# questions

title: what-is-the-area-of-the-triangle-whose-vertices-are-0-0-0-1-2-3-and-3-2-1
slug: what-is-the-area-of-the-triangle-whose-vertices-are-0-0-0-1-2-3-and-3-2-1-91734
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle whose vertices are $(0,0,0)$, $(1,2,3)$ and $(-3,-2,1) ?$</div>

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

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

answer:
<div class="solution">Let $\mathrm{A}(0,0,0), \mathrm{B}(1,2,3)$ and $\mathrm{C}(-3,-2,1)$ be the vertices of a triangle.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/bkC-QG5kT5ziPP_VfYX6zdVnH3jirATTAYc7QvR0wfQ.original.fullsize.png"/><br/><br/>Area of $\triangle \mathrm{ABC}=\frac{1}{2}\left\|\begin{array}{ccc}\mathrm{i} &amp; \mathrm{j} &amp; \mathrm{k} \\ 1 &amp; 2 &amp; 3 \\ -3 &amp; -2 &amp; 1\end{array}\right\|$<br/>$=\frac{1}{2}[\hat{i}(2+6)-\hat{j}(1+9)+\hat{k}(-2+6)]$<br/>$=\frac{1}{2}|8 \hat{\mathrm{i}}-10 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}|=\frac{1}{2} \sqrt{64+16+100}$<br/>$=\frac{1}{2}(6 \sqrt{5})=3 \sqrt{5}$</div>


title: what-is-the-area-of-the-triangle-whose-vertices-are-3-0-0-4-and-3-4
slug: what-is-the-area-of-the-triangle-whose-vertices-are-3-0-0-4-and-3-4-33674
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle whose vertices are $(3,0),(0,4)$ and $(3,4) ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">6 sq. unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$7.5 \mathrm{sq}$. unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9 sq. unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">12 sq. unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">6 sq. unit</span> </div>

answer:
<div class="solution">Required Area<br/>$=\frac{1}{2}\left|\begin{array}{lll}3 &amp; 0 &amp; 1 \\ 0 &amp; 4 &amp; 1 \\ 3 &amp; 4 &amp; 1\end{array}\right|=\frac{1}{2}[3(4-4)+1(0-12)]=6$</div>


title: what-is-the-area-of-the-triangle-with-vertices-0-2-2-2-0-1-and-3-4-0
slug: what-is-the-area-of-the-triangle-with-vertices-0-2-2-2-0-1-and-3-4-0-50513
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle with vertices $(0,2,2),(2,0,-1)$ and $(3,4,0)$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{15}{2} \mathrm{sq}$ unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">15 sq unit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{2}$ sq unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7 sq unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{15}{2} \mathrm{sq}$ unit</span> </div>

answer:
<div class="solution">Let $A=(0,2,2), \mathrm{B}=(2,0,-1)$ and $C=(3,4,0)$ $\overrightarrow{A B}=(2-0,0-2,-1-2)$ and $\overrightarrow{A C}=(3-0,4-2,0-2)$<br/>$\Rightarrow \overrightarrow{A B}=(2,-2,-3)$ and $\overrightarrow{A C}=(3,2,-2)$<br/>$\therefore$ Area of triangle $=\frac{1}{2} \times$ magnitude of $\overrightarrow{A B} \times \overrightarrow{A C}$<br/>$$<br/>\begin{array}{l}<br/>=\frac{1}{2}\|\overrightarrow{A B} \times \overrightarrow{A C}\| \\<br/>=\frac{1}{2}\|\hat{\mathbf{i}} \quad \hat{\mathbf{j}} \quad \hat{\mathbf{k}}\| \\<br/>=\frac{1}{2} \mid[\hat{\mathbf{i}}(4+6)-\hat{\mathbf{j}}(-4+9)+\hat{\mathbf{k}}(4+6)] \\<br/>=\frac{1}{2}|10 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+10 \hat{\mathbf{k}}| \\<br/>\begin{array}{ccc}<br/>-2 &amp; -3 \\<br/>3 &amp; 2 &amp; -2<br/>\end{array} \| \\<br/>=\frac{1}{2} \sqrt{(10)^{2}+(5)^{2}+(10)^{2}}=\frac{1}{2} \sqrt{225}=\frac{15}{2}<br/>\end{array}<br/>$$</div>


title: what-is-the-area-of-the-triangle-with-vertices-1-2-3-2-5-1-and-1-1-2
slug: what-is-the-area-of-the-triangle-with-vertices-1-2-3-2-5-1-and-1-1-2-76215
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle with vertices $(1,2,3),(2,5,-1)$ and $(-1,1,2)$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{155}}{2}$ square units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{175}}{2}$ square units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{155}}{4}$ square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{175}}{4}$ square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sqrt{155}}{2}$ square units</span> </div>

answer:
<div class="solution">Let $\overrightarrow{O A}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{O B}=2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and<br/>$\overrightarrow{O C}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ be three position vectors.<br/>Now, $\overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$<br/>and $\overrightarrow{A C}=\overrightarrow{O C}-\overrightarrow{O A}=-2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$<br/>$\therefore$ Area of $\Delta \mathrm{ABC}=\frac{1}{2}|\overrightarrow{A B} \times \overrightarrow{A C}|$<br/>Now, $\overrightarrow{A B} \times \overrightarrow{A C}=\left|\begin{array}{ccc}\hat{\mathrm{i}} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ 1 &amp; 3 &amp; -4 \\ -2 &amp; -1 &amp; -1\end{array}\right|$<br/>$=\hat{\mathrm{i}}(-3-4)-\hat{\mathrm{j}}(-1-8)+\hat{\mathrm{k}}(-1+6)$<br/>$=-7 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$<br/>Now, $|\overrightarrow{A B} \times \overrightarrow{A C}|=\sqrt{155}$<br/>Required Area $=\frac{\sqrt{155}}{2}$</div>


title: what-is-the-area-of-the-triangle-with-vertices-x-1-x-1-1-x-2-x-2-1-x-3-x-3-1
slug: what-is-the-area-of-the-triangle-with-vertices-x-1-x-1-1-x-2-x-2-1-x-3-x-3-1-46914
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area of the triangle with vertices <br/>$\left(x_{1}, \frac{1}{x_{1}}\right),\left(x_{2}, \frac{1}{x_{2}}\right),\left(x_{3}, \frac{1}{x_{3}}\right) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left|\frac{\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{3}-x_{1}\right)}{2 x_{1} x_{2} x_{3}}\right|$</span> </div>

answer:
<div class="solution">Area $=\frac{1}{2}\left|x_{1}\left(\frac{1}{x_{2}}-\frac{1}{x_{3}}\right)+x_{2}\left(\frac{1}{x_{3}}-\frac{1}{x_{1}}\right)+x_{3}\left(\frac{1}{x_{1}}-\frac{1}{x_{2}}\right)\right|$<br/>$=\frac{1}{2}\left|\frac{x_{1}\left(x_{3}-x_{2}\right)}{x_{2} x_{3}}+\frac{x_{2}\left(x_{1}-x_{3}\right)}{x_{1} x_{3}}+\frac{x_{3}\left(x_{2}-x_{1}\right)}{x_{1} x_{2}}\right|$<br/>$=\frac{1}{2}\left|\frac{-x_{1}^{2}\left(x_{2}-x_{3}\right)-x_{2}^{2}\left(x_{3}-x_{1}\right)-x_{3}^{2}\left(x_{1}-x_{2}\right)}{x_{1} x_{2} x_{3}}\right|$<br/>$=\left|\frac{\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{3}-x_{1}\right)}{2\left(x_{1} x_{2} x_{3}\right)}\right|$</div>


title: what-is-the-area-under-the-curve-f-x-x-e-x-above-the-x-axis-and-between-the-lines-x-0-and-x-1
slug: what-is-the-area-under-the-curve-f-x-x-e-x-above-the-x-axis-and-between-the-lines-x-0-and-x-1-98877
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area under the curve $f(x)=x e^{x}$ above the $x$ -axis and between the lines $x=0$ and $x=1 ? \quad$</div>

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

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

answer:
<div class="solution">Given curve is $f(x)=x e^{x}, x=0$ and $x=1$.<br/>So, Required Area $=\int_{0}^{1} f(x) d x=\int_{0}^{1} x e^{x} d x$<br/>Let $x$ be the first function and $e^{x}$ be the second function then by parts<br/>$=\left[x e^{x}-\int e^{x} d x\right]_{0}^{1}=\left[x e^{x}-e^{x}\right]_{0}^{1}$<br/>$=(e-e)-(0-1)=1$ sq unit</div>


title: what-is-the-area-under-the-curve-y-x-x-1-between-x-0-and-x-1
slug: what-is-the-area-under-the-curve-y-x-x-1-between-x-0-and-x-1-10235
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the area under the curve $\mathrm{y}=|\mathrm{x}|+|\mathrm{x}-1|$ between $\mathrm{x}=0$ and $\mathrm{x}=1 ?$</div>

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

answer:
<div class="solution">$|\mathrm{x}|$ for $\mathrm{x} \geq 0$<br/>$=\mathrm{x}$ and $|\mathrm{x}-1|$ for $\mathrm{x} \leq 1$<br/>$=-(\mathrm{x}-1)$<br/>so, $\int_{0}^{1}(|\mathrm{x}|+|\mathrm{x}-| 1 \mid)=$ required area<br/>$\mathrm{a}=\int_{0}^{1} \mathrm{x} \mathrm{dx}-\int_{0}^{1}(\mathrm{x}-1) \mathrm{d} \mathrm{x}$<br/>$=\left[\frac{\mathrm{x}^{2}}{2}\right]_{0}^{1}-\left[\frac{\mathrm{x}^{2}}{2}-\mathrm{x}\right]_{0}^{1}=\frac{1}{2}-\left(\frac{1}{2}-1\right)=1 \mathrm{sq}$ units</div>


title: what-is-the-argument-of-1-sin-i-cos-i-1
slug: what-is-the-argument-of-1-sin-i-cos-i-1-62651
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the argument of $(1-\sin \theta)+\mathrm{i} \cos \theta$ ?<br/>$(i=\sqrt{-1})$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{2}-\frac{\theta}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}+\frac{\theta}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}-\frac{\theta}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{4}+\frac{\theta}{2}$</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">$\frac{\pi}{4}+\frac{\theta}{2}$</span> </div>

answer:
<div class="solution">Given complex number is $(1-\sin \theta)+i \cos \theta \equiv a+i b$<br/>Argument $\equiv \tan \theta=\frac{b}{a}$<br/>$\Rightarrow \tan \theta=\frac{\cos \theta}{1-\sin \theta}$<br/>$=\frac{\cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2}}{\sin ^{2} \frac{\theta}{2}+\cos ^{2} \frac{\theta}{2}-2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}$<br/>$=\frac{\left(\cos \frac{\theta}{2}-\sin \frac{\theta}{2}\right)\left(\cos \frac{\theta}{2}+\sin \frac{\theta}{2}\right)}{\left(\sin \frac{\theta}{2}-\cos \frac{\theta}{2}\right)^{2}}$<br/>$=\frac{\cos \frac{\theta}{2}+\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}-\sin \frac{\theta}{2}}=\frac{1+\tan \frac{\theta}{2}}{1-\tan \frac{\theta}{2}}=\frac{\tan \frac{\pi}{4}+\tan \frac{\theta}{2}}{1-\tan \frac{\pi}{4} \tan \frac{\theta}{2}}$<br/>$\tan \theta=\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)$<br/>Hence, argument $=\frac{\pi}{4}+\frac{\theta}{2}$</div>


title: what-is-the-argument-of-the-complex-number-1-i-where-i-1
slug: what-is-the-argument-of-the-complex-number-1-i-where-i-1-83450
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the argument of the complex number $(-1-\mathrm{i})$ where $\mathrm{i}=\sqrt{-1}$ ?</div>

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

answer:
<div class="solution">Let $z=-1-i$ Since Real part of $z   &lt; 0$ and Imaginary part of $z &lt; 0$ therefore argument lies in IIIrd quadrant<br/>$\operatorname{rg}(\mathrm{z})=\pi+\tan ^{-1}\left|\frac{-1}{-1}\right|=\pi+\frac{\pi}{4}=\frac{5 \pi}{4}$</div>


title: what-is-the-argument-of-the-complex-number-3-i-1-i-2-i-where-i-1
slug: what-is-the-argument-of-the-complex-number-3-i-1-i-2-i-where-i-1-95659
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the argument of the complex number $\frac{(1+i)(2+i)}{3-i}$ where $i=\sqrt{-1} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{\pi}{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2}$</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> </ul>

answer:
<div class="solution">$\frac{(1+\mathrm{i})(2+\mathrm{i})}{3-\mathrm{i}}=\frac{1+3 \mathrm{i}}{3-\mathrm{i}}$<br/>$=\frac{1+3 \mathrm{i}}{3-\mathrm{i}} \times \frac{3+\mathrm{i}}{3+\mathrm{i}}=\frac{10 \mathrm{i}}{10}=\mathrm{i}$ or $0+\mathrm{i}$<br/>argument, $\theta=\tan ^{-1}\left(\frac{1}{0}\right)=\tan ^{-1}\left(\tan \frac{\pi}{2}\right)=\frac{\pi}{2}$</div>


title: what-is-the-arithmetic-mean-of-first-16-natural-numbers-with-weights-being-the-number-itself
slug: what-is-the-arithmetic-mean-of-first-16-natural-numbers-with-weights-being-the-number-itself-26009
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the arithmetic mean of first 16 natural numbers with weights being the number itself?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{17}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{33}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">11</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{187}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{17}{2}$</span> </div>

answer:
<div class="solution">Given natural numbers are $1,2,3,4,5, \ldots \ldots \ldots, 16$<br/>This is an A.P. with first term $=1$ and common difference $=1, n=16$<br/>$\therefore$<br/>By using sum of 16 natural numbers<br/>i.c., $\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}]$<br/>We have<br/>$\mathrm{S}_{16}=\frac{16}{2}[2(1)+15(1)]=8(17)=136$<br/>$\therefore \mathrm{AM}=\frac{136}{16}=\frac{17}{2}$</div>


title: what-is-the-arithmetic-mean-of-the-series-n-c-0-n-c-1-n-c-n
slug: what-is-the-arithmetic-mean-of-the-series-n-c-0-n-c-1-n-c-n-70618
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the arithmetic mean of the series ${ }^{\mathrm{n}} \mathrm{C}_{0},{ }^{\mathrm{n}} \mathrm{C}_{1}, \ldots \ldots{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}, ?$</div>

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

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

answer:
<div class="solution">Since, expansion contains $(\mathrm{n}+1)$ terms,<br/>Required mean $=\frac{{ }^{\mathrm{n}} \mathrm{C}_{0}+{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}+\ldots \ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}}{(\mathrm{n}+1)}$<br/>$=\frac{2^{\mathrm{n}}}{\mathrm{n}+1}$</div>


title: what-is-the-arithmetic-mean-of-the-series-n-c-1-n-c-2-n-c-3-n-c-n
slug: what-is-the-arithmetic-mean-of-the-series-n-c-1-n-c-2-n-c-3-n-c-n-15286
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the arithmetic mean of the series ${ }^{n} C_{1},{ }^{n} C_{2},{ }^{n} C_{3} \ldots{ }^{n} C_{n}$ ?</div>

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

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

answer:
<div class="solution">Arithmetic mean of the series<br/>$=\frac{{ }^{n} C_{1}+{ }^{n} C_{2}+{ }^{n} C_{3}+\ldots+{ }^{n} C_{n}}{n}=\frac{2^{n}-1}{n}$</div>


title: what-is-the-atomic-mass-of-fe-given-abundance-of-54-fe-10-56-fe-85-57-fe-5
slug: what-is-the-atomic-mass-of-fe-given-abundance-of-54-fe-10-56-fe-85-57-fe-5-67057
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'AP EAMCET', 'AP EAMCET 2022 (06 Jul Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">55.85</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the atomic mass of $\mathrm{Fe}$ ?<br/>Given abundance of ${ }^{54} \mathrm{Fe}=10 \%$<br/>$$<br/>\begin{aligned}<br/>&amp; { }^{56} \mathrm{Fe}=85 \% \\<br/>&amp; { }^{57} \mathrm{Fe}=5 \%<br/>\end{aligned}<br/>$$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">55.65</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">55.75</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">55.85</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">55.95</span> </li> </ul>

answer:
<div class="solution">Average atomic mass of $\mathrm{Fe}=54 \times 0.1+56 \times 0.85+$ $57 \times 0.05=55.85$</div>


title: what-is-the-binary-equivalent-of-decimal-number-08125-10
slug: what-is-the-binary-equivalent-of-decimal-number-08125-10-40708
question: <div class="question">What is the binary equivalent of decimal number $(0.8125)_{10}$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(0.1101)_{2}$</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">$(0.1001)_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(0.1111)_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(0.1011)_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(0.1101)_{2}$</span> </div>

answer:
<div class="solution">Work with option $(0.1101)_{2}=1 \times 2^{-1}+1 \times 2^{-2}+1 \times 2^{-4}$<br/>$=\frac{1}{2}+\frac{1}{4}+\frac{1}{16}=\frac{13}{16}=(0.8125)_{10}$<br/>Hence option (a) gives the binary equivalent of given decimal number.</div>


title: what-is-the-binary-equivalent-of-the-decimal-number-03125
slug: what-is-the-binary-equivalent-of-the-decimal-number-03125-24175
question: <div class="question">What is the binary equivalent of the decimal number $0.3125$ ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.0111$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.1010$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.0101$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.1101$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.0101$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/6qsxkw6Z419qojNqc8VTqaQIsVCTlYDCdxDBmU8TSEM.original.fullsize.png"/><br/><br/>$(0.3125)_{10}=(0.0101)_{2}$</div>


title: what-is-the-binary-number-equivalent-of-the-decimal-number-3225
slug: what-is-the-binary-number-equivalent-of-the-decimal-number-3225-32876
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the binary number equivalent of the decimal number<br/>$32.25$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100010.10$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">100000,10</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$100010.01$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$100000.0$</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">$100000.0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/AeEHwWTjSRu9JlI8_gGnUkgCNmrRgqx-gSZRGWU5lCE.original.fullsize.png"/><br/><br/>$\therefore$ Required binary number equivalent to $32.25$ is $100000.01$</div>


title: what-is-the-boiling-temperature-called-at-1-bar-pressure
slug: what-is-the-boiling-temperature-called-at-1-bar-pressure-52962
question: <div class="question">What is the boiling temperature called at 1 bar pressure?</div>
tags: ['Chemistry', 'States of Matter', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Standard boiling point</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Evaporation temperature</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Normal boiling point</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Superheated point</span> </li> </ul>

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

answer:
<div class="solution">If the pressure is 1 bar, the boiling point is called the standard boiling point of the liquid.</div>


title: what-is-the-centre-of-the-sphere-a-x-2-b-y-2-c-z-2-6-x-0-if-the-radius-is-1-unit
slug: what-is-the-centre-of-the-sphere-a-x-2-b-y-2-c-z-2-6-x-0-if-the-radius-is-1-unit-65375
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the centre of the sphere $a x^{2}+b y^{2}+c z^{2}-6 x=0$ if the radius is 1 unit?</div>

