathrm{y}=\cos ^{-1}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^{2}}\right)$<br/>Put $\mathrm{x}=\tan \theta \Rightarrow \theta=\tan ^{-1} \mathrm{x}$<br/>$\mathrm{y}=\cos ^{-1}\left(\frac{2 \tan \theta}{\left(1+\tan ^{2} \theta\right)}\right)=\cos ^{-1}(\sin 2 \theta)$<br/>$=\cos ^{-1}\left(\cos \left(\frac{\pi}{2}-2 \theta\right)\right)=\frac{\pi}{2}-2 \theta .$<br/>$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\pi}{2}-2 \theta\right)$<br/>$=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\pi}{2}-2 \tan ^{-1} \mathrm{x}\right)$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-2}{1+\mathrm{x}^{2}}$, when $|\mathrm{x}|   &lt; 1$</div>


title: if-y-cos-t-and-x-sin-t-then-what-is-d-x-d-y-equal-to
slug: if-y-cos-t-and-x-sin-t-then-what-is-d-x-d-y-equal-to-11220
question: <div class="question">If $y=\cos t$ and $x=\sin t$, then what is $\frac{d y}{d x}$ equal to?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{xy}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{x} / \mathrm{y}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\mathrm{y} / \mathrm{x}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-x / y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Let $y=\cos t, x=\sin t$<br/>$\frac{\mathrm{dy}}{\mathrm{dt}}=-\sin \mathrm{t}, \frac{\mathrm{dx}}{\mathrm{dt}}=\cos \mathrm{t}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{dx} / \mathrm{dt}}=-\frac{\sin \mathrm{t}}{\cos \mathrm{t}}=-\frac{\mathrm{x}}{\mathrm{y}}$</div>


title: if-y-cos-x-c-o-s-x-c-o-s-x-then-dx-dy-is-equal-to
slug: if-y-cos-x-c-o-s-x-c-o-s-x-then-dx-dy-is-equal-to-20193
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=(\cos \mathrm{x})^{(\cos \mathrm{x})^{(\cos \mathrm{x})^{...\infty}}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{y^{2} \tan x}{1-y \operatorname{In}(\cos x)}$</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">B</span> <span class="option-data">$\frac{y^{2} \tan x}{1+y \operatorname{In}(\cos x)}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{y^{2} \tan x}{1-y \ln (\sin x)}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{y^{2} \sin x}{1+y \operatorname{In}(\sin x)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{y^{2} \tan x}{1-y \operatorname{In}(\cos x)}$</span> </div>

answer:
<div class="solution">$\mathrm{y}=(\cos \mathrm{x})^{(\cos \mathrm{x})^{(\cos \mathrm{x})^{...\infty}}}$<br/>$\Rightarrow \mathrm{y}=(\cos \mathrm{x})^{\mathrm{y}}$<br/>$\Rightarrow \log \mathrm{y}=\mathrm{y} \cdot \log \cos \mathrm{x}$<br/>Differentiating on both sides, we get<br/>$\frac{1}{y} \cdot \frac{d y}{d x}=y(-\tan x)+\log \cos x \cdot \frac{d y}{d x}$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}\left(\frac{1}{\mathrm{y}}-\log \cos \mathrm{x}\right)=-\mathrm{y} \tan \mathrm{x}$<br/>$\Rightarrow \frac{d y}{d x}=\frac{-y \tan x}{\frac{1}{y}-\log \cos x}=\frac{-y^{2} \tan x}{1-y \log \cos x}$</div>


title: if-y-cot-1-1-s-i-n-x-1-s-i-n-x-1-s-i-n-x-1-s-i-n-x-where-0-x-2-then-dx-dy-is-equal-to
slug: if-y-cot-1-1-s-i-n-x-1-s-i-n-x-1-s-i-n-x-1-s-i-n-x-where-0-x-2-then-dx-dy-is-equal-to-18285
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$, where $0   &lt; x &lt; \frac{\pi}{2}$, then<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sin x+\cos x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sin x-\cos x$</span> </li> </ul>

answer:
<div class="solution">$\mathrm{y}=\cot ^{-1}\left[\frac{\sqrt{1+\sin \mathrm{x}}+\sqrt{1-\sin \mathrm{x}}}{\sqrt{1+\sin \mathrm{x}}-\sqrt{1-\sin \mathrm{x}}}\right]$<br/>$\mathrm{y}=\cot ^{-1}$ <br/>$\left[\begin{array}{l}\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}+2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}} \\ +\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}-2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}{\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}+2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}} \\ -\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}-2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}\end{array}\right]$<br/>$y=\cot ^{-1}\left[\frac{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}}+\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}}-\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}\right]$<br/>$y=\cot ^{-1}\left[\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\cos \frac{x}{2}-\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}+\sin \frac{x}{2}}\right]$<br/>$y=\cot ^{-1}\left[\frac{2 \cos \frac{x}{2}}{2 \sin \frac{x}{2}}\right]=\cot ^{-1}\left(\cot \frac{x}{2}\right)=\frac{x}{2}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}$</div>


title: if-y-e-x-2-sin-2-x-then-what-is-dx-dy-at-x-equal-to
slug: if-y-e-x-2-sin-2-x-then-what-is-dx-dy-at-x-equal-to-95281
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\mathrm{e}^{\mathrm{x}^{2}} \sin 2 \mathrm{x}$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\pi$ equal to?</div>

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

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

answer:
<div class="solution">$y=e^{x^{2}} \cdot \sin 2 x$<br/>$\frac{d y}{d x}=2 \cdot e^{x^{2}} \cdot \cos 2 x+2 x e^{x^{2}} \cdot \sin 2 x$<br/>$=2 e^{x^{2}}(\cos 2 x+x \sin 2 x)$<br/>$\left.\frac{d y}{d x}\right|_{x=\pi}=2 e^{\pi^{2}}(\cos 2 \pi+\pi \cdot \sin 2 \pi)$<br/>$=2 e^{\pi^{2}}(1+0)$<br/>$=2 \cdot e^{\pi^{2}}$</div>


title: if-y-f-x-p-dx-dy-and-q-dx-2-d-2-y-then-what-is-dy-2-d-2-x-equal-to
slug: if-y-f-x-p-dx-dy-and-q-dx-2-d-2-y-then-what-is-dy-2-d-2-x-equal-to-43763
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\mathrm{f}(\mathrm{x}), \mathrm{p}=\frac{\mathrm{dy}}{\mathrm{dx}}$ and $\mathrm{q}=\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$, then what is $\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dy}^{2}}$ equal<br/>to?</div>

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

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

answer:
<div class="solution">As given, $y=f(x), p=\frac{d y}{d x}$ and $q=\frac{d^{2} y}{d x^{2}}$<br/>$\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{1}{\mathrm{p}} \Rightarrow \frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dy}^{2}}=\frac{-1}{\mathrm{p}^{2}} \cdot \frac{\mathrm{dp}}{\mathrm{dy}}$<br/>$\frac{\mathrm{dp}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{q}$<br/>$\frac{d p}{d y}=\frac{d p}{d x} \cdot \frac{d x}{d y}=q \cdot \frac{1}{p}=\frac{q}{p}$<br/>$\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{d} \mathrm{y}^{2}}=\Rightarrow-\frac{1}{\mathrm{p}^{2}} \times \frac{\mathrm{q}}{\mathrm{p}}=\frac{-\mathrm{q}}{\mathrm{p}^{3}}$</div>


title: if-y-l-o-g-10-x-1-then-what-is-dx-dy-equal-to
slug: if-y-l-o-g-10-x-1-then-what-is-dx-dy-equal-to-38019
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\frac{1}{\log _{10} \mathrm{x}}$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{\left(\log _{x} 10\right)^{2}\left(\log _{10} e\right)}{x}$</span> </div>

answer:
<div class="solution">Differentiating the given function, $\mathrm{y}=\frac{1}{\log _{10} \mathrm{x}}$<br/>We get, $\frac{d y}{d x}=-\frac{1}{\left(\log _{10} x\right)^{2}} \cdot \frac{1}{x} \log _{10} e$<br/>$\Rightarrow \frac{d y}{d x}=-\frac{\left(\log _{x} 10\right)^{2} \cdot \log _{10} e}{x}$</div>


title: if-y-lo-g-10-x-lo-g-x-10-lo-g-x-x-lo-g-10-10-then-what-is-dx-dy-x-10-equal-to
slug: if-y-lo-g-10-x-lo-g-x-10-lo-g-x-x-lo-g-10-10-then-what-is-dx-dy-x-10-equal-to-54046
tags: ['Mathematics', 'Application of Derivatives', '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
question:
<div class="question">If $\mathrm{y}=\log _{10} \mathrm{x}+\log _{\mathrm{x}} 10+\log _{\mathrm{x}} \mathrm{x}+\log _{10} 10$ then what is<br/>$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{x}=10}$ equal to?</div>

