# questions

title: what-is-the-equation-to-the-sphere-whose-centre-is-at-2-3-4-and-radius-is-6-units
slug: what-is-the-equation-to-the-sphere-whose-centre-is-at-2-3-4-and-radius-is-6-units-48152
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation to the sphere whose centre is at $(-2,3,4)$ and radius is 6 units?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}+z^{2}+4 x-6 y-8 z=7$</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">$x^{2}+y^{2}+z^{2}+6 x-4 y-8 z=7$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}+z^{2}+4 x-6 y-8 z=4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}+z^{2}+4 x+6 y+8 z=4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+y^{2}+z^{2}+4 x-6 y-8 z=7$</span> </div>

answer:
<div class="solution">Given, centre of sphere $(\mathrm{h}, \mathrm{k}, 1)=(-2,3,4)$ radius $(\mathrm{r})=6$ units. Equation of sphere is $(\mathrm{x}-\mathrm{h})^{2}+(\mathrm{y}-\mathrm{k})^{2}+(\mathrm{z}-\ell)^{2}=\mathrm{r}^{2}$<br/>$\Rightarrow(x+2)^{2}+(y-3)^{2}+(z-4)^{2}=6^{2}$<br/>$\Rightarrow x^{2}+4 x+4+y^{2}-6 y+9+z^{2}-8 z+16=36$<br/>$\Rightarrow x^{2}+y^{2}+z^{2}+4 x-6 y-8 z=7$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-equation-to-the-straight-line-joining-the-origin-to-the-point-of-intersection-of-the-lines-a-x-b-y-1-and-b-x-a-y-1

slug:
what-is-the-equation-to-the-straight-line-joining-the-origin-to-the-point-of-intersection-of-the-lines-a-x-b-y-1-and-b-x-a-y-1-87316

question:
<div class="question">What is the equation to the straight line joining the origin<br/>to the point of intersection of the lines $\frac{x}{a}+\frac{y}{b}=1$ and<br/>$\frac{x}{b}+\frac{y}{a}=1 ?$</div>

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

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

answer:
<div class="solution">We know that the equation of straight line passing<br/>through the intersection point of two lines $\frac{x}{a}+\frac{y}{b}=1$<br/>and $\frac{x}{b}+\frac{y}{a}=1$, is<br/>$$<br/>\left(\frac{x}{a}+\frac{y}{b}-1\right)+\lambda\left(\frac{x}{b}+\frac{y}{a}-1\right)=0 ...(i)<br/>$$<br/>This line passes through the origin. $\therefore \quad(0-1)+\lambda(0-1)=0 \Rightarrow \lambda=-1$<br/>On putting the value of $\lambda$ in Eq. (i), we get<br/>$$<br/>\begin{aligned}<br/>&amp;\left(\frac{x}{a}+\frac{y}{b}-1\right)-1\left(\frac{x}{b}+\frac{y}{a}-1\right)=0 \\<br/>\Rightarrow &amp; \frac{x}{a}+\frac{y}{b}-1-\frac{x}{b}-\frac{y}{a}+1=0 \\<br/>\Rightarrow &amp; x\left(\frac{1}{a}-\frac{1}{b}\right)-y\left(\frac{1}{a}-\frac{1}{b}\right)=0 \\<br/>\Rightarrow &amp; x-y=0<br/>\end{aligned}<br/>$$</div>


title: what-is-the-equation-to-the-straight-line-passing-through-5-2-and-4-7
slug: what-is-the-equation-to-the-straight-line-passing-through-5-2-and-4-7-32162
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equation to the straight line passing through $(5,-2)$ and $(-4,7) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 5 x-2 y=4$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-4 x+7 y=9$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x+y=3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x-y=-1$</span> </li> </ul>

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

answer:
<div class="solution">Equation of line<br/>$\mathrm{y}+2=\frac{7+2}{-4-5}(\mathrm{x}-5)$<br/>$\Rightarrow \mathrm{y}+2=-\mathrm{x}+5$<br/>$\Rightarrow x+y=3$</div>


title: what-is-the-equation-to-the-straight-line-passing-through-a-b-c-and-parallel-to-zaxis
slug: what-is-the-equation-to-the-straight-line-passing-through-a-b-c-and-parallel-to-zaxis-90156
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 straight line passing through $(\mathrm{a}, \mathrm{b}, \mathrm{c})$ and parallel to z-axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{x-a}{1}=\frac{y-b}{0}=\frac{z-c}{0}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}$</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">$\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{0}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}$</span> </div>

answer:
<div class="solution">We know that dr's of z-axis are $(0,0,1)$ So, dr's of the required line are 0,0 and 1<br/>Now, equation of the line passing through $(a, b, c)$ and having dr's 0,0 and 1 is<br/>$\frac{\mathrm{x}-\mathrm{a}}{0}=\frac{\mathrm{y}-\mathrm{b}}{0}=\frac{\mathrm{z}-\mathrm{c}}{1}$</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-the-equilibrium-constant-k-c-for-the-reaction-cus-2agaq-cu2aq-2ags-if-the-standard-cell-potential-eo-cell-is-046-v

slug:
what-is-the-equilibrium-constant-k-c-for-the-reaction-cus-2agaq-cu2aq-2ags-if-the-standard-cell-potential-eo-cell-is-046-v-68248

question:
<div class="question">What is the equilibrium constant (K<sub>c</sub>)  for the reaction: Cu(s) + 2Ag+(aq) â Cu2+(aq) + 2Ag(s) if the standard cell potential (Eo<sub>(cell)</sub> is 0.46 V)?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.46</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3.19</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1015 Ã 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">3.92 Ã 1015</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">3.92 Ã 1015</span> </div>

answer:
<div class="solution">Eo<sub>(cell) </sub>= 0.059V Ã· 2 log K<sub>C</sub> = 0.46 or log K<sub>C</sub>=  (0.46V x 2) Ã· 0.059V = 15.6 <sub>Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  </sub>K<sub>C </sub>= 3.92  x 1015</div>


title: what-is-the-equivalent-binary-number-of-the-decimal-number-13625
slug: what-is-the-equivalent-binary-number-of-the-decimal-number-13625-30305
question: <div class="question">What is the equivalent binary number of the decimal number $13.625$ ?</div>
tags: ['Mathematics', 'Sets and Relations', '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">$1101.111$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1111.101$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1101.101$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$1111.111$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1101.101$</span> </div>

answer:
<div class="solution">$13.625$<br/>\begin{array}{c|c|c}<br/>2 &amp; 13 &amp; \\<br/>\hline 2 &amp; 6 &amp; 1 \\<br/>\hline 2 &amp; 3 &amp; 0 \\<br/>\hline &amp; 1 &amp; 1<br/>\end{array}<br/>$0.625 \times 2=1.250$<br/>ot $0.5 \times 2=1$<br/>$0.250 \times 2=0.5$<br/>1<br/>0<br/>1 $13.625=1101.101$</div>


title: what-is-the-equivalent-definition-of-the-function-given-by-f-x-2-x-x-0-0-x-0
slug: what-is-the-equivalent-definition-of-the-function-given-by-f-x-2-x-x-0-0-x-0-19363
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the equivalent definition of the function given by<br/>$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}2 \mathrm{x}, \mathrm{x} \geq 0 \\ 0, \mathrm{x}   &lt; 0\end{array} ?\right.$</div>

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

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

answer:
<div class="solution">The given function is $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}2 \mathrm{x}, \mathrm{x} \geq 0 \\ 0, \mathrm{x}   &lt; 0\end{array}\right.$<br/>The equation can be re-written as<br/>$f(x)=\left\{\begin{array}{ll}x+x, &amp; x \geq 0 \\ -x+x &amp; x &lt; 0\end{array}\right.$<br/>Hence, equivalent definition of given function is $\mathrm{f}(\mathrm{x})=|\mathrm{x}|+\mathrm{x}$</div>


title: what-is-the-fate-of-primary-xylem-in-a-dicot-root-showing-extensive-secondary-growth
slug: what-is-the-fate-of-primary-xylem-in-a-dicot-root-showing-extensive-secondary-growth-61971
tags: ['Biology', 'Anatomy of Flowering Plants']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the fate of primary xylem in a dicot root showing extensive secondary growth?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>It is retained in the centre of the axis<br/></span> </div>


title: what-is-the-focal-distance-of-any-point-p-x-1-y-1-on-the-parabola-y-2-4-a-x
slug: what-is-the-focal-distance-of-any-point-p-x-1-y-1-on-the-parabola-y-2-4-a-x-78914
tags: ['Mathematics', 'Parabola', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the focal distance of any point $P\left(x_{1}, y_{1}\right)$ on the parabola $y^{2}=4 a x ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x_{1}+y_{1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x_{1} y_{1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a x_{1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$a+x_{1}$</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> </ul>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/P8k2lQIVKxFCakjYE-KSsbCGF62LKV661D1dg2F33xo.original.fullsize.png"/><br/><br/>Focal-Distans The distance between a point on a parabola and its focus is called its focal distance Let $F(a, 0)$ be a focus on parabola $y^{2}=4 a x$. Since, $P\left(x_{1}, y_{1}\right)$ on $y^{2}=4 a x$<br/>$y_{1}^{2}=4 a x_{1}$<br/>Now, Focal distance<br/>Hence, focal distance $=a+x_{1}$.</div>


title: what-is-the-foot-of-the-perpendicular-from-the-point-2-3-on-the-line-x-y-11-0
slug: what-is-the-foot-of-the-perpendicular-from-the-point-2-3-on-the-line-x-y-11-0-56282
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(1,10)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(5,6)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(6,5)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(7,4)$</span> </li> </ul>

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

answer:
<div class="solution">We know a line perpendicular to a given line $a x-b y+c=0$ is $b x+a y+k=0$<br/>The equation of line perpendicular to given line<br/>$$<br/>x+y-11=0 ...(i)<br/>$$<br/>is $-x+y+\lambda=0$ ...(ii)<br/>Since, this equation passes through $(2,3)$. Therefore $(2,3)$ satisfies the equation of line $\therefore \quad-2+3+\lambda=0$<br/>$\Rightarrow \quad \lambda=-1$<br/>From Eq. (ii), $-x+y-1=0$<br/>$\Rightarrow \quad y=x+1$<br/>And from eq. (i), $x+x+1-11=0$<br/>$\Rightarrow \quad 2 x=10$<br/>$\Rightarrow \quad x=5$<br/>Hence, coordinates of foot of perpendicular from $(2,3)$ to given line is $(5,6)$</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2020 (21 Sep Shift 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-formula-for-finding-coefficient-of-variation-given-sigma-standard-deviation-and-barx-mean-neq-0

slug:
what-is-the-formula-for-finding-coefficient-of-variation-given-sigma-standard-deviation-and-barx-mean-neq-0-46340

question:
<div class="question">What is the formula for finding coefficient of variation, given \(\sigma=\) standard deviation and \(\bar{x}=\) mean \(\neq 0\) ?</div>

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

answer:
<div class="solution">Coefficient of variation, having standard deviation \(\sigma\) and mean \(\bar{x} \neq 0\), is<br/>\(\frac{\sigma}{\bar{x}} \times 100\)<br/>Hence, option (d) is correct.</div>


title: what-is-the-foundation-of-collision-theory
slug: what-is-the-foundation-of-collision-theory-77843
question: <div class="question">What is the foundation of collision theory?</div>
tags: ['Chemistry', 'Chemical Kinetics', 'NEET']
answer: <div class="solution">Collision theory is based on the kinetic theory of gases.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Quantum theory</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Thermodynamics</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Kinetic theory of gases</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Chemical equilibrium</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Kinetic theory of gases</span> </div>


title: what-is-the-fourth-term-of-an-ap-of-n-terms-whose-sum-is-n-n-1
slug: what-is-the-fourth-term-of-an-ap-of-n-terms-whose-sum-is-n-n-1-77960
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the Fourth term of an AP of $n$ terms whose sum is $n$ $(\mathrm{n}+1) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">20</span> </li> </ul>

answer:
<div class="solution">Given, sum of n terms $\left(\mathrm{S}_{\mathrm{n}}\right)=\mathrm{n}(\mathrm{n}+1)$<br/>We know, $\mathrm{T}_{\mathrm{n}}=\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1}$<br/>$=[\mathrm{n}(\mathrm{n}+1)]-[(\mathrm{n}-1)(\mathrm{n}-1+1)]$<br/>$=\mathrm{n}(\mathrm{n}+1)-(\mathrm{n}-1) \mathrm{n}$<br/>$=\mathrm{n}(\mathrm{n}+1-\mathrm{n}+1)$<br/>$=2 \mathrm{n}$<br/>Fourth term, $\mathrm{T}_{4}=2(4)=8$</div>


title: what-is-the-full-form-of-efb
slug: what-is-the-full-form-of-efb-40884
question: <div class="question">What is the full form of EFB?</div>
tags: ['Biology', 'Biotechnology - Principles and Processes', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Enzyme Federation of Biochemistry</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Ecological Foundation of Biodiversity</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">European Federation of Biotechnology</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">European Foundation of Botany</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">European Federation of Biotechnology</span> </div>

answer:
<div class="solution">The European Federation of Biotechnology was established by European scientists in 1978. It is a non-profit federation of national biotechnology associations, learned societies, universities, scientific institutes, biotechnology companies and individual biotechnologists working to promote biotechnology throughout Europe and beyond.</div>


title: what-is-the-function-of-androgens-secreted-by-the-testis
slug: what-is-the-function-of-androgens-secreted-by-the-testis-77831
question: <div class="question">What is the function of androgens secreted by the testis?</div>
tags: ['Biology', 'Chemical Coordination and Integration', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Maturation and functions of male accessory sex organs</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Development of female accessory sex organs</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Maintenance of pregnancy</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Stimulation of digestive juices secretion</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Maturation and functions of male accessory sex organs</span> </div>

answer:
<div class="solution">Androgens secreted by the testis stimulate the development, maturation, and functions of the male accessory sex organs, appearance of male secondary sex characters, spermatogenesis, male sexual behavior, anabolic pathways, and erythropoiesis.</div>


title: what-is-the-function-of-neutrophils-in-the-body
slug: what-is-the-function-of-neutrophils-in-the-body-47712
question: <div class="question">What is the function of neutrophils in the body?</div>
tags: ['Biology', 'Body Fluids and Circulation', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Transport of respiratory gases</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Secrete histamine, serotonin and heparin</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Destroy foreign organisms entering the body</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Resisting infections and associated with allergic reactions</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Destroy foreign organisms entering the body</span> </div>

answer:
<div class="solution">Neutrophils are the most abundant cells (60-65%) of the total WBCs and they are phagocytic cells which destroy foreign organisms entering the body.</div>


title: what-is-the-function-of-starch-in-plants
slug: what-is-the-function-of-starch-in-plants-65033
question: <div class="question">What is the function of starch in plants?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">To form helical secondary structures</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">To hold I2 molecules</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">To act as a storehouse of energy</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">To act as a reducing end</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">To act as a storehouse of energy</span> </div>

answer:
<div class="solution">Starch is described as present in plant tissues as a storehouse of energy.</div>


title: what-is-the-function-of-the-polar-nuclei-in-an-embryosac
slug: what-is-the-function-of-the-polar-nuclei-in-an-embryosac-67141
question: <div class="question">What is the function of the polar nuclei in an embryo-sac?</div>
tags: ['Biology', 'Plant Kingdom', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Produce an egg cell</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Enclose ovules</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Assist in fertilization</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Produce a diploid secondary nucleus</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Produce a diploid secondary nucleus</span> </div>

answer:
<div class="solution">The polar nuclei in an embryo-sac eventually fuse to produce a diploid secondary nucleus, which is necessary for the development of the endosperm, the tissue that nourishes the developing embryo.</div>


title: what-is-the-function-of-the-synergids-in-an-embryosac
slug: what-is-the-function-of-the-synergids-in-an-embryosac-73461
question: <div class="question">What is the function of the synergids in an embryo-sac?</div>
tags: ['Biology', 'Plant Kingdom', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Produce an egg cell</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Enclose ovules</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Assist in fertilization</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Produce a diploid secondary nucleus</span> </li> </ul>

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

answer:
<div class="solution">Synergids are cells present within an embryo-sac that assist in fertilization by releasing chemical signals that attract pollen tubes to the egg cell.</div>


title: what-is-the-fundamental-characteristic-of-animals-belonging-to-the-phylum-chordata
slug: what-is-the-fundamental-characteristic-of-animals-belonging-to-the-phylum-chordata-22975
tags: ['Biology', 'Animal Kingdom', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the fundamental characteristic of animals belonging to the phylum Chordata?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">presence of a notochord</span> </div>

answer:
<div class="solution">The fundamental characteristic of animals belonging to the phylum Chordata is the presence of a notochord, which is a flexible rod-like structure located between the gut and the nerve cord.</div>


title: what-is-the-fundamental-idea-behind-the-crystal-field-theory-cft
slug: what-is-the-fundamental-idea-behind-the-crystal-field-theory-cft-75272
question: <div class="question">What is the fundamental idea behind the Crystal Field Theory (CFT)?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">It assumes that metal-ligand bonds are purely covalent.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">It considers the metal-ligand bond as purely ionic due to electrostatic interactions.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">It assumes that the d orbitals in metal atoms are non-degenerate.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It explains the color exhibited by coordination compounds.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It considers the metal-ligand bond as purely ionic due to electrostatic interactions.</span> </div>

answer:
<div class="solution">The Crystal Field Theory (CFT) is based on the assumption that the metal-ligand bond is purely ionic due to electrostatic interactions between the metal ion and ligands.</div>


tags: ['Chemistry', 'Solutions', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-the-fundamental-principle-behind-the-solubility-rule-like-dissolves-like-in-relation-to-the-intermolecular-forces-present-in-solutions

slug:
what-is-the-fundamental-principle-behind-the-solubility-rule-like-dissolves-like-in-relation-to-the-intermolecular-forces-present-in-solutions-23640

question:
<div class="question">What is the fundamental principle behind the solubility rule "like dissolves like" in relation to the intermolecular forces present in solutions?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Solubility is determined by the similarity of intermolecular forces between the solute and solvent.</span> </div>

answer:
<div class="solution">Correct Option is : (A)<br/>Solubility is determined by the similarity of intermolecular forces between the solute and solvent.</div>


title: what-is-the-fundamental-structural-and-functional-unit-of-all-living-organisms
slug: what-is-the-fundamental-structural-and-functional-unit-of-all-living-organisms-80185
tags: ['Biology', 'Cell - The Unit of Life', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Cell</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the fundamental structural and functional unit of all living organisms?</div>