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

answer:
<div class="solution">In the given equation, there are three unknown parameters and no equation has been given to evaluate those, hence centre of sphere cannot be determined.</div>


title: what-is-the-characteristic-of-the-vascular-bundles-in-monocot-stem
slug: what-is-the-characteristic-of-the-vascular-bundles-in-monocot-stem-14359
question: <div class="question">What is the characteristic of the vascular bundles in monocot stem?</div>
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
answer: <div class="solution">The vascular bundles in monocot stem are conjoint and closed.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Open and radial</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Open and endarch</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Conjoin and closed</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Conjoin and endarch</span> </li> </ul>

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


title: what-is-the-coefficient-of-the-middle-term-in-the-binomial-expansion-of-2-3-x-4
slug: what-is-the-coefficient-of-the-middle-term-in-the-binomial-expansion-of-2-3-x-4-17276
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">216</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of the middle term in the binomial expansion of $(2+3 \mathrm{x})^{4} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">108</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">216</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Middle term in the expansion of $(x+y)^{n}$<br/>$=\left(\frac{n+1}{2}\right)^{\text {th }}$ term, if $n$ is odd<br/>$=\left(\frac{n}{2}+1\right)^{\text {th }}$ term, if $n$ is even<br/>Here $n=4$<br/>$\therefore$ Middle term is $\left(\frac{4}{2}+1\right)^{\text {th }}=3^{\mathrm{rd}}$ term $\begin{aligned} 4 c_{2} \times 2^{2} \times 3^{2} &amp;=6 \times 4 \times 9 \\ &amp;=216 \\ 62 . \quad \text { (c) } \alpha={ }^{2 n} \mathrm{C}_{n} \\ \beta=^{2 n-1} \mathrm{C}_{n} \\ \gamma=^{2 n-1} \mathrm{C}_{n-1} \end{aligned}$<br/>$\beta+\gamma={ }^{2 n-1} \mathrm{C}_{n}+{ }^{2 n-1} \mathrm{C}_{n-1}={ }^{2 n} \mathrm{C}_{n}=\alpha$</div>


title: what-is-the-coefficient-of-x-3-in-1-3-x-3-3-2-x
slug: what-is-the-coefficient-of-x-3-in-1-3-x-3-3-2-x-78090
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-918$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of $\mathrm{x}^{3}$ in $\frac{(3-2 \mathrm{x})}{(1+3 \mathrm{x})^{3}}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-272$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-540$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-870$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-918$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{(3-2 \mathrm{x})}{(1+3 \mathrm{x})^{3}}=(3-2 \mathrm{x})(1+3 \mathrm{x})^{-3}$<br/>$=(3-2 \mathrm{x})\left(1-9 \mathrm{x}+\frac{(-3)(-4)}{2 !} \cdot 9 \mathrm{x}^{2}\right.$<br/>$\left.\quad+\frac{(-3)(-4)(-5)}{3 !} \cdot 27 \mathrm{x}^{3}+\ldots .\right)$<br/>$\left[\operatorname{Expanding}(1+3 \mathrm{x})^{-3}\right]$<br/>$=(3-2 \mathrm{x})\left(1-9 \mathrm{x}+54 \mathrm{x}^{2}-270 \mathrm{x}^{3}+\ldots \ldots \ldots\right)$<br/>$\therefore$ Coefficient of $\mathrm{x}^{3}=-270 \times 3-2 \times 54$<br/>$=-810-108=-918$</div>


title: what-is-the-coefficient-of-x-3-y-4-in-2-x-3-y-2-5
slug: what-is-the-coefficient-of-x-3-y-4-in-2-x-3-y-2-5-28084
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">720</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of $x^{3} y^{4}$ in $\left(2 x+3 y^{2}\right)^{5}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">240</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">360</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">720</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1080</span> </li> </ul>

answer:
<div class="solution">$\mathrm{T}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}(2 \mathrm{x})^{\mathrm{r}-1}\left(3 \mathrm{y}^{2}\right)^{\mathrm{n}-\mathrm{r}+1}$<br/>$\mathrm{~T}_{4}=={ }^{5} \mathrm{C}_{3}(2 \mathrm{x})^{3}\left(3 \mathrm{y}^{2}\right)^{2}$<br/>$=\frac{5 !}{3 ! 2 !} 2^{3} \cdot \mathrm{x}^{3} \cdot 9 \mathrm{y}^{4}=\frac{5.4}{2.1} \times 8 \times 9 \times \mathrm{x}^{3} \mathrm{y}^{4}=720 \mathrm{x}^{3} \mathrm{y}^{4}$<br/>$\therefore$ Coefficient of $x^{3} y^{4}=720$</div>


title: what-is-the-coefficient-of-x-4-in-the-expansion-of-1-2-x-3-x-2-4-x-3-12
slug: what-is-the-coefficient-of-x-4-in-the-expansion-of-1-2-x-3-x-2-4-x-3-12-84327
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of $x^{4}$ in the expansion of $\left(1+2 x+3 x^{2}+4 x^{3}+\ldots .\right)^{1 / 2} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1/4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1/16</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1/128</span> </li> </ul>

answer:
<div class="solution">Consider $\left(1+2 \mathrm{x}+3 \mathrm{x}^{2}+4 \mathrm{x}^{3}+\ldots\right)^{1 / 2}=\left(1-\mathrm{x}^{-2}\right)^{1 / 2}$<br/>As we know that $(1-\mathrm{x})^{-2}=1+2 \mathrm{x}+3 \mathrm{x}^{2}+4 \mathrm{x}^{3}+\ldots$<br/>$\Rightarrow(1-x)^{-1}=1+x+x^{2}+x^{3}+x^{4}+\ldots$<br/>$\therefore$ Required coefficient of $x^{4}$ is 1 .</div>


title: what-is-the-coefficient-of-x-4-in-the-expansion-of-1-x-1-x-2
slug: what-is-the-coefficient-of-x-4-in-the-expansion-of-1-x-1-x-2-44664
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of $x^{4}$ in the expansion of $\left(\frac{1-x}{1+x}\right)^{2}$ ?</div>

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

answer:
<div class="solution">Consider $\left(\frac{1-x}{1+x}\right)^{2}=(1-x)^{2}(1+x)^{-2}$<br/>$=\left(1-2 x+x^{2}\right)(1+x)^{-2}$<br/>$=\left(1-2 x+x^{2}\right)\left(1-2 x+3 x^{2}-4 x^{3}+5 x^{4}-\ldots . .\right)$<br/>$\therefore \quad$ Coefficient of $x^{4}$ in $\left(\frac{1-x}{1+x}\right)^{2}=5+8+3=16$</div>


title: what-is-the-coefficient-of-x-5-in-the-expansion-1-2-x-3-x-2-4-x-3-5
slug: what-is-the-coefficient-of-x-5-in-the-expansion-1-2-x-3-x-2-4-x-3-5-14593
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of $\mathrm{x}^{5}$ in the expansion $\left(1-2 \mathrm{x}+3 \mathrm{x}^{2}-4 \mathrm{x}^{3}+\ldots \ldots . \infty\right)^{-5} ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(10 !) /(5 !)^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5^{-5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5^{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10 ! /\{6 !)(4 !)\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(10 !) /(5 !)^{2}$</span> </div>

answer:
<div class="solution">$1-2 \mathrm{x}+3 \mathrm{x}^{2}-4 \mathrm{x}^{3}+\ldots$<br/>$=(1+\mathrm{x})^{-2}, \mathrm{so},\left(1-2 \mathrm{x}+3 \mathrm{x}^{2}-4 \mathrm{x}^{3}+\ldots \infty\right)^{-5}$<br/>$=\left((1+\mathrm{x})^{-2}\right)^{-5}=(1+\mathrm{x})^{10} \Rightarrow \mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}}$<br/>Putting $\mathrm{r}=5$, coefficient of $\mathrm{x}^{5}={ }^{10} \mathrm{C}_{5}=\frac{10 !}{5 ! 5 !}=\frac{10 !}{(5 !)^{2}}$</div>


title: what-is-the-coefficient-of-x-17-in-the-expansion-of-3-x-6-x-3-9
slug: what-is-the-coefficient-of-x-17-in-the-expansion-of-3-x-6-x-3-9-88305
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the coefficient of $x^{17}$ in the expansion of $\left(3 x-\frac{x^{3}}{6}\right)^{9} ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{189}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{567}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{21}{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{189}{8}$</span> </div>

answer:
<div class="solution">Given expansion is<br/>$\left(3 x-\frac{x^{3}}{6}\right)^{9}$ where $a=3 x, \mathrm{~b}=\frac{-x^{3}}{6}, \mathrm{n}=9$<br/>Now, General Term $=\mathrm{T}_{r+1}={ }^{n} C_{r}(\mathrm{a})^{\mathrm{n}-\mathrm{r}} \cdot b^{2}$<br/>$={ }^{9} \mathrm{C}_{r}(3 x)^{9-\mathrm{r}}\left(\frac{-x^{3}}{6}\right)^{r}={ }^{9} \mathrm{C}_{r} \cdot 3^{9-r} \mathrm{x}^{9-r} \cdot \frac{(-1)^{r} x^{3 r}}{6^{r}}$<br/>$={ }^{9} C_{r} 3^{9-r}(-1)^{r} \frac{x^{9+2 r}}{6^{r}}$<br/>We can get coeff of $x^{17}$ when $9+2 r=17$<br/>$\Rightarrow \quad 2 r=17-9$<br/>$\Rightarrow \quad r=\frac{8}{2}=4$<br/>Hence, required coefficient<br/>$={ }^{9} \mathrm{C}_{4} \frac{3^{5}}{6^{4}}=\frac{126 \times 3}{16}=\frac{189}{8}$</div>


title: what-is-the-condition-that-one-root-of-the-equation-a-x-2-b-c-0-a-0-should-be-double-the-other
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-condition-that-one-root-of-the-equation-a-x-2-b-c-0-a-0-should-be-double-the-other-12391

question:
<div class="question">What is the condition that one root of the equation $a x^{2}+b$<br/>$+c=0, a \neq 0$ should be double the other?</div>

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

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

answer:
<div class="solution">Let the roots of the equation $a x^{2}+b x+c=0$ be $\alpha$ and $2 \alpha$<br/>Sum $=\alpha+2 \alpha=\frac{-b}{a}$ and product $=\alpha \cdot 2 \alpha=\frac{c}{a}$<br/>$\Rightarrow \alpha=\frac{-b}{3 a}$ and $\alpha^{2}=\frac{c}{2 a}$<br/>$\Rightarrow\left(\frac{-b}{3 a}\right)^{2}=\frac{c}{2 a} \Rightarrow \frac{b^{2}}{9 a^{2}}=\frac{c}{2 a}$<br/>$\Rightarrow 2 b^{2}=9 a c$</div>


title: what-is-the-condition-under-which-a-substance-always-remains-in-one-phase
slug: what-is-the-condition-under-which-a-substance-always-remains-in-one-phase-16320
tags: ['Chemistry', 'States of Matter', 'NEET']
answer: <div class="solution">At the critical temperature, a substance always remains in one phase.</div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the condition under which a substance always remains in one phase?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">At the critical temperature.</span> </div>


title: what-is-the-conjugate-of-2-i-1-2-i-2
slug: what-is-the-conjugate-of-2-i-1-2-i-2-87365
question: <div class="question">What is the conjugate of $\left(\frac{1+2 i}{2+i}\right)^{2}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{25}+i \frac{24}{25}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\frac{7}{25}-i \frac{24}{25}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{7}{25}+i \frac{24}{25}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{7}{25}-i \frac{24}{25}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{25}-i \frac{24}{25}$</span> </div>

answer:
<div class="solution">Consider $\left(\frac{1+2 i}{2+i}\right)^{2}=\left(\frac{(1+2 i)(2-i)}{(2+i)(2-i)}\right)^{2}$<br/>$=\left(\frac{2+4 i-i-2 i^{2}}{4-i^{2}}\right)^{2}=\left(\frac{4+3 i}{5}\right)^{2}$<br/>$=\frac{1}{25}\left(16+9 i^{2}+24 i\right)=\frac{7}{25}+\frac{24 i}{25}$<br/>$\therefore \quad$ Conjugate of $\left(\frac{1+2 i}{2+i}\right)^{2}=\frac{7}{25}-\frac{24}{25} i$<br/>$(\because$ conjugate of $a+i b=a-i b)$</div>


title: what-is-the-conventionally-chosen-standard-hydrogen-electrode-represented-as
slug: what-is-the-conventionally-chosen-standard-hydrogen-electrode-represented-as-89556
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the conventionally chosen standard hydrogen electrode represented as?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Pt(s) |H2(g) |H+(aq)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Pt(s) | H+(aq) |H2(g)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">H2(g) |Pt(s) |H+(aq)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">H+(aq) |H2(g) |Pt(s)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Pt(s) |H2(g) |H+(aq)</span> </div>

answer:
<div class="solution">The representation of the standard hydrogen electrode is Pt(s)| H2(g)| H+(aq).</div>


title: what-is-the-correct-sequence-of-sperm-formation
slug: what-is-the-correct-sequence-of-sperm-formation-13637
question: <div class="question">What is the correct sequence of sperm formation?</div>
tags: ['Biology', 'Human Reproduction', 'NEET', 'NEET 2013 (All India)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Spermatid, Spermatocyte, Spermatogonia, Spermatozoa</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Spermatogonia,<br/>Spermatocyte, Spermatozoa, Spermatid</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Spermatogonia, Spermatozoa, Spermatocyte, Spermatid</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Spermatogonia, Spermatocyte, Spermatid, Spermatozoa</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Spermatogonia, Spermatocyte, Spermatid, Spermatozoa</span> </div>

answer:
<div class="solution">Spermatogonia $\rightarrow$ Spermatocyte $\rightarrow$ Spermatid $\rightarrow$ Spermatozoa<br/>Spermatogonia are present on the inside wall of seminiferous tubule multiplied by mitotic division and increase in number. Spermatocytes are some of the spermatogonia, which periodically undergo meiosis. The secondary spermatocytes undergo the second meiatic division to produce four, equal haploid spermatids. The spermatids are transformed into spermatozoa (sperm).</div>


title: what-is-the-correct-sequence-of-the-following-values
slug: what-is-the-correct-sequence-of-the-following-values-27679
question: <div class="question">What is the correct sequence of the following values?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3&gt;2&gt;1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1&gt;2&gt;3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1&gt;3&gt;2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3&gt;1&gt;2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3&gt;2&gt;1$</span> </div>

answer:
<div class="solution">We work out the given statements.<br/>1. $\sin \frac{\pi}{12}=\sin 15^{\circ}=\frac{\sqrt{3}-1}{2 \sqrt{2}}$<br/>2. $\cos \frac{\pi}{12}=\cos 15^{\circ}=\frac{\sqrt{3}+1}{2 \sqrt{2}}$<br/>3. $\cot \frac{\pi}{12}=\cot 15^{\circ}=2+\sqrt{3}$<br/>So, correct sequence is $3&gt;2&gt;1$.</div>


title: what-is-the-cosine-of-angle-between-the-planes-x-y-z-l-0-and-2-x-2-y-2-z-1-0
slug: what-is-the-cosine-of-angle-between-the-planes-x-y-z-l-0-and-2-x-2-y-2-z-1-0-73850
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the cosine of angle between the planes $\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{l}=0$ and $2 \mathrm{x}-2 \mathrm{y}+2 \mathrm{z}+1=0 ?$</div>

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

answer:
<div class="solution">If $\theta$ be the angle between the planes $a_{1} x+b_{1} y+c_{1} z+d_{1}=0$ and $a_{2} x+b_{2} y+c_{2} z+d_{2}=0$<br/>then $\cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \cdot \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$<br/>$\begin{aligned} \cos \theta &amp;=\frac{1(2)+1(-2)+1(2)}{\sqrt{1^{2}+1^{2}+1^{2}}} \cdot \sqrt{2^{2}+(-2)^{2}+2^{2}} \\ &amp;=\frac{2}{\sqrt{3} 2 \sqrt{3}}=\frac{1}{3} \end{aligned}$</div>


title: what-is-the-cumulative-frequency-curve-of-statistical-data-commonly-called
slug: what-is-the-cumulative-frequency-curve-of-statistical-data-commonly-called-17901
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Ogive</span> </div>
answer: <div class="solution">The cumulative frequency curve of statistical data is called Ogive.</div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the cumulative frequency curve of statistical data commonly called?</div>

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


title: what-is-the-curve-which-passes-through-the-point-1-1-and-whose-slope-is-x-2-y
slug: what-is-the-curve-which-passes-through-the-point-1-1-and-whose-slope-is-x-2-y-87105
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the curve which passes through the point $(1,1)$ and<br/>whose slope is $\frac{2 y}{x} ?$</div>

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

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

answer:
<div class="solution">$\frac{d y}{d x}=\frac{2 y}{x}$<br/>$\Rightarrow \frac{d y}{y}=2 \frac{d x}{x}$<br/>On integration<br/>$\int \frac{d y}{y}=2 \int \frac{d x}{x}$<br/>$\Rightarrow \log y=2 \log x+\log a$<br/>$\Rightarrow \log y=\log x^{2}+\log a$<br/>$\Rightarrow \log y=\log \left(x^{2} \cdot a\right)$<br/>$\Rightarrow y=x^{2} a$<br/>at $(1,1) ; a=1$<br/>$\Rightarrow x^{2}=y=4\left(\frac{1}{4}\right) y$<br/>$\Rightarrow$ the curve is parabola.</div>


title: what-is-the-decimal-equivalent-of-101101
slug: what-is-the-decimal-equivalent-of-101101-12534
question: <div class="question">What is the decimal equivalent of $(101.101)$, ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(5.225)_{10}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(5.525)_{10}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(5.625)_{10}$</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">$(5.65)_{10}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(5.625)_{10}$</span> </div>

answer:
<div class="solution">Consider $(101.101)_{2}=1 \times 2^{2}+0 \times 2^{1}+1 \times 2^{0}+$<br/>$=4+1+\frac{1}{2}+\frac{1}{8}=\frac{40+4+1}{8}=\frac{45}{8}=(5.625)_{10}$</div>


title: what-is-the-decimal-number-representation-of-the-binary
slug: what-is-the-decimal-number-representation-of-the-binary-65117
question: <div class="question">What is the decimal number representation of the binary</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30.125$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$29.025$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$29.125$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$28.02^{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$29.125$</span> </div>

answer:
<div class="solution">$\begin{aligned} 11101.001 &amp;=1 \times 2^{4}+1 \times 2^{3}+1 \times 2^{2}+0 \times 2^{1}+1 \times 2^{0} \cdot 0 \times 2^{-1} \\+0 \times 2^{-2}+1 \times 2^{-3} \end{aligned}$<br/>$=(16+8+4+0+1) \cdot\left(0+0+\frac{1}{8}\right)=29.125$</div>


title: what-is-the-definition-of-the-term-primary-structure
slug: what-is-the-definition-of-the-term-primary-structure-55469
question: <div class="question">What is the definition of the term âPrimary Structureâ?</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Primary Structure refers to the first level of protein structure where amino acids are arranged in a specific sequence.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Primary Structure refers to the overall three-dimensional shape of a protein.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Primary Structure refers to the interactions between different protein subunits.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Primary Structure refers to the secondary structure elements, such as alpha helices and beta sheets, in a protein.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Primary Structure refers to the first level of protein structure where amino acids are arranged in a specific sequence.</span> </div>

answer:
<div class="solution">Correct Option is : (A)<br/>Primary Structure refers to the first level of protein structure where amino acids are arranged in a specific sequence.</div>


title: what-is-the-definition-of-twisted-aestivation
slug: what-is-the-definition-of-twisted-aestivation-31878
question: <div class="question">What is the definition of twisted aestivation?</div>
tags: ['Biology', 'Morphology of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">When the margins of sepals or petals overlap one another but not in any particular direction</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">When sepals or petals overlap one another in a particular direction</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">When sepals or petals just touch one another at the margin, without overlapping</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">When one margin of the appendage overlaps that of the next one and so on</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">When one margin of the appendage overlaps that of the next one and so on</span> </div>

answer:
<div class="solution">Twisted aestivation refers to a situation where one margin of the sepals or petals overlaps that of the next one, and so on.</div>


title: what-is-the-degree-of-the-differential-equation-1-d-x-d-y-4-d-x-2-d-2-y-2
slug: what-is-the-degree-of-the-differential-equation-1-d-x-d-y-4-d-x-2-d-2-y-2-44231
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation <br/>$\left(1+\frac{d y}{d x}\right)^{4}=\left(\frac{d^{2} y}{d x^{2}}\right)^{2} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">The given differential equation is<br/>$\left(1+\frac{d y}{d x}\right)^{4}=\left(\frac{d^{2} y}{d x^{2}}\right)^{2}$<br/>From above it is clear that degree of given differential equation is 2 .<br/>Because degree is the power of highest order derivative.</div>


title: what-is-the-degree-of-the-differential-equation-1-dx-dy-2-32-k-dx-2-d-2-y
slug: what-is-the-degree-of-the-differential-equation-1-dx-dy-2-32-k-dx-2-d-2-y-31497
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation<br/>$\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{3 / 2}=\mathrm{k} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}} ? \quad$</div>

