# questions

tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2020 (08 Jan Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">106</span> </div>
source_db: MarksBatch1_P1.db
title:
a-one-metre-long-both-ends-open-organ-pipe-is-kept-in-a-gas-that-has-double-the-density-of-air-at-stp-assuming-the-speed-of-sound-in-air-at-stp-is-300

slug:
a-one-metre-long-both-ends-open-organ-pipe-is-kept-in-a-gas-that-has-double-the-density-of-air-at-stp-assuming-the-speed-of-sound-in-air-at-stp-is-300-36594

question:
<div class="question">A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi><mo>,</mo></math> the frequency difference between the fundamental and second harmonic of this pipe is __________ Hz.</div>

answer:
<div class="solution"><blockquote><p><math><mi>V</mi><mo>=</mo><msqrt><mfrac><mrow><mi>B</mi></mrow><mrow><mi>ρ</mi></mrow></mfrac></msqrt></math><br/><math><mfrac><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi><mi>i</mi><mi>p</mi><mi>e</mi></mrow></msub></mrow><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>a</mi><mi>i</mi><mi>r</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msqrt><mfrac><mrow><mi>B</mi></mrow><mrow><mn>2</mn><mi>ρ</mi></mrow></mfrac></msqrt></mrow><mrow><msqrt><mfrac><mrow><mi>B</mi></mrow><mrow><mi>ρ</mi></mrow></mfrac></msqrt></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></math><br/><math><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi><mi>i</mi><mi>p</mi><mi>e</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>a</mi><mi>i</mi><mi>r</mi></mrow></msub></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></math><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi><mi>i</mi><mi>p</mi><mi>e</mi></mrow></msub></mrow><mrow><mn>2</mn><mi mathvariant="script">l</mi></mrow></mfrac></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mo>-</mo><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mfrac><msub><mi>V</mi><mrow><mi>p</mi><mi>i</mi><mi>p</mi><mi>e</mi></mrow></msub><mrow><mn>2</mn><mi mathvariant="script">l</mi></mrow></mfrac><mo>=</mo><mfrac><mn>300</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>=</mo><mn>105.75</mn><mo> </mo><mi>H</mi><mi>z</mi><mo> </mo><mo> </mo><mfenced separators="|"><mrow><mi>i</mi><mi>f</mi><mo> </mo><msqrt><mn>2</mn></msqrt><mo>=</mo><mn>1.41</mn></mrow></mfenced></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>106.05</mn><mo> </mo><mi>H</mi><mi>z</mi><mo> </mo><mo> </mo><mfenced separators="|"><mrow><mi>i</mi><mi>f</mi><mo> </mo><msqrt><mn>2</mn></msqrt><mo>=</mo><mn>1.414</mn></mrow></mfenced></math></p></blockquote></div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-pair-of-fair-dice-is-rolled-what-is-the-probability-that-the-second-dice-lands-on-a-higher-value-than-does-the-first

slug:
a-pair-of-fair-dice-is-rolled-what-is-the-probability-that-the-second-dice-lands-on-a-higher-value-than-does-the-first-15576

question:
<div class="question">A pair of fair dice is rolled. What is the probability that the second dice lands on a higher value than does the first ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{6}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5}{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5}{18}$</span> </li> </ul>

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

answer:
<div class="solution">Possibilities of having higher number on second dice. $\begin{array}{ll}\text { First dice } &amp; \text { Second dice }\end{array}$<br/>$1 \quad 2,3,4,5,6 \rightarrow 5$ possibilities<br/>$2 \quad 3,4,5,6 \rightarrow 4$ possibilities<br/>$3 \quad 4,5,6 \rightarrow 3$ possibilities<br/>$4 \quad 5,6 \rightarrow 2$ possibilities $5 \quad 6 \rightarrow 1$ possibility Total number of possibilities $=15$ Total number of events $=36$<br/>$\therefore$ Probability $=\frac{15}{36}=\frac{5}{12}$</div>


tags: ['Physics', 'Wave Optics', 'MHT CET', 'MHT CET 2022 (06 Aug Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-parallel-monochromatic-beam-of-light-is-incident-normally-on-a-narrow-slit-a-diffraction-pattern-is-formed-on-a-screen-placed-perpendicular-to-the-d-2

slug:
a-parallel-monochromatic-beam-of-light-is-incident-normally-on-a-narrow-slit-a-diffraction-pattern-is-formed-on-a-screen-placed-perpendicular-to-the-d-2-29058

question:
<div class="question">A parallel monochromatic beam of light is incident normally on a narrow slit. A diffraction pattern is formed on a screen placed perpendicular to the direction of the incident beam. At the first maximum of the diffraction pattern, the phase difference between the rays coming from the edges of the slit is</div>

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

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

answer:
<div class="solution">For nth secondary maxima, the path difference $x$ is the odd multiple of the half of the wavelength, i.e.,<br/>$x_n=(2 n+1) \frac{\lambda}{2}$<br/>So, for the first maxima, the phase difference would be:<br/>$\phi_1=\frac{2 \pi}{\lambda} x_1$<br/>$\therefore \phi_1=\frac{2 \pi}{\lambda}\left(\frac{3 \lambda}{2}\right)=3 \pi$</div>


tags: ['Mathematics', 'Vector Algebra', 'JEE Main', 'JEE Main 2004']
source_db: MarksBatch1_P1.db
title:
a-particle-is-acted-upon-by-constant-forces-4-i-j-3-k-and-3-i-j-k-which-displace-it-from-a-point-i-2-j-3-k-to-the-point-5-i-4-j-k-the-work-done-in-sta

slug:
a-particle-is-acted-upon-by-constant-forces-4-i-j-3-k-and-3-i-j-k-which-displace-it-from-a-point-i-2-j-3-k-to-the-point-5-i-4-j-k-the-work-done-in-sta-68071

question:
<div class="question">A particle is acted upon by constant forces $4 I+J-3 k$ and $3 I+J-k$ which displace it from a point $\hat{i}+2 \hat{j}+3 \hat{k}$ to the point $5 \hat{i}+4 \hat{j}+\hat{k}$. The work done in standard units by the forces is given by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>40<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>30<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>25<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>15</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>40<br/></span> </div>

answer:
<div class="solution">Work done by the forces $\vec{F}_1$ and $\vec{F}_2$ is $\left(\vec{F}_1+\vec{F}_2\right) \cdot \vec{d}$, where $\vec{d}$ is displacement According to question $\vec{F}_1+\vec{F}_2=(4 \hat{i}+\hat{j}-3 \hat{k})+(3 \hat{i}+\hat{j}-\hat{k})=7 \hat{i}+2 \hat{j}-4 \hat{k}$ and $\vec{d}=(5 \hat{i}+4 \hat{j}+\hat{k})-(\hat{i}+2 \hat{j}+3 \hat{k})=4 \hat{i}+2 \hat{j}-2 \hat{k}$. Hence $\left(\vec{F}_1+\vec{F}_2\right) \cdot \vec{d}$ is 40</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-particle-is-acted-upon-by-following-forces-i-2-i-3-j-5-k-ii-5-i-4-j-3-k-and-iii-3-i-7-k-in-which-plane-does-it-move

slug:
a-particle-is-acted-upon-by-following-forces-i-2-i-3-j-5-k-ii-5-i-4-j-3-k-and-iii-3-i-7-k-in-which-plane-does-it-move-39049

question:
<div class="question">A particle is acted upon by following forces:<br/>(i) $\quad 2 \hat{i}+3 \hat{j}+5 \hat{k}$,<br/>(ii) $-5 \hat{i}+4 \hat{j}-3 \hat{k}$ and (iii) $3 \hat{i}-7 \hat{k}$<br/>In which plane does it move?</div>

