# questions

tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2019 (21 Apr Shift 2)']
source_db: MarksBatch1_P2.db
title:
for-a-group-of-100-students-the-mean-x-1-and-the-standard-deviation-1-of-their-marks-were-found-to-be-40-and-15-respectively-later-it-was-observed-tha

slug:
for-a-group-of-100-students-the-mean-x-1-and-the-standard-deviation-1-of-their-marks-were-found-to-be-40-and-15-respectively-later-it-was-observed-tha-73107

question:
<div class="question">For a group of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> students, the mean <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub></math> and the standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mn>1</mn></msub></math> of their marks were found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> respectively. Later it was observed that the scores <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> were misread as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> respectively. If the mean and the standard deviation with the corrected observations of the scores are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mn>2</mn></msub></math> respectively, then</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"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>;</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>=</mo><msub><mi>σ</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>;</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>σ</mi><mn>2</mn></msub></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"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>;</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>&gt;</mo><msub><mi>σ</mi><mn>2</mn></msub></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"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>&gt;</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>;</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>=</mo><msub><mi>σ</mi><mn>2</mn></msub></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"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>;</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>&gt;</mo><msub><mi>σ</mi><mn>2</mn></msub></math></span> </div>

answer:
<div class="solution">It is given that.<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><mn>40</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><mn>15</mn></math></p><p>Now, the mean is,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>40</mn><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>100</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>4000</mn></math></p><p>Corrected summation,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mo>'</mo></msubsup><mo>=</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>30</mn><mo>-</mo><mn>60</mn><mo>+</mo><mn>40</mn><mo>+</mo><mn>50</mn><mo>=</mo><mn>4000</mn></math></p><p>The corrected mean,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mo>'</mo></msubsup></mrow><mn>100</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msub><mo>=</mo><mfrac><mn>4000</mn><mn>100</mn></mfrac><mo>=</mo><mn>40</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><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><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>225</mn><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>100</mn></mfrac><mo>-</mo><msup><mfenced><mfrac><mn>4000</mn><mn>100</mn></mfrac></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>182500</mn></math></p><p>Corrected summation,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>182500</mn><mo>-</mo><msup><mn>30</mn><mn>2</mn></msup><mo>-</mo><msup><mn>60</mn><mn>2</mn></msup><mo>+</mo><msup><mn>40</mn><mn>2</mn></msup><mo>+</mo><msup><mn>50</mn><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>182100</mn></math></p><p>The corrected standard deviation,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><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><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>182500</mn><mn>100</mn></mfrac><mo>-</mo><msup><mfenced><mfrac><mn>4000</mn><mn>100</mn></mfrac></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mn>1821</mn><mo>-</mo><mn>1600</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>σ</mi><mn>2</mn></msub><mo>=</mo><mn>14</mn><mo>.</mo><mn>86</mn></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi mathvariant="italic">x</mi><mo mathvariant="italic">¯</mo></mover><mn mathvariant="italic">1</mn></msub><mo mathvariant="italic">=</mo><msub><mover><mi mathvariant="italic">x</mi><mo mathvariant="italic">¯</mo></mover><mn mathvariant="italic">2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mn mathvariant="italic">1</mn></msub><mo mathvariant="italic">&gt;</mo><msub><mi>σ</mi><mn mathvariant="italic">2</mn></msub></math>.</p></div>


tags: ['Chemistry', 'Electrochemistry', 'TS EAMCET', 'TS EAMCET 2019 (04 May Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.348</span> </div>
source_db: MarksBatch1_P2.db
title:
for-a-halfcell-containing-a-pt-rod-immersed-in-a-solution-of-1-mh-a-o-2-g-is-bubbled-at-1-atm-the-standard-reduction-potential-for-water-formation-is-

slug:
for-a-halfcell-containing-a-pt-rod-immersed-in-a-solution-of-1-mh-a-o-2-g-is-bubbled-at-1-atm-the-standard-reduction-potential-for-water-formation-is-48898

question:
<div class="question">For a half-cell containing a Pt rod immersed in a solution of $1 \mathrm{M} \mathrm{H} A, \mathrm{O}_2(g)$ is bubbled at $1 \mathrm{~atm}$. The standard reduction potential for water formation is $1.23 \mathrm{~V}$. Given a dissociation constant, $K_a=1 \times 10^{-4}$ for $\mathrm{H} A$, what is $E_{\text {Half-cell }}$ at $298 \mathrm{~K}$ in V ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.289</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.171</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1.348</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">1.112</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/D8Bg6HztsT914LeaMnX3GTxRv6XJ_CKQlYcs0eSXXrI.original.fullsize.png"/><br/><br/>Given, Standard reduction potential for water formation $=1.23 \mathrm{~V}$<br/>Dissociation constant $\left(K_a\right)$ for $\mathrm{H} A=1 \times 10^{-4}$ $E_{\text {half-cell }}$ at $298 \mathrm{~K}=$ ?<br/>Number of electrons $=2$<br/>$$<br/>\begin{aligned}<br/>&amp; E=E^{\circ}-\frac{0.0592}{2} \log Q\left(K_a\right) \\<br/>&amp; E=1.23-\frac{0.0592}{2} \log \left(1 \times 10^{-4}\right) \\<br/>&amp; E=1.23-\frac{0.0592}{2} \times-4 \\<br/>&amp; E=1.348 \mathrm{~V}<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch1_P2.db
title:
for-a-particular-reversible-reaction-at-temperature-t-h-and-s-were-found-to-be-both-ve-if-t-e-is-the-temperature-at-equilibrium-the-reaction-would-be-

slug:
for-a-particular-reversible-reaction-at-temperature-t-h-and-s-were-found-to-be-both-ve-if-t-e-is-the-temperature-at-equilibrium-the-reaction-would-be-89418

question:
<div class="question">For a particular reversible reaction at temperature $T, \Delta H$ and $\Delta S$ were found to be both +ve. If $T_{e}$ is the temperature at equilibrium, the reaction would be spontaneous when</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$T&gt;T_{e}$</span> </div>

answer:
<div class="solution">At equilibrium $\Delta G=0$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\text { Hence, } \Delta G=\Delta H-T_{e} \Delta S=0 \\
<br/> <br/>\therefore \Delta H=T_{e} \Delta S \text { or } T_{e}=\frac{\Delta H}{\Delta S}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>For a spontaneous reaction $\Delta G$ must be negative which is possible only if $\Delta H-T \Delta S&lt;0$
<br/> <br/>$\therefore \Delta H<t s=""></t> <br/>or $T&gt;\frac{\Delta H}{\Delta S} ; T_{e}<t> <!-- /Solution --> <!-- Call to Action --> <section class="call-to-action" data-pagefind-ignore=""> <div class="call-to-action-header"> <svg fill="none" height="24" viewbox="0 0 25 24" width="25" xmlns="http://www.w3.org/2000/svg"> <path d="M19.9999 13.5C20.0018 13.8058 19.9089 14.1047 19.734 14.3555C19.559 14.6063 19.3107 14.7968 19.023 14.9006L14.1874 16.6875L12.4062 21.5269C12.3007 21.8134 12.1098 22.0608 11.8594 22.2355C11.6089 22.4101 11.3109 22.5038 11.0055 22.5038C10.7002 22.5038 10.4022 22.4101 10.1517 22.2355C9.90124 22.0608 9.71039 21.8134 9.60491 21.5269L7.8124 16.6875L2.97303 14.9062C2.68646 14.8008 2.43915 14.6099 2.26445 14.3595C2.08976 14.109 1.99609 13.811 1.99609 13.5056C1.99609 13.2003 2.08976 12.9022 2.26445 12.6518C2.43915 12.4013 2.68646 12.2105 2.97303 12.105L7.8124 10.3125L9.59366 5.47313C9.69914 5.18656 9.88998 4.93924 10.1404 4.76455C10.3909 4.58985 10.6889 4.49619 10.9943 4.49619C11.2996 4.49619 11.5977 4.58985 11.8481 4.76455C12.0986 4.93924 12.2894 5.18656 12.3949 5.47313L14.1874 10.3125L19.0268 12.0938C19.3146 12.1986 19.5628 12.3901 19.7371 12.642C19.9114 12.8939 20.0033 13.1937 19.9999 13.5ZM14.7499 4.5H16.2499V6C16.2499 6.19891 16.3289 6.38968 16.4696 6.53033C16.6102 6.67098 16.801 6.75 16.9999 6.75C17.1988 6.75 17.3896 6.67098 17.5302 6.53033C17.6709 6.38968 17.7499 6.19891 17.7499 6V4.5H19.2499C19.4488 4.5 19.6396 4.42098 19.7802 4.28033C19.9209 4.13968 19.9999 3.94891 19.9999 3.75C19.9999 3.55109 19.9209 3.36032 19.7802 3.21967C19.6396 3.07902 19.4488 3 19.2499 3H17.7499V1.5C17.7499 1.30109 17.6709 1.11032 17.5302 0.96967C17.3896 0.829018 17.1988 0.75 16.9999 0.75C16.801 0.75 16.6102 0.829018 16.4696 0.96967C16.3289 1.11032 16.2499 1.30109 16.2499 1.5V3H14.7499C14.551 3 14.3602 3.07902 14.2196 3.21967C14.0789 3.36032 13.9999 3.55109 13.9999 3.75C13.9999 3.94891 14.0789 4.13968 14.2196 4.28033C14.3602 4.42098 14.551 4.5 14.7499 4.5ZM22.9999 7.5H22.2499V6.75C22.2499 6.55109 22.1709 6.36032 22.0302 6.21967C21.8896 6.07902 21.6988 6 21.4999 6C21.301 6 21.1102 6.07902 20.9696 6.21967C20.8289 6.36032 20.7499 6.55109 20.7499 6.75V7.5H19.9999C19.801 7.5 19.6102 7.57902 19.4696 7.71967C19.3289 7.86032 19.2499 8.05109 19.2499 8.25C19.2499 8.44891 19.3289 8.63968 19.4696 8.78033C19.6102 8.92098 19.801 9 19.9999 9H20.7499V9.75C20.7499 9.94891 20.8289 10.1397 20.9696 10.2803C21.1102 10.421 21.301 10.5 21.4999 10.5C21.6988 10.5 21.8896 10.421 22.0302 10.2803C22.1709 10.1397 22.2499 9.94891 22.2499 9.75V9H22.9999C23.1988 9 23.3896 8.92098 23.5302 8.78033C23.6709 8.63968 23.7499 8.44891 23.7499 8.25C23.7499 8.05109 23.6709 7.86032 23.5302 7.71967C23.3896 7.57902 23.1988 7.5 22.9999 7.5Z" fill="#1589EE"></path> </svg> <h3>Looking for more such questions to practice?</h3> </div> <p>
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title: for-a-periodic-motion-represented-by-the-equation-y-sin-t-cos-t-the-amplitude-of-the-motion-is
tags: ['Physics', 'Oscillations', 'JEE Main', 'JEE Main 2023 (10 Apr Shift 2)']
source_db: MarksBatch1_P2.db
slug:
for-a-periodic-motion-represented-by-the-equation-y-sin-t-cos-t-the-amplitude-of-the-motion-is-84841

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

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

answer:
<div class="solution"><blockquote><p>Given the equation of motion is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>sinω</mi><mi>t</mi><mo>+</mo><mi>cosω</mi><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math></p><p>Simplify equation (1) to obtain the standard form:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right center left" columnspacing="0px"><mtr><mtd><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd><msqrt><mn>2</mn></msqrt><mfenced><mrow><mi>sin</mi><mi>ω</mi><mi>t</mi><mi>cos</mi><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>+</mo><mi>cos</mi><mi>ω</mi><mi>t</mi><mi>sin</mi><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd><msqrt><mn>2</mn></msqrt><mi>sin</mi><mfenced separators="|"><mrow><mi>ω</mi><mi>t</mi><mo>+</mo><mfrac><mstyle displaystyle="true"><mi>π</mi></mstyle><mstyle displaystyle="true"><mn>4</mn></mstyle></mfrac></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>2</mn></mfenced></mtd></mtr></mtable></math></p><p>Comparing equation (2) with standard equation of SHM <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mfenced><mrow><mi>ω</mi><mi>t</mi><mo>+</mo><mi>ϕ</mi></mrow></mfenced></math>, we get amplitude of the given oscillation as <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt></math>.</p></blockquote></div>


title: for-a-phase-change-h-2-o-l-s
slug: for-a-phase-change-h-2-o-l-s-73004
question: <div class="question">For a phase change<br/>$\mathrm{H}_2 \mathrm{O}$<br/>(l)<br/>(s)</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'AIIMS', 'AIIMS 2006']
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\Delta G=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\Delta S=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\Delta H=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\Delta U=0$.</span> </li> </ul>

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

answer:
<div class="solution">$\Delta G^{\circ}=-R T \ln K$<br/>[ $\Delta G^{\circ}=$ standard free energy change, $K=$ equilibrium constant]<br/>If a substance is in equilibrium between two phases at constant temperature and pressure, its chemical potential must have the same value in both the phases.<br/>$$<br/>\therefore \Delta G^{\circ}=0 \text {. }<br/>$$</div>


title: for-a-point-of-inflection-of-y-f-x-which-one-of-the-following-is-correct
slug: for-a-point-of-inflection-of-y-f-x-which-one-of-the-following-is-correct-23982
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P2.db
question:
<div class="question">For a point of inflection of $y=f(x)$, which one of the following is correct?</div>

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

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

answer:
<div class="solution">Lety $=\mathrm{f}(\mathrm{x})$ Now, for a point of inflection of $\mathrm{y}=\mathrm{f}(\mathrm{x})$,<br/>$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ must be zero.</div>


title: for-a-positive-integer-n-what-is-the-value-of-i-4-n-1
slug: for-a-positive-integer-n-what-is-the-value-of-i-4-n-1-38237
question: <div class="question">For a positive integer $n$, what is the value of $i^{4 n+1}$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">i</span> </div>
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">D</span> <span class="option-data">-i</span> </li> </ul>

answer:
<div class="solution">$i^{4 n+1}=\left(i^{4}\right)^{n} \times i=(1)^{n} \times i=1 \times i=i$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">50</span> </div>
source_db: MarksBatch1_P2.db
title:
for-a-reaction-a-b-p-the-following-data-are-provided-the-rate-constant-for-this-reaction-in-standard-unit-is-1

slug:
for-a-reaction-a-b-p-the-following-data-are-provided-the-rate-constant-for-this-reaction-in-standard-unit-is-1-37292

question:
<div class="question">For a reaction $A+B \longrightarrow P$, the following data are provided<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/-ovVcYOc1GCvzNRgUnddZh-nitn4ZCctRVPh2GLMQT4.original.fullsize.png"/><br/><br/>The rate constant for this reaction in standard unit is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.4 \times 10^{-4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">50</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">(d) $A+B \longrightarrow P$<br/>According to rate law expression<br/>$$<br/>\text { Rate }=k[A]^\alpha[B]^\beta(K \text { is constant }) \ldots (i)<br/>$$<br/>Taking ratio of (1) and (2) entry,<br/>$$<br/>\begin{aligned}<br/>\frac{4 \times 10^{-2}}{2 \times 10^{-2}} &amp; =\frac{[0.04]^\beta[0.02]^\alpha}{[0.02]^\beta[0.02]^\alpha} \\<br/>2 &amp; =[2]^\beta \\<br/>{[2]^1 } &amp; =[2]^\beta \\<br/>\beta &amp; =1<br/>\end{aligned}<br/>$$<br/>Taking ratio of (2) and (3) entry,<br/>$$<br/>\begin{aligned}<br/>\frac{8 \times 10^{-2}}{4 \times 10^{-2}} &amp; =\frac{[0.04]^\alpha[0.04]^\beta}{[0.02]^\alpha[0.04]^\beta} \\<br/>{[2] } &amp; =[2]^\alpha \Rightarrow \alpha=1<br/>\end{aligned}<br/>$$<br/>Put value of $\alpha$ and $\beta$ in (1) entry<br/>$$<br/>\begin{aligned}<br/>2 \times 10^{-2} &amp; =k[A]^1[B]^1 \\<br/>k &amp; =\frac{2 \times 10^{-2}}{0.02 \times 0.02} \\<br/>&amp; =\frac{2 \times 10^{-2} \times 10^4}{4}=\frac{100}{2}=50<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'TS EAMCET', 'TS EAMCET 2021 (06 Aug Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">50</span> </div>
source_db: MarksBatch1_P2.db
title:
for-a-reaction-a-b-p-the-following-data-are-provided-the-rate-constant-for-this-reaction-in-standard-unit-is

slug:
for-a-reaction-a-b-p-the-following-data-are-provided-the-rate-constant-for-this-reaction-in-standard-unit-is-34169

question:
<div class="question">For a reaction $A+B \longrightarrow P$, the following data are provided<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/-ovVcYOc1GCvzNRgUnddZh-nitn4ZCctRVPh2GLMQT4.original.fullsize.png"/><br/><br/>The rate constant for this reaction in standard unit is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.4 \times 10^{-4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">50</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">(d) $A+B \longrightarrow P$<br/>According to rate law expression<br/>$$<br/>\text { Rate }=k[A]^\alpha[B]^\beta(K \text { is constant }) \ldots (i)<br/>$$<br/>Taking ratio of (1) and (2) entry,<br/>$$<br/>\begin{aligned}<br/>\frac{4 \times 10^{-2}}{2 \times 10^{-2}} &amp; =\frac{[0.04]^\beta[0.02]^\alpha}{[0.02]^\beta[0.02]^\alpha} \\<br/>2 &amp; =[2]^\beta \\<br/>{[2]^1 } &amp; =[2]^\beta \\<br/>\beta &amp; =1<br/>\end{aligned}<br/>$$<br/>Taking ratio of (2) and (3) entry,<br/>$$<br/>\begin{aligned}<br/>\frac{8 \times 10^{-2}}{4 \times 10^{-2}} &amp; =\frac{[0.04]^\alpha[0.04]^\beta}{[0.02]^\alpha[0.04]^\beta} \\<br/>{[2] } &amp; =[2]^\alpha \Rightarrow \alpha=1<br/>\end{aligned}<br/>$$<br/>Put value of $\alpha$ and $\beta$ in (1) entry<br/>$$<br/>\begin{aligned}<br/>2 \times 10^{-2} &amp; =k[A]^1[B]^1 \\<br/>k &amp; =\frac{2 \times 10^{-2}}{0.02 \times 0.02} \\<br/>&amp; =\frac{2 \times 10^{-2} \times 10^4}{4}=\frac{100}{2}=50<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'COMEDK', 'COMEDK 2018']
source_db: MarksBatch1_P2.db
title:
for-a-reversible-reaction-x-g-3-y-g-2-z-g-delta-h40-mathrmkj-t-h-es-t-an-d-a-r-d-e-n-t-ro-p-i-eso-f-x-y-an-d-z-a-re-60-40-an-d-50-mathrmjk1-mathrmmol1

slug:
for-a-reversible-reaction-x-g-3-y-g-2-z-g-delta-h40-mathrmkj-t-h-es-t-an-d-a-r-d-e-n-t-ro-p-i-eso-f-x-y-an-d-z-a-re-60-40-an-d-50-mathrmjk1-mathrmmol1-76413

question:
<div class="question">For a reversible reaction,<br/>$$<br/>X(g)+3 Y(g) \rightleftharpoons 2 Z(g)<br/>$$<br/>$\Delta H=-40 \mathrm{~kJ}$ the standard entropies of $X, Y$ and $Z$ are 60,40 and $50 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ respectively.<br/>The temperature at which the above reaction attains equilibrium is about</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$400 \mathrm{~K}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$500 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$273 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$373 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$500 \mathrm{~K}$</span> </div>

answer:
<div class="solution">$\Delta H=-40 \mathrm{~kJ}=-40 \times 10^{3} \mathrm{~J}$<br/>$\begin{aligned} \Delta S &amp;=\Delta S_{\text {Product }}-\Delta S_{\text {Reactant }} \\ &amp;=2(50)-3(40)-60 \\ &amp;=100-120-60=-80 \\ \Delta G &amp;=\Delta H-T \Delta S \\ 0 &amp;=-40 \times 10^{3}-T(-80) \\ T &amp;=\frac{40 \times 10^{3}}{80}=500 \mathrm{~K} \end{aligned}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
for-a-set-a-consider-the-following-statements-1-a-p-a-p-a-2-a-p-a-a-3-p-a-a-p-a-where-p-denotes-power-set-which-of-the-statements-given-above-isare-co

slug:
for-a-set-a-consider-the-following-statements-1-a-p-a-p-a-2-a-p-a-a-3-p-a-a-p-a-where-p-denotes-power-set-which-of-the-statements-given-above-isare-co-60226

question:
<div class="question">For a set $\mathrm{A}$, consider the following statements: <br/>$\begin{array}{lll}\text { 1. } \mathrm{A} \cup \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{A}) &amp; \text { 2. } &amp; \{\mathrm{A}\} \cap \mathrm{P}(\mathrm{A})=\mathrm{A}\end{array}$<br/>3.$\mathrm{P}(\mathrm{A})-\{\mathrm{A}\}=\mathrm{P}(\mathrm{A})$<br/>where P denotes power set. <br/>Which of the statements given above is/are correct?</div>

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

answer:
<div class="solution">Since, Power set is the collection of all the subsets of the set A therefore $\mathrm{AUP}(\mathrm{A})=\mathrm{P}(\mathrm{A})$<br/>$\therefore$ statement (1) iscorrect.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
for-a-set-of-discrete-numbers-three-measures-of-central-tendency-are-given-below-1-arithmetic-mean-2-median-3-geometric-mear-which-of-the-above-measur

slug:
for-a-set-of-discrete-numbers-three-measures-of-central-tendency-are-given-below-1-arithmetic-mean-2-median-3-geometric-mear-which-of-the-above-measur-38519

question:
<div class="question">For a set of discrete numbers, three measures of central $\begin{array}{ll}\text { tendency are given below } &amp; {}\end{array}$<br/> 1 . Arithmetic mean 2 Median<br/> 3 . Geometric mear Which of the above measures may not have a meaningful definition?</div>