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

answer:
<div class="solution">$y=\log _{10} x+\log _{x} 10+\log _{x} x+\log _{10} 10$<br/>$y=\log _{10} x+\log _{x} 10+1+1$<br/>Differentiating equation w.r.t. $\mathrm{x}$<br/>$\frac{d y}{d x}=\frac{1}{x \log _{e} 10}-\frac{1}{\left(\log _{10} x\right)^{2}} \cdot \frac{1}{(x \log 10)}$<br/>$=\frac{1}{x \log _{e} 10}\left[1-\frac{1}{\left(\log _{10} x\right)^{2}}\right]$<br/>$\left(\frac{d y}{d x}\right)_{x=10}=\frac{1}{10 \log _{e} 10}[1-1]=0$<br/>$\left[\begin{array}{l}\text { Note: } \log _{x} 10=\frac{\log _{10} 10}{\log _{10} x}=\frac{1}{\log _{10} x} \\ \frac{d}{d x}\left[\frac{1}{\log _{10} x}\right]=-\left(\log _{10} x\right)^{-2} \times \frac{1}{x \log _{e} 10} \\ =-\frac{1}{\left(\log _{10} x\right)^{2} x \log _{e} 10}\end{array}\right]$</div>


title: if-y-n-e-m-x-e-m-x-then-what-is-d-x-d-y-at-x-0-equal-to
slug: if-y-n-e-m-x-e-m-x-then-what-is-d-x-d-y-at-x-0-equal-to-48386
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\ell n\left(e^{m x}+e^{-m x}\right)$, then what is $\frac{d y}{d x}$ at $x=0$ equal to?</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">$\mathrm{y}=\ln (\mathrm{emx}+\mathrm{e}-\mathrm{mx})$<br/>$\mathbf{p}$ $\quad \frac{d y}{d x}=\frac{1}{e^{m x}+e^{-m x}} \cdot \frac{d}{d x}(\mathrm{emx}+\mathrm{e}-\mathrm{mx})$<br/>$=\frac{m e^{m x}-m e^{-m x}}{e^{m n}+e^{-m x}}=\frac{m\left(e^{m x}-e^{-m x}\right)}{e^{m x}+e^{-m x}}$<br/>$=\frac{m\left[e^{m n}-\frac{1}{e^{m x}}\right]}{e^{m x}+\frac{1}{x^{n x}}}=\frac{m\left[e^{2 m x}-1\right]}{e^{2 m x}+1}$<br/>so, $\left.\frac{d y}{d x}\right|_{x=0}=\frac{m\left(e^{0}-1\right)}{e^{0}+1}=m(0)=0$</div>


title: if-y-n-tan-x-then-what-is-the-value-of-d-x-d-y-at-x-4
slug: if-y-n-tan-x-then-what-is-the-value-of-d-x-d-y-at-x-4-86717
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\ell \mathrm{n} \sqrt{\tan x}$, then what is the value of $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$?</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">$1 / 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $\mathrm{y}=\ln \sqrt{\tan x}$<br/>Differentiate both side w.r.t $^{\prime} x^{\prime}$<br/>$\frac{d y}{d x}=\frac{1}{\sqrt{\tan x}} \cdot \frac{1}{2 \sqrt{\tan x}} \cdot \sec ^{2} x$<br/>Now, $\frac{d y}{d x}$ at $x=\pi / 4$<br/>$=\frac{1}{\sqrt{\tan \frac{\pi}{4}}} \times \frac{1}{2 \sqrt{\tan \frac{\pi}{4}}} \times \frac{1}{\cos ^{2}(\pi / 4)}$<br/>$=\frac{1}{2} \times 1 \times \frac{1}{\left(\frac{1}{\sqrt{2}}\right)^{2}}=\frac{1}{2} \times 1 \times 2=1$</div>


title: if-y-sec-1-x-1-x-1-sin-1-x-1-x-1-then-d-x-d-y-is-equal-to
slug: if-y-sec-1-x-1-x-1-sin-1-x-1-x-1-then-d-x-d-y-is-equal-to-12745
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sec ^{-1}\left(\frac{x+1}{x-1}\right)+\sin ^{-1}\left(\frac{x-1}{x+1}\right)$, then $\frac{d y}{d x}$ is equal to</div>

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

answer:
<div class="solution">$\mathrm{y}=\sec ^{-1}\left(\frac{\mathrm{x}+1}{\mathrm{x}-1}\right)+\sin ^{-1}\left(\frac{\mathrm{x}-1}{\mathrm{x}+1}\right)$<br/>$=\cos ^{-1}\left(\frac{\mathrm{x}-1}{\mathrm{x}+1}\right)+\sin ^{-1}\left(\frac{\mathrm{x}+1}{\mathrm{x}-1}\right)$<br/>$=\frac{\pi}{2}\left(\because \sin ^{-1} \mathrm{x}+\cos ^{-1} \mathrm{x}=\frac{\pi}{2}\right) \quad \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=0$</div>


title: if-y-sec-2-cos-2-where-0-2-then-which-one-of-the-following-is-correct
slug: if-y-sec-2-cos-2-where-0-2-then-which-one-of-the-following-is-correct-89900
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sec ^{2} \theta+\cos ^{2} \theta$, where $0   &lt; \theta &lt; \frac{\pi}{2}$, then which one of<br/>the following is correct?</div>

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

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

answer:
<div class="solution">Since, $\cos ^{2} \theta$ lies between 0 and 1 therefore,<br/>$\sec ^{2} \hat{\theta}+\cos ^{2} \hat{\theta}&gt;2, \forall 0   &lt; \dot{\theta} &lt; \frac{\pi}{2}$<br/>$\therefore \quad y \geq 2$</div>


title: if-y-sin-1-1-4-x-2-4-x-then-what-is-d-x-d-y-equal-to
slug: if-y-sin-1-1-4-x-2-4-x-then-what-is-d-x-d-y-equal-to-53662
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sin ^{-1}\left(\frac{4 x}{1+4 x^{2}}\right)$, then what is $\frac{d y}{d x}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{1+4 x^{2}}$</span> </div>

answer:
<div class="solution">Let $y=\sin ^{-1}\left(\frac{4 x}{1+4 x^{2}}\right)=\sin ^{-1}\left(\frac{2 \cdot 2 x}{1+(2 x)^{2}}\right)$<br/>Put $2 x=\tan \theta \Rightarrow \theta=\tan ^{-1} 2 x$<br/>$\therefore y=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)$<br/>$=\sin ^{-1}(\sin 2 \theta)=2 \theta\left(\because \sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)$<br/>$=2 \tan ^{-1} 2 x$<br/>On differentiating w.r.t. $x$, we get<br/>$\frac{d y}{d x}=\frac{2}{1+(2 x)^{2}} \cdot 2=\frac{4}{1+4 x^{2}}$<br/>ALTERNATE SOLUTION<br/>$y=\sin ^{-1}\left(\frac{4 x}{1+4 x^{2}}\right)$<br/>$\frac{d y}{d x}=\frac{1}{\sqrt{1-\left(\frac{4 x}{1+4 x^{2}}\right)^{2}} \times \frac{\left(1+4 x^{2}\right) 4-4 x(8 x)}{\left(1+4 x^{2}\right)^{2}}}$<br/>$=\frac{1+4 x^{2}}{\sqrt{\left(1+4 x^{2}\right)^{2}-16 x^{2}}} \times \frac{4+16 x^{2}-32 x^{2}}{\left(1+4 x^{2}\right)^{2}}$<br/>$=\frac{4-16 x^{2}}{\left(1+4 x^{2}\right) \sqrt{1-8 x^{2}+16 x^{4}}}$<br/>$=\frac{4-16 x^{2}}{\left(1+4 x^{2}\right)\left(1-4 x^{2}\right)}$<br/>$=\frac{(2)^{2}-(4 x)^{2}}{\left(1+4 x^{2}\right)(1-2 x)(1+2 x)}=\frac{(2+4 x) 2}{\left(1+4 x^{2}\right)(1+2 x)}=\frac{4}{1+4 x^{2}}$</div>


title: if-y-sin-1-x-sin-1-1-x-2-what-is-d-x-d-y-equal-to
slug: if-y-sin-1-x-sin-1-1-x-2-what-is-d-x-d-y-equal-to-50551
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}}$, what is $\frac{d y}{d x}$ equal to?</div>

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

answer:
<div class="solution">Given function is:<br/>$y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}}$<br/>On differentiating, w.r.t. $\mathrm{x}$, we get<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\sqrt{1-\mathrm{x}^{2}}}+\frac{1}{\sqrt{1-1+\mathrm{x}^{2}}} \cdot \frac{1}{2 \sqrt{1-\mathrm{x}^{2}}}(-2 \mathrm{x})$<br/>$=\frac{1}{\sqrt{1-\mathrm{x}^{2}}}-\frac{1}{\sqrt{1-\mathrm{x}^{2}}}=0$</div>


title: if-y-sin-a-x-b-then-what-is-d-x-2-d-2-y-at-x-a-b-where-a-b-are-constants-and-a-0
slug: if-y-sin-a-x-b-then-what-is-d-x-2-d-2-y-at-x-a-b-where-a-b-are-constants-and-a-0-37048
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sin (a x+b)$, then what is $\frac{d^{2} y}{d x^{2}}$ at $x=-\frac{b}{a}$, where $a, b$ are constants and $\mathrm{a} \neq 0$?</div>

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

answer:
<div class="solution">Let $y=\sin (a x+b)$<br/>$\Rightarrow \frac{d y}{d x}=a \cos (a x+b)$<br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=-a^{2} \sin (a x+b)$<br/>Now, $\frac{d^{2} y}{d x^{2}}$ at $x=-\frac{b}{a}$ is<br/>$-a^{2} \sin \left(a\left(-\frac{b}{a}\right)+b\right)=-a^{2} \sin 0=0$</div>


title: if-y-sin-m-sin-1-x-what-is-the-value-of-d-2-y-d-x-2-at-x-0
slug: if-y-sin-m-sin-1-x-what-is-the-value-of-d-2-y-d-x-2-at-x-0-34283
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sin \left(m \sin ^{-1} x\right)$, what is the value of $d^{2} y / d x^{2}$ at $x=0$?</div>