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

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

answer:
<div class="solution">Three forces are given by, say, $\mathrm{F}_{1}, \mathrm{~F}_{2}$ and $\mathrm{F}_{3}$<br/>$\overrightarrow{\mathrm{F}}_{1}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}, \overrightarrow{\mathrm{F}}_{2}=-5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$<br/>and $\vec{F}_{3}=3 \hat{i}-7 \hat{k}$<br/>Total resultant force, $\overrightarrow{\mathrm{F}}$ is given by<br/>$\overrightarrow{\mathrm{F}}=\overrightarrow{\mathrm{F}}_{1}+\overrightarrow{\mathrm{F}}_{2}+\overrightarrow{\mathrm{F}}_{3}$<br/>$=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}-5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}+3 \hat{\mathrm{i}}-7 \hat{\mathrm{k}}=7 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$<br/>This show that the resultant force is in the $\mathrm{y}-\mathrm{z}$ plane. Thus, it moves in the $\mathrm{y}-\mathrm{z}$ plane.</div>


tags: ['Physics', 'Motion In One Dimension', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-particle-moves-in-a-straight-line-with-a-constant-acceleration-it-changes-its-velocity-from-10-m-s-1-to-20-m-s-1-while-passing-through-a-distance-13

slug:
a-particle-moves-in-a-straight-line-with-a-constant-acceleration-it-changes-its-velocity-from-10-m-s-1-to-20-m-s-1-while-passing-through-a-distance-13-89622

question:
<div class="question">A particle moves in a straight line with a constant acceleration. It changes its velocity from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mtext>â</mtext><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup></math>  to  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mtext>â</mtext><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup></math> while passing through a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>135</mn><mo> </mo><mi mathvariant="normal">m</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds. The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</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>1.8</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>12</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"><mn>9</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </div>

answer:
<div class="solution">Initial velocity,  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup></math><br/><br/>Final velocity ,  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mn>20</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup></math><br/><br/>Distance ,  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mn>135</mn><mo> </mo><mi mathvariant="normal">m</mi></math><br/><br/>Let , acceleration = <i>a</i><br/><br/>Using the formula ,  <math> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo><mn>2</mn><mi>a</mi><mi>s</mi> </mrow> </math><br/><br/><math> <mrow> <mo>â</mo><mtext>â</mtext><mtext>â</mtext><mtext>â</mtext><mi>a</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>â</mo><msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn><mi>s</mi> </mrow> </mfrac> </mrow> </math><br/><br/><math> <mrow> <mi>or</mi><mtext>â</mtext><mtext>â</mtext><mi>a</mi><mo>=</mo><mtext>â</mtext><mfrac> <mrow> <msup> <mrow> <mo stretchy="false">(</mo><mn>20</mn><mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>â</mo><msup> <mrow> <mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn><mo>Ã</mo><mn>135</mn> </mrow> </mfrac> <mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac> <mrow> <mn>400</mn><mo>â</mo><mn>100</mn> </mrow> <mrow> <mn>2</mn><mo>Ã</mo><mn>135</mn> </mrow> </mfrac> </mrow> </math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>or</mi><mtext>â</mtext><mtext>â</mtext><mi>a</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac><mn>300</mn><mrow><mn>2</mn><mo>Ã</mo><mn>135</mn></mrow></mfrac><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac><mn>150</mn><mn>135</mn></mfrac><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac><mn>10</mn><mn>9</mn></mfrac><mtext> </mtext><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>â</mo><mn>2</mn></mrow></msup></math><br/><br/>Now , using the relation ,  <math> <mrow> <mi>v</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mi>u</mi><mo>+</mo><mi>a</mi><mi>t</mi> </mrow> </math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac><mrow><mi>v</mi><mo>â</mo><mi>u</mi></mrow><mi>a</mi></mfrac><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac><mrow><mn>20</mn><mo>â</mo><mn>10</mn></mrow><mfrac><mn>10</mn><mn>9</mn></mfrac></mfrac><mtext>â</mtext><mo>=</mo><mtext>â</mtext><mfrac><mn>10</mn><mn>10</mn></mfrac><mo>Ã</mo><mn>9</mn><mo>=</mo><mtext>â</mtext><mn>9</mn><mo> </mo><mi mathvariant="normal">s</mi><mo>.</mo></math> <!-- /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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tags: ['Physics', 'Motion In One Dimension', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-particle-moving-along-x-axis-has-acceleration-f-at-time-t-given-by-f-f-0-1-t-t-where-f-0-and-t-are-constants-the-particle-at-t-0-and-the-instant-whe-1

slug:
a-particle-moving-along-x-axis-has-acceleration-f-at-time-t-given-by-f-f-0-1-t-t-where-f-0-and-t-are-constants-the-particle-at-t-0-and-the-instant-whe-1-40650

question:
<div class="question">A particle moving along <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-axis has acceleration <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> given by<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> are constants. The particle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> and the instant when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>0</mn></math>, the particle's velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>x</mi></msub></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><msup><mi>T</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><msup><mi>T</mi><mn>2</mn></msup></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><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</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> </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><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></math></span> </div>

answer:
<div class="solution"><p> Acceleration</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>v</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced><mo>·</mo><mi>d</mi><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo> </mo><mi>E</mi><mi>q</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></p><p>Integrating Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math> on both sides, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mi>t</mi><mo>-</mo><mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></mfrac><mo>·</mo><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mi>C</mi></math></p><p>After applying boundary conditions </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math></p><p>We get, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>v</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mi>t</mi><mo>-</mo><mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></mfrac><mo>·</mo><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo> </mo><mi>E</mi><mi>q</mi><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></math>   </p><p>As <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced></math></p><p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>=</mo><mi>T</mi></math></p><p>Substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>T</mi></math> in Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math>, then velocity</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>x</mi></msub><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi><mo>-</mo><mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></mfrac><mo>·</mo><mfrac><msup><mi>T</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></math></p></div>


tags: ['Physics', 'Motion In One Dimension', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch1_P1.db
title:
a-particle-moving-along-x-axis-has-acceleration-f-at-time-t-given-by-f-f-0-1-t-t-where-f-0-and-t-are-constants-the-particle-at-t-0-and-the-instant-whe

slug:
a-particle-moving-along-x-axis-has-acceleration-f-at-time-t-given-by-f-f-0-1-t-t-where-f-0-and-t-are-constants-the-particle-at-t-0-and-the-instant-whe-64218

question:
<div class="question">A particle moving along <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-axis has acceleration <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> given by<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> are constants. The particle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> and the instant when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mn>0</mn></math>, the particle's velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>x</mi></msub></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><msup><mi>T</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><msup><mi>T</mi><mn>2</mn></msup></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><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</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> </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><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></math></span> </div>

answer:
<div class="solution"><p> Acceleration</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>v</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced><mo>·</mo><mi>d</mi><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo> </mo><mi>E</mi><mi>q</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></p><p>Integrating Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math> on both sides, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mi>t</mi><mo>-</mo><mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></mfrac><mo>·</mo><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mi>C</mi></math></p><p>After applying boundary conditions </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math></p><p>We get, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>v</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mi>t</mi><mo>-</mo><mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></mfrac><mo>·</mo><mfrac><msup><mi>t</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo> </mo><mi>E</mi><mi>q</mi><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></math>   </p><p>As <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>t</mi><mi>T</mi></mfrac></mrow></mfenced></math></p><p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>=</mo><mi>T</mi></math></p><p>Substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>T</mi></math> in Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math>, then velocity</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>x</mi></msub><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi><mo>-</mo><mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></mfrac><mo>·</mo><mfrac><msup><mi>T</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>f</mi><mn>0</mn></msub><mi>T</mi></math></p></div>


tags: ['Physics', 'Oscillations', 'JEE Main', 'JEE Main 2023 (29 Jan Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">40</span> </div>
source_db: MarksBatch1_P1.db
title:
a-particle-of-mass-250-g-executes-a-simple-harmonic-motion-under-a-periodic-force-f-25-x-n-the-particle-attains-a-maximum-speed-of-4-m-s-1-during-its-

slug:
a-particle-of-mass-250-g-executes-a-simple-harmonic-motion-under-a-periodic-force-f-25-x-n-the-particle-attains-a-maximum-speed-of-4-m-s-1-during-its-68750

question:
<div class="question">A particle of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>250</mn><mo> </mo><mi mathvariant="normal">g</mi></math> executes a simple harmonic motion under a periodic force <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mo>(</mo><mo>–</mo><mn>25</mn><mi>x</mi><mo>)</mo><mo> </mo><mi mathvariant="normal">N</mi></math>. The particle attains a maximum speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> during its oscillation. The amplitude of the motion is ______<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cm</mi></math>.</div>

answer:
<div class="solution"><p>Given, force <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mo>(</mo><mo>–</mo><mn>25</mn><mi>x</mi><mo>)</mo><mo> </mo><mi mathvariant="normal">N</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>0</mn><mo>.</mo><mn>25</mn><mi>a</mi><mo>=</mo><mo>-</mo><mn>25</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mo>-</mo><mn>100</mn><mi>x</mi></math></p><p>Now comparing it with standard equation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mi>x</mi></math> we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>10</mn></math>.</p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>ω</mi><mi>A</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>ω</mi><mi>A</mi><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>A</mi><mo>=</mo><mfrac><mn>4</mn><mn>10</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>A</mi><mo>=</mo><mn>40</mn><mo> </mo><mi>cm</mi></math>.</p></div>