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

answer:
<div class="solution">(i) Arithmetic mean $=\frac{\text { Sum of all observations }}{\text { Total no. of obseryation }}$<br/>(ii) Median = The midpoint of the data after being ranked (arranged in ascending order)<br/>(iii) Geometric mean $=$ If $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are $n$ values of $a$ variate $x$, none of them being zero, then the geometric mean $G$ is defined as $G=\left(x_{1} x_{2} x_{3} \ldots x_{n}\right)^{x}$ Thuse, all of them are meaningfully defined.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
for-a-square-matrix-a-which-of-the-following-properties-hold-1-a-1-1-a-2-det-a-1-det-a-1-3-a-1-a-1-where-is-a-scalar-select-the-correct-answer-using-t

slug:
for-a-square-matrix-a-which-of-the-following-properties-hold-1-a-1-1-a-2-det-a-1-det-a-1-3-a-1-a-1-where-is-a-scalar-select-the-correct-answer-using-t-55784

question:
<div class="question">For a square matrix $\mathrm{A}$, which of the following properties hold? <br/>$1.$ $\left(\mathrm{A}^{-1}\right)^{-1}=\mathrm{A}$<br/>$2 \quad \operatorname{det}\left(\mathrm{A}^{-1}\right)=\frac{1}{\operatorname{det} \mathrm{A}}$<br/>$3.$ $(\lambda \mathrm{A})^{-1} \lambda \mathrm{A}^{-1}$ where $\lambda$ is a scalar Select the correct answer using the code given below:</div>

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

answer:
<div class="solution">Statement 1 and 2 are correct. Statement 3 is incorrect because<br/>$(\lambda \mathrm{A})^{-1}=\frac{1}{\lambda} \mathrm{A}^{-1}, \lambda \neq 0$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2022 (06 Aug Shift 2)']
source_db: MarksBatch1_P2.db
title:
for-a-stationary-wave-y-10-sin-15-x-cos-48-t-cm-the-distance-between-a-node-and-the-successive-antinode-is

slug:
for-a-stationary-wave-y-10-sin-15-x-cos-48-t-cm-the-distance-between-a-node-and-the-successive-antinode-is-56462

question:
<div class="question">For a stationary wave, $Y=10 \sin \left(\frac{\pi x}{15}\right) \cos (48 \pi t) \mathrm{cm}$, the distance between a node and the successive antinode is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$7.5 \mathrm{~cm}$</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">$30 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$60 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7.5 \mathrm{~cm}$</span> </div>

answer:
<div class="solution">Comparing given equation with standard equation $y=2 a \sin \left(\frac{2 \pi x}{\lambda}\right) \cos (2 \pi f t)$<br/>where, $a$ is the amplitude, $k=\frac{2 \pi}{\lambda}$ the propagation constant and $f$ the frequency.<br/>Therefore, we obtain: $\frac{2 \pi}{\lambda}=\frac{\pi}{15} \mathrm{~cm}^{-1}$<br/>$\Rightarrow \lambda=30 \mathrm{~cm}$<br/>Distance between nearest node and antinode $=\frac{\lambda}{4}=\frac{30}{4}=7.5 \mathrm{~cm}$</div>


tags: ['Physics', 'Alternating Current', 'KCET', 'KCET 2013']
source_db: MarksBatch1_P2.db
title:
for-a-transformed-the-turns-ratio-is-3-and-its-efficiency-is-075-the-current-flowing-in-the-primary-coil-is-2-a-and-the-voltage-applied-to-it-is-100-v

slug:
for-a-transformed-the-turns-ratio-is-3-and-its-efficiency-is-075-the-current-flowing-in-the-primary-coil-is-2-a-and-the-voltage-applied-to-it-is-100-v-34926

question:
<div class="question">For a transformed, the turns ratio is 3 and its efficiency is $0.75$. The current flowing in the primary coil is $2 \mathrm{~A}$ and the voltage applied to it is $100 \mathrm{~V}$. Then the voltage and the current flowing in the secondary coil are ... respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$150 \mathrm{~V}, 1.5 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$300 \mathrm{~V}, 0.5 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$300 \mathrm{~V}, 1.5 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$150 \mathrm{~V}, 0.5 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$300 \mathrm{~V}, 0.5 \mathrm{~A}$</span> </div>

answer:
<div class="solution">Given, $\frac{n_{s}}{n_{P}}=3 ; \quad \eta=0.75 ; I_{P}=2 \mathrm{~A}$ and $V_{P}=100 \mathrm{~V}$<br/>We know that,<br/>$\begin{aligned}<br/>&amp;\qquad \eta=\frac{V_{S}}{V_{P}} \frac{I_{S}}{I_{P}} \\<br/>&amp;\text { or } \quad \eta=\left(\frac{n_{s}}{n_{P}}\right) \frac{I_{s}}{I_{P}} \quad\left(\because \frac{n_{s}}{n_{p}}=\frac{V_{s}}{V_{p}}\right) \\<br/>&amp;\Rightarrow \quad 0.75=\frac{3 \times I_{s}}{2} \\<br/>&amp;\Rightarrow \quad I_{s}=0.5 \mathrm{~A} \\<br/>&amp;\text { By the relation, } \frac{n_{s}}{n_{P}}=\frac{V_{s}}{V_{P}} \\<br/>&amp;\Rightarrow \quad 3=\frac{V_{s}}{100} \Rightarrow \quad V_{s}=300 \mathrm{~V}<br/>\end{aligned}$</div>


title: for-all-n-n-2-4-n-15-n-1-is-divisible-by
slug: for-all-n-n-2-4-n-15-n-1-is-divisible-by-76457
question: <div class="question">For all $n \in N, 2^{4 n}-15 n-1$ is divisible by</div>
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">225</span> </div>
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">125</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">225</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">450</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{P}(n): 2^{4 n}-15 n-1$<br/>Put $n=2$ $\mathrm{P}(2)=2^{8}-30-1=225$ which is divisible by $225 .$ Let us assume,<br/>$\mathrm{P}(n)$ is true for $n=k$ is $\mathrm{P}(\mathrm{k}): 2^{4 \mathrm{k}}-15 \mathrm{k}-1$ is divisible by<br/>$225 .$<br/>$\Rightarrow 2^{4 k}-15 k-1=225 \lambda, \lambda \in R, k \in N$<br/>To prove for $n=k+1$<br/>Consider<br/>$2^{4 k+4}-15 k-15-1=2^{4 k} \cdot 2^{4}-15 k-16$<br/>$=2^{4}\left[\begin{array}{ll}225 \lambda+1+15 k]-15 k-16 &amp; (\text { from }(i))\end{array}\right.$<br/>$=2^{4} \cdot 225 \lambda \cdot 2^{4}+15 \cdot 2^{4} \cdot k-15 k-16$<br/>$=24 \cdot 225 \lambda+225 k$<br/>$=225\left[2^{4} \lambda+k\right]$<br/>$=225 r$ where $r=2^{4} \lambda+k$ is a constant<br/>Hence, $2^{4 n}-15 n-1$ is divisible by 225.</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
for-an-ap-with-first-term-u-and-common-difference-v-the-p-th-term-is-15-uv-more-than-the-q-th-term-which-one-of-the-following-is-correct

slug:
for-an-ap-with-first-term-u-and-common-difference-v-the-p-th-term-is-15-uv-more-than-the-q-th-term-which-one-of-the-following-is-correct-22279

question:
<div class="question">For an AP with first term u and common difference $v$, the $\mathrm{p}^{\text {th }}$ term is 15 uv more than the $\mathrm{q}^{\text {th }}$ term. Which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{p}=\mathrm{q}+15 \mathrm{v}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}=\mathrm{q}+15 \mathrm{u}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{p}=\mathrm{q}+14 \mathrm{v}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{p}=\mathrm{q}+14 \mathrm{u}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{p}=\mathrm{q}+15 \mathrm{u}$</span> </div>

answer:
<div class="solution">Since, first term and common difference of an $\mathrm{AP}$ are u and $\mathrm{v}$ respectively. $\mathrm{p}^{\text {th }}$ term, $\mathrm{Tp}=\mathrm{u}+(\mathrm{p}-1) \mathrm{v}$<br/>and $q^{\text {th }}$ term, $\mathrm{T}_{\mathrm{q}}=\mathrm{u}+(\mathrm{q}-1) \mathrm{v}$ According to condition given in question, ...(ii)<br/>$\Rightarrow \quad \mathrm{T}_{\mathrm{p}}=\mathrm{T}_{\mathrm{q}}+15 \mathrm{uv}$<br/>$\Rightarrow \quad T_{n}^{r}-T_{0}^{4}=15 \mathrm{uv}$<br/>$\Rightarrow \quad \mathrm{u}^{\mathrm{p}}+(\mathrm{p}-1) \mathrm{v}-\mathrm{u}-(\mathrm{q}-1) \mathrm{v}=15 \mathrm{uv}$<br/>$\Rightarrow \mathrm{v}(\mathrm{p}-1-\mathrm{q}+1)=15 \mathrm{uv}$<br/>$\Rightarrow \mathrm{v}(\mathrm{p}-\mathrm{q})=15 \mathrm{uv}$<br/>$\Rightarrow \mathrm{p}-\mathrm{q}=15 \mathrm{u} \Rightarrow \mathrm{p}=\mathrm{q}+15 \mathrm{u}$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
for-an-ideal-gas-the-instantaneous-change-in-pressure-p-with-volume-v-is-given-by-the-equation-d-p-d-v-a-p-if-p-p-0-at-v-0-is-the-given-boundary-condi-1

slug:
for-an-ideal-gas-the-instantaneous-change-in-pressure-p-with-volume-v-is-given-by-the-equation-d-p-d-v-a-p-if-p-p-0-at-v-0-is-the-given-boundary-condi-1-40894

question:
<div class="question">For an ideal gas the instantaneous change in pressure <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> with volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> is given by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>P</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>a</mi><mi>P</mi><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>0</mn></math> is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is the gas constant)</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"><mfrac><mrow><mi>a</mi><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>e</mi><mi>R</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">infinity</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>°</mo><mi mathvariant="normal">C</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>a</mi><mi>e</mi><mi>R</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>a</mi><mi>e</mi><mi>R</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>a</mi><mi>P</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><msub><mi>P</mi><mn>0</mn></msub><mi>P</mi></msubsup><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mi>P</mi></mfrac><mo>=</mo><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mi>v</mi></msubsup><mi>a</mi><mi>d</mi><mi>V</mi><mo>⇒</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mo>=</mo><mo>-</mo><mi>a</mi><mi>V</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mi>a</mi><mi>V</mi></mrow></msup></math><br/>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mi>R</mi><mi>T</mi></math>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math>)<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>+</mo><mi>V</mi><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mi>R</mi><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac></math><br/>[ for maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>,</mo><mo> </mo><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>]<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>-</mo><mi>a</mi><mi>P</mi><mi>V</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac></math><br/>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mrow><mi>P</mi><mi>V</mi></mrow><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>R</mi><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>e</mi><mi>R</mi><mi>a</mi></mrow></mfrac></math></div>


tags: ['Physics', 'Kinetic Theory of Gases', 'JEE Main', 'JEE Main 2021 (31 Aug Shift 1)']
source_db: MarksBatch1_P2.db
title:
for-an-ideal-gas-the-instantaneous-change-in-pressure-p-with-volume-v-is-given-by-the-equation-d-p-d-v-a-p-if-p-p-0-at-v-0-is-the-given-boundary-condi

slug:
for-an-ideal-gas-the-instantaneous-change-in-pressure-p-with-volume-v-is-given-by-the-equation-d-p-d-v-a-p-if-p-p-0-at-v-0-is-the-given-boundary-condi-78781

question:
<div class="question">For an ideal gas the instantaneous change in pressure <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> with volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> is given by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>P</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>a</mi><mi>P</mi><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>0</mn></math> is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is the gas constant)</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"><mfrac><mrow><mi>a</mi><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>e</mi><mi>R</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">infinity</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>°</mo><mi mathvariant="normal">C</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>a</mi><mi>e</mi><mi>R</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>a</mi><mi>e</mi><mi>R</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>a</mi><mi>P</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><msub><mi>P</mi><mn>0</mn></msub><mi>P</mi></msubsup><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mi>P</mi></mfrac><mo>=</mo><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mi>v</mi></msubsup><mi>a</mi><mi>d</mi><mi>V</mi><mo>⇒</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mo>=</mo><mo>-</mo><mi>a</mi><mi>V</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mi>a</mi><mi>V</mi></mrow></msup></math><br/>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mi>R</mi><mi>T</mi></math>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math>)<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>+</mo><mi>V</mi><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mi>R</mi><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac></math><br/>[ for maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>,</mo><mo> </mo><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>V</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>]<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>-</mo><mi>a</mi><mi>P</mi><mi>V</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac></math><br/>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mrow><mi>P</mi><mi>V</mi></mrow><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>R</mi><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>e</mi><mi>R</mi><mi>a</mi></mrow></mfrac></math></div>


tags: ['Mathematics', 'Determinants', 'AP EAMCET', 'AP EAMCET 2020 (21 Sep Shift 2)']
source_db: MarksBatch1_P2.db
title:
for-any-a-b-c-in-mathbfr-the-determinant-leftbeginarraylllb-c-bc-1-c-a-ca-1-a-b-ab-1endarrayright-is-equal-to

slug:
for-any-a-b-c-in-mathbfr-the-determinant-leftbeginarraylllb-c-bc-1-c-a-ca-1-a-b-ab-1endarrayright-is-equal-to-38594

question:
<div class="question">For any \(a, b, c \in \mathbf{R}\), the determinant \(\left|\begin{array}{lll}b c &amp; b+c &amp; 1 \\ c a &amp; c+a &amp; 1 \\ a b &amp; a+b &amp; 1\end{array}\right|\) is equal to</div>

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

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

answer:
<div class="solution">For any \(a, b, c \in \mathbf{R}\), the given determinant<br/>\(\Delta=\left|\begin{array}{lll}<br/>b c &amp; b+c &amp; 1 \\<br/>c a &amp; c+a &amp; 1 \\<br/>a b &amp; a+b &amp; 1<br/>\end{array}\right|\)<br/>On applying \(R_2 \rightarrow R_2-R_1\) and \(R_3 \rightarrow R_3-R_1\), we have<br/>\(\begin{gathered}<br/>\Delta=\left|\begin{array}{ccc}<br/>b c &amp; b+c &amp; 1 \\<br/>c(a-b) &amp; a-b &amp; 0 \\<br/>b(a-c) &amp; a-c &amp; 0<br/>\end{array}\right| \\<br/>=(a-b)(a-c)\left|\begin{array}{ccc}<br/>b c &amp; b+c &amp; 1 \\<br/>c &amp; 1 &amp; 0 \\<br/>b &amp; 1 &amp; 0<br/>\end{array}\right|<br/>\end{gathered}\)<br/>On applying \(R_3 \rightarrow R_3-R_2\), we have<br/>\(\begin{aligned}<br/>\Delta &amp; =(a-b)(a-c)\left|\begin{array}{ccc}<br/>b c &amp; b+c &amp; 1 \\<br/>c &amp; 1 &amp; 0 \\<br/>b-c &amp; 0 &amp; 0<br/>\end{array}\right| \\<br/>&amp; =(a-b)(a-c)[0-(b-c)]=(a-b)(b-c)(c-a)<br/>\end{aligned}\)<br/>Hence option (c) is correct.</div>


title: for-any-positive-integer-n-if-4-n-3-n-is-divided-by-9-then-what-is-the-remainder
slug: for-any-positive-integer-n-if-4-n-3-n-is-divided-by-9-then-what-is-the-remainder-73146
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch1_P2.db
question:
<div class="question">For any positive integer $\mathrm{n}$, if $4^{\mathrm{n}}-3 \mathrm{n}$ is divided by 9 , then what is the remainder?</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">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Using binomial theorem. $4^{\mathrm{n}}-3 \mathrm{n}=(1+3)^{\mathrm{n}}-3 \mathrm{n}$<br/>$=1+\mathrm{n} .3+\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} 3^{2}+\ldots-3 \mathrm{n}$<br/>$=1+\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \cdot 3^{2}+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{3 !} \cdot 3^{3}+\ldots$<br/>$\Rightarrow 4^{\mathrm{n}}-3 \mathrm{n}=9\left\{\frac{\mathrm{n}(\mathrm{n}-1)}{2 !}+\ldots\right\}+1$<br/>Thus, when $4^{\mathrm{n}}-3 \mathrm{n}$ is divided by 9 , the remainder is 1</div>


title: for-any-two-events-a-and-b-which-one-of-the-following-holds
slug: for-any-two-events-a-and-b-which-one-of-the-following-holds-84895
question: <div class="question">For any two events $A$ and $B$, which one of the following holds ?</div>
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{P}(\mathrm{A} \cap \mathrm{B}) \leq \mathrm{P}(\mathrm{A}) \leq \mathrm{P}(\mathrm{A} \cup \mathrm{B}) \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{P}(\mathrm{A} \cup \mathrm{B}) \leq \mathrm{P}(\mathrm{A}) \leq \mathrm{P}(\mathrm{A} \cap \mathrm{B}) \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{P}(\mathrm{A} \cup \mathrm{B}) \leq \mathrm{P}(\mathrm{B}) \leq \mathrm{P}(\mathrm{A} \cap \mathrm{B}) \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{P}(\mathrm{A} \cap \mathrm{B}) \leq \mathrm{P}(\mathrm{B}) \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \leq \mathrm{P}(\mathrm{A} \cup \mathrm{B})$</span> </li> </ul>

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

answer:
<div class="solution">Clearly, $\mathrm{A} \cap \mathrm{B} \subseteq \mathrm{A}$<br/>$\Rightarrow P(A \cap B) \leq \bar{P}(A)$<br/>$\mathrm{A} \subset \mathrm{A} \cup \mathrm{B}$<br/>$\Rightarrow \mathrm{P}(\mathrm{A}) \leq \mathrm{P}(\mathrm{A} \cup \mathrm{B})$<br/>We know that, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B}) \leq \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$<br/>From eqs. (i), (ii) and (iii), we get $P(A \cap B) \leq P(A) \leq P(A \cup B) \leq P(A)+P(B)$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
for-any-two-vectors-a-and-b-consider-the-following-statement-1-a-b-a-b-a-b-are-orthogonal-2-a-b-a-b-a-b-are-orthogonal-3-a-b-2-a-2-b-2-a-b-are-orthogo

slug:
for-any-two-vectors-a-and-b-consider-the-following-statement-1-a-b-a-b-a-b-are-orthogonal-2-a-b-a-b-a-b-are-orthogonal-3-a-b-2-a-2-b-2-a-b-are-orthogo-16761

question:
<div class="question">For any two vectors $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ consider the following<br/>statement:<br/>$1.$ $\quad|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}| \Leftrightarrow \overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ are orthogonal.<br/>$2.$ $\quad|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}}|+|\overrightarrow{\mathbf{b}}| \Leftrightarrow \overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ are orthogonal.<br/>$3.$ $\quad|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2} \Leftrightarrow \overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ are orthogonal.<br/>Which of the above statements is/are correct?</div>

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

answer:
<div class="solution">(i) $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$<br/>Squaring both the sides<br/>$|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}$<br/>or, $|\vec{a}|^{2}+|\vec{b}|^{2}+2 \vec{a} \cdot \vec{b}=|\vec{a}|^{2}+|\vec{b}|^{2}-2 \vec{a} \vec{b}$<br/>or, $4 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$<br/>$\Rightarrow \vec{a}$ and $\vec{b}$ are orthogonal.<br/>(ii) $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}|$<br/>Squaring both the sides<br/>$|\vec{a}+\vec{b}|=(|\vec{a}|+|\vec{b}|)^{2}$<br/>or, $|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+2|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|$<br/>or, $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}| \cdot|\overrightarrow{\mathrm{b}}|$<br/>or, $|\overrightarrow{\mathrm{a}}| \cdot|\overrightarrow{\mathrm{b}}| \cdot \cos \theta=|\overrightarrow{\mathrm{a}} \| \overrightarrow{\mathrm{b}}|$<br/>$\cos \theta=1 \Rightarrow \theta=0$<br/>$\vec{a}$ and $\vec{b}$ are parallel, and not orthogonal.<br/>(iii) $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}$<br/>$|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}$<br/>$2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$<br/>$\Rightarrow \vec{a}$ and $\overrightarrow{\mathrm{b}}$ are orthogonal. Statements 1 and 3 are correct.</div>


title: for-any-vector-a-a-i-2-a-j-2-a-k-2-is-equal-to
slug: for-any-vector-a-a-i-2-a-j-2-a-k-2-is-equal-to-40737
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch1_P2.db
question:
<div class="question">For any vector $\overrightarrow{\mathrm{a}}|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}|^{2}+|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}|^{2}+|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}|^{2}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$|\vec{a}|^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2|\overrightarrow{\mathrm{a}}|^{2}$</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">C</span> <span class="option-data">$3|\overrightarrow{\mathrm{a}}|^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4|\overrightarrow{\mathrm{a}}|^{2}$</span> </li> </ul>

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

answer:
<div class="solution">Let $\overrightarrow{\mathrm{a}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$<br/>$|\vec{a} \times \hat{i}|^{2}=y^{2}+z^{2}$<br/>$|\vec{a} \times \hat{j}|^{2}=x^{2}+z^{2}$<br/>$|\vec{a} \times \hat{k}|^{2}=x^{2}+y^{2}$<br/>$\therefore|\vec{a} \times \hat{i}|^{2}+|\vec{a} \times j|^{2}+|\vec{a} \times \hat{k}|^{2}$<br/>$=y^{2}+z^{2}+x^{2}+z^{2}+x^{2}+y^{2}$<br/>$=2 x^{2}+2 y^{2}+2 z^{2}$<br/>$=2\left(x^{2}+y^{2}+z^{2}\right)$<br/>$=2|\vec{a}|^{2}$</div>


title: for-any-vector-what-is-i-i-j-j-k-k-equal-to
slug: for-any-vector-what-is-i-i-j-j-k-k-equal-to-60968
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch1_P2.db
question:
<div class="question">For any vector $\vec{\alpha}$, what is $(\vec{\alpha} \cdot \hat{i}) \hat{i}+(\vec{\alpha} \cdot \hat{j}) \hat{j}+(\vec{\alpha} \cdot \hat{k}) \hat{k}$ equal<br/>to ?</div>

