# questions

tags: ['Chemistry', 'Thermodynamics (C)', 'AP EAMCET', 'AP EAMCET 2017 (25 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-891$</span> </div>
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch2_P1.db
title:
if-standard-enthalpy-of-formation-f-h-of-co-2-h-2-o-and-ch-4-are-393-286-and-740-kj-mol-1-respectively-the-standard-enthalpy-of-combustion-of-methane-

slug:
if-standard-enthalpy-of-formation-f-h-of-co-2-h-2-o-and-ch-4-are-393-286-and-740-kj-mol-1-respectively-the-standard-enthalpy-of-combustion-of-methane-28831

question:
<div class="question">If standard enthalpy of formation $\left(\Delta_{\mathrm{f}} \mathrm{H}^{\circ}\right)$ of $\mathrm{CO}_2, \mathrm{H}_2 \mathrm{O}$ and $\mathrm{CH}_4$ are $-393,-286$ and $-74.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively, the standard enthalpy of combustion of methane in $\mathrm{kJ} \mathrm{mol}^{-1}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-753$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-105$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-605$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-891$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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>


tags: ['Chemistry', 'Thermodynamics (C)', 'AP EAMCET', 'AP EAMCET 2023 (19 May Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-standard-molar-enthalpy-change-and-standard-molar-internal-energy-change-measured-in-bomb-calorimeter-are-equal-which-one-of-the-following-statemen

slug:
if-standard-molar-enthalpy-change-and-standard-molar-internal-energy-change-measured-in-bomb-calorimeter-are-equal-which-one-of-the-following-statemen-39911

question:
<div class="question">If standard molar enthalpy change and standard molar internal energy change measured in bomb calorimeter are equal, which one of the following statements is correct?</div>

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

answer:
<div class="solution">$\Delta \mathrm{H}^{\circ}=\Delta \mathrm{U}^{\circ}$ (given)<br/>$$<br/>\begin{aligned}<br/>&amp; \Delta \mathrm{H}^{\circ}=\Delta \mathrm{U}^{\circ}+\mathrm{P} \Delta \mathrm{V} \\<br/>&amp; =\Delta \mathrm{U}^{\circ}+\Delta \mathrm{nRT} \\<br/>&amp; =\Delta \mathrm{U}^{\circ}<br/>\end{aligned}<br/>$$<br/>$\Rightarrow \Delta \mathrm{nRT}=0$ which means $\Delta \mathrm{n}=0$ with constant pressure.</div>


title: if-sum-of-squares-of-the-roots-of-the-equation-x-2-k-x-b-0-is-2-b-what-is-k-equal-to
slug: if-sum-of-squares-of-the-roots-of-the-equation-x-2-k-x-b-0-is-2-b-what-is-k-equal-to-16272
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If sum of squares of the roots of the equation $x^{2}+k x-b=0$ is $2 b$, what is $k$ equal to?</div>

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

answer:
<div class="solution">Let the roots of the equation $x^{2}+k x-b=0$ be $\alpha$ and $\beta$. Sum : $\alpha+\beta=-k$ and Product : $\alpha \beta=-b$<br/>According to the question, we have<br/>$\quad \alpha^{2}+\beta^{2}=2 b$<br/>$\Rightarrow(\alpha+\beta)^{2}-2 \alpha \beta=2 b$<br/>$\Rightarrow k^{2}+2 b=2 b \Rightarrow k=0$</div>


title: if-sum-of-the-roots-of-3-x-2-3-p-1-x-p-5-0-is-equal-to-their-product-then-what-is-the-value-of-p
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-sum-of-the-roots-of-3-x-2-3-p-1-x-p-5-0-is-equal-to-their-product-then-what-is-the-value-of-p-76400

question:
<div class="question">If sum of the roots of $3 x^{2}+(3 p+1) x-(p+5)=0$ is equal to their product, then what is the value of $p ?$</div>

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

answer:
<div class="solution">Let $\alpha$ and $\beta$ are the roots of the given equation.<br/>$\therefore$<br/>$\alpha+\beta=\frac{-(3 p+1)}{3}$<br/>and $\quad \alpha \beta=\frac{-(p+5)}{3}$<br/>Now, $\frac{-(3 p+1)}{3}=\frac{-(p+5)}{3}$<br/>$\Rightarrow 3 p+1=p+5$<br/>$\therefore \quad 2 p=4$<br/>$\therefore \quad p=2$</div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main', 'JEE Main 2019 (08 Apr Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-surface-tension-s-moment-of-inertia-i-and-plancks-constant-h-were-to-be-taken-as-the-fundamental-units-the-dimensional-formula-for-linear-momentum-

slug:
if-surface-tension-s-moment-of-inertia-i-and-plancks-constant-h-were-to-be-taken-as-the-fundamental-units-the-dimensional-formula-for-linear-momentum-85276

question:
<div class="question">If Surface tension<math><mi> </mi><mo>(</mo><mi mathvariant="normal">S</mi><mo>)</mo></math>, Moment of Inertia <math><mo>(</mo><mi mathvariant="normal">I</mi><mo>)</mo></math> and Planck's constant <math><mo>(</mo><mi mathvariant="normal">h</mi><mo>)</mo><mo>,</mo></math> were to be taken as the fundamental units, the dimensional formula for linear momentum would be:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">h</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><math><mi mathvariant="normal">P</mi><mo>â</mo><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mi mathvariant="normal">h</mi></mrow><mrow><mi>c</mi></mrow></msup></math> <br/><math><mo>[</mo><mi mathvariant="normal">M</mi><mi mathvariant="normal">L</mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>C</mi></mrow></msup></math> <br/><math><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">L</mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mo>[</mo><msup><mrow><mi mathvariant="normal">M</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></msup><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>2</mn><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mi>c</mi></mrow></msup><mo>]</mo></math> <br/>From comparing the power on both side <br/><math><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>1</mn></math> <br/><math><mn>2</mn><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi><mo>=</mo><mn>1</mn></math> <br/><math><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <br/>From the above three equations we will get <br/><math><mi>a</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi>Â </mi><mi>b</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mtext>â</mtext><mo>&amp;</mo><mi>Â </mi><mi>c</mi><mo>=</mo><mn>0</mn></math> <br/><math><mi mathvariant="normal">P</mi><mo>â</mo><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi mathvariant="normal">h</mi></mrow><mrow><mn>0</mn></mrow></msup></math></div>


tags: ['Physics', 'Units and Dimensions', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-surface-tension-s-moment-of-inertia-i-and-plancks-constant-h-were-to-be-taken-as-the-fundamental-units-the-dimensional-formula-units-the-dimensiona

slug:
if-surface-tension-s-moment-of-inertia-i-and-plancks-constant-h-were-to-be-taken-as-the-fundamental-units-the-dimensional-formula-units-the-dimensiona-48188

question:
<div class="question">If Surface tension<math><mi> </mi><mo>(</mo><mi mathvariant="normal">S</mi><mo>)</mo></math>, Moment of Inertia <math><mo>(</mo><mi mathvariant="normal">I</mi><mo>)</mo></math> and Planck's constant <math><mo>(</mo><mi mathvariant="normal">h</mi><mo>)</mo><mo>,</mo></math> were to be taken as the fundamental units, the dimensional formula units, the dimensional formula for linear momentum would be:</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">h</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><math><mi mathvariant="normal">P</mi><mo>â</mo><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mi mathvariant="normal">h</mi></mrow><mrow><mi>c</mi></mrow></msup></math> <br/><math><mo>[</mo><mi mathvariant="normal">M</mi><mi mathvariant="normal">L</mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>=</mo><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>C</mi></mrow></msup></math> <br/><math><mfenced close="]" open="[" separators="|"><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">L</mi><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mo>[</mo><msup><mrow><mi mathvariant="normal">M</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></msup><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mn>2</mn><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></msup><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mi>c</mi></mrow></msup><mo>]</mo></math> <br/>From comparing the power on both side <br/><math><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>1</mn></math> <br/><math><mn>2</mn><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi><mo>=</mo><mn>1</mn></math> <br/><math><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <br/>From the above three equations we will get <br/><math><mi>a</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi>Â </mi><mi>b</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mtext>â</mtext><mo>&amp;</mo><mi>Â </mi><mi>c</mi><mo>=</mo><mn>0</mn></math> <br/><math><mi mathvariant="normal">P</mi><mo>â</mo><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi mathvariant="normal">h</mi></mrow><mrow><mn>0</mn></mrow></msup></math></div>


title: if-t-r-is-the-rth-term-in-the-expansion-of-1-x-101-then-what-is-the-ratio-t-19-t-20-equal-to
slug: if-t-r-is-the-rth-term-in-the-expansion-of-1-x-101-then-what-is-the-ratio-t-19-t-20-equal-to-26619
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{t}_{\mathrm{r}}$ is the rth term in the expansion of $(1+\mathrm{x})^{101}$, then what is<br/>the ratio $\frac{\mathrm{t}_{20}}{\mathrm{t}_{19}}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{20 \mathrm{x}}{19}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$83 \mathrm{x}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$19 \mathrm{x}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{83 x}{19}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{83 x}{19}$</span> </div>

answer:
<div class="solution">We find $r_{n}$ term:<br/>$\mathrm{t}_{\mathrm{r}}$ is the rth term in the expansion of $(1+\mathrm{x})^{101}$. $\mathrm{t}_{\mathrm{r}}=101 \mathrm{C}_{\mathrm{r}-1} \cdot(\mathrm{x})^{(\mathrm{r}-1)}$<br/>$\therefore \frac{\mathrm{t}_{20}}{\mathrm{t}_{19}}=\frac{101}{{ }^{01} \mathrm{C}_{19}} \cdot \frac{\mathrm{x}^{19}}{\mathrm{x}^{18}}=\frac{101}{{ }^{101} \mathrm{C}_{19} \mathrm{x}}{\mathrm{C}_{18}}=\frac{\frac{101 !}{19 ! 82 !}}{\frac{101 !}{18 ! 83 !}} \mathrm{x}=\frac{83 \mathrm{x}}{19}$</div>


title: if-tan-1-2-tan-1-3-are-two-angles-of-a-triangle-then-what-is-the-third-angle
slug: if-tan-1-2-tan-1-3-are-two-angles-of-a-triangle-then-what-is-the-third-angle-18523
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan ^{-1} 2, \tan ^{-1} 3$ are two angles of a triangle, then what is the third angle?</div>

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

answer:
<div class="solution">Let $\mathrm{A}=\tan ^{-1} 2, \mathrm{~B}=\tan ^{-1} 3$ and $\mathrm{C}$ be the angles of a<br/>triangle.<br/>By angle sum property, we have<br/>$\tan ^{-1} 2+\tan ^{-1} 3+\mathrm{C}=180^{\circ}$<br/>$\Rightarrow \tan ^{-1}\left(\frac{5}{-5}\right)=180^{\circ}-\mathrm{C}$<br/>$\Rightarrow \tan ^{-1}(-1)=180^{\circ}-\mathrm{C}$<br/>$\Rightarrow \frac{3 \pi}{4}=\pi-\mathrm{C} \Rightarrow \mathrm{C}=\pi-\frac{3 \pi}{4}=\frac{\pi}{4}$<br/>Hence, third angle is $\frac{\pi}{4}$.</div>


title: if-tan-2-1-and-tan-3-1-then-what-is-the-value-of
slug: if-tan-2-1-and-tan-3-1-then-what-is-the-value-of-82004
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan \theta=\frac{1}{2}$ and $\tan \varphi=\frac{1}{3}$, then what is the value of $?</div>

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

answer:
<div class="solution">\tan \theta=\frac{1}{2} \\ &amp; \tan \phi=\frac{1}{3} \end{aligned}$<br/>$\tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}$<br/>$=\frac{\frac{1}{2}+\frac{1}{3}}{1-\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)}=\frac{\frac{5}{6}}{\frac{5}{6}}=1$<br/>$\therefore \theta+\phi=\tan ^{-1}(1)=\frac{\pi}{4}$</div>


title: if-tan-2-2-tan-2-1-then-which-one-of-the-following-is-correct
slug: if-tan-2-2-tan-2-1-then-which-one-of-the-following-is-correct-45391
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan ^{2} \theta=2 \tan ^{2} \phi+1$, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\cos (2 \theta)=[\cos (2 \phi)-1] / 2$</span> </div>

answer:
<div class="solution">Work with option,<br/>$\cos (2 \phi)-1=\frac{1-\tan ^{2} \phi}{1+\tan ^{2} \phi}-1$<br/>$=-\frac{2 \tan ^{2} \phi}{1+\tan ^{2} \phi}=\frac{-\left(\tan ^{2} \theta-1\right)}{1+\frac{\tan ^{2} \theta-1}{2}}$<br/>$=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta} \times 2=\cos (2 \theta) 2$<br/>Thus, $\cos 2 \theta=\frac{\cos (2 \phi)-1}{2}$</div>


title: if-tan-2-and-tan-1-then-tan-2-is-equal-to
slug: if-tan-2-and-tan-1-then-tan-2-is-equal-to-93851
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-3$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If tan $(\alpha+\beta)=2$ and $\tan (\alpha-\beta)=1$, then $\tan (2 \alpha)$ is equal<br/>to</div>

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

answer:
<div class="solution">$\tan (\alpha+\beta)=2$<br/>$\tan (\alpha-\beta)=1$<br/>$\tan 2 \alpha=\tan [(\alpha+\beta)+(\alpha-\beta)]$<br/>$=\frac{\tan (\alpha+\beta)+\tan (\alpha-\beta)}{1-\tan (\alpha+\beta) \cdot \tan (\alpha-\beta)}$<br/>$=\frac{2+1}{1-2 \cdot 1}=\frac{3}{-1}=-3$</div>


title: if-tan-2-b-1-s-i-n-a-1-s-i-n-a-then-what-is-the-value-of-a-2-b
slug: if-tan-2-b-1-s-i-n-a-1-s-i-n-a-then-what-is-the-value-of-a-2-b-18207
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan ^{2} \mathrm{~B}=\frac{1-\sin \mathrm{A}}{1+\sin \mathrm{A}}$ then what is the value of $\mathrm{A}+2 \mathrm{~B}$ ?</div>

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

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

answer:
<div class="solution">Given equation is $\tan ^{2} \mathrm{~B}=\frac{1-\sin \mathrm{A}}{1+\sin \mathrm{A}}$<br/>$\Rightarrow$ Applying componendo and dividendo $\frac{1+\tan ^{2} \mathrm{~B}}{1-\tan ^{2} \mathrm{~B}}=\frac{2}{2 \sin \mathrm{A}}$<br/>$\Rightarrow \sin \mathrm{A}=\frac{1-\tan ^{2} \mathrm{~B}}{1+\tan ^{2} \mathrm{~B}} \Rightarrow \sin \mathrm{A}=\cos 2 \mathrm{~B}$<br/>$\Rightarrow \sin A=\sin \left(\frac{\pi}{2}-2 \mathrm{~B}\right)$<br/>$\Rightarrow \mathrm{A}=\frac{\pi}{2}-2 \mathrm{~B} \Rightarrow \mathrm{A}+2 \mathrm{~B}=\frac{\pi}{2}$</div>


title: if-tan-a-2-1-and-tan-b-3-1-then-what-is-the-value-of-a-b
slug: if-tan-a-2-1-and-tan-b-3-1-then-what-is-the-value-of-a-b-92217
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan \mathrm{A}=\frac{1}{2}$ and $\tan \mathrm{B}=\frac{1}{3}$, then what is the value of $(\mathrm{A}+\mathrm{B})$ ?</div>

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

answer:
<div class="solution">Let tan $A=\frac{1}{2}$ and $\tan B=\frac{1}{3}$<br/>We know, $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$<br/>$=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \cdot \frac{1}{3}}=\frac{\frac{5}{6}}{\frac{5}{6}}=1=\tan \pi / 4$<br/>$\Rightarrow A+B=\pi / 4$</div>


title: if-tan-a-12-and-tan-b-13-then-what-is-the-value-of-4-a-4-b
slug: if-tan-a-12-and-tan-b-13-then-what-is-the-value-of-4-a-4-b-87765
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan A=1 / 2$ and $\tan B=1 / 3$, then what is the value of $4 A+4 B ?$</div>

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

answer:
<div class="solution">Let tan $A=\frac{1}{2}, \tan B=\frac{1}{3}$<br/>We know,<br/>$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}}=\frac{5}{6} \times \frac{6}{5}$<br/>$\tan (A+B)=1$<br/>$\Rightarrow A+B=\tan ^{-1}(1)=\frac{\pi}{4}$<br/>Multiply by 4 on both side,<br/>$4(A+B)=\frac{\pi}{4} \times 4 \Rightarrow 4 A+4 B=\pi$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', '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_P1.db
title:
if-tan-a-34-and-tan-b-125-then-how-many-values-can-cot-a-b-have-depending-on-the-actual-values-of-a-and-b

slug:
if-tan-a-34-and-tan-b-125-then-how-many-values-can-cot-a-b-have-depending-on-the-actual-values-of-a-and-b-42913

question:
<div class="question">If $\tan A=3 / 4$ and $\tan B=-12 / 5$, then how many values can cot $(A-B)$ have depending on the actual values of $A$ and $B ?$</div>

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

answer:
<div class="solution">$\tan \mathrm{A}=\frac{3}{4}$ and $\tan \mathrm{B}=-\frac{12}{5}$<br/>$\therefore \quad \cot (\mathrm{A}-\mathrm{B})=\frac{1}{\tan (\mathrm{A}-\mathrm{B})}=\frac{1+\tan \mathrm{A} \tan \mathrm{B}}{\tan \mathrm{A}-\tan \mathrm{B}}$</div>


title: if-tan-a-tan-b-x-and-cot-b-cot-a-y-then-what-is-cot-a-b-equal-to
slug: if-tan-a-tan-b-x-and-cot-b-cot-a-y-then-what-is-cot-a-b-equal-to-24882
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan A-\tan B=x$ and $\cot B-\cot A=y$, then what is cot<br/>$(A-B)$ equal to?</div>

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

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

answer:
<div class="solution">Let $\tan A-\tan B=x$ and $\cot B-\cot A=y$.<br/>$\Rightarrow \frac{1}{\tan B}-\frac{1}{\tan A}=y$<br/>$\Rightarrow \frac{\tan A-\tan B}{\tan A \tan B}=y \Rightarrow \frac{x}{\tan A \tan B}=y$<br/>Consider $\cot (A-B)=\left(\frac{1}{\tan (A-B)}\right)$<br/>$=\frac{1+\tan A \tan B}{\tan A-\tan B}=\frac{1+\frac{x}{y}}{x}=\frac{y+x}{x y}=\frac{1}{x}+\frac{1}{y}$.</div>


title: if-tan-a-x-1-and-tan-b-x-1-then-x-2-tan-a-b-has-the-value
slug: if-tan-a-x-1-and-tan-b-x-1-then-x-2-tan-a-b-has-the-value-30097
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan \mathrm{A}=\mathrm{x}+1$ and $\tan \mathrm{B}=\mathrm{x}-1$, then $\mathrm{x}^{2} \tan (\mathrm{A}-\mathrm{B})$ has the<br/>value:</div>

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

answer:
<div class="solution">$x^{2} \tan (A-B)$<br/>$=x^{2}\left(\frac{\tan A-\tan B}{1+\tan A \tan B}\right)=x^{2}\left(\frac{(x+1)-(x-1)}{1+\left(x^{2}-1\right)}\right)$<br/>$=x^{2}\left(\frac{2}{x^{2}}\right)=2$</div>


title: if-tan-m-where-m-is-nonsquare-natural-number-then-sec-2-is
slug: if-tan-m-where-m-is-nonsquare-natural-number-then-sec-2-is-58991
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan \theta=\sqrt{m}$, where $\mathrm{m}$ is non-square natural<br/>number, then $\sec 2 \theta$ is</div>