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

answer:
<div class="solution">$y=\sin \left(m \sin ^{-1} x\right)$<br/>Then, $\frac{d y}{d x}=\cos \left(m \sin ^{-1} x\right) \frac{m}{\sqrt{1-x^{2}}}$<br/>$\therefore \frac{d^{2} y}{d x^{2}}=\cos \left(m \sin ^{-1} x\right) \cdot m\left\{\frac{-1}{2} \cdot \frac{(-2 x)}{\left(1-x^{2}\right)^{3 / 2}}\right\}$<br/>$+\frac{m}{\sqrt{1-x^{2}}}\left\{-\sin \left(m \sin ^{-1} x\right)\right\} \cdot \frac{m}{\sqrt{1-x^{2}}}$<br/>$=\frac{m}{\sqrt{1-x^{2}}}\left[\frac{x}{\left(1-x^{2}\right)} \cos \left(m \sin ^{-1} x\right)\right]$<br/>$\left.-\frac{m}{\sqrt{1-x^{2}}} \sin \left(m \sin ^{-1} x\right)\right]$<br/>Now, $\frac{d^{2} y}{d x^{2}}$ at $x=0$ is $m[0-0]=0 \quad\left(\because \sin ^{-1} 0=0\right)$</div>


title: if-y-sin-nx-then-which-one-of-the-following-is-correct
slug: if-y-sin-nx-then-which-one-of-the-following-is-correct-34048
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\sin (\ell \mathrm{nx})$, 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">$\frac{d^{2} y}{d x^{2}}+y=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{d^{2} y}{d x^{2}}=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2} \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$</span> </div>

answer:
<div class="solution">(c) $y=\sin (\log x)$<br/>$\Rightarrow \frac{d y}{d x}=\frac{\cos (\log x)}{x}$<br/>$\Rightarrow \quad x\left(\frac{d y}{d x}\right)=\cos (\log x)$<br/>Again differentiating, $x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=-\frac{\sin \log x}{x}=\frac{-y}{x}$<br/>$\Rightarrow \quad x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$</div>


title: if-y-sin-x-x-then-what-is-the-value-of-dx-dy-at-x-6
slug: if-y-sin-x-x-then-what-is-the-value-of-dx-dy-at-x-6-41253
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=|\sin \mathrm{x}| \mathrm{x} \mid$, then what is the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{\pi}{6} ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2^{-\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}$</span> </div>

answer:
<div class="solution">$\begin{aligned} y &amp;=(-\sin x)^{-x} \\ &amp; \frac{d y}{d x}=(-\sin )^{-x}\left[\frac{x}{\sin x}(-\cos x)+\log (-\sin x) \cdot(-1)\right] \\ &amp;\left[\frac{d y}{d x}\right] x=-\frac{\pi}{6}=\left(\frac{1}{2}\right)^{\pi / 6}\left[-\frac{\pi}{6} \sqrt{3}-\log \frac{1}{2}\right] \\ &amp;=2^{-\pi / 6}\left(\frac{6 \log 2-\sqrt{3} \pi}{6}\right) \end{aligned}$</div>


title: if-y-tan-1-2-5-t-a-n-x-5-2-t-a-n-x-then-what-is-dx-dy-equal-to
slug: if-y-tan-1-2-5-t-a-n-x-5-2-t-a-n-x-then-what-is-dx-dy-equal-to-67589
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\tan ^{-1}\left(\frac{5-2 \tan \sqrt{\mathrm{x}}}{2+5 \tan \sqrt{\mathrm{x}}}\right)$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to?</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; y=\tan ^{-1}\left(\frac{5-2 \tan \sqrt{x}}{2+5 \tan \sqrt{x}}\right) \\ &amp;=\tan ^{-1}\left(\frac{\frac{5}{2}-\tan \sqrt{x}}{1+\left(\frac{5}{2}\right) \tan \sqrt{x}}\right) \\ &amp;=\tan ^{-1} \frac{5}{2}-\tan ^{-1} \tan \sqrt{x} \\ &amp;=\tan ^{-1} \frac{5}{2}-\sqrt{x} \\ \therefore &amp; \frac{d y}{d x}=-\frac{1}{2 \sqrt{x}} \end{aligned}$</div>


title: if-y-x-1-x-1-then-what-is-dx-dy-equal-to
slug: if-y-x-1-x-1-then-what-is-dx-dy-equal-to-91235
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{y}=\frac{\mathrm{x}+1}{\mathrm{x}-1}$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to?</div>

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

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

answer:
<div class="solution">Let $\mathrm{y}=\frac{\mathrm{x}+1}{\mathrm{x}-1}$<br/>Differentiate both the side w.r.t ' $x$ '<br/>$\begin{aligned} \frac{\mathrm{dy}}{\mathrm{dx}} &amp;=\frac{(\mathrm{x}-1)(1)-(\mathrm{x}+1) 1}{(\mathrm{x}-1)^{2}} \\ &amp;=\frac{\mathrm{x}-1-\mathrm{x}-1}{(\mathrm{x}-1)^{2}}=\frac{-2}{(\mathrm{x}-1)^{2}} \end{aligned}$</div>


title: if-y-x-e-x-then-what-is-d-y-2-d-2-x-equal-to
slug: if-y-x-e-x-then-what-is-d-y-2-d-2-x-equal-to-44930
question: <div class="question">If $y=x+e^{x}$, then, what is $\frac{d^{2} x}{d y^{2}}$ equal to?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{e}^{\mathrm{x}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{\mathrm{e}^{\mathrm{x}}}{\left(1+\mathrm{e}^{\mathrm{x}}\right)^{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{e^{x}}{\left(1+e^{x}\right)}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{\mathrm{e}^{\mathrm{x}}}{\left(1+\mathrm{e}^{\mathrm{x}}\right)^{2}}$</span> </li> </ul>

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

answer:
<div class="solution">Given that $y=x+e^{x}$ Differentiating w. r. t. $\mathrm{x}$<br/>$\frac{\mathrm{d} \mathrm{y}}{\mathrm{dx}}=1+\mathrm{e}^{\mathrm{x}}$<br/>$\Rightarrow \frac{\mathrm{dx}}{\mathrm{dy}}=\frac{1}{1+\mathrm{e}^{\mathrm{x}}}$<br/>Differentiating w. r.t. $\mathrm{y}$<br/>$\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dy}^{2}}=-\frac{(1)\left(\mathrm{e}^{\mathrm{x}}\right)}{\left(1+\mathrm{e}^{\mathrm{x}}\right)^{2}} \cdot \frac{\mathrm{dx}}{\mathrm{dy}}$<br/>$=-\frac{\mathrm{e}^{\mathrm{x}}}{\left(1+\mathrm{e}^{\mathrm{x}}\right)^{2}} \cdot \frac{1}{\left(1+\mathrm{e}^{\mathrm{x}}\right)}=-\frac{\mathrm{e}^{\mathrm{x}}}{\left(1+\mathrm{e}^{\mathrm{x}}\right)^{3}}$</div>


title: if-y-x-ln-x-x-e-x-then-what-is-the-value-of-d-x-d-y-at-x-1
slug: if-y-x-ln-x-x-e-x-then-what-is-the-value-of-d-x-d-y-at-x-1-83447
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=x \ln x+x e^{x}$, then what is the value of $\frac{d y}{d x}$ at $x=1$?</div>

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

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

answer:
<div class="solution">$y=x \ln x+x e^{x}$<br/>After differentiating both sides with respect to $\mathrm{x}$,<br/>we get $\frac{d y}{d x}=x \cdot \frac{1}{x}+\log x+x e^{x}+e^{x}$<br/>or $=1+\log x+x e^{x}+e^{x}$<br/>Therefore $\left(\frac{d y}{d x}\right)_{x=1}=1+\log 1+1 . e^{1}+e^{1}=1+2 e$<br/>$[\because \log 1=0]$</div>


title: if-y-x-x-2-x-3-up-to-infinite-terms-where-x-1-then-which-one-of-the-following-is-correct
slug: if-y-x-x-2-x-3-up-to-infinite-terms-where-x-1-then-which-one-of-the-following-is-correct-61307
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=x+x^{2}+x^{3}+\ldots$ up to infinite terms where $x   &lt; 1$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{x}=\frac{\mathrm{y}}{1+\mathrm{y}}$</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">$\mathrm{x}=\frac{\mathrm{y}}{1-\mathrm{y}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{x}=\frac{1+\mathrm{y}}{\mathrm{y}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{x}=\frac{1-\mathrm{y}}{\mathrm{y}}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{y}=\mathrm{x}+\mathrm{x}^{2}+\mathrm{x}^{3}+\ldots$ upto infinite terms. $\mathrm{x}+\mathrm{x}^{2}+\mathrm{x}^{3}+\ldots$ is a G.P. with $\mathrm{a}=\mathrm{x}$ and $\mathrm{r}=\mathrm{x}$<br/>$\therefore \mathrm{y}=\frac{\mathrm{x}}{1-\mathrm{x}}$<br/>$\Rightarrow \mathrm{y}-\mathrm{xy}=\mathrm{x}$<br/>$\Rightarrow \mathrm{x}+\mathrm{xy}=\mathrm{y}$<br/>$\Rightarrow \mathrm{x}(1+\mathrm{y})=\mathrm{y}$<br/>$\Rightarrow \mathrm{x}=\frac{\mathrm{y}}{1+\mathrm{y}}$</div>