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

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

answer:
<div class="solution">Let $\vec{\alpha}=\vec{a} \hat{i}+\vec{b} \hat{j}+\vec{c} \hat{k}$<br/>Now, $\vec{\alpha} \cdot \hat{\mathrm{i}}=(\overrightarrow{\mathrm{a}} \hat{\mathrm{i}}+\overrightarrow{\mathrm{b}} \hat{\mathrm{j}}+\overrightarrow{\mathrm{c}} \hat{\mathrm{k}}) \cdot \hat{\mathrm{i}}=\overrightarrow{\mathrm{a}}$<br/>$\vec{\alpha} \cdot \hat{j}=(\vec{a} \hat{i}+\vec{b} \hat{j}+\vec{c} \hat{k}) \cdot \hat{j}=\vec{b}$<br/>$\vec{\alpha} \cdot \hat{\mathrm{k}}=(\overrightarrow{\mathrm{a}} \hat{\mathrm{i}}+\overrightarrow{\mathrm{b}} \hat{\mathrm{j}}+\overrightarrow{\mathrm{c}} \hat{\mathrm{k}}) \cdot \hat{\mathrm{k}}=\overrightarrow{\mathrm{c}}$<br/>Now, $\vec{a} \hat{i}+b \hat{j}+c \hat{k}=\vec{\alpha}$<br/>Thus, Required expression $=\vec{\alpha}$.</div>


title: for-being-functional-a-protein-has-to-acquire-at-least
slug: for-being-functional-a-protein-has-to-acquire-at-least-42075
question: <div class="question">For being functional, a protein has to acquire at least</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Primary structure</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Secondary structure</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Tertiary structure</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Quaternary structure.</span> </li> </ul>

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

answer:
<div class="solution">Most of the proteins are highly functional in their tertiary structure. The long chain of peptides is folded upon itself to form a hollow ball-like structure. The disulfide linkages that are present in the proteins give the domain structure and that enables their functional activity.<br/><br/> </div>


tags: ['Mathematics', 'Circle', 'AP EAMCET', 'AP EAMCET 2022 (07 Jul Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1$</span> </div>
source_db: MarksBatch1_P2.db
title:
for-different-real-nonzero-numbers-x-1-x-2-x-3-and-x-4-suppose-the-points-x-1-x-1-1-x-2-x-2-1-x-3-x-3-1-and-x-4-x-4-1-lie-on-the-boundary-of-a-circle-

slug:
for-different-real-nonzero-numbers-x-1-x-2-x-3-and-x-4-suppose-the-points-x-1-x-1-1-x-2-x-2-1-x-3-x-3-1-and-x-4-x-4-1-lie-on-the-boundary-of-a-circle-31053

question:
<div class="question">For different real non-zero numbers $x_1, x_2, x_3$ and $x_4$, suppose the points $\left(x_1, \frac{1}{x_1}\right),\left(x_2, \frac{1}{x_2}\right),\left(x_3, \frac{1}{x_3}\right)$ and $\left(x_4, \frac{1}{x_4}\right)$ lie on the boundary of a circle of radius 4 . Then, the value of $x_1 x_2 x_3 x_4$ is</div>

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

answer:
<div class="solution">Let the equation of the circle be $x^2+y^2+2 g x+2 f y+c=0$<br/>$\left(x_i, \frac{1}{x_i}\right)$ lies on the above circle.<br/>$\therefore \quad x_i^2+\frac{1}{x_i^2}+2 g x_i+\frac{2 f}{x_i}+C=0$<br/>$\begin{aligned} &amp; \Rightarrow \quad x_i^4+2 g x_i^3+c x_i^2+2 f x_i+1=0 \\ &amp; \therefore \quad x_1 x_2 x_3 x_4=1\end{aligned}$</div>


title: for-each-nonzeroreal-number-x-let-f-x-x-x-the-range-of-f-is
slug: for-each-nonzeroreal-number-x-let-f-x-x-x-the-range-of-f-is-93733
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
question:
<div class="question">For each non-zeroreal number $\mathrm{x}$, let $f(x)=\frac{x}{|x|}$. The range of $f$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a set consisting of two elements</span> </div>

answer:
<div class="solution">For a non-zero real number $\mathrm{X}$<br/>$f(x)=\frac{x}{|x|}$<br/>domain $f(x)=\frac{x}{|x|}$ range<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/P2Arc6g5srt7XDZhiSVoJZIj0OwtjyUERTloRzjt3R4.original.fullsize.png"/><br/><br/>for any value of $\mathrm{x}$ in domain a set consisting of two dementsi.e. $(-1,+1)$ $\therefore$ Option (c) is correct.</div>


title: if-1-tan-1-tan-2-then-what-is-equal-to
slug: if-1-tan-1-tan-2-then-what-is-equal-to-17285
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $(1+\tan \theta)(1+\operatorname{tan\phi})=2$, then what is $(\theta+\phi)$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">45^{\circ}</span> </div>

answer:
<div class="solution">Given, $(1+\tan \theta)(1+\tan \phi)=2$<br/>$\Rightarrow 1+\tan \theta+\tan \phi+\tan \theta \tan \phi=2$<br/>$\Rightarrow \tan \theta+\tan \phi=1-\tan \theta \tan \phi$<br/>$\Rightarrow \frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}=1$<br/>$\Rightarrow \tan (\theta+\phi)=\tan 45^{\circ}$<br/>$\Rightarrow \quad \theta+\phi=45^{\circ}$</div>


title: if-1-x-2-1-y-2-a-then-what-is-d-x-d-y-equal-to
slug: if-1-x-2-1-y-2-a-then-what-is-d-x-d-y-equal-to-91572
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a$, then what is $\frac{d y}{d x}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{\left(1-x^{2}\right)\left(1-y^{2}\right)}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{\frac{1-y^{2}}{1-x^{2}}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{\frac{1-x^{2}}{1-y^{2}}}$</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">Let $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a$<br/>On differentiating w.r.t. $x$, we get<br/>$\frac{1}{2 \sqrt{1-x^{2}}}(-2 x)+\frac{1(-2 y)}{2 \sqrt{1-y^{2}}} \frac{d y}{d x}=0$<br/>$\Rightarrow \frac{-x}{\sqrt{1-x^{2}}}-\frac{y}{\sqrt{1-y^{2}}} \frac{d y}{d x}=0$<br/>$\Rightarrow \frac{-x}{\sqrt{1-x^{2}}}=\frac{y}{\sqrt{1-y^{2}}} \frac{d y}{d x}$<br/>$\Rightarrow \frac{d y}{d x}=-\frac{x}{y} \sqrt{\frac{1-y^{2}}{1-x^{2}}}$</div>


title: if-1-x-y-z-16-are-in-geometric-progression-then-what-is-the-value-of-x-y-z
slug: if-1-x-y-z-16-are-in-geometric-progression-then-what-is-the-value-of-x-y-z-57473
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $1, x, y, z, 16$ are in geometric progression, then what is the value of $x+y+z$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">14</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

answer:
<div class="solution">As given $1, \mathrm{x}, \mathrm{y}, \mathrm{z} 16$ are in geometric progression. Let common ratio be $\mathrm{r}$, $\mathrm{x}=1 . \mathrm{r}=\mathrm{r}$<br/>$\mathrm{y}=1 \cdot \mathrm{r}^{2}=\mathrm{r}^{2}$<br/>$\mathrm{z}=1 \cdot \mathrm{r}^{3}=\mathrm{r}^{3}$<br/>and $16=1 \cdot r^{4} \Rightarrow 16=r^{4}$<br/>$\Rightarrow \mathrm{r}=2$<br/>$\therefore \quad \mathrm{x}=1 . \mathrm{r}=2, \mathrm{y}=1 \cdot \mathrm{r}^{2}=4$<br/>$\mathrm{z}=1 \cdot \mathrm{r}^{3}=8$<br/>$\therefore \quad x+y+z=2+4+8=14$</div>


title: if-2-0-0-4-5-0-0-16-1-p-20-then-what-is-the-value-of-p
slug: if-2-0-0-4-5-0-0-16-1-p-20-then-what-is-the-value-of-p-50124
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{ccc}2 &amp; 4 &amp; 0 \\ 0 &amp; 5 &amp; 16 \\ 0 &amp; 0 &amp; 1+p\end{array}\right|=20$, then what is the value of $p$?</div>

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

answer:
<div class="solution">$\left|\begin{array}{ccc}2 &amp; 4 &amp; 0 \\ 0 &amp; 5 &amp; 16 \\ 0 &amp; 0 &amp; 1+p\end{array}\right|=20$<br/>On expanding along $\mathrm{C}_{1}$, $2\{5(1+p)-0\}=20$<br/>$\Rightarrow 1+p=2$<br/>$\Rightarrow p=1$</div>


title: if-2-0-is-the-vertex-and-the-y-axis-is-the-directrix-of-a-parabola-then-where-is-its-focus
slug: if-2-0-is-the-vertex-and-the-y-axis-is-the-directrix-of-a-parabola-then-where-is-its-focus-48649
tags: ['Mathematics', 'Parabola', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $(2,0)$ is the vertex and the $y$ -axis is the directrix of a parabola, then where is its focus?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(4,0)$</span> </div>

answer:
<div class="solution">Vertec is $(2,0)$. Since, $y$ -axis is the directrix of a parabola. Equation directrix is $x=0 .$ So, axis of parabolais $x$ -axis. Let the focus be $(\mathrm{a}, 0)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/NEqEp3kmuMMnywso_5i7kcmP_H9dewvKkZLyHdLBxTM.original.fullsize.png"/><br/><br/>Distance of the vertex of a parabola from directrix = its distance from focus<br/>So, $\mathrm{OV}=\mathrm{VF} \Rightarrow(2-0)^{2}=(\mathrm{a}-2)^{2}$<br/>$a^{2}=4 a b a=4$<br/>$\Rightarrow$ Focus is $(4,0)$</div>


title: if-2-1-i-3-then-what-is-the-value-of-1-8-16-24-32
slug: if-2-1-i-3-then-what-is-the-value-of-1-8-16-24-32-69434
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha=\frac{1+\mathrm{i} \sqrt{3}}{2}$, then what is the value of $1+\alpha^{8}+\alpha^{16}+$ $\alpha^{24}+\alpha^{32} ?$</div>

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

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

answer:
<div class="solution">$\frac{-1+\mathrm{i} \sqrt{3}}{2}$ and $\frac{-1-\mathrm{i} \sqrt{3}}{2}$<br/>are complex cube roots of unity $\omega$ and $\omega^{2}$<br/>$\begin{aligned} &amp; \alpha=\frac{1+\mathrm{i} \sqrt{3}}{2}=-\omega^{2} \\ \therefore \quad &amp; 1+\alpha^{8}+\alpha^{16}+\alpha^{24}+\alpha^{32} \\=&amp; 1+\omega^{16}+\omega^{32}+\omega^{48}+\omega^{64} \\=&amp; 1+\omega+\omega^{2}+1+\omega=0+1+\omega=-\omega^{2} \end{aligned}$</div>


title: if-2-4-3-1-5-3-2-1-1-17-1-then-what-is-equal-to
slug: if-2-4-3-1-5-3-2-1-1-17-1-then-what-is-equal-to-92806
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-7$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left(\begin{array}{ll}2 &amp; 3 \\ 4 &amp; 1\end{array}\right) \times\left(\begin{array}{cc}5 &amp; -2 \\ -3 &amp; 1\end{array}\right)=\left(\begin{array}{cc}1 &amp; -1 \\ 17 &amp; \lambda\end{array}\right)$, then what is $\lambda$ equal to?</div>

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

answer:
<div class="solution">$\left(\begin{array}{ll}2 &amp; 3 \\ 4 &amp; 1\end{array}\right) \times\left(\begin{array}{cc}5 &amp; -2 \\ -3 &amp; 1\end{array}\right)=\left(\begin{array}{cc}1 &amp; -1 \\ 17 &amp; \lambda\end{array}\right)$<br/>$\left(\begin{array}{cc}1 &amp; -1 \\ 17 &amp; -7\end{array}\right)=\left(\begin{array}{cc}1 &amp; -1 \\ 17 &amp; \lambda\end{array}\right)$<br/>$\lambda=-7$</div>


title: if-2-5-f-x-dx-4-and-0-5-1-f-x-dx-7-then-what-is-2-0-f-x-dx-equal-to
slug: if-2-5-f-x-dx-4-and-0-5-1-f-x-dx-7-then-what-is-2-0-f-x-dx-equal-to-43864
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2016 (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 $\int_{-2}^{5} \mathrm{f}(\mathrm{x}) \mathrm{dx}=4$ and $\int_{0}^{5}\{1+\mathrm{f}(\mathrm{x})\} \mathrm{dx}=7$, then what is<br/>$\int_{-2}^{0} \mathrm{f}(\mathrm{x}) \mathrm{dx}$ equal to?</div>

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

answer:
<div class="solution">$\int_{-2}^{5} f(x) d x=4$ and $\int_{0}^{5}\{1+f(x)\} d x=7$<br/>$\int_{0}^{5} f(x) d x=7-\int_{0}^{5} 1 d x=7-5=2$<br/>$\int_{-2}^{0} f(x) d x=?$<br/>$\int_{-2}^{0} f(x) d x+\int_{0}^{5} f(x) d x=4$<br/>$\int_{-2}^{0} f(x) d x=4-\int_{0}^{5} f(x) d x$<br/>$=4-2=2 . \quad[$ Using eqn. $(1)]$<br/>$\int_{-2}^{0} f(x) d x=2$</div>


title: if-2-a-2-3-1-2-then-what-is-a-1-equal-to
slug: if-2-a-2-3-1-2-then-what-is-a-1-equal-to-54880
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $2 A=\left(\begin{array}{ll}2 &amp; 1 \\ 3 &amp; 2\end{array}\right)$, then what is $A^{-1}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\begin{array}{cc}2 &amp; -1 \\ -3 &amp; 2\end{array}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}\left(\begin{array}{cc}2 &amp; -1 \\ -3 &amp; 2\end{array}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}\left(\begin{array}{cc}2 &amp; -1 \\ -3 &amp; 2\end{array}\right)$</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">Given, $2 \mathrm{~A}=\left[\begin{array}{ll}2 &amp; 1 \\ 3 &amp; 2\end{array}\right]$<br/>$\mathrm{A}=\left(\begin{array}{cc}1 &amp; \frac{1}{2} \\ \frac{3}{2} &amp; 1\end{array}\right)$<br/>$|\mathrm{A}|=\frac{1}{4}$<br/>$\operatorname{adj} \mathrm{A}=\left(\begin{array}{cc}1 &amp; -\frac{1}{2} \\ \frac{-3}{2} &amp; 1\end{array}\right)$<br/>$\mathrm{A}^{-1}=4\left(\begin{array}{cc}1 &amp; -\frac{1}{2} \\ \frac{-3}{2} &amp; 1\end{array}\right)=\left(\begin{array}{cc}4 &amp; -2 \\ -6 &amp; 4\end{array}\right)$</div>


title: if-2-and-then-which-one-of-the-following-is-correct
slug: if-2-and-then-which-one-of-the-following-is-correct-55550
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha+\beta=\frac{\pi}{2}$ and $\beta+\gamma=\alpha$; 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">$2 \tan \beta+\tan \gamma=\tan \alpha$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\tan \beta+2 \tan \gamma=\tan \alpha$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan \beta+2 \tan \gamma=\tan \alpha$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2(\tan \beta+\tan \gamma)=\tan \alpha$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan \beta+2 \tan \gamma=\tan \alpha$</span> </div>

answer:
<div class="solution">As given, $\alpha+\beta=\frac{\pi}{2}$ and $\beta+\gamma=\alpha$<br/>$\Rightarrow \tan (\beta+\gamma)=\tan \alpha$<br/>$\Rightarrow \frac{\tan \beta+\tan \gamma}{1-\tan \beta \tan \gamma}=\tan \alpha$<br/>$\Rightarrow \tan \beta+\tan \gamma=\tan \alpha-\tan \alpha \tan \beta \tan \gamma$<br/>$\Rightarrow \tan \beta+\tan \gamma=\tan \alpha-\tan \alpha \cot \alpha \tan \gamma$<br/>$\left(\because \beta+\alpha=\frac{\pi}{2} \Rightarrow \beta=\pi / 2-\alpha \Rightarrow \tan (\pi / 2-\alpha)=\cot \alpha\right)$<br/>$\Rightarrow \tan \beta+\tan \gamma=\tan \alpha-\tan \gamma$<br/>$\Rightarrow \tan \beta+2 \tan \gamma=\tan \alpha$</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6,-4,1</span> </div>
source_db: MarksBatch2_P1.db
title:
if-2-is-one-of-the-roots-of-a-x-2-b-x-c-0-where-a-b-c-are-real-then-what-are-the-values-of-a-b-c-respectively

slug:
if-2-is-one-of-the-roots-of-a-x-2-b-x-c-0-where-a-b-c-are-real-then-what-are-the-values-of-a-b-c-respectively-58568

question:
<div class="question">If $\overline{ }_{-2}$ is one of the roots of $a x^{2}+b x+c=0$, where $a$,<br/>$b, c$ are real, then what are the values of $a, b, c$ respectively?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">6,-4,1</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">4,6,-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3,-2,1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6,4,1$</span> </li> </ul>

answer:
<div class="solution">Given quadratic equation is $a x^{2}+b x+c=0$ whose<br/>one root is $\frac{1}{2-\sqrt{-2}}$<br/>Consider $\frac{1}{2-\sqrt{-2}}=\frac{1}{2-\sqrt{2} i} \times \frac{2+\sqrt{2} i}{2+\sqrt{2} i}$<br/>$=\frac{2+\sqrt{2} i}{4+2}=\frac{2+\sqrt{2} i}{6}$<br/>$\therefore$ Another root will be $\frac{2-\sqrt{2} i}{6}$<br/>$(\because$ complex roots always occurs in pairs)<br/>Thus, sum of roots $=\frac{2+\sqrt{2} i}{6}+\frac{2-2 \sqrt{2} i}{6}=\frac{4}{6}$<br/>and product of roots $=\left(\frac{2+\sqrt{2} i}{6}\right)\left(\frac{2-\sqrt{2} i}{6}\right)$<br/>$=\frac{4+2}{36}=\frac{1}{6}$<br/>$\therefore$ Requiredequation is $x^{2}-($ sum of roots $) x+($ product of roots $)=0$<br/>$x^{2}-\frac{4}{6} x+\frac{1}{6}=0$<br/>$\Rightarrow 6 x^{2}-4 x+1=0$<br/>Thus, the values of $a, b, c$ are $6,-4,1$ respectively</div>


title: if-2-p-3-q-18-and-4-p-2-4-pq-3-q-2-36-0-then-what-2-p-q-equal-to
slug: if-2-p-3-q-18-and-4-p-2-4-pq-3-q-2-36-0-then-what-2-p-q-equal-to-56631
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $2 p+3 q=18$ and $4 p^{2}+4 p q-3 q^{2}-36=0$, then what $(2 p+q)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">20</span> </li> </ul>

answer:
<div class="solution">$4 p^{2}+4 p q-3 q^{2}-36=0$<br/>$\Rightarrow\left(4 p^{2}+4 p q+q^{2}\right)-4 q^{2}-36=0$<br/>$(2 p+q)^{2}=4 q^{2}+36$<br/>$2 p+3 q=18$<br/>$p=\frac{18-3 q}{2}$<br/>$\Rightarrow\left[\frac{18-3 \mathrm{q}}{2} \times 2+\mathrm{q}\right]^{2}=4 \mathrm{q}^{2}+36$<br/>$\Rightarrow(18-3 \mathrm{q}+\mathrm{q})^{2}$<br/>$\begin{aligned} &amp;(18-2 q)^{2}=4 q^{2}+36 \\ &amp; 324+4 q^{2}-72 q=4 q^{2}+36 \\ 72 q=&amp; 324-36=288 \end{aligned}$<br/>$\mathrm{q}=\frac{288}{72}=4$<br/>Putting the value of $\mathrm{q}$ in $2 \mathrm{p}+3 \mathrm{q}=18$ $2 p+3 \times 4=18$<br/>$2 p=18-12$<br/>$\mathrm{p}=\frac{6}{2}=3$<br/>$(2 p+q)=2 \times 3+4=10$<br/>Option (c) is correct</div>


title: if-02-x-2-and-lo-g-10-2-03010-then-what-is-the-value-of-x-to-the-nearest-tenth
slug: if-02-x-2-and-lo-g-10-2-03010-then-what-is-the-value-of-x-to-the-nearest-tenth-87586
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-0.4$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $(0.2)^{x}=2$ and $\log _{10} 2=0.3010$, then what is the value of x to the nearest tenth?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-10.0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-0.5$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-0.4$</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">$-0.2$</span> </li> </ul>

answer:
<div class="solution">$(0.2)^{x}=2$<br/>Taking log on both sides,<br/>$x \log _{10} \frac{2}{10}=\log _{10} 2$<br/>$x\left[\log _{10} 2-\log _{10} 10\right]=\log _{10} 2$<br/>$x[0.3010-1]=0.3010$<br/>$x=-\frac{0.3010}{0.6990}=-0.43$</div>


title: if-2-x-3-3-y-2-7-what-is-dx-dy-equal-to-y-0
slug: if-2-x-3-3-y-2-7-what-is-dx-dy-equal-to-y-0-56396
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $2 \mathrm{x}^{3}-3 \mathrm{y}^{2}=7$, what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ equal to $(\mathrm{y} \neq 0)?$</div>