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

answer:
<div class="solution">The given differential equation is<br/>$$<br/>\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{3 / 2}=\mathrm{k} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}<br/>$$<br/>To express it as a polynomial of derivatives we square both side,<br/>$$<br/>\Rightarrow\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{3}=\mathrm{k}^{2}\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}\right)^{2}<br/>$$<br/>Highest derivative has power $=2$ Degree of differential equation $=2$.</div>


title: what-is-the-degree-of-the-differential-equation-d-x-2-d-2-y-1-d-x-d-y-3-0
slug: what-is-the-degree-of-the-differential-equation-d-x-2-d-2-y-1-d-x-d-y-3-0-40508
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation<br/>$$<br/>\frac{d^{2} y}{d x^{2}}-\sqrt{1+\left(\frac{d y}{d x}\right)^{3}}=0 ?<br/>$$</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">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">Given differential equation is $\frac{d^{2} y}{d x^{2}}-\sqrt{1+\left(\frac{d y}{d x}\right)^{3}}=0$<br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=\sqrt{1+\left(\frac{d y}{d x}\right)^{3}}$<br/>On squaring both the sides,<br/>$\left(\frac{d^{2} y}{d x^{2}}\right)^{2}=1+\left(\frac{d y}{d x}\right)^{3}$<br/>Since, degree of the differential equation is the power of highest order derivative. Therefore from above it is clear that degree of equation is 2 .</div>


title: what-is-the-degree-of-the-differential-equation-d-x-3-d-3-y-23-4-3-d-x-2-d-2-y-5-d-x-d-y-0
slug: what-is-the-degree-of-the-differential-equation-d-x-3-d-3-y-23-4-3-d-x-2-d-2-y-5-d-x-d-y-0-72020
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation<br/>$\left(\frac{d^{3} y}{d x^{3}}\right)^{2 / 3}+4-3\left(\frac{d^{2} y}{d x^{2}}\right)+5\left(\frac{d y}{d x}\right)=0 ? \quad$</div>

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

answer:
<div class="solution">Degree of a differential equation is the power to which the highest derivative is raised when it is expressed as polynomial of derivatives. Given equation is<br/>$\left(\frac{d^{3} y}{d x^{3}}\right)^{2 / 3}-3\left(\frac{d^{2} y}{d x^{2}}\right)+5\left(\frac{d y}{d x}\right)+4=0$<br/>$\Rightarrow\left(\frac{d^{3} y}{d x^{3}}\right)^{2 / 3}=3 \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}-4$<br/>Cube on both side,<br/>$\left(\frac{d^{3} y}{d x^{3}}\right)^{2}=\left[3 \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}-4\right]^{3}$<br/>Hence, degree $=2$</div>


title: what-is-the-degree-of-the-differential-equation-d-x-3-d-3-y-32-d-x-2-d-2-y-2
slug: what-is-the-degree-of-the-differential-equation-d-x-3-d-3-y-32-d-x-2-d-2-y-2-46363
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation <br/>$\left(\frac{d^{3} y}{d x^{3}}\right)^{3 / 2}=\left(\frac{d^{2} y}{d x^{2}}\right)^{2} ?$</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">Consider the given differential equation,<br/>$\left(\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\right)^{\frac{3}{2}}=\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2}$<br/>In order to find degree, differential equation should be free from fractional indices.<br/>Now, squaring eqn (i) both sides, we get<br/>$\left(\frac{d^{3} y}{d x^{3}}\right)^{3}=\left(\frac{d^{2} y}{d x^{2}}\right)^{4}$<br/>Since, the power of highest order derivative is 3 , therefore degree $=3$.</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P3.db
title:
what-is-the-degree-of-the-differential-equation-d-x-4-d-4-y-35-5-d-x-3-d-3-y-6-d-x-2-d-2-y-8-d-x-d-y-5-0

slug:
what-is-the-degree-of-the-differential-equation-d-x-4-d-4-y-35-5-d-x-3-d-3-y-6-d-x-2-d-2-y-8-d-x-d-y-5-0-36866

question:
<div class="question">What is the degree of the differential equation<br/>$\left(\frac{d^{4} y}{d x^{4}}\right)^{3 / 5}-5 \frac{d^{3} y}{d x^{3}}+6 \frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}+5=0 ?$</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">4</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">2</span> </li> </ul>

answer:
<div class="solution">Consider differential equation<br/>$\left(\frac{d^{4} y}{d x^{4}}\right)^{\frac{3}{5}}-5 \frac{d^{3} y}{d x^{3}}+6 \frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}+5=0$<br/>$\Rightarrow\left(\frac{d^{4} y}{d x^{4}}\right)^{\frac{3}{5}}=5 \frac{d^{3} y}{d x^{3}}-6 \frac{d^{2} y}{d x^{2}}+8 \frac{d y}{d x}-5$<br/>$\Rightarrow\left(\frac{\mathrm{d}^{4} \mathrm{y}}{\mathrm{d} \mathrm{x}^{4}}\right)^{3}=\left(5 \frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{d} \mathrm{x}^{3}}-6 \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+8 \frac{\mathrm{dy}}{\mathrm{dx}}-5\right)^{5}$<br/>So, highest order derivative $=4$, degree $=3$</div>


title: what-is-the-degree-of-the-differential-equation-dx-3-d-3-y-2-dx-2-d-2-y-2-dx-dy-y-0
slug: what-is-the-degree-of-the-differential-equation-dx-3-d-3-y-2-dx-2-d-2-y-2-dx-dy-y-0-26868
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation $\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}+2\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2}-\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</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">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">Given differential equation<br/>$$<br/>\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}+2\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)-\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0<br/>$$<br/>Since exponent of highest order derivative is 1 therefore degree $=1$</div>


title: what-is-the-degree-of-the-differential-equation-k-dx-2-d-2-y-1-dx-dy-3-32-where-k-is-a-constant
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
slug:
what-is-the-degree-of-the-differential-equation-k-dx-2-d-2-y-1-dx-dy-3-32-where-k-is-a-constant-55244

question:
<div class="question">What is the degree of the differential equation<br/>$\mathrm{k} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{3}\right]^{3 / 2}$, where $\mathrm{k}$ is a constant?</div>

answer:
<div class="solution">In the given equation,<br/>$\mathrm{K} \cdot \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{3}\right]^{3 / 2}$<br/>Squaring both the sides,<br/>$\mathrm{K}^{2}\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{3}\right]^{3}$<br/>Degree of a differential equation is the highest power of the highest derivative in equation when derivatives are expressed as polynomial. Here degree of differential equation is $2 .$</div>


title: what-is-the-degree-of-the-differential-equation-y-x-dx-dy-dx-dy-1
slug: what-is-the-degree-of-the-differential-equation-y-x-dx-dy-dx-dy-1-80300
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the differential equation<br/>$\mathrm{y}=\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{-1} ?$</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">$-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Degree does not exist.</span> </li> </ul>

answer:
<div class="solution">Given differential equation is<br/>$$<br/>\mathrm{y}=\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{-1}<br/>$$<br/>Multiply by $\frac{\mathrm{dy}}{\mathrm{dx}}$<br/>$$<br/>\mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}+1<br/>$$<br/>Since power of highest order derivative is 2 . $\therefore$ degree $=2$</div>


title: what-is-the-degree-of-the-equation-dx-2-d-2-y-y-dx-dy-2-4-1
slug: what-is-the-degree-of-the-equation-dx-2-d-2-y-y-dx-dy-2-4-1-47654
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the degree of the equation<br/>$\left[\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right]=\left[\mathrm{y}+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{\frac{1}{4}} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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">The given differential equation is:<br/>$$<br/>\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\left[\mathrm{y}+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{1 / 4}<br/>$$<br/>This can be re-written as by squaring both the sides to the power 4 to make it a polynomial of derivative.<br/>$$<br/>\left[\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right]^{4}=\left[\mathrm{y}+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]<br/>$$<br/>Power of highest derivatives 4 , So, degree of the equation is 4 .</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
title:
what-is-the-degree-of-the-following-differential-equation-d-x-3-d-3-y-23-4-3-d-x-2-d-2-y-5-d-x-d-y-0

slug:
what-is-the-degree-of-the-following-differential-equation-d-x-3-d-3-y-23-4-3-d-x-2-d-2-y-5-d-x-d-y-0-62038

question:
<div class="question">What is the degree of the following differential equation? <br/>$\left(\frac{d^{3} y}{d x^{3}}\right)^{2 / 3}+4-3 \frac{d^{2} y}{d x^{2}}+5 \frac{d y}{d x}=0$</div>

answer:
<div class="solution">The given differential equation can be rewritten as<br/>$$<br/>\left(\frac{d^{3} y}{d x^{3}}\right)^{2}=\left(3 \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}-4\right)^{3}<br/>$$<br/>$\therefore$ Degree of differential equation is $2 .$ (\because Degree is the power of the highest order derivative)</div>


title: what-is-the-derivative-of-1-c-o-s-x-1-c-o-s-x-1
slug: what-is-the-derivative-of-1-c-o-s-x-1-c-o-s-x-1-74497
question: <div class="question">What is the derivative of $\sqrt{\frac{1+\cos x}{1-\cos x}}$ ?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \sec ^{2} \frac{x}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2} \operatorname{cosec}^{2} \frac{x}{2}$</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">$-\operatorname{cosec}^{2} \frac{x}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{y}=\sqrt{\frac{1+\cos \mathrm{x}}{1-\cos \mathrm{x}}}=\frac{\sqrt{2} \cos \frac{\mathrm{x}}{2}}{\sqrt{2} \sin \frac{\mathrm{x}}{2}}=\cot \frac{\mathrm{x}}{2}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=-\operatorname{cosec}^{2} \frac{\mathrm{x}}{2} \cdot \frac{1}{2}=-\frac{1}{2} \operatorname{cosec}^{2} \frac{\mathrm{x}}{2}$</div>


title: what-is-the-derivative-of-1-c-o-s-x-1-c-o-s-x
slug: what-is-the-derivative-of-1-c-o-s-x-1-c-o-s-x-64080
question: <div class="question">What is the derivative of $\sqrt{\frac{1+\cos x}{1-\cos x}}$ ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \sec ^{2} \frac{x}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2} \operatorname{cosec}^{2} \frac{x}{2}$</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">$-\operatorname{cosec}^{2} \frac{x}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{y}=\sqrt{\frac{1+\cos \mathrm{x}}{1-\cos \mathrm{x}}}$<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=-\operatorname{cosec}^{2} \frac{\mathrm{x}}{2} \cdot \frac{1}{2}=-\frac{1}{2} \operatorname{cosec}^{2} \frac{\mathrm{x}}{2}$</div>


title: what-is-the-derivative-of-cos-1-13-2-c-o-s-x-3-s-i-n-x
slug: what-is-the-derivative-of-cos-1-13-2-c-o-s-x-3-s-i-n-x-49661
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $\cos ^{-1}\left(\frac{2 \cos x+3 \sin x}{\sqrt{13}}\right)$?</div>

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

answer:
<div class="solution">The given function $f(x)=\cos ^{-1}\left(\frac{2 \cos x+3 \sin x}{\sqrt{13}}\right\}$<br/>can be written as<br/>$\cos ^{-1}\left(\cos x \cdot \frac{2}{\sqrt{13}}+\sin x \cdot \frac{3}{\sqrt{13}}\right\}$ ...(i)<br/>Let $\frac{2}{\sqrt{13}}=\cos \theta$ and $\frac{3}{\sqrt{13}}=\sin \theta$<br/>$\mathrm{So},(\mathrm{i}), \cos ^{-1}(\cos \mathrm{x} \cos \theta+\sin \mathrm{x} \sin \theta)$<br/>where $\theta=\tan ^{-1}\left(\frac{3}{2}\right)$<br/>$=\cos ^{-1}(\cos (x-\theta))=x-\theta$<br/>hence, $f^{\prime}(x)=1 \quad(\because \theta$ is a constant $)$</div>


title: what-is-the-derivative-of-f-x-1-x-2-with-respect-to-g-x-sin-1-x-where-x-1
slug: what-is-the-derivative-of-f-x-1-x-2-with-respect-to-g-x-sin-1-x-where-x-1-32135
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $f(x)=\sqrt{1-x^{2}}$ with respect to $\mathrm{g}(\mathrm{x})=\sin ^{-1} \mathrm{x}$, where $|\mathrm{x}| \neq 1 ?$</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">$-\mathrm{x}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x}{1-x^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{x}{1-x^{2}}$</span> </li> </ul>

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

answer:
<div class="solution">Here, $\frac{\mathrm{d}\left(\sqrt{1-\mathrm{x}^{2}}\right)}{\mathrm{d}\left(\sin ^{-1} \mathrm{x}\right)}=\frac{\frac{\mathrm{d}}{\mathrm{dx}}\left(\sqrt{1-\mathrm{x}^{2}}\right)}{\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1} \mathrm{x}\right)}$ is to be found.<br/>$f(x)=\sqrt{1-x^{2}}$<br/>So, $f^{\prime}(x)=\frac{d}{d x}\left(\sqrt{1-x^{2}}\right)=f^{\prime}(x) \frac{1}{2 \sqrt{1-x^{2}}}(-2 x)$<br/>$\Rightarrow f^{\prime}(x)=\frac{-x}{\sqrt{1-x^{2}}}$  ...(i)<br/>Also, $g(x)=\sin ^{-1} x$<br/>$\mathrm{g}^{\prime}(\mathrm{x})=\frac{1}{\sqrt{1-\mathrm{x}^{2}}}$ ...(ii)<br/>$\therefore \frac{\mathrm{f}^{\prime}(\mathrm{x})}{\mathrm{g}^{\prime}(\mathrm{x})}=\frac{-\mathrm{x} / \sqrt{1-\mathrm{x}^{2}}}{1 / \sqrt{1-\mathrm{x}^{2}}}$ $\quad$ (using Eqs. (i) and (ii))<br/>$=-\mathrm{x}$</div>


title: what-is-the-derivative-of-f-x-2-x-1-x-3-7-x
slug: what-is-the-derivative-of-f-x-2-x-1-x-3-7-x-40765
question: <div class="question">What is the derivative of $f(x)=\frac{7 x}{(2 x-1)(x+3)}$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{3}{(x+3)^{2}}-\frac{2}{(2 x-1)^{2}}$</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">B</span> <span class="option-data">$-\frac{3}{(x+3)^{2}}-\frac{1}{(2 x-1)^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{(x+3)^{2}}+\frac{1}{(2 x-1)^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{(x+3)^{2}}+\frac{2}{(2 x-1)^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{3}{(x+3)^{2}}-\frac{2}{(2 x-1)^{2}}$</span> </div>

answer:
<div class="solution">Given function is $f(x)=\frac{7 x}{(2 x-1)(x+3)}$<br/>Breaking into partial fraction<br/>We get, $f(x)=\frac{1}{2 x-1}+\frac{3}{x+3}$<br/>Differentiating w.r.t. $\mathrm{x}$, we get<br/>$f^{\prime}(x)=-\frac{2}{(2 x-1)^{2}}-\frac{3}{(x+3)^{2}}$</div>


title: what-is-the-derivative-of-f-x-x-x
slug: what-is-the-derivative-of-f-x-x-x-33433
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2|x|$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $\mathrm{f}(\mathrm{x})=\mathrm{x}|\mathrm{x}|?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$|\mathrm{x}|+\mathrm{x}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{x}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2|x|$</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">$-2|x|$</span> </li> </ul>

answer:
<div class="solution">Given that $f(x)=x|x|$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\mathrm{x} \cdot \frac{|\mathrm{x}|}{\mathrm{x}}+|\mathrm{x}|\left[\right.$ Since $\left.\mathrm{f}^{\prime}(\mathrm{uv})=\mathrm{u} . \mathrm{f}^{\prime}(\mathrm{v})+\mathrm{v} . \mathrm{f}^{\prime}(\mathrm{u})\right]$<br/>$=|\mathrm{x}|+|\mathrm{x}|=2|\mathrm{x}|$<br/>Here, $\mathrm{u}=\mathrm{x}, \mathrm{v}=|\mathrm{x}|$</div>


title: what-is-the-derivative-of-lo-g-10-5-x-2-3-with-respect-to-x
slug: what-is-the-derivative-of-lo-g-10-5-x-2-3-with-respect-to-x-87813
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $\log _{10}\left(5 \mathrm{x}^{2}+3\right)$ with respect to $\mathrm{x}$?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{10 \mathrm{x} \log _{10} \mathrm{e}}{5 \mathrm{x}^{2}+3}$</span> </div>

answer:
<div class="solution">$y=\log _{10}\left(5 x^{2}+3\right)$<br/>$\frac{d y}{d x}=\frac{d}{d x}\left(\log _{10}\left(5 x^{2}+3\right)\right)=\frac{1}{5 x^{2}+3} \times \log _{10} e \times 10 x$<br/>$=\frac{10 x \log _{10} e}{5 x^{2}+3}$</div>