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

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

answer:
<div class="solution">Let $\tan \theta=\sqrt{m}$, where $\mathrm{m}$ is a non-square natural num-<br/>ber.<br/>$\Rightarrow \sin \theta=\sqrt{m} \cos \theta$<br/>Consider, $\sec 2 \theta=\frac{1}{\cos 2 \theta}=\frac{1}{\cos ^{2} \theta-\sin ^{2} \theta}$<br/>$=\frac{1}{\cos ^{2} \theta-m \cos ^{2} \theta}=\frac{1}{\cos ^{2} \theta(1-m)}$<br/>$=\frac{\sec ^{2} \theta}{1-m}=\frac{1+\tan ^{2} \theta}{1-m}=\frac{1+m}{1-m}$<br/>$=\frac{(1+m)(1-m)}{(1-m)(1-m)}=\frac{\left(1-m^{2}\right)}{(1-m)^{2}}$<br/>Numerator will always be negative and denominator will always be positive<br/>Hence, $\sec 2 \theta=\frac{1-m^{2}}{(1-m)^{2}}$ is a negative number</div>


title: if-tan-sec-4-then-what-is-the-value-of-sin-theta
slug: if-tan-sec-4-then-what-is-the-value-of-sin-theta-24645
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\tan \theta+\sec \theta=4$, then what is the value of $\sin \theta ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 / 17$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$8 / 15$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$15 / 17$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$23 / 32$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$15 / 17$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \tan \theta+\sec \theta=4 \\ &amp; \Rightarrow \frac{\sin \theta}{\cos \theta}+\frac{1}{\cos \theta}=4 \\ &amp; \Rightarrow 1+\sin \theta=4 \cos \theta \end{aligned}$<br/>Squaring on both side, $(1+\sin \theta)^{2}=16 \cos ^{2} \theta=16\left(1-\sin ^{2} \theta\right)$<br/>$(1+\sin \theta)^{2}=16(1-\sin \theta)(1+\sin \theta)$<br/>$1+\sin \theta=16-16 \sin \theta$<br/>$17 \sin \theta=16-1$<br/>$\sin \theta=\frac{15}{17}$</div>


title: if-the-10-th-term-of-a-gp-is-9-and-4-th-term-is-4-then-what-is-its-7th-term
slug: if-the-10-th-term-of-a-gp-is-9-and-4-th-term-is-4-then-what-is-its-7th-term-51667
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the 10 th term of a GP is 9 and 4 th term is 4 , then what is its 7th term?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">14</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$27 / 14$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$56 / 15$</span> </li> </ul>

answer:
<div class="solution">General term of G.P= ar $^{\text {1-1 }}$. Given : $\mathrm{a}_{10}=9$ and $\mathrm{a}_{4}=4$<br/>$\Rightarrow \mathrm{ar}^{9}=9$ and $\mathrm{ar}^{3}=4$<br/>On dividing we get<br/>$\begin{aligned} \frac{a r^{9}}{a r^{3}}=&amp; \frac{9}{4} \Rightarrow r^{6}=\frac{9}{4} \\ a r^{3}=4 &amp; \Rightarrow\left(a r^{3}\right)^{2}=16 \\ &amp; \Rightarrow a^{2} r^{6}=16 \\ &amp; \Rightarrow a^{2} \times \frac{9}{4}=16 \Rightarrow a^{2}=\frac{64}{9} \Rightarrow a=\frac{8}{3} \\ \text { Thus, } a_{7}=a r^{6}=\frac{8}{3} \times \frac{9}{4}=6 \end{aligned}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-am-and-gm-between-two-number-are-in-the-ratio-m-n-then-what-is-the-ratio-between-the-two-numbers

slug:
if-the-am-and-gm-between-two-number-are-in-the-ratio-m-n-then-what-is-the-ratio-between-the-two-numbers-51948

question:
<div class="question">If the AM and GM between two number are in the ratio $m: n$, then what is the ratio between the two numbers?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{m+\sqrt{m^{2}-n^{2}}}{m-\sqrt{m^{2}-n^{2}}}$</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">$\frac{m+n}{m-n}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{m^{2}-n^{2}}{m^{2}+n^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{m^{2}+n^{2}-m n}{m^{2}+n^{2}+m n}$</span> </li> </ul>

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

answer:
<div class="solution">Let 'a' and 'b' two numbers.<br/>A.M $=\frac{a+b}{2}$ and $G \cdot M=\sqrt{a b}$<br/>According to the question, $A: G=m: n$<br/>$\Rightarrow \frac{a+b}{2 \sqrt{a b}}=\frac{m}{n} \Rightarrow \frac{(a+b)^{2}}{4 a b}=\frac{m^{2}}{n^{2}}$ ...(i)<br/>and $\frac{(a+b)^{2}-4 a b}{4 a b}=\frac{m^{2}-n^{2}}{n^{2}}$<br/>$\Rightarrow \frac{(a-b)^{2}}{4 a b}=\frac{m^{2}-n^{2}}{n^{2}}$ ...(ii)<br/>Since, on dividing Equation (i) and (ii), we get<br/>$\frac{(a+b)^{2}}{(a-b)^{2}}=\frac{m^{2}}{m^{2}-n^{2}} \Rightarrow \frac{(a+b)}{(a-b)}=\frac{m}{\sqrt{m^{2}-n^{2}}}$<br/>$\Rightarrow \frac{(a+b)+(a-b)}{(a+b)-(a-b)}=\frac{m+\sqrt{m^{2}-n^{2}}}{m-\sqrt{m^{2}-n^{2}}}$<br/>(Using componendo dividendo rule)<br/>$\Rightarrow \quad \frac{2 a}{2 b}=\frac{a}{b}=\frac{m+\sqrt{m^{2}-n^{2}}}{m-\sqrt{m^{2}-n^{2}}}$</div>


title: if-the-am-and-gm-of-two-numbers-are-5-and-4-respectively-then-what-is-the-hm-of-those-numbers
slug: if-the-am-and-gm-of-two-numbers-are-5-and-4-respectively-then-what-is-the-hm-of-those-numbers-61008
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the AM and GM of two numbers are 5 and 4 respectively, then what is the HM of those numbers?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5}{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{16}{5}$</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">$\frac{9}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{16}{5}$</span> </div>

answer:
<div class="solution">We know, $\mathrm{HM}=\frac{(\mathrm{GM})^{2}}{\mathrm{AM}}$<br/>$\therefore \mathrm{HM}=\frac{16}{5}$</div>


title: if-the-am-and-hm-of-two-numbers-are-27-and-12-respectively-then-what-is-their-gm-equal-to-2010-i
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">18</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-am-and-hm-of-two-numbers-are-27-and-12-respectively-then-what-is-their-gm-equal-to-2010-i-16812

question:
<div class="question">If the AM and HM of two numbers are 27 and 12 respectively, then what is their GM equal to? $\quad[2010-I]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">18</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">27</span> </li> </ul>

answer:
<div class="solution">Given $\mathrm{AM}=27$ and $\mathrm{HM}=12$ and we know that $(\mathrm{GM})^{2}=(\mathrm{AM})(\mathrm{HM})=27 \times 12$<br/>$\Rightarrow G M=\sqrt{27 \times 12}=\sqrt{3 \times 3 \times 3 \times 3 \times 2 \times 2}$<br/>$\Rightarrow \quad \mathrm{GM}=3 \times 3 \times 2=18$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-angle-between-the-lines-whose-direction-ratios-are-2-1-2-and-x-3-5-is-4-then-the-smaller-value-of-x-is

slug:
if-the-angle-between-the-lines-whose-direction-ratios-are-2-1-2-and-x-3-5-is-4-then-the-smaller-value-of-x-is-37446

question:
<div class="question">If the angle between the lines whose direction ratios are<br/>$(2,-1,2)$ and $(x, 3,5)$ is $\frac{\pi}{4}$, then the smaller value of $x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">52</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Given direction ratios are $(2,-1,2)$ and $(x, 3,5)$ We know that the angle between the lines whose direction ratios are $\left(a_{1}, b_{1}, c_{1}\right)$ and $\left(a_{2}, b_{2}, c_{2}\right)$ is<br/>$\cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$<br/>$\Rightarrow \cos \frac{\pi}{4}=\frac{2 x-3+10}{\sqrt{4+1+4} \sqrt{x^{2}+9+25}}=\frac{2 x+7}{\sqrt{9} \cdot \sqrt{x^{2}+34}}$<br/>$\Rightarrow \frac{1}{\sqrt{2}}=\frac{2 x+7}{3 \sqrt{x^{2}+34}} \Rightarrow 2 x+7=3 \sqrt{\frac{x^{2}+34}{2}}$<br/>$\Rightarrow 4 x^{2}+49+28 x=\frac{9\left(x^{2}+34\right)}{2}$ (Squaring on both<br/>sides)<br/>$\Rightarrow 2\left(4 x^{2}+49+28 x\right)=9 x^{2}+306$<br/>$\Rightarrow 8 x^{2}+98+56 x=9 x^{2}+306$<br/>$\Rightarrow x^{2}-56 x+208=0$<br/>$\therefore \mathrm{x}=\frac{56 \pm \sqrt{3136-812}}{2}=\frac{56 \pm 48}{2}=28 \pm 24$<br/>$=4,52 .$<br/>Smaller value $=4$.</div>


title: if-the-angle-between-the-vectors-a-and-b-is-3-what-is-the-angle-between-5-a-and-6-b
slug: if-the-angle-between-the-vectors-a-and-b-is-3-what-is-the-angle-between-5-a-and-6-b-63694
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the angle between the vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ is $\frac{\pi}{3}$, what is the<br/>angle between $-5 \overrightarrow{\mathbf{a}}$ and $6 \overrightarrow{\mathbf{b}}$ ?</div>

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

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

answer:
<div class="solution">From the figure it is clear that the angle between<br/>$6 \overrightarrow{\mathbf{b}}$ and $-5 \overrightarrow{\mathbf{a}}$ is $120^{\circ}$ or $\frac{2 \pi}{3}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/wHXl-DAjqaWaLjDurY6zEVtbIqq2O2wwOmyaV8zc0C0.original.fullsize.png"/><br/><br/>ALTERNATE SOLUTION:<img src="https://cdn-question-pool.getmarks.app/pyq/nda/f3BatEhDJjGlnphGaZ5C5brkP7snovJDXlpy_BiG604.original.fullsize.png"/><br/><br/>$\theta=\pi-\frac{\pi}{3}=\frac{2 \pi}{3}$</div>


title: if-the-angle-between-the-vectors-i-m-j-and-j-k-is-3-then-what-is-the-value-of-m
slug: if-the-angle-between-the-vectors-i-m-j-and-j-k-is-3-then-what-is-the-value-of-m-85773
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the angle between the vectors $\hat{i}-m \hat{j}$ and $\hat{j}+\hat{k}$ is $\frac{\pi}{3}$, then what is the value of $m ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</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">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">$\cos \frac{\pi}{3}=\frac{(\hat{\mathrm{i}}-\mathrm{m} \hat{\mathrm{j}}) \cdot(\hat{\mathrm{j}}+\hat{\mathrm{k}})}{\left|\sqrt{1+\mathrm{m}^{2}}\right|\left|\sqrt{1^{2}+1^{2}}\right|}$<br/>$\frac{1}{2}=\frac{-\mathrm{m}}{\sqrt{1+\mathrm{m}^{2}} \cdot \sqrt{2}}$</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-angles-of-a-triangle-abc-are-in-the-ratio-1-2-3-then-the-corresponding-sides-are-in-the-ratio

slug:
if-the-angles-of-a-triangle-abc-are-in-the-ratio-1-2-3-then-the-corresponding-sides-are-in-the-ratio-70676

question:
<div class="question">If the angles of a triangle $\mathrm{ABC}$ are in the ratio $1: 2: 3$. then the corresponding sides are in the ratio</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1: \sqrt{3}: 2$</span> </div>

answer:
<div class="solution">Given, angles of triangle are in ratio $1: 2: 3$ Consider, $\mathrm{A}=30^{\circ}, \mathrm{B}=60^{\circ}$ and $\mathrm{C}=90^{\circ}$<br/>We know, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$<br/>$\Rightarrow \frac{a}{\sin 30^{\circ}}=\frac{b}{\sin 60^{\circ}}=\frac{c}{\sin 90^{\circ}}$<br/>$\Rightarrow \frac{a}{\frac{1}{2}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{c}{1}$<br/>$\Rightarrow a: b: c=\frac{1}{2}: \frac{\sqrt{3}}{2}: 1=1: \sqrt{3}: 2$</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-angles-of-a-triangle-are-3-0-and-4-5-and-the-included-side-is-3-1-then-what-is-the-area-of-the-tringle

slug:
if-the-angles-of-a-triangle-are-3-0-and-4-5-and-the-included-side-is-3-1-then-what-is-the-area-of-the-tringle-76915

question:
<div class="question">If the angles of a triangle are $30^{\circ}$ and $45^{\circ}$ and the included<br/>side is $(\sqrt{3}+1)$, then what is the area of the tringle?</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/MtR667Rl3a4cNLsiLQzNsxZt84TxrHEdWivgJQ8WzyQ.original.fullsize.png"/><br/><br/>From $\Delta \mathrm{ADB}, \mathrm{AD}=\mathrm{BD}=\mathrm{x}$<br/>In $\Delta \mathrm{ADC}$<br/>$\tan 30^{\circ}=\frac{\mathrm{x}}{\sqrt{3}+1-\mathrm{x}}$<br/>$\Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{x}}{\sqrt{3}+1-\mathrm{x}} \Rightarrow \sqrt{3} \mathrm{x}=\sqrt{3}+1-\mathrm{x}$<br/>$\Rightarrow(\sqrt{3}+1) \mathrm{x}=\sqrt{3}+1$<br/>$x=\frac{\sqrt{3}+1}{\sqrt{3}+1}=1$<br/>Area of $\triangle \mathrm{ABC}=\frac{1}{2} \times(\sqrt{3}+1) \times 1=\frac{\sqrt{3}+1}{2}$</div>


title: if-the-area-of-a-triangle-with-vertices-3-0-3-0-and-0-k-is-9-sq-unit-then-what-is-the-value-of-k
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-area-of-a-triangle-with-vertices-3-0-3-0-and-0-k-is-9-sq-unit-then-what-is-the-value-of-k-64354

question:
<div class="question">If the area of a triangle with vertices $(-3,0),(3,0)$ and $(0, k)$ is 9 sq unit, then what is the value of $k ?$</div>

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

answer:
<div class="solution">Let the vertices of the $\Delta \mathrm{ABC}$ be $\mathrm{A}(-3,0), \mathrm{B}(3,0)$ and $\mathrm{C}(0, \mathrm{k})$<br/>$\therefore \quad$ Area of $\Delta \mathrm{ABC}=\frac{1}{2}\left|\begin{array}{ccc}-3 &amp; 0 &amp; 1 \\ 3 &amp; 0 &amp; 1 \\ 0 &amp; \mathrm{k} &amp; 1\end{array}\right|$<br/>Given, area is 9<br/>$\Rightarrow \quad 9=\frac{1}{2}\{-3(-\mathrm{k})+1(3 \mathrm{k})\}$<br/>$\Rightarrow \quad 18=3 \mathrm{k}+3 \mathrm{k}$<br/>$\Rightarrow \quad \mathrm{k}=\frac{18}{6}=3$</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">12</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-arithmetic-and-geometric-means-of-two-numbers-are-10-8-respectively-then-one-number-exceeds-the-other-number-by

slug:
if-the-arithmetic-and-geometric-means-of-two-numbers-are-10-8-respectively-then-one-number-exceeds-the-other-number-by-90647

question:
<div class="question">If the arithmetic and geometric means of two numbers are 10 , 8 respectively, then one number exceeds the other number by</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">10</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">Let a and $b$ be two numbers such that<br/>A.M. $=\frac{a+b}{2}=10$<br/>$\Rightarrow \mathrm{a}+\mathrm{b}=20$ ...(i)<br/>and G.M. $=\sqrt{a b}=8 \Rightarrow \mathrm{ab}=64 \quad$...(2)<br/>From (1) and (2), we have $a^{2}-20 a+64=0 \Rightarrow(a-4)(a-16)=0$<br/>$\Rightarrow \mathrm{a}=4,16$<br/>Thus, when $\mathrm{a}=4, \mathrm{~b}=16$ and when $\mathrm{a}=16, \mathrm{~b}=4$<br/>Hence, one number exceeds the other number by $12 .$</div>


tags: ['Biology', 'Ecosystem']
source_db: MarksBatch2_P1.db
title:
if-the-carbon-atoms-fixed-by-producers-already-have-passed-through-three-species-the-trophic-level-of-the-last-species-would-be

slug:
if-the-carbon-atoms-fixed-by-producers-already-have-passed-through-three-species-the-trophic-level-of-the-last-species-would-be-33338

question:
<div class="question">If the carbon atoms fixed by producers already have passed through three species, the trophic level of the last species would be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>scavenger<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>tertiary producer<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>tertiary consumer<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">D</span> <span class="option-data"><br/>secondary consumer</span> </li> </ul>

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

answer:
<div class="solution">Producer $\rightarrow$ Ist Trophic level $\rightarrow$<br/>(Primary consumer)<br/>IInd Trophic level $\rightarrow$ IIIrd Trophic level (Secondary consumer) (Tertiary consumer)<br/>Scavengers are the detrivores that feeds on the tissue of dead animals (e.g., vultures) and plays no role in carbon fixation. Producers fix the carbon dioxide which is passed through the different trophic level.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-cardinality-of-a-set-a-is-4-and-that-of-a-set-b-is-3-then-what-is-the-cardinality-of-the-set-a-b

slug:
if-the-cardinality-of-a-set-a-is-4-and-that-of-a-set-b-is-3-then-what-is-the-cardinality-of-the-set-a-b-96152

question:
<div class="question">If the cardinality of a set $A$ is 4 and that of a set $B$ is 3 , then what is the cardinality of the set $A \Delta B$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Cannot be determined as the sets $A$ and $B$ are not given</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Cannot be determined as the sets $A$ and $B$ are not given</span> </div>

answer:
<div class="solution">Given, $n(A)=4, n(B)=3$ Since, the sets A and B are not known, then cardinality of the set $\mathrm{A} \Delta \mathrm{B}$ cannot be determined.</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-centre-of-the-circle-passing-through-the-origin-is-3-4-then-the-intercepts-cut-off-by-the-circle-on-x-axis-and-y-axis-respectively-are-2015-ii

slug:
if-the-centre-of-the-circle-passing-through-the-origin-is-3-4-then-the-intercepts-cut-off-by-the-circle-on-x-axis-and-y-axis-respectively-are-2015-ii-14914

question:
<div class="question">If the centre of the circle passing through the origin is $(3,4)$, then the intercepts cut off by the circle on $\mathrm{x}$ -axis and $\mathrm{y}$ -axis respectively are $\quad[2015-I I]$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">6 unit and 8 unit</span> </div>

answer:
<div class="solution">Equation of circle having radius $\mathrm{r}$ and centre $(3,4)$ is $=(x-3)^{2}+(y-4)^{2}=r^{2}$<br/>if it is passing through $(0,0)$ $\therefore(0-3)^{2}+(0-4)^{2}=\mathrm{r}^{2}$<br/>$\Rightarrow \mathrm{r}^{2}=25$<br/>equation of circle is<br/>$$<br/>(x-3)^{2}+(y-4)^{2}=25<br/>$$<br/>putting $\mathrm{y}=0$ $\therefore \mathrm{x}=6$ unit $=$ interception $\mathrm{x}$ -axis<br/>intercept on y axis (putting $\mathrm{x}=0$ ) is $y-8$ unil</div>


title: if-the-centre-of-the-sphere-a-x-2-b-y-2-c-z-2-2-x-4-y-2-z-3-0-is-12-1-12-what-is-the-value-of-b
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-centre-of-the-sphere-a-x-2-b-y-2-c-z-2-2-x-4-y-2-z-3-0-is-12-1-12-what-is-the-value-of-b-16803

question:
<div class="question">If the centre of the sphere $a x^{2}+b y^{2}+c z^{2}-2 x+4 y+2 z-3=0$ is $(1 / 2,-1,-1 / 2)$, what<br/>is the value of $\mathrm{b}$ ?</div>