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

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

answer:
<div class="solution">Let $2 x^{3}-3 y^{2}=7$<br/>Differentiate both side, w.r.t. 'x'<br/>$6 x^{2}-6 y \frac{d y}{d x}=0$<br/>$\Rightarrow \frac{d y}{d x}=\frac{x^{2}}{y}$</div>


title: if-2-x-3-5-i-then-what-is-the-value-of-2-x-3-2-x-2-7-x-72
slug: if-2-x-3-5-i-then-what-is-the-value-of-2-x-3-2-x-2-7-x-72-87135
question: <div class="question">If $2 x=3+5 i$, then what is the value of $2 x^{3}+2 x^{2}-7 x+72$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-8</span> </li> </ul>

answer:
<div class="solution">Given 2 x=3+5i<br/>$\Rightarrow \quad x=\frac{3+5 i}{2}$<br/>Consider $x^{3}=\frac{27+125 i^{3}+225 i^{2}+135 i}{8}$<br/>$=\frac{27-125 i-225+135 i}{8} \quad\left(\begin{array}{r}\because i^{2}=-1 \\ i^{3}=-i\end{array}\right)$<br/>$=\frac{-198+10 i}{8}=\frac{-99+5 i}{4}$<br/>and $x^{2}=\frac{9+25 i^{2}+30 i}{4}$<br/>$=\frac{9-25+30 i}{4}=\frac{-8+15 i}{2}$<br/>Now, Consider $2 x^{3}+2 x^{2}-7 x+72$<br/>$=\left(\frac{-99+5 i}{2}\right)+(-8+15 i)-\frac{7(3+5 i)}{2}+72$<br/>$=-\frac{99}{2}+\frac{5 i}{2}-8+15 i-\frac{21}{2}-\frac{35}{2} i+72$<br/>$=\left(-\frac{99}{2}-8-\frac{21}{2}+72\right)+\left(\frac{5}{2}+15-\frac{35}{2}\right) i$<br/>$=\frac{-99-16-21+144}{2}=\frac{8}{2}=4$</div>


title: if-2-x-3-y-1-2-2-then-what-is-x-2-y-x-y-equal-to
slug: if-2-x-3-y-1-2-2-then-what-is-x-2-y-x-y-equal-to-26916
question: <div class="question">If $2^{x}=3^{y}=12^{2}$, then what is $(x+2 y) /(x y)$ equal to ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{z}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{z}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \underline{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{z}}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Given that $2^{\mathrm{x}}=3^{\mathrm{y}}=12^{\mathrm{z}}=\mathrm{k}$<br/>Taking $\log _{2}$ on both the sides $x=\log _{2} k, y=\log _{3} k$ and $z=\log _{12} k$<br/>$\frac{x+2 y}{x y}=\frac{\log _{2} k+2 \log _{3} k}{\log _{2} k \log _{3} k}$<br/>$=\frac{1}{\log _{3} \mathrm{k}}+\frac{2}{\log _{2} \mathrm{k}}$<br/>$=\log _{\mathrm{k}} 3+2 \log _{\mathrm{k}} 2=\log _{\mathrm{k}} 3+\log _{\mathrm{k}} 4$<br/>$=\log _{\mathrm{k}} 12=\frac{1}{\log _{12} \mathrm{k}}=\frac{1}{\mathrm{z}}$</div>


title: if-2-x-3-y-17-and-2-x-2-3-y-1-5-then-what-is-the-value-of-x
slug: if-2-x-3-y-17-and-2-x-2-3-y-1-5-then-what-is-the-value-of-x-59821
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2009 (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 $2^{x}+3^{y}=17$ and $2^{x+2}-3^{y+1}=5$, then what is the value of $x ?$</div>

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

answer:
<div class="solution">$A=-3$ and $B=8$<br/>Given, $2^{x}+3 y=17$<br/>and $2^{x+2}-3^{y+1}=5$<br/>$\Rightarrow 4.2^{x}-3.3^{y}=5$<br/>From equation (i) and (ii), we get $2^{x}=8$ and $3 y=9$<br/>$\Rightarrow x=3$ and $y=2$</div>


tags: ['Mathematics', 'Determinants', 'KCET', 'KCET 2017']
source_db: MarksBatch2_P1.db
title:
if-2leftbeginarrayll1-3-0-xendarrayrightleftbeginarraylly-0-1-2endarrayrightleftbeginarrayll5-6-1-8endarrayright-then-the-value-of-x-and-y-are

slug:
if-2leftbeginarrayll1-3-0-xendarrayrightleftbeginarraylly-0-1-2endarrayrightleftbeginarrayll5-6-1-8endarrayright-then-the-value-of-x-and-y-are-57855

question:
<div class="question">If \( 2\left[\begin{array}{ll}1 &amp; 3 \\ 0 &amp; x\end{array}\right]+\left[\begin{array}{ll}y &amp; 0 \\ 1 &amp; 2\end{array}\right]=\left[\begin{array}{ll}5 &amp; 6 \\ 1 &amp; 8\end{array}\right] \) then the value of \( x \) and \( y \) are</div>

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

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

answer:
<div class="solution">Given that,<br/>$2\left[\begin{array}{cc}1 &amp; 3 \\ 0 &amp; x\end{array}\right]+\left[\begin{array}{ll}y &amp; 0 \\ 1 &amp; 2\end{array}\right]=\left[\begin{array}{ll}5 &amp; 6 \\ 1 &amp; 8\end{array}\right]$<br/>$\left[\begin{array}{cc}2+y &amp; 6 \\ 1 &amp; 2 x+2\end{array}\right]=\left[\begin{array}{ll}5 &amp; 6 \\ 1 &amp; 8\end{array}\right]$<br/>Equating first elements, we have<br/>$2+y=5 \Rightarrow y=3$<br/>Equating last elements, we have<br/>$2 x+2=8 \Rightarrow x=3$<br/>Therefore, $x=3, y=3$.</div>


title: if-3-2-f-x-dx-3-7-and-3-9-f-x-dx-6-5-then-what-is-the-value-of-2-9-f-x-dx
slug: if-3-2-f-x-dx-3-7-and-3-9-f-x-dx-6-5-then-what-is-the-value-of-2-9-f-x-dx-89750
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int_{-3}^{2} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\frac{7}{3}$ and $\int_{-3}^{9} \mathrm{f}(\mathrm{x}) \mathrm{dx}=-\frac{5}{6}$, then what is the<br/>value of $\int_{2}^{9} \mathrm{f}(\mathrm{x}) \mathrm{dx}$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{19}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{19}{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{3}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{19}{6}$</span> </div>

answer:
<div class="solution">Value of the integral $\int_{2}^{9} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$<br/>$=\int_{-3}^{9} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}-\int_{-3}^{2} \mathrm{f}(\mathrm{x}) \mathrm{dx}$<br/>Given, $\int_{-3}^{9} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{-5}{6}$ and $\int_{-3}^{2} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{7}{3}$<br/>Putting these values in equation (i)<br/>$\int_{2}^{9} f(x) d x=\frac{-5}{6}-\frac{7}{3}=-\frac{19}{6}$</div>


title: if-3-a-b-a-3-b-k-a-b-then-what-is-the-value-of-k
slug: if-3-a-b-a-3-b-k-a-b-then-what-is-the-value-of-k-29195
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $(3 \vec{a}-\vec{b}) \times(\vec{a}+3 \vec{b})=k \vec{a} \times \vec{b}$ 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">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-8$</span> </li> </ul>

answer:
<div class="solution">$\quad \begin{aligned} (3 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}) \\ &amp;=(3 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times \overrightarrow{\mathbf{a}}+(3 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times 3 \overrightarrow{\mathbf{b}} \\ &amp;=3 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{a}} \times 3 \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{b}} \times 3 \overrightarrow{\mathbf{b}} \\ &amp;=0-(-\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+9 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}-0 \\ &amp;=10 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \\ &amp; \therefore k=10 \end{aligned}$</div>


tags: ['Mathematics', 'Matrices', 'KCET', 'KCET 2019']
source_db: MarksBatch2_P1.db
title:
if-3-a4-bprimeleftbeginarrayccc7-10-17-0-6-31endarrayright-and-2-b3-aprimeleftbeginarraycc1-18-4-0-5-7endarrayright-then-b

slug:
if-3-a4-bprimeleftbeginarrayccc7-10-17-0-6-31endarrayright-and-2-b3-aprimeleftbeginarraycc1-18-4-0-5-7endarrayright-then-b-76842

question:
<div class="question">If \( 3 A+4 B^{\prime}=\left[\begin{array}{ccc}7 &amp; -10 &amp; 17 \\ 0 &amp; 6 &amp; 31\end{array}\right] \) and \( 2 B-3 A^{\prime}=\left[\begin{array}{cc}-1 &amp; 18 \\ 4 &amp; 0 \\ 5 &amp; -7\end{array}\right] \) then \( B= \)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\( \left[\begin{array}{cc}1 &amp; -3 \\ -1 &amp; 1 \\ 2 &amp; 4\end{array}\right] \)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\( \left[\begin{array}{cc}1 &amp; 3 \\ -1 &amp; 1 \\ 2 &amp; -4\end{array}\right] \)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\( \left[\begin{array}{cc}1 &amp; 3 \\ -1 &amp; 1 \\ 2 &amp; 4\end{array}\right] \)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\( \left[\begin{array}{cc}-1 &amp; -18 \\ 4 &amp; -16 \\ -5 &amp; -7\end{array}\right] \)</span> </li> </ul>

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

answer:
<div class="solution">(C)<br/>\( 3 A+4 B^{\prime}=\left[\begin{array}{ccc}7 &amp; -10 &amp; 17 \\ 0 &amp; 6 &amp; 31\end{array}\right] \ldots .(1) \) \( \left(2 B-3 A^{\prime}\right)^{\prime}=(2 B)^{\prime}-\left(3 A^{\prime}\right)^{\prime}=2 B^{\prime}-3 A \) \( \Rightarrow 2 B^{\prime}-3 A=\left[\begin{array}{ccc}-1 &amp; 4 &amp; -5 \\ 18 &amp; 0 &amp; -7\end{array}\right] \ldots .(2) \)<br/>$(1)$<br/>(2)<br/>Adding (1) and (2) we get, \( 6 \mathrm{~B}^{\prime}=\left[\begin{array}{ccc}6 &amp; -6 &amp; 12 \\ 18 &amp; 6 &amp; 24\end{array}\right] \)<br/>\[<br/>\begin{array}{l}<br/>\Rightarrow B^{\prime}=\left[\begin{array}{ccc}<br/>1 &amp; -1 &amp; 2 \\<br/>3 &amp; 1 &amp; 4<br/>\end{array}\right] \\<br/>\therefore B=\left[\begin{array}{ccc}<br/>1 &amp; 3 &amp; \\<br/>-1 &amp; 1 &amp; \\<br/>2 &amp; 4 &amp;<br/>\end{array}\right]<br/>\end{array}<br/>\]</div>


title: if-3-is-the-root-of-the-equation-x-2-8-x-k-0-then-what-is-the-val-ue-of-k
slug: if-3-is-the-root-of-the-equation-x-2-8-x-k-0-then-what-is-the-val-ue-of-k-45416
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If 3 is the root of the equation $x^{2}-8 x+k=0$, then what is the val ue of $k$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-15$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">15</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">24</span> </li> </ul>

answer:
<div class="solution">Since, 3 is the root of the equation $x^{2}-8 x+k=0$ 3 satisfies the equation $x^{2}-8 x+k=0$ $9-24+k=0 \Rightarrow k=1 \mathrm{k}$</div>


title: if-3-tan-4-0-where-2-then-what-is-the-value-of-2-cot-5-cos-sin
slug: if-3-tan-4-0-where-2-then-what-is-the-value-of-2-cot-5-cos-sin-60711
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $3 \tan \theta+4=0$, where $(\pi / 2)   &lt; \theta &lt; \pi$, then what is the value of $2 \cot \theta-5 \cos \theta+\sin \theta ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{53}{10}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7}{10}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{23}{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{37}{10}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{23}{10}$</span> </div>

answer:
<div class="solution">As given, $3 \tan \theta+4=0 \Rightarrow \tan \theta=-\frac{4}{3}$<br/>$\left[\theta\right.$ lies in second quadrant i.e., $\left.\frac{\pi}{2}   &lt; \theta &lt; \pi\right]$<br/>$\therefore \quad \cot \theta=-\frac{3}{4} \Rightarrow \cos \theta=-\frac{3}{5}$ and $\sin \theta=\frac{4}{5}$<br/>Now, $2 \cot \theta-5 \cos \theta+\sin \theta$<br/>$=-\frac{6}{4}+\frac{15}{5}+\frac{4}{5}=\frac{-30+60+16}{20}</div>


title: if-3-x-1-3-x-1-30-then-what-is-the-value-of-3-x-2-3-x
slug: if-3-x-1-3-x-1-30-then-what-is-the-value-of-3-x-2-3-x-89860
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">90</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $3^{(x-1)}+3^{(x+1)}=30$, then what is the value of $3^{(x+2)}+3^{x}$ ?</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">60</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">81</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">90</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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: $3^{(x-1)}+3^{(x+1)}=30$<br/>$\Rightarrow \frac{3^{\mathrm{x}}}{3}+3.3^{\mathrm{x}}=30$<br/>Multiplying both the sides by 3 in equation (i) $3^{x}+3^{2} .3^{x}=90$<br/>$\Rightarrow 3^{\mathrm{x}}+3^{\mathrm{x}+2}=90$</div>


title: if-3-x-3-y-3-x-y-then-what-is-d-x-d-y-equal-to
slug: if-3-x-3-y-3-x-y-then-what-is-d-x-d-y-equal-to-38101
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $3^{\mathrm{x}}+3^{\mathrm{y}}=3^{\mathrm{x}+\mathrm{y}}$ then what is $\frac{d y}{d x}$ equal to?</div>

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

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

answer:
<div class="solution">$3^{x}+3^{y}=3^{x+y}$<br/>On differentiating w.r.t. $x$, we get<br/>$3^{x} \log 3+3^{y} \log 3 \frac{d y}{d x}=3^{x+y} \log 3\left(1+\frac{d y}{d x}\right)$<br/>$\Rightarrow \log 3\left[3^{x}+3^{y} \frac{d y}{d x}\right]=\log 3 \cdot 3^{x+y}\left(1+\frac{d y}{d x}\right)$<br/>$\Rightarrow \frac{d y}{d x}\left(-3^{x+y}+3^{y}\right)=3^{x+y}-3^{x}$<br/>$=3^{x} \cdot 3^{y}-3^{x}=3^{x}\left(3^{y}-1\right)$<br/>$\Rightarrow \frac{d y}{d x}=\frac{3^{x}\left(3^{y}-1\right)}{3^{y}\left(1-3^{x}\right)}=\frac{3^{x-y}\left(3^{y}-1\right)}{\left(1-3^{x}\right)}$</div>


tags: ['Mathematics', 'Ellipse', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-4-0-and-4-0-are-the-foci-of-an-ellipse-and-the-semi-minor-axis-is-3-then-the-ellipse-passes-through-which-one-of-the-following-points

slug:
if-4-0-and-4-0-are-the-foci-of-an-ellipse-and-the-semi-minor-axis-is-3-then-the-ellipse-passes-through-which-one-of-the-following-points-91353

question:
<div class="question">If $(4,0)$ and $(-4,0)$ are the foci of an ellipse and the semi-<br/>minor axis is 3 , then the ellipse passes through which one of the following points?</div>

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

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

answer:
<div class="solution">Foci of an ellipse are $(4,0)$ and $(-4,0)$. (Given) $2 a e=8 \Rightarrow a e=4$<br/>and semiminor axis is $3 \quad \therefore b=3$<br/>We know that, $e=\sqrt{1-\frac{b^{2}}{a^{2}}}$<br/>$\Rightarrow\left(\frac{4}{a}\right)^{2}=\left(1-\frac{9}{a^{2}}\right)\left(\because e=\frac{4}{a}, b=3\right)$<br/>$\Rightarrow \frac{16}{a^{2}}=\frac{a^{2}-9}{a^{2}}$<br/>$\Rightarrow 16=a^{2}-9 \Rightarrow a^{2}=25 \Rightarrow a=5$<br/>Now, standard equation of an ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ Thus, the equation of an ellipse is<br/>$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$<br/>which is satisfied by $(5,0)$. Hence the ellipse passes through $(5,0)$.</div>


title: if-4-sin-2-1-where-0-2-how-many-values-does-take-1
slug: if-4-sin-2-1-where-0-2-how-many-values-does-take-1-96773
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $4 \sin ^{2} \theta=1$, where $0   &lt; \theta &lt; 2 \pi$, how many values does $\theta$ take?</div>

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

answer:
<div class="solution">$\tan \left(-585^{\circ}\right)=-\tan 585^{\circ}$<br/>$=-\tan \left[540^{\circ}+45^{\circ}\right]=-\tan \left[3 \pi+\frac{\pi}{4}\right]$<br/>$\quad=-\left[\frac{\tan 3 \pi+\tan \frac{\pi}{4}}{1-\tan 3 \pi \tan \frac{\pi}{4}}\right]=-\left[\frac{0+1}{1-0 \times 1}\right]=-1$</div>


title: if-4-sin-2-1-where-0-2-how-many-values-does-take
slug: if-4-sin-2-1-where-0-2-how-many-values-does-take-31998
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $4 \sin ^{2} \theta=1$, where $0   &lt; \theta &lt; 2 \pi$, how many values does $\theta$ take?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">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">None of the above</span> </li> </ul>

answer:
<div class="solution">$4 \sin ^{2} \theta=1$<br/>$\Rightarrow \sin ^{2} \theta=\frac{1}{4} \Rightarrow \sin \theta=\pm \frac{1}{2} \Rightarrow \theta=\pm \frac{\pi}{6}$<br/>Hence $\theta$ take 4 values. (one value for each quadrant)</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-4-x-5-y-3-0-x-9-y-107-are-two-lines-of-regression-then-what-are-the-values-of-x-and-v-respectively

slug:
if-4-x-5-y-3-0-x-9-y-107-are-two-lines-of-regression-then-what-are-the-values-of-x-and-v-respectively-67073

question:
<div class="question">If $4 x-5 y+3 ?$ $0 x-9 y=107$ are two lines of<br/>regression, then what are the values of $\overline{\mathrm{x}}$ and $\overline{\mathrm{v}}$<br/>respectively? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12 and 18</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">18 and 12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">13 and 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">17 and 13</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">13 and 17</span> </div>

answer:
<div class="solution">$4 \mathrm{x}-5 \mathrm{y}+33=0$<br/>$20 \mathrm{x}-9 \mathrm{y}=107$<br/>$20 x-25 y+165=0$<br/>$(1) \times 5 \Rightarrow \quad 20 x-9 y-107=0$<br/>$(2) \Rightarrow \frac{y-(+)(+)}{-16 \mathrm{y}+272=0} \Rightarrow 16 \mathrm{y}=272 \Rightarrow \mathrm{y}=17$<br/>$(1) \Rightarrow 4 x-5(17)+33=0$<br/>$\Rightarrow 4 x-85+33=0$<br/>$\Rightarrow 4 x=52 \Rightarrow x=13$</div>


title: if-4-x-6-2-x-8-0-then-the-values-of-x-are
slug: if-4-x-6-2-x-8-0-then-the-values-of-x-are-69365
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1,2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $4^{\mathrm{x}}-6.2^{\mathrm{x}}+8=0$, then the values of $\mathrm{x}$ are</div>

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

answer:
<div class="solution">Given equation $4^{x}-6.2^{x}+8=0$<br/>$\Rightarrow\left(2^{x}\right)^{2}-6.2^{x}+8=0$<br/>Put $2^{\mathrm{x}}=\mathrm{y}$, we have<br/>$y^{2}-6 y+8=0$<br/>$(y-4)(y-2)=0$<br/>$y=2$,<br/>$\mathrm{So}_{2} \quad 2^{\mathrm{x}}=2 \Rightarrow \mathrm{x}=$<br/>and $2^{x}=2^{2} \Rightarrow x=2$</div>


title: if-5-0-0-7-1-x-y-1-2-then-which-one-of-the-following-is-correct
slug: if-5-0-0-7-1-x-y-1-2-then-which-one-of-the-following-is-correct-12473
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left[\begin{array}{cc}5 &amp; 0 \\ 0 &amp; 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-1 \\ 2\end{array}\right]$, 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">$x=5, y=14$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=-5, y=14$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x=-5, y=-14$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad x=5, y=-14$</span> </li> </ul>