title: if-y-x-x-what-is-d-x-d-y-at-x-1-equal-to
slug: if-y-x-x-what-is-d-x-d-y-at-x-1-equal-to-37059
question: <div class="question">If $y=x^{x}$, what is $\frac{d y}{d x}$ at $x=1$ equal to?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{y}=\mathrm{x}^{\mathrm{x}}$<br/>Take $\log$ on both the sides $\Rightarrow \ell \mathrm{ny}=\mathrm{x} \cdot \ln \mathrm{x}$<br/>$\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x}=1+\ell n x \quad \Rightarrow \frac{1}{y} \cdot \frac{d y}{d x}=\ell n e+\ell n x$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \ell \mathrm{n} \mathrm{ex}==\left(\mathrm{x}^{\mathrm{x}}\right) \ln \mathrm{ex}$<br/>$\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{\mathrm{x}=1}=1 . \quad$ $\ell n\mathrm{e}=1$</div>


title: if-y-z-x-x-y-z-y-z-x-y-z-x-0-then-which-one-of-the-following-is-correct
slug: if-y-z-x-x-y-z-y-z-x-y-z-x-0-then-which-one-of-the-following-is-correct-12873
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{lll}y &amp; x &amp; y+z \\ z &amp; y &amp; x+y \\ x &amp; z &amp; z+x\end{array}\right|=0$, then which one of the following is<br/>correct?</div>

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

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

answer:
<div class="solution">Given, $\left|\begin{array}{ccc}y &amp; x &amp; y+z \\ z &amp; y &amp; x+y \\ x &amp; z &amp; z+x\end{array}\right|=0$<br/>Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}$<br/>$\Rightarrow\left|\begin{array}{ccc}x+y+ &amp; x+y+z &amp; 2(x+y+z) \\ z &amp; y &amp; x+y \\ x &amp; z &amp; z+x\end{array}\right|=0$<br/>$\Rightarrow(x+y+z)\left|\begin{array}{ccc}1 &amp; 1 &amp; 2 \\ z &amp; y &amp; x+y \\ x &amp; z &amp; z+x\end{array}\right|=0$<br/>Applying $\mathrm{C}_{2} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}, \mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-2 \mathrm{C}_{1}$<br/>$\Rightarrow(x+y+z)\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ z &amp; z-y &amp; x+y-2 z \\ x &amp; z-x &amp; z-x\end{array}\right|=0$<br/>$\Rightarrow(x+y+z)\left|\begin{array}{cc}z-y &amp; x+y-2 z \\ z-x &amp; z-x\end{array}\right|=0$<br/>$\Rightarrow(x+y+z)(z-x)(z-y-x-y+2 z)=0$<br/>$\Rightarrow x+y=-z$ or $z=x$</div>


title: if-yleftbeginarraycccfx-gx-hx-l-m-n-a-b-cendarrayright-then-fracd-yd-x-is-equal-to
slug: if-yleftbeginarraycccfx-gx-hx-l-m-n-a-b-cendarrayright-then-fracd-yd-x-is-equal-to-90725
tags: ['Mathematics', 'Vector Algebra', 'KCET', 'KCET 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \( y=\left|\begin{array}{ccc}f(x) &amp; g(x) &amp; h(x) \\ l &amp; m &amp; n \\ a &amp; b &amp; c\end{array}\right| \), then \( \frac{d y}{d x} \) is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\( \left|\begin{array}{ccc}f^{\prime}(x) &amp; g^{\prime}(x) &amp; h^{\prime}(x) \\ l &amp; m &amp; n \\ a &amp; b &amp; c\end{array}\right| \)</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">\( \left|\begin{array}{ccc}l &amp; m &amp; n \\ f^{\prime}(x) &amp; g^{\prime}(x) &amp; h^{\prime}(x) \\ a &amp; b &amp; c\end{array}\right| \)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\( \left|\begin{array}{lll}f^{\prime}(x) &amp; l &amp; a \\ g^{\prime}(x) &amp; m &amp; b \\ h^{\prime}(x) &amp; n &amp; c\end{array}\right| \)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( \left|\begin{array}{ccc}l &amp; m &amp; n \\ a &amp; b &amp; c \\ f^{\prime}(x) &amp; g^{\prime}(x) &amp; h^{\prime}(x)\end{array}\right| \)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( \left|\begin{array}{ccc}f^{\prime}(x) &amp; g^{\prime}(x) &amp; h^{\prime}(x) \\ l &amp; m &amp; n \\ a &amp; b &amp; c\end{array}\right| \)</span> </div>

answer:
<div class="solution">Given that, \( y=\left|\begin{array}{ccc}f(x) &amp; g(x) &amp; h(x) \\ l &amp; m &amp; n \\ a &amp; b &amp; c\end{array}\right| \)<br/>So,<br/>\( \frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) &amp; g^{\prime}(x) &amp; h^{\prime}(x) \\ l &amp; m &amp; n \\ a &amp; b &amp; c\end{array}\right|+\left|\begin{array}{ccc}f(x) &amp; g(x) &amp; h(x) \\ 0 &amp; 0 &amp; 0 \\ a &amp; b &amp; c\end{array}\right|+ \)<br/>\( \left|\begin{array}{ccc}f(x) &amp; g(x) &amp; h(x) \\ l &amp; m &amp; n \\ 0 &amp; 0 &amp; 0\end{array}\right| \)<br/>\( =\left|\begin{array}{ccc}f^{\prime}(x) &amp; g^{\prime}(x) &amp; h^{\prime}(x) \\ l &amp; m &amp; n \\ a &amp; b &amp; c\end{array}\right| \)</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Mode</span> </div>
source_db: MarksBatch2_P1.db
title:
if-you-want-to-measure-the-intelligence-of-a-group-of-students-which-one-of-the-following-measures-will-be-more-suitable

slug:
if-you-want-to-measure-the-intelligence-of-a-group-of-students-which-one-of-the-following-measures-will-be-more-suitable-61107

question:
<div class="question">If you want to measure the intelligence of a group of students, which one of the following measures will be more suitable?</div>

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

answer:
<div class="solution">To measure the intelligence of a group of students mode will be more suitable</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">None</span> </div>
source_db: MarksBatch2_P1.db
title:
if-z-1-and-z-2-are-complex-numbers-with-z-1-z-2-then-which-of-the-following-isare-correct-1-z-1-z-2-2-real-part-of-z-1-real-part-of-z-2-3-imaginary-pa

slug:
if-z-1-and-z-2-are-complex-numbers-with-z-1-z-2-then-which-of-the-following-isare-correct-1-z-1-z-2-2-real-part-of-z-1-real-part-of-z-2-3-imaginary-pa-49840

question:
<div class="question">If $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ are complex numbers with $\left|\mathrm{z}_{1}\right|=\left|\mathrm{z}_{2}\right|$, then which of the following is/are correct? <br/>1- $z_{1}=z_{2}$<br/>2- Real part of $\mathrm{z}_{1}=$ Real part of $\mathrm{z}_{2}$<br/>3- Imaginary part of $z_{1}=$ Imaginary part of $z_{2}$ Select the correct answer using the code given below:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3 only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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 $Z_{1}=a_{1}+i b_{1}$ $\mathrm{Z}_{2}=\mathrm{a}_{2}+\mathrm{ib}_{2}$<br/>$\left|\mathrm{Z}_{1}\right|=\left|\mathrm{Z}_{2}\right|$<br/>$\sqrt{\left(\mathrm{a}_{1}\right)^{2}+\left(\mathrm{b}_{1}\right)^{2}}=\sqrt{\left(\mathrm{a}_{2}\right)^{2}+\left(\mathrm{b}_{2}\right)^{2}}$<br/>It is true for many values of $\mathrm{a}_{1}, \mathrm{a}_{2} \&amp; \mathrm{~b}_{1}, \mathrm{~b}_{2}$. So $\mathrm{a}_{1}$ must not equal to $\mathrm{a}_{2}$, and $\mathrm{b}_{1}$ must not equal to $\mathrm{b}_{2}$.</div>


title: if-z-1-cos-5-i-sin-5-then-what-is-z-equal-to
slug: if-z-1-cos-5-i-sin-5-then-what-is-z-equal-to-59887
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z=1+\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}$, then what is $|z|$ equal to ?</div>

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

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

answer:
<div class="solution">$\begin{aligned}|z| &amp;=\sqrt{1^{2}+\cos ^{2} \frac{\pi}{5}+2 \cos \frac{\pi}{5}+\sin ^{2} \frac{\pi}{5}} \\ &amp;=\sqrt{1^{2}+1+2 \cos \frac{\pi}{5}}=\sqrt{2\left(1+\cos \frac{\pi}{5}\right)} \\ &amp;=\sqrt{2\left(1+2 \cos ^{2} \frac{\pi}{10}-1\right)}=\sqrt{2\left(2 \cos ^{2} \frac{\pi}{10}\right)} \\ &amp;=2 \cos \frac{\pi}{10} \end{aligned}$</div>


title: if-z-1-i-tan-where-2-3-then-what-is-z-equal-to
slug: if-z-1-i-tan-where-2-3-then-what-is-z-equal-to-56909
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z=1+i \tan \alpha$ where $\pi   &lt; \alpha &lt; \frac{3 \pi}{2}$, then what is $|z|$ equal to?</div>