title: what-is-the-derivative-of-lo-g-t-a-n-x-cot-x-lo-g-c-o-t-x-tan-x-1-at-x-4
slug: what-is-the-derivative-of-lo-g-t-a-n-x-cot-x-lo-g-c-o-t-x-tan-x-1-at-x-4-70248
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">${0}$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of<br/>$\left(\log _{\tan x} \cot x\right)\left(\log _{\cot x} \tan x\right)^{-1}$ at $x=\frac{\pi}{4} ?$</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">$\frac{1}{2}$</span> </li> </ul>

answer:
<div class="solution">The given function, $f(x)=\left(\log _{\tan x} \cot x\right)\left(\log _{\cot x} \tan x\right)^{-1}$<br/>$=\left(\frac{\log \cot x}{\log \tan x}\right)\left(\frac{\log \tan x}{\log \cot x}\right)^{-1}$<br/>$=\left(\frac{\log \cot x}{\log \tan x}\right)\left(\frac{\log \cot x}{\log \tan x}\right)$<br/>$=\left(\frac{\log \cot x}{\log \tan x}\right)^{2}=\left(\frac{\log \frac{1}{\tan x}}{\log \tan x}\right)^{2}=\left(-\frac{\log \tan x}{\log \tan x}\right)^{2}=1$<br/>$\Rightarrow \mathrm{f}(\mathrm{x})=1 \quad$ (constant function).<br/>$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=0$<br/>and this is true for $0   &lt; x &lt; \frac{\pi}{2}$<br/>$\Rightarrow f^{\prime}\left(\frac{\pi}{4}\right)=0$</div>


title: what-is-the-derivative-of-lo-g-x-5-with-respect-to-lo-g-5-x
slug: what-is-the-derivative-of-lo-g-x-5-with-respect-to-lo-g-5-x-36662
question: <div class="question">What is the derivative of $\log _{x} 5$ with respect to $\log _{5} x$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\left(\log _{5} x\right)^{-2}$</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">B</span> <span class="option-data">$\left(\log _{5} x\right)^{-2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\left(\log _{x} 5\right)^{-2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\log _{x} 5\right)^{-2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\left(\log _{5} x\right)^{-2}$</span> </div>

answer:
<div class="solution">Let $u_{1}=\log _{x} 5$ and $u_{2}=\log _{5} x$<br/>$\Rightarrow u_{1}=\frac{\log _{e} 5}{\log _{e} x}$ and $u_{2}=\frac{\log _{e} x}{\log _{e} 5}$<br/>On differentiating w.r.t. $x$, we get<br/>$\frac{d u_{1}}{d x}=\left[\frac{\log _{e} x(0)-\left(\frac{1}{x}\right)}{\left(\log _{e} x\right)^{2}}\right] \log _{e} 5=-\frac{\log _{e} 5}{x\left(\log _{e} x\right)^{2}}$<br/>and $\frac{d u_{2}}{d x}=\frac{1}{x \log _{e} 5}$<br/>$\therefore \frac{d u_{1}}{d u_{2}}=\frac{d u_{1} / d x}{d u_{2} / d x}=-\frac{\log _{e} 5}{x\left(\log _{e} x\right)^{2}} \times x \log _{e} 5$<br/>$=-\left(\frac{\log _{e} 5}{\log _{e} x}\right)^{2}=-\left(\log _{x} 5\right)^{2}=-\left(\log _{5} x\right)^{-2}$</div>


title: what-is-the-derivative-of-sec-2-tan-1-x-with-respect-to-x
slug: what-is-the-derivative-of-sec-2-tan-1-x-with-respect-to-x-94011
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $\sec ^{2}\left(\tan ^{-1} x\right)$ with respect to $x$ ?</div>

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

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

answer:
<div class="solution">Let $\mathrm{y}=\sec ^{2}\left(\tan ^{-1} \mathrm{x}\right)$<br/>Let $\tan ^{-1} x=\theta \Rightarrow x=\tan \theta$<br/>$y=\sec ^{2} \theta=1+\tan ^{2} \theta=1+x^{2}$<br/>$\therefore \quad \frac{d y}{d x}=\frac{d}{d x}\left(1+x^{2}\right)=2 x$</div>


title: what-is-the-derivative-of-sin-2-x-with-respect-to-cos-2-x
slug: what-is-the-derivative-of-sin-2-x-with-respect-to-cos-2-x-18296
question: <div class="question">What is the derivative of $\sin ^{2} x$ with respect to $\cos ^{2} x ?$</div>
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_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan ^{2} x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cot ^{2} x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Let $u=\sin ^{2} x$ and $v=\cos ^{2} x$<br/>$\Rightarrow \quad \frac{d u}{d x}=2 \sin x \cos x=\sin 2 x$<br/>and $\frac{d v}{d x}=-2 \sin x \cos x=-\sin 2 x$<br/>$\therefore \quad \frac{d u}{d v}=\frac{d u / d x}{d v / d x}=\frac{-\sin 2 x}{\sin 2 x}=-1$</div>


title: what-is-the-derivative-of-sin-sin-x
slug: what-is-the-derivative-of-sin-sin-x-86159
question: <div class="question">What is the derivative of $\sin (\sin x)$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos (\cos x)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos (\sin x)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\cos (\sin x) \cos x$</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">D</span> <span class="option-data">$\cos (\cos x) \cos x$</span> </li> </ul>

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

answer:
<div class="solution">$\frac{\mathrm{d}}{\mathrm{dx}} \sin (\sin \mathrm{x})=\cos (\sin \mathrm{x}) \cdot \cos \mathrm{x}$</div>


title: what-is-the-derivative-of-tan-1-1-x-32-x-x-at-x-1
slug: what-is-the-derivative-of-tan-1-1-x-32-x-x-at-x-1-74156
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $\tan ^{-1}\left(\frac{\sqrt{x}-x}{1+x^{3 / 2}}\right)$ at $x=1 ?$</div>

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

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

answer:
<div class="solution">Let $y=\tan ^{-1}\left(\frac{\sqrt{x}-x}{1+x^{3 / 2}}\right)=\tan ^{-1} \frac{\sqrt{x}-x}{1+\sqrt{x}, x}$<br/>$=\tan ^{-1} \sqrt{\mathrm{x}}-\tan ^{-1} \mathrm{x}$<br/>On differentiating w.r.t.x, we get<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}} \cdot \frac{1}{2 \sqrt{\mathrm{x}}}-\frac{1}{1+\mathrm{x}^{2}}$<br/>Now, $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{x}=1}=\frac{1}{1+1} \cdot \frac{1}{2}-\frac{1}{1+1}=\frac{1}{4}-\frac{1}{2}=-\frac{1}{4}$.</div>


title: what-is-the-derivative-of-tan-1-x-1-x-2-1-with-respect-to-tan-1-x
slug: what-is-the-derivative-of-tan-1-x-1-x-2-1-with-respect-to-tan-1-x-74153
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan ^{-1} \mathrm{x} ?$</div>

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

answer:
<div class="solution">Let $\mathrm{y}=\tan ^{-1}\left[\frac{\sqrt{1+\mathrm{x}^{2}}-1}{\mathrm{x}}\right]$ and $\mathrm{u}=\tan ^{-1} \mathrm{x}$<br/>Put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$<br/>Then, $y=\tan ^{-1}\left[\frac{\sqrt{1+\tan ^{2} \theta}-1}{\tan \theta}\right]$<br/>$=\tan ^{-1}\left[\frac{\sqrt{\sec ^{2} \theta}-1}{\tan \theta}\right]$<br/>$=\tan ^{-1}\left[\frac{\sec \theta-1}{\tan \theta}\right]=\tan ^{-1}\left[\frac{\frac{1}{\cos \theta}-1}{\frac{\sin \theta}{\cos \theta}}\right]$<br/>$=\tan ^{-1}\left[\frac{1-\cos \theta}{\sin \theta}\right]=\tan ^{-1}\left[\frac{2 \sin ^{2} \frac{\theta}{2}}{2 \sin \frac{\theta}{2}, \cos \frac{\theta}{2}}\right]$<br/>$\left(\begin{array}{r}\because 1-\cos \theta=2 \sin ^{2} \frac{\theta}{2} \text { and } \\ \sin x=2 \sin \frac{x}{2} \cdot \cos \frac{x}{2}\end{array}\right)$<br/>$=\tan ^{-1}\left[\tan \frac{\theta}{2}\right]$<br/>$\Rightarrow y=\frac{\theta}{2} \Rightarrow y=\frac{\tan ^{-1} x}{2} \quad\left[\because \theta=\tan ^{-1} x\right]$<br/>$\Rightarrow y=\frac{u}{2}$<br/>$\frac{\mathrm{dy}}{\mathrm{du}}=\frac{1}{2}$<br/>$\therefore$ Option (b) is correct.</div>


title: what-is-the-derivative-of-the-function-2018-i-f-x-e-t-a-n-x-ln-sec-x-e-nx-at-x-4
slug: what-is-the-derivative-of-the-function-2018-i-f-x-e-t-a-n-x-ln-sec-x-e-nx-at-x-4-90658
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of the function $\quad[2018-I]$<br/>$\mathrm{f}(\mathrm{x})=\mathrm{e}^{\tan \mathrm{x}}+\ln (\sec \mathrm{x})-\mathrm{e}^{\ell \mathrm{n} \mathrm{x}}$ at $\mathrm{x}=\frac{\pi}{4} ?$</div>

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

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{e}^{\tan \mathrm{x}}+\ln (\sec \mathrm{x})-\mathrm{e}^{\mathrm{fn} \mathrm{x}}$<br/>$\begin{array}{l}f^{\prime}(\mathrm{x})=\mathrm{e}^{\tan \mathrm{x}} \sec ^{2} \mathrm{x}+\frac{1}{\sec \mathrm{x}} \cdot \sec x \tan \mathrm{x}-1 \\ \left(\because \mathrm{e}^{\ln \mathrm{x}}=\mathrm{x} \text { and } \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x})=1\right)\end{array}$<br/>$f^{\prime}\left(\frac{\pi}{4}\right)=\mathrm{e}^{\tan \frac{\pi}{4}} \cdot \sec ^{2} \frac{\pi}{4}+\tan \frac{\pi}{4}-1$<br/>$=2 \mathrm{e}+1-1$<br/>$=2 \mathrm{e}$</div>


title: what-is-the-derivative-of-x-1-at-x-2
slug: what-is-the-derivative-of-x-1-at-x-2-68192
question: <div class="question">What is the derivative of $|x-1|$ at $x=2$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.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">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Derivative does not exist</span> </li> </ul>

answer:
<div class="solution">$f(x)=|x-1|$<br/>Redefined the function $\mathrm{f}(\mathrm{x})$<br/>$f(x)=\left\{\begin{array}{ll}1-x, &amp; x   &lt; 1 \\ x-1, &amp; x&gt;1\end{array}\right.$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\left\{\begin{array}{ll}-1 ; &amp; \mathrm{x} &lt; 1 \\ 1 ; &amp; \mathrm{x}&gt;1\end{array}\right.$<br/>$\therefore \mathrm{f}^{\prime}(2)=1$</div>


title: what-is-the-derivative-of-x-3-with-respect-to-x-2
slug: what-is-the-derivative-of-x-3-with-respect-to-x-2-68701
question: <div class="question">What is the derivative of $x^{3}$ with respect to $x^{2} ?$</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 x^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{3 x}{2}$</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">$x$</span> </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">$\frac{3 x}{2}$</span> </div>

answer:
<div class="solution">$\mathrm{U}=\mathrm{x}^{3}$<br/>$\frac{\mathrm{dU}}{\mathrm{dx}}=3 \mathrm{x}^{2}$ ...(1)<br/>$\mathrm{V}=\mathrm{x}^{2}$<br/>$\frac{\mathrm{dV}}{\mathrm{dx}}=2 \mathrm{x}$ ...(2)<br/>From (1) and (2)<br/>$\frac{\mathrm{dU}}{\mathrm{dV}}=\frac{3 \mathrm{x}^{2}}{2 \mathrm{x}}=\frac{3}{2} \mathrm{x}$</div>


title: what-is-the-derivative-of-x-a-2-x-2-a-2-sin-1-a-x
slug: what-is-the-derivative-of-x-a-2-x-2-a-2-sin-1-a-x-83887
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the derivative of $x \sqrt{a^{2}-x^{2}}+a^{2} \sin ^{-1}\left(\frac{x}{a}\right)$?</div>

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

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

answer:
<div class="solution">Let $y=x \sqrt{a^{2}-x^{2}}+a^{2} \sin ^{-1}\left(\frac{x}{a}\right)$<br/>Differentiate both side w.r.t ' $x$ ' we get<br/>$\begin{aligned} \frac{d y}{d x} &amp;=x \frac{1}{2 \sqrt{a^{2}-x^{2}}} \cdot(-2 x)+\frac{a^{2}}{\sqrt{1-\frac{x^{2}}{a^{2}}}} \cdot \frac{1}{a} \\ &amp;=\frac{-x^{2}}{\sqrt{a^{2}-x^{2}}}+\frac{a^{2} \cdot a}{\sqrt{a^{2}-x^{2}}} \cdot \frac{1}{a} \\ &amp;=\frac{a^{2}-x^{2}}{\sqrt{a^{2}-x^{2}}}=\sqrt{a^{2}-x^{2}} \end{aligned}$</div>


tags: ['Mathematics', 'Differentiation', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-derivative-of-y-as-a-function-of-x-with-respect-to-x-for-x-0-consider-the-equation-x-y-2-y

slug:
what-is-the-derivative-of-y-as-a-function-of-x-with-respect-to-x-for-x-0-consider-the-equation-x-y-2-y-87925

question:
<div class="question">What is the derivative of y as a function of $\mathrm{x}$ with respect to $\mathrm{x}$ for $\mathrm{x}   &lt; 0 ?$<br/>Consider the equation $\mathrm{x}+|\mathrm{y}|=2 \mathrm{y}$</div>

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

answer:
<div class="solution">$y=\frac{1}{3} x$ for $x   &lt; 0$.<br/>Hence $\frac{d y}{d x}=\frac{1}{3}$<br/>Option (d) is correct.</div>


title: what-is-the-determinant-bc-c-a-ab-a-b-c-a-2-b-2-c-2-equal-to
slug: what-is-the-determinant-bc-c-a-ab-a-b-c-a-2-b-2-c-2-equal-to-47442
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the determinant<br/>$\left|\begin{array}{lll}\text { bc } &amp; a &amp; a^{2} \\ c a &amp; b &amp; b^{2} \\ a b &amp; c &amp; c^{2}\end{array}\right|$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left|\begin{array}{ccc}1 &amp; a^{2} &amp; a^{3} \\ 1 &amp; b^{2} &amp; b^{3} \\ 1 &amp; c^{2} &amp; c^{3}\end{array}\right|$</span> </div>

answer:
<div class="solution">$\left|\begin{array}{lll}b c &amp; a &amp; a^{2} \\ c a &amp; b &amp; b^{2} \\ a b &amp; c &amp; c^{2}\end{array}\right|$<br/>$\mathrm{R}_{1} \rightarrow \mathrm{aR}_{1}, \mathrm{R}_{2} \rightarrow \mathrm{bR}_{2}, \mathrm{R}_{3} \rightarrow \mathrm{cR}_{3}$ and divide whole<br/>determinant by abc<br/>$=\frac{1}{a b c}\left|\begin{array}{ccc}a b c &amp; a^{2} &amp; a^{3} \\ a b c &amp; b^{2} &amp; b^{3} \\ a b c &amp; c^{2} &amp; c^{2}\end{array}\right|$<br/>$=\frac{a b c}{a b c}\left|\begin{array}{ccc}1 &amp; a^{2} &amp; a^{3} \\ 1 &amp; b^{2} &amp; b^{3} \\ 1 &amp; b^{2} &amp; c^{2}\end{array}\right|=\left|\begin{array}{ccc}1 &amp; a^{2} &amp; a^{3} \\ 1 &amp; b^{2} &amp; b^{3} \\ 1 &amp; c^{2} &amp; c^{3}\end{array}\right|$</div>


title: what-is-the-determinant-of-the-matrix-x-z-y-y-z-z-y-z-z-x-x-y
slug: what-is-the-determinant-of-the-matrix-x-z-y-y-z-z-y-z-z-x-x-y-53489
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the determinant of the matrix $\left(\begin{array}{lll}\mathrm{x} &amp; \mathrm{y} &amp; \mathrm{y}+\mathrm{z} \\ \mathrm{z} &amp; \mathrm{z} &amp; \mathrm{z}+\mathrm{x} \\ \mathrm{y} &amp; \mathrm{z} &amp; \mathrm{x}+\mathrm{y}\end{array}\right)$?</div>