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

answer:
<div class="solution">The given equation is : $\left[\begin{array}{ll}5 &amp; 0 \\ 0 &amp; 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-1 \\ 2\end{array}\right]$<br/>Multiplying both sides by $\left[\begin{array}{ll}5 &amp; 0 \\ 0 &amp; 7\end{array}\right]$, we get<br/>$\left[\begin{array}{ll}5 &amp; 0 \\ 0 &amp; 7\end{array}\right]\left[\begin{array}{ll}5 &amp; 0 \\ 0 &amp; 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{ll}5 &amp; 0 \\ 0 &amp; 7\end{array}\right]\left[\begin{array}{c}-1 \\ 2\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]\left[\begin{array}{l}-\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{l}-5 \\ 14\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-5 \\ 14\end{array}\right] \quad \Rightarrow \mathrm{x}=-5$ and $\mathrm{y}=-14$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P1.db
title:
if-5-4-divides-the-line-segment-between-the-coordinate-axes-in-the-ratio-1-2-then-what-is-its-equation

slug:
if-5-4-divides-the-line-segment-between-the-coordinate-axes-in-the-ratio-1-2-then-what-is-its-equation-75256

question:
<div class="question">If $(-5,4)$ divides the line segment between the coordinate axes in the ratio $1: 2$, then what is its equation?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 x+5 y+20=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 x+8 y-7=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$8 x-5 y+60=0$</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">$5 x-8 y+57=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 x-5 y+60=0$</span> </div>

answer:
<div class="solution">Let $A(a, 0)$ and $B(0, b)$ be two points on $x$ -axis and $y$ axis respectively<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/GHYGs6jhbMMK2sMoufGPiuWXbHvO7bchggcFXzwZHss.original.fullsize.png"/><br/><br/>Given $(-5,4)$ divides line $A B$ in the ratio $1: 2$. By section formula we have<br/>$$<br/>\begin{aligned}<br/>&amp;-5=\frac{1 \times 0+2 \times a}{3} \\<br/>\Rightarrow &amp; a=\frac{-15}{2} \text { and } 4=\frac{1 \times b+2 \times 0}{3} \\<br/>\Rightarrow &amp; b=12<br/>\end{aligned}<br/>$$<br/>Thus, $A=\left(\frac{-15}{2}, 0\right)$ and $B=(0,12)$<br/>Hence, equation of line joining $\left(\frac{-15}{2}, 0\right)$ and $(0,12)$ is<br/>$$<br/>\begin{array}{l}<br/>(y-0)=\frac{12-0}{0+\frac{15}{2}}\left(x+\frac{15}{2}\right) \\<br/>\Rightarrow \quad y=\frac{4}{5}(2 x+15)<br/>\end{array}<br/>$$<br/>$\Rightarrow 5 y=(8 x+60) \Rightarrow 8 x-5 y+60=0$</div>


title: if-5-and-7-are-the-roots-of-the-equation-x-7-5-4-x-8-5-7-x-0-then-what-is-the-third-root
slug: if-5-and-7-are-the-roots-of-the-equation-x-7-5-4-x-8-5-7-x-0-then-what-is-the-third-root-14107
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-12$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If 5 and 7 are the roots of the equation<br/>$\left|\begin{array}{lll}x &amp; 4 &amp; 5 \\ 7 &amp; x &amp; 7 \\ 5 &amp; 8 &amp; x\end{array}\right|=0$, then what is the third root?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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">B</span> <span class="option-data">9</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">13</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">14</span> </li> </ul>

answer:
<div class="solution">Given, $\left|\begin{array}{lll}x &amp; 4 &amp; 5 \\ 7 &amp; x &amp; 7 \\ 5 &amp; 8 &amp; x\end{array}\right|=0$<br/>$\Rightarrow x\left(x^{2}-56\right)-4(7 x-35)+5(56-5 x)=0$<br/>$\Rightarrow x^{3}-56 x-28 x+140+280-25 x=0$<br/>$\Rightarrow x^{3}-109 x+420=0$<br/>$\Rightarrow(x-5)(x-7)(x+12)=0$<br/>$\Rightarrow x=-12$<br/>Hence, the third root is $-12$.</div>


title: if-5-x-2-y-k-0-is-a-tangent-to-the-parabola-y-2-6-x-then-their-point-of-contact-is-1
slug: if-5-x-2-y-k-0-is-a-tangent-to-the-parabola-y-2-6-x-then-their-point-of-contact-is-1-74565
tags: ['Mathematics', 'Application of Derivatives', 'AP EAMCET', 'AP EAMCET 2018 (25 Apr Shift 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>k</mi><mo>=</mo><mn>0</mn></math> is a tangent to the parabola <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>6</mn><mi>x</mi><mo>,</mo></math> then their point of contact 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"><mfenced><mrow><mfrac><mn>6</mn><mn>5</mn></mfrac><mo>,</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>6</mn><mn>5</mn></mfrac><mo>,</mo><mfrac><mn>6</mn><mn>25</mn></mfrac></mrow></mfenced></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"><mfenced><mrow><mfrac><mn>6</mn><mn>25</mn></mfrac><mo>,</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></mfenced></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"><mfenced><mrow><mfrac><mn>6</mn><mn>25</mn></mfrac><mo>,</mo><mfrac><mn>6</mn><mn>25</mn></mfrac></mrow></mfenced></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"><mfenced><mrow><mfrac><mn>6</mn><mn>25</mn></mfrac><mo>,</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>6</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>y</mi><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>6</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mfrac><mn>3</mn><msqrt><mn>6</mn><mi>x</mi></msqrt></mfrac></mfenced><mo>→</mo><mtext>slope of tangent</mtext></math></p><p>Also,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>k</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>y</mi><mo>=</mo><mfenced><mfrac><mn>5</mn><mn>2</mn></mfrac></mfenced><mi>x</mi><mo>+</mo><mfrac><mi>k</mi><mn>2</mn></mfrac></math></p><p>Comparing with the standard form of line in slope intercept form, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>→</mo><mtext>slope</mtext></math></p><p>Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>6</mn><mi>x</mi></msqrt></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>3</mn><mrow><msqrt><mn>2</mn><mo>×</mo><mn>3</mn></msqrt><mo>·</mo><msqrt><mi>x</mi></msqrt></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><mi>x</mi></msqrt></mfrac><mo>=</mo><mfrac><mn>5</mn><msqrt><mn>2</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>5</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mn>6</mn><mn>25</mn></mfrac></math></p><p>And, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></math>.</p><p>Hence, point of contact is  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>6</mn><mn>25</mn></mfrac><mo>,</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></mfenced></math>.</p><p> </p><p> </p></div>


title: if-5-x-9-0-is-the-directrix-of-the-hyperbola-16-x-2-9-y-2-144-then-its-corresponding-focus-is
slug: if-5-x-9-0-is-the-directrix-of-the-hyperbola-16-x-2-9-y-2-144-then-its-corresponding-focus-is-48185
tags: ['Mathematics', 'Hyperbola', 'JEE Main', 'JEE Main 2019 (10 Apr Shift 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math> is the directrix of the hyperbola <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>144</mn><mo>,</mo></math> then its corresponding focus 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"><mfenced separators="|"><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mi mathvariant="normal"> </mi><mn>0</mn></mrow></mfenced><mi mathvariant="normal"> </mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mrow><mn>5</mn><mo>,</mo><mi mathvariant="normal"> </mi><mn>0</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mrow><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>,</mo><mi mathvariant="normal"> </mi><mn>0</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfenced separators="|"><mrow><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mi mathvariant="normal">Â </mi><mn>0</mn></mrow></mfenced></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"><mfenced separators="|"><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mi mathvariant="normal"> </mi><mn>0</mn></mrow></mfenced><mi mathvariant="normal"> </mi></math></span> </div>

answer:
<div class="solution"><p>The equation of a hyperbola in standard form is <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> and its foci are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>±</mo><mi>a</mi><mi>e</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> and directrices are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>±</mo><mfrac><mi>a</mi><mi>e</mi></mfrac><mo>.</mo></math></p><p>The given equation of hyperbola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>144</mn><mo>,</mo><mo> </mo><mo> </mo><mo>⇒</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>9</mn></mfrac><mo>-</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mn>16</mn></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>9</mn><mo>,</mo><mi> </mi><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>16</mn></math></p><p>Given, directrix of the hyperbola is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mo>⇒</mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>9</mn><mn>5</mn></mfrac><mo>.</mo></math></p><p>Hence, on comparing the directrix with the standard equation, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>e</mi></mfrac><mo>=</mo><mfrac><mn>9</mn><mn>5</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>3</mn><mi>e</mi></mfrac><mo>=</mo><mfrac><mn>9</mn><mn>5</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>e</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math></p><p>And, hence the required focus is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mrow><mo>-</mo><mi>a</mi><mi>e</mi><mo>,</mo><mi> </mi><mn>0</mn></mrow></mfenced><mo>=</mo><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mi> </mi><mn>0</mn><mo>)</mo><mo>.</mo></math></p></div>


title: if-6-i-4-20-3-i-3-i-3-1-1-i-x-i-y-where-i-1-then-what-is-x-equal-to
slug: if-6-i-4-20-3-i-3-i-3-1-1-i-x-i-y-where-i-1-then-what-is-x-equal-to-37780
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{ccc}6 i &amp; -3 i &amp; 1 \\ 4 &amp; 3 i &amp; -1 \\ 20 &amp; 3 &amp; i\end{array}\right|=x+i y$, where $i=\sqrt{-1}$, then what is $x$ equal to?</div>

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

answer:
<div class="solution">$\left|\begin{array}{lcc}6 \mathrm{i} &amp; -3 \mathrm{i} &amp; 1 \\ 4 &amp; 3 \mathrm{i} &amp; -1 \\ 20 &amp; 3 &amp; \mathrm{i}\end{array}\right|$<br/>$=6 \mathrm{i}\left[3 \mathrm{i}^{2}+3\right]+3 \mathrm{i}[4 \mathrm{i}+20]+1[12-60 \mathrm{i}]$<br/>$=6 \mathrm{i}[-3+3]+12 \mathrm{i}^{2}+60 \mathrm{i}+12-60 \mathrm{i}$<br/>$=-12+12=0=\mathrm{x}+\mathrm{iy}$<br/>$\therefore \mathrm{x}=0$</div>


title: if-7-points-out-of-12-are-in-the-same-straight-line-then-what-is-the-number-of-triangles-formed
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">185</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-7-points-out-of-12-are-in-the-same-straight-line-then-what-is-the-number-of-triangles-formed-51684

question:
<div class="question">If 7 points out of 12 are in the same straight line, then what is the number of triangles formed ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">84</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">175</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">185</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">201</span> </li> </ul>

answer:
<div class="solution">Number of triangles formed from 12 point $={ }^{12} \mathrm{C}_{3}$ Since 7 parts are collinear, then ${ }^{7} \mathrm{C}_{3}$ triangles will not be formed so. $={ }^{12} \mathrm{C}_{3}-{ }^{7} \mathrm{C}_{3}$<br/>$=\frac{12 !}{3 ! 9 !}-\frac{7 !}{3 ! 4 !}=\frac{12.11 .10}{3.2 .1}-\frac{7.6 .5}{3.2 .1}$<br/>$=220-35=185$</div>


title: if-8-5-6-5-x-3-1-1-1-2-then-what-is-the-value-of-x
slug: if-8-5-6-5-x-3-1-1-1-2-then-what-is-the-value-of-x-69018
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{ccc}8 &amp; -5 &amp; 1 \\ 5 &amp; \mathrm{x} &amp; 1 \\ 6 &amp; 3 &amp; 1\end{array}\right|=2$ then what is the value of $\mathrm{x}$ ?</div>

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

answer:
<div class="solution">Since $\left|\begin{array}{ccc}8 &amp; -5 &amp; 1 \\ 5 &amp; x &amp; 1 \\ 6 &amp; 3 &amp; 1\end{array}\right|=2$<br/>$\Rightarrow 8(x-3)+5(5-6)+1(15-6 x)=2$<br/>$\Rightarrow 8 x-24-5+15-6 x=2$<br/>$\Rightarrow 2 x=16$<br/>$\Rightarrow x=8$</div>


title: if-8-then-what-is-the-value-of-2-cos-1-10-2-cos-2-1-10-2-cos-1-10-2-cos-4-1-10
slug: if-8-then-what-is-the-value-of-2-cos-1-10-2-cos-2-1-10-2-cos-1-10-2-cos-4-1-10-23039
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\theta=\frac{\pi}{8}$, then what is the value of $(2 \cos \theta+1)^{10}(2 \cos 2 \theta-1)^{10}(2 \cos \theta-1)^{10}(2 \cos 4 \theta-1)^{10 ?}$</div>

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

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

answer:
<div class="solution">$[(2 \cos \theta+1)(2 \cos \theta-1)]^{10}[2 \cos 2 \theta-1]^{10}[2 \cos 4 \theta-1]^{10}$<br/>when, $\theta=\frac{\pi}{8}$ (given)<br/>$=\left(4 \cos ^{2} \frac{\pi}{8}-1\right)^{10}\left(2 \times \frac{1}{\sqrt{2}}-1\right)^{10}(-1)^{10}$<br/>$=\left(4 \times \frac{2+\sqrt{2}}{4}-1\right)^{10}(\sqrt{2}-1)^{10}$<br/>$=[(\sqrt{2}+1)(\sqrt{2}-1)]^{10}=1^{10}=1$</div>


title: if-8-what-is-the-value-of-cos-cos-2-cos-4
slug: if-8-what-is-the-value-of-cos-cos-2-cos-4-38211
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha=\frac{\pi}{8}$, what is the value of $\cos \alpha \cos 2 \alpha \cos 4 \alpha$ ?</div>

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

answer:
<div class="solution">As given : $\alpha=\frac{\pi}{8}$<br/>$\cos \alpha \cos 2 \alpha \cos 4 \alpha=\cos \frac{\pi}{8} \cdot \cos \frac{\pi}{4} \cos \frac{\pi}{2}=0$<br/>$\left(\because \cos \frac{\pi}{2}=0\right)$</div>


title: if-8-x-9-y-20-and-7-x-10-y-9-then-what-is-2-x-y-equal-to
slug: if-8-x-9-y-20-and-7-x-10-y-9-then-what-is-2-x-y-equal-to-96960
question: <div class="question">If $8 x-9 y=20$ and $7 x-10 y=9$, then what is $2 x-y$ equal to?</div>
tags: ['Mathematics', 'Basic of Mathematics', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">11</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">13</span> </li> </ul>

answer:
<div class="solution">$8 x-9 y=20$ or $80 x-90 y=200$...(1)<br/>$7 x-10 y=9$ or $63 x-90 y=81$...(2)<br/>Subtracting $(2)$ from $(1)$, we get $17 \mathrm{x}=119$<br/>$\mathrm{x}=7$<br/>$8 \times 7-9 y=20$<br/>$9 y=36$<br/>$y=4$<br/>$2 x-y=2 \times 7-4=10$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-10-x-010-2-11-y-1-2-10-z-11-2-then-what-are-the-possible-values-of-the-binary-digits-x-y-z-respectively

slug:
if-10-x-010-2-11-y-1-2-10-z-11-2-then-what-are-the-possible-values-of-the-binary-digits-x-y-z-respectively-10584

question:
<div class="question">If $(10 x 010)_{2}-(11 y 1)_{2}=(10 z 11)_{2}$, then what are the possible values of the binary digits $x, y, z$ respectively?</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \therefore 133=(10000101)_{2} \\ &amp;(10 x 010)_{2}-(11 y 1)_{2}=(10 z 11)_{2} \\ &amp; \Rightarrow\left(2^{5} \times 1+0+x \times 2^{3}+0 \times 2^{2}+1 \times 2^{1}+0\right) \\ &amp;=\left(2^{3} \times 1+2^{2} \times 1+y \times 2^{1}+1 \times 2^{0}\right) \\ &amp;=2^{4} \times 1+0+2^{2} \times z+2^{1} \times 1+2^{0} \end{aligned}$<br/>$\Rightarrow(34+8 x)-(13+2 y)=19+4 z$<br/>$\Rightarrow 2=-8 x+2 y+4 z$<br/>$\Rightarrow x=0, y=1, z=0$</div>


title: if-13-2-3-2-3-3-3-n-3-n-3-2-n-1-3-a-b-then-a-and-b-are-respectively
slug: if-13-2-3-2-3-3-3-n-3-n-3-2-n-1-3-a-b-then-a-and-b-are-respectively-10042
tags: ['Mathematics', 'Mathematical Induction', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $1.3+2.3^{2}+3.3^{3}+\ldots .+\mathrm{n} .3^{n}=\frac{(2 \mathrm{n}-1) 3^{\mathrm{a}}+\mathrm{b}}{3}$ then a and b<br/>are respectively</div>

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

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

answer:
<div class="solution">$1.3+2.3^{2}+3.3^{3}+\ldots .+\mathrm{n} .3^{\mathrm{n}}=\frac{(2 \mathrm{n}-1) 3^{\mathrm{a}}+\mathrm{b}}{4}$<br/>Let us put $3=\mathrm{x}$. L.H.S: $S=x+2 x^{2}+3 x^{3}+\ldots .+n \cdot x^{n}$<br/>$x s=x^{2}+2 x^{3}+3 x^{4}+\ldots+n \cdot x^{n+1}$<br/>$(1)-(2) \Rightarrow S-x S=\left(x+2 x^{2}+3 x^{3}+\ldots .+n \cdot x^{n}\right)-\left(x^{2}+2 x^{3}\right.$<br/>$\left.+3 x^{4}+\ldots+n \cdot x^{n+1}\right)$<br/>$\Rightarrow S(1-x)=x+x^{2}+x^{3}+\ldots .+x^{n}-n x^{n+1}$<br/>$\Rightarrow S(1-x)=\frac{x\left(1-x^{n}\right)}{1-x}-n x^{n+1}$<br/>$\Rightarrow S=\left(\frac{1}{x-1}\right)\left(\frac{-x\left(x^{n}-1\right)+n x^{n+1}(x-1)}{x-1}\right)$<br/>Put $\mathrm{x}=3$<br/>$\Rightarrow S=\frac{1}{2}\left(\frac{-3^{n+1}+3+2 n \cdot 3^{n+1}}{2}\right)=\left(\frac{3^{n+1}(2 n-1)+3}{4}\right)$</div>


title: if-14-1-x-110-are-in-hp-then-what-is-the-value-of-x
slug: if-14-1-x-110-are-in-hp-then-what-is-the-value-of-x-93706
question: <div class="question">If $1 / 4,1 / x, 1 / 10$ are in HP, then what is the value of $x$?</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">Since $\frac{1}{4}, \frac{1}{x}, \frac{1}{10}$ are in H.P.<br/>$\therefore 4, \mathrm{x}, 10$ are in $\mathrm{AP}$<br/>$\Rightarrow 2 \mathrm{x}=4+10 \Rightarrow \mathrm{x}=\frac{14}{2}=7$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'MHT CET', 'MHT CET 2020 (16 Oct Shift 1)']
source_db: MarksBatch2_P1.db
title:
if-38-55-kj-of-heat-is-absorbed-when-6-0-g-of-o-2-react-with-cif-according-to-reaction-2-cif-g-o-2-g-cl-2-o-g-0-f-2-g-what-is-the-standard-enthalpy-of

slug:
if-38-55-kj-of-heat-is-absorbed-when-6-0-g-of-o-2-react-with-cif-according-to-reaction-2-cif-g-o-2-g-cl-2-o-g-0-f-2-g-what-is-the-standard-enthalpy-of-60413

question:
<div class="question">If $38 \cdot 55 \mathrm{~kJ}$ of heat is absorbed when $6 \cdot 0 \mathrm{~g}$ of $\mathrm{O}_{2}$ react with CIF according to reaction $2 \mathrm{CIF}_{(\mathrm{g})}+\mathrm{O}_{2(\mathrm{~g})} \longrightarrow \mathrm{Cl}_{2} \mathrm{O}_{(\mathrm{g})}+0 \mathrm{~F}_{2}(\mathrm{~g})$<br/>What is the standard enthalpy of reaction?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$72.28 \mathrm{~K}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$205.6 \mathrm{k} \mathrm{J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$102.8 \mathrm{k}]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$49.80 \mathrm{k}]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$205.6 \mathrm{k} \mathrm{J}$</span> </div>

answer:
<div class="solution">(A)<br/>$\begin{array}{l}<br/>2 \mathrm{ClF}_{(\mathrm{g})}+\mathrm{O}_{2(\mathrm{~g})} \longrightarrow \mathrm{Cl}_{2} \mathrm{O}_{(\mathrm{g})}+\mathrm{OF}_{2(\mathrm{~g})} \\<br/>\text { For } 6.0 \mathrm{~g} \text { of } \mathrm{O}_{2} \Rightarrow \Delta \mathrm{H}=+38.55 \mathrm{~kJ} \\<br/>\therefore \text { For } 32.0 \mathrm{~g} \text { of } \mathrm{O}_{2} \Rightarrow \Delta \mathrm{H}^{0}=\frac{38.55 \times 32}{6}=205.6 \mathrm{~kJ}<br/>\end{array}$</div>


title: if-11101011-is-converted-decimal-system-then-the-resulting-number-is
slug: if-11101011-is-converted-decimal-system-then-the-resulting-number-is-75143
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">235</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If (11101011) is converted decimal system, then the<br/>resulting number is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">235</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">175</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">160</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">126</span> </li> </ul>

answer:
<div class="solution">$(11101011)_{2}$<br/>$=\left(1 \times 2^{7}+1 \times 2^{6}+1 \times 2^{5}+0 \times 2^{4}+1 \times 2^{3}\right.$<br/>$+0 \times 2^{2}+1 \times 2$</div>


title: if-a-0-0-b-and-1-1-are-collinear-what-is-a-b-ab-equal-to
slug: if-a-0-0-b-and-1-1-are-collinear-what-is-a-b-ab-equal-to-98952
tags: ['Mathematics', 'Straight Lines', '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 $(a, 0),(0, b)$ and $(1,1)$ are collinear, what is $(a+b-a b)$ equal to?</div>

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

answer:
<div class="solution">Since, $(a, 0),(0, b)$ and $(1,1)$ are collinear.<br/>$\left|\begin{array}{lll}a &amp; 0 &amp; 1 \\ 0 &amp; b &amp; 1 \\ 1 &amp; 1 &amp; 1\end{array}\right|=0$<br/>$\Rightarrow a(b-1)+1(0-b)=0$<br/>$\Rightarrow a b-a-b=0$<br/>$\Rightarrow \quad a+b-a b=0$</div>


title: if-a-0-0-where-is-cube-root-of-unity-then-what-is-a-100-equal-to
slug: if-a-0-0-where-is-cube-root-of-unity-then-what-is-a-100-equal-to-41229
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$A$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left[\begin{array}{cc}\omega &amp; 0 \\ 0 &amp; \omega\end{array}\right]$, where $\omega$ is cube root of unity, then what is $A^{100}$ equal to?</div>

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

answer:
<div class="solution">Given, $A=\left[\begin{array}{ll}\omega &amp; 0 \\ 0 &amp; \omega\end{array}\right]$<br/>Now, $A^{2}=\left[\begin{array}{ll}\omega &amp; 0 \\ 0 &amp; \omega\end{array}\right]\left[\begin{array}{ll}\omega &amp; 0 \\ 0 &amp; \omega\end{array}\right]=\left[\begin{array}{cc}\omega^{2} &amp; 0 \\ 0 &amp; \omega^{2}\end{array}\right]$<br/>$A^{3}=\left[\begin{array}{cc}\omega^{2} &amp; 0 \\ 0 &amp; \omega^{2}\end{array}\right]\left[\begin{array}{ll}\omega &amp; 0 \\ 0 &amp; \omega\end{array}\right]=\left[\begin{array}{cc}\omega^{3} &amp; 0 \\ 0 &amp; \omega^{3}\end{array}\right]$<br/>$\begin{aligned} \text { Similarly, } A^{100} &amp;=\left[\begin{array}{cc}\omega^{100} &amp; 0 \\ 0 &amp; \omega^{100}\end{array}\right] \\ &amp;=\left[\begin{array}{cc}\left(\omega^{3}\right)^{33} \cdot \omega^{1} &amp; 0 \\ 0 &amp; \left(\omega^{3}\right)^{33} \cdot \omega^{1}\end{array}\right] \\ &amp;=\left[\begin{array}{cc}\omega &amp; 0 \\ 0 &amp; \omega\end{array}\right]\left(\because \omega^{3}=1\right)=\mathrm{A} \end{aligned}$</div>


title: if-a-0-1-1-0-b-i-0-0-i-c-0-i-i-0-then-which-one-of-the-following-is-not-correct
slug: if-a-0-1-1-0-b-i-0-0-i-c-0-i-i-0-then-which-one-of-the-following-is-not-correct-30030
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left[\begin{array}{cc}0 &amp; 1 \\ -1 &amp; 0\end{array}\right], B=\left[\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right], C=\left[\begin{array}{cc}0 &amp; -i \\ -i &amp; 0\end{array}\right]$,<br/>then which one of the following is not correct?</div>