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

answer:
<div class="solution">The given equation of sphere $a x^{2}+b y^{2}+c z^{2}-2 x+4 y+2 z-3=0$<br/>This equation represents a equation of sphere, if coefficient of $x^{2}, y^{2}$ and $z^{2}$ is same.<br/>i.e., $a=b=c$<br/>Equation of sphere can be re-written as $b x^{2}+b y^{2}+b z^{2}-2 x+4 y+2 z-3=0$<br/>$\Rightarrow x^{2}+y^{2}+z^{2}-\frac{2 x}{b}+\frac{4 y}{b}+\frac{2 z}{b}-\frac{3}{b}=0$<br/>The centre of this sphere is $\left(\frac{1}{b}, \frac{-2}{b}, \frac{-1}{b}\right)$<br/>Given that the centre of sphere is $\left(\frac{1}{2},-1,-\frac{1}{2}\right)$<br/>$\frac{1}{b}=\frac{1}{2} \Rightarrow b=2$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5,2</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-centroid-of-a-triangle-formed-by-7-x-y-6-and-910-is-6-3-then-the-values-of-x-and-y-are-respectively

slug:
if-the-centroid-of-a-triangle-formed-by-7-x-y-6-and-910-is-6-3-then-the-values-of-x-and-y-are-respectively-60994

question:
<div class="question">If the centroid of a triangle formed by $(7, x),(y,-6)$ and (9,10) is $(6,3)$, then the values of $x$ and $y$ are respectively</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">2,5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1,0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0,0</span> </li> </ul>

answer:
<div class="solution">Centroid of given triangle $=\left(\frac{7+\mathrm{y}+9}{3}, \frac{\mathrm{x}-6+10}{3}\right)$<br/>$\Rightarrow\left(\frac{16+\mathrm{y}}{3}, \frac{\mathrm{x}+4}{3}\right)=(6,3)$<br/>$\Rightarrow 16+\mathrm{y}=18 ; \mathrm{x}+4=9$<br/>$\Rightarrow \mathrm{y}=2 ; \mathrm{x}=5$</div>


tags: ['Physics', 'Units and Dimensions', 'AP EAMCET', 'AP EAMCET 2019 (20 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-charge-of-electron-e-mass-of-electron-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-then-the-permeabi

slug:
if-the-charge-of-electron-e-mass-of-electron-m-speed-of-light-in-vacuum-c-and-plancks-constant-h-are-taken-as-fundamental-quantities-then-the-permeabi-57296

question:
<div class="question">If the charge of electron \(e\), mass of electron \(m\), speed of light in vacuum \(c\) and Planck's constant \(h\) are taken as fundamental quantities, then the permeability of vacuum \(\mu_0\) can be expressed as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{h}{m c^2}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{h c}{m e^2}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{h}{c e^2}\)</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">\(\frac{m c^2}{h e^2}\)</span> </li> </ul>

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

answer:
<div class="solution">We can expressed the permeability of vacuum,<br/>\(\mu_0 \propto e^a m^b c^c h^d\)<br/>or \(\mu_0=k e^a m^b c^c h^d\) ...(i)<br/>Where, \(k\) is a dimensional constant.<br/>As we know that,<br/>Dimension of<br/>\(\begin{aligned}<br/>\mu_0 &amp; =\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2} \mathrm{~A}^{-2}\right], \\<br/>e &amp; =[\mathrm{AT}], \\<br/>m &amp; =[\mathrm{M}], \\<br/>c &amp; =\left[\mathrm{L} \mathrm{T}^{-1}\right], \\<br/>h &amp; =\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1}\right],<br/>\end{aligned}\)<br/>and<br/>Putting the dimension of various physical quantities in Eq. (i), we get<br/>\(\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2} \mathrm{~A}^{-2}\right]=k[\mathrm{~A} \mathrm{~T}]^a\left[\mathrm{M}^b\left[\mathrm{~L} \mathrm{~T}^{-1}\right]^c\right]\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1}\right]^d\)<br/>\(\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2} \mathrm{~A}^{-2}\right]=k[\mathrm{M}]^{b+d}[\mathrm{~L}]^{c+2 d}[\mathrm{~T}]^{a-c-d}[\mathrm{~A}]^a\)<br/>Comparing the powers of \(\mathrm{M}, \mathrm{L}, \mathrm{T}\) and \(\mathrm{A}\) on the both sides, we get<br/>\(\begin{aligned}<br/>b+d &amp; =1 \quad \ldots (ii) \\<br/>c+2 d &amp; =1 \quad \ldots (iii)  \\<br/>a-c-d &amp; =-2 \quad \ldots (iv)  \\<br/>a &amp; =-2 \quad \ldots (v) <br/>\end{aligned}\)<br/>After solving the Eqs. (ii), (iii), (iv) and (v), we get<br/>\(\begin{aligned}<br/>a &amp; =-2, \\<br/>b &amp; =0, \\<br/>c &amp; =-1, \\<br/>\text{and } d &amp; =1, \\<br/>\therefore \mu_0 &amp; =\frac{h}{c e^2}<br/>\end{aligned}\)<br/>So, the permeability of vacuum \(\mu_0\) can be expressed as, \(\frac{h}{c e^2}\).</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-circle-x-2-y-2-2-gx-2-f-y-c-0-c-0-touches-the-y-axis-then-which-one-of-the-following-is-correct

slug:
if-the-circle-x-2-y-2-2-gx-2-f-y-c-0-c-0-touches-the-y-axis-then-which-one-of-the-following-is-correct-69179

question:
<div class="question">If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0(c&gt;0)$ touches the $y$ axis, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">As given, the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ touches y-axis. then $\mathrm{r}=\pm \mathrm{g}$ and $\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}=\mathrm{g}^{2}$<br/>$\Rightarrow \mathrm{f}^{2}-\mathrm{c}=0 \Rightarrow \mathrm{f}^{2}=\mathrm{c} \Rightarrow \mathrm{f}=\pm \sqrt{\mathrm{c}}$</div>


tags: ['Physics', 'Current Electricity', 'VITEEE', 'VITEEE 2007']
source_db: MarksBatch2_P1.db
title:
if-the-coefficient-of-mutual-induction-of-the-primary-and-secondary-coils-of-an-induction-coil-is-5-h-and-a-current-of-10-a-is-cut-off-in-5-1-0-4-seco

slug:
if-the-coefficient-of-mutual-induction-of-the-primary-and-secondary-coils-of-an-induction-coil-is-5-h-and-a-current-of-10-a-is-cut-off-in-5-1-0-4-seco-10069

question:
<div class="question">If the coefficient of mutual induction of the primary and secondary coils of an induction coil is $5 \mathrm{H}$ and a current of $10 \mathrm{~A}$ is cut off in $5 \times 10^{-4}$ second, the $e m f$ induced(in volt) in the secondary coil is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \times 10^{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1 \times 10^{5}$</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">$25 \times 10^{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 \times 10^{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 \times 10^{5}$</span> </div>

answer:
<div class="solution">$\mathrm{M}=5 \mathrm{H}, \mathrm{I}=10 \mathrm{~A}, \mathrm{t}=5 \times 10^{-4} \mathrm{~s}$.<br/>Now, emf induced in secondary is<br/>$\begin{array}{l}<br/>\mathrm{e}=-\mathrm{M} \frac{\mathrm{d}}{\mathrm{dt}}, \quad \text { (-ve sign shows direction) } \\<br/>\Rightarrow \mathrm{e}=5 \times \frac{10}{5 \times 10^{-4}}=1 \times 10^{+5} \mathrm{~V}<br/>\end{array}$</div>


tags: ['Mathematics', 'Linear Programming', 'KCET', 'KCET 2014']
source_db: MarksBatch2_P1.db
title:
if-the-coefficient-of-variation-and-standard-deviation-are-60-and-21-respectively-the-arithmetic-mean-of-distribution-is

slug:
if-the-coefficient-of-variation-and-standard-deviation-are-60-and-21-respectively-the-arithmetic-mean-of-distribution-is-57772

question:
<div class="question">If the coefficient of variation and standard deviation are \( 60 \) and \( 21 \) respectively, the arithmetic<br/>mean of distribution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( 30 \)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\( 21 \)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\( 60 \)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\( 35 \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\( 35 \)</span> </div>

answer:
<div class="solution">We know that, coefficient of variation \(=\frac{\sigma}{\mu} \times 100 \%\)<br/>\(\Rightarrow 60=\frac{21}{\mu} \times 100\), substitute the given values<br/>\(\therefore \mu=\frac{2100}{60}=35\)</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-coefficients-of-a-m-and-a-n-in-the-expansion-of-1-a-m-n-are-and-then-which-one-of-the-following-is-correct

slug:
if-the-coefficients-of-a-m-and-a-n-in-the-expansion-of-1-a-m-n-are-and-then-which-one-of-the-following-is-correct-21987

question:
<div class="question">If the coefficients of $\mathrm{a}^{\mathrm{m}}$ and $\mathrm{a}^{\mathrm{n}}$ in the expansion of $(1+a)^{\mathrm{m}+n}$ are $\alpha$ and $\beta$, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">$(1+a)^{m+11}$<br/>$\alpha=$ coefficient of $a^{\mathrm{m}}={ }^{m+n} C_{\mathrm{m}}$<br/>$\beta=$ coefficient of $a^{\mathrm{n}}={ }^{m+n} C_{n}$<br/>We know, ${ }^{n} C_{r}={ }^{n} C_{n-r}$<br/>$\therefore \beta={ }^{m+n} C_{n}={ }^{m+n} C_{m+n-n}={ }^{m+n} C_{m}=\alpha$<br/>$\therefore \alpha=\beta$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch2_P1.db
title:
if-the-coefficients-of-variation-of-two-distributions-are-60-and-70-and-their-standard-deviations-are-21-and-16-respectively-then-their-arithmetic-mea-1

slug:
if-the-coefficients-of-variation-of-two-distributions-are-60-and-70-and-their-standard-deviations-are-21-and-16-respectively-then-their-arithmetic-mea-1-11604

question:
<div class="question">If the coefficients of variation of two distributions are 60 and 70 and their standard deviations are 21 and 16 respectively, then their arithmetic means are respectively</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$35,22.85$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$32,25.85$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$35,28.25$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$35,25.25$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$35,22.85$</span> </div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2017 (25 Apr Shift 2)']
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch2_P1.db
title:
if-the-coefficients-of-variation-of-two-distributions-are-60-and-70-and-their-standard-deviations-are-21-and-16-respectively-then-their-arithmetic-mea

slug:
if-the-coefficients-of-variation-of-two-distributions-are-60-and-70-and-their-standard-deviations-are-21-and-16-respectively-then-their-arithmetic-mea-87713

question:
<div class="question">If the coefficients of variation of two distributions are 60 and 70 and their standard deviations are 21 and 16 respectively, then their arithmetic means are respectively</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$35,22.85$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$32,25.85$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$35,28.25$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$35,25.25$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$35,22.85$</span> </div>


title: if-the-components-of-b-along-and-perpendicular-to-a-are-a-and-b-a-respectively-what-is-equal-to
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-components-of-b-along-and-perpendicular-to-a-are-a-and-b-a-respectively-what-is-equal-to-24913

question:
<div class="question">If the components of $\overrightarrow{\mathrm{b}}$ along and perpendicular to $\overrightarrow{\mathrm{a}}$ are<br/>$\lambda \vec{a}$ and $\vec{b}-\lambda \vec{a}$ respectively, what is $\lambda$ equal to ?</div>

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

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

answer:
<div class="solution">We know that the components of $\overrightarrow{\mathrm{b}}$ along $\overrightarrow{\mathrm{a}}$ is $\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}$ and perpendicular to $\overrightarrow{\mathrm{a}}$ is $\overrightarrow{\mathrm{b}}-\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}$<br/>As given : $\left\{\begin{array}{l} \left.\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^{2}}\right\} \vec{a}=\lambda \vec{a}\end{array}\right.$<br/>and $\overrightarrow{\mathrm{b}}-\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}-\lambda \overrightarrow{\mathrm{a}}$<br/>$\Rightarrow \lambda=\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">81</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-correlation-coefficient-between-x-and-y-is-06-covariance-is-27-and-variance-of-y-is-25-then-what-is-the-variance-of-x

slug:
if-the-correlation-coefficient-between-x-and-y-is-06-covariance-is-27-and-variance-of-y-is-25-then-what-is-the-variance-of-x-64265

question:
<div class="question">If the correlation coefficient between $\mathrm{x}$ and $\mathrm{y}$ is $0.6$, covariance is 27 and variance of $y$ is 25 , then what is the variance of x? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{9}{5}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{81}{25}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">81</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, $r=0.6$, covariance $=27, \sigma_{(y)}^{2}=25 \Rightarrow \sigma(y)=5$<br/>We know, $\mathrm{r}=\frac{\text { covariance }(\mathrm{x}, \mathrm{y})}{\sigma(\mathrm{x}) \cdot \sigma(\mathrm{y})}$<br/>$\Rightarrow \sigma(\mathrm{x})=\frac{\operatorname{covariance}(\mathrm{x}, \mathrm{y})}{\mathrm{r} . \sigma(\mathrm{y})}=\frac{27}{\left(\frac{6}{10}\right) \cdot 5}$<br/>$=\frac{27 \times 2}{6}=9$<br/>$\Rightarrow \sigma^{2}(\mathrm{x})=81 .$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.5$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-covariance-between-x-and-y-is-30-variance-of-x-is-25-and-variance-of-y-is-144-then-what-is-the-correlation-coefficient

slug:
if-the-covariance-between-x-and-y-is-30-variance-of-x-is-25-and-variance-of-y-is-144-then-what-is-the-correlation-coefficient-94464

question:
<div class="question">If the covariance between $x$ and $y$ is 30 , variance of $x$ is 25 and variance of $y$ is 144 , then what is the correlation coefficient? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.5$</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">$0.6$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.7$</span> </li> </ul>

answer:
<div class="solution">$\operatorname{cov}(x, y)=30$<br/>$$<br/>\begin{array}{l}<br/>\operatorname{var}(x)=25 ; \operatorname{var}(y)=144 \\<br/>\Rightarrow r(x, y)=\frac{\operatorname{cov}(x, y)}{\sqrt{\operatorname{var}(x) \cdot \operatorname{var}(y)}} \\<br/>\Rightarrow r(x, y)=\frac{30}{\sqrt{25 \times 144}}=\frac{30}{5 \times 12}=0.5<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'JIPMER', 'JIPMER 2004']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.05 H</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-current-3-a-flowing-in-the-primary-coil-is-made-zero-in-01-sec-the-emf-induced-in-the-secondary-coil-is-15-volt-the-mutual-inductance-between-t

slug:
if-the-current-3-a-flowing-in-the-primary-coil-is-made-zero-in-01-sec-the-emf-induced-in-the-secondary-coil-is-15-volt-the-mutual-inductance-between-t-74781

question:
<div class="question">If the current 3 A flowing in the primary coil is made zero in 0.1 sec, the emf induced in the secondary coil is 1.5 volt. The mutual inductance between the coils is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.05 H</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1.05 H</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.1 H</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.2 H</span> </li> </ul>

answer:
<div class="solution">Here: $e=1.5$ volt<br/>$\Delta i=0-3=-3 \mathrm{~A}, \Delta t=0.1 \mathrm{sec}$<br/>Now the emf induced in secondary coil<br/>$e=\frac{M \Delta i}{\Delta t} \quad$ (numerically)<br/>or $\quad 1.5=\frac{M \times 3}{0.1}$<br/>or $\quad M=\frac{1.5 \times 0.1}{3}=0.05 \mathrm{H}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-data-are-moderately-nonsymmetrical-then-which-onc-of-the-following-empirical-relationships-is-correct

slug:
if-the-data-are-moderately-nonsymmetrical-then-which-onc-of-the-following-empirical-relationships-is-correct-58632

question:
<div class="question">If the data are moderately non-symmetrical, then which onc of the following empirical relationships is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \times$ Standard deviation $=5 \times$ Mean deviation</span> </div>

answer:
<div class="solution">We know, $\frac{4}{5}$ standard deviation = Mean deviation $\Rightarrow 4 \times$ standard deviation $=5 \times$ mean deviation</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2019 (12 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-data-x-1-x-2-x-10-is-such-that-the-mean-of-first-four-of-these-is-11-the-mean-of-the-remaining-six-is-16-and-the-sum-of-squares-of-all-of-these

slug:
if-the-data-x-1-x-2-x-10-is-such-that-the-mean-of-first-four-of-these-is-11-the-mean-of-the-remaining-six-is-16-and-the-sum-of-squares-of-all-of-these-64629

question:
<div class="question">If the data <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub></math> is such that the mean of first four of these is <math><mn>11</mn><mo>,</mo></math> the mean of the remaining six is <math><mn>16</mn></math> and the sum of squares of all of these is <math><mn>2000</mn></math>, then the standard deviation of this data is:</div>

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

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

answer:
<div class="solution">Let the numbers are <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub><mfenced separators="|"><mrow><mi>N</mi><mo>=</mo><mn>10</mn></mrow></mfenced></math><br/>Given <math><mfenced close="}" open="" separators="|"><mrow><mtable><mtr><mtd><mrow><maligngroup></maligngroup><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mn>4.11</mn><mo>=</mo><mn>44</mn></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup></maligngroup><msub><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>…</mo><mo>.</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>=</mo><mn>6.16</mn><mo>=</mo><mn>96</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>⇒</mo><mrow><mo stretchy="false">∑</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>140</mn></mrow></mrow></math><br/><math><mo>⇒</mo><mi> </mi></math> variance <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mi>N</mi></mfrac><mo>-</mo><msup><mfenced separators="|"><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>N</mi></mfrac></mfenced><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>2000</mn><mn>10</mn></mfrac><mo>-</mo><msup><mfenced separators="|"><mfrac><mn>140</mn><mn>10</mn></mfrac></mfenced><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mn>200</mn><mo>-</mo><mn>196</mn><mo>=</mo><mn>4</mn></math><br/><math><mo>⇒</mo><mi> </mi></math> Standard deviation <math><mi>σ</mi><mo>=</mo><mn>2</mn><mo>.</mo></math></div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-derivative-of-the-function-f-x-a-x-2-b-b-x-2-a-x-4-x-1-x-1-is-every-where-continuous-then-what-are-the-values-of-a-and-b

slug:
if-the-derivative-of-the-function-f-x-a-x-2-b-b-x-2-a-x-4-x-1-x-1-is-every-where-continuous-then-what-are-the-values-of-a-and-b-95620

question:
<div class="question">If the derivative of the function<br/>$f(x)=\left\{\begin{array}{cc}a x^{2}+b &amp; x   &lt; -1 \\ b x^{2}+a x+4 &amp; x \geq-1\end{array}\right.$<br/>is every where continuous, then what are the values of a and $\mathrm{b}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad a=2, b=3$</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">$a=3, b=2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a=-2, b=-3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a=-3, b=-2$</span> </li> </ul>

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

answer:
<div class="solution">Derivative of $f(x)=\left\{\begin{array}{ll}a x^{2}+b &amp; x   &lt; -1 \\ b x^{2}+a x+a &amp; x \geq-1\end{array}\right.$ is<br/>$f^{\prime}(x)=\left\{\begin{array}{ll}2 a x &amp; x &lt; -1 \\ 2 b x+a, &amp; x \geq-1\end{array}\right.$<br/>If $\mathrm{f}^{\prime}(\mathrm{x})$ is continuous everywhere then it is also continuous at $\mathrm{x}=-1$<br/>$\left.\mathrm{f}^{\prime}(\mathrm{x})\right|_{\mathrm{x}=-1}=-2 \mathrm{a}=-2 \mathrm{~b}+\mathrm{a}$<br/>or, $3 \mathrm{a}=2 \mathrm{~b}$<br/>From the given choice $\mathrm{a}=2, \mathrm{~b}=3$ satisfied this equation.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-difference-between-the-roots-of-a-x-2-b-x-c-0-is-1-then-which-one-of-the-following-is-correct

slug:
if-the-difference-between-the-roots-of-a-x-2-b-x-c-0-is-1-then-which-one-of-the-following-is-correct-28298

question:
<div class="question">If the difference between the roots of $a x^{2}+b x+c=0$ is 1 , then which one of the following is correct?</div>

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

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

answer:
<div class="solution">Let $\alpha$ and $\beta$ be the roots of $a x^{2}+b x+c=0$ $\alpha+\beta=-\mathrm{b} / \mathrm{a}$<br/>$\alpha \beta=\mathrm{c} / \mathrm{a}$<br/>Given $\alpha-\beta=1$<br/>Consider $(\alpha+\beta)^{2}-(\alpha-\beta)^{2}=4 \alpha \beta$<br/>$\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}-1=\frac{4 \mathrm{c}}{\mathrm{a}}$<br/>$b^{2}-a^{2}=4 a c$<br/>$b^{2}=a(a+4 c)$</div>


tags: ['Physics', 'Oscillations', 'VITEEE', 'VITEEE 2019']
source_db: MarksBatch2_P1.db
title:
if-the-differential-equation-for-a-simple-harmonic-motion-is-d-t-2-d-2-y-2-y-0-the-timeperiod-of-the-motion-is

slug:
if-the-differential-equation-for-a-simple-harmonic-motion-is-d-t-2-d-2-y-2-y-0-the-timeperiod-of-the-motion-is-29611

question:
<div class="question">If the differential equation for a simple harmonic motion is $\frac{d^{2} y}{d t^{2}}+2 y=0$, the time-period of the motion is</div>