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

answer:
<div class="solution">A cell is the fundamental structural and functional unit of all living organisms. It is capable of independent existence and performing the essential functions of life.</div>


title: what-is-the-general-solution-of-1-e-x-y-d-y-e-x-d-x
slug: what-is-the-general-solution-of-1-e-x-y-d-y-e-x-d-x-59924
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the general solution of $\left(1+e^{x}\right) y d y=e^{x} d x ? \quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y^{2}=\ln \left[c^{2}\left(e^{x}+1\right)^{2}\right]$</span> </div>

answer:
<div class="solution">The given differential equation is<br/>$$<br/>\left(1+e^{x}\right) y d y=e^{x} d x<br/>$$<br/>By separating the variable, we get<br/>$$<br/>y d y=\frac{e^{x}}{1+e^{x}} d x<br/>$$<br/>Integrating on both the sides,<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \quad \int y d y=\int\left(\frac{e^{x}}{1+e^{x}}\right) d x \\<br/>\Rightarrow \quad \frac{y^{2}}{2}=\log \left(1+e^{x}\right)+\log c \\<br/>\Rightarrow \quad y^{2}=2 \log \left[c\left(1+e^{x}\right)\right] \\<br/>(\because \log m+\log n=\log m n)<br/>\end{array}<br/>$$<br/>$\Rightarrow \quad y^{2}=\log \left[c^{2}\left(1+e^{x}\right)^{2}\right]$</div>


title: what-is-the-general-solution-of-the-differential-equation-d-x-d-y-y-x-0
slug: what-is-the-general-solution-of-the-differential-equation-d-x-d-y-y-x-0-73104
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the general solution of the differential equation<br/>$\frac{d y}{d x}+\frac{x}{y}=0 ?$</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \frac{d y}{d x}+\frac{x}{y}=0 \\ &amp; \Rightarrow \frac{d y}{d x}=\frac{-x}{y} \\ &amp; \Rightarrow y d y+x d x=0 \\ &amp; \Rightarrow \frac{y^{2}}{2}+\frac{x^{2}}{2}+c=0 \\ &amp; \Rightarrow \mathrm{y}^{2}+\mathrm{x}^{2}+2 \mathrm{c}=0 \\ &amp; \Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{c} . \end{aligned}$</div>


title: what-is-the-general-solution-of-the-differential-equation-e-x-tan-ydx-1-e-x-sec-2-y-dy-0
slug: what-is-the-general-solution-of-the-differential-equation-e-x-tan-ydx-1-e-x-sec-2-y-dy-0-56245
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the general solution of the differential equation $\mathrm{e}^{\mathrm{x}} \tan \mathrm{y} \mathrm{d} \mathrm{x}+\left(1-\mathrm{e}^{\mathrm{x}}\right) \sec ^{2} \mathrm{y} \mathrm{dy}=0 ?$</div>

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

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

answer:
<div class="solution">Given diff equation can be written as<br/>$\frac{e^{x}}{1-e^{x}} d x=-\frac{\sec ^{2} y}{\tan y} d y$<br/>On integrating both the sides, we get<br/>$\int \frac{\mathrm{e}^{\mathrm{x}}}{1-\mathrm{e}^{\mathrm{x}}} \mathrm{dx}=-\int \frac{\mathrm{sec}^{2} \mathrm{y}}{\tan \mathrm{y}} \mathrm{dy}$<br/>$\Rightarrow-\log \left(1-\mathrm{e}^{\mathrm{x}}\right)=-\log (\tan \mathrm{y})+\log \mathrm{c}$<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \log (\tan y)=\log c+\log \left(1-e^{x}\right) \\<br/>\Rightarrow \quad \log (\tan y)=\log \left[c\left(1-e^{x}\right)\right] \\<br/>\Rightarrow \quad \tan y=c\left(1-e^{x}\right)<br/>\end{array}<br/>$$<br/>Where ' $\mathrm{c}$ ' is the constant of integration.</div>


title: what-is-the-general-solution-of-the-differential-equation-x-2-dy-y-2-dx-0
slug: what-is-the-general-solution-of-the-differential-equation-x-2-dy-y-2-dx-0-23571
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the general solution of the differential equation $\mathrm{x}^{2} \mathrm{dy}+\mathrm{y}^{2} \mathrm{dx}=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+y=c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x y=c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$c(x+y)=x y$</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><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above where $c$ is the constant of integration.</span> </li> </ul>

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

answer:
<div class="solution">Given differential equation is $\mathrm{x}^{2} \mathrm{dy}+\mathrm{y}^{2} \mathrm{dx}=0$ $\Rightarrow \quad x^{2} d y=-y^{2} d x$<br/>$\Rightarrow \quad \frac{d y}{y^{2}}+\frac{d x}{x^{2}}=0$<br/>$\Rightarrow \quad \int \mathrm{y}^{-2} \mathrm{dy}+\int \mathrm{x}^{-2} \mathrm{dx}=0$<br/>$\Rightarrow \frac{\mathrm{y}^{-2+1}}{-2+1}+\frac{\mathrm{x}^{-2+1}}{-2+1}=\mathrm{a}$ where 'a' is a constant of<br/>integration<br/>$\Rightarrow \quad-\frac{1}{y}-\frac{1}{x}=a$<br/>$-(x+y)=a x y \Rightarrow c(x+y)=x y$<br/>where $\mathrm{c}=-\frac{1}{\mathrm{a}}$ is a constant of integration.</div>


title: what-is-the-general-solution-of-the-differential-equation-x-d-y-y-d-x-y-2
slug: what-is-the-general-solution-of-the-differential-equation-x-d-y-y-d-x-y-2-71370
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the general solution of the differential equation $x$ $d y-y d x=y^{2} ? \quad$</div>

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

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

answer:
<div class="solution">Differential equation $x \mathrm{dy}-\mathrm{y} \mathrm{dx}=\mathrm{y}^{2}$ $=(\mathrm{y} \mathrm{dx}-\mathrm{x} \mathrm{dy})=\mathrm{y}^{2}$<br/>$\therefore \mathrm{d}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)=0$<br/>$\frac{x}{y}=C \therefore x=C y$<br/>$\left(\mathrm{x}^{2}+\mathrm{x}+1\right) \mathrm{dy}+\left(\mathrm{y}^{2}+\mathrm{y}+1\right) \mathrm{dx}=0$<br/>$\left(\mathrm{x}^{2}+\mathrm{x}+1\right) \mathrm{d} \mathrm{y}=-\left(\mathrm{y}^{2}+\mathrm{y}+1\right) \mathrm{d} \mathrm{x}$<br/>$\quad \frac{\mathrm{d} \mathrm{x}}{\left(1+\mathrm{x}+\mathrm{x}^{2}\right)}+\frac{\mathrm{dy}}{\left(1+\mathrm{y}+\mathrm{y}^{2}\right)}=0$<br/>$\Rightarrow \int \frac{\mathrm{dx}}{\left(\mathrm{x}+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}+\int \frac{\mathrm{dy}}{\left(\mathrm{y}+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}=0$<br/>$\Rightarrow \frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \mathrm{x}+1}{\sqrt{3}}\right)+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \mathrm{y}+1}{\sqrt{3}}\right)$<br/>$=\frac{2}{\sqrt{3}} \tan ^{-1}\left(\mathrm{C}_{1}\right)$<br/>$\Rightarrow \tan ^{-1}\left\{\frac{\left(\frac{2 \mathrm{x}+1}{\sqrt{3}}\right)+\left(\frac{2 \mathrm{y}+1}{\sqrt{3}}\right)}{1-\left(\frac{2 \mathrm{x}+1}{\sqrt{3}}\right)\left(\frac{2 \mathrm{y}+1}{\sqrt{3}}\right)}\right\}=\tan ^{-1} \mathrm{C}_{1}$<br/>$\left[\because \tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{y}=\tan ^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)\right]$<br/>$\Rightarrow \frac{\sqrt{3}[(2 \mathrm{x}+1)+(2 \mathrm{y}+1)]}{3-(2 \mathrm{x}+1)(2 \mathrm{y}+1)}=\mathrm{C}_{1}$<br/>$\Rightarrow \frac{2 \sqrt{3}(\mathrm{x}+\mathrm{y}+1)}{-4 \mathrm{xy}-2 \mathrm{y}-2 \mathrm{x}+2}=\mathrm{C}_{1}$<br/>$\Rightarrow 2 \sqrt{3}(\mathrm{x}+\mathrm{y}+1)=\mathrm{C}_{1}(2-2 \mathrm{x}-2 \mathrm{y}-4 \mathrm{xy})$<br/>$\Rightarrow 2 \sqrt{3}(\mathrm{x}+\mathrm{y}+1)=2 \mathrm{C}_{1}(1-\mathrm{x}-\mathrm{y}-2 \mathrm{xy})$<br/>$\Rightarrow(\mathrm{x}+\mathrm{y}+1)=\frac{\mathrm{C}_{1}}{\sqrt{3}}(1-\mathrm{x}-\mathrm{y}-2 \mathrm{xy})$<br/>$(x+y+1)=A(1+B x+C y+D x y)$</div>


title: what-is-the-general-solution-of-the-differential-equation-ydx-x-2-y-2-dy-0
slug: what-is-the-general-solution-of-the-differential-equation-ydx-x-2-y-2-dy-0-13452
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the general solution of the differential equation $\mathrm{ydx}-\left(\mathrm{x}+2 \mathrm{y}^{2}\right) \mathrm{dy}=0 ? \quad$</div>

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

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

answer:
<div class="solution">ydx $-\left(x+2 y^{2}\right) d y=0$<br/>$\quad \Rightarrow y d x-x d y-2 y^{2} d y=0$<br/>$\Rightarrow y-x \cdot \frac{d y}{d x}=2 y^{2} \cdot \frac{d y}{d x}$<br/>$\Rightarrow \frac{y-x \cdot \frac{d y}{d x}}{y^{2}}=2 \cdot \frac{d y}{d x}$<br/>Integrating<br/>$\Rightarrow \frac{x}{y}=2 y+c$<br/>$\Rightarrow \mathrm{x}=2 \mathrm{y}^{2}+\mathrm{cy}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 1)']
answer: <div class="solution">Both statements (1) and (2) are correct.</div>
source_db: MarksBatch2_P3.db
title:
what-is-the-geometric-interpretation-of-the-identity-a-b-a-b-2-a-b-1-if-the-diagonals-of-a-given-parallelogram-are-used-as-sides-of-a-second-parallelo

slug:
what-is-the-geometric-interpretation-of-the-identity-a-b-a-b-2-a-b-1-if-the-diagonals-of-a-given-parallelogram-are-used-as-sides-of-a-second-parallelo-86402

question:
<div class="question">What is the geometric interpretation of the identity<br/>$(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})$ ?<br/>$1.$ If the diagonals of a given parallelogram are used as sides of a second parallelogram, then the area of the second parallelogram is twice that of the given parallelogram.<br/>$2.$ If the semi-diagonals of a given parallelogram are used as sides of a second parallelogram, then the area of the second parallelogram is half that of the given parallelogram. Select the correct answer using the code given below</div>


title: what-is-the-geometric-mean-of-1040-and-60-appox
slug: what-is-the-geometric-mean-of-1040-and-60-appox-64155
question: <div class="question">What is the geometric mean of 10,40 and 60 (appox)?</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">28</span> </div>
answer: <div class="solution">G.M= $(10 \times 40 \times 60)^{1 / 3}=28.84 \simeq 28$</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">28</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$29.6$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">70</span> </li> </ul>


title: what-is-the-geometric-mean-of-the-data-2-4-8-16-32
slug: what-is-the-geometric-mean-of-the-data-2-4-8-16-32-79480
question: <div class="question">What is the geometric mean of the data $2,4,8,16,32 ?$</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 1)']
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">4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">Required geometric mean $=\sqrt[5]{2.4 .8 .16 .32}$ $=\left(32^{3}\right)^{1 / 5}=\left(2^{15}\right)^{1 / 5}=2^{3}=8$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-geometric-mean-of-the-ratio-of-corresponding-terms-of-two-series-where-g-1-and-g-2-are-geometric-means-of-the-two-series

slug:
what-is-the-geometric-mean-of-the-ratio-of-corresponding-terms-of-two-series-where-g-1-and-g-2-are-geometric-means-of-the-two-series-82603

question:
<div class="question">What is the geometric mean of the ratio of corresponding terms of two series where $\mathrm{G}_{1}$ and $\mathrm{G}_{2}$ are geometric means of the two series?</div>

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

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

answer:
<div class="solution">Let series be a, $\mathrm{G}_{1}$, b and $\mathrm{a}^{\prime}, \mathrm{G}_{2}, \mathrm{~b}^{\prime} \mathrm{so}, \mathrm{G}_{1}=\sqrt{\mathrm{ab}}$ and<br/>$\mathrm{G}_{2}=\sqrt{\mathrm{a}^{\prime} \mathrm{b}^{\prime}}$<br/>Series formed by ratio of the corresponding terms are :<br/>$\frac{\mathrm{a}}{\mathrm{a}^{\prime}}, \frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}, \frac{\mathrm{b}}{\mathrm{b}^{\prime}}$<br/>Geometric means of this series $=\sqrt{\frac{a}{a^{\prime}} \cdot \frac{b}{b^{\prime}}}$<br/>$=\sqrt{\frac{\mathrm{ab}}{\mathrm{a}^{\prime} \mathrm{b}^{\prime}}}=\frac{\sqrt{\mathrm{ab}}}{\sqrt{\mathrm{a}^{\prime} \mathrm{b}^{\prime}}}=\frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}$<br/>So, geometric mean of the ratio of corresponding term of two series where $\mathrm{G}_{1}$ and $\mathrm{G}_{2}$ are geometric means of<br/>two series is $\frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}$</div>


title: what-is-the-geometric-mean-of-the-sequence-1-2-4-8-2-n
slug: what-is-the-geometric-mean-of-the-sequence-1-2-4-8-2-n-80429
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the geometric mean of the sequence $1,2,4,8, \ldots . .$ $2^{\mathrm{n}} ?$</div>

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

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

answer:
<div class="solution">Geometric mean $=\sqrt[n+1]{1.2 .4 .8 \ldots 2^{n}}$<br/>$\left(\because\right.$ there are $(n+1)$ multiples from $2^{\circ}$ to $\left.2^{n}\right)$<br/>$=\sqrt[n+1]{2^{0} \cdot 2^{1} \cdot 2^{2} \cdot 2^{3} \ldots 2^{n}}$<br/>$=n+1]{2^{1+2+3+\ldots+n}}=\sqrt[n+1]{2 \frac{n(n+1)}{2}}$<br/>$=\left[2 \frac{n(n+1)}{2}\right]^{\frac{1}{n+1}}=2^{\frac{n}{2}}$</div>


title: what-is-the-graafian-follicle
slug: what-is-the-graafian-follicle-23886
question: <div class="question">What is the Graafian follicle ?</div>
tags: ['Biology', 'Human Reproduction', 'NEET']
answer: <div class="solution">The Graafian follicle now transforms into the mature follicle.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The mature follicle</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The primary follicle</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The secondary follicle</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The tertiary follicle</span> </li> </ul>

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


title: what-is-the-greatest-integer-among-the-following-by-which-the-number-5-5-7-5-is-divisible
slug: what-is-the-greatest-integer-among-the-following-by-which-the-number-5-5-7-5-is-divisible-90438
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the greatest integer among the following by which the number $5^{5}+7^{5}$ is divisible?</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">8</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">11</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">We know, $a^{n}+b^{n}$ is divisible by $(a+b)$, if $n$ is odd. Here, $\mathrm{n}=5$ is odd $\therefore 5^{5}+7^{5}$ is divisible by $5+7=12$</div>


title: what-is-the-greatest-value-of-the-positive-integer-n-satisfying-2-1-4-1-8-1-2-n-1-1-2-1000-1
slug: what-is-the-greatest-value-of-the-positive-integer-n-satisfying-2-1-4-1-8-1-2-n-1-1-2-1000-1-21958
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the greatest value of the positive integer $\mathrm{n}$ satisfying <br/>$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots \ldots+\frac{1}{2^{n-1}}   &lt; 2-\frac{1}{1000} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">11</span> </li> </ul>

answer:
<div class="solution">$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots \ldots \ldots . .+\frac{1}{2^{\mathrm{n}-1}}   &lt; 2-\frac{1}{1000}$<br/>$\frac{1-\frac{1}{2^{n}}}{1-\frac{1}{2}} &lt; 2-\frac{1}{1000}$<br/>$\Rightarrow \quad 2-\frac{1}{2^{D-1}} &lt; 2-\frac{1}{1000}$<br/>$2^{n-1} &lt; 1000$<br/>Now, $(2)^{9}=512 \&amp;(2)^{10}=1024$<br/>$\therefore \quad n-1=9$<br/>$\Rightarrow \quad n=10 .$</div>


title: what-is-the-image-of-the-point-1-2-on-the-line-3-x-4-y-1-0
slug: what-is-the-image-of-the-point-1-2-on-the-line-3-x-4-y-1-0-30086
question: <div class="question">What is the image of the point $(1,2)$ on the line $3 x+4 y-1=0$ ?</div>
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(-\frac{7}{5},-\frac{6}{5}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{7}{8}, \frac{1}{2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{7}{8},-\frac{1}{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(-\frac{7}{5}, \frac{1}{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(-\frac{7}{5},-\frac{6}{5}\right)$</span> </div>

answer:
<div class="solution">Let $(a, b)$ be the image of point $(1,2)$ w.r.t. line $3 x+4 y-1=0$<br/>$\therefore\left(\frac{a+1}{2}, \frac{b+2}{2}\right)$ will be on the line $3 x+4 y-1=0$<br/>$\Rightarrow 3\left(\frac{a+1}{2}\right)+4\left(\frac{b+2}{2}\right)-1=0$<br/>$\Rightarrow \quad 3 a+3+4 b+8-2=0$<br/>$\Rightarrow \quad 3 a+4 b+9=0$<br/>Now, co-ordinates given in option 'a' satisfies the equation $3 a+4 b+9=0$ Thus, the image of point $(1,2)$ is $\left(-\frac{7}{5},-\frac{6}{5}\right)$.</div>


title: what-is-the-image-of-the-point-2-3-in-the-line-y-x
slug: what-is-the-image-of-the-point-2-3-in-the-line-y-x-80590
question: <div class="question">What is the image of the point $(2,3)$ in the line $y=-x$ ?</div>
tags: ['Mathematics', 'Straight Lines', '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">$(-3,-2)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-3,2)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-2,-3)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(3,2)$</span> </li> </ul>