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

answer:
<div class="solution">Let $\mathrm{A}=\left[\begin{array}{cc}0 &amp; 1 \\ -1 &amp; 0\end{array}\right], \mathrm{B}=\left[\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right], \mathrm{C}=\left[\begin{array}{cc}0 &amp; -i \\ -i &amp; 0\end{array}\right]$<br/>Now, $A^{2}=\left[\begin{array}{cc}0 &amp; 1 \\ -1 &amp; 0\end{array}\right]\left[\begin{array}{cc}0 &amp; 1 \\ -1 &amp; 0\end{array}\right]=\left[\begin{array}{cc}-1 &amp; 0 \\ 0 &amp; -1\end{array}\right] \ldots . .$ (1)<br/>$\mathrm{B}^{2}=\left[\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right]\left[\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right]=\left[\begin{array}{cc}-1 &amp; 0 \\ 0 &amp; -1\end{array}\right] \ldots \ldots . .(2)$<br/>From (1) and (2), we have $\mathrm{A}^{2}=\mathrm{B}^{2}$<br/>Now, $C^{2}=\left[\begin{array}{cc}0 &amp; -i \\ -i &amp; 0\end{array}\right]\left[\begin{array}{cc}0 &amp; -i \\ -i &amp; 0\end{array}\right]=\left[\begin{array}{cc}-1 &amp; 0 \\ 0 &amp; -1\end{array}\right]$<br/>$\ldots \ldots . .(3)$<br/>From (2) and (3), we have $\mathrm{B}^{2}=\mathrm{C}^{2}$<br/>Now, $\mathrm{AB}=\left[\begin{array}{cc}0 &amp; 1 \\ -1 &amp; 0\end{array}\right]\left[\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right]=\left[\begin{array}{cc}0 &amp; -i \\ -i &amp; 0\end{array}\right]=\mathrm{C}$<br/>Now, we find<br/>$\mathrm{BA}=\left[\begin{array}{cc}i &amp; 0 \\ 0 &amp; -i\end{array}\right]\left[\begin{array}{cc}0 &amp; 1 \\ -1 &amp; 0\end{array}\right]=\left[\begin{array}{ll}0 &amp; i \\ i &amp; 0\end{array}\right] \neq \mathrm{C}$<br/>Hence $\mathrm{AB} \neq \mathrm{BA}$</div>


title: if-a-0-1-1-0-then-the-matrix-a-is-an
slug: if-a-0-1-1-0-then-the-matrix-a-is-an-93105
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}0 &amp; 1 \\ 1 &amp; 0\end{array}\right]$, then the matrix $\mathrm{A}$ is $/ \mathrm{an}$</div>

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

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

answer:
<div class="solution">$A=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$\mathrm{A}^{2}=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{ll}(1)(1)+(0)(0) &amp; (1)(0)+(0)(1) \\ (0)(1)+(1)(0) &amp; (0)(0)+(1)(1)\end{array}\right]$<br/>$=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]=\mathrm{A}$<br/>$\therefore$ It is involuntary matrix.</div>


title: if-a-0-1-1-0-then-the-value-of-a-4-is
slug: if-a-0-1-1-0-then-the-value-of-a-4-is-62435
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{cc}0 &amp; 1 \\ 1 &amp; 0\end{array}\right]$, then the value of $\mathbf{A}^{4}$ is</div>

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

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

answer:
<div class="solution">$\mathrm{A}=\left[\begin{array}{ll}0 &amp; 1 \\ 1 &amp; 0\end{array}\right]$<br/>$\mathrm{A}^{4}=\mathrm{A}^{2} \cdot \mathrm{A}^{2}$<br/>$=\left[\begin{array}{ll}0 &amp; 1 \\ 1 &amp; 0\end{array}\right]\left[\begin{array}{ll}0 &amp; 1 \\ 1 &amp; 0\end{array}\right] \cdot\left[\begin{array}{ll}0 &amp; 1 \\ 1 &amp; 0\end{array}\right]\left[\begin{array}{ll}0 &amp; 1 \\ 1 &amp; 0\end{array}\right]$<br/>$=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right] \cdot\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$</div>


title: if-a-0-1-and-b-1-0-then-what-is-a-b-equal-to
slug: if-a-0-1-and-b-1-0-then-what-is-a-b-equal-to-33075
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\{0,1\}$ and $\mathrm{B}=\{1,0\}$, then what is $\mathrm{A} \times \mathrm{B}$ equal to?</div>

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

answer:
<div class="solution">Let $A=\{0,1\}, B=\{1,0\}$<br/>$\mathrm{A} \times \mathrm{B}=\{(0,1),(1,1),(0,0),(1,0)\}$<br/>$\mathrm{A} \times \mathrm{A}=\{(0,0),(0,1),(1,1),(1,0)\}$<br/>$\therefore \mathrm{A} \times \mathrm{B}=\mathrm{A} \times \mathrm{A}$</div>


title: if-a-0-b-b-a-0-0-b-a-0-then-which-one-of-the-following-is-correct
slug: if-a-0-b-b-a-0-0-b-a-0-then-which-one-of-the-following-is-correct-97807
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|\begin{array}{lll}a &amp; b &amp; 0 \\ 0 &amp; a &amp; b \\ b &amp; 0 &amp; a\end{array}\right|=0$, then which one of the following is correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a}{b}$ is one of the cube roots of $-1$.</span> </div>

answer:
<div class="solution">$\left|\begin{array}{lll}\mathrm{a} &amp; \mathrm{b} &amp; 0 \\ 0 &amp; \mathrm{a} &amp; \mathrm{b} \\ \mathrm{b} &amp; 0 &amp; \mathrm{a}\end{array}\right|$<br/>$=a\left[a^{2}-0\right]-b\left[-b^{2}\right]+0$<br/>$=a^{3}+b^{3}=0$<br/>$\Rightarrow a^{3}=-b^{3}$<br/>$\Rightarrow\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{3}=-1$<br/>Hence, $\frac{\mathrm{a}}{\mathrm{b}}$ is one of the cube roots of $-1$</div>


title: if-a-1-0-1-and-b-1-2-0-1-such-that-a-2-b-then-what-is-the-value-of
slug: if-a-1-0-1-and-b-1-2-0-1-such-that-a-2-b-then-what-is-the-value-of-45007
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2011 (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 $\mathrm{A}=\left[\begin{array}{ll}\alpha &amp; 0 \\ 1 &amp; 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 &amp; 0 \\ 2 &amp; 1\end{array}\right]$<br/>such that $A^{2}=B$, then what is the value of $\alpha$?</div>

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

answer:
<div class="solution">Let $\mathrm{A}=\left[\begin{array}{ll}\alpha &amp; 0 \\ 1 &amp; 1\end{array}\right]$<br/>$\Rightarrow \quad \mathrm{A}^{2}=\left[\begin{array}{ll}\alpha &amp; 0 \\ 1 &amp; 1\end{array}\right]\left[\begin{array}{ll}\alpha &amp; 0 \\ 1 &amp; 1\end{array}\right]$<br/>$\Rightarrow \quad \mathrm{A}^{2}=\left[\begin{array}{cc}\alpha^{2} &amp; 0 \\ \alpha+1 &amp; 1\end{array}\right]$<br/>But it is given that<br/>$A^{2}=B$<br/>$\Rightarrow \quad\left[\begin{array}{ll}\alpha^{2} &amp; 0 \\ \alpha+1 &amp; 1\end{array}\right]=\left[\begin{array}{ll}1 &amp; 0 \\ 2 &amp; 1\end{array}\right]$<br/>$\Rightarrow \quad \alpha+1=2$<br/>$\Rightarrow \quad \alpha=1$</div>


title: if-a-1-0-2-3-is-a-2-2-matrix-and-f-x-x-2-x-2-is-a-polynomial-then-what-is-f-a
slug: if-a-1-0-2-3-is-a-2-2-matrix-and-f-x-x-2-x-2-is-a-polynomial-then-what-is-f-a-97252
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]$ is a $2 \times 2$ matrix and $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-\mathrm{x}+2$ is a polynomial, then what is $\mathrm{f}(\mathrm{A})$?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{ll}2 &amp; 6 \\ 0 &amp; 8\end{array}\right]$</span> </div>

answer:
<div class="solution">Given that, $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]$<br/>$\mathrm{A}^{2}=\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]=\left[\begin{array}{lr}1 &amp; 2+6 \\ 0 &amp; 9\end{array}\right]=\left[\begin{array}{ll}1 &amp; 8 \\ 0 &amp; 9\end{array}\right]$<br/>Since, $f(x)=x^{2}-x+2$<br/>Putting A in place of $x$ $f(A)=A^{2}-A+2 I$<br/>$=\left[\begin{array}{ll}1 &amp; 8 \\ 0 &amp; 9\end{array}\right]-\left[\begin{array}{ll}1 &amp; 2 \\ 0 &amp; 3\end{array}\right]+\left[\begin{array}{ll}2 &amp; 0 \\ 0 &amp; 2\end{array}\right]=\left[\begin{array}{ll}1-1+2 &amp; 8-2+0 \\ 0-0+0 &amp; 9-3+2\end{array}\right]$<br/>$=\left[\begin{array}{ll}2 &amp; 6 \\ 0 &amp; 8\end{array}\right]$</div>


title: if-a-1-1-2-1-and-b-0-1-1-2-then-what-is-b-1-a-1-equal-to
slug: if-a-1-1-2-1-and-b-0-1-1-2-then-what-is-b-1-a-1-equal-to-48978
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 1 &amp; 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}0 &amp; -1 \\ 1 &amp; 2\end{array}\right]$, then what is $\mathrm{B}^{-1} \mathrm{~A}^{-1}$ equal to?</div>

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

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

answer:
<div class="solution">$|\mathrm{A}|=-1,|\mathrm{~B}|=1$<br/>$\mathrm{~A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{Adj} \mathrm{A}=\frac{1}{-1}\left[\begin{array}{cc}1 &amp; -1 \\ -2 &amp; 1\end{array}\right]^{\mathrm{T}}=\left[\begin{array}{cc}-1 &amp; 2 \\ 1 &amp; -1\end{array}\right]$<br/>$\mathrm{B}^{-1}=\frac{1}{\mid \mathrm{B}} \mid \mathrm{adj} \mathrm{B}=\frac{1}{1}\left[\begin{array}{cc}2 &amp; -1 \\ 1 &amp; 0\end{array}\right]^{\mathrm{T}}=\left[\begin{array}{cc}2 &amp; 1 \\ -1 &amp; 0\end{array}\right]$<br/>$\mathrm{B}^{-1} \mathrm{~A}^{-1}=\left[\begin{array}{cc}2 &amp; 1 \\ -1 &amp; 0\end{array}\right]\left[\begin{array}{cc}-1 &amp; 2 \\ 1 &amp; -1\end{array}\right]=\left[\begin{array}{cc}-1 &amp; 3 \\ 1 &amp; -2\end{array}\right]$</div>


title: if-a-1-1-2-2-2-what-is-det-a
slug: if-a-1-1-2-2-2-what-is-det-a-43218
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}^{-1}=\left[\begin{array}{cr}1 &amp; -2 \\ -2 &amp; 2\end{array}\right]$, what is det (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">$-2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">We know that det $(\mathrm{A})=\frac{1}{\operatorname{det}\left(\mathrm{A}^{-1}\right)}$<br/>$\mathbf{A}^{-1}=\left[\begin{array}{rr}1 &amp; -2 \\ -2 &amp; 2\end{array}\right]$<br/>$\Rightarrow\left|\mathrm{A}^{-1}\right|=\left|\begin{array}{rr}1 &amp; -2 \\ -2 &amp; 2\end{array}\right|=2-4=-2 \Rightarrow \operatorname{det}(\mathrm{A})=-\frac{1}{2}$</div>


title: if-a-1-2-0-3-2-4-then-the-matrix-x-for-which-2-x-3-a-0-holds-true-is
slug: if-a-1-2-0-3-2-4-then-the-matrix-x-for-which-2-x-3-a-0-holds-true-is-64584
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ccc}1 &amp; 0 &amp; -2 \\ 2 &amp; -3 &amp; 4\end{array}\right]$, then the matrix $\mathrm{X}$ for which $2 \mathrm{X}+3 \mathrm{~A}=$ 0 holds true is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{ccc}-\frac{3}{2} &amp; 0 &amp; -3 \\ -3 &amp; -\frac{9}{2} &amp; -6\end{array}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{ccc}\frac{3}{2} &amp; 0 &amp; -3 \\ 3 &amp; -\frac{9}{2} &amp; -6\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{ccc}\frac{3}{2} &amp; 0 &amp; 3 \\ 3 &amp; \frac{9}{2} &amp; 6\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left[\begin{array}{ccc}-\frac{3}{2} &amp; 0 &amp; 3 \\ -3 &amp; \frac{9}{2} &amp; -6\end{array}\right]$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{ccc}-\frac{3}{2} &amp; 0 &amp; 3 \\ -3 &amp; \frac{9}{2} &amp; -6\end{array}\right]$</span> </div>

answer:
<div class="solution">$\because 2 X+3 A=0$<br/>$\Rightarrow x=\frac{-3}{2} A$<br/>$\Rightarrow x=\frac{-3}{2}\left[\begin{array}{ccc}1 &amp; 0 &amp; -2 \\ 2 &amp; -3 &amp; 4\end{array}\right]$<br/>$\Rightarrow X=\left[\begin{array}{ccc}-\frac{3}{2} &amp; 0 &amp; 3 \\ -3 &amp; \frac{9}{2} &amp; -6\end{array}\right]$</div>


title: if-a-1-2-2-3-and-a-2-ka-i-2-o-where-i-2-is-the-2-2-identity-matrix-then-what-is-the-value-of-k
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-a-1-2-2-3-and-a-2-ka-i-2-o-where-i-2-is-the-2-2-identity-matrix-then-what-is-the-value-of-k-46027

question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right]$ and $\mathbf{A}^{2}-\mathrm{kA}-\mathrm{I}_{2}=\mathrm{O}$, where $\mathrm{I}_{2}$ is the $2 \times 2$ identity matrix, then what is the value of $\mathrm{k}$ ?</div>

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

answer:
<div class="solution">$\mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right]$<br/>$\mathrm{A}^{2}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right]\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right]=\left[\begin{array}{ll}1+4 &amp; 2+6 \\ 2+6 &amp; 4+9\end{array}\right]=\left[\begin{array}{cc}5 &amp; 8 \\ 8 &amp; 13\end{array}\right]$<br/>$\mathrm{A}^{2}-\mathrm{kA}-\mathrm{I}_{2}=\left[\begin{array}{cc}5 &amp; 8 \\ 8 &amp; 13\end{array}\right]-\left[\begin{array}{cc}\mathrm{k} &amp; 2 \mathrm{k} \\ 2 \mathrm{k} &amp; 3 \mathrm{k}\end{array}\right]-\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{cc}5-k-1 &amp; 8-2 k-0 \\ 8-2 k-0 &amp; 13-3 k-1\end{array}\right]$<br/>$=\left[\begin{array}{cc}4-k &amp; 8-2 k \\ 8-2 k &amp; 12-3 k\end{array}\right]$<br/>Given, $A^{2}-k A-I_{2}=0$<br/>$\therefore 4-k=0 \Rightarrow k=4$</div>


title: if-a-1-2-2-3-and-b-1-1-0-0-then-what-is-determinant-of-ab
slug: if-a-1-2-2-3-and-b-1-1-0-0-then-what-is-determinant-of-ab-87309
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left(\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right)$ and $\mathrm{B}=\left(\begin{array}{ll}1 &amp; 0 \\ 1 &amp; 0\end{array}\right)$ then what is determinant of AB?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">20</span> </li> </ul>

answer:
<div class="solution">$\begin{array}{l}\text { Let } \mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{ll}1 &amp; 0 \\ 1 &amp; 0\end{array}\right] \\ \mathrm{AB}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 3\end{array}\right]\left[\begin{array}{ll}1 &amp; 0 \\ 1 &amp; 0\end{array}\right] \Rightarrow \mathrm{AB}=\left[\begin{array}{ll}3 &amp; 0 \\ 5 &amp; 0\end{array}\right] \\ \text { So, }|\mathrm{AB}|=0 &amp; (\because \text { one column is zero })\end{array}$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-1-2-3-1-3-2-1-4-3-and-b-1-6-5-2-12-10-1-6-5-then-which-of-the-following-isare-correct-1-a-and-b-commute-2-a-b-is-a-null-matrix-select-the-correct

slug:
if-a-1-2-3-1-3-2-1-4-3-and-b-1-6-5-2-12-10-1-6-5-then-which-of-the-following-isare-correct-1-a-and-b-commute-2-a-b-is-a-null-matrix-select-the-correct-18542

question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ccc}1 &amp; 1 &amp; -1 \\ 2 &amp; -3 &amp; 4 \\ 3 &amp; -2 &amp; 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}-1 &amp; -2 &amp; -1 \\ 6 &amp; 12 &amp; 6 \\ 5 &amp; 10 &amp; 5\end{array}\right]$ then which of the following is/are correct? <br/>$1.$ $A$ and $B$ commute.<br/>$2. A B$ is a null matrix. Select the correct answer using the code given below:</div>

answer:
<div class="solution">$\mathrm{A}=\left[\begin{array}{ccc}1 &amp; 1 &amp; -1 \\ 2 &amp; -3 &amp; 4 \\ 3 &amp; -2 &amp; 3\end{array}\right] ; \mathrm{B}=\left[\begin{array}{ccc}-1 &amp; -2 &amp; -1 \\ 6 &amp; 12 &amp; 6 \\ 5 &amp; 10 &amp; 5\end{array}\right]$<br/>$\mathrm{AB}=\left[\begin{array}{lll}0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0\end{array}\right]$<br/>$\mathrm{BA}=\left[\begin{array}{ccc}-8 &amp; 7 &amp; -10 \\ 48 &amp; -42 &amp; 60 \\ 40 &amp; -35 &amp; 50\end{array}\right]$<br/>as $\mathrm{AB} \neq \mathrm{BA}$<br/>So $\mathrm{A}$ and $\mathrm{B}$ are not commute.</div>


title: if-a-1-2-3-2-1-2-3-2-1-then-which-one-of-the-following-is-correct
slug: if-a-1-2-3-2-1-2-3-2-1-then-which-one-of-the-following-is-correct-15180
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left[\begin{array}{ccc}1 &amp; -2 &amp; -3 \\ 2 &amp; 1 &amp; -2 \\ 3 &amp; 2 &amp; 1\end{array}\right]$, then which one of the following is<br/>correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A$ is non-singular matrix</span> </div>

answer:
<div class="solution">Here we see that its diagonal elements are not zero, so it is not anti-symmetric matrix. Now, $|\mathrm{A}|=1(1+4)+2(2+6)-3(4-3)$<br/>$=5+16-3=18 \neq 0$<br/>Hence, it is non-singular matrix.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-1-2-3-4-and-r-1-1-1-3-2-2-3-1-3-4-4-3-4-4-is-a-relation-on-a-a-then-which-one-of-the-following-is-correct

slug:
if-a-1-2-3-4-and-r-1-1-1-3-2-2-3-1-3-4-4-3-4-4-is-a-relation-on-a-a-then-which-one-of-the-following-is-correct-17367

question:
<div class="question">If $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{R}=\{(1,1),(1,3),(2,2),(3,1),$,<br/>$(3,4),(4,3),(4,4)\}$ is a relation on $A \times A$, 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">$R$ is reflexive</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$R$ is symmetric and transitive</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$R$ istransitive but not reflexive</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$R$ is neither reflexive nor transitive</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$R$ is neither reflexive nor transitive</span> </div>

answer:
<div class="solution">Since, $3 \in \mathbf{A}$<br/>but $(3,3) \notin \mathrm{R}$<br/>So, it is notreflexive. and $(3,4) \in \mathrm{R}$ and $(4,3) \in \mathrm{R}$<br/>but $(3,3) \notin \mathrm{R}$<br/>So, it is also not transitive. Hence, $R$ is neither reflexive nor transitive.</div>


title: if-a-1-2-5-6-and-b-1-2-3-then-what-is-a-b-b-a-equal-to
slug: if-a-1-2-5-6-and-b-1-2-3-then-what-is-a-b-b-a-equal-to-18273
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\{1,2,5,6\}$ and $B=\{1,2,3\}$, then what is $(A \times B) \cap(B \times A)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{(1,1),(2,1),(6,1),(3,2)\}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\{(1,1),(1,2),(2,1),(2,2)\}$</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">$\{(1,1),(2,2)\}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\{(1,1),(1,2),(2,5),(2,6)\}$</span> </li> </ul>