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

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

answer:
<div class="solution">The differential equation of simple harmonic motion is
<br/>$$
<br/>\frac{d^{2} y}{d t^{2}}+2 y=0 \text { or } \frac{d^{2} y}{d t^{2}}=-2 y......(i)
<br/>$$
<br/>Standard equation of simple harmonic motion is
<br/>$$
<br/>\frac{d^{2} y}{d t^{2}}=-\omega^{2} y......(ii)
<br/>$$
<br/>Comparing eq. (i) and (ii), $\omega^{2}=2$ or $\omega=\sqrt{2}$
<br/>As we know, $\omega=\frac{2 \pi}{T}$
<br/>$\therefore$ Time period, $T=\frac{2 \pi}{\omega}=\frac{2 \pi}{\sqrt{2}}=\pi \sqrt{2} \mathrm{sec}$</div>


tags: ['Mathematics', 'Permutation Combination', 'JEE Main', 'JEE Main 2021 (22 Jul Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">96</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-digits-are-not-allowed-to-repeat-in-any-number-formed-by-using-the-digits-0-2-4-6-8-then-the-number-of-all-numbers-greater-than-10-000-is-equal

slug:
if-the-digits-are-not-allowed-to-repeat-in-any-number-formed-by-using-the-digits-0-2-4-6-8-then-the-number-of-all-numbers-greater-than-10-000-is-equal-45841

question:
<div class="question">If the digits are not allowed to repeat in any number formed by using the digits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo></math> then the number of all numbers greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>,</mo><mn>000</mn></math> is equal to __________.</div>

answer:
<div class="solution"><p>We have to form a number greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>,</mo><mn>000</mn></math> by using the digits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo></math> hence, the first place of the number can't be zero.</p><p>Thus, for the first place we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> choices i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>8</mn></math> and the remaining places can be filled with any of the digits, thus we have respectively <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>1</mn></math> choices.</p><p>Hence, using the fundamental principle of multiplication, the total numbers which are greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>,</mo><mn>000</mn></math> are</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/d71239ae-9ae1-4f58-9f50-1318ef46fcaf-image.png" style="width: 222px; height: 73px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>96</mn><mo>.</mo></math></p></div>


tags: ['Biology', 'Plant Kingdom']
source_db: MarksBatch2_P1.db
title:
if-the-diploid-number-of-a-flowering-plant-is-36-what-would-be-the-chromosome-number-in-its-endosperm

slug:
if-the-diploid-number-of-a-flowering-plant-is-36-what-would-be-the-chromosome-number-in-its-endosperm-90680

question:
<div class="question">If the diploid number of a flowering plant is 36 . What would be the chromosome number in its endosperm?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>36<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>18<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>54<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">D</span> <span class="option-data"><br/>72</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>54<br/></span> </div>

answer:
<div class="solution">Endosperm is a product of triple fusion. One male nuclei $(n=18)$ fuses with diploid secondary nucleus $(2 n=$ $36)$, so it becomes triploid $(3 n=54)$. Thus, ploidy of endosperm is $(3 n)$ and chromosomes will be 54.</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-direction-ratios-of-the-normal-to-a-plane-are-1-m-n-and-the-length-of-the-normal-is-p-then-what-is-the-sum-of-intercepts-cutoff-by-the-plane-fr

slug:
if-the-direction-ratios-of-the-normal-to-a-plane-are-1-m-n-and-the-length-of-the-normal-is-p-then-what-is-the-sum-of-intercepts-cutoff-by-the-plane-fr-35178

question:
<div class="question">If the direction ratios of the normal to a plane are $   &lt; 1, \mathrm{~m}, \mathrm{n}&gt;$ and the length of the normal is $\mathrm{p}$, then what is the sum of intercepts cut-off by the plane from the coordinate axes?</div>

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

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

answer:
<div class="solution">Since, the direction ratio's of normal to a plane are $   &lt; l, \mathrm{~m}, \mathrm{n}&gt;$ and the length of normal is $\mathrm{p}$, then intercept on $\mathrm{x}$ -axis is $\mathrm{p} / /$ and that on $\mathrm{y}$ -axis is $\mathrm{p} / \mathrm{m}$ and on $\mathrm{z}$ -axis it is $\mathrm{p} / \mathrm{n}$, hence sum of intercepts cut off by the plane from the coordinate axes<br/>$=p\left(\frac{1}{\ell}+\frac{1}{m}+\frac{1}{n}\right)$</div>


title: if-the-distance-between-the-points-7-1-3-and-4-5-is-13-units-then-what-is-one-of-the-values-of
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-distance-between-the-points-7-1-3-and-4-5-is-13-units-then-what-is-one-of-the-values-of-57775

question:
<div class="question">If the distance between the points $(7,1,-3)$ and $(4,5, \lambda)$ is 13 units, then what is one of the values of $\lambda$ ? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">We have,<br/>$13=\sqrt{(4-7)^{2}+(5-1)^{2}+(\lambda+3)^{2}}$<br/>$169=9+16+\lambda^{2}+9+6 \lambda$<br/>$\Rightarrow \quad \lambda^{2}+6 \lambda-135=0$<br/>$\Rightarrow \quad \lambda^{2}+15 \lambda-9 \lambda-135=0$<br/>$\Rightarrow \quad \lambda(\lambda+15)-9(\lambda+15)=0$<br/>$\Rightarrow(\lambda+15)(\lambda-9)=0$<br/>$\Rightarrow \quad \lambda=-15$ or $\lambda=9$</div>


tags: ['Biology', 'Molecular Basis of Inheritance', 'NEET']
source_db: MarksBatch2_P1.db
title:
if-the-dna-strand-undergoing-transcription-has-a-following-base-pair-sequence-what-would-be-the-sequence-of-mrna-formed-from-this-strand-3-gat-ctt-agc

slug:
if-the-dna-strand-undergoing-transcription-has-a-following-base-pair-sequence-what-would-be-the-sequence-of-mrna-formed-from-this-strand-3-gat-ctt-agc-83685

question:
<div class="question">If the DNA strand undergoing transcription has a following base pair sequence. What would be the sequence of mRNA formed from this strand?<br/> <br/>
3' GAT CTT AGC GGG AAT TCG 5'</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5' GAU CUU AGC GGG AAU UCG 3'</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5' CAU GAA UGC CCC UUA AGC 3'</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">5' CUA GAA UCG CCC UUA AGC 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">3' CUA GAA UGG CCC UUA AGC 5'</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">5' CUA GAA UCG CCC UUA AGC 3'</span> </div>

answer:
<div class="solution">Protein synthesis is the process of manufacturing protein from DNA involving two steps - transcription and translation. Transcription is the process of making mRNA from the DNA and this mRNA is translated into a long chain polypeptide with help of ribosome and tRNA.<br/><br/>The strand of NDA from which the new mRNA is synthesized is called as template strand or non-coding or anti-sense strand. It is in the 3' <math><mo>â</mo></math>  5' direction. The mRNA is synthesized on it in the 5' <math><mo>â</mo></math>  3' with the help of RNA polymerase enzyme which leads in the same direction.<br/><br/>The new mRNA sequence transcribed from the DNA sequence - 3' GAT CTT AGC GGG AAT TCG 5' will be 5' CUA GAA UCG CCC UUA AGC 3'</div>


tags: ['Mathematics', 'Hyperbola', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-eccentricity-and-length-of-latus-rectum-of-a-hyperbola-are-3-13-and-3-10-units-respectively-then-what-is-the-length-of-the-transverse-axis

slug:
if-the-eccentricity-and-length-of-latus-rectum-of-a-hyperbola-are-3-13-and-3-10-units-respectively-then-what-is-the-length-of-the-transverse-axis-77816

question:
<div class="question">If the eccentricity and length of latus rectum of a hyperbola<br/>are $\frac{\sqrt{13}}{3}$ and $\frac{10}{3}$ units respectively, then what is the length<br/>of the transverse axis?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{2}$ unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12 unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{15}{2}$ unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{15}{4}$ unit</span> </li> </ul>

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

answer:
<div class="solution">Length of latus rectum of a hyperbola is $\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}$ where<br/>a is the half of the distance bet ween two vertex of the hyperbola.<br/>Latus rectum $=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=\frac{10}{3}$<br/>or, $\mathrm{b}^{2}=\frac{5 \mathrm{a}}{3}$...(1)<br/>In case of hyperbola, $b^{2}=a^{2}\left(e^{2}-1\right)$...(2)<br/>Putting value of $b^{2}$ from equation (1) and e $=\frac{\sqrt{13}}{3}$ in<br/>equation (2),<br/>$\frac{5 a}{3}=a^{2}\left(\frac{13}{9}-1\right)$<br/>or, $\frac{5 a}{3}=\frac{4 a^{2}}{9}$<br/>$\Rightarrow 4 a^{2}-15 a=0$ or $a(4-15 a)=0$<br/>$a \neq 0$, hence, $a=\frac{15}{4}$<br/>Length of transverse axis $=2 a=2 \times \frac{15}{4}=\frac{15}{2}$</div>


tags: ['Mathematics', 'Hyperbola', 'JEE Main', 'JEE Main 2019 (08 Apr Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-the-eccentricity-of-the-standard-hyperbola-passing-through-the-point-46-is-2-then-the-equation-of-the-tangent-to-the-hyperbola-at-46-is

slug:
if-the-eccentricity-of-the-standard-hyperbola-passing-through-the-point-46-is-2-then-the-equation-of-the-tangent-to-the-hyperbola-at-46-is-55416

question:
<div class="question">If the eccentricity of the standard hyperbola passing through the point <math><mo>(</mo><mn>4,6</mn><mo>)</mo></math> is <math><mn>2</mn><mo>,</mo></math> then the equation of the tangent to the hyperbola at <math><mo>(</mo><mn>4,6</mn><mo>)</mo></math> is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><p>Let us suppose equation of hyperbola is<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mn>2</mn></math> </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mfenced><mrow><msup><mi>e</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mi>a</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup></math><br/>It is passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mn>4,6</mn></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>16</mn><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><mn>36</mn><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>,</mo><mtext> </mtext><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn></math><br/>So, equation of tangent is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><msub><mi>x</mi><mn>1</mn></msub></mrow><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><mrow><mi>y</mi><msub><mi>y</mi><mn>1</mn></msub></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p>Tangent at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></math> is<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>-</mo><mfrac><mi>y</mi><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></math><br/><math><mo>⇒</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p></div>


tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-ellipse-9-x-2-16-y-2-144-intercepts-the-line-3-x-4-y-12-then-what-is-the-length-of-the-chord-so-formed

slug:
if-the-ellipse-9-x-2-16-y-2-144-intercepts-the-line-3-x-4-y-12-then-what-is-the-length-of-the-chord-so-formed-96519

question:
<div class="question">If the ellipse $9 \mathrm{x}^{2}+16 \mathrm{y}^{2}=144$ intercepts the line $3 \mathrm{x}+4 \mathrm{y}=12$. then what is the length of the chord so formed?</div>

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

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

answer:
<div class="solution">Here,<br/>$9 x^{2}+16 y^{2}=144$ and $3 x+4 y=12$<br/>$\Rightarrow x=\frac{12-4 y}{3}$<br/>$9\left(\frac{12-4 y}{3}\right)^{2}+16 y^{2}=144$<br/>On solving we get, $y=0,3$ For $y=0 ; x=4$<br/>For $y=3 ; x=0$<br/>$\Rightarrow$ Length of chord $=\sqrt{(0-3)^{2}+(4-0)^{2}}=\sqrt{9+16}$<br/>$=\sqrt{25}=5$ units</div>


tags: ['Mathematics', 'Pair of Lines', 'MHT CET', 'MHT CET 2021 (20 Sep Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-10</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-equation-3-x-2-k-x-y-3-y-2-0-represents-the-bisectors-of-angles-between-the-lines-x-2-3-x-y-4-y-2-0-then-value-of-k-is

slug:
if-the-equation-3-x-2-k-x-y-3-y-2-0-represents-the-bisectors-of-angles-between-the-lines-x-2-3-x-y-4-y-2-0-then-value-of-k-is-23036

question:
<div class="question">If the equation $3 x^2-k x y-3 y^2=0$ represents the bisectors of angles between the lines $x^2-3 x y-4 y^2=0$, then value of $k$ is</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">-10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">We have $x^2-3 x y-4 y^2=0$ and comparing it with standard equation, we write<br/>$$<br/>\mathrm{A}=1, \mathrm{H}=\frac{-3}{2}, \mathrm{~B}=-4<br/>$$<br/>Equation of bisector of angle of this line is<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{x^2-y^2}{A-B}=\frac{x y}{H} \Rightarrow \frac{x^2-y^2}{1+4}=\frac{x y}{\left(\frac{-3}{2}\right)} \\<br/>&amp; \therefore-3 x^2+3 y^2=10 x y \Rightarrow 3 x^2+10 x y-3 y^2=0<br/>\end{aligned}<br/>$$<br/>Comparing with given equation, we get $\mathrm{k}=-10$</div>


tags: ['Mathematics', 'Circle', 'AP EAMCET', 'AP EAMCET 2022 (06 Jul Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-the-equation-a-x-2-b-y-2-2-h-x-y-2-gx-2-f-y-c-0-represents-a-circle-passing-through-the-origin-then

slug:
if-the-equation-a-x-2-b-y-2-2-h-x-y-2-gx-2-f-y-c-0-represents-a-circle-passing-through-the-origin-then-61544

question:
<div class="question">If the equation $a x^2+b y^2+2 h x y+2 g x+2 f y+c=0$ represents a circle passing through the origin, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a=b, c=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|a|=|b|, h=0=c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$a=b, h=c=0$</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">$a=b, h=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a=b, h=c=0$</span> </div>

answer:
<div class="solution">The standard equation of circle which passes through origin is $x^2+y^2+2 g x+2 f y=0$<br/>If the equation $a x^2+b y^2+2 h x y+2 f y+c=0$ represent a circle passing through origin, then $a=b, h=0, c=0$.</div>


title: if-the-equation-x-2-b-x-1-0-does-not-possess-real-roots-then-which-one-of-the-following-is-correct
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-the-equation-x-2-b-x-1-0-does-not-possess-real-roots-then-which-one-of-the-following-is-correct-78290

question:
<div class="question">If the equation $x^{2}-b x+1=0$ does not possess real roots, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">Given quadratic equation is $x^{2}-b x+1=0$ It has no real roots. It means, equation has imaginary roots. Which is possible when $B^{2}-4 A C   &lt; 0$ Here, $B=-b, A=1, C=1$<br/>$\Rightarrow b^{2}-4 &lt; 0 \Rightarrow b^{2} &lt; 4 \Rightarrow-2 &lt; b &lt; 2$</div>


title: if-the-equation-x-2-k-2-2-k-1-x-has-equal-roots-then-what-is-the-value-of-k
slug: if-the-equation-x-2-k-2-2-k-1-x-has-equal-roots-then-what-is-the-value-of-k-35767
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the equation $x^{2}+k^{2}=2(k+1) x$ has equal roots, then what is the value of $\mathrm{k} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2}$</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">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

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

answer:
<div class="solution">$x^{2}+k^{2}=2(k+1) x$<br/>$\Rightarrow x^{2}-2(k+1) x+k^{2}=0$<br/>For roots to beequal discriminant $=0$ So, $\{-2(k+1)\}^{2}-4 k^{2}=0$<br/>or $4(k+1)^{2}-4 k^{2}=0$<br/>or $(\mathrm{k}+1)^{2}-\mathrm{k}^{2}=0$<br/>or, $2 \mathrm{k}+1=0$<br/>$k=-\frac{1}{2}$</div>


title: if-the-equation-x-2-k-x-1-0-has-the-roots-and-then-what-is-the-value-of-1-1
slug: if-the-equation-x-2-k-x-1-0-has-the-roots-and-then-what-is-the-value-of-1-1-38925
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the equation $x^{2}+k x+1=0$ has the roots $\alpha$ and $\beta$, then what is the value of $(\alpha+\beta) \times\left(\alpha^{-1}+\beta^{-1}\right) ?$</div>

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

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

answer:
<div class="solution">As given: Roots of the equations $x^{2}+k x+1=0$ are $\alpha$ and $\beta .$ Given expression $(\alpha+\beta)\left(\alpha^{-1}+\beta^{-1}\right)=(\alpha+\beta)\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)$<br/>$=(\alpha+\beta)\left(\frac{\alpha+\beta}{\alpha \beta}\right)=\frac{(\alpha+\beta)^{2}}{\alpha \beta}=\frac{(-k)^{2}}{1}=\mathrm{k}^{2}$</div>


title: if-the-equations-x-2-k-x-64-0-and-x-2-8-x-k-0-have-real-roots-then-what-is-the-value-of-k
slug: if-the-equations-x-2-k-x-64-0-and-x-2-8-x-k-0-have-real-roots-then-what-is-the-value-of-k-81793
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the equations $x^{2}+k x+64=0$ and $x^{2}-8 x+k=0$ have real roots, then what is the value of $k ?$</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">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">16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Givenequations are $x^{2}+k x+64=0$ and $x^{2}-8 x+k=0$...(i)<br/>Since both the eqns have real roots, discriminant $\geq 0$...(ii)<br/>$=b^{2} \geq 4 a c$<br/>from eq $^{\mathrm{n}}(\mathrm{i})$, we have<br/>$k^{2} \geq 4(64) \Rightarrow k^{2} \geq 256$<br/>$\Rightarrow k \geq 16$...(A)<br/>and from $\mathrm{eq}^{\mathrm{n}}$ (ii), we have<br/>$\Rightarrow \quad 64 \geq 4 k \Rightarrow 4 k \leq 64$<br/>$\Rightarrow k \leq 16$$\ldots(\mathrm{B})$<br/>Hence, from eq $^{\mathrm{n}}(\mathrm{A})$ and $(\mathrm{B})$, we have $\mathrm{k}=16$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-equations-x-2-p-x-q-0-and-x-2-a-x-b-0-have-a-common-root-and-the-roots-of-the-second-equation-are-equal-then-which-one-of-the-following-is-corr

slug:
if-the-equations-x-2-p-x-q-0-and-x-2-a-x-b-0-have-a-common-root-and-the-roots-of-the-second-equation-are-equal-then-which-one-of-the-following-is-corr-96870

question:
<div class="question">If the equations $x^{2}-p x+q=0$ and $x^{2}-a x+b=0$ have a common root and the roots of the second equation are equal, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a p=2(b+q)$</span> </div>

answer:
<div class="solution">Given equations are $x^{2}-p x+q=0$...(1)<br/>and $x^{2}-a x+b=0$...(2)<br/>Root of second equation is<br/>$x=\frac{a \pm \sqrt{a^{2}-4 b}}{2}$<br/>Since, Roots of second equation are equal discriminant<br/>$=0$<br/>$\sqrt{a^{2}-4 b}=0 \Rightarrow a^{2}=4 b$...(3)<br/>$x=\frac{a}{2}$<br/>Since, Equations (1) and (2) have common roots.<br/>$x=\frac{a}{2}$ is the root of equation (1) also.<br/>Thus, $x=\frac{a}{2}$ satisfies (1)<br/>$\Rightarrow\left(\frac{a}{2}\right)^{2}-p\left(\frac{a}{2}\right)+q=0$<br/>$\Rightarrow \frac{a^{2}}{4}=\frac{p a}{2}-q \Rightarrow 2 b=p a-2 q \quad($ from $(3)$<br/>$\Rightarrow a p=2(b+q)$</div>


title: if-the-equilibrium-constant-k-is-measured-directly-in-the-laboratory-what-can-be-calculated
slug: if-the-equilibrium-constant-k-is-measured-directly-in-the-laboratory-what-can-be-calculated-49989
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the equilibrium constant (K) is measured directly in the laboratory, what can be calculated?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The value of âGÂ° at any  other temperature</span> </div>

answer:
<div class="solution">If the equilibrium  constant (K) is measured directly in the laboratory, the value of âGÂ° at  any other temperature can be calculated using the relationship between â<sub>r</sub>GÂ° and K.</div>