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

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

answer:
<div class="solution">Given : $z=1+i \tan \alpha, \pi   &lt; \alpha &lt; \frac{3 \pi}{2}$<br/>$|z|=\sqrt{(1)^{2}+(\tan \alpha)^{2}}=\sqrt{\sec ^{2} \alpha}$<br/>$|z|=\sec \alpha .$<br/>Since, $\pi &lt; \alpha &lt; \frac{3 \pi}{2}$<br/>$\therefore$ Sec $\alpha$ lies in III $^{\mathrm{rd}}$ quadrant <br/>In $3^{\mathrm{rd}}$ quadrant, $\mathrm{sec} \alpha$ is negative.</div>


title: if-z-1-i-then-what-is-the-inverse-of-z-2
slug: if-z-1-i-then-what-is-the-inverse-of-z-2-60542
question: <div class="question">If $z=1+i$, then what is the inverse of $z^{2}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{i}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">i</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{i}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{2}$</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">$-\frac{1}{2}$</span> </div>

answer:
<div class="solution">Given that $z=1+i$ $\Rightarrow z^{2}=(1+i)^{2}=1+i^{2}+2 i=1-1+2 i=2 i$<br/>Inverse of $z^{2}=\frac{1}{2 i}=-\frac{i^{2}}{2 \mathrm{i}}\left[\right.$ Since $\left.i^{2}=-1 ;-\mathrm{i}^{2}=1\right]$<br/>$=-\frac{\mathrm{i}}{2}$</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-z-1-z-2-are-any-two-complex-numbers-such-that-z-1-z-2-z-1-z-2-which-one-of-the-following-is-correct

slug:
if-z-1-z-2-are-any-two-complex-numbers-such-that-z-1-z-2-z-1-z-2-which-one-of-the-following-is-correct-48728

question:
<div class="question">If $z_{1}, z_{2}$ are any two complex numbers such that $\left|z_{1}+z_{2}\right|=$ $\left|z_{1}\right|+\left|z_{2}\right|$, which one of the following is correct ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{z}_{1}=\alpha \mathrm{z}_{2}$ with $\alpha \in \mathrm{R}$</span> </div>

answer:
<div class="solution">Let $z_{1}=r_{1}(\cos \theta+i \sin \theta)$<br/>Squaring both the sides $\mathrm{r}_{1}^{2}+\mathrm{r}_{2}^{2}+2 \mathrm{r}_{1} \mathrm{r}_{2} \cos (\theta-\phi)=\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)^{2}$<br/>$=\mathrm{r}_{1}{ }^{2}+\mathrm{r}_{2}{ }^{2}+2 \mathrm{r}_{1} \mathrm{r}_{2}$<br/>or, $2 \mathrm{r}_{1} \mathrm{r}_{2} \cos (\theta-\phi)=2 \mathrm{r}_{1} \mathrm{r}_{2}$<br/>or, $\cos (\theta-\phi)=1$<br/>$\theta-\phi=0 \Rightarrow \theta=\phi$<br/>$\operatorname{Arg}\left(z_{1}\right)=\operatorname{Arg}\left(z_{2}\right)$ so, $z_{1}=\alpha z_{2}$<br/>where $\alpha \in \mathrm{R}$</div>


title: if-z-2-3-2-i-107-2-3-2-i-107-then-what-is-the
slug: if-z-2-3-2-i-107-2-3-2-i-107-then-what-is-the-57624
tags: ['Mathematics', 'Complex Number', '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
question:
<div class="question">If $\mathrm{z}=\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)^{107}+\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)^{107}$, then what is the</div>

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

answer:
<div class="solution">$z=\left[\frac{\sqrt{3}}{2}+\frac{i}{2}\right]^{107}+\left[\frac{\sqrt{3}}{2}-\frac{i}{2}\right]^{107} .$<br/>$\because \cos \frac{\pi}{6}=\frac{\sqrt{3}}{2} \&amp; \sin \frac{\pi}{6}=\frac{1}{2}$<br/>$\Rightarrow \quad \mathrm{z}=\left[\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right]^{107}+\left[\cos \frac{\pi}{6}-\mathrm{i} \sin \frac{\pi}{6}\right]^{107}$<br/>$\mathrm{Also},(\cos \theta+\mathrm{i} \sin \theta)^{\mathrm{n}}=\cos n \theta+\mathrm{i} \sin \mathrm{n} \theta$<br/>$\Rightarrow \quad \mathrm{z}=\cos \frac{107 \pi}{6}+\mathrm{i} \sin \frac{107 \pi}{6}+\cos \frac{107 \pi}{6}-\mathrm{i} \sin \frac{107 \mathrm{t}}{6} .$<br/>$\Rightarrow \quad \operatorname{Im}(\mathrm{z})=0$</div>


title: if-z-2-i-1-2-i-1-2-i-2-i-then-what-is-the-value-of-z-2-z-z-i-1
slug: if-z-2-i-1-2-i-1-2-i-2-i-then-what-is-the-value-of-z-2-z-z-i-1-73645
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z=\frac{1+2 i}{2-i}-\frac{2-i}{1+2 i}$, then what is the value of $z^{2}+z \bar{z}$ ?<br/>$(i=\sqrt{-1})$</div>

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

answer:
<div class="solution">$z=\frac{1+2 i}{2-i}-\frac{2-i}{1+2 i}$<br/>$z=\frac{(1+2 i)^{2}-(2-i)^{2}}{(2-i)(1+2 i)}=\frac{1+4 i^{2}+4 i-4-i^{2}+4 i}{2+4 i-i-2 i^{2}}$<br/>$=\frac{-3-4+8 i+1}{4+3 i}=\frac{-6+8 i}{4+3 i}=\frac{(-6+8 i)(4-3 i)}{16+9}$<br/>$=\frac{-24+18 i+32 i-24 i^{2}}{25}=\frac{50 i}{25}=2 i$<br/>Consider,<br/>$z^{2}+z \bar{z}=(2 i)^{2}+(2 i)(-2 i)=4 i^{2}-4 i^{2}=0$</div>


title: if-z-3-i-2-1-2-i-where-i-1-then-argument-of-z-is
slug: if-z-3-i-2-1-2-i-where-i-1-then-argument-of-z-is-32349
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z=\frac{-2(1+2 i)}{3+i}$ where $i=\sqrt{-1}$, then argument<br/>$\theta(-\pi   &lt; \theta \leq \pi)$ of $z$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3 \pi}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{5 \pi}{6}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-\frac{3 \pi}{4}$</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>

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

answer:
<div class="solution">$\begin{aligned} &amp; z=\frac{-2(1+2 i)}{3+i} \\ &amp;=\frac{-2-4 i}{3+i}=\frac{-2-4 i}{3+i} \times \frac{3-i}{3-i} \\ &amp;=\frac{-6+2 i-12 i+4 i^{2}}{10} \\ &amp;=\frac{-6-10 i-4}{10}=\frac{-10-10 \mathrm{i}}{10}=-1-i \\ &amp; z=-1-i=r(\cos \theta+i \sin \theta) \end{aligned}$<br/>On comparing real and imaginary part on both sides, we get $r \cos \theta=-1$<br/>$r \sin \theta=-1$<br/>On dividing eq. (ii) by (i), we get<br/>$\frac{\mathrm{r} \sin \theta}{\mathrm{r} \cos \theta}=\frac{-1}{-1}$<br/>$\tan \theta=1=\tan \frac{\pi}{4}$<br/>$\Rightarrow \quad \theta=\frac{\pi}{4}$<br/>$\therefore \quad \theta=\frac{\pi}{4}$<br/>$\therefore$ Option (b) is correct.</div>


title: if-z-5-3-then-the-maximum-value-of-z-1-is
slug: if-z-5-3-then-the-maximum-value-of-z-1-is-43891
question: <div class="question">If $|z+5| \leq 3$, then the maximum value of $|z+1|$ is</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</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">10</span> </li> </ul>

answer:
<div class="solution">$|z+4| \leq 3$<br/>$|z+1|=|z+4+(-3)|$<br/>We know, $\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|$<br/>So, $|z+4+(-3)| \leq|z+4|+|-3|$<br/>$\leq|3|+|-3|$<br/>$\leq 6$<br/>So, maximum value $=6$.</div>


title: if-z-f-o-f-x-x-2-where-f-x-x-2-then-what-is-d-x-d-z-equal-to
slug: if-z-f-o-f-x-x-2-where-f-x-x-2-then-what-is-d-x-d-z-equal-to-24561
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z=f o f(x)=x^{2}$ where $f(x)=x^{2}$, then what is $\frac{d z}{d x}$ equal to?</div>

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

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

answer:
<div class="solution">$\mathrm{z}=$ fof $(\mathrm{x})=\mathrm{f}\left(\mathrm{x}^{2}\right)=\mathrm{x}^{4}$<br/>$\frac{\mathrm{dz}}{\mathrm{dx}}=4 \mathrm{x}^{3}$</div>


title: if-z-is-a-complex-number-such-that-z-z-1-1-then-what-is-the-value-of-z-99-z-99
slug: if-z-is-a-complex-number-such-that-z-z-1-1-then-what-is-the-value-of-z-99-z-99-24353
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z$ is a complex number such that $z+z^{-1}=1$, then what is the value of $z^{99}+z^{-99} ?$</div>