tags: ['Physics', 'Magnetic Effects of Current', 'JEE Advanced', 'JEE Advanced 2008 (Paper 1)']
source_db: MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-moving-with-velocity-v-enters-region-ii-normal-to-the-boundary-as-shown-in-the-figure-region-ii-has-a-uniform-magnet

slug:
a-particle-of-mass-m-and-charge-q-moving-with-velocity-v-enters-region-ii-normal-to-the-boundary-as-shown-in-the-figure-region-ii-has-a-uniform-magnet-86730

question:
<div class="question">A particle of mass $m$ and charge $q$, moving with velocity $v$ enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field $B$ perpendicular to the plane of the paper. The length of the Region II is $l$. Choose the correct choice(s)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/2R-9CzzQsKQiIWeWGqRQfQoFeKBfJB3YsXFFqvgUbGU.original.fullsize.png"/><br/></div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>The particle enters Region III only if its velocity $v&gt;\frac{q l B}{m}$<br/>, <br/>Path length of the particle in Region II is maximum when velocity $v=\frac{q I B}{m}$<br/>, <br/>Time spent in Region II is same for any velocity $v$ as long as the particle returns to Region I</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathbf{V}} \perp \overrightarrow{\mathbf{B}}$ in region II. Therefore, path of particle is circle in region II. Particle enters in region III if, radius of circular path, $r&gt;1$ or $\frac{m v}{B q}&gt;1$ or $v&gt;\frac{B q l}{m}$<br/>If $v=\frac{B q l}{m}, r=\frac{m v}{B q}=l$, particle will turn back and path length will be maximum as shown in figure in region II. If particle returns to region I, time spent in region II will be $t=\frac{T}{2}=\frac{\pi m}{B q}$, which is independent of $v$.<br/>$\therefore$ correct options are (a), (c) and (d).<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_advanced/0u3oe3umQiRM7QrzjiMfVEpq5UCFuBEPfFEcz8oRLs4.original.fullsize.png"/><br/></div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2021']
source_db: MarksBatch1_P1.db
title:
a-particle-of-mass-m-and-charge-q-moving-with-velocity-v-enters-regionb-form-regiona-along-the-normal-to-the-boundary-as-shown-is-the-figure-regionb-h

slug:
a-particle-of-mass-m-and-charge-q-moving-with-velocity-v-enters-regionb-form-regiona-along-the-normal-to-the-boundary-as-shown-is-the-figure-regionb-h-20977

question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/vortu4tbfcKuUfrfiJLJTx2yGEtKmdFW4vY7gY8hRfo.original.fullsize.png"/><br/> <br/>A particle of mass $\mathrm{m}$ and charge $\mathrm{q}$ moving with velocity $\mathrm{v}$ enters region-b form region-a along the normal to the boundary as shown is the figure. Region-b has a uniform magnetic field $B$ perpendicular to the plane of the paper. Also, region-b has length L. Choose the correct statements :</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">The particle enters region-c only if $\mathrm{v}&gt;\frac{\mathrm{qLB}}{\mathrm{m}}$, Path of the particle is a circle in region-b, Time spent in region -b is independent of velocity v</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/kONNlzMjqI7mEKR3f9dGf47hzwufQFI2TV7WvsmFoHI.original.fullsize.png"/><br/> <br/>Particle enter region 'c' if
<br/>Radius is greater than ' $L$ ' $\Rightarrow \frac{m V}{q B}&gt;L \Rightarrow V&gt;\frac{q B L}{m}$
<br/>$\because$ B is uniform and $\overrightarrow{\mathrm{V}} \perp \overrightarrow{\mathrm{B}}$ so.
<br/>Path is circular in region b
<br/>$$
<br/>\text { Time spent }=\frac{\text { Angle cov ered }}{\omega} \quad \omega=\frac{\mathrm{qB}}{\mathrm{m}}
<br/>$$
<br/>Now angle covered depens on whether particle goes to region c or comes back to region 'a'</div>


tags: ['Physics', 'Magnetic Effects of Current', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-particle-with-charge-q-and-mass-m-enters-a-magnetic-field-of-magnitude-b-existing-only-to-the-right-of-the-boundary-y-z-the-direction-of-the-motion-

slug:
a-particle-with-charge-q-and-mass-m-enters-a-magnetic-field-of-magnitude-b-existing-only-to-the-right-of-the-boundary-y-z-the-direction-of-the-motion-38067

question:
<div class="question">A particle with charge <math><mo>+</mo><mi>Q</mi></math> and mass <math><mi>m</mi></math> enters a magnetic field of magnitude <math><mi>B</mi><mo>,</mo></math> existing only to the right of the boundary <math><mi>Y</mi><mi>Z</mi><mo>.</mo></math> The direction of the motion of the particle is perpendicular to the direction of <math><mi>B</mi><mo>.</mo></math> Then, the time spent by the particle in the field will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mn>2</mn><mi>Ï</mi><mi>m</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mi>Ï</mi><mi>m</mi></mrow><mrow><mi>Q</mi><mi>B</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><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mi>m</mi><mi>v</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Infinite</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><mi>Ï</mi><mi>m</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/zzmodsgecdsubvhpbuobnhemxawrsnbzwrdsrnhtuvtkmisygp.jpeg" style="max-width:100%; object-fit:contain"/><br/><br/><math><mover accent="true"><mrow><mi>F</mi></mrow><mo>â</mo></mover><mo>=</mo><mi>Q</mi><mfenced separators="|"><mrow><mover accent="true"><mrow><mi>V</mi></mrow><mo>â</mo></mover><mo>Ã</mo><mover accent="true"><mrow><mi>B</mi></mrow><mo>â</mo></mover></mrow></mfenced></math><br/><br/><math><mi>F</mi><mo>=</mo><mi>Q</mi><mfenced separators="|"><mrow><mi>V</mi><mi>B</mi></mrow></mfenced><mi> </mi></math><br/><br/>Since this force is constant and is perpendicular to particle's direction of motion. Particle will move in a circle.<br/><br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>Q</mi><mi>V</mi><mi>B</mi><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>R</mi></mrow></mfrac></math><br/><br/><math><mi>R</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>V</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></math><br/><br/><math><mo>â´</mo><mi> </mi><mi> </mi></math> Time required to travel <math><mi>Ï</mi><mi>R</mi></math> distance<br/><br/><math><mi>t</mi><mo>=</mo><mfrac><mrow><mi>Ï</mi><mi>R</mi></mrow><mrow><mi>V</mi></mrow></mfrac></math><br/><br/><math><mo>â</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>t</mi><mo>=</mo><mfrac><mrow><mi>Ï</mi><mo>Ã</mo><mfrac><mrow><mi>m</mi><mi>V</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></mrow><mrow><mi>v</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>Ï</mi><mi>m</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></math><br/><br/><math><mi>t</mi><mo>=</mo><mfrac><mrow><mi>Ï</mi><mi>m</mi></mrow><mrow><mi>Q</mi><mi>B</mi></mrow></mfrac></math></div>


title: a-passenger-in-an-aeroplane-shall
slug: a-passenger-in-an-aeroplane-shall-31910
question: <div class="question">A passenger in an aeroplane shall</div>
tags: ['Physics', 'Ray Optics']
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>never see a rainbow<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>may see a primary and a secondary rainbow as concentric circles<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>may see a primary and a secondary rainbow as concentric arcs<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>shall never see a secondary rainbow</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>may see a primary and a secondary rainbow as concentric circles<br/></span> </div>

answer:
<div class="solution">When a passenger in an aeroplane then he may see primary and seoondary rainbow such as concentric circles.</div>


tags: ['Biology', 'Principles of Inheritance and Variation', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-patient-comes-to-the-clinic-with-a-genetic-disorder-characterized-by-a-webbed-neck-and-the-presence-of-rudimentary-ovaries-this-person-most-likely-h

slug:
a-patient-comes-to-the-clinic-with-a-genetic-disorder-characterized-by-a-webbed-neck-and-the-presence-of-rudimentary-ovaries-this-person-most-likely-h-16514

question:
<div class="question">A patient comes to the clinic with a genetic disorder characterized by a webbed neck and the presence of rudimentary ovaries. This person most likely has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">two X and one Y sex chromosomes</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">only one sex chromosome which is 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">three 21<sup>st </sup>chromosomes</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">three X chromosomes</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">only one sex chromosome which is X</span> </div>