tags: ['Chemistry', 'Electrochemistry', 'AP EAMCET', 'AP EAMCET 2019 (22 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-etext-cell-circ-of-an-equilibrium-reaction-as2-b2a-q-rightleftharpoons-a2a-q2-bs-at-298-mathrmk-is-059-mathrmv-the-equilibrium-constant-kc-is

slug:
if-the-etext-cell-circ-of-an-equilibrium-reaction-as2-b2a-q-rightleftharpoons-a2a-q2-bs-at-298-mathrmk-is-059-mathrmv-the-equilibrium-constant-kc-is-39404

question:
<div class="question">If the \(E_{\text {cell }}^{\circ}\) of an equilibrium reaction \(A(s)+2 B^{2+}(a q) \rightleftharpoons A^{2+}(a q)+2 B(s)\)<br/>at \(298 \mathrm{~K}\) is \(0.59 \mathrm{~V}\), the equilibrium constant \(K_c\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(1.0 \times 10^{10}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(1.0 \times 10^2\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1.0 \times 10^{-20}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(1.0 \times 10^{20}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">\(1.0 \times 10^{20}\)</span> </div>

answer:
<div class="solution">For the reaction,<br/>\(\begin{gathered}<br/>A+2 B^{2+}(a q) \rightleftharpoons A^{2+}(a q)+2 B(s) \\<br/>\Delta G^{\circ}=-R T \ln K_c=-n F E_{\text {cell }}^{\circ} \quad \ldots (i)<br/>\end{gathered}\)<br/>where,<br/>\(\begin{aligned}<br/>\Delta G^{\circ} &amp; =\text { Gibbs free energy } \\<br/>n &amp; =\text { number of electrons involved }(=2) \\<br/>F &amp; =\text { Faraday's constant }=96500 \mathrm{C} \text { charge. } \\<br/>E^{\circ} &amp; =\text { standard electrode cell potential }(=0.59 \mathrm{~V}) \\<br/>R &amp; =\text { gas constant } \\<br/>T &amp; =\text { temperature }(=298 \mathrm{~K})<br/>\end{aligned}\)<br/>\(\therefore\) From Eq. (i), we have<br/>\(\begin{aligned}<br/>\text {or } \log K_c &amp; =\frac{n F E^{\circ}}{2.303 R T} \\<br/>&amp; =\frac{2 \times 96500 \times 0.59}{2303 \times 8.314 \times 298}=-19.96 \approx 20 \\<br/>\therefore \quad K_c &amp; =1.0 \times 10^{20}<br/>\end{aligned}\)<br/>Hence, option (c) is the correct answer.<br/>Alternatively<br/>At, \(T=298 \mathrm{~K}\)<br/>\(\ln \quad K_c=\frac{n E_{\text {ccll }}^{\circ}}{0.059}=\frac{2 \times 0.59}{0.059}=20\)<br/>\(\therefore \quad K_c=1 \times 10^{20}\)</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-extremities-of-a-diameter-of-a-circle-are-0-0-and-a-3-1-a-3-then-the-circle-passes-through-which-one-of-the-following-points

slug:
if-the-extremities-of-a-diameter-of-a-circle-are-0-0-and-a-3-1-a-3-then-the-circle-passes-through-which-one-of-the-following-points-15240

question:
<div class="question">If the extremities of a diameter of a circle are $(0,0)$ and<br/>$\left(\mathrm{a}^{3}, 1 / \mathrm{a}^{3}\right)$, then the circle passes through which one of the<br/>following points?</div>

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

answer:
<div class="solution">Given the extremities of a diameter of a circle as<br/>$(0,0)$ and $\left(\mathrm{a}^{3}, \frac{1}{\mathrm{a}}\right)$, equation of circle is<br/>$(x-0)\left(x-a^{3}\right)+(y-0)\left(y-\frac{1}{a^{3}}\right)=0$<br/>$\Rightarrow \quad x^{2}-x a^{3}+y^{2}-\frac{y}{a^{3}}=0$<br/>$\Rightarrow \quad x^{2}+y^{2}-x a^{3}-\frac{y}{a^{3}}=0$<br/>Putting $\mathrm{x}=\frac{1}{\mathrm{a}}$ and $\mathrm{y}=\mathrm{a}$, the equation is satisfied.<br/>Thus, the circle passes through the point $\left(\frac{1}{\mathrm{a}}, \mathrm{a}\right)$.</div>


tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-foci-of-the-conics-a-2-x-2-7-y-2-1-and-144-x-2-81-y-2-25-1-were-to-coincide-then-what-is-the-value-of-a

slug:
if-the-foci-of-the-conics-a-2-x-2-7-y-2-1-and-144-x-2-81-y-2-25-1-were-to-coincide-then-what-is-the-value-of-a-49506

question:
<div class="question">If the foci of the conics $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{7}=1$ and $\frac{x^{2}}{144}-\frac{y^{2}}{81}=\frac{1}{25}$<br/>were to coincide, then what is the value of a ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">The equation of ellipse is given as<br/>$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{7}=1$<br/>Eccentricity is given by<br/>$e=\sqrt{1-\frac{7}{a^{2}}}$<br/>Therefore, foci of ellipse are $(\pm \mathrm{ae}, 0) \mathrm{ie}$,<br/>$\left(\pm a \sqrt{1-\frac{7}{a^{2}}}, 0\right)$<br/>Now, the equation of given hyper bola is<br/>$\frac{x^{2}}{144}-\frac{y^{2}}{81}=\frac{1}{25} \Rightarrow \frac{x^{2}}{\frac{144}{25}}-\frac{y^{2}}{\frac{81}{25}}=1$<br/>So, a $=\frac{12}{5}$ and $b=\frac{9}{5}$<br/>$\therefore \quad e^{\prime}=\sqrt{1+\frac{81 / 25}{144 / 25}}=\sqrt{\frac{144+81}{144}}=\sqrt{\frac{225}{144}}$<br/>$\quad=\frac{15}{12}$<br/>Foci of hyper bola are $\left(\pm \frac{12}{5}, \frac{15}{12}, 0\right)$ ie, $(\pm 3,0)$.<br/>Since these foci coincides<br/>$\Rightarrow 3=a \sqrt{1-\frac{7}{a^{2}}}$<br/>$\Rightarrow \frac{3}{\mathrm{a}}=\sqrt{1-\frac{7}{\mathrm{a}^{2}}}$<br/>$\Rightarrow \frac{9}{\mathrm{a}^{2}}=1-\frac{7}{\mathrm{a}^{2}}$<br/>$\Rightarrow \frac{16}{a^{2}}=1 \Rightarrow a=4$</div>


title: if-the-free-energy-change-of-a-reaction-is-greater-than-zero-then-the-reaction-is
slug: if-the-free-energy-change-of-a-reaction-is-greater-than-zero-then-the-reaction-is-94162
tags: ['Biology', 'Biomolecules (B)', 'JIPMER', 'JIPMER 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the free energy change of a reaction is greater than zero, then the reaction is</div>

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

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

answer:
<div class="solution">An endergonic reaction (also called a non-spontaneous reaction) is a chemical reaction in which the standard change in free energy is positive and energy is absorbed.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-function-f-x-2-x-t-a-n-1-x-2-x-s-i-n-1-x-is-continuous-at-each-point-in-its-domain-then-what-is-the-value-of-f-0

slug:
if-the-function-f-x-2-x-t-a-n-1-x-2-x-s-i-n-1-x-is-continuous-at-each-point-in-its-domain-then-what-is-the-value-of-f-0-77394

question:
<div class="question">If the function $\mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}-\sin ^{-1} \mathrm{x}}{2 \mathrm{x}+\tan ^{-1} \mathrm{x}}$ is continuous at each<br/>point in its domain, then what is the value of $\mathrm{f}(0)$ ?</div>

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

answer:
<div class="solution">$f(0)=\lim _{x \rightarrow 0} \frac{2 x-\sin ^{-1} x}{2 x+\tan ^{-1} x}$<br/>$=\lim _{x \rightarrow 0} \frac{2-\frac{\sin ^{-1} x}{x}}{2+\frac{\tan ^{-1} x}{x}}=\frac{2-1}{2+1}=\frac{1}{3}$</div>


title: if-the-function-f-x-x-2-4-x-x-2-x-2-i-sco-n-t-in-u-o-u-s-a-t-x2-t-h-e-n-w-ha-t-i-s-f2-equal-to
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-the-function-f-x-x-2-4-x-x-2-x-2-i-sco-n-t-in-u-o-u-s-a-t-x2-t-h-e-n-w-ha-t-i-s-f2-equal-to-47332

question:
<div class="question">If the function <br/>$$<br/>f(x)=\frac{x(x-2)}{x^{2}-4}, x \neq \pm 2<br/>$$<br/>is continuous at $x=2$, then what is $f(2)$ equal to?</div>

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

answer:
<div class="solution">Let $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}(\mathrm{x}-2)}{\mathrm{x}^{2}-4}=\frac{\mathrm{x}(\mathrm{x}-2)}{(\mathrm{x}-2)(\mathrm{x}+2)}=\frac{x}{x+2}$<br/>Since $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$<br/>$\therefore \quad \lim _{x \rightarrow 2} f(x)=f(2)$<br/>$\Rightarrow \quad \lim _{x \rightarrow 2} \frac{x}{x+2}=f(2)$<br/>$\Rightarrow \quad \mathrm{f}(2)=\frac{2}{4}=\frac{1}{2}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'AP EAMCET', 'AP EAMCET 2019 (20 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-function-f11-rightarrow-r-defined-by-fxleftbeginarraycc2x1-text-for-x-in10-1-text-for-x0-2x1-text-for-x-in01endarrayright-then-in-11-fx-has

slug:
if-the-function-f11-rightarrow-r-defined-by-fxleftbeginarraycc2x1-text-for-x-in10-1-text-for-x0-2x1-text-for-x-in01endarrayright-then-in-11-fx-has-10576

question:
<div class="question">If the function \(f:[-1,1] \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{cc}2^x+1, &amp; \text { for } x \in[-1,0) \\ 1, &amp; \text { for } x=0 \\ 2^x-1, &amp; \text { for } x \in(0,1]\end{array}\right.\) then, in \([-1,1], f(x)\) has</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">neither maximum nor minimum</span> </div>

answer:
<div class="solution">The given function \(f:[-1,1] \rightarrow R\)<br/>defined by \(f(x)=\left[\begin{array}{cc}2^x+1, &amp; \text { for } x \in[-1,0) \\ 1, &amp; \text { for } x=0 \\ 2^x-1, &amp; \text { for } x \in(0,1]\end{array}\right.\)<br/>The given function \(f(x)\) in \(x \in(-1,0)\) strictly increasing and \(f\left(0^{-}\right) \rightarrow 2\)<br/>but \(f(0)=1\)<br/>and in interval \(x \in(0,1)\), again it is strictly increasing, but \(f\left(0^{+}\right)=0\)<br/>So, function has neither maximum nor minimum. Hence, option (4) is correct.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-graph-of-a-quadratic-polynomial-lies-entirely-above-x-axis-then-which-one-of-the-following-is-correct

slug:
if-the-graph-of-a-quadratic-polynomial-lies-entirely-above-x-axis-then-which-one-of-the-following-is-correct-19408

question:
<div class="question">If the graph of a quadratic polynomial lies entirely above $\mathrm{x}$ -axis, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both the roots are complex</span> </div>

answer:
<div class="solution">Since the graph is not meeting the $\mathrm{x}$ -axis at all, roots are Complex numbers.</div>


title: if-the-inverse-of-1-0-0-p-x-0-q-0-1-is-1-0-0-p-1-0-q-0-1-then-what-is-the-value-of-x
slug: if-the-inverse-of-1-0-0-p-x-0-q-0-1-is-1-0-0-p-1-0-q-0-1-then-what-is-the-value-of-x-78829
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the inverse of $\left[\begin{array}{lll}1 &amp; \mathrm{p} &amp; \mathrm{q} \\ 0 &amp; \mathrm{x} &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$ is $\left[\begin{array}{ccc}1 &amp; -\mathrm{p} &amp; -\mathrm{q} \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$ then what is the value of $\mathrm{x}$?</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">Zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{\mathrm{p}}+\frac{1}{\mathrm{q}}$</span> </li> </ul>

answer:
<div class="solution">Let $A=\left[\begin{array}{lll}1 &amp; p &amp; q \\ 0 &amp; x &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$ and $A^{-1}=\left[\begin{array}{ccc}1 &amp; -p &amp; -q \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>$\mathrm{AA}^{-1}=\mathrm{I}$<br/>$\Rightarrow\left[\begin{array}{lll}1 &amp; \mathrm{p} &amp; \mathrm{q} \\ 0 &amp; \mathrm{x} &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]\left[\begin{array}{ccc}1 &amp; -\mathrm{p} &amp; -\mathrm{q} \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]=\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; \mathrm{x} &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]=\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$<br/>$\Rightarrow \mathrm{x}=1$</div>


title: if-the-latus-rectum-of-an-ellipse-is-equal-to-half-of-the-minor-axis-then-what-is-its-eccentricity
tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-the-latus-rectum-of-an-ellipse-is-equal-to-half-of-the-minor-axis-then-what-is-its-eccentricity-87763

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

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

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

answer:
<div class="solution">Length of minor axis $=2 \mathrm{~b}$ and latus rectum $=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}$<br/>According to given condition $\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=\mathrm{b}$<br/>$\Rightarrow 2 \mathrm{~b}=\mathrm{a}$<br/>Now, $\mathrm{e}=\sqrt{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}=\sqrt{1-\frac{\mathrm{b}^{2}}{4 \mathrm{~b}^{2}}}=\sqrt{\frac{3}{4}}$<br/>$\Rightarrow \mathrm{e}=\frac{\sqrt{3}}{2}$</div>


tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-latus-rectum-of-an-ellipse-is-equal-to-one-half-its-minor-axis-what-is-the-eccentricity-of-the-ellipse

slug:
if-the-latus-rectum-of-an-ellipse-is-equal-to-one-half-its-minor-axis-what-is-the-eccentricity-of-the-ellipse-26469

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

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{3}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{15}}{4}$</span> </li> </ul>

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

answer:
<div class="solution">Length of latus rectum of an ellipse is $\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}$ where bis<br/>semi minor axis and a is semi-major axis. As given, $\frac{2 b^{2}}{a}=b$<br/>$\Rightarrow 2 \mathrm{~b}=\mathrm{a} \Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\frac{1}{2}$<br/>Weknow that eccentricity $\mathrm{e}=\sqrt{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2}$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-least-number-of-zeroes-in-a-lower-triangular-matrix-is-10-then-what-is-the-order-of-the-matrix

slug:
if-the-least-number-of-zeroes-in-a-lower-triangular-matrix-is-10-then-what-is-the-order-of-the-matrix-71366

question:
<div class="question">If the least number of zeroes in a lower triangular matrix is 10, then what is the order of the matrix?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \times 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4 \times 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \times 5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10 \times 10$</span> </li> </ul>

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

answer:
<div class="solution">Number of zeroes in a lower triangular matrix of order $\mathrm{n} \times \mathrm{n}$ is<br/>$1+2+3+\ldots .+\mathrm{n}=\frac{\mathrm{n}(\mathrm{n}+1)}{2}$<br/>Number of zeros $=10$<br/>$\Rightarrow \frac{\mathrm{n}(\mathrm{n}+1)}{2}=10 \quad \Rightarrow \mathrm{n}^{2}+\mathrm{n}-20=0$<br/>$\Rightarrow(n+5)(n-4)=0 \quad \Rightarrow n=4$ or $-5$<br/>$(-5$ is meaningless $)$ $\Rightarrow \mathrm{n}=4 .$ <br/>$\Rightarrow$ order of the matrix is $4 \times 4$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2003']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-1

slug:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-1-18590

question:
<div class="question">If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$, then the ratio of the final and initial fundamental frequencies is :</div>

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

answer:
<div class="solution">Fundamental frequency, $n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>Initial frequency, $n_1=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>New length, $l^{\prime}=l-\frac{l \times 40}{100}=0.6 l$<br/>New tension, $T^{\prime}=T+\frac{T \times 44}{100}$<br/>$=1.44 \mathrm{~T}$<br/>New frequency $n_2=\frac{1}{2 l^{\prime}} \sqrt{\frac{T^{\prime}}{m}}$<br/>$\therefore \quad \frac{n_2}{n_1}=\frac{\frac{1}{2 l^{\prime}} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}$<br/>$\frac{n_2}{n_1}=\frac{l}{l^{\prime}} \sqrt{\left(\frac{T^{\prime}}{T}\right)}$<br/>$=\frac{l}{0.6 l} \sqrt{\frac{1.44 T}{T}}$<br/>$=\frac{1}{0.6} \times 1.2$<br/>$=2$<br/>$\therefore \quad n_2: n_1=2: 1$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-2

slug:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-2-48808

question:
<div class="question">If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$, then the ratio of the final and initial fundamental frequencies is</div>

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

answer:
<div class="solution">Fundamental frequency in case of string is<br/>$n=\frac{1}{2 l} \sqrt{\frac{T}{m}} \Rightarrow n \propto \frac{\sqrt{T}}{1} \Rightarrow \frac{n^{\prime}}{n}=\sqrt{\frac{T^{\prime}}{T}} \times \frac{1}{r}$<br/>putting $T^{\prime}=T+0.44 T=\frac{144}{100} T$ and $\left.l^{\prime}=|-0.4|=\frac{3}{5} \right\rvert\,$ <br/>We get $\frac{n^{\prime}}{n}=\frac{2}{1}$.</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-3

slug:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-3-19120

question:
<div class="question">If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$, then the ratio of the final and initial fundamental frequencies is :</div>