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

answer:
<div class="solution">Let there be a point $\mathrm{P}(2,3)$ on cartesian plane. Image of this point in the line $\mathrm{y}=-\mathrm{x}$ will lie on a line which is perpendicular to this line and distance of this point from $\mathrm{y}=-\mathrm{x}$ will be equal to distance of the image from this line.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/891pmtYrdJFSUqDACqzBDcXOYmiYQFQU2nAzgXQVZVc.original.fullsize.png"/><br/><br/>Let $\mathrm{Q}$ be the image of $\mathrm{p}$ and let the co-ordinate of $\mathrm{Q}$ be $(\mathrm{h}, \mathrm{k})$<br/>Slope of line $\mathrm{y}=-\mathrm{x}$ is $-1$ Line joining $P, Q$ will be perpendicular to $\mathrm{y}=-\mathrm{x} \mathrm{so}$, its slope $=1$ Let the equation of the line be $\mathrm{y}=\mathrm{x}+\mathrm{c}$ since this passes through point $(2,3)$ $3=2+c \Rightarrow c=1$<br/>and the equation $\mathrm{y}=\mathrm{x}+1$ The point of intersection $\mathrm{R}$ lies in the middle of $\mathrm{P} \&amp; \mathrm{Q}$. Point of intersection of line $\mathrm{y}=-\mathrm{x}$ and $\mathrm{y}=\mathrm{x}+1$ is $2 \mathrm{y}=1, \Rightarrow \mathrm{y}=\frac{1}{2}$ and $\mathrm{x}=-\frac{1}{2}$<br/>Hence, $\frac{\mathrm{h}+2}{2}=-\frac{1}{2}$ and $\frac{\mathrm{k}+3}{2}=\frac{1}{2}$<br/>$\Rightarrow \quad \mathrm{h}=-3$ and $\mathrm{k}=-2$<br/>So, the image of the point $(2,3)$ in the line $\mathrm{y}=-\mathrm{x}$ is $(-3,-2)$</div>


title: what-is-the-importance-of-cleanliness-and-hygiene-in-dairy-farming
slug: what-is-the-importance-of-cleanliness-and-hygiene-in-dairy-farming-87945
question: <div class="question">What is the importance of cleanliness and hygiene in dairy farming?</div>
tags: ['Biology', 'Strategies for Enhancement in Food Production', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">It helps to reduce the chance of direct contact between the produce and the handler</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">It is mandatory for regular inspections</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">It is not important</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It helps to identify and rectify problems early</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It helps to reduce the chance of direct contact between the produce and the handler</span> </div>

answer:
<div class="solution">Maintaining stringent cleanliness and hygiene is important in dairy farming to reduce the chance of direct contact between the produce and the handler.</div>


title: what-is-the-importance-of-record-keeping-in-dairy-farming
slug: what-is-the-importance-of-record-keeping-in-dairy-farming-54711
question: <div class="question">What is the importance of record keeping in dairy farming?</div>
tags: ['Biology', 'Strategies for Enhancement in Food Production', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">It helps to identify and rectify problems early</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">It is mandatory for regular inspections</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">It is not important</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">It helps to reduce the chance of direct contact between the produce and the handler</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It helps to identify and rectify problems early</span> </div>

answer:
<div class="solution">Proper record keeping in dairy farming is important to help identify and rectify problems early.</div>


title: what-is-the-inclination-of-the-line-3-x-y-1-0
slug: what-is-the-inclination-of-the-line-3-x-y-1-0-87008
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the inclination of the line $\sqrt{3} \mathrm{x}-\mathrm{y}-1=0 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$60^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$135^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$150^{\circ}$</span> </li> </ul>

answer:
<div class="solution">Given equation can be written as $\mathrm{y}=\sqrt{3} \mathrm{x}-1$ on comparing with $\mathrm{y}=\mathrm{mx}+\mathrm{c}$<br/>We get $\tan \theta=\sqrt{3} \Rightarrow \theta=60^{\circ}$</div>


tags: ['Chemistry', 'States of Matter', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-the-initial-reason-a-clear-boundary-is-visible-between-the-liquid-and-vapor-phases-when-heating-a-liquid-in-a-closed-vessel

slug:
what-is-the-initial-reason-a-clear-boundary-is-visible-between-the-liquid-and-vapor-phases-when-heating-a-liquid-in-a-closed-vessel-43131

question:
<div class="question">What is the initial reason a clear boundary is visible between the liquid and vapor phases when heating a liquid in a closed vessel?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The liquid becomes less dense</span> </div>

answer:
<div class="solution">Initially, a clear boundary is visible because the liquid becomes less dense as it is heated.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-the-interior-acute-angle-of-the-parallelogram-whose-sides-are-represented-by-the-vectors-2-1-i-2-1-j-k-and-2-1-i-2-1-j-k

slug:
what-is-the-interior-acute-angle-of-the-parallelogram-whose-sides-are-represented-by-the-vectors-2-1-i-2-1-j-k-and-2-1-i-2-1-j-k-41563

question:
<div class="question">What is the interior acute angle of the parallelogram whose<br/>sides are represented by the vectors $\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and<br/>$\frac{1}{\sqrt{2}} \hat{\mathrm{i}}-\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\hat{\mathrm{k}} ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$60^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$15^{\circ}$</span> </li> </ul>

answer:
<div class="solution">Let $\overrightarrow{\mathrm{a}}=\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\hat{k}$<br/>and $\overrightarrow{\mathrm{b}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}-\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\therefore \cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$<br/>$=\frac{\left(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\hat{k}\right) \cdot\left(\frac{1}{\sqrt{2}} \hat{i}-\frac{1}{\sqrt{2}} \hat{j}+\hat{k}\right)}{\sqrt{\frac{1}{2}+\frac{1}{2}+1} \sqrt{\frac{1}{2}+\frac{1}{2}+1}}$<br/>$=\frac{1}{2}\left[\frac{1}{2}-\frac{1}{2}+1\right]=\frac{1}{2}=\cos 60^{\circ}$<br/>$\therefore \theta=60^{\circ}$</div>


title: what-is-the-interval-in-which-the-function-f-x-9-x-2-is-increasing-f-x-0
slug: what-is-the-interval-in-which-the-function-f-x-9-x-2-is-increasing-f-x-0-32836
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the interval in which the function $\mathrm{f}(\mathrm{x})=\sqrt{9-\mathrm{x}^{2}}$ is increasing? $(f(x)&gt;0)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0   &lt; x &lt; 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-3   &lt; x &lt; 0$</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">$\quad 0   &lt; \mathrm{x} &lt; 9$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3   &lt; x &lt; 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-3   &lt; x &lt; 0$</span> </div>

answer:
<div class="solution">$f(x)=\sqrt{9-x^{2}}$<br/>$f^{\prime}(x)=\frac{1}{2 \sqrt{9-x^{2}}} \times(-2 x)=-\frac{x}{\sqrt{9-x^{2}}}$<br/>For function to be increasing<br/>$-\frac{x}{\sqrt{9-x^{2}}}&gt;0$<br/>or $-x&gt;0$ or $x   &lt; 0$<br/>but $\sqrt{9-\mathrm{x}^{2}}$ is defined only when $9-x^{2}&gt;0$ or $x^{2}-9 &lt; 0$<br/>$(x+3)(x-3) &lt; 0$<br/>ie $-3 &lt; x &lt; 3$<br/>$-3 &lt; x &lt; 3 \cap x &lt; 0$<br/>$\Rightarrow-3 &lt; x &lt; 0$</div>


title: what-is-the-interval-over-which-the-function-f-x-6-x-x-2-x-0-is-increasing
slug: what-is-the-interval-over-which-the-function-f-x-6-x-x-2-x-0-is-increasing-15308
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the interval over which the function $f(x)=6 x-x^{2}, x&gt;0$ is increasing $? \quad$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(0,3)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(3,6)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(6,9)$</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">$(0,3)$</span> </div>

answer:
<div class="solution">Given $f(x)=6 x-x^{2}, x&gt;0$ Differentiate w.r.t. $x$ both side we get $f^{\prime}(x)=6-2 x$<br/>As we know $f(x)$ will be increasing function, if $f^{\prime}(x)&gt;0 \quad \therefore 6-2 x&gt;0 \Rightarrow x   &lt; 3$<br/>Thus, required interval is $(0,3)$.</div>


title: what-is-the-inverse-of-0-0-1-0-1-0-1-0-0
slug: what-is-the-inverse-of-0-0-1-0-1-0-1-0-0-99688
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the inverse of $\left[\begin{array}{lll}0 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 0\end{array}\right]$?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{lll}0 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 0\end{array}\right]$</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">$\left[\begin{array}{ccc}-1 &amp; 0 &amp; 1 \\ 0 &amp; -1 &amp; 0 \\ 0 &amp; 0 &amp; -1\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\begin{array}{ccc}0 &amp; 0 &amp; -1 \\ 0 &amp; -1 &amp; 0 \\ -1 &amp; 0 &amp; 0\end{array}\right]$</span> </li> </ul>

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

answer:
<div class="solution">$A=\left[\begin{array}{ccc}0 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 0\end{array}\right]$<br/>$\therefore \quad|A|=[0(0)-0(0)+1(-1)]=-1$<br/>and $\operatorname{adj} A=\left[\begin{array}{rrr}0 &amp; 0 &amp; -1 \\ 0 &amp; -1 &amp; 0 \\ -1 &amp; 0 &amp; 0\end{array}\right]$<br/>Hence, $A^{-1}=\frac{1}{|A|} \operatorname{adj} A$<br/>$=-\frac{1}{1}\left[\begin{array}{rrr}0 &amp; 0 &amp; -1 \\ 0 &amp; -1 &amp; 0 \\ -1 &amp; 0 &amp; 0\end{array}\right]=\left[\begin{array}{lll}0 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 0\end{array}\right]$</div>


title: what-is-the-inverse-of-a-1-i-1-i-1-i-1-i
slug: what-is-the-inverse-of-a-1-i-1-i-1-i-1-i-10801
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the inverse of $\mathrm{A}=\left[\begin{array}{cc}1+\mathrm{i} &amp; 1+\mathrm{i} \\ -1+\mathrm{i} &amp; 1-\mathrm{i}\end{array}\right]$?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{4}\left[\begin{array}{cc}1-\mathrm{i} &amp; -1-\mathrm{i} \\ 1-\mathrm{i} &amp; 1+\mathrm{i}\end{array}\right]$</span> </div>

answer:
<div class="solution">We know that<br/>If $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} &amp; \mathrm{b} \\ \mathrm{c} &amp; \mathrm{d}\end{array}\right], \operatorname{adj} \mathrm{A}=\left[\begin{array}{cc}\mathrm{d} &amp; -\mathrm{b} \\ -\mathrm{c} &amp; \mathrm{a}\end{array}\right]$<br/>and $|\mathrm{A}|=\mathrm{ad}-\mathrm{bc}$<br/>$\mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}$ if $|\mathrm{A}| \neq 0$<br/>So, $\mathrm{A}^{-1}=\frac{1}{\mathrm{ad}-\mathrm{bc}}\left[\begin{array}{cc}\mathrm{d} &amp; -\mathrm{b} \\ -\mathrm{c} &amp; \mathrm{a}\end{array}\right]=\frac{1}{|\mathrm{~A}|}\left[\begin{array}{cc}\mathrm{d} &amp; -\mathrm{b} \\ -\mathrm{c} &amp; \mathrm{a}\end{array}\right]$<br/>Given that $\mathrm{A}=\left[\begin{array}{cc}1+\mathrm{i} &amp; 1+\mathrm{i} \\ -1+\mathrm{i} &amp; 1-\mathrm{i}\end{array}\right]$<br/>Then adj $\mathrm{A}=\left[\begin{array}{cc}1-\mathrm{i} &amp; -1-\mathrm{i} \\ 1-\mathrm{i} &amp; 1+\mathrm{i}\end{array}\right]$<br/>and, $|\mathrm{A}|=\left|\begin{array}{cc}1+\mathrm{i} &amp; 1+\mathrm{i} \\ -1+\mathrm{i} &amp; 1-\mathrm{i}\end{array}\right|$<br/>$=\left(1-i^{2}\right)-\left(i^{2}-1\right)=\left(1-i^{2}-i^{2}+1\right)=1+1+1+1=4$<br/>$\Rightarrow \quad \mathrm{A}^{-1}=\frac{1}{4}\left[\begin{array}{cc}1-\mathrm{i} &amp; -1-\mathrm{i} \\ 1-\mathrm{i} &amp; 1+\mathrm{i}\end{array}\right]$</div>


title: what-is-the-inverse-of-the-function-y-5-l-o-g-x
slug: what-is-the-inverse-of-the-function-y-5-l-o-g-x-76301
question: <div class="question">What is the inverse of the function $\mathrm{y}=5^{\log \mathrm{x}}$ ?</div>
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=5^{1 / \log y}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x=y^{1 / \log 5}$</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">$x=5^{\log y}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=y^{\log 5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x=y^{1 / \log 5}$</span> </div>

answer:
<div class="solution">Given equation is $\mathrm{y}=5 \log \mathrm{x}$ To get the inverse, we express $\mathrm{x}$ in terms of $\mathrm{y}$ Taking log on both the sides, $\begin{aligned} &amp; \log y=\log x \cdot \log 5 \\ \Rightarrow &amp; \log x=\frac{\log y}{\log 5}=\log y^{\frac{1}{\log 5}} \\ \Rightarrow &amp; x=y^{\frac{1}{\log 5}} \end{aligned}$</div>


title: what-is-the-inverse-of-the-matrix-a-cos-sin-0-sin-cos-0-0-0-1
slug: what-is-the-inverse-of-the-matrix-a-cos-sin-0-sin-cos-0-0-0-1-42120
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the inverse of the matrix <br/>$A=\left(\begin{array}{ccc}\cos \theta &amp; \sin \theta &amp; 0 \\ -\sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right) ?$</div>

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

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

answer:
<div class="solution">$A=\left[\begin{array}{ccc}\cos \theta &amp; \sin \theta &amp; 0 \\ -\sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>We know, $\mathrm{AA}^{-1}=\mathrm{I}$<br/>Let us take first option (a) as $\overline{\mathrm{A}}^{1}$.<br/>$\left[\begin{array}{ccc}\cos \theta &amp; \sin \theta &amp; 0 \\ -\sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]\left[\begin{array}{ccc}\cos \theta &amp; -\sin \theta &amp; 0 \\ \sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{ccc}\cos ^{2} \theta+\sin ^{2} \theta &amp; -\sin \theta \cos \theta+\cos \theta \sin \theta &amp; 0 \\ -\sin \theta \cos \theta+\cos \theta \sin \theta &amp; \sin ^{2} \theta+\cos ^{2} \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]=\mathrm{I}$</div>


title: what-is-the-iupac-name-of-ch-3-ch-ch-2-ch-3-cho
slug: what-is-the-iupac-name-of-ch-3-ch-ch-2-ch-3-cho-15061
tags: ['Chemistry', 'General Organic Chemistry', 'AP EAMCET', 'AP EAMCET 2021 (19 Aug Shift 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>IUPAC</mi></math> name of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mi>CH</mi><mfenced><mrow><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>CH</mi><mn>3</mn></msub></mrow></mfenced><mi>CHO</mi></math>?</div>

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

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

answer:
<div class="solution"><p>The given compound is:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi mathvariant="normal">C</mi><mn>4</mn></munder><msub><mi mathvariant="normal">H</mi><mn>3</mn></msub><munder><mi mathvariant="normal">C</mi><mn>3</mn></munder><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><munder><mi mathvariant="normal">C</mi><mn>2</mn></munder><mi mathvariant="normal">H</mi><mo>(</mo><msub><mi>CH</mi><mn>3</mn></msub><mo>)</mo><munder><mi mathvariant="normal">C</mi><mn>1</mn></munder><mi>HO</mi></math></p><p>There are four carbon in the longest chain, hence, the root word is<strong> but</strong>. All carbon in the longest chain are connected with single bonds, hence, <strong>an</strong> is the primary suffix. Aldehyde is the major functional group, hence, <strong>al </strong>is the secondary suffix. Methyl substituent is attached to the second carbon so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>-</mo></math>methyl will be used as prefix. </p><p>Hence, by combing all of these things <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>IUPAC</mi></math> name of this compound is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>-</mo></math>methylbutanal.</p></div>


title: what-is-the-key-difference-between-primary-and-secondary-sewage-treatment-1
slug: what-is-the-key-difference-between-primary-and-secondary-sewage-treatment-1-64381
tags: ['Biology', 'Microbes in Human Welfare', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the key difference between primary and secondary sewage treatment?</div>