answer:
<div class="solution">Turner's Syndrome is a disorder is caused due to the absence of one of the X chromosomes. These patients are females with only 45 chromosomes. Such females are sterile as ovaries are rudimentary. They lack other secondary sexual characters. They have a short neck with a webbed appearance, low hairline at the back of the neck, low-set ears, hands, and feet that are swollen or puffy at birth.</div>


tags: ['Physics', 'Waves and Sound', 'JEE Advanced', 'JEE Advanced 2012 (Paper 1)']
source_db: MarksBatch1_P1.db
title:
a-person-blows-into-openend-of-a-long-pipe-as-a-result-a-high-pressure-pulse-of-air-travels-down-the-pipe-when-this-pulse-reaches-the-other-end-of-the

slug:
a-person-blows-into-openend-of-a-long-pipe-as-a-result-a-high-pressure-pulse-of-air-travels-down-the-pipe-when-this-pulse-reaches-the-other-end-of-the-49024

question:
<div class="question">A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe.
<br/> <br/>When this pulse reaches the other end of the pipe,</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is open., a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed.</span> </div>

answer:
<div class="solution">When a sound pulse is reflected from open end of a pipe, phase changes by $180^{\circ}$. A high pressure pulse i.e., compression is reflected as a low pressure pulse i.e., rarefaction. When sound pulse is reflected through a rigid boundary (closed end of a pipe), no phase change occurs so a high pressure pulse is reflected as a high pressure pulse.</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch1_P1.db
title:
a-person-is-to-count-4500-notes-let-a-n-denote-the-number-of-notes-he-counts-in-the-nth-minute-if-a-1-a-2-a-3-a-10-150-and-a-10-a-11-a-12-are-in-apwit

slug:
a-person-is-to-count-4500-notes-let-a-n-denote-the-number-of-notes-he-counts-in-the-nth-minute-if-a-1-a-2-a-3-a-10-150-and-a-10-a-11-a-12-are-in-apwit-63856

question:
<div class="question">A person is to count 4500 notes. Let $a_{n}$ denote the number of notes he counts in the nth minute. If $a_{1}=a_{2}=a_{3}=\ldots .$ $=a_{10}=150$, and $a_{10}, a_{11}, a_{12}, \ldots$ are in APwith the common difference $-2$, then the time taken by him to count all the notes is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">24 minutes</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">34 minutes</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">125 minutes</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">135 minutes</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">34 minutes</span> </div>

answer:
<div class="solution">Given, $\mathrm{a}_{1}=\mathrm{a}_{2}=\mathrm{a}_{3}=\ldots \ldots . .=\mathrm{a}_{10}=150$<br/>Also, $a_{10}, a_{11}, a_{12}, \ldots$ are in A.P. and $d=-2$ since, $\mathrm{a}_{10}=150$, A.P. is $150,148,146, \ldots$<br/>For the first 10 minutes, he has counted $150 \times 10=1500$ notes. Time taken to count remaining 3000 notes<br/>$\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}]$<br/>$\Rightarrow 3000=\frac{\mathrm{n}}{2}[2 \times 148+(\mathrm{n}-1)(-2)]$<br/>$\Rightarrow 3000=\frac{\mathrm{n}}{2} \times 2(148-\mathrm{n}+1)$<br/>$\Rightarrow 3000=148 n-n^{2}+n$<br/>$\Rightarrow n^{2}-149 n+3000=0$<br/>$\Rightarrow(\mathrm{n}-24)(\mathrm{n}-125)$<br/>$\Rightarrow \mathrm{n}=24$, or $\mathrm{n}=125 .$<br/>Since he has taken 10 minutes to count 1500 notes, he will not take 125 min to count 3000 notes. $\mathrm{So}, \mathrm{n}=24$<br/>$\therefore$ Total time $=10+24=34$ minutes.</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch1_P1.db
title:
a-person-standing-on-the-bank-of-a-river-observes-that-the-angle-subtended-by-a-tree-on-the-opposite-of-bank-is-6-0-when-he-retires-40-m-from-the-bank

slug:
a-person-standing-on-the-bank-of-a-river-observes-that-the-angle-subtended-by-a-tree-on-the-opposite-of-bank-is-6-0-when-he-retires-40-m-from-the-bank-13659

question:
<div class="question">A person standing on the bank of a river observes that the angle subtended by a tree on the opposite of bank is $60^{\circ}$. When he retires $40 \mathrm{~m}$. from the bank, he finds the angle to be $30^{\circ} .$ What is the breadth of the river?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$20 \mathrm{~m}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/wb9mfV29PwJUGG_gKCeB2PMQoj-QgXW9VZS_NpSmPJM.original.fullsize.png"/><br/><br/>$\operatorname{In} \Delta \mathrm{ABC}, \tan 60^{\circ}=\frac{\mathrm{h}}{\mathrm{x}}$<br/>$\Rightarrow \sqrt{3}=\frac{\mathrm{h}}{\mathrm{x}} \Rightarrow \mathrm{h}=\sqrt{3} \mathrm{x}$<br/>$\operatorname{In} \Delta \mathrm{ABD}, \tan 30^{\circ}=\frac{\mathrm{h}}{\mathrm{x}+40}$<br/>$\Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{h}}{\mathrm{x}+40} \Rightarrow \mathrm{x}+40=\sqrt{3} \mathrm{~h}$<br/>Putting value of h from equation (1), we get $x+40=3 x$<br/>$\mathrm{x}=20 \mathrm{~m}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2020 (22 Sep Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-piano-wire-with-a-diameter-of-090-mm-is-replaced-by-another-wire-of-diameter-093-mm-of-the-same-material-if-tension-of-wire-is-kept-the-same-then-th

slug:
a-piano-wire-with-a-diameter-of-090-mm-is-replaced-by-another-wire-of-diameter-093-mm-of-the-same-material-if-tension-of-wire-is-kept-the-same-then-th-65714

question:
<div class="question">A piano wire with a diameter of $0.90 \mathrm{~mm}$ is replaced by another wire of diameter $0.93 \mathrm{~mm}$ of the same material. If tension of wire is kept the same, then the percentage change in frequency of fundamental tone is</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">$-3 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$+3.2 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-3.2 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-3.2 \%$</span> </div>

answer:
<div class="solution">Given, diameter, $d_1=0.9 \mathrm{~mm}=0.9 \times 10^{-3} \mathrm{~m}$<br/>$$<br/>\begin{aligned}<br/>&amp; =9 \times 10^{-4} \mathrm{~m} \\<br/>d_2=0.93 \mathrm{~mm} &amp; =9.3 \times 10^{-4} \mathrm{~m}<br/>\end{aligned}<br/>$$<br/><br/>Frequency of fundamental tone is given as<br/>$$<br/>f=\frac{1}{2 l} \sqrt{\frac{T}{m}}<br/>$$<br/>where, $m=$ mass per unit length.<br/>$$<br/>=\frac{M}{l}=\frac{V \cdot \rho}{l}=\frac{A \cdot l}{l} \cdot \rho=A \rho<br/>$$<br/><br/><br/>$$<br/>\Rightarrow \quad m=\frac{\pi d^2 \rho}{4}<br/>$$<br/><br/>From Eqs. (i) and (ii), we get<br/>$$<br/>\begin{aligned}<br/>&amp; f=\frac{1}{2 l} \sqrt{\frac{T}{\pi d^2 \rho / 4}}=\frac{1}{2 l} \sqrt{\frac{4 T}{\pi d^2 \rho}}=\frac{1}{l} \sqrt{\frac{T}{\pi d^2 \rho}} \\<br/>\Rightarrow &amp; f=\frac{1}{l d} \sqrt{\frac{T}{\pi \rho}} \Rightarrow f \propto \frac{1}{d} \\<br/>\therefore &amp; \frac{f_1}{f_2}=\frac{d_2}{d_1}=\frac{9.3 \times 10^{-4}}{9 \times 10^{-4}}=\frac{9.3}{9} \Rightarrow \frac{f_2}{f_1}=\frac{9}{9.3}<br/>\end{aligned}<br/>$$<br/>$\therefore$ Percentage change in frequency<br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{f_2-f_1}{f_1} \times 100=\left(\frac{f_2}{f_1}-1\right) \times 100 \\<br/>&amp; =\left(\frac{9}{9.3}-1\right) \times 100=-3.2 \%<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
a-pipe-20-cm-long-is-closed-at-one-end-which-harmonic-mode-of-the-pipe-is-resonantly-excited-by-a-source-of-12375-hz-sound-velocity-in-air-330-ms-1