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

answer:
<div class="solution">Fundamental frequency, $n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>Initial frequency, $n_1=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>New length, $l^{\prime}=l-\frac{l \times 40}{100}=0.6 l$<br/>New tension, $T^{\prime}=T+\frac{T \times 44}{100}$<br/>$=1.44 \mathrm{~T}$<br/>New frequency $n_2=\frac{1}{2 l^{\prime}} \sqrt{\frac{T^{\prime}}{m}}$<br/>$\therefore \quad \frac{n_2}{n_1}=\frac{\frac{1}{2 l^{\prime}} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}$<br/>$\frac{n_2}{n_1}=\frac{l}{l^{\prime}} \sqrt{\left(\frac{T^{\prime}}{T}\right)}$<br/>$=\frac{l}{0.6 l} \sqrt{\frac{1.44 T}{T}}$<br/>$=\frac{1}{0.6} \times 1.2$<br/>$=2$<br/>$\therefore \quad n_2: n_1=2: 1$</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2003']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc

slug:
if-the-length-of-a-stretched-string-is-shortened-by-40-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenc-75403

question:
<div class="question">If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$, then the ratio of the final and initial fundamental frequencies is :</div>

answer:
<div class="solution">Fundamental frequency, $n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>Initial frequency, $n_1=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>New length, $l^{\prime}=l-\frac{l \times 40}{100}=0.6 l$<br/>New tension, $T^{\prime}=T+\frac{T \times 44}{100}$<br/>$=1.44 \mathrm{~T}$<br/>New frequency $n_2=\frac{1}{2 l^{\prime}} \sqrt{\frac{T^{\prime}}{m}}$<br/>$\therefore \quad \frac{n_2}{n_1}=\frac{\frac{1}{2 l^{\prime}} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}$<br/>$\frac{n_2}{n_1}=\frac{l}{l^{\prime}} \sqrt{\left(\frac{T^{\prime}}{T}\right)}$<br/>$=\frac{l}{0.6 l} \sqrt{\frac{1.44 T}{T}}$<br/>$=\frac{1}{0.6} \times 1.2$<br/>$=2$<br/>$\therefore \quad n_2: n_1=2: 1$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2018 (23 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">40</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-length-of-a-stretched-string-is-shortened-by-x-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenci

slug:
if-the-length-of-a-stretched-string-is-shortened-by-x-and-the-tension-is-increased-by-44-then-the-ratio-of-the-final-and-initial-fundamental-frequenci-32060

question:
<div class="question">If the length of a stretched string is shortened by $x \%$ and the tension is increased by $44 \%$. Then the ratio of the final and initial fundamental frequencies is $1: 2$, then the value of $x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">30</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">40</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">60</span> </li> </ul>

answer:
<div class="solution">Fundamental frequency for string, $n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>So, $n \propto \sqrt{T} / l$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/t53noBhxrZgOIQF4cHsysrAfjsSmuSdIpC3zzhelz6c.original.fullsize.png"/><br/><br/>Now, $\quad T^{\prime}=T+0.44 T=\frac{144 T}{100}$<br/>and $\quad l^{\prime}=l-\frac{x}{100} l=\left(\frac{100-x}{100}\right) l$<br/>Also, $\quad \frac{n^{\prime}}{n}=\frac{2}{1}$<br/>$\frac{2}{1}=\sqrt{\frac{144 T}{100 T}} \times \frac{l}{\left(\frac{100-x}{100}\right) l}=\frac{12}{10} \times \frac{100}{100-x}$<br/>or $(100-x)=60$ or $x=40$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2018 (25 Apr Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-length-of-a-stretched-string-is-shortened-by-x-and-the-tension-is-increased-by-44-then-the-ratio-of-the-initial-and-final-fundamental-frequenci

slug:
if-the-length-of-a-stretched-string-is-shortened-by-x-and-the-tension-is-increased-by-44-then-the-ratio-of-the-initial-and-final-fundamental-frequenci-90508

question:
<div class="question">If the length of a stretched string is shortened by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>%</mo></math> and the tension is increased by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>44</mn><mo>%</mo></math> then the ratio of the initial and final fundamental frequencies is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>2</mn></math>, the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi>x</mi><mo>'</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math></span> </div>

answer:
<div class="solution"><p>Recall the formula of fundamental frequency in terms of tension and mass per unit length,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><msqrt><mfrac><mi>T</mi><mi>μ</mi></mfrac></msqrt></math></p><p>it means,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>f</mi><mn>1</mn></msub><msub><mi>f</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>l</mi><mn>1</mn></msub></mrow></mfrac><msqrt><mfrac><msub><mi>T</mi><mn>1</mn></msub><mi>μ</mi></mfrac></msqrt></mrow><mrow><mi>f</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>l</mi><mn>2</mn></msub></mrow></mfrac><msqrt><mfrac><msub><mi>T</mi><mn>2</mn></msub><mi>μ</mi></mfrac></msqrt></mrow></mfrac><mo>=</mo><mfrac><msub><mi>l</mi><mn>2</mn></msub><msub><mi>l</mi><mn>1</mn></msub></mfrac><mo>×</mo><msqrt><mfrac><msub><mi>T</mi><mn>1</mn></msub><msub><mi>T</mi><mn>2</mn></msub></mfrac></msqrt></math></p><p>Put given data,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>l</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>100</mn></mfrac></mstyle></mrow></mfenced></mrow><mi>l</mi></mfrac><mo>×</mo><msqrt><mfrac><mi>T</mi><mrow><mi>T</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>44</mn><mn>100</mn></mfrac></mstyle></mrow></mfenced></mrow></mfrac></msqrt><mspace linebreak="newline"></mspace><mo>⇒</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mfrac><mfenced><mrow><mn>100</mn><mo>-</mo><mstyle displaystyle="true"><mi>x</mi></mstyle></mrow></mfenced><mn>10</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><mn>12</mn></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mn>60</mn><mo>=</mo><mn>100</mn><mo>-</mo><mi>x</mi><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>40</mn></math></p></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2023 (10 May Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-length-of-stretched-string-is-reduced-by-40-and-tension-is-increased-by-44-then-the-ratio-of-final-to-initial-frequencies-of-stretched-string-i

slug:
if-the-length-of-stretched-string-is-reduced-by-40-and-tension-is-increased-by-44-then-the-ratio-of-final-to-initial-frequencies-of-stretched-string-i-44946

question:
<div class="question">If the length of stretched string is reduced by $40 \%$ and tension is increased by $44 \%$ then the ratio of final to initial frequencies of stretched string is</div>

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

answer:
<div class="solution">Let the initial length and tension be $l$ and $\mathrm{T}$ respectively.<br/>After shortening,<br/>The new length $l_{\text {new }}=l-\frac{40}{100} l=\frac{3}{5} l$<br/>After increase in tension, the new tension $\mathrm{T}_{\text {new }}=\mathrm{T}+\frac{44}{100} \mathrm{~T}=\frac{144 \mathrm{~T}}{100}$<br/>Fundamental frequency of a vibrating string is given by<br/>$\begin{aligned}<br/>\mathrm{n} &amp; =\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}} \\<br/>\therefore \quad \mathrm{n}_1 &amp; =\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}} \\<br/>\mathrm{n}_2 &amp; =\frac{1}{2 l} \sqrt{\frac{\mathrm{T}_{\mathrm{new}}}{\mathrm{m}}} \\<br/>\therefore \quad \frac{\mathrm{n}_1}{\mathrm{n}_2} &amp; =\frac{l \text { aew }}{l} \times \frac{\sqrt{\mathrm{T}}}{\sqrt{\mathrm{T}_{\mathrm{new}}}} \\<br/>&amp; =\frac{\frac{3}{5} l}{l} \times \sqrt{\frac{100 \mathrm{~T}}{144 \mathrm{~T}}} \\<br/>&amp; =\frac{3}{5} \times \frac{10}{12}=\frac{1}{2} \\<br/>\therefore \quad \frac{\mathrm{n}_2}{\mathrm{n}_1} &amp; =\frac{2}{1}<br/>\end{aligned}$</div>


title: if-the-letters-of-the-word-bazar-are-arranged-in-dictionary-order-then-what-is-the-50th-word
slug: if-the-letters-of-the-word-bazar-are-arranged-in-dictionary-order-then-what-is-the-50th-word-90082
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the letters of the word BAZAR are arranged in dictionary order, then what is the 50th word?</div>

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

answer:
<div class="solution">With Aat first place, rest 4 places will be arranged in $4 !$ ways so, Number of words begin with $\mathrm{A}=4 !=24$ Similarly with $\mathrm{B}$ at first place,<br/>Number of words begin with $\mathrm{B}=\frac{4 !}{2 !}=12$<br/>[As there are two $\left.\mathrm{A}_{\mathrm{s}}\right]$<br/><br/>Number of words begin with $\mathrm{R}=\frac{4 !}{2 !}=12$<br/>Thus, 48 words have starting letter $\mathrm{A}, \mathrm{B}$ and $\mathrm{R}$. So, 49 th word will be ZAABR and 50 th word will be ZAARB.</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-line-through-the-points-a-k-1-1-and-b-2-k-0-2-is-perpendicular-to-the-line-through-the-points-b-and-c-2-2-k-k-1-then-what-is-the-value-of-k

slug:
if-the-line-through-the-points-a-k-1-1-and-b-2-k-0-2-is-perpendicular-to-the-line-through-the-points-b-and-c-2-2-k-k-1-then-what-is-the-value-of-k-48375

question:
<div class="question">If the line through the points $A(k, 1,-1)$ and $B(2 k, 0,2)$ is perpendicular to the line through the points $B$ and $C(2+2 k, k, 1)$, then what is the value of $k ? \quad$</div>

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

answer:
<div class="solution">Givenpoints are $A(k, 1,-1), B(2 k, 0,2)$ and $C(2+2 k, k, 1)$<br/>Let $r_{1}=$ length of line<br/>$A B=\sqrt{(2 k-k)^{2}+(0-1)^{2}+(2+1)^{2}}=\sqrt{k^{2}+10}$<br/>and $r_{2}=$ length of line $B C=\sqrt{(2)^{2}+k^{2}+(-1)^{2}}$<br/>$=\sqrt{k^{2}+5}$<br/>Now, let $\ell_{1}, m_{1}, n_{1}$ be direction-cosines of line $A B$ and<br/>$\ell_{2}, m_{2}, n_{2}$ be the direction cosines of $B C$.<br/>Since $\mathrm{AB}$ is perpendicular to $\mathrm{BC}$<br/>$\ell_{1} \ell_{2}+m_{1} m_{2}+n_{1} n_{2}=0$<br/>Now, $\ell_{1}=\frac{k}{\sqrt{k^{2}+10}}, m_{1}=\frac{-1}{\sqrt{k^{2}+10}}, n_{1}=\frac{3}{\sqrt{k^{2}+10}}$<br/>and $\ell_{2}=\frac{2}{\sqrt{k^{2}+5}}, m_{2}=\frac{k}{\sqrt{k^{2}+5}}, n_{2}=\frac{-1}{\sqrt{k^{2}+5}}$<br/>So, $\ell_{1} \ell_{2}+m_{1} m_{2}+n_{1} n_{2}=0$<br/>$\Rightarrow \frac{2 k}{\sqrt{k^{2}+10} \sqrt{k^{2}+5}}-\frac{k}{\sqrt{k^{2}+10} \sqrt{k^{2}+5}}-\frac{3}{\sqrt{k^{2}+10} \sqrt{k^{2}+5}}=0$<br/>$\Rightarrow 2 k-k-3=0$<br/>$\Rightarrow k=3$<br/>For $k=3, A B$ is perpendicular to $B C$.</div>


title: if-the-line-x10-is-the-directrix-of-the-parabola-y2k-x80-then-one-of-the-values-of-k-is
slug: if-the-line-x10-is-the-directrix-of-the-parabola-y2k-x80-then-one-of-the-values-of-k-is-72147
tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the line \( x-1=0 \) is the directrix of the parabola \( y^{2}-k x+8=0 \), then one of the values of \( k \) is</div>

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

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

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>k</mi><mi>x</mi><mo>-</mo><mn>8</mn><mo> </mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>8</mn><mi>k</mi></mfrac></mrow></mfenced></math></p><p>Shifting the origin</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mn>2</mn></msup><mo>=</mo><mi>k</mi><mi>X</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mo> </mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mn>8</mn><mi>k</mi></mfrac></math></p><p>Directrix of standard parabola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mo>-</mo><mfrac><mi>k</mi><mn>4</mn></mfrac></math></p><p>Directrix of original parabola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>8</mn><mi>k</mi></mfrac><mo>-</mo><mfrac><mi>k</mi><mn>4</mn></mfrac></math></p><p>Now,  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> also coincides with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>8</mn><mi>k</mi></mfrac><mo>-</mo><mfrac><mi>k</mi><mn>4</mn></mfrac></math></p><p>On solving, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>8</mn></math></p></div>


title: if-the-lines-3-y-4-x-1-y-x-5-and-5-y-b-x-3-are-ooncurrent-then-what-is-the-value-of-b
slug: if-the-lines-3-y-4-x-1-y-x-5-and-5-y-b-x-3-are-ooncurrent-then-what-is-the-value-of-b-29980
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the lines $3 y+4 x=1, y=x+5$ and $5 y+b x=3$ are ooncurrent, then what is the value of $b ?$</div>

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

answer:
<div class="solution">The equation of given lines are <br/>$3 y+4 x=1$ ...(i)<br/>$y=x+5$ ...(ii)<br/>and $5 y+b x=3$ ...(iii)<br/>On solving Eqs. (i) and (ii), we get $x=-2$ and $y=3$<br/>If these lines are concurrent, then these values must satisfy the third equation $15-2 b=3 \Rightarrow 2 b=12 \Rightarrow b=6$<br/>$ALTERNATESOLUTION$:<br/>Given equation of lines are $3 y+4 x-1=0$<br/>$x+5-y=0$ and<br/>$b x+5 y-3=0$<br/>Since, the given lines are concurrent $\therefore$ The value of determinant made by coeff of equations is $0 .$<br/>ie, $\left|\begin{array}{ccc}4 &amp; 3 &amp; -1 \\ 1 &amp; -1 &amp; 5 \\ b &amp; 5 &amp; -3\end{array}\right|=0$<br/>$\Rightarrow 4(3-25)-3(-3-5 b)-1(5+b)=0$<br/>$\Rightarrow-88+4+14 b=0$<br/>$\Rightarrow-84=-14 b$<br/>$\Rightarrow b=6 .$</div>


title: if-the-lines-3-y-4-x-1-y-x-5-and-5-y-bx-3-are-concurrent-then-what-is-the-value-of-b
slug: if-the-lines-3-y-4-x-1-y-x-5-and-5-y-bx-3-are-concurrent-then-what-is-the-value-of-b-81707
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the lines $3 \mathrm{y}+4 \mathrm{x}=1, \mathrm{y}=\mathrm{x}+5$ and $5 \mathrm{y}+\mathrm{bx}=3$ are concurrent, then what is the value of b?</div>

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

answer:
<div class="solution">Given lines, $3 \mathrm{y}+4 \mathrm{x}=1 \Rightarrow 4 \mathrm{x}+3 \mathrm{y}-1=0$<br/>$y=x+5 \Rightarrow x-y+5=0$<br/>$5 y+b x=3 \Rightarrow b x+5 y-3=0$<br/>Since, these lines are concurrent,<br/>$\left|\begin{array}{ccc}4 &amp; 3 &amp; -1 \\ 1 &amp; -1 &amp; 5 \\ b &amp; 5 &amp; -3\end{array}\right|=0$<br/>$\Rightarrow 4(3-25)-3(-3-5 b)-1(5+b)=0$<br/>$\Rightarrow 4(-22)+9+15 b-5-b=0$<br/>$\Rightarrow-88+4+14 b=0$<br/>$\Rightarrow-84+14 b=0$<br/>$\Rightarrow \mathrm{b}=6$</div>


title: if-the-magnitude-of-a-b-equals-to-a-b-then-which-one-of-the-following-is-correct
slug: if-the-magnitude-of-a-b-equals-to-a-b-then-which-one-of-the-following-is-correct-41723
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the magnitude of $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ equals to $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}$, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is $45^{\circ}$</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">$\vec{a}$ is parallel to $\vec{b}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\vec{a}$ is perpendicular to $\overrightarrow{\mathrm{b}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is $45^{\circ}$</span> </div>

answer:
<div class="solution">Since magnitude of $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=$ magnitude of $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}$<br/>$\therefore|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|$<br/>$\Rightarrow|\vec{a}||\vec{b}| \sin \theta=|\vec{a}||\vec{b}| \cos \theta$ (By Definition)<br/>$\Rightarrow \tan \theta=1$<br/>$\Rightarrow \theta=\frac{\pi}{4}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 unit</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-magnitude-of-difference-of-two-unit-vectors-is-3-then-the-magnitude-of-sum-of-the-two-vectors-is

slug:
if-the-magnitude-of-difference-of-two-unit-vectors-is-3-then-the-magnitude-of-sum-of-the-two-vectors-is-52362

question:
<div class="question">If the magnitude of difference of two unit vectors is $\sqrt{3}$, then the magnitude of sum of the two vectors is</div>

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

answer:
<div class="solution">$|\hat{a}-\hat{b}|=\sqrt{3}$<br/>$\Rightarrow|\hat{a}-\hat{b}|^{2}=(\sqrt{3})^{2}$<br/>$\Rightarrow \hat{a} . \hat{a}+\hat{b} \cdot \hat{b}-2 \hat{a} \cdot \hat{b}=3$<br/>$\Rightarrow 2 \hat{a} \cdot \hat{b}=-1$<br/>Now; $|\hat{a}+\hat{b}|^{2}=\hat{a} \cdot \hat{a}+\hat{b} \cdot \hat{b}+2 \hat{a} \cdot \hat{b}=1+1-1$<br/>$\Rightarrow|\hat{a}+\hat{b}|^{2}=1$<br/>$\Rightarrow|\hat{a}+\hat{b}|=1$</div>


title: if-the-magnitudes-of-two-vectors-a-and-b-are-equal-then-which-one-of-the-following-is-correct
slug: if-the-magnitudes-of-two-vectors-a-and-b-are-equal-then-which-one-of-the-following-is-correct-27716
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the magnitudes of two vectors a and $\mathrm{b}$ are equal then which one of the following is correct?</div>

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

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

answer:
<div class="solution">Given $|\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}}|$<br/>Consider $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})$<br/>$=\vec{a} \cdot \vec{a}-\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{a}-\vec{b} \cdot \vec{b}$<br/>$=|\overrightarrow{\mathrm{a}}|^{2}-|\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}-|\overrightarrow{\mathrm{a}}|^{2}=0$<br/>Hence $(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$.</div>


tags: ['Physics', 'Motion In Two Dimensions', 'KCET', 'KCET 2015']
source_db: MarksBatch2_P1.db
title:
if-the-mass-of-a-body-is-m-on-the-surface-of-the-earth-the-mass-of-the-same-body-on-the-surface-of-the-moon-is

slug:
if-the-mass-of-a-body-is-m-on-the-surface-of-the-earth-the-mass-of-the-same-body-on-the-surface-of-the-moon-is-26669

question:
<div class="question">If the mass of a body is \( M \) on the surface of the earth, the mass of the same body on the<br/>surface of the moon is</div>

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

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

answer:
<div class="solution">Mass is fundamental property and it does not depend on external conditions. So mass of an object always remains<br/>same.<br/>Therefore, if \( M \) is mass of body on surface of Earth, then mass of same body on the surface of moon is \( M \).</div>


title: if-the-matrix-3-1-2-0-1-2-4-1-is-not-invertible-then
slug: if-the-matrix-3-1-2-0-1-2-4-1-is-not-invertible-then-77468
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the matrix $\left[\begin{array}{ccc}\alpha &amp; 2 &amp; 2 \\ -3 &amp; 0 &amp; 4 \\ 1 &amp; -1 &amp; 1\end{array}\right]$ is not invertible, then:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\alpha=-5$</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">$\alpha=5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \alpha=1$</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{A}=\left[\begin{array}{ccc}\alpha &amp; 2 &amp; 2 \\ -3 &amp; 0 &amp; 4 \\ 1 &amp; -1 &amp; 1\end{array}\right]$<br/>$|\mathrm{A}|=\left|\begin{array}{ccc}\alpha &amp; 2 &amp; 2 \\ -3 &amp; 0 &amp; 4 \\ 1 &amp; -1 &amp; 1\end{array}\right|$<br/>$|A|=\alpha(0+4)-2(-3-4)+2(3-0)=4 \alpha+20$<br/>Since $\mathrm{A}^{-1}$ does not exist, $\therefore|\mathrm{A}|=0$<br/>$4 \alpha+20=0$<br/>$4 \alpha=-20$<br/>$\alpha=-5$</div>


title: if-the-matrix-a-2-x-1-1-1-3-x-3-1-0-x-is-singular-then-what-is-the-solution-set-s
slug: if-the-matrix-a-2-x-1-1-1-3-x-3-1-0-x-is-singular-then-what-is-the-solution-set-s-86418
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the matrix<br/>$A=\left[\begin{array}{ccc}2-x &amp; 1 &amp; 1 \\ 1 &amp; 3-x &amp; 0 \\ -1 &amp; -3 &amp; -x\end{array}\right]$<br/>is singular, then what is the solution set $S ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{S}=\{0,2,3\}$</span> </div>

answer:
<div class="solution">Let $A=\left[\begin{array}{ccc}2-x &amp; 1 &amp; 1 \\ 1 &amp; 3-x &amp; 0 \\ -1 &amp; -3 &amp; -x\end{array}\right]$<br/>Since, this matrix is singular. <br/>$\therefore |A|=0$<br/>$\Rightarrow\left|\begin{array}{ccc}2-x &amp; 1 &amp; 1 \\ 1 &amp; 3-x &amp; 0 \\ -1 &amp; -3 &amp; -x\end{array}\right|=0$<br/>Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}+\mathrm{R}_{3}$<br/>$\Rightarrow\left|\begin{array}{ccc}2-x &amp; 1 &amp; 1 \\ 0 &amp; -x &amp; -x \\ -1 &amp; -3 &amp; -x\end{array}\right|=0$<br/>$\Rightarrow(2-x)\left(x^{2}-3 x\right)-1(-x)+1(-x)=0$<br/>$\Rightarrow(2-x)(x)(x-3)=0$<br/>$\Rightarrow x=2,0,3$<br/>Hence, solution set $S=\{0,2,3\}$</div>


title: if-the-matrix-a-is-such-that-1-0-3-1-a-1-0-1-1-then-what-is-a-equal-to
slug: if-the-matrix-a-is-such-that-1-0-3-1-a-1-0-1-1-then-what-is-a-equal-to-82135
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the matrix $A$ is such that $\left(\begin{array}{ll}1 &amp; 3 \\ 0 &amp; 1\end{array}\right) A=\left(\begin{array}{cc}1 &amp; 1 \\ 0 &amp; -1\end{array}\right)$, then what is $A$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\begin{array}{cc}1 &amp; 4 \\ 0 &amp; -1\end{array}\right)$</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">$\left(\begin{array}{ll}1 &amp; 4 \\ 0 &amp; 1\end{array}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\begin{array}{cc}-1 &amp; 4 \\ 0 &amp; -1\end{array}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\begin{array}{ll}1 &amp; -4 \\ 0 &amp; -1\end{array}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\begin{array}{cc}1 &amp; 4 \\ 0 &amp; -1\end{array}\right)$</span> </div>

answer:
<div class="solution">$\because\left(\begin{array}{ll}1 &amp; 3 \\ 0 &amp; 1\end{array}\right) \mathrm{A}=\left(\begin{array}{rr}1 &amp; 1 \\ 0 &amp; -1\end{array}\right)$<br/>Let $\mathrm{B} \quad=\left(\begin{array}{ll}1 &amp; 3 \\ 0 &amp; 1\end{array}\right)$ and $|\mathrm{B}|=1$<br/>$\therefore \mathrm{B}^{-1}=\left(\begin{array}{rr}1 &amp; -3 \\ 0 &amp; 1\end{array}\right)$ $\quad\left(\because \mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj} \mathrm{A}\right)$<br/>$\therefore \mathrm{A}=\left(\begin{array}{rr}1 &amp; -3 \\ 0 &amp; 1\end{array}\right)\left(\begin{array}{rr}1 &amp; 1 \\ 0 &amp; -1\end{array}\right)=\left(\begin{array}{rr}1 &amp; 4 \\ 0 &amp; -1\end{array}\right)$</div>


title: if-the-matrix-ab-is-a-zero-matrix-then-which-one-of-the-following-is-correct
slug: if-the-matrix-ab-is-a-zero-matrix-then-which-one-of-the-following-is-correct-59208
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the matrix $\mathrm{AB}$ is a zero matrix, then which one of the following is correct?</div>