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

answer:
<div class="solution">The key difference between primary and secondary sewage treatment is In primary sewage treatment, there is mechanical or physical removal of small and large substances while the secondary sewage treatment includes the removal of organic matter by microbes.</div>


title: what-is-the-key-difference-between-primary-and-secondary-sewage-treatment
slug: what-is-the-key-difference-between-primary-and-secondary-sewage-treatment-34430
tags: ['Biology', 'Microbes in Human Welfare']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the key difference between primary and secondary sewage treatment?</div>

answer:
<div class="solution">Primary $\left(1^{\circ}\right)$ treatment is a physical process that involves for removal of particulate as settelable particle.<br/>Secondary $\left(2^{\circ}\right)$ treatment is purely a biological treatment involving microbial oxidation.</div>


title: what-is-the-largest-value-of-a-third-order-determinant-whose-elements-are-0-or-1
slug: what-is-the-largest-value-of-a-third-order-determinant-whose-elements-are-0-or-1-15670
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the largest value of a third order determinant whose elements are 0 or 1?</div>

answer:
<div class="solution">In the third order determinant $\left|\begin{array}{lll}1 &amp; 0 &amp; 1 \\ 1 &amp; 1 &amp; 0 \\ 0 &amp; 1 &amp; 1\end{array}\right|$ which<br/>satisfied these conditions. Its value is 2 , which is largest if the elements of a determinant are 0 or 1.</div>


title: what-is-the-last-digit-of-3-4-4-n-1-where-n-is-a-natural-number
slug: what-is-the-last-digit-of-3-4-4-n-1-where-n-is-a-natural-number-46044
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the last digit of $3^{4^{4 \mathrm{n}}}+1$, where $n$ is a natural number?</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">7</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">In $3^{\text {n }}$, last digit is 3 , if $n=1,9$ if $n=2,7$ if $n=3$ and 1 if $\mathrm{n}=4$ and it is repeated after than<br/>Given expression is $3^{3^{4 \mathrm{n}}}+1$<br/>Let $x=3^{3^{4 n}}+1=3^{81 n}+1$<br/>$\Rightarrow \mathrm{x}=3^{80 \mathrm{n}} \cdot 3^{\mathrm{n}}+1$<br/>Last digit of $x$ will be decided by $3^{\text {n }}$ since $3^{80 \text { n }}$ has power multiple of 4 . $\operatorname{If} n=1$ last digit is $3+1=4$ $\mathrm{n}=2$ last digit is $3^{2}+1=9+1=10$ So, last digit is zero. $\mathrm{n}=3$ last digit is $3^{3}+1=27+1=28$<br/>last digit is 8 If $n=4$ last digit is $3^{4}+1=81+1=82$<br/>last digit is 2 . So, there is no definite value of last digit.</div>


title: what-is-the-least-positive-integer-n-for-which-1-i-1-i-n-1
slug: what-is-the-least-positive-integer-n-for-which-1-i-1-i-n-1-99187
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 2)']
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 least positive integer $n$ for which $\left(\frac{1+i}{1-i}\right)^{n}=1$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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">Consider $\frac{1+i}{1-i}=\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{(1+i)^{2}}{1-i^{2}}=\frac{(1+i)^{2}}{1+1}$<br/>$=\frac{1+i^{2}+2 i}{2}=\frac{2 i}{2}=i$<br/>$\therefore\left(\frac{1+i}{1-i}\right)^{n}=i^{n}$<br/>Now, $i^{n}=1$ is possible for $n=4$.</div>


title: what-is-the-least-value-of-25-cosec-2-x-36-sec-2-x
slug: what-is-the-least-value-of-25-cosec-2-x-36-sec-2-x-89562
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">121</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the least value of $25 \operatorname{cosec}^{2} x+36 \sec ^{2} x$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\overline{1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">11</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">120</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">121</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; 25 \operatorname{cosec}^{2} x+36 \sec ^{2} x . \\ &amp; \text { Minimum value }=(\sqrt{25}+\sqrt{36})^{2} \\ &amp;=(5+6)^{2}=(11)^{2}=121 \end{aligned}$</div>


title: what-is-the-least-value-of-the-standard-deviation-of-5-integers-no-two-of-which-are-equal
slug: what-is-the-least-value-of-the-standard-deviation-of-5-integers-no-two-of-which-are-equal-33861
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the least value of the standard deviation of 5 integers no two of which are equal?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{5}$</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">$\sqrt{2}$</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">D</span> <span class="option-data">No such least value can be computed</span> </li> </ul>

answer:
<div class="solution">Let us consider any five integers which are $3,4,5,6,7$.<br/>mean $=\frac{3+4+5+6+7}{5}=\frac{25}{5}=5$<br/>$S D=\sqrt{\frac{(5-3)^{2}+(5-4)^{2}+(5-5)^{2}+(5-6)^{2}+(5-7)^{2}}{5}}$<br/>$\quad=\sqrt{\frac{4+1+0+1+4}{5}=\sqrt{2}}$<br/>Hence, the least value of the standard deviation of $^{5}$ integers is $\sqrt{2}$</div>


title: what-is-the-least-vlaue-of-f-x-2-x-3-3-x-2-12-x-1-on-2-25
slug: what-is-the-least-vlaue-of-f-x-2-x-3-3-x-2-12-x-1-on-2-25-19205
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-19$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the least vlaue of $f(x)=2 x^{3}-3 x^{2}-12 x+1$ on $[-2,2.5] ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-19$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$-16.5$</span> </li> </ul>

answer:
<div class="solution">Let $f(x)=2 x^{3}-3 x^{2}-12 x+1$ $\Rightarrow f^{\prime}(x)=6 x^{2}-6 x-12$<br/>Put $f^{\prime}(x)=0$ for maxima or minima<br/>$\Rightarrow 6 x^{2}-6 x-12=0 \Rightarrow x^{2}-x-2=0$<br/>$\Rightarrow(x-2)(x+1)=0$<br/>$\Rightarrow \quad x=2, x=-1$<br/>Now, $f^{\prime \prime}(x)=12 x-6$<br/>$\Rightarrow f^{\prime \prime}(2)=24-6=18&gt;0$<br/>Since, $f^{\prime \prime}(x)$ is $+v e$ at $x=2$<br/>$\therefore f(x)$ is minimum at $x=2$. Hence, minimum value is $f(2)=2(2)^{3}-3(2)^{2}-12 \times 2+1$<br/>$=16-12-24+1=-19$</div>


title: what-is-the-length-of-arc-of-a-circle-of-radius-5-cm-subtending-a-central-angle-measuring-1-5
slug: what-is-the-length-of-arc-of-a-circle-of-radius-5-cm-subtending-a-central-angle-measuring-1-5-77004
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the length of arc of a circle of radius $5 \mathrm{~cm}$ subtending a central angle measuring $15^{\circ} ?$</div>

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

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

answer:
<div class="solution">We know length of arc of a circle $=2 \pi r \frac{\theta}{360}$ where ' $r$ ' is the radius and $\theta$ is the central angle. So, $r=5, \theta=15^{\circ}$<br/>$\therefore$ Length $=2 \pi \times 5 \times \frac{15^{\circ}}{360^{\circ}}=\frac{5 \pi}{12} \mathrm{~cm}$</div>


title: what-is-the-length-of-the-intercept-made-on-the-x-axis-by-the-circle-x2y22-g-x2-f-yc0
slug: what-is-the-length-of-the-intercept-made-on-the-x-axis-by-the-circle-x2y22-g-x2-f-yc0-84978
tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the length of the intercept made on the $x$ -axis by the circle $x^{2}+y^{2}+2 g x+2 f y+c=0 ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{\left(\mathrm{g}^{2}-\mathrm{c}\right)}}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{\left(\mathrm{g}^{2}-4 \mathrm{c}\right)}}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \sqrt{\left(\mathrm{g}^{2}-4 \mathrm{c}\right)}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2 \sqrt{\left(\mathrm{g}^{2}-\mathrm{c}\right)}$</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">$2 \sqrt{\left(\mathrm{g}^{2}-\mathrm{c}\right)}$</span> </div>

answer:
<div class="solution">Let the circle $\mathrm{S} \equiv \mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$ cut $\mathrm{x}$ -axis at<br/>two points $\mathrm{A}$, and $\mathrm{B}$ at $\mathrm{x}=\mathrm{a}$ and $\mathrm{x}=\mathrm{b}$. Let $C$ be the centre and $\mathrm{AB}$ the chord.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/3DGjEdA2c5x3ADlRZFLOJlwKArkgSaqqYEvxKtIwB_k.original.fullsize.png"/><br/><br/>Length of chord $\mathrm{AB}=\mathrm{b}-\mathrm{a}$. Perpendicular from centre c on chord bisect this chord at point M. Radius $r=\sqrt{g^{2}+f^{2}-c}$ also radius $A C=$<br/>$r=\sqrt{A M^{2}+C M^{2}}$<br/>$\mathrm{AM}=\frac{\mathrm{AB}}{2}=\frac{\mathrm{b}-\mathrm{a}}{2}$<br/>So, $\mathrm{r}=\sqrt{\left(\frac{\mathrm{b}-\mathrm{a}}{2}\right)^{2}+(-\mathrm{f})^{2}}$<br/>Thus, $\sqrt{g^{2}+f^{2}-c}=\sqrt{\left(\frac{b-a}{2}\right)^{2}+f^{2}}$<br/>or $\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}=\left(\frac{\mathrm{b}-\mathrm{a}}{2}\right)^{2}+\mathrm{f}^{2}$<br/>or $\left(\frac{\mathrm{b}-\mathrm{a}}{2}\right)^{2}=\mathrm{g}^{2}-\mathrm{c}$<br/>$\frac{\mathrm{b}-\mathrm{a}}{2}=\sqrt{\mathrm{g}^{2}-\mathrm{c}}$<br/>Hence, $\mathrm{b}-\mathrm{a}=\mathrm{AB}=2 \sqrt{\mathrm{g}^{2}-\mathrm{c}}$<br/>The length of the intercept made on the $x$ -axis by the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is $2 \sqrt{g^{2}-c}$.</div>


title: what-is-the-length-of-the-latus-rectum-of-the-ellipse-25-x-2-16-y-2-400
slug: what-is-the-length-of-the-latus-rectum-of-the-ellipse-25-x-2-16-y-2-400-74887
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$32/5$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the length of the latus rectum of the ellipse $25 x^{2}+16 y^{2}=400 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$25 / 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">25/4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$16 / 5$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$32/5$</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 ellipse is $25 x^{2}+16 y^{2}=400$ $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-length-of-the-longest-interval-in-which-the-function-f-x-3-sin-x-4-sin-3-x-is-increasing

slug:
what-is-the-length-of-the-longest-interval-in-which-the-function-f-x-3-sin-x-4-sin-3-x-is-increasing-33312

question:
<div class="question">What is the length of the longest interval in which the function $f(\mathrm{x})=3 \sin \mathrm{x}-4 \sin ^{3} \mathrm{x}$ is increasing?</div>

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=3 \sin \mathrm{x}-4 \sin ^{3} \mathrm{x}=\sin 3 \mathrm{x}$<br/>$\sin 3 x$ increases in $\left[\frac{-\pi}{6}, \frac{\pi}{6}\right]$<br/>So, interval length $=\frac{\pi}{3}$</div>


title: what-is-the-length-of-the-perpendicular-from-the-origin-to-the-plane-a-x-b-y-2-ab-z-1
slug: what-is-the-length-of-the-perpendicular-from-the-origin-to-the-plane-a-x-b-y-2-ab-z-1-81338
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the length of the perpendicular from the origin to<br/>the plane $a x+b y+\sqrt{2 a b} z=1 ?$</div>

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

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

answer:
<div class="solution">Length of perpendicular from the origin to the plane $a x+b y+\sqrt{2 a b} z-1=0$ is<br/>$=\left|\frac{0+0+0-1}{\sqrt{a^{2}+b^{2}+2 a b}}\right|=\frac{1}{\sqrt{(a+b)^{2}}}=\frac{1}{(a+b)}$</div>


title: what-is-the-length-of-the-vector-1-1
slug: what-is-the-length-of-the-vector-1-1-19682
question: <div class="question">What is the length of the vector $(1,1)$ ?</div>
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">$\frac{1}{2}$</span> </li> </ul>

answer:
<div class="solution">Given, vector is $(1,1)$.<br/>$\therefore \quad$ Length of vector $=\sqrt{1^{2}+1^{2}+0^{2}}=\sqrt{2}$.</div>


title: what-is-the-locus-of-a-point-which-is-equidistant-from-the-point-m-n-n-m-and-the-point-m-n-n-m
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-locus-of-a-point-which-is-equidistant-from-the-point-m-n-n-m-and-the-point-m-n-n-m-87629

question:
<div class="question">What is the locus of a point which is equidistant from the point $(m+n, n-m)$ and the point $(m-n, n+m)$ ?</div>

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

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

answer:
<div class="solution">Let the locus of a point be $(h, k)$ Let the given points be $P(m+n, n-m)$ and $\mathrm{Q}(m-n, n+m)$<br/>$\therefore$ By distance formula, we have $\sqrt{[h-(m+n)]^{2}+[k-(n-m)]^{2}}$<br/>$\quad=\sqrt{[h-(m-n)]^{2}+[k-(n+m)]^{2}}$<br/>$\Rightarrow h^{2}+(m+n)^{2}-2 h(m+n)+k^{2}$<br/>$+h^{2}+(m-n)^{2}-2 h(m-n)+k^{2}+(n-m)^{2}-2 k(n-m)$<br/>$\Rightarrow 2 h(m-n-m-n)+2 k(n+m-n+m)=0$<br/>$\Rightarrow-4 h n+4 k m=0 \Rightarrow m k=n h$<br/>Hence, locus of a point is $n x=m y$.</div>


title: what-is-the-locus-of-a-point-which-is-equidistant-from-the-points-a-b-a-b-and-b-a-a-b
slug: what-is-the-locus-of-a-point-which-is-equidistant-from-the-points-a-b-a-b-and-b-a-a-b-40268
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the locus of a point which is equidistant from the points $(a+b, a-b)$ and $(b-a, a+b) ?$</div>

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

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

answer:
<div class="solution">Let the coordinates of the point are $(x, y)$ $\therefore \sqrt{\{x-(a+b)\}^{2}+\{y-(a-b)\}^{2}}$<br/>$=\sqrt{\{x-(b-a)\}^{2}+\{y-(a+b)\}^{2}}$<br/>Squaring both sides, we get $\Rightarrow x^{2}+(a+b)^{2}-2 x(a+b)+y^{2}+(a-b)^{2}-2 y(a-b)$<br/>$=x^{2}+(b-a)^{2}-2 x(b-a)+y^{2}+(a+b)^{2}-2 y(a+b)$<br/>$\Rightarrow 2 x(a+b)+2 y(a-b)=2 x(b-a)+2 y(a+b)$<br/>$\Rightarrow x\{(a+b)-(b-a)\}+y\{(a-b)-(a+b)\}=0$<br/>$\Rightarrow 2 a x+(-2 b y)=0$<br/>$\Rightarrow a x-b y=0$<br/>$\Rightarrow-a x+b y=0$</div>


title: what-is-the-locus-of-a-point-which-moves-equidistant-from-the-coordinate-axes
slug: what-is-the-locus-of-a-point-which-moves-equidistant-from-the-coordinate-axes-66658
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the locus of a point which moves equidistant from the coordinate axes?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x \pm y=0$</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">$x+2 y=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x+y=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

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

answer:
<div class="solution">We have, $y=x$ ...(i)<br/>$y=-x$ ...(ii)<br/>$-x=-y$ ...(iii)<br/>$x=-y$ ...(iv)<br/>Thus, the required locus of a point which moves equidistant from the co-ordinate axes is $x \pm v=0$</div>


title: what-is-the-locus-of-points-of-intersection-of-a-sphere-and-a-plane
slug: what-is-the-locus-of-points-of-intersection-of-a-sphere-and-a-plane-98177
question: <div class="question">What is the locus of points of intersection of a sphere and a plane?</div>
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Circle</span> </div>
answer: <div class="solution">Locus of points of intersection of a sphere and a plane is circle.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Circle</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Ellipse</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Parabola</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Hyperbola</span> </li> </ul>


title: what-is-the-locus-of-points-the-difference-of-whose-distances-from-two-points-being-constant
slug: what-is-the-locus-of-points-the-difference-of-whose-distances-from-two-points-being-constant-49914
tags: ['Mathematics', 'Hyperbola', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the locus of points, the difference of whose distances from two points being constant?</div>

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

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

answer:
<div class="solution">We know that the locus of the difference of whose distances from two points being constant, is a hyperbola.</div>


title: what-is-the-locus-of-the-point-of-intersection-of-the-straight-lines-x-a-y-b-m-and-x-a-y-b-1-m
tags: ['Mathematics', 'Pair of Lines', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-locus-of-the-point-of-intersection-of-the-straight-lines-x-a-y-b-m-and-x-a-y-b-1-m-52024

question:
<div class="question">What is the locus of the point of intersection of the straight lines $(\mathrm{x} / \mathrm{a})+(\mathrm{y} / \mathrm{b})=\mathrm{m}$ and $(\mathrm{x} / \mathrm{a})-(\mathrm{y} / \mathrm{b})=1 / \mathrm{m}$ ?</div>

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

answer:
<div class="solution">The given equation of straight lines are<br/>$\frac{x}{a}+\frac{y}{b}=m$...(i)<br/>and $\frac{x}{a}-\frac{y}{b}=\frac{1}{m} \quad \ldots$ (ii)<br/>From Eqs. (1) and (2), we get $\left(\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}\right)\left(\frac{\mathrm{x}}{\mathrm{a}}-\frac{\mathrm{y}}{\mathrm{b}}\right)=\mathrm{m} \times \frac{1}{\mathrm{~m}}=1$<br/>$\Rightarrow \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$<br/>which represents a Hyperbola and is the locus of point of intersection the given straight lines.</div>


title: what-is-the-locus-of-the-point-which-is-at-a-distance-8-units-to-the-left-of-y-axis
slug: what-is-the-locus-of-the-point-which-is-at-a-distance-8-units-to-the-left-of-y-axis-25816
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the locus of the point which is at a distance 8 units to the left of $Y$ -axis?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad X=-8$</span> </div>

answer:
<div class="solution">Required locus is $X=-8$ which is at a distance of 8 units to the left of $Y$ -axis.</div>


title: what-is-the-locus-of-the-point-x-y-for-which-the-vectors-i-x-j-2-k-and-2-i-j-y-k-are-orthogonal
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
what-is-the-locus-of-the-point-x-y-for-which-the-vectors-i-x-j-2-k-and-2-i-j-y-k-are-orthogonal-56967

question:
<div class="question">What is the locus of the point $(\mathrm{x}, \mathrm{y})$ for which the vectors $(\hat{i}-x \hat{j}-2 \hat{k})$ and $(2 \hat{i}+\hat{j}+y \hat{k})$ are orthogonal ?</div>