slug:
a-pipe-20-cm-long-is-closed-at-one-end-which-harmonic-mode-of-the-pipe-is-resonantly-excited-by-a-source-of-12375-hz-sound-velocity-in-air-330-ms-1-23727

question:
<div class="question">A pipe $20 \mathrm{~cm}$ long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source of $1237.5$ $\mathrm{Hz}$ ? (sound velocity in air $=330 \mathrm{~ms}^{-1}$ )</div>

answer:
<div class="solution">As given that,<br/>Length of pipe, $l=20 \mathrm{~cm}=20 \times 10^{-2} \mathrm{~m}$<br/>$$<br/>v=1237.5 \mathrm{~Hz}, v=330 \mathrm{~m} / \mathrm{s}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/Kes5AbyvghKyDEJLyukgujWt7MmXOBETCeCIp6UiKp8.original.fullsize.png"/><br/><br/>$l=\frac{\lambda}{4}$ or $\lambda=4 l$<br/>$$<br/>\begin{aligned}<br/>&amp;\left(v_{\text {funda }}\right)_1=\frac{v}{4 l}=\frac{330}{4 \times 20 \times 10^{-2}} \\<br/>&amp;\left(v_{\text {funda }}\right)_1=\frac{330 \times 100}{80}=412.5 \mathrm{~Hz} \\<br/>&amp;\frac{\left(v_{\text {given }}\right)}{\left(v_{\text {funda }}\right)_1}=\frac{1237.5}{412.5}=\frac{3}{1}<br/>\end{aligned}<br/>$$<br/>The frequency in Ist, IInd, IIIrd .... harmonic are in the ratio $1: 2: 3: 4 \ldots \ldots$ at one end open organ pipe.</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (15 Oct Shift 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P1.db
title:
a-pipe-closed-at-one-end-has-lengh-0-8-cm-at-its-open-end-a-05-m-long-uniform-string-is-vibrating-in-its-second-harmonic-and-it-resonates-with-the-fun

slug:
a-pipe-closed-at-one-end-has-lengh-0-8-cm-at-its-open-end-a-05-m-long-uniform-string-is-vibrating-in-its-second-harmonic-and-it-resonates-with-the-fun-52116

question:
<div class="question">A pipe closed at one end has lengh $0 \cdot 8 \mathrm{~cm}$. At its open end a $0.5 \mathrm{~m}$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is $50 \mathrm{~N}$ and the speed of sound is $320 \mathrm{~m} / \mathrm{s}$, the mass of the string is</div>

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

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


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2021 (21 Sep Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-pipe-closed-at-one-end-has-length-08-m-at-its-open-end-05-m-long-uniform-string-is-vibrating-in-its-2-nd-harmonic-and-it-resonates-with-the-fundamen

slug:
a-pipe-closed-at-one-end-has-length-08-m-at-its-open-end-05-m-long-uniform-string-is-vibrating-in-its-2-nd-harmonic-and-it-resonates-with-the-fundamen-70576

question:
<div class="question">A pipe closed at one end has length $0.8 \mathrm{~m}$. At its open end $0.5 \mathrm{~m}$ long uniform string is vibrating in its $2^{\text {nd }}$ harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is $50 \mathrm{~N}$ and the speed of sound is $320 \mathrm{~m} / \mathrm{s}$, the mass of the string is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">20 gram</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">10 gram</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">5 gram</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">15 gram</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp; 2 \times\left[\frac{1}{2 \ell_1} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}\right]=\frac{v}{4 \ell_2} \\ &amp; \frac{1}{0.5} \sqrt{\frac{50}{\mathrm{~m}}}=\frac{320}{4 \times 0.8} \\ &amp; \therefore \mathrm{m}=0.02 \mathrm{~kg} / \mathrm{m} \\ &amp; \therefore \text { Total mass of the string } \\ &amp; =0.02 \times 0.5 \mathrm{~kg}=10 \mathrm{gm}\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2018']
source_db: MarksBatch1_P1.db
title:
a-pipe-closed-at-one-end-has-length-83-cm-the-number-of-possible-natural-oscillations-of-air-column-whose-frequencies-lie-below-1000-hz-are-velocity-o-1

slug:
a-pipe-closed-at-one-end-has-length-83-cm-the-number-of-possible-natural-oscillations-of-air-column-whose-frequencies-lie-below-1000-hz-are-velocity-o-1-83349

question:
<div class="question">A pipe closed at one end has length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>83</mn><mo> </mo><mi>cm</mi></math>. The number of possible natural oscillations of air column whose frequencies lie below <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn><mo> </mo><mi>Hz</mi></math> are (velocity of sound in air <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>332</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>)</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>3</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>4</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> </div>

answer:
<div class="solution">Fundamental frequency<br/><math> <mrow> <mi>n</mi><mo>=</mo><mfrac> <mi>V</mi> <mrow> <mn>4</mn><mi>L</mi></mrow> </mfrac> <mo>=</mo><mfrac> <mrow> <mn>332</mn></mrow> <mrow> <mn>4</mn><mo>Ã</mo><mn>83</mn><mo>Ã</mo><msup> <mrow> <mn>10</mn></mrow> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> </mrow> </mfrac><mo>=</mo><mn>100</mn> </mrow> </math> .<br/>Hence possible frequencies are <math> <mrow> <mi>n</mi><mo>=</mo><mn>100</mn></mrow> </math> .<br/><math> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>=</mo><mn>300</mn></mrow></math>.<br/><math> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>=</mo><mn>500</mn></mrow></math>.<br/><math> <mrow> <msub> <mi>n</mi> <mn>3</mn> </msub> <mo>=</mo><mn>700.</mn></mrow> </math><br/><math> <mrow> <msub> <mi>n</mi> <mn>5</mn> </msub> <mo>=</mo><mn>900</mn></mrow></math>.<br/> </div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-pipe-closed-at-one-end-has-length-83-cm-the-number-of-possible-natural-oscillations-of-air-column-whose-frequencies-lie-below-1000-hz-are-velocity-o

slug:
a-pipe-closed-at-one-end-has-length-83-cm-the-number-of-possible-natural-oscillations-of-air-column-whose-frequencies-lie-below-1000-hz-are-velocity-o-34464

question:
<div class="question">A pipe closed at one end has length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>83</mn><mo> </mo><mi>cm</mi></math>. The number of possible natural oscillations of air column whose frequencies lie below <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn><mo> </mo><mi>Hz</mi></math> are (velocity of sound in air <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>332</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>)</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>3</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>4</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> </div>

answer:
<div class="solution">Fundamental frequency<br/><br/><math> <mrow> <mi>n</mi><mo>=</mo><mfrac> <mi>V</mi> <mrow> <mn>4</mn><mi>L</mi></mrow> </mfrac> <mo>=</mo><mfrac> <mrow> <mn>332</mn></mrow> <mrow> <mn>4</mn><mo>Ã</mo><mn>83</mn><mo>Ã</mo><msup> <mrow> <mn>10</mn></mrow> <mrow> <mo>â</mo><mn>2</mn></mrow> </msup> </mrow> </mfrac><mo>=</mo><mn>100</mn> </mrow> </math> .<br/><br/>Hence possible frequencies are <math> <mrow> <mi>n</mi><mo>=</mo><mn>100</mn></mrow> </math> .<br/><br/><math> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>=</mo><mn>300</mn></mrow></math>.<br/><br/><math> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>=</mo><mn>500</mn></mrow></math>.<br/><br/><math> <mrow> <msub> <mi>n</mi> <mn>3</mn> </msub> <mo>=</mo><mn>700.</mn></mrow> </math><br/><br/><math> <mrow> <msub> <mi>n</mi> <mn>5</mn> </msub> <mo>=</mo><mn>900</mn></mrow></math>.<br/> </div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2022 (06 Aug Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-pipe-closed-at-one-end-length-08-m-at-its-open-end-a-05-mlong-uniform-string-is-vibrating-in-its-second-harmonic-and-it-resonates-with-the-fundament

slug:
a-pipe-closed-at-one-end-length-08-m-at-its-open-end-a-05-mlong-uniform-string-is-vibrating-in-its-second-harmonic-and-it-resonates-with-the-fundament-12462

question:
<div class="question">A pipe closed at one end length $0.8 \mathrm{~m}$. At its open end a $0.5 \mathrm{mlong}$ uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire in $50 \mathrm{~N}$ and the speed of sound is $320 \mathrm{~m} / \mathrm{s}$, the mass of the string used is</div>