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

answer:
<div class="solution">LetA $=\{1,2,5,6\}$ and $\mathrm{B}=\{1,2,3\}$ $\mathrm{A} \times \mathrm{B}=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)$<br/>$(5,1),(5,2),(5,3),(6,1),(6,2),(6,3)\}$<br/>and $\mathrm{B} \times \mathrm{A}=\{(1,1),(1,2),(1,5),(1,6),(2,1),(2,2)$,<br/>$(2,5),(2,6),(3,1),(3,2),(3,5),(3,6)\}$<br/>$\Rightarrow \quad(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{B} \times \mathrm{A})$<br/>$=\{(1,1),(1,2),(2,1),(2,2)\}$</div>


title: if-a-1-2-b-2-3-and-c-3-4-then-what-is-the-cardinality-of-a-b-a-c
slug: if-a-1-2-b-2-3-and-c-3-4-then-what-is-the-cardinality-of-a-b-a-c-76197
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\{1,2\}, B=\{2,3\}$ and $C=\{3,4\}$, then what is the cardinality of $(A \times B) \cap(A \times C)$</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">6</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$\mathrm{A} \times \mathrm{B}=\{(1,2),(1,3),(2,2),(2,3)\}$<br/>$\mathrm{A} \times \mathrm{C}=\{(1,3),(1,4),(2,3),(2,4)\}$<br/>Number of elementsin $(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{A} \times \mathrm{C})=2$</div>


title: if-a-1-3-5-7-then-what-is-the-cardinality-of-the-power-set-p-a
slug: if-a-1-3-5-7-then-what-is-the-cardinality-of-the-power-set-p-a-71435
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (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 $A=\{1,3,5,7\}$, then what is the cardinality of the power $\operatorname{set} P(A)$?</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">15</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">17</span> </li> </ul>

answer:
<div class="solution">Number of element in set A is Cordinality of the power set $\mathrm{P}(\mathrm{A})=2^{4}=16$</div>


title: if-a-1-b-1-c-1-0-such-that-1-a-1-1-1-1-b-1-1-1-1-c-then-what-is-equal-to
slug: if-a-1-b-1-c-1-0-such-that-1-a-1-1-1-1-b-1-1-1-1-c-then-what-is-equal-to-69639
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a^{-1}+b^{-1}+c^{-1}=0$ such that $\left|\begin{array}{ccc}1+a &amp; 1 &amp; 1 \\ 1 &amp; 1+b &amp; 1 \\ 1 &amp; 1 &amp; 1+c\end{array}\right|=\lambda$,<br/>then what is $\lambda$ equal to?</div>

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

answer:
<div class="solution">Let $a^{-1}+b^{-1}+c^{-1}=0$<br/>$\Rightarrow \quad \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$<br/>$\Rightarrow \quad \frac{b c+a c+a b}{a b c}=0$<br/>$\Rightarrow a b+b c+c a=0$ ...(1)<br/>Consider $\left|\begin{array}{ccc}1+a &amp; 1 &amp; 1 \\ 1 &amp; 1+b &amp; 1 \\ 1 &amp; 1 &amp; 1+c\end{array}\right|=\lambda$<br/>$\Rightarrow(1+a)[(1+b)(1+c)-1]-(1+c-1)+(1-1-b)=\lambda$<br/>$\Rightarrow(1+a)(c+b+b c)-c-b=\lambda$<br/>$\Rightarrow \mathrm{bc}+\mathrm{ac}+\mathrm{ab}+\mathrm{abc}=\lambda$<br/>$\Rightarrow$ abc $=\lambda \quad$ (using (1))</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
title:
if-a-1-x-b-1-y-c-1-a-2-x-b-2-y-c-2-1-a-1-a-2-b-1-b-2-2-b-1-b-2-c-1-c-2-3-c-1-c-2-a-1-a-2-then-x-y-is-equal-to-which-one-of-the-following

slug:
if-a-1-x-b-1-y-c-1-a-2-x-b-2-y-c-2-1-a-1-a-2-b-1-b-2-2-b-1-b-2-c-1-c-2-3-c-1-c-2-a-1-a-2-then-x-y-is-equal-to-which-one-of-the-following-28413

question:
<div class="question">If $\left(a_{1} / x\right)+\left(b_{1} / y\right)=c_{1},\left(a_{2} / x\right)+\left(b_{2} / y\right)=c_{2}$<br/>$\Delta_{1}=\left|\begin{array}{ll}a_{1} &amp; b_{1} \\ a_{2} &amp; b_{2}\end{array}\right|, \Delta_{2}=\left|\begin{array}{ll}b_{1} &amp; c_{1} \\ b_{2} &amp; c_{2}\end{array}\right|, \quad \Delta_{3}=\left|\begin{array}{ll}c_{1} &amp; a_{1} \\ c_{2} &amp; a_{2}\end{array}\right|$,<br/>then $(x, y)$ is equal to which one of the following?</div>

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

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

answer:
<div class="solution">Let $\frac{1}{x}=u, \frac{1}{y}=v$<br/>$\therefore a_{1} u+b_{1} v=c_{1}$ and $a_{2} u+b_{2} v=c_{2}$<br/>Using the method of cross multiplication $\frac{u}{b_{1} c_{2}-b_{2} c_{1}}=\frac{v}{c_{1} a_{2}-c_{2} a_{1}}=\frac{-1}{a_{1} b_{2}-a_{2} b_{1}}$<br/>$\Rightarrow \frac{\frac{1}{x}}{\left|\begin{array}{ll}b_{1} &amp; c_{1} \\ b_{2} &amp; c_{2}\end{array}\right|}=\frac{\frac{1}{y}}{\left|\begin{array}{ll}c_{1} &amp; a_{1} \\ c_{2} &amp; a_{2}\end{array}\right|}$ $=\frac{-1}{\left|\begin{array}{ll}a_{1} &amp; b_{1} \\ a_{2} &amp; b_{2}\end{array}\right|}$<br/>$\begin{aligned} &amp; \frac{\frac{1}{x}}{\Delta_{2}}=\frac{\frac{1}{y}}{\Delta_{3}}=-\frac{1}{\Delta_{1}} \\ \therefore &amp; \frac{1}{x}=-\frac{\Delta_{2}}{\Delta_{1}} \\ \text { and } \frac{1}{y}=-\frac{\Delta_{3}}{\Delta_{1}} \\ \Rightarrow x &amp;=-\frac{\Delta_{1}}{\Delta_{2}} \text { and } y=-\frac{\Delta_{1}}{\Delta_{3}} \end{aligned}$</div>


tags: ['Mathematics', 'Determinants', 'JEE Main']
answer: <div class="solution">It is based on fundamental concept.</div>
source_db: MarksBatch2_P1.db
title:
if-a-1-x-b-1-y-c-1-z-0-a-2-x-b-2-y-c-2-z-0-a-3-x-b-3-y-c-3-z-0-and-a-1-a-2-a-3-b-1-b-2-b-3-c-1-c-2-c-3-0-then-the-given-system-has

slug:
if-a-1-x-b-1-y-c-1-z-0-a-2-x-b-2-y-c-2-z-0-a-3-x-b-3-y-c-3-z-0-and-a-1-a-2-a-3-b-1-b-2-b-3-c-1-c-2-c-3-0-then-the-given-system-has-39961

question:
<div class="question">If $a_1 x+b_1 y+c_1 z=0, a_2 x+b_2 y+c_2 z=0 \quad a_3 x+b_3 y+c_3 z=0$ and $\left|\begin{array}{lll}a_1 &amp; b_1 &amp; c_1 \\ a_2 &amp; b_2 &amp; c_2 \\ a_3 &amp; b_3 &amp; c_3\end{array}\right|=0$, then the given system has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">One trivial and one non-trivial solution</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">No solution</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">One solution</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Infinite solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Infinite solution</span> </div>


title: if-a-2-1-1-b-1-1-0-c-5-1-1-then-what-is-the-unit-vector-parallel-to-a-b-c-in-the-opposite-direction
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-a-2-1-1-b-1-1-0-c-5-1-1-then-what-is-the-unit-vector-parallel-to-a-b-c-in-the-opposite-direction-69508

question:
<div class="question">If $\overrightarrow{\mathrm{a}}=(2,1,-1), \overrightarrow{\mathrm{b}}=(1,-1,0), \overrightarrow{\mathrm{c}}=(5,-1,1)$, then what is the<br/>unit vector parallel to $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}$ in the opposite direction?</div>

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

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

answer:
<div class="solution">Let $\overrightarrow{\mathrm{a}}=(2,1,-1), \overrightarrow{\mathrm{b}}=(1,-1,0), \overrightarrow{\mathrm{c}}=(5,-1,1)$<br/>$\therefore \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}=(-2,1,-2)$<br/>Let $\overrightarrow{\mathrm{n}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$ be the unit vector which is $\|$ to $(-2$<br/>$1,-2$ ) in the opposite direction.<br/>$\therefore \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1$ and $\left|\begin{array}{ccc}\hat{\mathrm{i}} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ \mathrm{x} &amp; \mathrm{y} &amp; \mathrm{z} \\ -2 &amp; 1 &amp; -2\end{array}\right|=0$<br/>$\Rightarrow x=-2 y, y=y, z=-2 y$<br/>$x^{2}+y^{2}+z^{2}=1 \Rightarrow 4 y^{2}+y^{2}+4 y^{2}=1 \Rightarrow y=\pm \frac{1}{3}$<br/>Hence, the Required vector<br/>$\hat{n}=\frac{2}{3} \hat{i}-\frac{\hat{j}}{3}+\frac{2}{3} \hat{k}$</div>


title: if-a-2-1-7-5-then-that-is-a-3-a-1-equal-to
slug: if-a-2-1-7-5-then-that-is-a-3-a-1-equal-to-99984
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$7 I$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left[\begin{array}{ll}2 &amp; 7 \\ 1 &amp; 5\end{array}\right]$ then that is $A+3 A^{-1}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 I$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 I$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$7 I$</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">D</span> <span class="option-data">None of these where $I$ is the identity matrix order $2 .$</span> </li> </ul>

answer:
<div class="solution">$\mathrm{A}=\left[\begin{array}{ll}2 &amp; 7 \\ 1 &amp; 5\end{array}\right]$<br/>Now, $\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})$<br/>$=\frac{1}{(10-7)}\left[\begin{array}{cc}5 &amp; -7 \\ -1 &amp; 2\end{array}\right]$<br/>$=\frac{1}{3}\left[\begin{array}{cc}5 &amp; -7 \\ -1 &amp; 2\end{array}\right]$<br/>$\begin{aligned} \Rightarrow \quad &amp; 3 \mathrm{~A}^{-1}=\left[\begin{array}{cc}5 &amp; -7 \\ -1 &amp; 2\end{array}\right] \\ &amp; \text { Now, } \mathrm{A}+3 \mathrm{~A}^{-1}=\left[\begin{array}{ll}2 &amp; 7 \\ 1 &amp; 5\end{array}\right]+\left[\begin{array}{cc}5 &amp; -7 \\ -1 &amp; 2\end{array}\right] \end{aligned}$<br/>$=\left[\begin{array}{ll}7 &amp; 0 \\ 0 &amp; 7\end{array}\right]=7\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]=7 I$ where $I$ is Identity Matrix.</div>


title: if-a-2-2-2-2-then-what-is-a-n-equal-to
slug: if-a-2-2-2-2-then-what-is-a-n-equal-to-37650
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}2 &amp; 2 \\ 2 &amp; 2\end{array}\right]$, then what is $\mathrm{A}^{\mathrm{n}}$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{cc}2^{n} &amp; 2^{n} \\ 2^{n} &amp; 2^{n}\end{array}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\begin{array}{cc}2 n &amp; 2 n \\ 2 n &amp; 2 n\end{array}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{ll}2^{2 \mathrm{n}-1} &amp; 2^{2 \mathrm{n}-1} \\ 2^{2 \mathrm{n}-1} &amp; 2^{2 \mathrm{n}-1}\end{array}\right]$</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">D</span> <span class="option-data">$\left[\begin{array}{cc}2^{2 n+1} &amp; 2^{2 n+1} \\ 2^{2 n+1} &amp; 2^{2 n+1}\end{array}\right]$</span> </li> </ul>

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

answer:
<div class="solution">Given matrix is:<br/>$\mathrm{A}=\left[\begin{array}{ll}2 &amp; 2 \\ 2 &amp; 2\end{array}\right]$<br/>$\mathrm{A}^{2}=\left[\begin{array}{ll}2 &amp; 2 \\ 2 &amp; 2\end{array}\right]\left[\begin{array}{ll}2 &amp; 2 \\ 2 &amp; 2\end{array}\right]=\left[\begin{array}{ll}4+4 &amp; 4+4 \\ 4+4 &amp; 4+4\end{array}\right]=\left[\begin{array}{ll}2^{3} &amp; 2^{3} \\ 2^{3} &amp; 2^{3}\end{array}\right]$<br/>$\begin{aligned} \mathrm{A}^{3} &amp;=\left[\begin{array}{ll}8 &amp; 8 \\ 8 &amp; 8\end{array}\right]\left[\begin{array}{ll}2 &amp; 2 \\ 2 &amp; 2\end{array}\right]=\left[\begin{array}{ll}16+16 &amp; 16+16 \\ 16+16 &amp; 16+16\end{array}\right] \\ &amp;=\left[\begin{array}{ll}32 &amp; 32 \\ 32 &amp; 32\end{array}\right]=\left[\begin{array}{cc}2^{5} &amp; 2^{5} \\ 2^{5} &amp; 2^{5}\end{array}\right] \end{aligned}$<br/>Going this way we get<br/>$\mathrm{A}^{4}=\left[\begin{array}{cc}2^{7} &amp; 2^{7} \\ 2^{7} &amp; 2^{7}\end{array}\right] \Rightarrow \mathrm{A}^{\mathrm{n}}=\left[\begin{array}{cc}2^{2 \mathrm{n}-1} &amp; 2^{2 \mathrm{n}-1} \\ 2^{2 \mathrm{n}-1} &amp; 2^{2 \mathrm{n}-1}\end{array}\right]$</div>


title: if-a-2-2-2-2-then-which-one-of-the-following-is-correct
slug: if-a-2-2-2-2-then-which-one-of-the-following-is-correct-43314
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left(\begin{array}{cc}-2 &amp; 2 \\ 2 &amp; -2\end{array}\right)$, 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">$A^{2}=-2 A$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$A^{2}=-4 A$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$A^{2}=-3 A$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{A}^{2}=4 \mathrm{~A}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{A}=\left(\begin{array}{cc}-2 &amp; 2 \\ 2 &amp; -2\end{array}\right)$<br/>$\mathrm{A}^{2}=\left(\begin{array}{cc}-2 &amp; 2 \\ 2 &amp; -2\end{array}\right)\left(\begin{array}{cc}-2 &amp; 2 \\ 2 &amp; -2\end{array}\right)$<br/>$=\left(\begin{array}{cc}4+4 &amp; -4-4 \\ -4-4 &amp; 4+4\end{array}\right)=\left(\begin{array}{cc}8 &amp; -8 \\ -8 &amp; 8\end{array}\right)$<br/>$=-4\left(\begin{array}{cc}-2 &amp; 2 \\ 2 &amp; -2\end{array}\right)=-4 \mathrm{~A}$</div>


title: if-a-2-2-and-det-a-3-125-then-is-equal-to
slug: if-a-2-2-and-det-a-3-125-then-is-equal-to-48081
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{cc}\alpha &amp; 2 \\ 2 &amp; \alpha\end{array}\right]$ and $\operatorname{det}\left(\mathrm{A}^{3}\right)=125$, then $\alpha$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pm 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pm 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\pm 3$</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">$\pm 5$</span> </li> </ul>

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

answer:
<div class="solution">$\quad A=\left[\begin{array}{ll}\alpha &amp; 2 \\ 2 &amp; \alpha\end{array}\right], \operatorname{det}\left(A^{3}\right)=125$<br/>$\left|A^{3}\right|=125 \Rightarrow|A|=5$<br/>$\therefore \alpha 2-4=5 \Rightarrow \alpha^{2}=9 \Rightarrow \alpha=\pm 3$</div>


title: if-a-2-3-b-1-4-c-0-2-and-d-x-y-are-the-vertices-of-a-parallelogram-then-what-is-the-value-of-x-y
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
slug:
if-a-2-3-b-1-4-c-0-2-and-d-x-y-are-the-vertices-of-a-parallelogram-then-what-is-the-value-of-x-y-98318

question:
<div class="question">If $\mathrm{A}(2,3), \mathrm{B}(1,4), \mathrm{C}(0-2)$ and $\mathrm{D}(\mathrm{x}, \mathrm{y})$ are the vertices of a<br/>parallelogram, then what is the value of $(\mathrm{x}, \mathrm{y})$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(1,-3)$</span> </div>

answer:
<div class="solution">As given : $\mathrm{A}(2,3), \mathrm{B}(1,4), \mathrm{C}(0,-2)$ and $\mathrm{D}(\mathrm{x}, \mathrm{y})$ are the<br/>vertices of a parallelogram. Diagonals of a parallelogram bisect each other So, mid-point are same for both diagonals $\mathrm{AC}$ and $\mathrm{BD}$.<br/>$\frac{2+0}{2}=\frac{1+x}{2}$ and $\frac{3-2}{2}=\frac{4+y}{2}$<br/>$\Rightarrow \quad x=1$ and $y=-3$<br/>$\Rightarrow \quad \mathrm{D}(\mathrm{x}, \mathrm{y})=(1,-3)$</div>


title: if-a-2-3-b-4-5-c-5-6-then-what-is-the-number-of-elements-of-a-b-c
slug: if-a-2-3-b-4-5-c-5-6-then-what-is-the-number-of-elements-of-a-b-c-44896
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\{2,3\}, B=\{4,5\}, C=\{5,6\}$, then what is the number of elements of $A \times(B \cap C)$ ?</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">4</span> </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">8</span> </li> </ul>

answer:
<div class="solution">Given $\mathrm{A}=\{2,3\}, \mathrm{B}=\{4,5\}, \mathrm{C}=\{5,6\}$<br/>: $B \cap C=\{5\}$<br/>$\Rightarrow \quad A \times(B \cap C)=\{2,3\} \times\{5\}=\{(2,5),(3,5)\}$<br/>Hence, required number of elements in $A \times(B \cap C)=2$</div>


title: if-a-2-a-2-3-a-3-are-in-gp-then-what-is-the-fourth-term-of-the-gp
slug: if-a-2-a-2-3-a-3-are-in-gp-then-what-is-the-fourth-term-of-the-gp-30019
question: <div class="question">If $a, 2 a+2,3 a+3$ are in GP, then what is the fourth term of the GP?</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-13.5$</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">$13.5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-27$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">27</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-13.5$</span> </div>

answer:
<div class="solution">Since, a, $2 \mathrm{a}+2$ and $3 \mathrm{a}+3$ are in GP $\therefore(2 a+2)^{2}=a(3 a+3)$<br/>$\Rightarrow 4 \mathrm{a}^{2}+4+8 \mathrm{a}=3 \mathrm{a}^{2}+3 \mathrm{a} \Rightarrow \mathrm{a}^{2}+5 \mathrm{a}+4=0$<br/>$\Rightarrow a(a+4)+1(a+4)=0 \Rightarrow(a+4)(a+1)=0$<br/>$\Rightarrow a+4=0$ or $a+1=0$<br/>$\Rightarrow a=-4$ or $-1$<br/>Let the fourth term be $\mathrm{x}$.<br/>$\frac{a}{2 a+2}=\frac{3 a+3}{x}$<br/>$\Rightarrow x=\frac{(3 a+3)(2 a+2)}{a}$<br/>When $\mathrm{a}=-4$<br/>$\mathrm{x}=-13.5$<br/>and $\mathrm{a}=-1, \mathrm{x}=0$<br/>So, the fourth term is $-13.5$</div>


title: if-a-2-and-b-3-then-a-b-2-a-b-2-is-equal-to
slug: if-a-2-and-b-3-then-a-b-2-a-b-2-is-equal-to-43824
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">36</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}|=2$ and $|\vec{b}|=3$, then $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">72</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">64</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">48</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">36</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \hat{n} \\ &amp; \therefore|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta=6 \sin \theta \\ &amp; \&amp;|\vec{a} \cdot \vec{b}|=|\vec{a}||\vec{b}| \cos \theta=6 \cos \theta \\ &amp;|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=(6 \sin \theta)^{2}+(6 \cos \theta)^{2} \\ &amp;=36\left(\sin ^{2} \theta+\cos ^{2} \theta\right) \\ &amp;=36 \end{aligned}$</div>


title: if-a-2-b-2-c-2-0-then-what-is-a-2-b-2-3-b-2-c-2-3-c-2-a-2-3-a-4-b-4-3-b-4-c-4-3-c-4-a-4-3-equal-to
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P1.db
slug:
if-a-2-b-2-c-2-0-then-what-is-a-2-b-2-3-b-2-c-2-3-c-2-a-2-3-a-4-b-4-3-b-4-c-4-3-c-4-a-4-3-equal-to-93009

question:
<div class="question">If $a^{2}+b^{2}+c^{2}=0$, then what is $\frac{\left(a^{4}-b^{4}\right)^{3}+\left(b^{4}-c^{4}\right)^{3}+\left(c^{4}-a^{4}\right)^{3}}{\left(a^{2}-b^{2}\right)^{3}+\left(b^{2}-c^{2}\right)^{3}+\left(c^{2}-a^{2}\right)^{3}}$<br/>equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{a}^{2} \mathrm{~b}^{2} \mathrm{c}^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-a^{2} b^{2} c^{2}$</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">$\mathrm{abc}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \mathrm{a}^{2} \mathrm{~b}^{2} \mathrm{c}^{2}$</span> </li> </ul>

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

answer:
<div class="solution">We know that if $\mathrm{x}+\mathrm{y}$ ' Hereboth in nominator and denominator $a^{4}-b^{4}+b^{4}-c^{4}+c^{4}-a^{4}=0$<br/>and $a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2}=0$<br/>Hence, $\frac{\left(a^{4}-b^{4}\right)^{3}+\left(b^{4}-c^{4}\right)^{3}+\left(c^{4}-a^{4}\right)^{3}}{\left(a^{2}-b^{2}\right)^{3}+\left(b^{2}-c^{2}\right)^{3}+\left(c^{2}-a^{2}\right)^{3}}$<br/>$=\frac{3\left(a^{4}-b^{4}\right)\left(b^{4}-c^{4}\right)\left(c^{4}-a^{4}\right)}{3\left(a^{2}-b^{2}\right)\left(b^{2}-c^{2}\right)\left(c^{2}-a^{2}\right)}$<br/>$=\left(a^{2}+b^{2}\right)\left(b^{2}+c^{2}\right)\left(c^{2}+a^{2}\right.$<br/>$=\left(-c^{2}\right)\left(-a^{2}\right)\left(-b^{2}\right)$<br/>$=-a^{2} b^{2} c^{2}$<br/>$\left(\because a^{2}+b^{2}+c^{2}=0\right)$</div>


title: if-a-2-b-3-and-a-b-6-then-what-is-a-b-equal-to
slug: if-a-2-b-3-and-a-b-6-then-what-is-a-b-equal-to-34160
tags: ['Mathematics', 'Vector Algebra', '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 $|\overrightarrow{\mathrm{a}}|=\sqrt{2},|\overrightarrow{\mathrm{b}}|=\sqrt{3}$ and $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=\sqrt{6}$, then what is $|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$<br/>equal to?</div>