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

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

answer:
<div class="solution">As given: $(\hat{i}-x \hat{j}-2 \hat{k})$ and $(2 \hat{i}+\hat{j}+y \hat{k})$ areorthogonal.<br/>So, there dot product $=0$<br/>$\Rightarrow(\hat{i}-x \bar{j}-2 \hat{k}) \cdot(2 \hat{i}+\hat{j}+y \hat{k})=0$<br/>$\Rightarrow 2-x-2 y=0$<br/>$\Rightarrow x+2 y=2$<br/>Which is an equation of straight line. Thus. the locus of the point $(\mathrm{x}, \mathrm{v})$ is a straight line.</div>


title: what-is-the-m-1-th-root-of-a-m-m-m-1-m-1-1
slug: what-is-the-m-1-th-root-of-a-m-m-m-1-m-1-1-39872
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">a</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the $(\mathrm{m}-1)^{\text {th }}$ root of $\left[\left(\mathrm{a}^{\mathrm{m}}\right)^{\mathrm{m}}-\left(\frac{1}{\mathrm{~m}}\right)\right]^{\frac{1}{\mathrm{~m}+1}}$ ?</div>

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

answer:
<div class="solution">The given expression $\left[\left(a^{m}\right)^{m-\frac{1}{m}}\right]^{\frac{1}{m+1}}$<br/>$=\left[\left(a^{m}\right)^{\frac{m^{2}-1}{m}}\right]^{\frac{1}{m+1}}=\left[\left(a^{m}\right)^{\frac{(m-1)(m+1)}{m}}\right]^{\frac{1}{m+1}}$<br/>$=\left[\mathrm{a}^{\mathrm{m}}\right]^{\frac{\mathrm{m}-1}{\mathrm{~m}}}=\mathrm{a}^{\mathrm{m}-1}$<br/>Its $(\mathrm{m}-1)^{t h}$ root $=\left(\mathrm{a}^{\mathrm{m}-1}\right)^{1 / \mathrm{m}-1}=\mathrm{a}$<br/>Ience, $(\mathrm{m}-1)^{\mathrm{th}}$ root of $\left[\left(\mathrm{a}^{\mathrm{m}}\right)^{\mathrm{m}-\left(\frac{1}{\mathrm{~m}}\right)}\right]^{\frac{1}{(\mathrm{~m}+1)}}$<br/>$=(\mathrm{m}-1)$ th root of $\mathrm{a}^{\mathrm{m}-1}=$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
what-is-the-magnitude-of-the-moment-of-the-couple-consisting-of-the-force-f-3-i-2-j-k-acting-through-the-point-i-j-k-and-and-f-acting-through-the-poin

slug:
what-is-the-magnitude-of-the-moment-of-the-couple-consisting-of-the-force-f-3-i-2-j-k-acting-through-the-point-i-j-k-and-and-f-acting-through-the-poin-95450

question:
<div class="question">What is the magnitude of the moment of the couple<br/>consisting of the force $\overrightarrow{\mathbf{F}}=3 \hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}-\hat{\mathbf{k}}$ acting through the<br/>point $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and and $-\overrightarrow{\mathbf{F}}$ acting through the<br/>point $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ ?</div>

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

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

answer:
<div class="solution">Here, $\overrightarrow{\mathrm{F}}=\hat{3} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$<br/>and $\overrightarrow{\mathrm{r}}_{1}-\overrightarrow{\mathrm{r}_{2}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$<br/>Moment of couple $=\left(\overrightarrow{\mathrm{r}_{1}}-\overrightarrow{\mathrm{r}_{2}}\right) \times \overrightarrow{\mathrm{F}}$<br/>$=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \times(\hat{3}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$<br/>$=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ -1 &amp; 2 &amp; 2 \\ 3 &amp; 2 &amp; -1\end{array}\right|=\hat{i}(-2-4)-\hat{j}(1-6)+\hat{k}(-2-6)$<br/>$=-6 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-8 \hat{\mathrm{k}}$<br/>Magnitude of the moment $=|-6 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-8 \hat{\mathrm{k}}|$<br/>$=\sqrt{36+25+64}=5 \sqrt{5}$</div>


title: what-is-the-main-concern-regarding-the-manipulation-of-living-organisms-by-the-human-race
slug: what-is-the-main-concern-regarding-the-manipulation-of-living-organisms-by-the-human-race-70664
tags: ['Biology', 'Biotechnology and Its Applications', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the main concern regarding the manipulation of living organisms by the human race?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The morality of human activities that might help or harm living organisms</span> </div>

answer:
<div class="solution">Some ethical standards are required to evaluate the morality of all human activities that might help or harm living organisms.</div>


title: what-is-the-main-purpose-of-the-international-code-for-botanical-nomenclature-icbn
slug: what-is-the-main-purpose-of-the-international-code-for-botanical-nomenclature-icbn-96880
tags: ['Biology', 'The Living World', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the main purpose of the International Code for Botanical Nomenclature (ICBN)?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">To provide scientific names for plants</span> </div>

answer:
<div class="solution">The International Code for Botanical Nomenclature (ICBN) provides principles and criteria on which scientific names for plants are based.</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-the-main-reason-for-the-different-reactivities-of-primary-secondary-and-tertiary-alcohols-in-the-lucas-test

slug:
what-is-the-main-reason-for-the-different-reactivities-of-primary-secondary-and-tertiary-alcohols-in-the-lucas-test-52177

question:
<div class="question">What is the main reason for the different reactivities of primary, secondary, and tertiary alcohols in the Lucas test?</div>

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

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

answer:
<div class="solution">The different reactivities of primary, secondary, and tertiary alcohols in the Lucas test are primarily due to steric hindrance caused by the number of alkyl groups attached to the carbon bearing the hydroxyl group. Tertiary alcohols have the least steric hindrance and react</div>


title: what-is-the-major-product-of-the-hofmann-rearrangement-of-a-primary-amide
slug: what-is-the-major-product-of-the-hofmann-rearrangement-of-a-primary-amide-21734
tags: ['Chemistry', 'Amines', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the major product of the Hofmann rearrangement of a primary amide?</div>

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

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

answer:
<div class="solution">Primary amine<br/><br/>Explanation: The Hofmann rearrangement involves the conversion of a primary amide to a primary amine with one fewer carbon atom. This reaction typically uses bromine $\left(\mathrm{Br}_2\right)$ and a strong base like $\mathrm{NaOH}$.</div>


title: what-is-the-major-product-of-the-reaction-between-an-amide-and-lialh-4-followed-by-hydrolysis
slug: what-is-the-major-product-of-the-reaction-between-an-amide-and-lialh-4-followed-by-hydrolysis-10253
tags: ['Chemistry', 'Amines', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the major product of the reaction between an amide and $\mathrm{LiAlH}_4$ followed by hydrolysis?</div>

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

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

answer:
<div class="solution">Primary amine<br/><br/>Explanation: $\mathrm{LiAlH}_4$ reduces the amide to a primary amine, and subsequent hydrolysis of the reduced product yields the primary amine.</div>


title: what-is-the-maximum-area-of-a-triangle-that-can-be-inscribed-in-a-circle-of-radius-a
slug: what-is-the-maximum-area-of-a-triangle-that-can-be-inscribed-in-a-circle-of-radius-a-48512
tags: ['Mathematics', 'Area Under Curves', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is the maximum area of a triangle that can be inscribed in a circle of radius a?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3 \sqrt{3} a^{2}}{4}$</span> </div>

answer:
<div class="solution">For area of triangle to be maximum, it should be equilateral triangle.<br/>Area of $\Delta \mathrm{OAB}=\frac{1}{2} \mathrm{ab} \sin \theta$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/T7bz_HMxOnZx7ukStpzMlOHYhV_H6xlKP7WrcdANvks.original.fullsize.png"/><br/><br/>$=\frac{1}{2} \cdot$ a $a \cdot \sin \theta$<br/>$=\frac{1}{2} \mathrm{a}^{2} \cdot \sin 120^{\circ}$<br/>$=\frac{\sqrt{3}}{4} \mathrm{a}^{2}$<br/>Area of triangle $\mathrm{OAB}, \mathrm{OBC}, \mathrm{OAC}$<br/>$=$ Area of triangle $\mathrm{ABC}$<br/>$=3 \times \frac{\sqrt{3}}{4} a^{2}=\frac{3 \sqrt{3}}{4} \mathrm{a}^{2}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
answer: <div class="solution">Required number of lines $={ }^{4} C_{2}=\frac{4 !}{2 ! 2 !}=6$</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-maximum-number-of-straight-lines-that-can-be-drawn-with-any-four-points-in-a-plane-such-that-each-line-contains-at-least-two-of-these-poin

slug:
what-is-the-maximum-number-of-straight-lines-that-can-be-drawn-with-any-four-points-in-a-plane-such-that-each-line-contains-at-least-two-of-these-poin-53007

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


title: what-is-the-maximum-point-on-the-curve-x-e-x-y
slug: what-is-the-maximum-point-on-the-curve-x-e-x-y-19794
question: <div class="question">What is the maximum point on the curve $x=e^{x} y$ ?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(1, e)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(1, e^{-1}\right)$</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">$(e, 1)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(e^{-1}, 1\right)$</span> </li> </ul>

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

answer:
<div class="solution">Given curve is $x=e^{x} y$<br/>Which can be rewritten as $y=x e^{-x}$ On differentiating w.r.t. $x$, we get<br/>$\frac{d y}{d x}=-x e^{-x}+e^{-x}$<br/>Put $\frac{d y}{d x}=0$ for maxima or minima<br/>$\Rightarrow-x e^{-x}+e^{-x}=0$<br/>$\Rightarrow e^{-x}(1-x)=0$<br/>Since $e^{-x}$ can not be zero $\therefore \quad 1-x=0 \Rightarrow x=1$<br/>Now, $\frac{d^{2} y}{d x^{2}}=-e^{-x}+x e^{-x}-e^{-x}=x e^{-x}-2 e^{-x}$<br/>$=e^{-x}(x-2)$<br/>$\Rightarrow\left(\frac{d^{2} y}{d x^{2}}\right)   &lt; 0$<br/>$\therefore y$ is maximum at $x=1$. Thus, when $x=1$ then $\mathrm{y}=e^{-1}$<br/>Hence, maximumpoint on the curve $x=e^{x} y$ is $\left(1, e^{-1}\right)$.</div>


title: what-is-the-maximum-slope-of-the-curve-y-x-3-3-x-2-2-x-27
slug: what-is-the-maximum-slope-of-the-curve-y-x-3-3-x-2-2-x-27-93266
question: <div class="question">What is the maximum slope of the curve $y=-x^{3}+3 x^{2}+2 x-27 ?$</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P4.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">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-23$</span> </li> </ul>

answer:
<div class="solution">The equation of curve is given as:<br/>$y=-x^{\frac{1}{3}}+3 x^{2}+2 x-27$<br/>On differentiating w.r.t.x, we get<br/>$\frac{d y}{d x}=-3 x^{2}+6 x+2$<br/>This represents slope of the curve at any point. Let $\quad A=\frac{d y}{d x}=-3 x^{2}+6 x+2$<br/>$\Rightarrow \frac{\mathrm{d} \mathrm{A}}{\mathrm{dx}}=-6 \mathrm{x}+6$<br/>and $\frac{d^{2} A}{d x^{2}}=-6$<br/>Put $\frac{\mathrm{d} \mathrm{A}}{\mathrm{dx}}=0$ for maxima or minima. $-6 x+6=0$<br/>$\Rightarrow \mathrm{x}=1$<br/>Now, $\left(\frac{\mathrm{d}^{2} \mathrm{~A}}{\mathrm{dx}^{2}}\right)_{\mathrm{x}=1} 1=-6   &lt; 0$<br/>$\therefore \quad$ A is maximumat $x=1$ $\therefore \quad$ Maximum slope of curve $=-3+6+2=5$</div>


title: what-is-the-maximum-value-of-3-cos-x-4-sin-x-5
slug: what-is-the-maximum-value-of-3-cos-x-4-sin-x-5-39894
question: <div class="question">What is the maximum value of $3 \cos x+4 \sin x+5$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P4.db
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">7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">12</span> </li> </ul>

answer:
<div class="solution">Maximum value of<br/>$$<br/>\begin{aligned}<br/>3 \cos x+4 \sin x+5 &amp;=\sqrt{(4)^{2}+(3)^{2}}+5=\sqrt{16+9}+5 \\<br/>&amp;=\sqrt{25}+5=5+5=10<br/>\end{aligned}<br/>$$</div>


title: what-is-the-maximum-value-of-16-sin-12-sin-2
slug: what-is-the-maximum-value-of-16-sin-12-sin-2-25682
question: <div class="question">What is the maximum value of $16 \sin \theta-12 \sin ^{2} \theta$ ?</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{4}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{16}{3}$</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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{16}{3}$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\theta)=16 \sin \theta-12 \sin ^{2} \theta$<br/>$f^{\prime}(\theta)=16 \cos \theta-24 \sin \theta \cos \theta$<br/>$f^{\prime}(\theta)=0 \Rightarrow \cos \theta(16-24 \sin \theta)=0$<br/>$\Rightarrow \cos \theta=0$ (or) $16-24 \sin \theta=0$<br/>$\Rightarrow \theta=\frac{\pi}{2}($ or $) \sin \theta=\frac{2}{3}$<br/>$\mathrm{f}\left(\frac{\pi}{2}\right)=16 \sin \frac{\pi}{2}-12 \sin ^{2} \frac{\pi}{2}=16-12=4$<br/>$\mathrm{f}\left(\sin \theta=\frac{2}{3}\right)=16\left(\frac{2}{3}\right)-12\left(\frac{2}{3}\right)^{2}$<br/>$=\frac{32}{3}-\frac{48}{9}=\frac{32}{3}-\frac{16}{3}=\frac{16}{3} .$</div>


title: what-is-the-maximum-value-of-sin-3-cos-2-cos-3-sin-2
slug: what-is-the-maximum-value-of-sin-3-cos-2-cos-3-sin-2-39053
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the maximum value of $\sin 3 \theta \cos 2 \theta+\cos 3 \theta \sin 2 \theta$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">Consider $\sin 3 \theta \cos 2 \theta+\cos 3 \theta \sin 2 \theta$<br/>$=\sin (3 \theta+2 \theta) \quad\left[\begin{array}{c}\because \sin \mathrm{A} \cos \mathrm{B}+\cos \mathrm{A} \sin \mathrm{B} \\ =\sin (\mathrm{A}+\mathrm{B})\end{array}\right]$<br/>$=\sin (5 \theta)$<br/>We know, $-1   &lt; \sin \theta &lt; 1$<br/>Hence, maximum value of given expression is $1 .$</div>


title: what-is-the-maximum-value-of-sin-theta-cos-theta
slug: what-is-the-maximum-value-of-sin-theta-cos-theta-92072
question: <div class="question">What is the maximum value of $\sin \theta \cos \theta ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 / 2$</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">$1 / \sqrt{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{3} / 2$</span> </li> </ul>

answer:
<div class="solution">Let $P=\sin \theta \cos \theta$<br/>Multiply and divide by 2 , we get $P=\frac{2 \sin \theta \cos \theta}{2}=\frac{\sin 2 \theta}{2}$<br/>The maximum value of $\sin 2 \theta$ is 1 .<br/>$\therefore$ Maximum value of $P=\frac{1}{2}$</div>


title: what-is-the-maximum-value-of-the-function-f-x-4-sin-2-x-1
slug: what-is-the-maximum-value-of-the-function-f-x-4-sin-2-x-1-46381
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the maximum value of the function $f(\mathrm{x})=4 \sin ^{2} \mathrm{x}+1 ?$</div>