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

answer:
<div class="solution">For fundamental harmonics of a closed end pipe, $\frac{\lambda}{4}=L$<br/>Velocity of sound is in air is given by, $v=\lambda f$<br/>For second harmonics of a string, $\lambda^{\prime}=L^{\prime}$ the wavelength is the length of the string.<br/>Speed of sound on the wave is related as, $v_{\mathrm{S}}=\lambda^{\prime} f$.<br/>Speed of sound is given by, $v_{\mathrm{S}}=\sqrt{\frac{T}{\mu}}, T$ is the tension in the wire and $\mu$ the mass per unit length of the string.<br/>Since, the string in second harmonics is in resonance with the closed end pipe in fundamental mode,<br/>$\begin{aligned} &amp; f=\frac{v}{4 L}=\frac{v_{\mathrm{S}}}{L^{\prime}}=\frac{1}{L^{\prime}} \sqrt{\frac{T}{\mu}} \\ &amp; \Rightarrow \frac{320 \mathrm{~m} / \mathrm{s}}{4 \times 0.8 \mathrm{~m}}=\frac{1}{0.5 \mathrm{~m}} \sqrt{\frac{50 \mathrm{~N}}{\mu}} \\ &amp; \Rightarrow \mu=\frac{1}{50} \mathrm{kgm}^{-1} \\ &amp; \Rightarrow m=\mu L^{\prime}=\frac{0.5 \mathrm{~m}}{50 \mathrm{~kg}^{-1} \mathrm{~m}}=10 \mathrm{~g}\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-pipe-of-1-m-length-is-closed-at-one-end-taking-the-speed-of-sound-in-air-as-320-m-s-1-the-air-column-in-the-pipe-cannot-resonate-for-the-frequency-i

slug:
a-pipe-of-1-m-length-is-closed-at-one-end-taking-the-speed-of-sound-in-air-as-320-m-s-1-the-air-column-in-the-pipe-cannot-resonate-for-the-frequency-i-34429

question:
<div class="question">A pipe of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> length is closed at one end. Taking the speed of sound in air as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>320</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, the air column in the pipe cannot resonate for the frequency <math><mfenced separators="|"><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal"> </mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi></mrow></mfenced></math></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>80</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>160</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>240</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>560</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>160</mn></math></span> </div>

answer:
<div class="solution">Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>320</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mi> </mi><mi>l</mi><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math><br/><br/>For an organ pipe whose one end is closed, only odd harmonics containing odd multiples of fundamental frequency are present.<br/><br/>Resonate frequency of the first mode, <math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>4</mn><msup><mrow><mi>l</mi></mrow><mrow><mi>′</mi></mrow></msup></mrow></mfrac></math><br/><br/>Second mode, <math><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>3</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo></math><br/><br/>Third mode, <math><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>5</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo></math><br/><br/>Fourth mode, <math><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mn>7</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo></math><br/><br/>Fifth mode, <math><msub><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>=</mo><mn>9</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></math><br/><br/>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>320</mn><mn>4</mn></mfrac><mo>=</mo><mn>80</mn><mo> </mo><mi>Hz</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub><mo>=</mo><mn>3</mn><mo>×</mo><mn>80</mn><mo>=</mo><mn>240</mn><mi> </mi><mi>Hz</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>3</mn></msub><mo>=</mo><mn>5</mn><mo>×</mo><mn>80</mn><mo>=</mo><mn>400</mn><mi> </mi><mi>Hz</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>4</mn></msub><mo>=</mo><mn>7</mn><mo>×</mo><mn>80</mn><mo>=</mo><mn>560</mn><mi> </mi><mi>Hz</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>5</mn></msub><mo>=</mo><mn>9</mn><mo>×</mo><mn>80</mn><mo>=</mo><mn>720</mn><mi mathvariant="normal"> </mi><mi>Hz</mi></math><br/><br/>So <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>160 </mi><mi>Hz</mi></math>  is not possible</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-pipe-of-length-1-5-m-closed-at-one-end-is-filled-with-gas-and-resonates-in-its-fundamental-mode-with-a-tuning-fork-another-open-organ-pipe-of-same-d-1

slug:
a-pipe-of-length-1-5-m-closed-at-one-end-is-filled-with-gas-and-resonates-in-its-fundamental-mode-with-a-tuning-fork-another-open-organ-pipe-of-same-d-1-98260

question:
<div class="question">A pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi></math> closed at one end is filled with gas and resonates in its fundamental mode with a tuning fork. Another open organ pipe of same dimensions filled with air resonates in its fundamental mode with the same tuning fork. If this experiment is performed at  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>Â°</mo><mo> </mo><mi mathvariant="normal">C</mi></math>, then the speed of sound at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo>Â°</mo><mi mathvariant="normal">C</mi></math> in the same gas is (Speed of sound in air is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math>)</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>580</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</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">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>880</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>743</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </div>

answer:
<div class="solution"><p>For open pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mi mathvariant="normal">v</mi><mrow><mn>2</mn><mi mathvariant="normal">â</mi></mrow></mfrac><mo>=</mo><mfrac><mn>360</mn><mrow><mn>2</mn><mo>Ã</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mn>120</mn><mo> </mo><mi>Hz</mi></math></p><p>For the closed at one end pipe, let the velocity of sound in gas at  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><msup><mo> </mo><mo>Â°</mo></msup><mi mathvariant="normal">C</mi></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mrow><mn>4</mn><mi mathvariant="normal">â</mi></mrow></mfrac><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mrow><mn>4</mn><mo>Ã</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mn>6</mn></mfrac></math></p><p>Since same tuning fork is used frequency will be same and thus,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mi>f</mi><mo>â</mo><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mo>=</mo><mn>720</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></p><p>Also </p><p><math><mi mathvariant="normal">v</mi><mo>â</mo><msqrt><mi mathvariant="normal">T</mi></msqrt><mo>â</mo><mi mathvariant="normal">v</mi><mo>=</mo><mn>720</mn><mo>Ã</mo><msqrt><mfrac><mn>273</mn><mn>303</mn></mfrac></msqrt><mo>=</mo><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s.</mi></math></p></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-pipe-of-length-1-5-m-closed-at-one-end-is-filled-with-gas-and-resonates-in-its-fundamental-mode-with-a-tuning-fork-another-open-organ-pipe-of-same-d-2

slug:
a-pipe-of-length-1-5-m-closed-at-one-end-is-filled-with-gas-and-resonates-in-its-fundamental-mode-with-a-tuning-fork-another-open-organ-pipe-of-same-d-2-55219

question:
<div class="question">A pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi></math> closed at one end is filled with gas and resonates in its fundamental mode with a tuning fork. Another open organ pipe of same dimensions filled with air resonates in its fundamental mode with the same tuning fork. If this experiment is performed at  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>Â°</mo><mo> </mo><mi mathvariant="normal">C</mi></math>, then the speed of sound at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo>Â°</mo><mi mathvariant="normal">C</mi></math> in the same gas is (Speed of sound in air is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math>)</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>580</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</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">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>880</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>743</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </div>

answer:
<div class="solution"><p>For open pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mi mathvariant="normal">v</mi><mrow><mn>2</mn><mi mathvariant="normal">â</mi></mrow></mfrac><mo>=</mo><mfrac><mn>360</mn><mrow><mn>2</mn><mo>Ã</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mn>120</mn><mo> </mo><mi>Hz</mi></math></p><p>For the closed at one end pipe, let the velocity of sound in gas at  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><msup><mo> </mo><mo>Â°</mo></msup><mi mathvariant="normal">C</mi></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mrow><mn>4</mn><mi mathvariant="normal">â</mi></mrow></mfrac><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mrow><mn>4</mn><mo>Ã</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mn>6</mn></mfrac></math></p><p>Since same tuning fork is used frequency will be same and thus,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mi>f</mi><mo>â</mo><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mo>=</mo><mn>720</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></p><p>Also </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">v</mi><mo>â</mo><msqrt><mi mathvariant="normal">T</mi></msqrt><mo>â</mo><mi mathvariant="normal">v</mi><mo>=</mo><mn>720</mn><mo>Ã</mo><msqrt><mfrac><mn>273</mn><mn>303</mn></mfrac></msqrt><mo>=</mo><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math>.</p></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-pipe-of-length-1-5-m-closed-at-one-end-is-filled-with-gas-and-resonates-in-its-fundamental-mode-with-a-tuning-fork-another-open-organ-pipe-of-same-d