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

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

answer:
<div class="solution">$\left|\begin{array}{ccc}x &amp; y &amp; y+z \\ z &amp; x &amp; z+x \\ y &amp; z &amp; x+y\end{array}\right|=\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}$<br/>$\left|\begin{array}{ccc}x+y+z &amp; x+y+z &amp; 2(x+y+z) \\ z &amp; x &amp; z+x \\ y &amp; z &amp; x+y\end{array}\right|$<br/>$\begin{aligned} &amp;=(x+y+z)\left|\begin{array}{ccc}1 &amp; 1 &amp; 2 \\ z &amp; x &amp; z+x \\ y &amp; z &amp; x+y\end{array}\right| \\ &amp;=(x+y+z)(z+x)^{2},(\text { replacing } z \operatorname{by} x) \end{aligned}$</div>


title: what-is-the-diameter-of-the-sphere-x-2-y-2-z-2-4-x-6-y-8-z-7-0
slug: what-is-the-diameter-of-the-sphere-x-2-y-2-z-2-4-x-6-y-8-z-7-0-38541
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the diameter of the sphere $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}-4 \mathrm{x}+6 \mathrm{y}-8 \mathrm{z}-7=0 ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">12 units.</span> </div>

answer:
<div class="solution">General equation of sphere is. $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+2 \mathrm{ux}+2 \mathrm{vy}+2 \mathrm{wz}+\mathrm{d}=0$<br/>On comparing with the given equation we have $\mathrm{u}=-2, \mathrm{v}=3, \mathrm{w}=-4, \mathrm{~d}=-7$<br/>$\begin{aligned} \text { Radius } &amp;=\sqrt{u^{2}+v^{2}+w^{2}-d} \\ &amp;=\sqrt{4+9+16+7}=\sqrt{36}=6 \end{aligned}$<br/>Diameter $=2 \times 6=12$</div>


title: what-is-the-difference-between-a-primary-oocyte-and-a-secondary-oocyte
slug: what-is-the-difference-between-a-primary-oocyte-and-a-secondary-oocyte-80494
question: <div class="question">What is the difference between a primary oocyte and a secondary oocyte?</div>
tags: ['Biology', 'Human Reproduction']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">Primary oocyte a diploid $(2 n)$ cell, formed by the transformation of primordial follicles in foetal ovary when the gamete mother cell, oogonia is arrested at prophase-1 of meiosis. Secondary oocyte the hapliod (n) cell formed from primary oocyte that completes its first meiotic division, during puberty and produces the female gamete ova(n).</div>


title: what-is-the-difference-between-primary-and-secondary-metabolites
slug: what-is-the-difference-between-primary-and-secondary-metabolites-85060
question: <div class="question">What is the difference between primary and secondary metabolites?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Primary metabolites have identifiable functions while secondary metabolites do not.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Secondary metabolites have identifiable functions while primary metabolites do not.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both primary and secondary metabolites have identifiable functions.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither primary nor secondary metabolites have identifiable functions.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Primary metabolites have identifiable functions while secondary metabolites do not.</span> </div>

answer:
<div class="solution">Primary metabolites have identifiable functions and play known roles in normal physiological processes, while the role or functions of all secondary metabolites in host organisms is not yet understood.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-difference-between-the-smallest-five-digit-binary-integer-and-the-lar-gest-four-digit-binary-integer

slug:
what-is-the-difference-between-the-smallest-five-digit-binary-integer-and-the-lar-gest-four-digit-binary-integer-46336

question:
<div class="question">What is the difference between the smallest five digit binary integer and the lar gest four digit binary integer?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The greatest one digit binary integer</span> </div>

answer:
<div class="solution">The largest four digit number in binary system is 1111 which isequivalent to 15 in decimal system and smallest five digit number in binary system is 10000 which is equivalent to 16 in decimal system. Difference between numbers $=16-15=1$ Which is the greatest one digit binary integer.</div>


title: what-is-the-difference-in-the-roots-of-the-equation-x-2-10-x-9-0
slug: what-is-the-difference-in-the-roots-of-the-equation-x-2-10-x-9-0-49817
question: <div class="question">What is the difference in the roots of the equation $x^{2}-10 x+9=0 ?$</div>
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P3.db
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">5</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">$x^{2}-10 x+9=0$<br/>$x^{2}-x-9 x+9=0$<br/>$x(x-1)-9(x-1)=0$<br/>$(\mathrm{x}-1)(\mathrm{x}-9)=0$<br/>$\mathrm{x}=1,9$<br/>Difference in roots $=9-1=8$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-3</span> </div>
source_db: MarksBatch2_P3.db
title:
what-is-the-differentiable-coefficient-of-f-x-at-x-3-let-f-x-be-a-function-defined-in-1-x-by-2014-i-f-x-2-x-3-x-x-2-for-1-x-2-for-x-2

slug:
what-is-the-differentiable-coefficient-of-f-x-at-x-3-let-f-x-be-a-function-defined-in-1-x-by-2014-i-f-x-2-x-3-x-x-2-for-1-x-2-for-x-2-23906

question:
<div class="question">What is the differentiable coefficient of $f(x)$ at $x=3$ ?<br/>Let $f(x)$ be a function defined in $1 \leq x   &lt; \infty$ by $\quad[2014-I]$<br/>$f(x)=\left\{\begin{array}{cc}2-x &amp; \text { for } 1 \leq x \leq 2 \\ 3 x-x^{2} &amp; \text { for } x&gt;2\end{array}\right.$</div>

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

answer:
<div class="solution">$f^{\prime}(x)=\left\{\begin{array}{ccc}-1 &amp; \text { for } &amp; 1 \leq x \leq 2 \\ 3-2 x &amp; \text { for } &amp; x&gt;2\end{array}\right.$<br/>$f(x)$ at $x=3$<br/>$f^{\prime}(3)=3-2(3)=3-6=-3$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-differential-coefficient-of-f-x-at-x-12-consider-the-function-f-x-x-2-5-x-13-x-3-x-3-2014-ii

slug:
what-is-the-differential-coefficient-of-f-x-at-x-12-consider-the-function-f-x-x-2-5-x-13-x-3-x-3-2014-ii-46427

question:
<div class="question">What is the differential coefficient of $f(x)$ at $x=12$ ?<br/>Consider the function $f(x)=\left\{\begin{array}{ll}x^{2}-5 &amp; x \leq 3 \\ \sqrt{x+13} &amp; x&gt;3\end{array} \quad[2014-I I]\right.$</div>

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

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

answer:
<div class="solution">Given, $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}+13}, \mathrm{x}&gt;3$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{2 \sqrt{\mathrm{x}+13}}$<br/>$\Rightarrow \mathrm{f}^{\prime}(12)=\frac{1}{2 \sqrt{12+13}}=\frac{1}{2 \times 5}=\frac{1}{10}$</div>


title: what-is-the-differential-coefficient-of-lo-g-x-x
slug: what-is-the-differential-coefficient-of-lo-g-x-x-82691
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_P3.db
question:
<div class="question">What is the differential coefficient of $\log _{\mathrm{x}} \mathrm{x}$?</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">$\frac{1}{x}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${x}$</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{y}=\log _{\mathrm{x}} \mathrm{x}$<br/>$\Rightarrow \mathrm{y}=1($ for $\mathrm{x}&gt;0$ and $\mathrm{x} \neq 1)$<br/>On differentiating both the sidew.r.t ' $x$ ', we get $\frac{\mathrm{dy}}{\mathrm{dx}}=0$</div>


title: what-is-the-differential-equation-corresponding-to-y-2-2-ay-x-2-a-2-by-eliminating-a
slug: what-is-the-differential-equation-corresponding-to-y-2-2-ay-x-2-a-2-by-eliminating-a-35912
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the differential equation corresponding to $\mathrm{y}^{2}-2 \mathrm{ay}+\mathrm{x}^{2}=\mathrm{a}^{2}$ by eliminating a? $\quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(x^{2}-2 y^{2}\right) p^{2}-4 p x y-x^{2}=0$</span> </div>

answer:
<div class="solution">$\mathrm{y}^{2}-2 \mathrm{ay}+\mathrm{x}^{2}=\mathrm{a}^{2}$<br/>$\Rightarrow y^{2}-2(y)(a)+a^{2}+x^{2}=a^{2}+a^{2}$<br/>$\Rightarrow(y-a)^{2}+x^{2}=2 a^{2}$<br/>Dift w.r.t x $2(\mathrm{y}-\mathrm{a}) \cdot \mathrm{y}^{1}+2 \mathrm{x}=0$<br/>$\Rightarrow(\mathrm{y}-\mathrm{a}) \cdot \mathrm{p}=-\mathrm{x} \quad\left(\right.$ Let $\left.\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}^{1}=\mathrm{P}\right)$<br/>$\Rightarrow \mathrm{y}-\mathrm{a}=\frac{-\mathrm{x}}{\mathrm{p}}$<br/>$\Rightarrow a=y+\frac{x}{p}$<br/>(i) $\Rightarrow\left(\frac{-x}{p}\right)^{2}+x^{2}=2\left(y+\frac{x}{p}\right)^{2}$<br/>(from (ii) (iii))<br/>$=2\left(\frac{(\mathrm{py}+\mathrm{x})^{2}}{\mathrm{p} 2}\right)$<br/>$\Rightarrow \frac{x^{2}+p^{2} x^{2}}{p^{2}}=\frac{2\left(p^{2} y^{2}+x^{2}+2 p x y\right)}{p^{2}}$<br/>$\Rightarrow x^{2}+p^{2} x^{2}-2 p^{2} y^{2}-2 x^{2}-4 p x y=0$<br/>$\Rightarrow\left(\mathrm{x}^{2}-2 \mathrm{y}^{2}\right) \mathrm{p}^{2}-4 \mathrm{pxy}-\mathrm{x}^{2}=0$</div>


title: what-is-the-differential-equation-for-y-2-4-a-x-a
slug: what-is-the-differential-equation-for-y-2-4-a-x-a-11860
question: <div class="question">What is the differential equation for $y^{2}=4 a(x-a)$ ?</div>
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y y^{\prime}-2 x y y^{\prime}+y^{2}=0 \quad$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y y^{\prime}\left(y y^{\prime}+2 x\right)+y^{2}=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y y^{\prime}\left(y y^{\prime}-2 x\right)+y^{2}=0$</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">$y y^{\prime}-2 x y y^{\prime}+y=0$</span> </li> </ul>

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

answer:
<div class="solution">Given curve is $\mathrm{y}^{2}=4 a(x-a)$ On differentiating w.r.t. $\mathrm{x}$, we get $2 y y^{\prime}=4 a$<br/>$\Rightarrow \quad a=\frac{y y^{\prime}}{2}$<br/>On putting the value of $a$ in Eq. (i), we get<br/>$y^{2}=4\left(\frac{y y^{\prime}}{2}\right)\left(x-\frac{y y^{\prime}}{2}\right)=y y^{\prime}\left(2 x-y y^{\prime}\right)$<br/>$\Rightarrow \quad y y^{\prime}\left(y y^{\prime}-2 x\right)+y^{2}=0$</div>


title: what-is-the-differential-equation-of-all-parabolas-whose-axes-are-parallel-to-yaxis
slug: what-is-the-differential-equation-of-all-parabolas-whose-axes-are-parallel-to-yaxis-36719
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the differential equation of all parabolas whose axes are parallel to Y-axis? $\quad$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{d^{3} y}{d x^{3}}=0$</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">$\frac{d^{2} x}{d y^{2}}=C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{d^{3} x}{d v^{3}}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{d^{3} y}{d r^{3}}=C$<br/>(where $\mathrm{C}$ is a constant).</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{d^{3} y}{d x^{3}}=0$</span> </div>

answer:
<div class="solution">The general equation of all parabolas where axes are parallel to $Y$ -axis, is<br/>$$<br/>y=\mathrm{A} x^{2}+\mathrm{B} x+\mathrm{C}<br/>$$<br/>where $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are arbitrary constants. On differentiating eq. (i) w.r.t. $x$, we get<br/>$$<br/>\frac{d y}{d x}=2 \mathrm{~A} x+\mathrm{B}<br/>$$<br/>(ii) On differentiating eq. (ii) w.r.t. $x$, we get<br/>$$<br/>\frac{d^{2} y}{d x^{2}}=2 \mathrm{~A}<br/>$$<br/>(iii) On differentiating eq. (iii) w.r.t. $x$, we get<br/>$$<br/>\frac{d^{3} y}{d x^{3}}=0<br/>$$</div>


title: what-is-the-differential-equation-of-the-curve-y-ax-2-bx
slug: what-is-the-differential-equation-of-the-curve-y-ax-2-bx-14200
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the differential equation of the curve $\mathrm{y}=\mathrm{ax}^{2}+\mathrm{bx}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0$</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">B</span> <span class="option-data">$x^{2} \frac{d^{2} y}{d x^{2}}-y\left(\frac{d y}{d x}\right)^{2}+2=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-\left(y \frac{d y}{d x}\right)^{2}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\begin{array}{ll}\text {  } \text { None of the above } &amp; {}\end{array}$</span> </li> </ul>

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

answer:
<div class="solution">Given, equation is:<br/>$\mathrm{y}=\mathrm{ax}^{2}+\mathrm{bx}$<br/>Differentiating w.r.t. $\mathrm{x}$,<br/>$\Rightarrow \quad \frac{d y}{d x}=2 a x+b$<br/>Differentiation g w.r.t. $\mathrm{x}$,<br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=2 a$<br/>$\Rightarrow \quad a=\frac{1}{2} \frac{d^{2} y}{d x^{2}}$<br/>From (2) and (3)<br/>$$<br/>\mathrm{b}=\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x} \cdot \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}<br/>$$<br/>Putting values of $a$ and $b$ in equation (1):<br/>$$<br/>\begin{array}{l}<br/>\mathrm{y}=\frac{1}{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}} \cdot \mathrm{x}^{2}+\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}\right) \\<br/>\Rightarrow \quad 2 \mathrm{y}=\mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}+2 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{d} \mathrm{x}}-2 \mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}} \\<br/>\Rightarrow \mathrm{x}^{2} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}-2 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{y}=0<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-differential-equation-to-family-of-parabolas-having-their-vertices-at-the-origin-and-foci-on-the-x-axis

slug:
what-is-the-differential-equation-to-family-of-parabolas-having-their-vertices-at-the-origin-and-foci-on-the-x-axis-50101

question:
<div class="question">What is the differential equation to family of parabolas having their vertices at the origin and foci on the $\mathrm{x}$ -axis?</div>

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

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

answer:
<div class="solution">Let the equation of parabola is $y^{2}=4 a x$<br/>On differentiating w.r.t.x, we get $2 \mathrm{yy}^{\prime}=4 \mathrm{a}$<br/>$\Rightarrow \quad \frac{1}{2} \mathrm{yy}^{\prime}=\mathrm{a}$<br/>put the value of 'a 'in equation (i), we get<br/>$$<br/>\mathrm{y}^{2}=\frac{4}{2} \mathrm{yy}^{\prime} \mathrm{x}<br/>$$<br/>$\Rightarrow \quad \mathrm{y}=2 \mathrm{xy}^{\prime}$</div>


title: what-is-the-differentiation-of-lo-g-x-x-with-respect-to-ln-x
slug: what-is-the-differentiation-of-lo-g-x-x-with-respect-to-ln-x-27379
question: <div class="question">What is the differentiation of $\log _{x} x$ with respect to $\ln x$?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 $[2010-I]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x$</span> </li> </ul>

answer:
<div class="solution">Let $u=\log _{x} x=1$ $\quad \left(\because \log _{a} a=1\right)$<br/>Differentiate w.r.t ' $x^{\prime}$<br/>$\frac{d u}{d x}=0$<br/>and Let $v=\ln x \Rightarrow \frac{d v}{d x}=\frac{1}{x}$<br/>$\therefore \quad \frac{d u}{d v}=\frac{d u / d x}{d v / d x}=0$</div>


title: what-is-the-distance-between-the-lines-3-x-4-y-9-and-6-x-8-y-18
slug: what-is-the-distance-between-the-lines-3-x-4-y-9-and-6-x-8-y-18-30301
tags: ['Mathematics', 'Straight Lines', '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_P3.db
question:
<div class="question">What is the distance between the lines $3 \mathrm{x}+4 \mathrm{y}=9$ and $6 x+8 y=18 ?$</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">3 units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9 units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">18 units</span> </li> </ul>

answer:
<div class="solution">Given lines are $3 x+4 y=9 \quad \ldots(1)$<br/>$2(3 x+4 y=9)$...(2)<br/>Which are parallel to each other. Distance between them $=\frac{|9-9|}{\sqrt{9+16}}=\frac{0}{5}=0$</div>


title: what-is-the-distance-between-the-planes-x-2-y-z-1-0-and-3-x-6-y-3-z-2-0
slug: what-is-the-distance-between-the-planes-x-2-y-z-1-0-and-3-x-6-y-3-z-2-0-19003
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the distance between the planes $x-2 y+z-1=0$ and $-3 x+6 y-3 z+2=0 ?$</div>

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

answer:
<div class="solution">Given planes are $x-2 y+z=1$<br/>and $-3 x+6 y-3 z=-2$<br/>$=x-2 y+z=\frac{2}{3}$<br/>Since both planes are parallel and $\mathrm{a}=1, \mathrm{~b}=-2, \mathrm{c}=1$<br/>and $\mathrm{d}_{1}=-1, \mathrm{~d}_{2}=\frac{-2}{3}$<br/>Distance $=\left|\frac{\mathrm{d}_{2}-\mathrm{d}_{1}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}}\right|$<br/>Distance $=\left|\frac{1-\frac{2}{3}}{\sqrt{1+4+1}}\right|=\frac{1}{3 \sqrt{6}}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-distance-between-the-points-which-divide-the-line-segment-joining-4-3-and-5-7-internally-and-externally-in-the-ratio-2-3

slug:
what-is-the-distance-between-the-points-which-divide-the-line-segment-joining-4-3-and-5-7-internally-and-externally-in-the-ratio-2-3-76879

question:
<div class="question">What is the distance between the points which divide the line segment joining $(4,3)$ and $(5,7)$ internally and externally in the ratio $2: 3 ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{12 \sqrt{17}}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{13 \sqrt{17}}{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{17}}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{6 \sqrt{17}}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{12 \sqrt{17}}{5}$</span> </div>

answer:
<div class="solution">Let $\mathrm{A}(\mathrm{x}, \mathrm{y})$ be the point that divides $(4,3)$ and $(5,7)$ internally is ratio $2: 3$<br/>$\mathrm{A}=(\mathrm{x}, \mathrm{y})=\left(\frac{\mathrm{m} \mathrm{x}_{2}+\mathrm{nx}_{1}}{\mathrm{~m}+\mathrm{n}}, \frac{\mathrm{my}_{2}+\mathrm{ny}_{1}}{\mathrm{~m}+\mathrm{n}}\right)$<br/>$=\left(\frac{10+12}{2+3}, \frac{14+9}{2+3}\right)=\left(\frac{22}{5}, \frac{23}{5}\right)$<br/>Let $\mathrm{B}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)$ be point that divides $(4,3)$ and $(5,7)$ externally in ratio $2: 3$<br/>$\mathrm{B}=\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)=\left(\frac{\mathrm{m} \mathrm{x}_{2}-\mathrm{nx}_{1}}{\mathrm{~m}-\mathrm{n}}, \frac{\mathrm{m} \mathrm{y}_{2}-\mathrm{ny}_{1}}{\mathrm{~m}-\mathrm{n}}\right)$<br/>$=\left(\frac{10-12}{2-3}, \frac{14-9}{2-3}\right)=(2,-5)$<br/>Distance between $\mathrm{A}$ and $\mathrm{B}=\sqrt{\left(\mathrm{x}^{\prime}-\mathrm{x}\right)^{2}+\left(\mathrm{y}^{\prime}-\mathrm{y}\right)^{2}}$<br/>$=\sqrt{\left(2-\frac{22}{5}\right)^{2}+\left(-5-\frac{23}{5}\right)^{2}}$<br/>$=\sqrt{\frac{(-12)^{2}}{25}+\frac{(-48)^{2}}{25}}$<br/>$=\frac{12}{5} \sqrt{1+4^{2}}=\frac{12}{5} \sqrt{17}$</div>


title: what-is-the-distance-between-the-straight-lines-3-x-4-y-9-and-6-x-8-y-15
slug: what-is-the-distance-between-the-straight-lines-3-x-4-y-9-and-6-x-8-y-15-66934
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the distance between the straight lines $3 \mathrm{x}+4 \mathrm{y}=9$ and $6 x+8 y=15 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{3}{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