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

answer:
<div class="solution">Given, $z+z^{-1}=1 \Rightarrow z+\frac{1}{z}=1$<br/>$\Rightarrow z^{2}-z+1=0 \Rightarrow z=-\omega$ and $-\omega^{2}$<br/>when $z=-\omega$<br/>$\therefore z^{99}+z^{-99}=(-\omega)^{99}+(-\omega)^{-99}=-1-1=-2$<br/>when $z=-\omega^{2}$<br/>$\therefore z^{99}+z^{-99}=\left(-\omega^{2}\right)^{99}+\left(-\omega^{2}\right)^{-99}$<br/>$\quad=-1-1=-2$</div>


title: if-z-x-iy-2-1-2-i-25-where-i-1-then-what-is-the-fundamental-amplitude-of-z-i-2-z-2
slug: if-z-x-iy-2-1-2-i-25-where-i-1-then-what-is-the-fundamental-amplitude-of-z-i-2-z-2-96522
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{z}=\mathrm{x}+\mathrm{iy}=\left(\frac{1}{\sqrt{2}}-\frac{\mathrm{i}}{\sqrt{2}}\right)^{-25}$, where $\mathrm{i}=\sqrt{-1}$, then what is<br/>the fundamental amplitude of $\frac{\mathrm{z}-\sqrt{2}}{\mathrm{z}-\mathrm{i} \sqrt{2}} ?$</div>

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

answer:
<div class="solution">$\begin{aligned} z &amp;=x+i y \\ &amp;=\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)^{-25} \\ &amp;=\left[\cos \frac{\pi}{4}-i \sin \frac{\pi}{4}\right]^{-25} \end{aligned}$<br/>$\because(\operatorname{Cos} \pi-\mathrm{i} \sin \pi)^{\mathrm{n}}=\operatorname{Cos} \mathrm{n} \pi-\mathrm{i} \operatorname{Sin} \mathrm{n} \pi$<br/>$\left[\begin{array}{l}\cos (-\theta)=\cos \theta \\ \sin (-\theta)=-\sin \theta\end{array}\right]$<br/>$\begin{array}{l}\mathrm{z}=\left[\cos \left(\frac{25 \pi}{4}\right)+i \sin \left(\frac{25 \pi}{4}\right)\right] \\ = &amp; {\left[\cos \left(6 \pi+\frac{\pi}{4}\right)+i \sin \left(6 \pi+\frac{\pi}{4}\right)\right]} \\ =\cos \frac{\pi}{4}+i \sin \frac{\pi}{4} \\ =\frac{1+i}{\sqrt{2}} \\ \frac{z-\sqrt{2}}{z-i \sqrt{2}}=\frac{(1+i-2) \sqrt{2}}{\sqrt{2}(1+i-2 i)} \\ =\frac{-1+i}{1-i}=-1 \\ \text { Amplitude }=\tan ^{-1} \frac{b}{a}=\tan ^{-1}\left(\frac{0}{-1}\right)=\tan ^{-1} 0 \\ =\tan ^{-1}(\tan \pi)=\pi\end{array}$<br/>Hence fundamental amplitude of $\left(\frac{z-\sqrt{2}}{z-i \sqrt{2}}\right)$ is $\pi$</div>


title: if-z-z-4-2-then-the-maximum-value-of-z-is-equal-to
slug: if-z-z-4-2-then-the-maximum-value-of-z-is-equal-to-88262
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $|z|$ is equal to</div>

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

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

answer:
<div class="solution">$\left|\mathrm{z}-\frac{4}{\mathrm{z}}\right|=2 .$<br/>We know $|\mathrm{a}-\mathrm{b}| \geq|\mathrm{a}|-|\mathrm{b}|$<br/>$\therefore\left|z-\frac{4}{z}\right| \geq|z|-\left|\frac{4}{z}\right|$<br/>$\Rightarrow 2 \geq|z|-\frac{4}{|z|} \Rightarrow|z|^{2}-2|z|-4 \leq 0$<br/>$\Rightarrow|z|=\frac{2 \pm \sqrt{4-4(1)(-4)}}{2(1)}=\frac{2 \pm \sqrt{20}}{2}=\frac{2 \pm 2 \sqrt{5}}{2}=1 \pm \sqrt{5}$</div>


title: if-z-z-then-which-one-of-the-following-is-correct
slug: if-z-z-then-which-one-of-the-following-is-correct-75326
question: <div class="question">If $z=-\bar{z}$, then which one of the following is correct?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">real part of $z$ is zero. $[2012-I]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The imaginary part of $\mathrm{z}$ is zero.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The real part of $z$ is equal to imaginary</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The sum of real and imaginary parts of z is z.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">real part of $z$ is zero. $[2012-I]$</span> </div>

answer:
<div class="solution">Let z $=\mathrm{x}+\mathrm{y}$ then $\overline{\mathrm{z}}=\mathrm{x}-\mathrm{iy}$<br/>Now $z=-\bar{z}$<br/>$\Rightarrow(\mathrm{x}+\mathrm{iy})=-(\mathrm{x}-\mathrm{iy}) \Rightarrow \mathrm{x}+\mathrm{iy}=-\mathrm{x}+\mathrm{iy}$<br/>$\Rightarrow 2 \mathrm{x}=0 \Rightarrow \operatorname{Re}(\mathrm{z})=0$</div>


title: if-z-z-z-z-then-the-locus-of-z-is
slug: if-z-z-z-z-then-the-locus-of-z-is-16277
question: <div class="question">If $|z+\bar{z}|=|z-\bar{z}|$, then the locus of $z$ is:</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">A pair of straight lines</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A line</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A set of four straight lines</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A circle</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A pair of straight lines</span> </div>

answer:
<div class="solution">Let $z=x+$ iy, $\bar{z}=x-i y$<br/>$|\mathrm{z}+\overline{\mathrm{z}}|=|\mathrm{z}-\overline{\mathrm{z}}|$<br/>$|(\mathrm{x}+\mathrm{iy})+(\mathrm{x}-\mathrm{iy})|=|(\mathrm{x}+\mathrm{iy})-(\mathrm{x}-\mathrm{iy})|$<br/>$|2 \mathrm{x}|=|2 \mathrm{y}|$<br/>$\mathrm{x}=\pm \mathrm{y}$</div>


title: if-zleftbeginarrayccc1-12-i-5-i-12-i-3-53-i-5-i-53-i-7endarrayright-then-isqrt1-1
slug: if-zleftbeginarrayccc1-12-i-5-i-12-i-3-53-i-5-i-53-i-7endarrayright-then-isqrt1-1-57789
tags: ['Mathematics', 'Determinants', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(z=\left(\begin{array}{ccc}1 &amp; 1+2 i &amp; -5 i \\ 1-2 i &amp; -3 &amp; 5+3 i \\ 5 i &amp; 5-3 i &amp; 7\end{array}\right)\) then \((i=\sqrt{-1})\)</div>

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

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hints : } z=\left|\begin{array}{ccc}<br/>1 &amp; 1+2 i &amp; -5 i \\<br/>1-2 i &amp; -3 &amp; 5+3 i \\<br/>5 i &amp; 5-3 i &amp; 7<br/>\end{array}\right|=1(-21-64)-((1-2 i)(7(1+2 i)+5 i(5-3 i)))+5 i((1+2 i)(5+3 i)-15 i) \\<br/>&amp; \text { = Real } \\<br/>&amp;<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
ifa-is-an-invertible-matrix-of-order-n-and-k-is-any-positive-real-number-then-the-value-of-det-ka-1-det-a-is

slug:
ifa-is-an-invertible-matrix-of-order-n-and-k-is-any-positive-real-number-then-the-value-of-det-ka-1-det-a-is-88168

question:
<div class="question">IfA is an invertible matrix of order $\mathrm{n}$ and $\mathrm{k}$ is any positive real number, then the value of $[\operatorname{det}(\mathrm{kA})]^{-1}$ det $\mathrm{A}$ is</div>

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

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

answer:
<div class="solution">If A is a matrix that is invertible then det- $(\mathrm{kA})$ will be $\mathrm{k}^{\mathrm{n}} \cdot \operatorname{det}(\mathrm{A})$, where $\mathrm{n}$ is the order.<br/>$\therefore [\operatorname{det}(\mathrm{KA})]^{-1} \operatorname{det}(\mathrm{A})$<br/>$=\left[(\mathrm{K})^{\mathrm{n}} \times \operatorname{det}(\mathrm{A})\right]^{-1} \cdot \operatorname{det}(\mathrm{A})$<br/>$=\mathrm{K}^{-\mathrm{n}} \times \frac{1}{\operatorname{det}(\mathrm{A})} \times \operatorname{det}(\mathrm{A})$<br/>$=\mathrm{K}^{-\mathrm{n}}$</div>


title: ifabcd-is-a-cyclic-quadrilateral-then-what-is-sin-a-sin-b-sin-c-sin-d-equal-to
slug: ifabcd-is-a-cyclic-quadrilateral-then-what-is-sin-a-sin-b-sin-c-sin-d-equal-to-22495
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">IfABCD is a cyclic quadrilateral then what is $\sin A+\sin B-\sin C-\sin D$ equal to?</div>