slug:
a-pipe-of-length-1-5-m-closed-at-one-end-is-filled-with-gas-and-resonates-in-its-fundamental-mode-with-a-tuning-fork-another-open-organ-pipe-of-same-d-69006

question:
<div class="question">A pipe of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi></math> closed at one end is filled with gas and resonates in its fundamental mode with a tuning fork. Another open organ pipe of same dimensions filled with air resonates in its fundamental mode with the same tuning fork. If this experiment is performed at  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>Â°</mo><mo> </mo><mi mathvariant="normal">C</mi></math>, then the speed of sound at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo>Â°</mo><mi mathvariant="normal">C</mi></math> in the same gas is (Speed of sound in air is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math>)</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>580</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</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">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>880</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>743</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></span> </div>

answer:
<div class="solution"><p>For open pipe <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mi mathvariant="normal">v</mi><mrow><mn>2</mn><mi mathvariant="normal">â</mi></mrow></mfrac><mo>=</mo><mfrac><mn>360</mn><mrow><mn>2</mn><mo>Ã</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mn>120</mn><mo> </mo><mi>Hz</mi></math></p><p>For the closed at one end pipe, let the velocity of sound in gas at  <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><msup><mo> </mo><mo>Â°</mo></msup><mi mathvariant="normal">C</mi></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mrow><mn>4</mn><mi mathvariant="normal">â</mi></mrow></mfrac><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mrow><mn>4</mn><mo>Ã</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mn>6</mn></mfrac></math></p><p>Since same tuning fork is used frequency will be same and thus,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mi>f</mi><mo>â</mo><msup><mi mathvariant="normal">v</mi><mo>'</mo></msup><mo>=</mo><mn>720</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math></p><p>Also </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">v</mi><mo>â</mo><msqrt><mi mathvariant="normal">T</mi></msqrt><mo>â</mo><mi mathvariant="normal">v</mi><mo>=</mo><mn>720</mn><mo>Ã</mo><msqrt><mfrac><mn>273</mn><mn>303</mn></mfrac></msqrt><mo>=</mo><mn>683</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math>.</p></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2014 (06 Apr)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch1_P1.db
title:
a-pipe-of-length-85-cm-is-closed-from-one-end-find-the-number-of-possible-natural-oscillations-of-air-column-in-the-pipe-whose-frequencies-lie-below-1

slug:
a-pipe-of-length-85-cm-is-closed-from-one-end-find-the-number-of-possible-natural-oscillations-of-air-column-in-the-pipe-whose-frequencies-lie-below-1-13984

question:
<div class="question">A pipe of length 85 cm is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below 1250 Hz. The velocity of sound in air is 340 m/s.</div>

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

answer:
<div class="solution"><p>fundamental frequency</p><p>       <img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/096b677d-0e21-44ef-b83f-8e831d1401d6-image.png" style="width: 309px; height: 147px;"/><br/><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>L</mtext><mo>=</mo><mfrac><mi>λ</mi><mn>4</mn></mfrac><mspace linebreak="newline"></mspace><msub><mi>v</mi><mn>1</mn></msub><mo>=</mo><mfrac><mtext>v</mtext><mrow><mn>4</mn><mtext>L</mtext></mrow></mfrac></math><br/><br/><br/>       <img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/a2efe8d3-c2a7-485a-ab8e-27e171714b46-image.png" style="width: 287px; height: 134px;"/><br/><br/><br/><math> <mrow> <mtext>L</mtext> <mo>=</mo> <mfrac> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>λ</mi> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mn>3</mn><mtext>v</mtext></mrow><mrow><mn>4</mn><mtext>L</mtext></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴   </mo><mi>v</mi><mo>=</mo><mtext>odd</mtext><mfenced><mfrac><mtext>v</mtext><mrow><mn>4</mn><mtext>L</mtext></mrow></mfrac></mfenced></math><br/><br/>Now <math> <mrow> <mtext>L</mtext> <mo>=</mo> <mn>8</mn> <mn>5</mn> <mo>cm</mo> <mo>=</mo> <mtext>0.85 m</mtext> </mrow> </math><br/><br/>        <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>v</mtext><mo>=</mo><mn>3</mn><mn>4</mn><mn>0</mn><mo>m/s</mo></math><br/><br/>         <math> <mrow> <mo>=</mo> <mtext>odd</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mfrac> <mrow> <mn>3</mn> <mn>4</mn> <mn>0</mn> </mrow> <mrow> <mn>4</mn> <mo>×</mo> <mtext>0.85</mtext> </mrow> </mfrac> </mrow> </mfenced> </mrow> </math><br/><br/>         <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>odd</mtext><mo>×</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo> </mo><mtext>Hz</mtext></math><br/><br/>Hence possible frequencies below 1250 Hz are<br/><br/>100 Hz, 300 Hz, 500 Hz...........1100 Hz<br/><br/>Hence 6 </p></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2016 (03 Apr)']
source_db: MarksBatch1_P1.db
title:
a-pipe-open-at-both-ends-has-a-fundamental-frequency-f-in-air-the-pipe-is-dipped-vertically-in-water-so-that-half-of-it-is-in-water-the-fundamental-fr

slug:
a-pipe-open-at-both-ends-has-a-fundamental-frequency-f-in-air-the-pipe-is-dipped-vertically-in-water-so-that-half-of-it-is-in-water-the-fundamental-fr-33191

question:
<div class="question">A pipe open at both ends has a fundamental frequency <math><mi>f</mi></math> in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>2</mn><mi>f</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mi>f</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">C</span> <span class="option-data"><math><mfrac><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>3</mn><mi>f</mi></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> </li> </ul>

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

answer:
<div class="solution"><br/><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_main/image4-6abc58e0-417e-4a4f-a8d6-55d209feebb0.png"/><br/><br/><math><mi>f</mi><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math><br/><br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_main/image5-c759526c-df86-46f8-93c2-4289c517bc7e.png"/><br/><br/><math><msup><mrow><mi>f</mi></mrow><mrow><mi>â²</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>4</mn><msup><mrow><mi>l</mi></mrow><mrow><mi>â²</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>4</mn><mfenced separators="|"><mrow><mfrac><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mfrac><mo>=</mo><mi>f</mi></math></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (12 Oct Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-pipe-open-at-one-end-has-length-08-m-at-the-open-end-of-the-tube-a-string-05-m-long-is-vibrating-in-its-1-st-overtone-and-resonates-with-fundamental

slug:
a-pipe-open-at-one-end-has-length-08-m-at-the-open-end-of-the-tube-a-string-05-m-long-is-vibrating-in-its-1-st-overtone-and-resonates-with-fundamental-84079

question:
<div class="question">A pipe open at one end has length $0.8 \mathrm{~m}$. At the open end of the tube a string $0.5 \mathrm{~m}$ long is vibrating in its $1^{\text {st }}$ overtone and resonates with fundamental frequency of pipe. If tension in the string is $50 \mathrm{~N}$, the mass of string is (speed of sound $=320 \mathrm{~m} / \mathrm{s}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">25 gram</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">15 gram</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">20 gram</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">10 gram</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">10 gram</span> </div>

answer:
<div class="solution">$2 \times\left[\frac{1}{2 \ell_{1}} \sqrt{\frac{T}{m}}\right]=\frac{v}{4 \ell_{2}}$<br/>$\frac{1}{0.5} \sqrt{\frac{50}{m}}=\frac{320}{4 \times 0.8}$<br/>$\therefore \quad$ Total mass of the string $=0.02 \times 0.5 \mathrm{~kg}=10 \mathrm{gm}$<br/>$\therefore \quad 02 \mathrm{~kg} / \mathrm{m}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-plane-p-passes-through-the-line-of-intersection-of-the-planes-2-x-y-3-z-2-x-y-z-1-and-the-point-1-0-1-if-the-plane-p-touches-the-sphere-x-2-y-2-z-2-

slug:
a-plane-p-passes-through-the-line-of-intersection-of-the-planes-2-x-y-3-z-2-x-y-z-1-and-the-point-1-0-1-if-the-plane-p-touches-the-sphere-x-2-y-2-z-2-79904

question:
<div class="question">A plane P passes through the line of intersection of the planes $2 x-y+3 z=2, x+y-z=1$ and the point $(1,0,1)$. If the plane $\mathrm{P}$ touches the sphere $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=\mathrm{r}^{2}$, then what is $r$ equal to?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{\sqrt{29}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{4}{\sqrt{29}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5}{\sqrt{29}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{\sqrt{29}}$</span> </div>