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

answer:
<div class="solution">If matrix $\mathrm{AB}=0$ then it is not necessary that either A is zero matrix or $\mathrm{B}$ is zero matrix.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-matrix-b-is-the-adjoint-of-the-square-matrix-a-and-is-the-value-of-the-determinant-of-a-then-what-is-ab-equal-to

slug:
if-the-matrix-b-is-the-adjoint-of-the-square-matrix-a-and-is-the-value-of-the-determinant-of-a-then-what-is-ab-equal-to-61323

question:
<div class="question">If the matrix $\mathrm{B}$ is the adjoint of the square matrix A and $\alpha$ is the value of the determinant of $\mathrm{A}$, then what is $\mathrm{AB}$ equal to ?</div>

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

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

answer:
<div class="solution">Since, adjoint of the square matrix A is $\mathrm{B} \Rightarrow \frac{\mathrm{B}}{|\mathrm{A}|}=\mathrm{A}^{-1}$<br/>$\Rightarrow \frac{\mathrm{AB}}{|\mathrm{A}|}=\mathrm{AA}^{-1}=\mathrm{I}$<br/>$\Rightarrow \mathrm{AB}=|\mathrm{A}| \mathrm{I} \Rightarrow \mathrm{AB}=\alpha \mathrm{I}$</div>


title: if-the-matrix-cos-sin-0-sin-cos-0-0-0-1-is-singular-then-what-is-one-of-the-values-of
slug: if-the-matrix-cos-sin-0-sin-cos-0-0-0-1-is-singular-then-what-is-one-of-the-values-of-46754
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the matrix $\left[\begin{array}{ccc}\cos \theta &amp; \sin \theta &amp; 0 \\ \sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$ is singular, then what is one of the values of $\theta$?</div>

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

answer:
<div class="solution">A matrix is singular if value of its determinant is zero.<br/>Given that matrix $\left[\begin{array}{ccc}\cos \theta &amp; \sin \theta &amp; 0 \\ \sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right]$ is singular,<br/>$\left|\begin{array}{ccc}\cos \theta &amp; \sin \theta &amp; 0 \\ \sin \theta &amp; \cos \theta &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right|=0$<br/>$\Rightarrow\left|\begin{array}{ll}\cos \theta &amp; \sin \theta \\ \sin \theta &amp; \cos \theta\end{array}\right|=0 \Rightarrow \cos ^{2} \theta-\sin ^{2} \theta=0=\cos \frac{\pi}{2}$<br/>$\Rightarrow \cos 2 \theta=\cos \frac{\pi}{2} \Rightarrow \theta=\frac{\pi}{4}$</div>


tags: ['Mathematics', 'Matrices', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-matrix-mathrmaleftbeginarrayll3-7-2-5endarrayright-satisfies-the-equation-mathrma2lambda-mathrmamu-mathrmimathrmo-leftright-where-ileftbeginarr

slug:
if-the-matrix-mathrmaleftbeginarrayll3-7-2-5endarrayright-satisfies-the-equation-mathrma2lambda-mathrmamu-mathrmimathrmo-leftright-where-ileftbeginarr-48926

question:
<div class="question">If the matrix \( \mathrm{A}=\left[\begin{array}{ll}3 &amp; 7 \\ 2 &amp; 5\end{array}\right] \) satisfies the equation \( \mathrm{A}^{2}+\lambda \mathrm{A}+\mu \mathrm{I}=\mathrm{O} \)<br/>\( \left(\right. \) where,\( I=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right] \) and \( \left.O=\left[\begin{array}{ll}0 &amp; 0 \\ 0 &amp; 0\end{array}\right]\right) \), then the value of \( \frac{\mu-\lambda}{3} \) is</div>

answer:
<div class="solution">This question can be easily done by using the concept of characteristic equation of a matrix.<br/>Consider the equation <math> <mrow> <mtext>det (A - kI)</mtext> <mo>=</mo> <mn>0</mn> </mrow> </math>, where k is a scalar.<br/>Now, when we solve the equation <math> <mrow> <mfenced close="|" open="|" separators=","> <mrow> <mtext>A</mtext> <mo>-</mo> <mtext>k</mtext> <mtext>I</mtext> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> </math>, then we get a polynomial in k.<br/>Here, if we put k = A(matrix), then the equation obtained is called as the characteristic equation of square matrix A.<br/>The characteristic polynomial of A is <math> <mrow> <mfenced close="|" open="|" separators=","> <mrow> <mtext>A</mtext> <mo>-</mo> <mtext>k</mtext> <mtext>I</mtext> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> </math>.<br/><math> <mrow> <mo>∴  </mo> <mfenced close="|" open="|" separators=","> <mrow> <mfenced close="]" open="[" separators=","> <mrow> <mtable> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>-</mo> <mtext>k</mtext> <mfenced close="]" open="[" separators=","> <mrow> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> </math><br/><math> <mrow> <mo>⇒  </mo> <mfenced close="|" open="|" separators=","> <mrow> <mfenced close="]" open="[" separators=","> <mrow> <mtable> <mtr> <mtd> <mn>3</mn> <mo>-</mo> <mtext>k</mtext> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> <mo>-</mo> <mtext>k</mtext> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> </math><br/><math> <mrow> <mo>⇒  </mo> <mfenced close="|" open="|" separators=","> <mrow> <mtable> <mtr> <mtd> <mn>3</mn> <mo>-</mo> <mtext>k</mtext> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> <mo>-</mo> <mtext>k</mtext> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </mrow> </math><br/><math> <mrow> <mo>⇒  </mo> <mfenced close=")" open="(" separators=","> <mrow> <mn>3</mn> <mo>-</mo> <mtext>k</mtext> </mrow> </mfenced> <mfenced close=")" open="(" separators=","> <mrow> <mn>5</mn> <mo>-</mo> <mtext>k</mtext> </mrow> </mfenced> <mo>-</mo> <mn>1</mn> <mn>4</mn> <mo>=</mo> <mn>0</mn> </mrow> </math><br/><math> <mrow> <mo>⇒  </mo> <msup> <mtext>k</mtext> <mn>2</mn> </msup> <mo>-</mo> <mn>8</mn> <mtext>k</mtext> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mrow> </math><br/><math> <mrow> <mo>∴  </mo> <msup> <mtext>A</mtext> <mn>2</mn> </msup> <mo>-</mo> <mn>8</mn> <mtext>A</mtext> <mo>+</mo> <mtext>I</mtext> <mo>=</mo> <mn>0</mn> </mrow> </math><br/>On comparing the above equation with the given equation <math> <mrow> <msup> <mtext>A</mtext> <mn>2</mn> </msup> <mo>+</mo> <mi>λ</mi> <mtext>A</mtext> <mo>+</mo> <mi>μ</mi> <mtext>I</mtext> <mo>=</mo> <mtext>O</mtext> </mrow> </math>, we get<br/><math> <mrow> <mi>λ</mi> <mo>=</mo> <mo>-</mo> <mn>8</mn> <mo>,</mo> <mi>μ</mi> <mo>=</mo> <mn>1</mn> <mo>⇒</mo> <mfrac> <mrow> <mi>μ</mi> <mo>-</mo> <mi>λ</mi> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> </mrow> </math></div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: MarksBatch2_P1.db
title:
if-the-matrix-mr-is-given-by-mrleftbeginarrayccr-r1-r1-rendarrayright-for-r123-ldots-then-operatornamedetleftmathrmm1rightoperatornamedetleftmathrmm2r

slug:
if-the-matrix-mr-is-given-by-mrleftbeginarrayccr-r1-r1-rendarrayright-for-r123-ldots-then-operatornamedetleftmathrmm1rightoperatornamedetleftmathrmm2r-42510

question:
<div class="question">If the matrix \(M_r\) is given by \(M_r=\left(\begin{array}{cc}r &amp; r-1 \\ r-1 &amp; r\end{array}\right)\) for \(r=1,2,3, \ldots\) then \(\operatorname{det}\left(\mathrm{M}_1\right)+\operatorname{det}\left(\mathrm{M}_2\right)+\ldots . .+\operatorname{det}\left(\mathrm{M}_{2008}\right)=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2007</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2008</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((2008)^2\)</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">\((2007)^2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((2008)^2\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint : } \operatorname{det}\left(M_r\right)=r^2-(r-1)^2=r^2-\left(r^2-2 r+1\right)=2 r-1 \\<br/>&amp; \begin{aligned}<br/>\therefore \sum_{r=1}^{2008} \operatorname{det}\left(M_r\right)=\sum_{r=1}^{2008}(2 r-1)= &amp; +3+5+\cdots+4015 \\<br/>= &amp; (2008)^2<br/>\end{aligned}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2019 (10 Jan Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-the-mean-and-standard-deviation-of-5-observations-x-1-x-2-x-3-x-4-x-5-are-10-and-3-respectively-then-the-variance-of-6-observations-x-1-x-2-x-5-and

slug:
if-the-mean-and-standard-deviation-of-5-observations-x-1-x-2-x-3-x-4-x-5-are-10-and-3-respectively-then-the-variance-of-6-observations-x-1-x-2-x-5-and-24156

question:
<div class="question">If the mean and standard deviation of <math><mn>5</mn></math> observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mi> </mi><msub><mi>x</mi><mn>3</mn></msub><mo>,</mo><mi> </mi><msub><mi>x</mi><mn>4</mn></msub><mo>,</mo><mi> </mi><msub><mi>x</mi><mn>5</mn></msub></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> and <math><mn>3</mn><mo>,</mo></math> respectively, then the variance of <math><mn>6</mn></math> observations <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi> </mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi> </mi><mo>…</mo><mo>,</mo><mi> </mi><msub><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msub></math> and <math><mo>-</mo><mn>50</mn></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>582.5</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mn>507.5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>509.5</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>586.5</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>507.5</mn></math></span> </div>

answer:
<div class="solution"><blockquote><p>We have,</p><p>Mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><msub><mi>x</mi><mn>4</mn></msub><mo>+</mo><msub><mi>x</mi><mn>5</mn></msub></mrow><mn>5</mn></mfrac><mo>=</mo><mn>10</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><msub><mi>x</mi><mn>4</mn></msub><mo>+</mo><msub><mi>x</mi><mn>5</mn></msub><mo>=</mo><mn>50</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced><mo> </mo></math><br/><br/>Variance</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Σ</mi><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>5</mn></mfrac><mo>-</mo><msup><mfenced separators="|"><mover><mi>x</mi><mo>¯</mo></mover></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn></math> <br/><br/><math><mo>⇒</mo><mfrac><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>5</mn></mrow></mfrac><mo>-</mo><mn>100</mn><mo>=</mo><mn>9</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>4</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>5</mn><mn>2</mn></msubsup><mo>=</mo><mn>545</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced><mo> </mo></math><br/><br/>Now,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">w</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><msub><mi>x</mi><mn>4</mn></msub><mo>+</mo><msub><mi>x</mi><mn>5</mn></msub><mo>-</mo><mn>50</mn></mrow><mn>6</mn></mfrac><mo>=</mo><mn>0</mn></math> <br/><br/>New variance</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>6</mn></munderover></mstyle><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>6</mn></mfrac><mo>-</mo><msup><mfenced separators="|"><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">w</mi></mrow></msub></mfenced><mn>2</mn></msup><mo> </mo></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>4</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>5</mn><mn>2</mn></msubsup><mo>+</mo><mn>2500</mn></mrow><mn>6</mn></mfrac><mo>-</mo><mn>0</mn></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>545</mn><mo>+</mo><mn>2500</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3045</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>=</mo><mn>507.5</mn></math></p></blockquote></div>


tags: ['Mathematics', 'Probability', 'AP EAMCET', 'AP EAMCET 2020 (23 Sep Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-the-mean-and-standard-deviation-of-a-binomial-distribution-are-20-and-4-respectively-then-the-number-of-trials-is

slug:
if-the-mean-and-standard-deviation-of-a-binomial-distribution-are-20-and-4-respectively-then-the-number-of-trials-is-71358

question:
<div class="question">If the mean and standard deviation of a binomial distribution are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>, respectively, then the number of trials is ________</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math></span> </div>

answer:
<div class="solution">Given, mean of a binomial distribution <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>20</mn></math><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>n</mi><mi>p</mi><mo>=</mo><mn>20</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>…</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>and standard deviation of a binomial distribution <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>n</mi><mi>p</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced><mo>=</mo><mn>4</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>…</mo><mfenced><mi>ii</mi></mfenced></math></p><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>p</mi><mo>=</mo><mn>20</mn></math> from equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced></math> to equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>20</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></math></p><p>Put the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> in the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>n</mi><mfenced><mfrac><mn>4</mn><mn>5</mn></mfrac></mfenced><mo>=</mo><mn>20</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>n</mi><mo>=</mo><mn>25</mn></math> </p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2020 (05 Sep Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-the-mean-and-the-standard-deviation-of-the-data-3-5-7-a-b-are-5-and-2-respectively-then-a-and-b-are-the-roots-of-the-equation

slug:
if-the-mean-and-the-standard-deviation-of-the-data-3-5-7-a-b-are-5-and-2-respectively-then-a-and-b-are-the-roots-of-the-equation-89980

question:
<div class="question">If the mean and the standard deviation of the data <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> respectively, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are the roots of the equation:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>20</mn><mi>x</mi><mo>+</mo><mn>19</mn><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>19</mn><mo>=</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>20</mn><mi>x</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>0</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>19</mn><mo>=</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>7</mn><mo>+</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>25</mn><mo>⇒</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>10</mn></math><br/>S.D. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><msup><mn>5</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mn>7</mn><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>-</mo><msup><mn>5</mn><mn>2</mn></msup></msqrt><mo>=</mo><mn>2</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>83</mn></mrow><mn>5</mn></mfrac><mo>-</mo><mn>25</mn><mo>=</mo><mn>4</mn><mo>⇒</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>62</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>=</mo><mn>62</mn><mo>⇒</mo><mi>a</mi><mi>b</mi><mo>=</mo><mn>19</mn></math><br/>so equation whose roots are a and b is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>19</mn><mo>=</mo><mn>0</mn></math></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">32.05</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-mean-of-50-observations-is-25-and-their-standard-deviation-is-4-and-the-sum-of-the-squares-of-all-the-observations-is-then-1000-is

slug:
if-the-mean-of-50-observations-is-25-and-their-standard-deviation-is-4-and-the-sum-of-the-squares-of-all-the-observations-is-then-1000-is-33003

question:
<div class="question">If the mean of <math><mn>50</mn></math> observations is <math><mn>25</mn></math> and their standard deviation is <math><mn>4</mn></math> and the sum of the squares of all the observations is <math><mi>Î»</mi></math>, then <math><mfrac><mrow><mi>Î»</mi></mrow><mrow><mn>1000</mn></mrow></mfrac></math> is</div>

answer:
<div class="solution">Variance <math><mo>=</mo><mfrac><mrow><mrow><munderover><mo stretchy="false">â</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>50</mn></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>â</mo><mn>16</mn><mo>=</mo><mfrac><mrow><mrow><munderover><mo stretchy="false">â</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>50</mn></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mn>50</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn>25</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>â</mo><mi></mi><mrow><munderover><mo stretchy="false">â</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>50</mn></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow><mo>=</mo><mn>641</mn><mo>Ã</mo><mn>50</mn></math><br/><math><mo>â´</mo><mi></mi><mrow><munderover><mo stretchy="false">â</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>50</mn></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow><mo>=</mo><mn>32050</mn><mo>=</mo><mi>Î»</mi></math></div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2020 (17 Sep Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">252500</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-mean-of-100-observations-is-50-and-their-standard-deviation-is-5-then-the-sum-of-squares-of-all-observations-is

slug:
if-the-mean-of-100-observations-is-50-and-their-standard-deviation-is-5-then-the-sum-of-squares-of-all-observations-is-50455

question:
<div class="question">If the mean of 100 observations is 50 and their standard deviation is 5 , then the sum of squares of all observations is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">50000</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">250000</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">252500</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">255000</span> </li> </ul>

answer:
<div class="solution">Given; \(\bar{x}=50\)<br/>\(n=100\) and \(S . D(\sigma)=5\)<br/>As, \(\bar{x}=\frac{\Sigma x_i}{n} \Rightarrow \Sigma x_i=n . \bar{x}=5000\).<br/>Also, S.D. is given by,<br/>\(\begin{aligned}<br/>\sigma &amp; =\sqrt{\frac{\Sigma x_i^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2} \\<br/>\Rightarrow \quad \sigma^2 &amp; =\frac{\Sigma x_i^2}{n}-(\bar{x})^2 \Rightarrow 25=\frac{\Sigma x_i^2}{100}-(50)^2 \\<br/>\Rightarrow \quad \Sigma x_i^2 &amp; =252500<br/>\end{aligned}\)</div>


title: if-the-mean-of-a-binomial-distribution-is-25-then-its-standard-deviation-lies-in-the-interval
slug: if-the-mean-of-a-binomial-distribution-is-25-then-its-standard-deviation-lies-in-the-interval-50833
tags: ['Mathematics', 'Probability', 'VITEEE', 'VITEEE 2016']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the mean of a binomial distribution is 25 , then its standard deviation lies in the interval</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$[0,5)$</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">$(0,5]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[0,25)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(0,25]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$[0,5)$</span> </div>

answer:
<div class="solution">Standard deviation $\sigma=\sqrt{\mathrm{npq}} \geq 0$
<br/>Now mean $=n p=25$ and $q &lt; 1$
<br/>So $\sigma=\sqrt{\mathrm{npq}} &lt; \sqrt{\mathrm{np}}=5$
<br/>$\therefore 0 \leq \sigma &lt; 5$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-mean-of-few-observations-is-40-and-standard-deviation-is-8-then-what-is-the-coefficient-of-variation

slug:
if-the-mean-of-few-observations-is-40-and-standard-deviation-is-8-then-what-is-the-coefficient-of-variation-25386

question:
<div class="question">If the mean of few observations is 40 and standard deviation is 8 , then what is the coefficient of variation? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$10 \%$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$20 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$30 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$20 \%$</span> </div>

answer:
<div class="solution">Mean $=40, \mathrm{S.D}=8$<br/>$\therefore \quad$ Coeft of variation $=\frac{8}{40} \times 100$<br/>$=\frac{1}{5} \times 100-20 \%$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2020 (10 Sep Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.58</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-mean-of-the-discrete-distribution-89-6-5-x-4-6-5-is-6-then-its-standard-deviation-nearest-to-two-decimal-places-is

slug:
if-the-mean-of-the-discrete-distribution-89-6-5-x-4-6-5-is-6-then-its-standard-deviation-nearest-to-two-decimal-places-is-54277

question:
<div class="question">If the mean of the discrete distribution 8,9 , $6,5, x, 4,6,5$ is 6 , then its standard deviation (nearest to two decimal places) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2.50</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1.58</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">0.51</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.41</span> </li> </ul>

answer:
<div class="solution">We have,<br/>$\begin{aligned}<br/>&amp; \text { Mean }=6 \\<br/>&amp; \therefore \quad \frac{8+9+6+5+x+4+6+5}{8}=6 \\<br/>&amp; \Rightarrow 43+x=48 \Rightarrow x=5<br/>\end{aligned}$<br/>So, data are $8,9,6,5,5,4,6,5$<br/>$\begin{array}{llrl}\therefore &amp; &amp; x_i=8,9,6,5,5,4,6,5 \\ \Rightarrow &amp; x_i{ }^2 &amp; =64,81,36,25,25,16,36,25\end{array}$<br/>$\begin{aligned} \therefore \quad \Sigma x_i^2 &amp; =64+81+36+25+25+16+36+25 \\ &amp; =308\end{aligned}$<br/>$\begin{aligned} \therefore \quad \operatorname{SD}(x) &amp; =\sqrt{\frac{1}{N} \Sigma x_i^2-(\bar{x})^2} \\ &amp; =\sqrt{\frac{1}{8} \times 308-(6)^2}=\sqrt{38.5-36} \\ &amp; =\sqrt{2.5}=1.58\end{aligned}$</div>


title: if-the-mid-point-between-the-points-a-b-a-b-and-a-b-lies-on-the-line-a-x-b-y-k-what-is-k-equal-to
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">ab</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-mid-point-between-the-points-a-b-a-b-and-a-b-lies-on-the-line-a-x-b-y-k-what-is-k-equal-to-34526

question:
<div class="question">If the mid point between the points $(a+b, a-b)$ and $(-a, b)$ lies on the line $a x+b y=k$, what is $k$ equal to?</div>