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

answer:
<div class="solution">$\quad$ We know, $-1 \leq \sin x \leq 1$<br/>$\Rightarrow 0 \leq \sin ^{2} x \leq 1$<br/>$\Rightarrow 0 \leq 4 \sin ^{2} x \leq 4$<br/>$\Rightarrow 1 \leq 4 \sin ^{2} x+1 \leq 5$</div>


title: what-is-the-maximum-value-of-the-function-lo-g-x-x
slug: what-is-the-maximum-value-of-the-function-lo-g-x-x-50574
question: <div class="question">What is the maximum value of the function $\log x-x ?$</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P4.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">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\infty$</span> </li> </ul>

answer:
<div class="solution">Let $y=\log x-x$ $\therefore \frac{d y}{d x}=\frac{1}{x}-1$ and $\frac{d^{2} y}{d x^{2}}=-\frac{1}{x^{2}}$<br/>For maximum and minimum value of $y$<br/>$\frac{d y}{d x}=\frac{1}{x}-1=0 \Rightarrow \frac{1}{x}=1 \Rightarrow x=1$<br/>For $x=1, \frac{d^{2} y}{d x^{2}}=-v e$<br/>Thus, the value of given function is maximum for $x=1$<br/>So, the maximum value of the function $=\log (1)-1=-1$</div>


title: what-is-the-maximum-value-of-x-y-subject-to-the-condition-x-y-8
slug: what-is-the-maximum-value-of-x-y-subject-to-the-condition-x-y-8-31522
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the maximum value of $\mathrm{x}$. y subject to the condition $\mathrm{x}+\mathrm{y}=8 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">24</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">32</span> </li> </ul>

answer:
<div class="solution">The given condition is:<br/>$x+y=8 \Rightarrow y=8-x$<br/>and let, the product of $\mathrm{x}$ and $\mathrm{y}$ be, $\mathrm{P}=\mathrm{xy}$<br/>$\Rightarrow P=x(8-x)=8 x-x^{2}$<br/>Differentiating w.r.t. $\mathrm{x}$<br/>$\Rightarrow \frac{\mathrm{dP}}{\mathrm{dx}}=8-2 \mathrm{x}$<br/>and $\frac{\mathrm{d}^{2} \mathrm{P}}{\mathrm{dx}^{2}}=-2   &lt; 0$<br/>Put $\frac{\mathrm{dP}}{\mathrm{dx}}=0$, for maxima or minima<br/>$8-2 x=0$<br/>$\Rightarrow x=\frac{8}{2}=4$ and $y=4$<br/>and $\left(\frac{\mathrm{d}^{2} \mathrm{P}}{\mathrm{dx}^{2}}\right) &lt; 0$ when $\mathrm{x}=\mathrm{y}$<br/>$\therefore$ P is maximumat $\mathrm{x}=4$ Maximum value of $\mathrm{P}=4 \cdot(8-4)=4.4=16$</div>


title: what-is-the-mean-deviation-about-the-mean-for-the-data-4-8-9-10-12-13-17
slug: what-is-the-mean-deviation-about-the-mean-for-the-data-4-8-9-10-12-13-17-49275
tags: ['Mathematics', 'Statistics', '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_P4.db
question:
<div class="question">What is the mean deviation about the mean for the data 4, $8,9,10,12,13,17 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">25</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Mean $=(\bar{x})-\frac{\sum x_{1}}{N}$<br/>Here $\bar{x}=\frac{4+7+8+9+10+12+13+17}{8}=16$<br/>Mean deviation about mean $=\frac{\sum\left|\mathrm{x}_{i}-\overline{\mathrm{x}}\right|}{\mathrm{N}}$<br/>$|4-10|+|7-10|+|8-10|+|9-10|+|10-10|$<br/>$=\frac{+|12-10|+|13-10|+|17-10|}{8}$<br/>$=\frac{6+3+2+1+0+2+3+7}{8}=\frac{24}{8}=3$</div>


title: what-is-the-mean-deviation-from-the-mean-of-the-numbers-10-9-21-16-24
slug: what-is-the-mean-deviation-from-the-mean-of-the-numbers-10-9-21-16-24-86442
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$5-2$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the mean deviation from the mean of the numbers $10,9,21,16,24 ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$5-2$</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">$5.0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4.0$</span> </li> </ul>

answer:
<div class="solution">Given numbers $-10,9,21,16,24$ Mean $=\frac{10+9+21+16+24}{5}=\frac{80}{5}=16$<br/>\begin{array}{|c|c|}<br/>\hline Numbers &amp; Distance (d) from mean (16) \\<br/>\hline 10 &amp; 6 \\<br/>\hline 9 &amp; 7 \\<br/>\hline 21 &amp; 5 \\<br/>\hline 16 &amp; 0 \\<br/>\hline 24 &amp; 8 \\<br/>\hline\Sigma d=26 &amp; \\<br/>\hline<br/>\end{array}<br/>Mean deviation $=\frac{\Sigma d}{5}=\frac{26}{5}=5.2$</div>


title: what-is-the-mean-deviation-of-the-data-2-9-9-3-6-9-4
slug: what-is-the-mean-deviation-of-the-data-2-9-9-3-6-9-4-58108
question: <div class="question">What is the mean deviation of the data $2,9,9,3,6,9,4 ?$</div>
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.23$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$257 \quad$</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">C</span> <span class="option-data">$3.23$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.57$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$257 \quad$</span> </div>

answer:
<div class="solution">Given data is $2,9,9,3,6,9,4$. We know,<br/>bservatuon Mean $=\frac{\text { Sum or an }}{\text { Total number of }}$<br/>Mean $=\frac{2+9+9+3+6+9+4}{7}=\frac{42}{7}=6$<br/>Mean deviation<br/>$=\frac{|2-6|+3|9-6|+|3-6|+|6-6|+|4-6|}{7}$<br/>$=\frac{4+9+3+0+2}{7}=\frac{18}{7}=2.57$</div>


title: what-is-the-mean-of-first-n-odd-natural-numbers
slug: what-is-the-mean-of-first-n-odd-natural-numbers-20448
question: <div class="question">What is the mean of first $\mathrm{n}$ odd natural numbers</div>
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{n}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{(\mathrm{n}+1)}{2} \quad$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{n(n+1)}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{n}+1$</span> </li> </ul>

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

answer:
<div class="solution">Sum of first n odd natural numbers $=\mathrm{n}^{2}$.<br/>Now, mean $=\frac{\mathrm{n}^{2}}{\mathrm{n}}=\mathrm{n}$</div>


title: what-is-the-mean-of-the-squares-of-the-first-20-natural-numbers
slug: what-is-the-mean-of-the-squares-of-the-first-20-natural-numbers-62310
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the mean of the squares of the first 20 natural numbers ? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$151.5$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$143.5$</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">65</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">72</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$143.5$</span> </div>

answer:
<div class="solution">Mean of the squares of the first 20 natural number<br/>$=\frac{(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}=\frac{21 \times 41}{6}=143.5$</div>


title: what-is-the-meaning-of-vexillary-or-papilionaceous-aestivation
slug: what-is-the-meaning-of-vexillary-or-papilionaceous-aestivation-31203
question: <div class="question">What is the meaning of vexillary or papilionaceous aestivation?</div>
tags: ['Biology', 'Morphology of Flowering Plants', 'NEET']
source_db: MarksBatch2_P4.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="correct"> <span class="option-label">C</span> <span class="option-data">When the largest petal overlaps the two lateral petals which in turn overlap the two smallest anterior petals</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">When sepals or petals just touch one another at the margin, without overlapping</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">When the largest petal overlaps the two lateral petals which in turn overlap the two smallest anterior petals</span> </div>

answer:
<div class="solution">Vexillary or papilionaceous aestivation refers to a situation where there are five petals in a flower, and the largest petal (standard) overlaps the two smallest anterior petals.</div>


title: what-is-the-measure-of-the-angle-11-4-3-5-3-0-in-radian
slug: what-is-the-measure-of-the-angle-11-4-3-5-3-0-in-radian-90049
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the measure of the angle $114^{\circ} 35^{\prime} 30^{\prime}$ in radian?</div>

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

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

answer:
<div class="solution">From relation between minute 6 seconds measure $60^{13}=\mathrm{I}^{\circ}$<br/>$\Rightarrow 30^{\prime \prime}=\frac{1^{\prime}}{2}$<br/>$\Rightarrow 35^{\prime} 30^{\prime \prime}=\left(35+\frac{1}{2}\right)^{\prime}=\left(\frac{71}{2}\right)^{\prime}$<br/>Also, $60^{\circ}=1^{\circ}$<br/>$\therefore 1^{\prime}=\left(\frac{1}{60}\right)^{\circ}$<br/>$\Rightarrow\left(\frac{71}{2}\right)^{\prime}=\left(\frac{71}{2} \times \frac{1}{60}\right)^{\circ}=\left(\frac{71}{120}\right)^{\circ}$<br/>$\therefore \quad 114^{\circ} 35^{\circ} 30^{\prime}=\left(114+\frac{71}{120}\right)^{\circ}=\left(\frac{13751}{120}\right)^{\circ}$<br/>We know that, $2 \pi \mathrm{rad}=360^{\circ}$<br/>$\Rightarrow 1^{\circ}=\frac{2 \pi}{360} \mathrm{rad}$<br/>$\Rightarrow\left(\frac{13751}{120}\right)^{\circ}=\frac{2 \pi}{360^{\circ}} \times \frac{13751}{120} \mathrm{rad}$<br/>$=\frac{2 \times 22 \times 13751}{7 \times 360 \times 120} \mathrm{rad}=2.0008069 \mathrm{rad}$<br/>$\Rightarrow 114^{\circ} 35^{\prime} 30^{\prime \prime}=2 \mathrm{rad}($ approx. $)$</div>


title: what-is-the-measured-emf-of-the-cell-with-the-representation-pts-h2g-1-bar-haq-1-m-cu2-aq1-m-cu
tags: ['Chemistry', 'Electrochemistry', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.34 V</span> </div>
source_db: MarksBatch2_P4.db
slug:
what-is-the-measured-emf-of-the-cell-with-the-representation-pts-h2g-1-bar-haq-1-m-cu2-aq1-m-cu-66361

question:
<div class="question">What is the measured emf of the cell with the representation Pt(s) |H2(g, 1 bar) |H+(aq, 1 M) | |Cu2+ (aq,1 M) | Cu?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.34 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">1.10 V</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-0.76 V</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.44 V</span> </li> </ul>

answer:
<div class="solution">The measured emf of the cell:Pt(s) |H2(g, 1 bar) |H+(aq, 1 M) | |Cu2+ (aq,1 M) | Cu is 0.34 V and it is also the value for the standard electrode potential of the half-cell corresponding to the reaction: Cu2+ (aq, 1M) + 2 e-  â Cu(s).</div>


title: what-is-the-measured-emf-of-the-cell-with-the-representation-pts-h2g-1-bar-haq-1-m-zn2-aq-1m-zn
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P4.db
slug:
what-is-the-measured-emf-of-the-cell-with-the-representation-pts-h2g-1-bar-haq-1-m-zn2-aq-1m-zn-55795

question:
<div class="question">What is the measured emf of the cell with the representation Pt(s) | H2(g, 1 bar) | H+(aq, 1 M) | | Zn2+ (aq, 1M) | Zn?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.34 V</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.10 V</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">-0.76 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">D</span> <span class="option-data">-1.44 V</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">-0.76 V</span> </div>

answer:
<div class="solution">The measured emf of the cell: Pt(s) | H2(g, 1 bar) | H+(aq, 1 M) | | Zn2+ (aq, 1M) | Zn is -0.76 V corresponding to the standard electrode potential of the half-cell reaction: Zn2+ (aq, 1 M) + 2e-  â Zn(s).</div>


title: what-is-the-median-of-the-distribution-3-7-6-9-5-4-2
slug: what-is-the-median-of-the-distribution-3-7-6-9-5-4-2-23624
question: <div class="question">What is the median of the distribution $3,7,6,9,5,4,2$ ?</div>
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">$\therefore$ Required Median $=\left(\frac{n+1}{2}\right)^{\mathrm{th}}$ obs $=4^{\mathrm{th}} \mathrm{obs}=5$</div>


title: what-is-the-median-of-the-numbers-46-0-93-48-76-2-127-35-82-61-39-52
slug: what-is-the-median-of-the-numbers-46-0-93-48-76-2-127-35-82-61-39-52-24656
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$4.9$</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the median of the numbers $4.6,0,9.3,-4.8,7.6,2$. $12.7,3.5,8.2,6.1,3.9,5.2 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3.8$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4.9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5.7$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6.0$</span> </li> </ul>

answer:
<div class="solution">By arranging the given numbers in ascending order, $-4.8,0,2.3,3.5,3.9, \frac{4.6,5.2}{\downarrow}, 6.1,7.6,8.2,9.3,12.7$<br/>Median $=\frac{4.6+5.2}{2}$<br/>$=\frac{9.8}{2}=4.9$</div>


title: what-is-the-middle-term-in-the-expansion-of-1-2-x-8
slug: what-is-the-middle-term-in-the-expansion-of-1-2-x-8-59337
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the middle term in the expansion of $\left(1-\frac{x}{2}\right)^{8}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{35 x^{4}}{8}$</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">$\frac{17 x^{5}}{8}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{35 x^{5}}{8}$</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">$\frac{35 x^{4}}{8}$</span> </div>

answer:
<div class="solution">Since $n=8$ is even number therefore middle term<br/>$=\left(\frac{n}{2}+1\right)^{\mathrm{th}}$ term $=(4+1)=5^{\mathrm{th}}$ term<br/>Hence, $T_{5}={ }^{8} C_{4}(1)^{4}\left(-\frac{x}{2}\right)^{4}$<br/>$=\frac{8 !}{4 ! 4 !} \times \frac{x^{4}}{16}=\frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \cdot \frac{x^{4}}{16}=\frac{70 x^{4}}{16}=\frac{35 x^{4}}{8}$</div>


title: what-is-the-middle-term-in-the-expansion-of-3-x-y-y-x-3-12
slug: what-is-the-middle-term-in-the-expansion-of-3-x-y-y-x-3-12-49371
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the middle term in the expansion of $\left(\frac{\mathrm{x} \sqrt{\mathrm{y}}}{3}-\frac{3}{\mathrm{y} \sqrt{\mathrm{x}}}\right)^{12}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{C}(12,7) \mathrm{x}^{3} \mathrm{y}^{-3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{C}(12,6) \mathrm{x}^{-3} \mathrm{y}^{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{C}(12,7) \mathrm{x}^{-3} \mathrm{y}^{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$C(12,6) \times^{3} y^{-3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$C(12,6) \times^{3} y^{-3}$</span> </div>

answer:
<div class="solution">In the expansion of $\left(\frac{\mathrm{x} \sqrt{\mathrm{y}}}{3}-\frac{3}{\mathrm{y} \sqrt{\mathrm{x}}}\right)^{12}$, then middle term<br/>is $\frac{12}{2}+1=7^{\text {th }}$ term. $(\mathrm{r}+1)_{\mathrm{th}}$ term<br/>$\mathrm{T}_{\mathrm{r}+1}={ }^{12} \mathrm{C}_{\mathrm{r}}\left[\frac{\mathrm{x} \sqrt{\mathrm{y}}}{3}\right]^{12-\mathrm{r}} \cdot\left(-\frac{3}{\mathrm{y} \sqrt{\mathrm{x}}}\right)^{\mathrm{r}}$<br/>$\therefore \quad \mathrm{T}_{7}=\mathrm{T}_{6+1}={ }^{12} \mathrm{C}_{6}\left(\frac{\mathrm{x} \sqrt{\mathrm{y}}}{3}\right)^{6}\left(-\frac{3}{\mathrm{y} \sqrt{\mathrm{x}}}\right)^{6}$<br/>$={ }^{12} \mathrm{C}_{6} \frac{\mathrm{x}^{6} \mathrm{y}^{3}}{\mathrm{y}^{6} \mathrm{x}^{3}}={ }^{12} \mathrm{C}_{6} \mathrm{x}^{3} \mathrm{y}^{-3}=\mathrm{C}(12,6) \mathrm{x}^{3} \mathrm{y}^{-3}$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-minimum-ph-required-to-prevent-the-precipitation-of-zns-in-a-solution-that-is-001-m-z-n-c-l-2-and-saturated-with-010-m-h-2-s-g-i-v-e-n-k-s

slug:
what-is-the-minimum-ph-required-to-prevent-the-precipitation-of-zns-in-a-solution-that-is-001-m-z-n-c-l-2-and-saturated-with-010-m-h-2-s-g-i-v-e-n-k-s-72360

question:
<div class="question">What is the minimum pH required to prevent the precipitation of ZnS in a solution that is <math><mn>0.01</mn><mtext mathvariant="normal"> </mtext><mtext mathvariant="normal">M</mtext><mtext mathvariant="normal"> </mtext><mtext mathvariant="normal">Z</mtext><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">C</mtext><msub><mrow><mtext mathvariant="normal">l</mtext></mrow><mrow><mn>2</mn></mrow></msub><mtext mathvariant="normal"> </mtext></math> and saturated with <math><mn>0.10</mn><mtext mathvariant="normal"> </mtext><mtext mathvariant="normal">M</mtext><mtext mathvariant="normal"> </mtext><msub><mrow><mtext mathvariant="normal">H</mtext></mrow><mrow><mn>2</mn></mrow></msub><mtext mathvariant="normal">S</mtext><mtext mathvariant="normal"> </mtext><mo>?</mo></math><br/> <br/> <math><mfenced close="]" open="[" separators="|"><mrow><mtext mathvariant="normal">G</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal"> </mtext><msub><mrow><mtext mathvariant="normal">K</mtext></mrow><mrow><mtext mathvariant="normal">s</mtext><mtext mathvariant="normal">p</mtext></mrow></msub><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>21</mn></mrow></msup><mo>,</mo><mtext mathvariant="normal"> </mtext><msub><mrow><mtext mathvariant="normal">K</mtext></mrow><mrow><msub><mrow><mtext mathvariant="normal">a</mtext></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>Ã</mo><msub><mrow><mtext mathvariant="normal">K</mtext></mrow><mrow><msub><mrow><mtext mathvariant="normal">a</mtext></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>20</mn></mrow></msup></mrow></mfenced></math></div>