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

answer:
<div class="solution">Consider $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}=2\left(|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}\right)$<br/>By putting the values of $|\overrightarrow{\mathrm{a}}|,|\overrightarrow{\mathrm{b}}|$ and $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|$, we get<br/>$6+|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}=2(2+3)$<br/>$\Rightarrow \quad|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|=2$</div>


title: if-a-2-b-3-c-0-and-a-b-b-c-c-a-b-c-then-what-is-the-value-of
slug: if-a-2-b-3-c-0-and-a-b-b-c-c-a-b-c-then-what-is-the-value-of-55621
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}=\overrightarrow{0}$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}=\lambda(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$, then<br/>what is the value of $\lambda ?$</div>

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

answer:
<div class="solution">$\vec{a}+2 \vec{b}+3 \vec{c}=\overrightarrow{0}$ ...(1)<br/>$\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\lambda(\vec{b} \times \vec{c})$ ...(2)<br/>$(1) \times \vec{b} \Rightarrow \vec{b} \times \vec{a}+2 \vec{b} \times \vec{b}+3 \vec{b} \times \vec{c}=0$<br/>$\Rightarrow-\vec{a} \times \vec{b}+0+3 \vec{b} \times \vec{c}=0$<br/>$\Rightarrow \vec{a} \times \vec{b}=3 \vec{b} \times \vec{c}$ ...(3)<br/>$(1) \times \vec{c} \Rightarrow \vec{c} \times \vec{a}+2 \vec{c} \times \vec{b}+3 \vec{c} \times \vec{c}=0$<br/>$\Rightarrow \vec{c} \times \vec{a}-2 \vec{b} \times \vec{c}+0=0$<br/>$\Rightarrow \vec{c} \times \vec{a}=2 \vec{b} \times \vec{c}$ ...(4)<br/>$\therefore(2) \Rightarrow \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}=\lambda(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$<br/>$\Rightarrow 3 \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+2(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\lambda(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$<br/>$\Rightarrow 6(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\lambda(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$<br/>$\therefore \lambda=6$</div>


title: if-a-2-b-5-and-a-b-8-then-what-is-the-value-of-a-b
slug: if-a-2-b-5-and-a-b-8-then-what-is-the-value-of-a-b-36444
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (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 $|\vec{a}|=2,|\vec{b}|=5$ and $|\vec{a} \times \vec{b}|=8$, then what is the value of<br/>$\vec{a} \cdot \vec{b} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">We know that<br/>$|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|^{2}+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2} \times|\overrightarrow{\mathbf{b}}|^{2}$<br/>$\therefore \quad 64+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=(4 \times 25)$<br/>$\Rightarrow|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^{2}=36$<br/>$\Rightarrow \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=6$</div>


title: if-a-2-b-5-and-a-b-then-what-is-a-b-equal-to
slug: if-a-2-b-5-and-a-b-then-what-is-a-b-equal-to-62693
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2014 (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 $|\vec{a}|=2,|\vec{b}|=5$ and $|\vec{a} \times \vec{b}|$, then what is $\vec{a} \cdot \vec{b}$ equal to ?</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">7</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9</span> </li> </ul>

answer:
<div class="solution">Given $|\mathrm{a}|=2,|\mathrm{~b}|=5$ and $|\mathrm{a} \times \mathrm{b}|=8$<br/>Also $|\mathrm{a} \times \mathrm{b}|=|\mathrm{a}| \cdot|\mathrm{b}| \cdot|\sin \theta|$<br/>$\Rightarrow|\sin \theta|=\frac{8}{2 \times 5}=\frac{4}{5}$<br/>$\Rightarrow|\cos \theta|=\frac{3}{5} \Rightarrow \cos \theta=\pm \frac{3}{5}$<br/>$\therefore \mathrm{a} \cdot \mathrm{b}=|\mathrm{a}| \cdot|\mathrm{b}| \cos \theta=2 \times 5 \times \frac{3}{5}=6$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-2-i-2-j-3-k-b-i-2-j-k-and-c-3-i-j-are-three-vectors-such-that-a-t-b-is-perpendicular-to-c-then-what-is-t-equal-to

slug:
if-a-2-i-2-j-3-k-b-i-2-j-k-and-c-3-i-j-are-three-vectors-such-that-a-t-b-is-perpendicular-to-c-then-what-is-t-equal-to-21685

question:
<div class="question">If $\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}$ are three<br/>vectors such that $\vec{a}+t \vec{b}$ is perpendicular to $\vec{c}$, then what is t equal to ?</div>

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

answer:
<div class="solution">$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=(2-\mathrm{t}) \hat{\mathrm{i}}+(2+2 \mathrm{t}) \hat{\mathrm{j}}+(3+\mathrm{t}) \hat{\mathrm{k}}$<br/>$(\vec{a}+t \vec{b})$ and $\vec{c}$ is perpendicular. Therefore,<br/>$(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{c}}=0$<br/>$3(2-\mathrm{t})+2+2 \mathrm{t}=0$<br/>$6-3 \mathrm{t}+2 \mathrm{t}+2=0$<br/>$\mathrm{t}=8$</div>


title: if-a-2-i-3-j-4-k-and-b-3-i-2-j-k-are-perpendicular-then-what-is-the-value-of
slug: if-a-2-i-3-j-4-k-and-b-3-i-2-j-k-are-perpendicular-then-what-is-the-value-of-97738
tags: ['Mathematics', 'Vector Algebra', '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 $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}$ are perpendicular,<br/>then what is the value of $\lambda ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

answer:
<div class="solution">$\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}$<br/>Given, $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ are $\perp^{\mathrm{r}}$.<br/>$\therefore \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0$<br/>$\Rightarrow \mathrm{a}_{1} \mathrm{a}_{2}+\mathrm{b}_{1} \mathrm{~b}_{2}+\mathrm{c}_{1} \mathrm{c}_{2}=0$<br/>$\Rightarrow 6+6-4 \lambda=0$<br/>$\Rightarrow 4 \lambda=12 \Rightarrow \lambda=3 .$</div>


title: if-a-2-i-3-j-k-b-i-4-j-2-k-then-what-is-a-b-a-b-equal-to
slug: if-a-2-i-3-j-k-b-i-4-j-2-k-then-what-is-a-b-a-b-equal-to-75740
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}} ; \quad$ then what is<br/>$(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$ equal to ?</div>

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

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

answer:
<div class="solution">Given vectors are:<br/>$\begin{aligned} &amp; \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-3 \overline{\mathrm{j}}-\hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{b}}=\overline{\mathrm{i}}+4 \overline{\mathrm{j}}-2 \overrightarrow{\mathrm{k}} \\ \Rightarrow \quad &amp; \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}-\hat{\mathrm{k}})+(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}) \\ &amp;=3 \overline{\mathrm{i}}+\overline{\mathrm{j}}-3 \hat{\mathrm{k}} \end{aligned}$<br/>and $\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}-3 \overline{\mathrm{j}}-\hat{\mathrm{k}})-(\overline{\mathrm{i}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$<br/>$=\overline{\mathrm{i}}-7 \overline{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\therefore(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \times(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})=\left|\begin{array}{ccc}\hat{\mathrm{i}} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ 3 &amp; 1 &amp; -3 \\ 1 &amp; -7 &amp; 1\end{array}\right|$<br/>$=\hat{\mathrm{i}}\left|\begin{array}{cc}1 &amp; -3 \\ -7 &amp; 1\end{array}\right|-\hat{\mathrm{j}}\left|\begin{array}{cc}3 &amp; -3 \\ 1 &amp; 1\end{array}\right|+\widehat{k}\left|\begin{array}{cc}3 &amp; 1 \\ 1 &amp; -7\end{array}\right|$<br/>$=\hat{\mathrm{i}}(1-21)-\hat{\mathrm{j}}(3+3)+\mathrm{k}(-21-1)$<br/>$=-20 \hat{\mathrm{i}}-6 \widehat{\mathrm{j}}-22 \hat{\mathrm{k}}$<br/>$=-2(10 \hat{\mathrm{i}}+3 \widehat{\mathrm{j}}+11 \widehat{\mathrm{k}})$<br/>Now, $\vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ 2 &amp; -3 &amp; -1 \\ 1 &amp; 4 &amp; -2\end{array}\right|$<br/>$=\hat{i}\left|\begin{array}{cc}-3 &amp; -1 \\ 4 &amp; -2\end{array}\right|-\hat{j}\left|\begin{array}{cc}2 &amp; -1 \\ 1 &amp; -2\end{array}\right|+\hat{k}\left|\begin{array}{cc}2 &amp; -3 \\ 1 &amp; 4\end{array}\right|$<br/>$=\widehat{\mathrm{i}}(6+4)-\hat{\mathrm{j}}(-4+1)+\mathrm{k}(8+3)$<br/>$=10 \hat{\mathrm{i}}+3 \widehat{\mathrm{j}}+11 \hat{\mathrm{k}}$<br/>Hence, $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})=-2(\vec{a} \times \vec{b})$</div>


title: if-a-2-x-x-0-x-and-a-1-1-1-0-2-then-what-is-the-value-of-x
slug: if-a-2-x-x-0-x-and-a-1-1-1-0-2-then-what-is-the-value-of-x-37519
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{cc}2 \mathrm{x} &amp; 0 \\ \mathrm{x} &amp; \mathrm{x}\end{array}\right]$ and $\mathrm{A}^{-1}=\left[\begin{array}{cc}1 &amp; 0 \\ -1 &amp; 2\end{array}\right]$, then what is the value of $\mathrm{x}$?</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">$\frac{1}{2}$</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">Given matrices are :<br/>$\mathrm{A}=\left[\begin{array}{cc}2 \mathrm{x} &amp; 0 \\ \mathrm{x} &amp; \mathrm{x}\end{array}\right]$ and $\mathrm{A}^{-1}=\left[\begin{array}{cc}1 &amp; 0 \\ -1 &amp; 2\end{array}\right]$<br/>$\mathrm{AA}^{-1}=\mathrm{I}$<br/>$\Rightarrow\left[\begin{array}{cc}2 \mathrm{x} &amp; 0 \\ \mathrm{x} &amp; \mathrm{x}\end{array}\right]\left[\begin{array}{cc}1 &amp; 0 \\ -1 &amp; 2\end{array}\right]=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{cc}2 x &amp; 0 \\ 0 &amp; 2 x\end{array}\right]=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right] \Rightarrow 2 x=1$</div>


title: if-a-3-1-2-4-then-what-is-a-adj-a-equal-to
slug: if-a-3-1-2-4-then-what-is-a-adj-a-equal-to-40935
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left[\begin{array}{ll}3 &amp; 2 \\ 1 &amp; 4\end{array}\right]$, then what is $A(\operatorname{adj} A)$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{cc}10 &amp; 0 \\ 0 &amp; 10\end{array}\right]$</span> </div>

answer:
<div class="solution">Let $A=\left[\begin{array}{ll}3 &amp; 2 \\ 1 &amp; 4\end{array}\right]$<br/>We have If $A$ is a square matrix of order $n$ then $A(\operatorname{adj} A)=|A| . \mathrm{I}_{\mathrm{n}}$<br/>Here, $n=2$ $\therefore A(\operatorname{adj} A)=\mathrm{I}_{2}|A|$<br/>$=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]\left|\begin{array}{ll}3 &amp; 2 \\ 1 &amp; 4\end{array}\right|=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right](12-2)=10\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$=\left[\begin{array}{ll}10 &amp; 0 \\ 0 &amp; 10\end{array}\right]$</div>


title: if-a-3-5-7-4-6-8-and-b-3-4-5-6-7-8-then-which-one-of-the-following-is-correct
slug: if-a-3-5-7-4-6-8-and-b-3-4-5-6-7-8-then-which-one-of-the-following-is-correct-64173
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}3 &amp; 4 \\ 5 &amp; 6 \\ 7 &amp; 8\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{lll}3 &amp; 5 &amp; 7 \\ 4 &amp; 6 &amp; 8\end{array}\right]$, 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">$\mathrm{B}$ is the inverse of $\mathrm{A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{B}$ is the adjoint of $\mathrm{A}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{B}$ is the transpose of A</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

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

answer:
<div class="solution">We have,<br/>$A=\left[\begin{array}{ll}3 &amp; 4 \\ 5 &amp; 6 \\ 7 &amp; 8\end{array}\right]$<br/>$\mathrm{A}^{\mathrm{T}}=\left[\begin{array}{lll}3 &amp; 5 &amp; 7 \\ 4 &amp; 6 &amp; 8\end{array}\right]=\mathrm{B}$<br/>$\therefore \mathrm{B}$ is the transpose of $\mathrm{A}$</div>


title: if-a-3-b-4-and-a-b-5-then-what-is-the-value-of-a-b
slug: if-a-3-b-4-and-a-b-5-then-what-is-the-value-of-a-b-23714
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{b}}|=4$ and $|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|=5$, then what is the value of<br/>$|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=$?</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">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \sqrt{2}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">$\quad|\vec{a}+\vec{b}|^{2}+|\vec{a}-\vec{b}|^{2}=2\left\{|\vec{a}|^{2}+|\vec{b}|^{2}\right\}=2 \times 25=50$<br/>$\Rightarrow \quad|\vec{a}+\vec{b}|^{2}+25=50$<br/>$\Rightarrow \quad|\vec{a}+\vec{b}|^{2}=25$<br/>$\Rightarrow \quad|\vec{a}+\vec{b}|=5$</div>


title: if-a-3-b-4-and-a-b-7-then-what-is-the-value-of-a-b
slug: if-a-3-b-4-and-a-b-7-then-what-is-the-value-of-a-b-68761
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}|=3,|\vec{b}|=4$ and $|\vec{a}-\vec{b}|=7$, then what is the value of<br/>$|\vec{a}+\vec{b}| ?$</div>

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

answer:
<div class="solution">Given, $|\vec{a}|=3,|\vec{b}|=4$ and $|\vec{a}-\vec{b}|=7$<br/>Since, $|\vec{a}+\vec{b}|^{2}+|\vec{a}-\vec{b}|^{2}=2\left[|\vec{a}|^{2}+|\vec{b}|^{2}\right]$<br/>$\therefore$ By putting the values of $|\vec{a}|,|\vec{b}|$ and $|\vec{a}-\vec{b}|$ we get<br/>$|\vec{a}+\vec{b}|^{2}+7^{2}=2\left[3^{2}+4^{2}\right]$<br/>$|\vec{a}+\vec{b}|^{2}=50-49 \Rightarrow|\vec{a}+\vec{b}|^{2}=1 \Rightarrow|\vec{a}+\vec{b}|=1$</div>


title: if-a-3-b-4-then-for-what-value-of-1-is-a-b-perpendicular-to-a-b
slug: if-a-3-b-4-then-for-what-value-of-1-is-a-b-perpendicular-to-a-b-47025
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\overrightarrow{\mathbf{a}}|=3,|\overrightarrow{\mathbf{b}}|=4$, then for what value of 1 is $(\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}})$<br/>perpendicular to $(\overrightarrow{\mathbf{a}}-\lambda \overrightarrow{\mathbf{b}})$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{4}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{9}{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{5}$</span> </li> </ul>

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

answer:
<div class="solution">$\because(\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}})$ is perpendicular to $(\overrightarrow{\mathrm{a}}-\lambda \overrightarrow{\mathrm{b}})$, their dot<br/>product is zero, so, $(\vec{a}+\lambda \vec{b}) \cdot(\vec{a}-\lambda \vec{b})=0$<br/>$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}-\lambda^{2}|\overrightarrow{\mathrm{b}}|^{2}-\lambda \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}}=0$<br/>$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}-\lambda^{2}|\overrightarrow{\mathrm{b}}|^{2}=0 \quad(\because \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b} \cdot \mathrm{a}})$<br/>$\Rightarrow 9-16 \lambda^{2}=0$<br/>$\Rightarrow \lambda=\pm \frac{3}{4} \quad \lambda=\frac{3}{4}$ matches with the given option.</div>


title: if-a-4-2-x-3-x-2-x-1-is-symmetric-then-what-is-x-equal-to
slug: if-a-4-2-x-3-x-2-x-1-is-symmetric-then-what-is-x-equal-to-29344
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left(\begin{array}{cc}4 &amp; x+2 \\ 2 x-3 &amp; x+1\end{array}\right)$ is symmetric, then what is $x$ equal<br/>to ?</div>

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

answer:
<div class="solution">$\because \mathrm{A}=\mathrm{A}^{\prime}$<br/>$\Rightarrow\left(\begin{array}{cc}4 &amp; x+2 \\ 2 x-3 &amp; x+1\end{array}\right)=\left(\begin{array}{cc}4 &amp; 2 x-3 \\ x+2 &amp; x+1\end{array}\right)$<br/>$\Rightarrow 2 x-3=x+2$<br/>$\therefore x=5$</div>


title: if-a-4-i-6-14-i-10-i-6-4-i-and-k-2-i-1-where-i-1-then-ka-is-equal-to
slug: if-a-4-i-6-14-i-10-i-6-4-i-and-k-2-i-1-where-i-1-then-ka-is-equal-to-41941
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\left[\begin{array}{cc}4 i-6 &amp; 10 i \\ 14 i &amp; 6+4 i\end{array}\right]$ and $k=\frac{1}{2 i}$ where $i=\sqrt{-1}$, then<br/>$\mathrm{kA}$ is equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{cc}2+3 \mathrm{i} &amp; 5 \\ 7 &amp; 2-3 \mathrm{i}\end{array}\right]$</span> </div>

answer:
<div class="solution">$A=\left[\begin{array}{cc}4 i-6 &amp; 10 i \\ 14 i &amp; 6+4 i\end{array}\right]$ and $K=\frac{1}{2 i}$<br/>$\mathrm{K}=\frac{1}{2 \mathrm{i}}=\frac{\mathrm{i}}{2 \mathrm{i}(\mathrm{i})}=\frac{\mathrm{i}}{2 \mathrm{i}^{2}}=\frac{-\mathrm{i}}{2} .$<br/>$\therefore \mathrm{KA}=\frac{-\mathrm{i}}{2}\left[\begin{array}{cc}4 \mathrm{i}-6 &amp; 10 \mathrm{i} \\ 14 \mathrm{i} &amp; 6+4 \mathrm{i}\end{array}\right]$<br/>$=\left[\begin{array}{cc}(4 \mathrm{i}-6)\left(\frac{-\mathrm{i}}{2}\right) &amp; 10 \mathrm{i}\left(\frac{-\mathrm{i}}{2}\right) \\ 14 \mathrm{i}\left(\frac{-\mathrm{i}}{2}\right) &amp; \left.(6+4 \mathrm{i})\left(\frac{-\mathrm{i}}{2}\right)\right]\end{array}\right.$<br/>$=\left[\begin{array}{cc}-2 \mathrm{i}^{2}+3 \mathrm{i} &amp; -5 \mathrm{i}^{2} \\ -7 \mathrm{i}^{2} &amp; -3 \mathrm{i}-2 \mathrm{i}^{2}\end{array}\right]=\left[\begin{array}{cc}2+3 \mathrm{i} &amp; 5 \\ 7 &amp; 2-3 \mathrm{i}\end{array}\right]$</div>


title: if-a-4-n-2-n-is-a-natural-number-and-b-3-n-n-is-a-natural-number-then-what-is-a-b-equal-to
slug: if-a-4-n-2-n-is-a-natural-number-and-b-3-n-n-is-a-natural-number-then-what-is-a-b-equal-to-11527
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\{4 n+2 \mid n$ is a natural number $\}$ and $\mathrm{B}=\{3 n\rfloor n$ is a natural number\}, then what is $(A \cap B)$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left\{12 n^{2}+6 n \mid n\right.$ is a natural number}</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{24 n-12 \mid n$ is a natural number}</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\{60 n+30 \mid n$ is a natural number}</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\{12 n-6 \mid n$ is a natural 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\{12 n-6 \mid n$ is a natural number}</span> </div>

answer:
<div class="solution">Let $A=\{4 n+2: n \in N\}$<br/>and $B=\{3 n: n \in N\}$<br/>$\Rightarrow \quad A=\{6,10,14,18,22,26,30,34,38,42, \ldots \ldots$<br/>and $\mathrm{B}=\{3,6,9,12,15,18,21,24,27,30, \ldots .\}$<br/>$\mathrm{A} \cap \mathrm{B}=\{6,18,30,42, \ldots . .\}$<br/>$=6+12 n-12=12 n-6 .$<br/>Hence, $\mathrm{A} \cap \mathrm{B}=\{12 n-6: n$ is a natural number $\}$.</div>


title: if-a-7-b-11-and-a-b-10-3-then-a-b-is-equal-to
slug: if-a-7-b-11-and-a-b-10-3-then-a-b-is-equal-to-46460
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $|\vec{a}|=7,|\vec{b}|=11$ and $|\vec{a}+\vec{b}|=10 \sqrt{3}$, then $|\vec{a}-\vec{b}|$ is equal<br/>to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">40</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \sqrt{10}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2 \sqrt{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">$|\overrightarrow{\mathrm{a}}|=7,|\overrightarrow{\mathrm{b}}|=11$ and $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=10 \sqrt{3}$<br/>Now $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+2|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta$<br/>$\begin{array}{ll}\therefore &amp; (10 \sqrt{3})^{2}=49+121+2 \times 7 \times 7 \cos \theta \\ \therefore &amp; 300=170+154 \cos \theta \\ &amp; \frac{300-170}{154}=\cos \theta \\ \therefore &amp; \frac{65}{77}=\cos 0 \\ &amp; \quad \text { Now, }|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}-2|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta \\ &amp; =(7)^{2}+(11)^{2}-2 \times 7 \times 11 \times \frac{65}{77} \\ &amp; =49+121-2 \times 65 \\ &amp; \equiv 170-130 \equiv 40 \\ \therefore &amp; |\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|=\sqrt{40}=2 \sqrt{10}\end{array}$<br/>$\therefore \quad$ Option (d) is correct.</div>