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

answer:
<div class="solution">Given lines, $L_{1}=3 x+4 y-9=0$<br/>$\mathrm{L}_{2}=6 \mathrm{x}+8 \mathrm{y}-15=0 \Rightarrow 3 \mathrm{x}+4 \mathrm{y}-\frac{15}{2}=0$<br/>Observe that the coefficients of $\mathrm{x}$ and $\mathrm{y}$ are same. $\therefore \mathrm{L}_{1}$ and $\mathrm{L}_{2}$ are parallel lines.<br/>Distance between parallel lines $=\frac{\left|c_{1}-c_{2}\right|}{\sqrt{a^{2}+b^{2}}}$<br/>$=\frac{\left|-9+\frac{15}{2}\right|}{\sqrt{3^{2}+4^{2}}}$<br/>$=\frac{\left|\frac{-18+15}{2}\right|}{\sqrt{25}}=\frac{3}{10}$</div>


title: what-is-the-distance-of-the-line-2-x-y-2-z-3-from-the-origin
slug: what-is-the-distance-of-the-line-2-x-y-2-z-3-from-the-origin-49141
question: <div class="question">What is the distance of the line $2 x+y+2 z=3$ from the origin ?</div>
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.5$ units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2 units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.5$ units</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{d}=\left|\frac{2 \times 0+0 \times 1+2 \times 0-3}{\sqrt{2^{2}+1^{2}+2^{2}}}\right|=\frac{3}{3}=1$ unit</div>


title: what-is-the-distance-of-the-origin-from-the-plane-2-x6-y3-z70
slug: what-is-the-distance-of-the-origin-from-the-plane-2-x6-y3-z70-73824
question: <div class="question">What is the distance of the origin from the plane $2 x+6 y-3 z+7=0 ?</div>
tags: ['Mathematics', 'Three Dimensional Geometry', '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_P3.db
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">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">Required distance $=\frac{|2(0)+6(0)-3(0)+7|}{\sqrt{(2)^{2}+(6)^{2}+(-3)^{2}}}$<br/>$=\frac{|7|}{\sqrt{4+36+9}}=\left|\frac{7}{7}\right|=1$</div>


title: what-is-the-distance-of-the-point-1-2-0-from-yzplane-is
slug: what-is-the-distance-of-the-point-1-2-0-from-yzplane-is-41030
question: <div class="question">What is the distance of the point $(1,2,0)$ from yz-plane is:</div>
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 unit</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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 units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3 units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4 units</span> </li> </ul>

answer:
<div class="solution">Equation of plane is $x=0$ $\therefore \quad$ Required distance from $(1,2,0)$ is<br/>$=\frac{1.1+2.0+0.0-0}{\sqrt{1^{2}+0^{2}+0^{2}}}=\frac{1}{\sqrt{1}}=1$ unit</div>


title: what-is-the-distance-of-the-point-2-3-4-from-the-plane-3-x-6-y-2-z-11
slug: what-is-the-distance-of-the-point-2-3-4-from-the-plane-3-x-6-y-2-z-11-55524
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 unit</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the distance of the point $(2,3,4)$ from the plane $3 \mathrm{x}$ $6 y+2 z+11=? \quad$</div>

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

answer:
<div class="solution">Distance $=\left|\frac{3 \times 2-6 \times 3+2 \times 4+11}{\sqrt{3^{2}+(-6)^{2}+(2)^{2}}}\right|$<br/>$=\left|\frac{6-18+8+11}{\sqrt{49}}\right|=1$</div>


title: what-is-the-domain-of-the-function-f-x-x-x-1
slug: what-is-the-domain-of-the-function-f-x-x-x-1-89110
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the domain of the function $\mathrm{f}(\mathrm{x})=\frac{1}{\sqrt{|\mathrm{x}|-\mathrm{x}}}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(-\infty, 0)$</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">$(0, \infty)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 0   &lt; x &lt; 1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x&gt;1$</span> </li> </ul>

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

answer:
<div class="solution">We know that<br/>$|x|=\left\{\begin{array}{cl}x, &amp; x \geq 0 \\ -x, &amp; x   &lt; 0\end{array}\right.$<br/>For domain, $|x|-x&gt;0$ Case $2: x &lt; 0$ Case $1: x&gt;0 \Rightarrow x-x=0$ (not possible)<br/>$-x-x&gt;0$<br/>$\Rightarrow-2 x&gt;0$<br/>$\Rightarrow x &lt; 0$<br/>So, $x \in(-\infty, 0)$</div>


title: what-is-the-eccentricity-of-an-ellipse-if-its-latusrectum-is-equal-toonehalf-of-its-minor-axis
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-eccentricity-of-an-ellipse-if-its-latusrectum-is-equal-toonehalf-of-its-minor-axis-25663

question:
<div class="question">What is the eccentricity of an ellipse, if its latusrectum is equal toone-half of its  minor axis?</div>

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

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

answer:
<div class="solution">Since, Latusrectum of an ellipse $=\frac{2 b^{2}}{a}$ and minor axis $=2 \mathrm{~b}$<br/>$b=\frac{2 b^{2}}{a} \Rightarrow a=2 b$<br/>Also, $e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{b^{2}}{4 b^{2}}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$</div>


title: what-is-the-eccentricity-of-rectangular-hyperbola
slug: what-is-the-eccentricity-of-rectangular-hyperbola-97293
question: <div class="question">What is the eccentricity of rectangular hyperbola?</div>
tags: ['Mathematics', 'Hyperbola', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{6}$</span> </li> </ul>

answer:
<div class="solution">Here, $b^{2}=a^{2}\left(e^{2}-1\right)$<br/>For rectangular hyperbola : $a=b$ $\Rightarrow b^{2}=b^{2}\left(e^{2}-1\right)$<br/>$\Rightarrow e^{2}-1=1$<br/>$\Rightarrow e^{2}=2 \Rightarrow e=\pm \sqrt{2}$<br/>For hyper bola, $\mathrm{e}&gt;1$<br/>Hence, $\mathrm{e}=\sqrt{2}$</div>


title: what-is-the-eccentricity-of-the-conic-4-x-2-9-y-2-144
slug: what-is-the-eccentricity-of-the-conic-4-x-2-9-y-2-144-96392
question: <div class="question">What is the eccentricity of the conic $4 x^{2}+9 y^{2}=144 ?$</div>
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{5}}{3} \quad$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{5}}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{\sqrt{5}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}$</span> </li> </ul>

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

answer:
<div class="solution">Given equation can be written as<br/>$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$<br/>Thisis an ellipse.<br/>$\Rightarrow a^{2}=36, b^{2}=16$<br/>$\quad e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{16}{36}}=\sqrt{\frac{20}{36}}=\frac{2 \sqrt{5}}{6}=\frac{\sqrt{5}}{3}$</div>


title: what-is-the-effect-of-denaturation-on-the-structure-of-proteins-1
slug: what-is-the-effect-of-denaturation-on-the-structure-of-proteins-1-99328
question: <div class="question">What is the effect of denaturation on the structure of proteins?</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Denaturation causes proteins to fold into their natural, functional conformation.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Denaturation has no effect on the structure of proteins.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">As a result of denaturation, globules get unfolded and helixes get uncoiled. Secondary and tertiary structures of protein are destroyed, but the primary structures remain unaltered</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Denaturation promotes the formation of more complex protein structures.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">As a result of denaturation, globules get unfolded and helixes get uncoiled. Secondary and tertiary structures of protein are destroyed, but the primary structures remain unaltered</span> </div>

answer:
<div class="solution">Correct Option is : (C)<br/>As a result of denaturation, globules get unfolded and helixes get uncoiled. Secondary and tertiary structures of protein are destroyed, but the primary structures remain unaltered</div>


title: what-is-the-effluent-from-the-primary-settling-tank-used-for
slug: what-is-the-effluent-from-the-primary-settling-tank-used-for-34060
question: <div class="question">What is the effluent from the primary settling tank used for?</div>
tags: ['Biology', 'Microbes in Human Welfare', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Primary treatment</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Secondary treatment</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Physical removal of particles</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Increase in pollution potential</span> </li> </ul>

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

answer:
<div class="solution">The effluent from the primary settling tank is taken for secondary treatment.</div>


tags: ['Chemistry', 'Electrochemistry', 'JIPMER', 'JIPMER 2009']
source_db: MarksBatch2_P3.db
title:
what-is-the-electrode-potential-in-v-of-the-following-electrode-at-2-5-c-ni-2-01-m-ni-s-standard-reaction-potential-of-ni-2-ni-is-025-v-f-2303-rt-006

slug:
what-is-the-electrode-potential-in-v-of-the-following-electrode-at-2-5-c-ni-2-01-m-ni-s-standard-reaction-potential-of-ni-2-ni-is-025-v-f-2303-rt-006-12845

question:
<div class="question">What is the electrode potential (in V ) of the following electrode at $25^{\circ} \mathrm{C}$ ?<br/>$\mathrm{Ni}^{2+}(0.1 \mathrm{M}) \mid \mathrm{Ni}(s)$<br/>(Standard reaction potential of $\mathrm{Ni}^{2+} \mid \mathrm{Ni}$ is $-0.25 \mathrm{~V}, \frac{2.303 R T}{F}=0.06$ )</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-0.28 V$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-0.34 V$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-0.82 V$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-0.22 V$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">-0.28 V$</span> </div>

answer:
<div class="solution">$\begin{aligned} E_{\text {cell }}^{\circ} &amp; =E_{\text {cell }}^{\circ}-\frac{2.303 R T}{n F} \log \left[\frac{[\mathrm{Ni}]}{\left[\mathrm{Ni}^{2+}\right]}\right] \\ &amp; =-0.25-\frac{0.06}{2} \log \left(\frac{1}{0.1}\right) \quad(\because n=2) \\ &amp; =-0.25-0.03 \times 1 \\ E_{\text {cell }} &amp; =-0.28 \mathrm{~V}\end{aligned}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-the-enthalpy-change-that-accompanies-the-melting-of-one-mole-of-a-solid-substance-in-its-standard-state-called

slug:
what-is-the-enthalpy-change-that-accompanies-the-melting-of-one-mole-of-a-solid-substance-in-its-standard-state-called-16707

question:
<div class="question">What is the enthalpy change that accompanies the melting of one mole of a solid substance in its standard state called?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Standard enthalpy of fusion or molar enthalpy of fusion</span> </div>

answer:
<div class="solution">The enthalpy change that accompanies the melting of one mole of a solid substance in its standard state is called the standard enthalpy of fusion or molar enthalpy of fusion.</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-a-circle-whose-centre-lies-on-the-x-axis-at-a-distance-h-from-the-origin-and-the-circle-passes-through-the-origin

slug:
what-is-the-equation-of-a-circle-whose-centre-lies-on-the-x-axis-at-a-distance-h-from-the-origin-and-the-circle-passes-through-the-origin-91806

question:
<div class="question">What is the equation of a circle, whose centre lies on the $\mathrm{x}$ -axis at a distance $\mathrm{h}$ from the origin and the circle passes through the origin?</div>

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

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

answer:
<div class="solution">Centre of the circle is $(\mathrm{h}, 0)$ and circle passes through the origin. In the general equation of circle:<br/>$x^{2}+y^{2}+2 g x+2 f x+c=0$<br/>$\mathrm{g}=-\mathrm{h}$ and $\mathrm{f}=0$<br/>so, $x^{2}+y^{2}-2 h x+0+c$<br/>$=x^{2}+y^{2}-2 h x+c=0$<br/>since circle passes through origin $(0,0)$ $0+0-0+\mathrm{c}=0 \Rightarrow \mathrm{c}=0$<br/>and equation (i) radius to $x^{2}+y^{2}-2 h x=0$</div>


tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-a-curve-passing-through-0-1-and-whose-differential-equation-is-given-by-d-y-y-tan-x-d-x

slug:
what-is-the-equation-of-a-curve-passing-through-0-1-and-whose-differential-equation-is-given-by-d-y-y-tan-x-d-x-25921

question:
<div class="question">What is the equation of a curve passing through $(0,1)$ and whose differential equation is given by $d y=y \tan x d x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=\cos x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y=\sin x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y=\sec x$</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">$y=\operatorname{cosec} x$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{dy}=\mathrm{y} \tan \mathrm{x} \mathrm{d} \mathrm{x}$<br/>$\begin{aligned} \int \frac{\mathrm{dy}}{\mathrm{y}} &amp;=\int \tan \mathrm{x} \mathrm{d} \mathrm{x} \\ \log |\mathrm{y}| &amp;=\log |\sec \mathrm{x}|+\log |\mathrm{c}| \\ \log |\mathrm{y}| &amp;=\log |\mathrm{c} \sec \mathrm{x}| \\ \mathrm{y} &amp;=\operatorname{csec} \mathrm{x} \\ \text { at } \mathrm{x} &amp;=0, \mathrm{y}=1 \\ \mathrm{y} &amp;=\mathrm{c} \end{aligned}$<br/>Solution is given by <br/>$\mathrm{y}=\sec x$</div>


title: what-is-the-equation-of-a-line-parallel-to-x-axis-at-a-distance-of-5-units-below-xaxis
slug: what-is-the-equation-of-a-line-parallel-to-x-axis-at-a-distance-of-5-units-below-xaxis-23976
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$y=-5$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of a line parallel to $\mathrm{x}$ -axis at a distance of 5 units below x-axis?</div>

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

answer:
<div class="solution">Equation of a line parallel to $\mathrm{x}$ -axis at a distance of 5 units below $\mathrm{x}$ -axis is $y=-5$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/luUl6vszqjZF-SnLcuxJ_YTHRY50bFdBs_3arvr61Pc.original.fullsize.png"/></div>


title: what-is-the-equation-of-a-plane-through-the-x-axis-and-passing-through-the-point-1-2-3
slug: what-is-the-equation-of-a-plane-through-the-x-axis-and-passing-through-the-point-1-2-3-34450
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=1$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of a plane through the $x$ -axis and passing through the point $(1,2,3)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+y+z=6$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x=1$</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">$y+z=5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$z+y=1$</span> </li> </ul>

answer:
<div class="solution">The equation of plane which is passing through $x$ -axis<br/>is<br/>$x=a$<br/>This plane also passes through $(1,2,3)$ By putting $x=1$ in eqn of plane, we get $a=1$ Hence $x=1$ is required equation of plane.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-a-straight-line-which-passes-through-3-4-and-sum-of-whose-x-and-y-intercepts-is-14

slug:
what-is-the-equation-of-a-straight-line-which-passes-through-3-4-and-sum-of-whose-x-and-y-intercepts-is-14-36939

question:
<div class="question">What is the equation of a straight line which passes through $(3,4)$ and sum of whose $x$ and $y$ intercepts is 14 ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$4 x+3 y=24$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+y=14$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 4 x-3 y=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x+4 y=25$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 x+3 y=24$</span> </div>

answer:
<div class="solution">Equation of a straight line in intercept form is<br/>$\frac{x}{a}+\frac{y}{b}=1 \Rightarrow \frac{3}{a}+\frac{4}{b}=1$<br/>Given, $a+b=14$ On solving (i) and (ii) we get<br/>$\frac{3}{a}+\frac{4}{14-a}=1 \Rightarrow a^{2}-13 a+42=0$<br/>$\Rightarrow(a-7)(a-6)=0$<br/>$\Rightarrow \mathrm{a}=6$ and $\mathrm{b}=8$<br/>or $\mathrm{a}=7$ and $\mathrm{b}=7$.<br/>- Required eqns are $4 \mathrm{x}+3 \mathrm{y}=24$ or $\mathrm{x}+\mathrm{y}=1$.</div>


title: what-is-the-equation-of-circle-which-touches-the-lines-x-0-y-0-and-x-2
slug: what-is-the-equation-of-circle-which-touches-the-lines-x-0-y-0-and-x-2-66335
tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of circle which touches the lines $\mathrm{x}=0$, $\mathrm{y}=0$ and $\mathrm{x}=2 ?$</div>

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

answer:
<div class="solution">Refer to the figure it is clear that coordinates of centre of circle are $(1,1)$ and diameter of circle $=2$ and hence radius of circle is 1 .<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/a2q_FVJ6_iT94HvyEyKSn406Iq7OsUrvUm-AYgP8Ar8.original.fullsize.png"/><br/><br/>- Equation of circle with centre $(1,1)$ and radius $=1$ is $(x-1)^{2}+(y-1)^{2}=1$<br/>$\Rightarrow x^{2}-2 x+1+y^{2}-2 y+1=1$<br/>$\Rightarrow \quad x^{2}+y^{2}-2 x-2 y+1=0$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-line-passing-through-0-1-and-making-an-angle-with-the-yaxis-equal-to-the-inclination-of-the-line-x-y-4-with-x-axis

slug:
what-is-the-equation-of-line-passing-through-0-1-and-making-an-angle-with-the-yaxis-equal-to-the-inclination-of-the-line-x-y-4-with-x-axis-81325

question:
<div class="question">What is the equation of line passing through $(0,1)$ and making an angle with the y-axis equal to the inclination of the line $\mathrm{x}-\mathrm{y}=4$ with $\mathrm{x}$ -axis?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y=x+1$</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">$x=y+1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x=y+2$</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">$y=x+1$</span> </div>

answer:
<div class="solution">Given line is $\mathrm{x}-\mathrm{y}=4$<br/>slope $=1$ i.e. $\mathrm{m}=1$ Since required line passes through $(0,1)$ $\therefore \mathrm{y}-1=\mathrm{m}(\mathrm{x}-0)$<br/>$\Rightarrow \mathrm{y}-1=1(\mathrm{x}) \Rightarrow \mathrm{y}=\mathrm{x}+1$</div>


title: what-is-the-equation-of-parabola-whose-verted-is-at-0-0-and-focus-is-at-0-2
slug: what-is-the-equation-of-parabola-whose-verted-is-at-0-0-and-focus-is-at-0-2-93345
tags: ['Mathematics', 'Parabola', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of parabola whose verted is at $(0,0)$ and focus is at $(0,-2)$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y^{2}+8 x=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y^{2}-8 x=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x^{2}+8 y=0$</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">$x^{2}-8 y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+8 y=0$</span> </div>

answer:
<div class="solution">Focus is $(0,-2)$ $-2$ and parabola is<br/>$x^{2}=-8 y$<br/>$x^{2}=4 a y$<br/>$x^{2}+8 y=0$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-straight-line-passing-through-the-point-4-3-and-making-equal-intercepts-on-the-coordinate-axes

slug:
what-is-the-equation-of-straight-line-passing-through-the-point-4-3-and-making-equal-intercepts-on-the-coordinate-axes-79994

question:
<div class="question">What is the equation of straight line passing through the point $(4,3)$ and making equal intercepts on the coordinate axes?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x+y=7$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 x+4 y=7$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x-y=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+y=7$</span> </div>

answer:
<div class="solution">Let equation of line be $\frac{x}{a}+\frac{y}{a}=1$ or $x+y=a$ line passing through $(4,3)$, then $\mathrm{a}=0$ Required equation, $x+y=7$</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-circle-which-passes-through-the-points-3-2-and-2-0-and-having-its-centre-on-the-line-2-x-y-3-0-1

slug:
what-is-the-equation-of-the-circle-which-passes-through-the-points-3-2-and-2-0-and-having-its-centre-on-the-line-2-x-y-3-0-1-52146

question:
<div class="question">What is the equation of the circle which passes through the points $(3,-2)$ and $(-2,0)$ and having its centre on the line $2 x-y-3=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+3 x+2=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}+3 x+12 y+2=0$</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">C</span> <span class="option-data">$x^{2}+y^{2}+2 x=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}=5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+3 x+12 y+2=0$</span> </div>