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

answer:
<div class="solution"><math><mtext>Z</mtext><mtext>n</mtext><mtext>S</mtext><mover><mrow><mo>â</mo></mrow></mover><mtext>Z</mtext><msup><mrow><mtext>n</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>+</mo><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></math><br/><br/><math><msub><mrow><mtext>K</mtext></mrow><mrow><mtext>s</mtext><mtext>p</mtext></mrow></msub><mo>=</mo><mfenced close="]" open="[" separators="|"><mrow><mtext>Z</mtext><msup><mrow><mtext>n</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup></mrow></mfenced><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></mrow></mfenced></math><br/><br/><math><mtext>Z</mtext><mtext>n</mtext><mtext>C</mtext><msub><mrow><mtext>l</mtext></mrow><mrow><mn>2</mn></mrow></msub><mo>â</mo><mtext>Z</mtext><msup><mrow><mtext>n</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>+</mo><mn>2</mn><mtext>C</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>-</mo></mrow></msup></math><br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/125512_solutiondanjkadasfasdasdf.jpg" style="width: 255px; height: 79px;"/><br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/125512_solutiondanjkad_344r_2.jpg" style="width: 160px; height: 88px;"/><br/><br/><math><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mtext>K</mtext></mrow><mrow><mtext>s</mtext><mtext>p</mtext></mrow></msub></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><mtext>Z</mtext><msup><mrow><mtext>n</mtext></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>21</mn></mrow></msup></mrow><mrow><mfenced separators="|"><mrow><mn>0.01</mn></mrow></mfenced></mrow></mfrac></math><br/><br/><math><mo>â</mo><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></mrow></mfenced><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>19</mn></mrow></msup></math><br/><br/><math><mtext>K</mtext><msub><mrow><mtext>a</mtext></mrow><mrow><mn>1</mn></mrow></msub><mo>.</mo><mtext>K</mtext><msub><mrow><mtext>a</mtext></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></mrow></mfenced></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msub><mrow><mtext>H</mtext></mrow><mrow><mn>2</mn></mrow></msub><mtext>S</mtext></mrow></mfenced></mrow></mfrac></math><br/><br/>Where, <math><msub><mrow><mtext>K</mtext></mrow><mrow><msub><mrow><mtext>a</mtext></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><mfrac><mrow><mo>[</mo><mtext>H</mtext><msup><mrow><mtext>S</mtext></mrow><mrow><mo>-</mo></mrow></msup><mo>]</mo><mo>[</mo><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup><mo>]</mo></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msub><mrow><mtext>H</mtext></mrow><mrow><mn>2</mn></mrow></msub><mtext>S</mtext></mrow></mfenced></mrow></mfrac></math><br/><br/><math><msub><mrow><mtext>K</mtext></mrow><mrow><msub><mrow><mtext>a</mtext></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><mfrac><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></mrow></mfenced></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><mtext>H</mtext><msup><mrow><mtext>S</mtext></mrow><mrow><mo>-</mo></mrow></msup></mrow></mfenced></mrow></mfrac></math><br/><br/>So, <math><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>20</mn></mrow></msup><mo>=</mo><mfrac><mrow><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>S</mtext></mrow><mrow><mn>2</mn><mo>-</mo></mrow></msup></mrow></mfenced></mrow><mrow><mfenced close="]" open="[" separators="|"><mrow><msub><mrow><mtext>H</mtext></mrow><mrow><mn>2</mn></mrow></msub><mtext>S</mtext></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="|"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>19</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><mn>0.1</mn></mrow></mfenced></mrow></mfrac></math><br/><br/><math><mo>â</mo><mtext> </mtext><mtext> </mtext><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced><mo>=</mo><mn>0.1</mn></math><br/><br/><math><mo>â</mo><mtext>p</mtext><mtext>H</mtext><mo>=</mo><mo>-</mo><mrow><mrow><mtext mathvariant="normal">log</mtext></mrow><mo>â¡</mo><mrow><mfenced close="]" open="[" separators="|"><mrow><msup><mrow><mtext>H</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfenced></mrow></mrow><mo>=</mo><mn>1</mn></math></div>


title: what-is-the-minimum-value-of-2-x-2-3-x-5
slug: what-is-the-minimum-value-of-2-x-2-3-x-5-60460
question: <div class="question">What is the minimum value of $2 x^{2}-3 x+5 ? \quad$</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$31 / 4$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$31 / 8$</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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$31 / 8$</span> </div>

answer:
<div class="solution">Let $y=2 x^{2}-3 x+5$<br/>$\Rightarrow \frac{d y}{d x}=4 x-3$<br/>and $\frac{d^{2} y}{d x^{2}}=4$<br/>For maximum and minimum value of $y$<br/>$\frac{d y}{d x}=0$<br/>$\Rightarrow 4 x-3=0$<br/>$\Rightarrow \quad x=\frac{3}{4}$<br/>Since, for every value of $x, \frac{d^{2} y}{d x^{2}}=+v e$ So, minimum value of $2 x^{2}-3 x+5$ is minimum<br/>$x=\frac{3}{4}$<br/>$\therefore \quad$ The minimum value $=2\left(\frac{3}{4}\right)^{2}-3\left(\frac{3}{4}\right)+5$<br/>$=\frac{9}{8}-\frac{9}{4}+5=\frac{9-18+40}{8}=\frac{31}{8}$</div>


title: what-is-the-minimum-value-of-a-2-x-b-2-y-where-x-y-c-2
slug: what-is-the-minimum-value-of-a-2-x-b-2-y-where-x-y-c-2-62525
question: <div class="question">What is the minimum value of $a^{2} x+b^{2} y$ where $x y=c^{2} ?$</div>
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{abc}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{abc}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 \mathrm{abc}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 a b c$</span> </li> </ul>

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

answer:
<div class="solution">Let $p=a^{2} x+b^{2} y$ and $x y=c^{2}$ <br/>$\Rightarrow y=\frac{c^{2}}{x}$ ...(1)<br/>$\Rightarrow P=a^{2} x+b^{2}\left(\frac{c^{2}}{x}\right)$<br/>Now, $\frac{d P}{d x}=0 \Rightarrow a^{2}-\frac{b^{2} c^{2}}{x^{2}}=0$<br/>$\therefore y=\frac{c^{2}}{\frac{b c}{a}}=\frac{a c^{2}}{b c}=\frac{a c}{b}$<br/>$\Rightarrow a^{2}=\frac{b^{2} c^{2}}{x^{2}}$<br/>$\Rightarrow x=\frac{b c}{a} \therefore P_{\min }=a^{2}\left(\frac{b c}{a}\right)+b^{2}\left(\frac{a c}{b}\right)$<br/>$=a b c+a b c=2 a b c .$</div>


title: what-is-the-minimum-value-of-cos-cos-2
slug: what-is-the-minimum-value-of-cos-cos-2-91636
question: <div class="question">What is the minimum value of $\cos \theta+\cos 2 \theta$ ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{9}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{9}{16}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{9}{8}$</span> </div>

answer:
<div class="solution">Let $\mathrm{A}=\cos \theta+\cos 2 \theta$<br/>$\therefore$ On differentiating w.r.t. to $\theta$, we get<br/>$\frac{\mathrm{d} \mathrm{A}}{\mathrm{d} \theta}=-\sin \theta-2 \sin 2 \theta$<br/>Put $\frac{\mathrm{dA}}{\mathrm{d} \theta}=0$ for maximaor minima. $\sin \theta+2 \sin 2 \theta=0 \Rightarrow \sin \theta+4 \sin \theta \cos \theta$<br/>$\Rightarrow \sin \theta(1+4 \cos \theta)=0$<br/>$\Rightarrow \sin \theta=0$, or $4 \cos \theta+1=0$<br/>$\Rightarrow \cos \theta=1$ or $\cos \theta=-\frac{1}{4}$<br/>Now, $\frac{\mathrm{d}^{2} \mathrm{~A}}{\mathrm{~d} \theta^{2}}=-\cos \theta-4 \cos 2 \theta$<br/>$=-\cos \theta-4\left(2 \cos ^{2} \theta-1\right)$<br/>For $\cos \bar{\theta}=1$<br/>$\frac{\mathrm{d}^{2} \mathrm{~A}}{\mathrm{~d} \theta^{2}}=-\cos \theta-4\left(2 \cos ^{2} \theta-1\right)$<br/>$=-1-4(2(1)-1)=-1-4=-5   &lt; 0$<br/>So, A is maximum at $\cos \theta=1$<br/>$\Rightarrow\left(\frac{\mathrm{d}^{2} \mathrm{~A}}{\mathrm{~d} \theta^{2}}\right)_{\cos \theta=\frac{-1}{4}}=\frac{1}{4}-4\left(2 \cdot \frac{1}{16}-1\right)&gt;0$<br/>$\quad\left[\right.$ Since $\left.\cos 2 \theta=2 \cos ^{2} \theta-1\right]$<br/>$\therefore \mathrm{A}$ is minimum at $\theta=\cos ^{-1}\left(\frac{-1}{4}\right)$<br/>Now minimum value of $\cos \theta+\cos 2 \theta$<br/>or of $\cos \theta+2 \cos ^{2} \theta-1$<br/>$=\left(\frac{-1}{4}\right)+2\left(\frac{1}{16}\right)-1$<br/>$=\frac{-1}{4}+\frac{1}{8}-1=\frac{-2+1-8}{8}=\frac{-9}{8}$</div>


title: what-is-the-minimum-value-of-px-qy-p-0-q-0-when-xy-r-2
slug: what-is-the-minimum-value-of-px-qy-p-0-q-0-when-xy-r-2-57989
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the minimum value of $\mathrm{px}+\mathrm{qy}(\mathrm{p}&gt;0, \mathrm{q}&gt;0)$ when $\mathrm{xy}=\mathrm{r}^{2} ?$</div>

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

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

answer:
<div class="solution">Given that $\mathrm{xy}=\mathrm{r}^{2}$ $\Rightarrow \quad \mathrm{y}=\frac{\mathrm{r}^{2}}{\mathrm{x}}$<br/>Let $S=p x+q y=p x+\frac{q r^{2}}{x}$<br/>$\Rightarrow \quad \frac{\mathrm{dS}}{\mathrm{dx}}=\mathrm{p}-\frac{\mathrm{qr}^{2}}{\mathrm{x}^{2}}$<br/>$\frac{\mathrm{dS}}{\mathrm{dx}}=0$ for maximum or minimum.<br/>So, $0=p-\frac{q r^{2}}{x^{2}}$<br/>$\Rightarrow \quad x^{2}=\frac{q r^{2}}{p} \Rightarrow x=\pm \sqrt{\frac{q}{p}} \cdot$<br/>Now, $\frac{d^{2} S}{d x^{2}}=\frac{2 q r^{2}}{x^{3}}$<br/>At $x=+\sqrt{\frac{q}{p}} \cdot r \quad \frac{d^{2} S}{d x^{2}}&gt;0$<br/>Hence, $\mathrm{S}$ is minimum at $\mathrm{x}=\sqrt{\frac{\mathrm{q}}{\mathrm{p}}} \mathrm{r}$<br/>$\Rightarrow \mathrm{y}=\frac{\mathrm{r}^{2}}{\sqrt{\frac{\mathrm{q}}{\mathrm{p}} \cdot \mathrm{r}}}=\sqrt{\frac{\mathrm{p}}{\mathrm{q}} \cdot \mathrm{r}}$<br/>Minimum value of $p x+q y=p \cdot \sqrt{\frac{q}{p}} \cdot r+q \sqrt{\frac{p}{q}} \cdot r$<br/>$=\sqrt{\mathrm{pq}} \mathrm{r}+\sqrt{\mathrm{pq}} \mathrm{r}=2 \mathrm{r} \sqrt{\mathrm{pq}}$</div>


title: what-is-the-minimum-value-of-x-x-1-1-3-where-0-x-1
slug: what-is-the-minimum-value-of-x-x-1-1-3-where-0-x-1-48918
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the minimum value of $[\mathrm{x}(\mathrm{x}-1)+1]^{3}$, where $0   &lt; \mathrm{x} &lt; 1$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{3}{4}\right)^{\frac{1}{3}}$</span> </div>

answer:
<div class="solution">$[x(x-1)+1]^{1 / 3}$ is minimum when $x^{2}-x+1$ is minimum, $0 \leq x \leq 1$<br/>$x^{2}-x+1=\left(x-\frac{1}{2}\right)^{2}+\left(1-\frac{1}{4}\right)$<br/>$=\left(x-\frac{1}{2}\right)^{2}+\frac{3}{4}$<br/>$\therefore \quad$ minimum value of $x^{2}-x+1=\frac{3}{4}$<br/>$\Rightarrow$ Required value $=\left(\frac{3}{4}\right)^{1 / 3}$</div>


title: what-is-the-minimum-value-of-x
slug: what-is-the-minimum-value-of-x-95145
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">Minimum value of $|\mathrm{x}|$ is 0 .</div>
source_db: MarksBatch2_P4.db
question:
<div class="question">$\begin{array}{ll}\text {What is the minimum value of }|\mathrm{x}| ? &amp; {}\end{array}$</div>


title: what-is-the-mode-for-the-data-20-20-20-21-21-21-21-21-22-22-22-22-22-22-22-23-23-23-23-23-24-24-25
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">22</span> </div>
answer: <div class="solution">Since observation 22 occurs maximum time. mode $=22$</div>
source_db: MarksBatch2_P4.db
slug:
what-is-the-mode-for-the-data-20-20-20-21-21-21-21-21-22-22-22-22-22-22-22-23-23-23-23-23-24-24-25-49030

question:
<div class="question">What is the mode for the data $20,20,20,21,21,21,21,21,22$, $22,22,22,22,22,22,23,23,23,23,23,24,24,25 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">21</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">22</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">25</span> </li> </ul>


title: what-is-the-modulus-of-1-1-i-2-1-2-i-equal-to
slug: what-is-the-modulus-of-1-1-i-2-1-2-i-equal-to-26000
question: <div class="question">What is the modulus of $\frac{1+2 i}{1-(1-i)^{2}}$ equal to?</div>
tags: ['Mathematics', 'Complex Number', '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_P4.db
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=""> <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">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $z=\frac{1+2 i}{1-(1-i)^{2}}$<br/>$\Rightarrow|z|=\frac{|1+2 i|}{\left|1-(1-i)^{2}\right|}=\frac{|1+2 i|}{\left|1-1-i^{2}+2 i\right|}=\frac{|1+2 i|}{|1+2 i|}=1$</div>


title: what-is-the-modulus-of-1-1-i-2-1-2-i
slug: what-is-the-modulus-of-1-1-i-2-1-2-i-40487
question: <div class="question">What is the modulus of $\left|\frac{1+2 i}{1-(1-i)^{2}}\right|$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P4.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">$\sqrt{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

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


title: what-is-the-modulus-of-2-i-2-i-where-i-1
slug: what-is-the-modulus-of-2-i-2-i-where-i-1-91610
tags: ['Mathematics', 'Complex Number', '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_P4.db
question:
<div class="question">What is the modulus of $\frac{\sqrt{2}+\mathrm{i}}{\sqrt{2}-\mathrm{i}}$ where $\mathrm{i}=\sqrt{-1}$ ?</div>

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

answer:
<div class="solution">Let $z=\frac{\sqrt{2}+i}{\sqrt{2}-i}=\frac{\sqrt{2}+i}{\sqrt{2}-i} \times \frac{\sqrt{2}+i}{\sqrt{2}+i}$<br/>$\quad=\frac{(\sqrt{2}+i)^{2}}{(\sqrt{2})^{2}-(i)^{2}}=\frac{2 \sqrt{2} i+1}{3}$<br/>$\quad$ Now, $|z|=\sqrt{\left(\frac{2 \sqrt{2}}{3}\right)^{2}+\left(\frac{1}{3}\right)^{2}}=1$</div>


tags: ['Chemistry', 'Kinetic Theory of Gases', 'NEET']
answer: <div class="solution">Correct Option is : (C)<br/>22.4 liters</div>
source_db: MarksBatch2_P4.db
title:
what-is-the-molar-volume-of-an-ideal-gas-at-standard-temperature-and-pressure-stp-which-is-defined-as-1-atmospheric-pressure-and-0-c

slug:
what-is-the-molar-volume-of-an-ideal-gas-at-standard-temperature-and-pressure-stp-which-is-defined-as-1-atmospheric-pressure-and-0-c-99251

question:
<div class="question">What is the molar volume of an ideal gas at standard temperature and pressure (STP), which is defined as 1 atmospheric pressure and 0 Â°C?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 liter</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10 liters</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">22.4 liters</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">100 liters</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">22.4 liters</span> </div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P4.db
title:
what-is-the-moment-about-the-point-i-2-j-3-k-of-a-force-represented-by-i-j-k-acting-through-the-point-2-i-3-j-k

slug:
what-is-the-moment-about-the-point-i-2-j-3-k-of-a-force-represented-by-i-j-k-acting-through-the-point-2-i-3-j-k-70417

question:
<div class="question">What is the moment about the point $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, of a<br/>force represented by $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$, acting through the point<br/>$-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 3 \hat{i}+\hat{j}-4 \hat{k}$</span> </div>

answer:
<div class="solution">Force, $\overrightarrow{\mathrm{F}}$ is given by $\overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\begin{aligned} \overrightarrow{\mathrm{OA}} &amp;=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{OB}}=-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\ \therefore \quad \overrightarrow{\mathrm{r}}=&amp; \overrightarrow{\mathrm{AB}}=-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \\ &amp;=-3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \end{aligned}$<br/>Moment $\overrightarrow{\mathrm{M}}$ about the point $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{M}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}=(-3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}) \times(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$<br/>$=\left|\begin{array}{ccc}\hat{\mathrm{i}} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ -3 &amp; 1 &amp; -2 \\ 1 &amp; 1 &amp; 1\end{array}\right|=\hat{\mathrm{i}}(1+2)-\hat{\mathrm{j}}(-3+2)+\hat{\mathrm{k}}(-3-1)$<br/>$=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-4 \hat{\mathrm{k}}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P4.db
title:
what-is-the-moment-about-the-point-i-2-j-k-of-a-force-represented-by-3-i-k-acting-through-the-point-2-i-j-3-k

slug:
what-is-the-moment-about-the-point-i-2-j-k-of-a-force-represented-by-3-i-k-acting-through-the-point-2-i-j-3-k-87780

question:
<div class="question">What is the moment about the point $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ of a force<br/>represented by $3 \hat{i}+\hat{k}$ acting through the point $2 \hat{i}-\hat{j}+3 \hat{k}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad-3 \hat{i}+11 \hat{j}+9 \hat{k}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \hat{i}+2 \hat{j}+9 \hat{k}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+9 \hat{\mathrm{k}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad-3 \hat{i}+11 \hat{j}+9 \hat{k}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \overrightarrow{\mathrm{r}} &amp;=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})-(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}) \\ &amp;=(\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \\ &amp; \text { Moment }(\tau)=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}} \\ &amp;=(\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times(3 \hat{\mathrm{i}}+\hat{\mathrm{k}}) \\ &amp;=-3 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}+9 \hat{\mathrm{k}} \end{aligned}$</div>


title: what-is-the-most-abundant-chemical-in-living-organisms
slug: what-is-the-most-abundant-chemical-in-living-organisms-95726
question: <div class="question">What is the most abundant chemical in living organisms?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Water</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Carbon</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Hydrogen</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Oxygen</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Water</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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 most abundant chemical in living organisms is water, not carbon, hydrogen, or oxygen.</div>


title: what-is-the-most-abundant-element-in-the-earths-crust-by-mass
slug: what-is-the-most-abundant-element-in-the-earths-crust-by-mass-62003
question: <div class="question">What is the most abundant element in the Earth's crust by mass?</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Oxygen</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Oxygen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Silicon</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Aluminum</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Boron</span> </li> </ul>

answer:
<div class="solution">Oxygen is the most abundant element in the Earth's crust by mass, with 45.5%.</div>


title: what-is-the-most-abundant-element-in-the-universe
slug: what-is-the-most-abundant-element-in-the-universe-78878
question: <div class="question">What is the most abundant element in the universe?</div>
tags: ['Chemistry', 'Hydrogen', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Oxygen</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Carbon</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Hydrogen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Helium</span> </li> </ul>