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

answer:
<div class="solution">We know in cyclic quadrilateral, ABCD $\mathrm{A}+\mathrm{C}=180^{\circ}, \mathrm{B}+\mathrm{D}=180^{\circ}$<br/>$\therefore \mathrm{A}=180^{\circ}-\mathrm{C}, \mathrm{B}=180^{\circ}-\mathrm{D}$<br/>$\sin \mathrm{A}+\sin \mathrm{B}-\sin \mathrm{C}-\sin \mathrm{D}$<br/>$=\sin \left(180^{\circ}-\mathrm{C}\right)+\sin \left(180^{\circ}-\mathrm{D}\right)-\sin \mathrm{C}-\sin \mathrm{D}$<br/>$=\sin \mathrm{C}+\sin \mathrm{D}-\sin \mathrm{C}-\sin \mathrm{D}=0$</div>


title: iff-r-r-such-that-f-x-13-x-then-what-is-the-value-off-1-x
slug: iff-r-r-such-that-f-x-13-x-then-what-is-the-value-off-1-x-58701
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Iff: $\mathrm{R} \rightarrow \mathrm{R}^{+}$ such that $\mathrm{f}(\mathrm{x})=(1 / 3)^{\mathrm{x}}$, then what is the value $\operatorname{off}^{-1}(\mathrm{x}) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{1 / 3} x$</span> </div>

answer:
<div class="solution">Given function is $\mathrm{f}(\mathrm{x})=\left(\frac{1}{3}\right)^{\mathrm{x}}$<br/>Let $f(x)=y$, so, $y=\left(\frac{1}{3}\right)^{x}$<br/>Taking $\log _{1 / 3}$ on both sides $\Rightarrow \mathrm{x} \times \log _{1 / 3}\left(\frac{1}{3}\right)=\log _{(1 / 3)} \mathrm{y}$<br/>$\mathrm{x}=\log _{(1 / 3)} \mathrm{y}$<br/>$\Rightarrow \mathrm{f}^{-1}(\mathrm{x})=\log _{(1 / 3)} \mathrm{x}$</div>


title: ifn-3-n-and-n-1-are-in-gp-then-the-value-of-n-will-be
slug: ifn-3-n-and-n-1-are-in-gp-then-the-value-of-n-will-be-31757
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Ifn! $3 \times(n !)$ and $(n+1) !$ are in GP, then the value of $n$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</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">10</span> </li> </ul>

answer:
<div class="solution">Let n!, $3(\mathrm{n} !)$ and $(\mathrm{n}+1) !$ are in G. $P$. Then, $[3(\mathrm{n} !)]^{2}=(\mathrm{n} !)(\mathrm{n}+1) !$<br/>$\Rightarrow 9 \times n ! \times n !=(n !)(n+1) n !$<br/>$\Rightarrow 9=(\mathrm{n}+1) \Rightarrow \mathrm{n}=8$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
ifo-pare-the-points-0-0-0-2-3-1-respectively-then-what-is-the-equation-to-the-plane-through-p-at-right-angles-to-op

slug:
ifo-pare-the-points-0-0-0-2-3-1-respectively-then-what-is-the-equation-to-the-plane-through-p-at-right-angles-to-op-97866

question:
<div class="question">IfO, Pare the points $(0,0,0),(2,3,-1)$ respectively, then what is the equation to the plane through $\mathrm{P}$ at right angles to $\mathrm{OP}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 x+3 y+z=16$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 x+3 y-z=14$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x+3 y+z=14$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x+3 y-z=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 x+3 y-z=14$</span> </div>

answer:
<div class="solution">Since, coordinates of points $\mathrm{O}$ and $\mathrm{P}$ are $(0,0,0)$ and $(2,3,-1)$ respectively. Direction ratios of $\mathrm{OP}$ are $   &lt; 2,3,-1\rangle$<br/>The plane is perpendicular to OP. So, its equation is $2 x+3 y-z+d=0$<br/>Since, this plane passes through $(2,3,-1) ; 2 \times 2+3 \times$<br/>$3-1 \times-1+d=0$<br/>$\Rightarrow 4+9+1+\mathrm{d}=0$<br/>$\Rightarrow \mathrm{d}=-14$<br/>On putting the value of $\mathrm{d}$ in Eq. (i) $2 x+3 y-z-14=0$<br/>$\Rightarrow 2 x+3 y-z=14$<br/>which is required equation of plane</div>


title: ifpe-denotes-the-probability-of-an-event-e-then-e-is-called-certain-event-if
slug: ifpe-denotes-the-probability-of-an-event-e-then-e-is-called-certain-event-if-13206
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2012 (Phase 2)']
answer: <div class="solution">Prob. (certain event) $=1$ $\Rightarrow \mathrm{P}(\mathrm{E})=1$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">IfP(E) denotes the probability of an event $\mathrm{E}$, then $\mathrm{E}$ is called certain event if:</div>

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

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


title: ifpq-are-2-5-7-3-2-1-respectively-then-what-are-the-direction-ratios-of-the-line-pq
slug: ifpq-are-2-5-7-3-2-1-respectively-then-what-are-the-direction-ratios-of-the-line-pq-69920
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">IfP,Q are $(2,5,-7),(-3,2,1)$ respectively, then what are the direction ratios of the line $\mathrm{PQ}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$   &lt; 10,6,-16&gt;$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$   &lt; 5,3,8&gt;$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$   &lt; -5,-3,-8&gt;$</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">Since, coordinates of points $\mathrm{P}$ and $\mathrm{Q}$ are $(2,5,-7)$ and $(-3,2,1)$ respectively. Direction ratios of $\mathrm{PQ}$ are $   &lt; -3-2,2-5,1+7&gt;$ i.e., $ &lt; -5,-3,8&gt;$</div>


title: ifthe-latus-rectum-of-an-ellipse-is-equal-to-half-its-minor-axis-then-what-is-its-eccentricity
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
ifthe-latus-rectum-of-an-ellipse-is-equal-to-half-its-minor-axis-then-what-is-its-eccentricity-57205

question:
<div class="question">Ifthe latus rectum of an ellipse is equal to half its minor axis, then what is its eccentricity?</div>

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

answer:
<div class="solution">Let the equation of ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$<br/>Length of minor axis $=2 \mathrm{~b}$ and length of latus rectum $=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}$<br/>According to the question,<br/>$$<br/>\frac{2 b^{2}}{a}=b \Rightarrow 2 b=a \Rightarrow 4 b^{2}=a^{2}<br/>$$<br/>Now, eccentricity of ellipse $e=\frac{\sqrt{a^{2}-b^{2}}}{a}$<br/>$$<br/>e=\frac{\sqrt{4 b^{2}-b^{2}}}{2 b}=\frac{\sqrt{3} b}{2 b}=\frac{\sqrt{3}}{2}<br/>$$<br/>$\Rightarrow \quad e=\frac{\sqrt{3}}{2}$</div>


title: ifthe-matrix-a-is-such-that-a-2-i-then-which-one-of-the-following-is-correct
slug: ifthe-matrix-a-is-such-that-a-2-i-then-which-one-of-the-following-is-correct-12905
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Ifthe matrix<br/>$A=\left[\begin{array}{ll}<br/>\alpha &amp; \beta \\<br/>\beta &amp; \alpha<br/>\end{array}\right]$<br/>is such that $A^{2}=I$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\alpha=0, \beta=1$ or $\alpha=1, \beta=0$</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">$\alpha=0, \beta \neq 1$ or $\alpha \neq 1, \beta=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha=1, \beta \neq 0$ or $\alpha \neq 1, \beta=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\alpha \neq 0, \beta \neq 0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha=0, \beta=1$ or $\alpha=1, \beta=0$</span> </div>

answer:
<div class="solution">$\begin{array}{ll}\text { Let } &amp; A=\left[\begin{array}{ll}\alpha &amp; \beta \\ \beta &amp; \alpha\end{array}\right] \\ \therefore  &amp; \mathrm{A}^{2}=\left[\begin{array}{ll}\alpha &amp; \beta \\ \beta &amp; \alpha\end{array}\right]\left[\begin{array}{ll}\alpha &amp; \beta \\ \beta &amp; \alpha\end{array}\right] \\ &amp; =\left[\begin{array}{rc}\alpha^{2}+\beta^{2} &amp; 2 \alpha \beta \\ 2 \alpha \beta &amp; \alpha^{2}+\beta^{2}\end{array}\right]\end{array}$<br/>Now, $\mathrm{A}^{2}=\mathrm{I}$<br/>$\Rightarrow \quad\left[\begin{array}{cc}\alpha^{2}+\beta^{2} &amp; 2 \alpha \beta \\ 2 \alpha \beta &amp; \alpha^{2}+\beta^{2}\end{array}\right]=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$\Rightarrow \quad \alpha^{2}+\beta^{2}=1, \quad \alpha \beta=0$<br/>$\Rightarrow \quad \alpha=0, \beta=1$<br/>or $\quad \beta=0, \alpha=1$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
ifthe-roots-of-a-quadratic-equation-a-x-2-b-x-c-0-are-and-then-the-quadratic-equation-having-roots-2-and-2-is

slug:
ifthe-roots-of-a-quadratic-equation-a-x-2-b-x-c-0-are-and-then-the-quadratic-equation-having-roots-2-and-2-is-34385

question:
<div class="question">Ifthe roots of a quadratic equation $a x^{2}+b x+c=0$ are $\alpha$ and $\beta$, then the quadratic equation having roots $\alpha^{2}$ and $\beta^{2}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a^{2} x^{2}-\left(b^{2}-2 a c\right) x+c^{2}=0$</span> </div>

answer:
<div class="solution">Quadratic equation is given by $\mathrm{x}^{2}-($ sum of roots $) \mathrm{x}+($ product of roots $)=0$ Required equation is $\mathrm{x}^{2}-\mathrm{x}\left(\alpha^{2}+\beta^{2}\right)+(\alpha \beta)^{2}=0$<br/>$\Rightarrow \mathrm{x}^{2}-\mathrm{x}\left[(\alpha+\beta)^{2}-2 \alpha \beta\right]+(\alpha \beta)^{2}=0$<br/>$\Rightarrow \mathrm{x}^{2}-\mathrm{x}\left(\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}-\frac{2 \mathrm{c}}{\mathrm{a}}\right)+\frac{\mathrm{c}^{2}}{\mathrm{a}^{2}}=0$<br/>$\Rightarrow \mathrm{a}^{2} \mathrm{x}^{2}-\mathrm{x}\left(\mathrm{b}^{2}-2 \mathrm{ac}\right)+\mathrm{c}^{2}=$</div>


title: ifthe-roots-of-a-quadratic-equation-are-m-n-and-m-n-then-the-quadratic-equation-will-be
slug: ifthe-roots-of-a-quadratic-equation-are-m-n-and-m-n-then-the-quadratic-equation-will-be-84454
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Ifthe roots of a quadratic equation are $\mathrm{m}+\mathrm{n}$ and $\mathrm{m}-\mathrm{n}$, then the quadratic equation will be</div>