answer:
<div class="solution">Given, centre of circle lies on line $2 x-y-3=0$ Let $x=h$ $2 h-y-3=0 \Rightarrow y=2 n-3$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/hTLPAYbZo8pB1M66fXEZeGlAoMcnVVtOs9n_bRPY8Os.original.fullsize.png"/><br/><br/>$\therefore$ centre of circle $=(\mathrm{h}, 2 \mathrm{~h}-3)$<br/>We know, OA $=$ OB $\Rightarrow(\mathrm{h}-3)^{2}+(2 \mathrm{~h}-3+2)^{2}=(\mathrm{h}+2)^{2}+(2 \mathrm{~h}-3)^{2}$<br/>$\Rightarrow \mathrm{h}^{2}-6 \mathrm{~h}+9+(2 \mathrm{~h}-1)^{2}=\mathrm{h}^{2}+4 h+4+4 \mathrm{~h}^{2}-12 \mathrm{~h}+9$<br/>$\Rightarrow y^{2}-6 h+9+4 h^{2}-4 h+1$<br/>$=h^{2}+4 h+4+\not 4 h^{2}-12 h+9$<br/>$\Rightarrow-10 h+10=-8 h+13$<br/>$\Rightarrow-2 h=3 \Rightarrow h=\frac{-3}{2}$<br/>$\therefore$ centre $=\left(\frac{-3}{2}, 2\left(\frac{-3}{2}\right)-3\right)=\left(\frac{-3}{2},-6\right)$<br/>$\therefore$ radius $=(\mathrm{h}+2)^{2}+(2 \mathrm{~h}-3)^{2}$<br/>$=\left(\frac{-3}{2}+2\right)^{2}+\left[2\left(\frac{-3}{2}\right)-3\right]^{2}=\frac{1}{4}+36$<br/>$\therefore$ Equation of circle<br/>$\Rightarrow\left(x+\frac{3}{2}\right)^{2}+(y+6)^{2}=\frac{1}{4}+36$<br/>$\Rightarrow x^{2}+y^{2}+3 x+12 y+2=0$</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-circle-which-passes-through-the-points-3-2-and-2-0-and-having-its-centre-on-the-line-2-x-y-3-0

slug:
what-is-the-equation-of-the-circle-which-passes-through-the-points-3-2-and-2-0-and-having-its-centre-on-the-line-2-x-y-3-0-64741

question:
<div class="question">What is the equation of the circle which passes through the points $(3,-2)$ and $(-2,0)$ and having its centre on the line $2 x-y-3=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+3 x+2=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}+3 x+12 y+2=0$</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">C</span> <span class="option-data">$x^{2}+y^{2}+2 x=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}=5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+3 x+12 y+2=0$</span> </div>

answer:
<div class="solution">Given, centre of circle lies on line $2 x-y-3=0$ Let $x=h$ $2 h-y-3=0 \Rightarrow y=2 n-3$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/hTLPAYbZo8pB1M66fXEZeGlAoMcnVVtOs9n_bRPY8Os.original.fullsize.png"/><br/><br/>$\therefore$ centre of circle $=(\mathrm{h}, 2 \mathrm{~h}-3)$<br/>We know, OA $=$ OB $\Rightarrow(\mathrm{h}-3)^{2}+(2 \mathrm{~h}-3+2)^{2}=(\mathrm{h}+2)^{2}+(2 \mathrm{~h}-3)^{2}$<br/>$\Rightarrow \mathrm{h}^{2}-6 \mathrm{~h}+9+(2 \mathrm{~h}-1)^{2}=\mathrm{h}^{2}+4 h+4+4 \mathrm{~h}^{2}-12 \mathrm{~h}+9$<br/>$\Rightarrow y^{2}-6 h+9+4 h^{2}-4 h+1$<br/>$=h^{2}+4 h+4+\not 4 h^{2}-12 h+9$<br/>$\Rightarrow-10 h+10=-8 h+13$<br/>$\Rightarrow-2 h=3 \Rightarrow h=\frac{-3}{2}$<br/>$\therefore$ centre $=\left(\frac{-3}{2}, 2\left(\frac{-3}{2}\right)-3\right)=\left(\frac{-3}{2},-6\right)$<br/>$\therefore$ radius $=(\mathrm{h}+2)^{2}+(2 \mathrm{~h}-3)^{2}$<br/>$=\left(\frac{-3}{2}+2\right)^{2}+\left[2\left(\frac{-3}{2}\right)-3\right]^{2}=\frac{1}{4}+36$<br/>$\therefore$ Equation of circle<br/>$\Rightarrow\left(x+\frac{3}{2}\right)^{2}+(y+6)^{2}=\frac{1}{4}+36$<br/>$\Rightarrow x^{2}+y^{2}+3 x+12 y+2=0$</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-curve-passing-through-the-origin-and-satisfying-the-differential-equation-d-y-y-tan-x-sec-x-d-x

slug:
what-is-the-equation-of-the-curve-passing-through-the-origin-and-satisfying-the-differential-equation-d-y-y-tan-x-sec-x-d-x-36131

question:
<div class="question">What is the equation of the curve passing through the origin and satisfying the differential equation $d y=(y \tan x+\sec x) d x ?$</div>

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

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

answer:
<div class="solution">The differential equation $\mathrm{dy}=(\mathrm{y} \tan \mathrm{x}+\sec \mathrm{x}) \mathrm{d} \mathrm{x}$ can be written as<br/>$\frac{d y}{d x}=y \tan x+\sec x$<br/>or, $\quad \frac{d y}{d x}-y \tan x=\sec x$<br/>which is of the form $\frac{d y}{d x}+P(x) \cdot y=Q(x)$ Here $\mathrm{P}(\mathrm{x})=-\tan \mathrm{x}$ and $\mathrm{Q}(\mathrm{x})=\sec \mathrm{x}$<br/>Integrating factor $\mathrm{IF}=\mathrm{e}^{\int \mathrm{P}(\mathrm{x}) \mathrm{dx}}$<br/>$I F=e^{\int-\tan x d x}=e^{\int-\frac{\sin x}{\cos x} d x}$<br/>Putting $\cos \mathrm{x}=\mathrm{t}$<br/>$-\sin x d x=d t$<br/>$I f=\mathrm{e}^{\int \frac{\mathrm{dt}}{\mathrm{t}}}=\mathrm{e}^{\log _{\mathrm{e}} \mathrm{t}}=\mathrm{t}=\cos \mathrm{x}$<br/>The solution is<br/>$\mathrm{y} \cdot \mathrm{Q}(\mathrm{x})=\int \mathrm{I} \mathrm{F} \cdot \mathrm{Q}(\mathrm{x}) \mathrm{d} \mathrm{x}+\mathrm{c}$<br/>or, $\mathrm{y} \cdot \sec \mathrm{x}=\int \cos \mathrm{x} \cdot \sec \mathrm{x} \mathrm{d} \mathrm{x}+\mathrm{c}$<br/>or, $\mathrm{y} \cdot \sec \mathrm{x}=\int \mathrm{d} \mathrm{x}+\mathrm{c}$<br/>or, $y \cdot \sec x=x+c$<br/>Since the curve passes through the origin. $0=0+\mathrm{c} \Rightarrow \mathrm{c}=0$<br/>and $y \sec x=x$<br/>or, $\mathrm{y}=\mathrm{x} \cos \mathrm{x}$</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-curve-passing-through-the-point-0-3-satisfying-the-differential-equation-sin-x-cos-y-d-x-cos-x-sin-y-d-y-0

slug:
what-is-the-equation-of-the-curve-passing-through-the-point-0-3-satisfying-the-differential-equation-sin-x-cos-y-d-x-cos-x-sin-y-d-y-0-70625

question:
<div class="question">What is the equation of the curve passing through the point $\left(0, \frac{\pi}{3}\right)$ satisfying the differential equation $\sin x \cos y d x+\cos x \sin y d y=0 ?$</div>

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

answer:
<div class="solution">Given differential equation is $\sin x \cos y d x+\cos x \sin y d y=0$<br/>$\Rightarrow \sin x \cos y d x=-\cos x \sin y d y$<br/>$\Rightarrow \frac{\sin x}{\cos x} d x=-\frac{\sin y}{\cos y} d y$<br/>Integrate on both side<br/>$\int \frac{\sin x}{\cos x} d x=-\int \frac{\sin y}{\cos y} d y$<br/>$\Rightarrow-\log (\cos x)=\log (\cos y)+\log c$<br/>where $\log c$ is constant of integration. $\Rightarrow-\log c=\log (\cos y)+\log (\cos x)$<br/>$\frac{1}{c}=\cos y \cos x$<br/>(1)<br/>Since, this curve passing through $\left(0, \frac{\pi}{3}\right)$ $\therefore$ it satisfies equation (1)<br/>So, $\frac{1}{c}=\cos \frac{\pi}{3} \cdot \cos 0$<br/>$\frac{1}{c}=\frac{1}{2} \times 1 \Rightarrow c=2$<br/>Hence, required equation of curve is $\cos x \cos y=\frac{1}{2}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-curve-whose-slope-at-any-point-is-equal-to-2-x-and-which-passes-through-the-origin

slug:
what-is-the-equation-of-the-curve-whose-slope-at-any-point-is-equal-to-2-x-and-which-passes-through-the-origin-26807

question:
<div class="question">What is the equation of the curve whose slope at any point is equal to $2 x$ and which passes through the origin?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y(1-x)=x^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y^{2}\left(1+x^{2}\right)=x^{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y^{2}=(x+1)^{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$y=x^{2}$</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">$y=x^{2}$</span> </div>

answer:
<div class="solution">Slope at the point $(x, y)=2 x$ $\therefore \frac{d y}{d x}=2 x$<br/>$\Rightarrow \int d y=\int 2 x d x$<br/>$\Rightarrow y=x^{2}+c$<br/>Given that, it passes through the origin. So, $0=(0)^{2}+c$<br/>$\Rightarrow c=0$<br/>Then, $y=x^{2}$</div>


title: what-is-the-equation-of-the-ellipse-having-foci-2-0-and-the-eccentricity-4-1
slug: what-is-the-equation-of-the-ellipse-having-foci-2-0-and-the-eccentricity-4-1-66430
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the ellipse having foci $(\pm 2,0)$ and<br/>the eccentricity $\frac{1}{4} ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{64}+\frac{y^{2}}{60}=1$</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">$\frac{x^{2}}{60}+\frac{y^{2}}{64}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x^{2}}{20}+\frac{y^{2}}{24}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x^{2}}{24}+\frac{y^{2}}{20}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x^{2}}{64}+\frac{y^{2}}{60}=1$</span> </div>

answer:
<div class="solution">foci: $(\pm 2,0), \mathrm{e}=\frac{1}{4}$<br/>$\mathrm{c}=2, \mathrm{e}=\frac{1}{4}=\frac{\mathrm{c}}{\mathrm{a}} \Rightarrow \frac{2}{\mathrm{a}}=\frac{1}{4} \Rightarrow \mathrm{a}=8$<br/>We know, $a^{2}-b^{2}=c^{2}$<br/>$\Rightarrow b^{2}=a^{2}-c^{2}=8^{2}-2^{2}=64-4=60$<br/>Eqn of ellipse $\Rightarrow \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$<br/>$\Rightarrow \frac{x^{2}}{64}+\frac{y^{2}}{60}=1$</div>


title: what-is-the-equation-of-the-ellipse-whose-vertices-are-5-0-and-foci-are-at-4-0
slug: what-is-the-equation-of-the-ellipse-whose-vertices-are-5-0-and-foci-are-at-4-0-12618
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the ellipse whose vertices are $(\pm 5,0)$ and foci are at $(\pm 4,0) ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$</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">$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$</span> </div>

answer:
<div class="solution">Given.<br/>vertices, $(5,0)$ and $(-5,0)$ foci $(4,0)$ and $(-4,0)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/jHRAlY3_AnrZWwZ5wb3ShGDEi2abGqZQxh9iW4_codU.original.fullsize.png"/><br/><br/>i.e., $\mathrm{ae}=4$<br/>$\Rightarrow \mathrm{e}=\frac{4}{\mathrm{a}}=\frac{4}{5}$<br/>we know, $b^{2}=a^{2}\left(1-e^{2}\right)$<br/>$=5^{2}\left(1-\left(\frac{4}{5}\right)^{2}\right)$<br/>$=25\left(\frac{25-16}{25}\right)=9$<br/>$\therefore \mathrm{a}^{2}=25, \mathrm{~b}^{2}=9$<br/>Equation of ellipse is $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$.</div>


title: what-is-the-equation-of-the-hyperbola-having-latus-rectum-and-eccentrieity-8-and-5-3-respectively
tags: ['Mathematics', 'Hyperbola', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-equation-of-the-hyperbola-having-latus-rectum-and-eccentrieity-8-and-5-3-respectively-16927

question:
<div class="question">What is the equation of the hyperbola having latus rectum<br/>and eccentrieity 8 and $\frac{3}{\sqrt{5}}$ respectively?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{25}-\frac{y^{2}}{20}=1$</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">$\frac{x^{2}}{40}-\frac{y^{2}}{20}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x^{2}}{40}-\frac{y^{2}}{30}=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x^{2}}{30}-\frac{y^{2}}{25}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x^{2}}{25}-\frac{y^{2}}{20}=1$</span> </div>

answer:
<div class="solution">Let the equation of hyperbola be $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$<br/>Latus rectum $=8=\frac{2 b^{2}}{a} \Rightarrow b^{2}=4 a$...(i)<br/>$\begin{aligned} &amp; \text { Also, } b^{2}=a^{2}\left(e^{2}-1\right) \\ \Rightarrow &amp; 4 a=a^{2}\left(e^{2}-1\right) \end{aligned} \quad[$ From(i)]<br/>$\Rightarrow 4 a=a^{2}\left[\left(\frac{3}{\sqrt{5}}\right)^{2}-1\right]$<br/>$\Rightarrow \quad a=5 \&amp; b^{2}=20$<br/>Equation is $\frac{x^{2}}{25}-\frac{y^{2}}{20}=1$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=y$</span> </div>
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-line-joining-the-origin-with-the-point-of-intersection-of-the-lines-4-x-3-y-12-and-3-x-4-y-12

slug:
what-is-the-equation-of-the-line-joining-the-origin-with-the-point-of-intersection-of-the-lines-4-x-3-y-12-and-3-x-4-y-12-98102

question:
<div class="question">What is the equation of the line joining the origin with the point of intersection of the lines $4 x+3 y=12$ and $3 x+4 y=12$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+y=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x-y=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 y=4 x$</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>

answer:
<div class="solution">The equation of given lines are<br/>$$<br/>4 x+3 y=12 ...(i)<br/>$$<br/>and $3 x+4 y=12$ ...(ii)<br/>On simplifying (i) and (ii), we get<br/>$x=\frac{12}{7}$ and $y=\frac{12}{7}$<br/>Point of intersection of given line is $\left(\frac{12}{7}, \frac{12}{7}\right)$. Hence, the equation of line passing through $(0,0)$ and $\left(\frac{12}{7}, \frac{12}{7}\right)$ is<br/>$\frac{y-0}{x-0}=\frac{\frac{12}{7}-0}{\frac{12}{7}-0} \Rightarrow y=x$</div>


title: what-is-the-equation-of-the-line-midway-between-the-lines-3-x-4-y-12-0-and-3-x-4-y-6
slug: what-is-the-equation-of-the-line-midway-between-the-lines-3-x-4-y-12-0-and-3-x-4-y-6-31732
tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the line mid-way between the lines $3 x-4 y+12=0$ and $3 x-4 y=6$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 x-4 y-9=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 x-4 y+9=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 3 x-4 y-3=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3 x-4 y+3=0$</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">$3 x-4 y+3=0$</span> </div>

answer:
<div class="solution">$3 x-4 y+12=0$ or $y=\frac{3}{4} x+3$<br/>$3 x-4 y=6 \quad$ or $\quad y=\frac{3}{4} x-\frac{3}{2}$<br/>Equation of line mid-way between these two lines $\mathrm{y}=\frac{3}{4} \mathrm{x}+\left(\frac{3-\frac{3}{2}}{2}\right)$<br/>$\mathrm{y}=\frac{3}{4} \mathrm{x}+\frac{3}{4}$<br/>$4 y=3 x+3$<br/>$3 x-4 y+3=0$</div>


title: what-is-the-equation-of-the-line-passing-through-2-3-and-parallel-to-yaxis
slug: what-is-the-equation-of-the-line-passing-through-2-3-and-parallel-to-yaxis-23286
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$X=2$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the line passing through $\begin{array}{ll}(2,-3) \text { and parallel to Y-axis? } &amp; {}\end{array}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad Y=-3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad Y=2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$X=2$</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">$X=-3$</span> </li> </ul>

answer:
<div class="solution">Since required line is parallel to $y$ -axis therefore its equation is $x=2 .(\because$ Required line passes through $(2,-3))$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-line-passing-through-the-point-of-intersection-of-the-lines-x-2-y-3-0-and-2-x-y-5-0-and-parallel-to-the-line-y-x-10-0

slug:
what-is-the-equation-of-the-line-passing-through-the-point-of-intersection-of-the-lines-x-2-y-3-0-and-2-x-y-5-0-and-parallel-to-the-line-y-x-10-0-89790

question:
<div class="question">What is the equation of the line passing through the point of intersection of the lines $\mathrm{x}+2 \mathrm{y}-3=0$ and $2 \mathrm{x}-\mathrm{y}+5=0$ and parallel to the line $\mathrm{y}-\mathrm{x}+10=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7 x-7 y+18=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 x-7 y+18=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$5 x-5 y+18=0$</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">$x-y+5=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 x-5 y+18=0$</span> </div>

answer:
<div class="solution">Given lines,<br/>$x+2 y-3=0$ ...(i)<br/>$2 x-y+5=0$ ...(ii)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Jcp5ysqaHfs3qUrTOdfMa6eLfOXb3XzrAvLuR8c6uyE.original.fullsize.png"/><br/><br/>(1) $\Rightarrow \mathrm{x}+2\left(\frac{11}{5}\right)-3=0$<br/>$\Rightarrow x+\frac{22}{5}-3=0 \Rightarrow x+\frac{7}{5}=0 \Rightarrow x=\frac{-7}{5}$<br/>Given, the required line is parallel to $\mathrm{y}-\mathrm{x}+10=0$. $\Rightarrow \mathrm{y}=\mathrm{x}-10$<br/>$\Rightarrow \mathrm{y}=(1) \mathrm{x}+(-10)$<br/>$\therefore$ slope $(\mathrm{m})=1$<br/>- Required line with slope 1 and passing through<br/>$\left(\frac{-7}{5}, \frac{11}{5}\right)$ is<br/>$\mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right)$<br/>$\Rightarrow \mathrm{y}-\frac{11}{5}=1\left(\mathrm{x}+\frac{7}{5}\right)$<br/>$\Rightarrow 5 \mathrm{y}-11=5 \mathrm{x}+7 \Rightarrow 5 \mathrm{x}-5 \mathrm{y}+18=0$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-line-through-1-2-so-that-the-segment-of-the-line-intercepted-between-the-axes-is-bisected-at-this-point

slug:
what-is-the-equation-of-the-line-through-1-2-so-that-the-segment-of-the-line-intercepted-between-the-axes-is-bisected-at-this-point-34537

question:
<div class="question">What is the equation of the line through $(1,2)$ so that the segment of the line intercepted between the axes is bisected at this point $?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 2 x-y=4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x-y+4=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 x+y=4$</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">$2 x+y+4=0$</span> </li> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/t5E0TCXJDT12WlqyDhZyRhSckyp63EoJkQ9Kqtz-i-c.original.fullsize.png"/><br/><br/>$\frac{0+\mathrm{x}}{2}=1 ; \frac{0+\mathrm{y}}{2}=2$<br/>$x=2 ; y=4$<br/>Equation of line passing through $(2,0)$ and $(0,4)$<br/>$\mathrm{y}-0=\frac{4-0}{0-2}(\mathrm{x}-2)$<br/>$y=-2 x+4$<br/>$2 x+y=4$</div>