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

answer:
<div class="solution">The paragraph states that dihydrogen is the most abundant element in the universe.</div>


title: what-is-the-most-abundant-isotope-of-nitrogen
slug: what-is-the-most-abundant-isotope-of-nitrogen-74258
question: <div class="question">What is the most abundant isotope of nitrogen?</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14N</span> </div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">14N</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">15N</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">16N</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">17N</span> </li> </ul>

answer:
<div class="solution">The paragraph states that nitrogen has two stable isotopes: 14N and 15N. 14N is the most abundant isotope of nitrogen, accounting for approximately 99.63% of naturally occurring nitrogen.</div>


title: what-is-the-most-abundant-organic-substance-in-the-plant-kingdom
slug: what-is-the-most-abundant-organic-substance-in-the-plant-kingdom-62240
question: <div class="question">What is the most abundant organic substance in the plant kingdom?</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
answer: <div class="solution">Cellulose is the most abundant organic substance in the plant kingdom.</div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Starch</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Cellulose</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Lignin</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Hemicellulose</span> </li> </ul>

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


title: what-is-the-most-abundant-plant-pigment-in-the-world
slug: what-is-the-most-abundant-plant-pigment-in-the-world-95538
question: <div class="question">What is the most abundant plant pigment in the world?</div>
tags: ['Biology', 'Photosynthesis in Higher Plants', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Chlorophyll b</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Xanthophylls</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Carotenoids</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Chlorophyll 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">Chlorophyll a</span> </div>

answer:
<div class="solution">The ability of chlorophyll a pigment to absorb lights of different wavelengths shows that chlorophyll a is the most abundant pigment.</div>


title: what-is-the-most-abundant-protein-in-animal-world
slug: what-is-the-most-abundant-protein-in-animal-world-60197
question: <div class="question">What is the most abundant protein in animal world?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
answer: <div class="solution">Collagen is the most abundant protein in animal world.</div>
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Collagen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Ribulose bisphosphate Carboxylase-Oxygenase (RuBisCO)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Enzymes</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Hormones</span> </li> </ul>

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


title: what-is-the-most-abundant-protein-in-the-animal-world
slug: what-is-the-most-abundant-protein-in-the-animal-world-86737
question: <div class="question">What is the most abundant protein in the animal world?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
source_db: MarksBatch2_P4.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Enzymes</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Hormones</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Collagen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">RuBisCO</span> </li> </ul>

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

answer:
<div class="solution">Collagen is the most abundant protein in the animal world. RuBisCO is the most abundant protein in the whole of the biosphere.</div>


title: what-is-the-most-probable-number-of-successes-in-10-trials-with-probability-of-success-23
slug: what-is-the-most-probable-number-of-successes-in-10-trials-with-probability-of-success-23-52980
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the most probable number of successes in 10 trials with probability of success $2 / 3$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">This is the question of Binomial Distribution. Number of $r$ success in $n$ trial is $P_{r}={ }^{n} C_{r} p^{r} q^{n-r}$<br/>Where $\mathrm{p}=$ prob of success $\mathrm{q}=$ prob. of failure $=1-\mathrm{p}$<br/>Given : $n=10, p=\frac{2}{3}, q=\frac{1}{3}$<br/>Fo $\quad r=10, P_{10}={ }^{10} C_{10} \quad\left(\frac{2}{3}\right)^{10}\left(\frac{1}{3}\right)^{0}=\frac{2^{10}}{3^{10}}$<br/>For $\quad r=7, P_{7}=120 \cdot \frac{2^{7}}{3^{10}}$<br/>For $\mathrm{r}=5, \mathrm{P}_{5}=252 \times \frac{2^{5}}{3^{10}}$<br/>For $\quad r=4, P_{4}=210 \cdot \frac{2^{4}}{3^{10}}$<br/>It is maximum for $\mathrm{r}=7$</div>


title: what-is-the-n-th-term-of-the-sequence-25-125-625-3125
slug: what-is-the-n-th-term-of-the-sequence-25-125-625-3125-18508
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the $n^{\text {th }}$ term of the sequence $25,-125,625,-3125, \ldots ?$</div>

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

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

answer:
<div class="solution">Given series $25,-125,625,-3125 \ldots \ldots$ is geometric progression. $a=t_{1}=25, t_{2}=-125$<br/>$\mathrm{r}=\frac{\mathrm{t}_{2}}{\mathrm{t}_{1}}=\frac{-125}{25}=-5$<br/>$\mathrm{n}^{\text {th }}$ term $\left(\mathrm{t}_{\mathrm{n}}\right)=\operatorname{ar}^{\mathrm{n}-1}=(25)(-5)^{\mathrm{n}-1}$<br/>$=(5)^{2}(-1)^{\mathrm{l}-1}(5)^{\mathrm{n}-1}=(-1)^{\mathrm{n}-1}(5)^{2+\mathrm{n}-1}$<br/>$=(-1)^{\mathrm{n}-1} 5^{\mathrm{n}+1}$</div>


tags: ['Biology', 'Plant Kingdom', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-name-of-the-process-by-which-two-male-gametes-fuse-with-the-egg-cell-and-diploid-secondary-nucleus-respectively

slug:
what-is-the-name-of-the-process-by-which-two-male-gametes-fuse-with-the-egg-cell-and-diploid-secondary-nucleus-respectively-17884

question:
<div class="question">What is the name of the process by which two male gametes fuse with the egg cell and diploid secondary nucleus, respectively?</div>

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

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

answer:
<div class="solution">Double fertilization is the unique process in angiosperms where two male gametes fuse with the egg cell and diploid secondary nucleus, respectively, resulting in the formation of a zygote and the triploid primary endosperm nucleus.</div>


tags: ['Biology', 'Cell - The Unit of Life', 'NEET']
source_db: MarksBatch2_P4.db
title:
what-is-the-name-of-the-small-fragment-that-appears-on-a-chromosome-due-to-nonstaining-secondary-constrictions-at-a-constant-location

slug:
what-is-the-name-of-the-small-fragment-that-appears-on-a-chromosome-due-to-nonstaining-secondary-constrictions-at-a-constant-location-96768

question:
<div class="question">What is the name of the small fragment that appears on a chromosome due to non-staining secondary constrictions at a constant location?</div>

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

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

answer:
<div class="solution">A satellite is a small fragment that appears on a chromosome due to non-staining secondary constrictions at a constant location.</div>


title: what-is-the-nth-term-of-the-sequence-1-5-9-13-17
slug: what-is-the-nth-term-of-the-sequence-1-5-9-13-17-59647
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the nth term of the sequence $1,5,9,13,17, \ldots \ldots \ldots$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 2 \mathrm{n}-1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{n}+1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4 \mathrm{n}-3$</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">None of the above</span> </li> </ul>

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

answer:
<div class="solution">Given sequence is $1,5,9,13,17, \ldots$<br/>Which is an A.P. Here $\mathrm{a}=1, \mathrm{~d}=4$<br/>$\therefore$ nth term $=a_{n}=a+(n-1) d=1+(n-1) 4$<br/>$=1+4 n-4=4 n-3$</div>


tags: ['Mathematics', 'Differential Equations', '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_P4.db
title:
what-is-the-number-of-arbitrary-constants-in-the-particular-solution-of-differential-equation-of-third-order

slug:
what-is-the-number-of-arbitrary-constants-in-the-particular-solution-of-differential-equation-of-third-order-44842

question:
<div class="question">What is the number of arbitrary constants in the particular solution of differential equation of third order?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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">Particular solution of $\mathrm{D} . \varepsilon$. of third order have three arbitary constant.</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4850</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-number-of-diagonals-which-can-be-drawn-by-joining-the-angular-points-of-a-polygon-of-100-sides

slug:
what-is-the-number-of-diagonals-which-can-be-drawn-by-joining-the-angular-points-of-a-polygon-of-100-sides-19869

question:
<div class="question">What is the number of diagonals which can be drawn by joining the angular points of a polygon of 100 sides?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">4850</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">4950</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">5000</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10000</span> </li> </ul>

answer:
<div class="solution">We have, $\mathrm{n}=100$<br/>Number of diagonals $=\frac{\mathrm{n}^{2}-3 \mathrm{n}}{2}$<br/>$=\frac{100^{2}-300}{2}=\frac{9700}{2}=4850$</div>


title: what-is-the-number-of-different-messages-that-can-be-represented-by-three-0s-and-two-1s
slug: what-is-the-number-of-different-messages-that-can-be-represented-by-three-0s-and-two-1s-59262
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the number of different messages that can be represented by three 0's and two 1's?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">9</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7</span> </li> </ul>

answer:
<div class="solution">Number of different messages that can be represented by three 0 's and two 1 's is 10 .<br/>Option (a) is correct.</div>


title: what-is-the-number-of-digits-in-the-numeral-form-of-8172-given-lo-g-10-2-03010
slug: what-is-the-number-of-digits-in-the-numeral-form-of-8172-given-lo-g-10-2-03010-60217
tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the number of digits in the numeral form of 8172 (Given $\left.\log _{10}{ }^{2}=0.3010\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">51</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">15</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">14</span> </li> </ul>

answer:
<div class="solution">$x=2^{5}$<br/>Taking log on both sides of above equation, we get $\log x=51 \log 2$<br/>$=51 \times 0.3010=15.381$<br/>Number ofdigits in $8^{17}=15+1=16$</div>


title: what-is-the-number-of-distinct-solutions-of-the-equation-z-2-z-0-where-z-is-a-complex-number
slug: what-is-the-number-of-distinct-solutions-of-the-equation-z-2-z-0-where-z-is-a-complex-number-34580
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Three</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the number of distinct solutions of the equation $\mathrm{z}^{2}+|\mathrm{z}|=0$ (where z is a complex number)?</div>

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

answer:
<div class="solution">Let $z=x+i y$ $z^{2}+|z|=0$<br/>$\Rightarrow x^{2}-y^{2}+2 i x y+\sqrt{x^{2}+y^{2}}=0$<br/>$\Rightarrow x^{2}-y^{2}+\sqrt{x^{2}+y^{2}}+i 2 x y=0+i 0$<br/>$\Rightarrow x^{2}-y^{2}+\sqrt{x^{2}+y^{2}}=0$<br/>and $2 x y=0 \Rightarrow x y=0 \Rightarrow x=0$ or $y=0$<br/>Now: For $y=0$ in eq. (1) we get :<br/>$x^{2}+\sqrt{x^{2}}=0$<br/>$\Rightarrow x^{2}+|x|=0$<br/>Clearly $x^{2}+|x|$ will always be greater than 0 for all $x&gt;0 .$<br/>Let, $x \leq 0$<br/>$x^{2}+|x|=0$<br/>$\Rightarrow x^{2}-x=0 \Rightarrow x(x-1)=0$<br/>$\Rightarrow x=0$ or $(x-1)=0$<br/>$\Rightarrow x=0 \quad(\because x \leq 0)$<br/>$\therefore \mathbf{z = 0}$<br/>For $x=0$ in eq. (1) we get,<br/>$-y^{2}+\sqrt{y^{2}}=0$<br/>$-y^{2}+|y|=0$<br/>If $y&gt;0$, then $-y^{2}+|y|=0$<br/>$\Rightarrow-y^{2}+y=0$<br/>$\Rightarrow y=0, y=1$<br/>$\Rightarrow y=1$<br/>$(\because y&gt;0)$<br/>$\therefore \mathrm{z}=\mathbf{i}$<br/>If $y   &lt; 0$, then $-y^{2}+|y|=0$<br/>$\Rightarrow-y^{2}-y=0$<br/>$\Rightarrow y=0, y=-1$<br/>$\Rightarrow y=-1$<br/>$(\because y &lt; 0)$<br/>$\therefore \mathrm{z}=-\mathbf{i}$<br/>$\therefore \quad$ There are only 3 distinct solutions.</div>


title: what-is-the-number-of-fivedigit-numbers-formed-with-01-2-3-4-without-any-repetition-of-digits
slug: what-is-the-number-of-fivedigit-numbers-formed-with-01-2-3-4-without-any-repetition-of-digits-90511
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">96</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the number of five-digit numbers formed with 0,1 , 2, 3, 4 without any repetition of digits?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">96</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">120</span> </li> </ul>

answer:
<div class="solution">Tomakea 5 digit number, 0 can not come in the bagining. So, it can be filled in 4 ways. Rest of the places can be filled in $4 !$ ways. So total number of digit formed $=4 \times 4 !$ $=4 \times 24=96$</div>


title: what-is-the-number-of-fourdigit-decimal-numbers-1-in-which-no-digit-is-repeated
slug: what-is-the-number-of-fourdigit-decimal-numbers-1-in-which-no-digit-is-repeated-90160
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4536</span> </div>
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the number of four-digit decimal numbers $(   &lt; 1)$ in which no digit is repeated?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3024</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4536</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">5040</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

answer:
<div class="solution">Let the given 4 digit decimal number is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/I6aowte79EsLdeuyDwRnhp79ZAM5-WwD7z6uNip4UDQ.original.fullsize.png"/><br/><br/>Places after decimal can be filled in the following ways:<img src="https://cdn-question-pool.getmarks.app/pyq/nda/790XuAGbuPscuQQzy5QPlW2o9Qp6WtnKWKj-jqABfzE.original.fullsize.png"/><br/><br/>Total number of ways $=7 \times 8 \times 9 \times 9=4536$<br/>#</div>


title: what-is-the-number-of-moles-of-secondary-carbon-atoms-in-n-mole-isopentane
slug: what-is-the-number-of-moles-of-secondary-carbon-atoms-in-n-mole-isopentane-77494
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'MHT CET', 'MHT CET 2023 (11 May Shift 2)']
source_db: MarksBatch2_P4.db
question:
<div class="question">What is the number of moles of secondary carbon atoms in $\mathrm{n}$ mole isopentane?</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/WVC1loaAeUpc9-dqnp2B2IjLPQeijCpyAK0xUglYYH4.original.fullsize.png"/><br/><br/>1 molecule of isopentane has 1 secondary carbon atom.<br/>$\therefore \quad 1 \mathrm{~mol}$ isopentane $=1 \mathrm{~mol}$ secondary carbon atoms<br/>$\therefore \quad \mathrm{n}$ mol isopentane $=\mathrm{n}$ mol secondary carbon atoms</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">854</span> </div>
source_db: MarksBatch2_P4.db
title:
what-is-the-number-of-natural-numbers-less-than-or-equal-to-1000-which-are-neither-divisible-by-10-nor-15-nor-2-5

slug:
what-is-the-number-of-natural-numbers-less-than-or-equal-to-1000-which-are-neither-divisible-by-10-nor-15-nor-2-5-27671

question:
<div class="question">What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor $25^{\circ}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">860</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">854</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">840</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">82</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{A}, \mathrm{B} \&amp; \mathrm{C}$ be the sets of numbers divisible by 10,15 \&amp; 25 respectively No. divisible by $10=100=\mathrm{n}(\mathrm{A})$ No. divisible by $15=66=\mathrm{n}(\mathrm{B})$ No. divisible by $25=40=\mathrm{n}(\mathrm{C})$<br/>No. divisible by $(10 \&amp; 15)=33=\mathrm{n}(\mathrm{A} \cap \mathrm{B})$<br/>No. divisible by $(15 \&amp; 25)=13=\mathrm{n}(\mathrm{B} \cap \mathrm{C})$<br/>No. divisible by $(25 \&amp; 10)=20=\mathrm{n}(\mathrm{A} \cap \mathrm{C})$<br/>No. divisible by $(10,15 \&amp; 25)=6=n\left(A \cap B_{\cap} C\right)$ No. divisible by 10,15 and $25=n(A \cup B \cup C)$ $=100+66+40-33-13-20+6=146$<br/>Thus, no. which are neither divisible by 10 nor 15 nor $25=1000-146=854 .$</div>


title: what-is-the-number-of-nonzero-terms-in-the-expansion-of-1-2-3-x-11-1-2-3-x-11-after-simplification
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P4.db
slug:
what-is-the-number-of-nonzero-terms-in-the-expansion-of-1-2-3-x-11-1-2-3-x-11-after-simplification-90854

question:
<div class="question">What is the number of non-zero terms in the expansion of $(1+2 \sqrt{3} \mathrm{x})^{11}+(1-2 \sqrt{3} \mathrm{x})^{11}$ (after simplification)?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">11</span> </li> </ul>

answer:
<div class="solution">We know, in the expansion of $(\mathrm{x}+\mathrm{y})^{\mathrm{n}}+(\mathrm{x}-\mathrm{y})^{\mathrm{n}}$, of<br/>$\mathrm{n}=$ even, then number of non zero terms is $\frac{\mathrm{n}}{2}+1$<br/>$\mathrm{n}=$ odd, then number of non zero terms in $\frac{\mathrm{n}+1}{2}$.<br/>Here, $\mathrm{n}=11$ which is odd.<br/>number of non zero terms $=\frac{11+1}{2}=6$.</div>