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

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

answer:
<div class="solution">Let $\alpha=\mathrm{m}+\mathrm{n}, \beta=\mathrm{m}-$<br/>Now, $\alpha+\beta=\mathrm{m}+\mathrm{n}+\mathrm{m}-\mathrm{n}=2 \mathrm{~m}$<br/>$\alpha \times \beta=(m+n)(m-n)=m^{2}-n^{2}$<br/>Now, Required Quadratic Equation will be, $\mathrm{x}^{2}-(\alpha+\beta) \mathrm{x}+\alpha \beta=0$<br/>$x^{2}-2 m x+m^{2}-n^{2}=0$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
ifthe-roots-of-the-equation-3-ax-2-2-bx-c-0-are-in-the-ratio-2-3-then-which-one-of-the-following-is-correct

slug:
ifthe-roots-of-the-equation-3-ax-2-2-bx-c-0-are-in-the-ratio-2-3-then-which-one-of-the-following-is-correct-52831

question:
<div class="question">Ifthe roots of the equation $3 \mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0$ are in the ratio $2: 3$, 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">$8 a c=25 b$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$8 a c=9 b^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 b^{2}=9 a c$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$8 b^{2}=25 a c$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 b^{2}=25 a c$</span> </div>

answer:
<div class="solution">Let roots of equation be $2 \alpha$ and $3 \alpha$<br/>$2 \alpha+3 \alpha=-\frac{2 b}{3 a}$<br/>$\Rightarrow \alpha=\frac{-2 b}{15 a}$...(i)<br/>$2 \alpha \cdot 3 \alpha=\frac{\mathrm{c}}{3 \mathrm{a}} \Rightarrow \alpha^{2}=\frac{\mathrm{c}}{18 \mathrm{a}}$...(ii)<br/>Now, put value of $\alpha$ in equation (ii) $\left(\frac{-2 b}{15 a}\right)^{2}=\frac{c}{18 a} \Rightarrow \frac{4 b^{2}}{225 a^{2}}=\frac{c}{18 a}$<br/>$\Rightarrow 8 b^{2}=25 a c$</div>


tags: ['Mathematics', 'Quadratic Equation', '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
title:
ifthe-roots-of-the-equation-x-2-b-x-c-0-are-two-consecutiv-integers-then-what-is-the-value-of-b-2-4-c

slug:
ifthe-roots-of-the-equation-x-2-b-x-c-0-are-two-consecutiv-integers-then-what-is-the-value-of-b-2-4-c-78266

question:
<div class="question">Ifthe roots of the equation $x^{2}-b x+c=0$ are two consecutiv integers, then what is the value of $b^{2}-4 c$ ?</div>

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

answer:
<div class="solution">Let $\alpha$ and $(\alpha+1)$ be the roots of equation $x^{2}-b x+c=0$<br/>$\therefore \alpha+(\alpha+1)=b$<br/>$\Rightarrow 2 \alpha+1=b$<br/>and $\alpha(\alpha+1)=c$<br/>Now, $b^{2}-4 c=(2 \alpha+1)^{2}-4[\alpha(\alpha+1)]$<br/>$=4 \alpha^{2}+1+4 \alpha-4 \alpha^{2}-4 \alpha=1$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
ifthe-roots-of-the-quadratic-equation-3-x-2-5-x-p-0-are-real-and-unequal-then-which-one-of-the-following-is-correct

slug:
ifthe-roots-of-the-quadratic-equation-3-x-2-5-x-p-0-are-real-and-unequal-then-which-one-of-the-following-is-correct-49682

question:
<div class="question">Ifthe roots of the quadratic equation $3 \mathrm{x}^{2}-5 \mathrm{x}+\mathrm{p}=0$ are real and unequal, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{p}   &lt; 25 / 12$</span> </div>

answer:
<div class="solution">The given equation is. $3 x^{2}-5 x+p=0$<br/>We have, $\mathrm{a}=3, \mathrm{~b}=-5, \mathrm{c}=$<br/>$\mathrm{D}=\mathrm{b}^{2}-4 \mathrm{ac}=25-12 \mathrm{p}$<br/>For Real and unequal, $\mathrm{D}&gt;0$ $\therefore 25-12 \mathrm{p}&gt;0$<br/>$\Rightarrow \quad 25&gt;12 \mathrm{p} \Rightarrow \mathrm{p}   &lt; \frac{25}{12}$</div>


title: ifx-and-y-are-real-numberssuch-that-x-y-and-x-y-then-which-one-of-the-following-is-correct
slug: ifx-and-y-are-real-numberssuch-that-x-y-and-x-y-then-which-one-of-the-following-is-correct-34070
tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">Ifx and y are real numberssuch that $x&gt;y$ and $|x|&gt;|y|$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x&gt;0$</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">$\mathrm{y}&gt;0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y   &lt; 0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{x}   &lt; 0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x&gt;0$</span> </div>

answer:
<div class="solution">As given: $\mathrm{x}$ and $\mathrm{y}$ are real numbers such that $\mathrm{x}&gt;\mathrm{y}$ and $|\mathrm{x}|&gt;|\mathrm{y}|$ then $\mathrm{x}$ can not be negative of zero $\Rightarrow \mathrm{x}&gt;0 .$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers']
options: None
correct_option: None
source_db: MarksBatch2_P1.db
title:
illustrate-with-examples-the-limitations-of-willamson-synthesis-for-the-preparation-of-certain-types-of-ethers

slug:
illustrate-with-examples-the-limitations-of-willamson-synthesis-for-the-preparation-of-certain-types-of-ethers-18202

question:
<div class="question">Illustrate with examples the limitations of Willamson synthesis for the preparation of certain types of ethers.</div>

answer:
<div class="solution">Williamson's synthesis is a versatile method for the synthesis of both symmetrical and unsymmetrical ethers. However, for the synthesis of unsymmetrical ethers, a proper choice of reactants is necessary. Since Williamson's synthesis occurs by $\mathrm{S}_{\mathrm{N}} 2$ mechanism and primary alkyl halides are most reactive in $\mathrm{S}_{\mathrm{N}} 2$ reaction, therefore, best yields of unsymmetrical ethers are obtained when the alkyl halides are primary and the alkoxide may be primary, secondary or tertiary. For example, tertbutylethyl ether is prepared by treating ethyl bromide with sodium tert-butoxide.<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/ytMIOiMvB5y8Q1c-iEOzWxw_gnSh5kzJUmk8k3RxdBw.original.fullsize.png"/><br/><br/>The above ether cannot be prepared by treating sodium ethoxide with tert-butyl chloride or bromide since under these condition an alkene, i.e., isobutylene is the main product.<img src="https://cdn-question-pool.getmarks.app/modules/cbse/ryWr3j6U95Ux83xUlhgS1lDTUrf5O5-gQ5PTvdvJvSo.original.fullsize.png"/><br/><br/>Aryl and vinyl halides cannot be used as substrates because they are less reactive in nucleophilic substitution.</div>


title: in-3-d-series-the-metal-having-the-highest-m-2-m-standard-electrode-potential-is
slug: in-3-d-series-the-metal-having-the-highest-m-2-m-standard-electrode-potential-is-36912
tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2022 (27 Jun Shift 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi mathvariant="normal">d</mi></math> series, the metal having the highest <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">M</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">M</mi></math> standard electrode potential is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Cr</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"><mi>Fe</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"><mi>Cu</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"><mi>Zn</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"><mi>Cu</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Cr</mi><mrow><mo>+</mo><mn>2</mn></mrow></msup><mo>/</mo><mi>Cr</mi><mo>→</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>90</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Fe</mi><mrow><mo>+</mo><mn>2</mn></mrow></msup><mo>/</mo><mi>Fe</mi><mo>→</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>44</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Cu</mi><mrow><mo>+</mo><mn>2</mn></mrow></msup><mo>/</mo><mi>Cu</mi><mo>→</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>34</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Zn</mi><mrow><mo>+</mo><mn>2</mn></mrow></msup><mo>/</mo><mi>Zn</mi><mo>→</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>76</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p>So Ans. <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Cu</mi><mrow><mo>+</mo><mn>2</mn></mrow></msup><mo>/</mo><mi>Cu</mi></math></p></div>