title: what-is-the-equation-of-the-line-which-passes-through-4-5-and-is-perpendicular-to-3-x4-y50
slug: what-is-the-equation-of-the-line-which-passes-through-4-5-and-is-perpendicular-to-3-x4-y50-48217
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the line which passes through $(4,-5)$ and is perpendicular to $3 x+4 y+5=0 ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad 4 x-3 y-31=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 x-4 y-41=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 4 x+3 y-1=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x+4 y+8=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 4 x-3 y-31=0$</span> </div>

answer:
<div class="solution">$3 \mathrm{x}+4 \mathrm{y}+5=0$ or $\mathrm{y}=\frac{-3}{4} \mathrm{x}+\frac{-5}{4}$<br/>Slope $=\frac{-3}{4}$<br/>Slope of required line, $\mathrm{m}=\frac{-1}{-\frac{3}{4}}=\frac{4}{3}$<br/>Also line passes through $(4,-5)$<br/>Equation of line, $\mathrm{y}+5=\frac{4}{3}(\mathrm{x}-4)$<br/>$\Rightarrow 3 y+15=4 x-16$<br/>$\Rightarrow 4 x-3 y-31=0$</div>


tags: ['Mathematics', 'Parabola', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-parabola-whose-vertex-and-focus-are-on-the-x-axis-at-distance-a-and-b-from-the-origin-respectively-b-a-0

slug:
what-is-the-equation-of-the-parabola-whose-vertex-and-focus-are-on-the-x-axis-at-distance-a-and-b-from-the-origin-respectively-b-a-0-92361

question:
<div class="question">What is the equation of the parabola, whose vertex and focus are on the $x$ -axis at distance a and b from the origin respectively? ( $\mathrm{b}&gt;\mathrm{a}&gt;0$ )</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y^{2}=4(b-a)(x-a)$</span> </div>

answer:
<div class="solution">The parabola's vertex and focus lie on $x$ -axis at points $(\mathrm{a}, 0)$ and $(\mathrm{b}, 0)$. Vertex and focus lie on the $\mathrm{x}$ -axis hence. the axis of parabola is $x$ -axis. Equation of parabola Vertex whose is a point $\left(x_{1}, y_{1}\right)$ then is $\left(y-y_{1}\right)^{2}=4 k\left(x-x_{1}\right)$<br/>$\mathrm{So}, \mathrm{y}_{1}=0$ and $\mathrm{x}_{1}=\mathrm{a}$ and $\mathrm{k}=$ distance between focus and vertex $=(\mathrm{b}-\mathrm{a})$ so the equation is $(y-0)^{2}=4(b-a) \cdot(x-a)$<br/>i.e., $y^{2}=4(b-a)(x-a)$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-plane-passing-through-the-point-1-1-1-and-perpendicular-to-each-of-the-planes-x-2-y-8-z-0-and-2-x-5-y-z-0

slug:
what-is-the-equation-of-the-plane-passing-through-the-point-1-1-1-and-perpendicular-to-each-of-the-planes-x-2-y-8-z-0-and-2-x-5-y-z-0-56186

question:
<div class="question">What is the equation of the plane passing through the point $(1,-1,-1)$ and perpendicular to each of the planes $x-2 y-8 z=0$ and $2 x+5 y-z=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 7 x-3 y+2 z=14$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x+5 y-3 z=12$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x-7 y+3 z=4$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$14 x-5 y+3 z=16$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$14 x-5 y+3 z=16$</span> </div>

answer:
<div class="solution">Since, the required plane passing through $(1,-1,-1)$ therefore only equation given in option 'd' satisfied by the point $(1,-1,-1)$<br/>Hence, Required equation of plane is $14 x-5 y+3 z=16$</div>


title: what-is-the-equation-of-the-plane-passing-through-the-points-2-6-6-3-10-9-and-5-0-6
slug: what-is-the-equation-of-the-plane-passing-through-the-points-2-6-6-3-10-9-and-5-0-6-35165
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the plane passing through the points $(-2,6,-6),(-3,10,-9)$ and $(-5,0,-6) ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad 2 x-y-2 z=2$</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">$2 x+y+3 z=3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x+y+z=6$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y-z=3$</span> </li> </ul>

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

answer:
<div class="solution">We know, the equation of plane passing thr<br/>points $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)$ and $\left(x_{3}, y_{3}, z_{3}\right)$ is<br/>So, the plane passing through points $(-2,6,-6),(-3$<br/>$10,-9)$ and $(-5,0,-6)$ is $\left|\begin{array}{ccc}x+2 &amp; y-6 &amp; z+6 \\ -1 &amp; 4 &amp; -3 \\ -3 &amp; -6 &amp; 0\end{array}\right|=0$<br/>$\Rightarrow(x+2)(-18)-(y-6)(-9)+(z+6)(6+12)=10$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-plane-passing-through-x-1-y-1-z-1-and-normal-to-the-line-with-a-b-c-as-direction-ratios

slug:
what-is-the-equation-of-the-plane-passing-through-x-1-y-1-z-1-and-normal-to-the-line-with-a-b-c-as-direction-ratios-78224

question:
<div class="question">What is the equation of the plane passing through $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and normal to the line with $   &lt; \mathrm{a}, \mathrm{b}, \mathrm{c}&gt;$ as direction<br/>ratios?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$a x+b y+c z=a x_{1}+b y_{1}+c z_{1}$</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">$a\left(x+x_{1}\right)+b\left(y+y_{1}\right)+c\left(z+z_{1}\right)=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a x+b y+c z=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a x+b y+c z=x_{1}+y_{1}+z_{1}=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a x+b y+c z=a x_{1}+b y_{1}+c z_{1}$</span> </div>

answer:
<div class="solution">The equation of the plane passing through $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and normal to the line with $   &lt; \mathrm{a}, \mathrm{b}, \mathrm{c}&gt;$ as direction ratios,<br/>is given by $a\left(x-x_{1}\right)+b\left(y-y_{1}\right)+c\left(z-z_{1}\right)=0$<br/>$a x-a x_{1}+b y-b y_{1}+c z-c z_{1}=0$<br/>$\Rightarrow \quad a x+b y+c z=a x_{1}+b y_{1}+c z_{1}$<br/>which is required equation of plane.</div>


title: what-is-the-equation-of-the-plane-through-z-axis-and-parallel
slug: what-is-the-equation-of-the-plane-through-z-axis-and-parallel-14586
question: <div class="question">What is the equation of the plane through $z$ -axis and parallel</div>
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x \cot \theta+y=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x \tan \theta-y=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x+y \cot \theta=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y \tan \theta=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x \tan \theta-y=0$</span> </div>

answer:
<div class="solution">Let equation of plane through z-axis is<br/>$a x+b y=0$<br/>It is given that this plane is parallel to the line<br/>$\frac{x-1}{\cos \theta}=\frac{y+2}{\sin \theta}=\frac{z-3}{0}$<br/>Since the plane parallel to the line<br/>$a \cos \theta+b \sin \theta=0$<br/>$\Rightarrow a \cos \theta=-b \sin \theta \Rightarrow a=-b \tan \theta$<br/>$-b \tan \theta x+b y=0$<br/>$\Rightarrow x \tan \theta-y=0(\because b \neq 0)$<br/>which is required equation of plane.</div>


title: what-is-the-equation-of-the-right-bisector-of-the-line-segment-joining-1-1-and-2-3
slug: what-is-the-equation-of-the-right-bisector-of-the-line-segment-joining-1-1-and-2-3-32826
tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the right bisector of the line segment joining $(1,1)$ and $(2,3)$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 x+4 y-11=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x-4 y-5=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x-4 y-11=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y+1=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 x+4 y-11=0$</span> </div>

answer:
<div class="solution">Equation of given line is $(y-3)=\left(\frac{3-1}{2-1}\right)(x-2)$<br/>$\Rightarrow y=2 x-1$<br/>Slope $m_{1}=2$<br/>and slope of perpendicular $=-\frac{1}{2}$ The perpendicular is also bisector, therefore it will pass through its mid-point. $\Rightarrow$ Coordinates of mid-point of given line are :<br/>$\left(\frac{2+1}{2}, \frac{3+1}{2}\right)$ or $\left(\frac{3}{2}, 2\right)$<br/>So, equation of perpendicular bisector is:<br/>$(y-2)=-\frac{1}{2}\left(x-\frac{3}{2}\right)$<br/>$\Rightarrow 2 x+4 y-11=0$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-sphere-which-has-its-centre-at-6-1-2-and-touches-the-plane-2-x-y-2-z-2-0

slug:
what-is-the-equation-of-the-sphere-which-has-its-centre-at-6-1-2-and-touches-the-plane-2-x-y-2-z-2-0-31983

question:
<div class="question">What is the equation of the sphere which has its centre at $(6,-1,2)$ and touches the plane $2 x-y+2 z-2=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+z^{2}+12 x-2 y+4 z+16=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+y^{2}+z^{2}+12 x-2 y+4 z-16=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}+z^{2}-12 x+2 y-4 z+16=0$</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">$x^{2}+y^{2}+z^{2}-12 x+2 y-4 z+25=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+z^{2}-12 x+2 y-4 z+16=0$</span> </div>

answer:
<div class="solution">Given centre of sphere is $(6,-1,2)$ and $e q^{n}$ of plane is $2 x-y+2 z-2=0$<br/>Since, sphere touches the plane therefore $\perp$ distance from centre to the plane is radius of the sphere.<br/>$\therefore$ Radius $=\frac{2(6)-1(-1)+2(2)-2}{\sqrt{4+1+4}}=\frac{15}{3}=5$<br/>$\therefore$ Required equation of sphere is $(x-6)^{2}+(y+1)^{2}+(z-2)^{2}=5^{2}$<br/>$\Rightarrow x^{2}+y^{2}+z^{2}-12 x+2 y-4 z+16=0$</div>


title: what-is-the-equation-of-the-sphere-with-unit-radius-having-centre-at-the-origin
slug: what-is-the-equation-of-the-sphere-with-unit-radius-having-centre-at-the-origin-62106
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation of the sphere with unit radius having centre at the origin?</div>

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

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

answer:
<div class="solution">$\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}+\left(z-z_{1}\right)^{2}=r^{2}$<br/>Centre $(0,0,0)$ and radius $=1$ $(x-0)^{2}+(y-0)^{2}+(z-0)^{2}=(1)^{2}$<br/>$x^{2}+y^{2}+z^{2}=1$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-straight-line-cutting-off-an-intercept-2-from-the-negative-direction-of-y-axis-and-inclined-at-3-0-with-the-positive-direc

slug:
what-is-the-equation-of-the-straight-line-cutting-off-an-intercept-2-from-the-negative-direction-of-y-axis-and-inclined-at-3-0-with-the-positive-direc-84256

question:
<div class="question">What is the equation of the straight line cutting off an intercept 2 from the negative direction of $y$ -axis and inclined at $30^{\circ}$ with the positive direction of $\mathrm{x}$ -axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x-2 \sqrt{3} y-3 \sqrt{2}=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+2 \sqrt{3} y-3 \sqrt{2}=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x+\sqrt{3} y-2 \sqrt{3}=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x-\sqrt{3} y-2 \sqrt{3}=0$</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">$x-\sqrt{3} y-2 \sqrt{3}=0$</span> </div>

answer:
<div class="solution">Given, angle $(\theta)=30^{\circ}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/HjYw5vu_MtVAt_RlgO21GCt-_iBXSXFXkt1N8xY6WwU.original.fullsize.png"/><br/><br/>$\therefore \mathrm{m}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$<br/>Point $=(-2,0)$ i.e., $\mathrm{C}=-2$. Slope-intercept form is $\mathrm{y}=\mathrm{mn}+\mathrm{c}$<br/>$\Rightarrow \mathrm{y}=\frac{1}{\sqrt{3}} \mathrm{x}-2$<br/>$\Rightarrow \sqrt{3} \mathrm{y}=\mathrm{x}-2 \sqrt{3}$<br/>$\Rightarrow x-\sqrt{3} y-2 \sqrt{3}=0$</div>


title: what-is-the-equation-of-the-straight-line-parallel-to-2-x-3-y-1-0-and-passes-through-the-point-1-2
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-equation-of-the-straight-line-parallel-to-2-x-3-y-1-0-and-passes-through-the-point-1-2-50155

question:
<div class="question">What is the equation of the straight line parallel to $2 \mathrm{x}+3 \mathrm{y}$ $+1=0$ and passes through the point $(-1,2) ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 x+3 y-4=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x+3 y-5=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x+y-1=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x-2 y+7=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 x+3 y-4=0$</span> </div>

answer:
<div class="solution">The equation of line parallel to $2 x+3 y+1=0$ is $2 x+3 y+K=0$<br/>It is passing through point $(-1,2)$ $\therefore 2(-1)+3(2)+\mathrm{K}=0$<br/>$\Rightarrow-2+6+\mathrm{K}=0 \Rightarrow \mathrm{K}=-4$<br/>Eqn. is $2 \mathrm{x}+3 \mathrm{y}-4=0$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-straight-line-passing-through-the-point-2-3-and-making-an-intercept-on-the-positive-yaxis-equal-to-twice-its-intercept-on-

slug:
what-is-the-equation-of-the-straight-line-passing-through-the-point-2-3-and-making-an-intercept-on-the-positive-yaxis-equal-to-twice-its-intercept-on-84758

question:
<div class="question">What is the equation of the straight line passing through the point $(2,3)$ and making an intercept on the positive y-axis equal to twice its intercept on the positive x-axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 2 x+y=5$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 x+y=7$</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">C</span> <span class="option-data">$x+2 y=7$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x-y=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 x+y=7$</span> </div>

answer:
<div class="solution">Given line passes through $(2,3)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/Se2w-9987ziF5hkO1Ulidwq_k3BRM63azbBRsTGN94Q.original.fullsize.png"/><br/><br/>Intercept form: $\frac{x}{a}+\frac{y}{b}=1$<br/>$\Rightarrow \frac{x}{a}+\frac{y}{2 a}=1$<br/>$\Rightarrow 2 \mathrm{x}+\mathrm{y}=2 \mathrm{a}$<br/>But this passes through $(2,3)$ $\therefore 2 \mathrm{a}=2(2)+3=7$<br/>$\Rightarrow a=\frac{7}{2}$<br/>$\therefore$ Equation of line is $2 \mathrm{x}+\mathrm{y}=2\left(\frac{7}{2}\right)$<br/>$\Rightarrow 2 x+y=7$</div>


title: what-is-the-equation-of-the-straight-line-which-is-perpendicular-to-y-x-and-passes-through-3-2
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-equation-of-the-straight-line-which-is-perpendicular-to-y-x-and-passes-through-3-2-48181

question:
<div class="question">What is the equation of the straight line which is perpendicular to $\mathrm{y}=\mathrm{x}$ and passes through $(3,2) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x-y=5$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x+y=5$</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">$x+y=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y=1$</span> </li> </ul>

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

answer:
<div class="solution">Given line, $\mathrm{y}=\mathrm{x}$ $\Rightarrow x-y=0$<br/>Slope of this line $=\frac{-1}{-1}=1$ Slope of line perpendicular to this line $=-1$ The perpendicular line passes through $(3,2)$ $\therefore$ Equation is $\mathrm{y}-2=-1(\mathrm{x}-3) \Rightarrow \mathrm{y}-2=-\mathrm{x}+3$<br/>$\Rightarrow x+y-5=0 \Rightarrow x+y=5$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-of-the-straight-line-which-passes-through-the-point-of-intersection-of-the-straight-lines-x-2-y-5-and-3-x-7-y-17-and-is-perpendic

slug:
what-is-the-equation-of-the-straight-line-which-passes-through-the-point-of-intersection-of-the-straight-lines-x-2-y-5-and-3-x-7-y-17-and-is-perpendic-79147

question:
<div class="question">What is the equation of the straight line which passes through the point of intersection of the straight lines $\mathrm{x}+2 \mathrm{y}=5$ and $3 \mathrm{x}+7 \mathrm{y}=17$ and is perpendicular to the straight line $3 \mathrm{x}+4 \mathrm{y}=10 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 x+3 y+2=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 x-y+2=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 x-3 y-2=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 x-3 y+2=0$</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">$4 x-3 y+2=0$</span> </div>

answer:
<div class="solution">Intersecting lines are $: x+2 y=5 \&amp; 3 x+7 y=17$ On solving these we get: $x=1$ \&amp; $y=2$ Equation of perpendicular line is<br/>$3 x+4 y=10$ or $y=\frac{-3}{4} x+10$<br/>So, slope $=\frac{-3}{4}$<br/>$\Rightarrow$ Slope of required line $=\frac{4}{3}$ Equation of given line is<br/>$(y-2)=\frac{4}{3}(x-1)$ or $4 x-3 y+2=0$</div>


title: what-is-the-equation-to-circle-which-touches-both-the-axes-and-has-centre-on-the-line-x-y-4
slug: what-is-the-equation-to-circle-which-touches-both-the-axes-and-has-centre-on-the-line-x-y-4-63658
tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation to circle which touches both the axes and has centre on the line $x+y=4$ ?</div>

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

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

answer:
<div class="solution">The equation of circle, which touches both the axes, is given by<br/>$$<br/>x^{2}+y^{2}-2 r x-2 r y+r^{2}=0<br/>$$<br/>$\ldots \ldots . .(\mathrm{i})$<br/>Now, the centre $(r, r)$ of this circle lies on the line<br/>$$<br/>\begin{array}{l}<br/>x+y=4 \\<br/>r+r=4 \Rightarrow r=2<br/>\end{array}<br/>$$<br/>$\therefore \quad$ Put value of $r$ in Eq. (i), we get<br/>$$<br/>x^{2}+y^{2}-4 x-4 y+4=0<br/>$$<br/>which is required equation of circle,</div>


title: what-is-the-equation-to-the-plane-through-1-2-3-parallel-to-3-x-4-y-5-z-0
slug: what-is-the-equation-to-the-plane-through-1-2-3-parallel-to-3-x-4-y-5-z-0-99755
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation to the plane through $(1,2,3)$ parallel to $3 x+4 y-5 z=0 ? \quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 x+4 y+5 z+4=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad 3 x+4 y-5 z+14=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad 3 x+4 y-5 z+4=0$</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">$3 x+4 y-5 z-4=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 3 x+4 y-5 z+4=0$</span> </div>

answer:
<div class="solution">The equation of a plane parallel to the plane $3 x+4 y-5 z=0$ is given by,<br/>$$<br/>3 x+4 y-5 z=d<br/>$$<br/>....(i) Sinceplane (i) passes through $(1,2,3)$ then, $3+8-15=\mathrm{d}$ $\Rightarrow \mathrm{d}=-4$<br/>$\therefore$ from(i)<br/>$$<br/>3 x+4 y-5 z+4=0<br/>$$</div>