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

answer:
<div class="solution">Given points are $(a+b, a-b)$ and $(-a, b)$ Mid point is $\left(\frac{a+b-a}{2}, \frac{a-b+b}{2}\right)=\left(\frac{b}{2}, \frac{a}{2}\right)$<br/>Since, it lies on $a x+b y=k$ $a\left(\frac{b}{2}\right)+b\left(\frac{a}{2}\right)=k \Rightarrow k=a b$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-the-monthly-expenditure-pattern-of-a-person-who-carns-monthlysalary-of-rs-15000-is-represented-in-a-pie-diagram-then-the-sector-angle-of-an-item-on

slug:
if-the-monthly-expenditure-pattern-of-a-person-who-carns-monthlysalary-of-rs-15000-is-represented-in-a-pie-diagram-then-the-sector-angle-of-an-item-on-15394

question:
<div class="question">If the monthly expenditure pattern of a person who carns monthlysalary of Rs 15000 is represented in a pie diagram then the sector angle of an item on transport expense:<br/>measures $15^{\circ}$. What is his monthly expenditure on transport</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Rs} 450$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Rs} 625$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\operatorname{Rs} 675$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cannot be computed from the given data</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Rs} 625$</span> </div>

answer:
<div class="solution">Since, monthly salary $=$ Rs 15000 and sector angle of expenses $=15^{\circ}$<br/>$\therefore \quad$ Amount $=\frac{15^{\circ}}{360^{\circ}} \times 15000=\mathrm{Rs} 625$</div>


title: if-the-n-th-term-of-an-arithmetic-progression-is-2-n-1-then-what-is-the-sum-upto-n-terms
slug: if-the-n-th-term-of-an-arithmetic-progression-is-2-n-1-then-what-is-the-sum-upto-n-terms-89052
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the $n$ th term of an arithmetic progression is $2 n-1$, then what is the sum upto $n$ terms?</div>

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

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

answer:
<div class="solution">Given $a_{n}=2 n-1$<br/>$\therefore S_{n}=\sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n}(2 n-1)$<br/>$=2 \sum_{k=1}^{n} n-n=2 \cdot \frac{n(n+1)}{2}-n=n^{2}+n-n=n^{2}$</div>


title: if-the-nth-term-of-an-arithmetic-progression-is-3-n-7-then-what-is-the-sum-of-its-first-50-terms
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4175</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-the-nth-term-of-an-arithmetic-progression-is-3-n-7-then-what-is-the-sum-of-its-first-50-terms-71913

question:
<div class="question">If the nth term of an arithmetic progression is $3 \mathrm{n}+7$, then what is the sum of its first 50 terms?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3925</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4100</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">4175</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">8200</span> </li> </ul>

answer:
<div class="solution">As given, $\mathrm{n}_{\text {th }}$ term $\mathrm{T}_{\mathrm{n}}=3 \mathrm{n}+7$<br/>Sum of n term, $\mathrm{S}_{\mathrm{n}}=\sum \mathrm{T}_{\mathrm{n}}$<br/>$=\sum(3 n+7)=3 \sum n+7 \sum 1$<br/>$=\frac{3 n(n+1)}{2}+7 n=n\left[\frac{3 n+3+14}{2}\right]=n\left[\frac{3 n+17}{2}\right]$<br/>Sum of 50 terms $S_{50}=50\left[\frac{3 \times 50+17}{2}\right]$ $=50\left[\frac{167}{2}\right]=25 \times 167=4175$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-number-235-in-decimal-system-is-converted-into-binary-system-then-what-is-the-resulting-number

slug:
if-the-number-235-in-decimal-system-is-converted-into-binary-system-then-what-is-the-resulting-number-71500

question:
<div class="question">If the number 235 in decimal system is converted into binary system, then what is the resulting number?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(11110011)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(11101011)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$(11110101)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(11011011)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(11101011)$</span> </div>

answer:
<div class="solution">\begin{array}{l|l|l}<br/>2 &amp; 235 &amp; 1 \\<br/>\hline 2 &amp; 117 &amp; 1 \\<br/>\hline 2 &amp; 58 &amp; 0 \\<br/>\hline 2 &amp; 29 &amp; 1 \\<br/>\hline 2 &amp; 14 &amp; 0 \\<br/>\hline 2 &amp; 7 &amp; 1 \\<br/>\hline 2 &amp; 3 &amp; 1 \\<br/>\hline &amp; 1 &amp;<br/>\end{array}<br/>So, $(235)_{10}=(11101011)_{2}$</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-number-of-terms-of-an-ap-is-2-n-1-then-what-is-the-ratio-of-the-sum-of-the-odd-terms-to-the-sum-of-even-terms

slug:
if-the-number-of-terms-of-an-ap-is-2-n-1-then-what-is-the-ratio-of-the-sum-of-the-odd-terms-to-the-sum-of-even-terms-36490

question:
<div class="question">If the number of terms of an A.P. is $(2 n+1)$, then what is the ratio of the sum of the odd terms to the sum of even terms?</div>

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

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

answer:
<div class="solution">Let the $\mathrm{AP}$ is $\mathrm{a}, \mathrm{a}+\mathrm{d}, \mathrm{a}+2 \mathrm{~d}, \ldots \ldots, \mathrm{a}+(2 \mathrm{n}-1) \mathrm{d}, \mathrm{a}+2 \mathrm{nd}$<br/>Series of even terms. $\mathrm{a}+\mathrm{d}, \mathrm{a}+3 \mathrm{~d}, \ldots \ldots \ldots, \mathrm{a}+(2 \mathrm{n}-1) \mathrm{d}$, has $\mathrm{n}$ terms<br/>$\begin{aligned} \text { Sum of even number } &amp;=\frac{n}{2}[(a+d)+\{a+(2 n-1) d\}] \\ &amp;=\frac{n}{2}[2 a+2 n d]=n[a+n d] \end{aligned}$<br/>Series of odd terms $\mathrm{a}, \mathrm{a}+2 \mathrm{~d}, \mathrm{a}+4 \mathrm{~d}, \ldots \ldots \ldots \ldots, \mathrm{a}+2 \mathrm{nd}$, has $(\mathrm{n}+1)$ terms.<br/>$\begin{aligned} \text { Sum of odd numbers } &amp;=\frac{n+1}{2}[a+(a+2 n d)] \\ &amp;=\frac{n+1}{2}(2 a+2 n d) \\ &amp;=(n+1)(a+n d) \end{aligned}$<br/>So, the required ratio $=\frac{n+1}{n}$</div>


title: if-the-numbers-n-3-4-n-2-5-n-1-are-in-ap-what-is-the-value-of-n
slug: if-the-numbers-n-3-4-n-2-5-n-1-are-in-ap-what-is-the-value-of-n-55898
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If the numbers $\mathrm{n}-3,4 \mathrm{n}-2,5 \mathrm{n}+1$ are in AP, what is the value of $\mathrm{n} ?$</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">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \because(\mathrm{n}-3), 4 \mathrm{n}-2,5 \mathrm{n}+1 \text { are in A.P. } \\ &amp; \Rightarrow(4 \mathrm{n}-2)-(\mathrm{n}-3)=(5 \mathrm{n}+1)-(4 n-2) \\ &amp; \Rightarrow 3 \mathrm{n}+1=\mathrm{n}+3 \\ &amp; \Rightarrow \mathrm{n}=1 \end{aligned}$</div>


tags: ['Mathematics', 'Probability', 'MHT CET', 'MHT CET 2020 (13 Oct Shift 2)']
source_db: MarksBatch2_P1.db
title:
if-the-p-m-f-of-a-random-variable-x-is-beginarraycccccc-hline-mathrmx-1-2-3-4-5-hline-mathrmpmathrmxx-k-frack3-frack4-frack2-frack2-hline-endarray-the

slug:
if-the-p-m-f-of-a-random-variable-x-is-beginarraycccccc-hline-mathrmx-1-2-3-4-5-hline-mathrmpmathrmxx-k-frack3-frack4-frack2-frack2-hline-endarray-the-96906

question:
<div class="question">If the p. m. f. of a random variable $\mathrm{X}$ is<br/>\begin{array}{|c|c|c|c|c|c|}<br/>\hline \mathrm{X} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\<br/>\hline \mathrm{P}(\mathrm{X}=x) &amp; k &amp; \frac{k}{3} &amp; \frac{k}{4} &amp; \frac{k}{2} &amp; \frac{k}{2} \\<br/>\hline<br/>\end{array}<br/>then $k=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{15}{31}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{12}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{11}{12}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{12}{31}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{12}{31}$</span> </div>

answer:
<div class="solution">Here $\mathrm{k}+\frac{\mathrm{k}}{3}+\frac{\mathrm{k}}{4}+\frac{\mathrm{k}}{2}+\frac{\mathrm{k}}{2}=1$<br/>$\therefore \mathrm{k}\left(\frac{12+4+3+6+6}{12}\right)=1 \Rightarrow \mathrm{k}\left(\frac{31}{12}\right)=1 \Rightarrow \mathrm{k}=\frac{12}{13}$</div>


title: if-the-perimeter-of-a-triangle-abc-is-30-cm-then-what-is-the-value-of-a-cos-2-c-2-c-cos-2-a-2
slug: if-the-perimeter-of-a-triangle-abc-is-30-cm-then-what-is-the-value-of-a-cos-2-c-2-c-cos-2-a-2-83998
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the perimeter of a triangle $\mathrm{ABC}$ is $30 \mathrm{~cm}$, then what is the value of $a \cos ^{2}(\mathrm{C} / 2)+c \cos ^{2}(\mathrm{~A} / 2) ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$15 \mathrm{~cm}$</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">$10 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{15}{2} \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$13 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$15 \mathrm{~cm}$</span> </div>

answer:
<div class="solution">We know from properties of triangle that<br/>$\cos \frac{A}{2}=\sqrt{\frac{s(s-a)}{b c}} \Rightarrow \cos ^{2} \frac{A}{2}=\frac{s(s-a)}{b c}$<br/>and $\cos \frac{\mathrm{C}}{2}=\sqrt{\frac{\mathrm{s}(\mathrm{s}-\mathrm{c})}{\mathrm{ab}}} \Rightarrow \cos ^{2} \frac{\mathrm{C}}{2}=\frac{\mathrm{s}(\mathrm{s}-\mathrm{c})}{\mathrm{ab}}$<br/>So, $a \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{A}{2}$<br/>$=\mathrm{a}\left(\frac{\mathrm{s}(\mathrm{s}-\mathrm{c})}{\mathrm{ab}}\right)+\mathrm{c}\left(\frac{\mathrm{s}(\mathrm{s}-\mathrm{a})}{\mathrm{bc}}\right)=\frac{\mathrm{s}(\mathrm{s}-\mathrm{c}+\mathrm{s}-\mathrm{a})}{\mathrm{b}}$<br/>$=\frac{s(2 s-a-c)}{b}=\frac{s(a+b+c-a-c)}{b}=\frac{s \cdot b}{b}=s$<br/>$=\frac{30}{2}=15 \mathrm{~cm}$ [given that $2 \mathrm{~s}=30$ ]</div>


title: if-the-pmf-of-a-rv-x-is-then-the-standard-deviation-of-x-is-given-p-q-1
slug: if-the-pmf-of-a-rv-x-is-then-the-standard-deviation-of-x-is-given-p-q-1-78307
tags: ['Mathematics', 'Probability', 'MHT CET', 'MHT CET 2020 (19 Oct Shift 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the p.m.f. of a r.v. $X$ is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/li7PIl9RuMg1C8RKfUKitEVg076Leh_wTw_0KrOSQLg.original.fullsize.png"/><br/><br/>then, the standard deviation of $\mathrm{X}$ is (given $p+q=1$ )</div>

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

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

answer:
<div class="solution">(B)<br/>\begin{array}{|c|c|c|c|}<br/>\hline \mathrm{x}_{\mathrm{i}} &amp; \mathrm{p}_{\mathrm{i}} &amp; \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} &amp; \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^{2} \\<br/>\hline 0 &amp; \mathrm{q}^{2} &amp; 0 &amp; 0 \\<br/>\hline 1 &amp; 2 \mathrm{pq} &amp; 2 \mathrm{pq} &amp; 2 \mathrm{pq} \\<br/>\hline 2 &amp; \mathrm{p}^{2} &amp; 2 \mathrm{p}^{2} &amp; 4 \mathrm{p}^{2} \\<br/>\hline Total &amp; &amp; 2 \mathrm{pq}+2 \mathrm{p}^{2} &amp; 2 \mathrm{pq}+4 \mathrm{p}^{2} \\<br/>\hline<br/>\end{array}<br/>$\operatorname{Mean}(\mu)=\mathrm{E}(\mathrm{x})=\Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \quad=2 \mathrm{p}(\mathrm{p}+\mathrm{q})=2 \mathrm{p} \quad \ldots[\because \mathrm{p}+\mathrm{q}=1$, given $]$<br/>Variance $\left(\sigma_{\mathrm{x}}^{2}\right)=\sum \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}{ }^{2}-\mu^{2}$<br/>$=2 \mathrm{pq}+4 \mathrm{p}^{2}-4 \mathrm{p}^{2}=2 \mathrm{pq}$<br/>Standard deviation $\left(\sigma_{x}\right)=\sqrt{2 \mathrm{pq}}$</div>


title: if-the-point-a-a-lies-between-the-lines-x-y-2-then-which-one-of-the-following-is-correct
slug: if-the-point-a-a-lies-between-the-lines-x-y-2-then-which-one-of-the-following-is-correct-73004
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the point $(a, a)$ lies between the lines $|x+y|=2$, then which one of the following is correct?</div>

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

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

answer:
<div class="solution">$|\mathrm{x}+\mathrm{y}|=2 \Rightarrow \mathrm{x}+\mathrm{y}=\pm 2$<br/>$\Rightarrow \quad x+y+2=0$ and $x+y-2=0$<br/>$\Rightarrow-2   &lt; 2 a &lt; 0$ and $-1 &lt; a &lt; 0$<br/>$\Rightarrow \quad|a| &lt; 1$</div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-the-point-z-1-1-i-where-i-1-is-the-reflection-of-a-point-z-2-x-iy-in-the-line-i-z-iz-5-then-the-point-z-2-is

slug:
if-the-point-z-1-1-i-where-i-1-is-the-reflection-of-a-point-z-2-x-iy-in-the-line-i-z-iz-5-then-the-point-z-2-is-79113

question:
<div class="question">If the point $z_{1}=1+i$ where $i=\sqrt{-1}$ is the reflection of a point $\mathrm{z}_{2}=\mathrm{x}+$ iy in the line $\mathrm{i} \overline{\mathrm{z}}-\mathrm{i} \mathrm{z}=5$, then the point $\mathrm{z}_{2}$ is</div>

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

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

answer:
<div class="solution">Let $z=a+b i$ 74<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/lLT04lkxcK4nyrdNGpbY7y827VzP9MXmKQQZtyb6uSA.original.fullsize.png"/><br/><br/>$\Rightarrow \bar{z}=\mathrm{a}-\mathrm{bi}$<br/>$\therefore \mathrm{i} \overline{\mathrm{z}}-\mathrm{iz}=\mathrm{i}[(\mathrm{a}-\mathrm{bi})-(\mathrm{a}+\mathrm{bi})]=5$<br/>$\Rightarrow \mathrm{i}[-2 \mathrm{bi}]=5$<br/>$\Rightarrow \mathrm{b}=\frac{5}{2}$<br/>So from figure it is clear that<br/>$x=1, y=\frac{5}{2}+\frac{3}{2}=4$<br/>$\mathrm{z}_{2}=1+4 \mathrm{i}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
title:
if-the-points-2-4-2-6-and-2-3-k-are-the-vertices-of-an-equilateral-triangle-then-what-is-the-value-of-k

slug:
if-the-points-2-4-2-6-and-2-3-k-are-the-vertices-of-an-equilateral-triangle-then-what-is-the-value-of-k-63721

question:
<div class="question">If the points $(2,4),(2,6)$ and $(2+\sqrt{3}, \mathrm{k})$ are the vertices of an equilateral triangle, then what is the value of $\mathrm{k}$ ?</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/nfOqFzNEwKJcmjTLsHe4ZMghLrs1dVkjKUWh0Qoassw.original.fullsize.png"/><br/><br/>Since, $\mathrm{ABC}$ is an equilateral $\Delta$. $\mathrm{AB}=\mathrm{BC}=\mathrm{CA}$<br/>Consider $\mathrm{BC}=\mathrm{AC}$<br/>$\Rightarrow \sqrt{3+(\mathrm{k}-6)^{2}}=\sqrt{3+(\mathrm{k}-4)^{2}}$<br/>$\Rightarrow \sqrt{39+\mathrm{k}^{2}-12 \mathrm{k}}=\sqrt{\mathrm{k}^{2}-8 \mathrm{k}+19}$<br/>$\Rightarrow \quad \mathrm{k}^{2}-12 \mathrm{k}+39=\mathrm{k}^{2}-8 \mathrm{k}+19$<br/>$\Rightarrow \quad-4 \mathrm{k}=-20$<br/>$\mathrm{k}=5$</div>


title: if-the-points-a-1-2-b-2-4-and-c-3-a-are-collinear-what-is-the-length-bc
slug: if-the-points-a-1-2-b-2-4-and-c-3-a-are-collinear-what-is-the-length-bc-52506
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If the points $\mathrm{A}(1,2), \mathrm{B}(2,4)$ and $C(3, \mathrm{a})$ are collinear, what is the length $\mathrm{BC}$ ?</div>

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

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

answer:
<div class="solution">Since the points are collinear.<br/>$\left|\begin{array}{lll}1 &amp; 2 &amp; 1 \\ 2 &amp; 4 &amp; 1 \\ 3 &amp; a &amp; 1\end{array}\right|=0$<br/>Expanding the determinant<br/>$\Rightarrow 1\left|\begin{array}{ll}4 &amp; 1 \\ \mathrm{a} &amp; 1\end{array}\right|-2\left|\begin{array}{ll}2 &amp; 1 \\ 3 &amp; 1\end{array}\right|+1\left|\begin{array}{ll}2 &amp; 4 \\ 3 &amp; \mathrm{a}\end{array}\right|=0$<br/>$\Rightarrow(4-a)-2(2-3)+1(2 a-12)=0$<br/>$\Rightarrow \quad 4-a+2+2 a-12=0$<br/>$\Rightarrow \quad a-6=0$<br/>$\Rightarrow \quad a=6$<br/>Thus, Coordinates of $\mathrm{C}$ are $(3,6)$. Thus, $\mathrm{BC}=\sqrt{(3-2)^{2}+(6-4)^{2}}$<br/>$=\sqrt{1+4}=\sqrt{5}$ unit</div>