answer:
<div class="solution">Plane $\mathrm{P}$ touches the sphere $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=\mathrm{r}^{2}$ then<br/>$\mathrm{r}=$ Distane between centre of sphere $(0,0,0)$ to plance $\mathrm{P}$.<br/>$\Rightarrow \mathrm{r}=\left|\frac{5(0)+2(0)-5}{\sqrt{5^{2}+2^{2}+(0)^{2}}}\right|$<br/>$\frac{5}{\sqrt{25+4}}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-plane-p-passes-through-the-line-of-intersection-of-the-planes-2-x-y-3-z-2-x-y-z-1-and-the-point-1-0-1-what-are-the-direction-ratios-of-the-line-of-i

slug:
a-plane-p-passes-through-the-line-of-intersection-of-the-planes-2-x-y-3-z-2-x-y-z-1-and-the-point-1-0-1-what-are-the-direction-ratios-of-the-line-of-i-74932

question:
<div class="question">A plane P passes through the line of intersection of the planes $2 x-y+3 z=2, x+y-z=1$ and the point $(1,0,1)$. What are the direction ratios of the line of intersection of the given planes?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\langle 2,-5,-3\rangle$</span> </div>

answer:
<div class="solution">Tofind, intersection point first put $z=0$ $2 x-y=2$<br/>$\frac{x+y=1}{3 x=3}$<br/>$x+y=$<br/>$y=1-$<br/>$\mathrm{At} x=1, v=0$<br/>$(x, y, z)=(1,0,0)$<br/>Putting $x=0$<br/>$y-z=1$<br/>$\frac{-y+3 z=2}{2 z=3}$<br/>$z=\frac{3}{2}$<br/>$y=1+z=1+\frac{3}{2}=\frac{5}{2}$<br/>$(x, y, z) \equiv\left(0, \frac{5}{2}, \frac{3}{2}\right)$<br/>Point of intersection $\left(x_{1}, y_{1}, z_{1}\right)=(1,0,0)$<br/>$\left(x_{2}, y_{2}, z_{2}\right)=\left(0, \frac{5}{2}, \frac{3}{2}\right)$<br/>Hence direction ratios of the line of intersection of given plane $   &lt; 2,-5,-3&gt;$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-plane-p-passes-through-the-line-of-intersection-of-the-planes-2-x-y-3-z-2-x-y-z-1-and-the-point-1-0-1-what-is-the-equation-of-the-plane-p

slug:
a-plane-p-passes-through-the-line-of-intersection-of-the-planes-2-x-y-3-z-2-x-y-z-1-and-the-point-1-0-1-what-is-the-equation-of-the-plane-p-21271

question:
<div class="question">A plane P passes through the line of intersection of the planes $2 x-y+3 z=2, x+y-z=1$ and the point $(1,0,1)$. What is the equation of the plane P? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 2 x+5 y-2=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 x+2 y-5=0$</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><li class=""> <span class="option-label">C</span> <span class="option-data">$x+z-2=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x-y-2 z=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 x+2 y-5=0$</span> </div>

answer:
<div class="solution">Eq. of plane through two given planes is:<br/>$(2 x-y+3 z-2)+\lambda(x+y-z-1)=0$<br/>It passes through $(1,0,1)$ $\therefore 3-\lambda=0 \Rightarrow \lambda=3$<br/>Eq. of plane is:<br/>$5 x+2 y-5=0$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'AP EAMCET', 'AP EAMCET 2018 (25 Apr Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-plane-passes-through-the-point-3-5-7-if-the-direction-ratios-of-its-normal-are-equal-to-the-intercepts-made-by-the-plane-x-3-y-2-z-9-with-the-coordi-1

slug:
a-plane-passes-through-the-point-3-5-7-if-the-direction-ratios-of-its-normal-are-equal-to-the-intercepts-made-by-the-plane-x-3-y-2-z-9-with-the-coordi-1-47521

question:
<div class="question">A plane passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>)</mo></math>. If the direction ratios of its normal are equal to the intercepts made by the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>9</mn></math> with the coordinate axes, then the equation of that plane is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>5</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>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>105</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>12</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>z</mi><mo>=</mo><mn>49</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"><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>49</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>49</mn></math></span> </div>

answer:
<div class="solution"><p>Given:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mi>x</mi><mn>9</mn></mfrac><mo>+</mo><mfrac><mi>y</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mi>z</mi><mfenced><mstyle displaystyle="true"><mfrac><mn>9</mn><mn>2</mn></mfrac></mstyle></mfenced></mfrac><mo>=</mo><mn>1</mn></math></p><p>Comparing with the standard form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mi>a</mi></mfrac><mo>+</mo><mfrac><mi>y</mi><mi>b</mi></mfrac><mo>+</mo><mfrac><mi>z</mi><mi>c</mi></mfrac><mo>=</mo><mn>1</mn></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>9</mn><mo>→</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mtext>intercept</mtext></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>3</mn><mo>→</mo><mfenced><mrow><mi>y</mi><mo>-</mo><mtext>intercept</mtext></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfenced><mfrac><mn>9</mn><mn>2</mn></mfrac></mfenced><mo>→</mo><mfenced><mrow><mi>z</mi><mo>-</mo><mtext>intercept</mtext></mrow></mfenced></math></p><p>So, the normal vector for the required plane is <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mn>9</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mover><mi>k</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mn>3</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>5</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mn>7</mn><mover><mi>k</mi><mo>^</mo></mover></math></p><p>Equation of required plane,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>→</mo></mover><mo>·</mo><mover><mi>n</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mn>9</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>5</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mn>7</mn><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced><mo>·</mo><mfenced><mrow><mn>9</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mi>r</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mn>9</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>27</mn><mo>+</mo><mn>15</mn><mo>+</mo><mfrac><mn>63</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced><mrow><mi>x</mi><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mi>y</mi><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mi>z</mi><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced><mo>·</mo><mfenced><mrow><mn>9</mn><mover><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi>j</mi><mo>^</mo></mover><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>42</mn><mo>+</mo><mfrac><mn>63</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mi>z</mi><mo>=</mo><mfrac><mn>147</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>z</mi><mo>=</mo><mfrac><mn>49</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>49</mn></math>.</p></div>


tags: ['Physics', 'Wave Optics', 'MHT CET', 'MHT CET 2011']
source_db: MarksBatch1_P1.db
title:
a-plane-wave-front-of-wavelength-is-incident-on-a-single-slit-of-width-b-what-is-the-angular-width-for-secondary-maximum

slug:
a-plane-wave-front-of-wavelength-is-incident-on-a-single-slit-of-width-b-what-is-the-angular-width-for-secondary-maximum-31213

question:
<div class="question">A plane wave front of wavelength $\lambda$ is incident on a single slit of width $b$. What is the angular width for secondary maximum?</div>

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

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

answer:
<div class="solution">$\beta=\frac{\lambda D}{d}$ and $\theta=\frac{\beta}{D}$<br/>$\therefore$<br/>$$<br/>\theta=\frac{\lambda}{d}<br/>$$<br/>Width of single slit $=b$<br/>So,<br/>$$<br/>\theta=\frac{\lambda}{b}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-point-is-chosen-at-random-inside-a-circle-what-is-the-probability-that-the-point-is-closer-to-the-centre-of-the-circle-than-to-its-boundary

slug:
a-point-is-chosen-at-random-inside-a-circle-what-is-the-probability-that-the-point-is-closer-to-the-centre-of-the-circle-than-to-its-boundary-85054

question:
<div class="question">A point is chosen at random inside a circle. What is the probability that the point is closer to the centre of the circle than to its boundary?</div>

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

answer:
<div class="solution">Probability $=\frac{\pi\left(\frac{\mathrm{r}}{2}\right)^{2}}{\pi \mathrm{r}^{2}}$<br/>$=\frac{\pi\left(\frac{r^{2}}{4}\right)}{A r^{2}}$<br/>$=\frac{r^{2}}{4 r^{2}}=\frac{1}{4}$</div>


tags: ['Mathematics', 'Probability', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-point-is-chosen-at-random-inside-a-rectangle-measuring-6-inches-by-5-inches-what-is-the-probability-that-the-randomly-selected-point-is-at-least-one-1

slug:
a-point-is-chosen-at-random-inside-a-rectangle-measuring-6-inches-by-5-inches-what-is-the-probability-that-the-randomly-selected-point-is-at-least-one-1-87047

question:
<div class="question">A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{2}{5}$</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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{5}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/UB2IOrOVkA1AvTb3MnnEdYO0zie6ZTLyoUKWQkPqXZ4.original.fullsize.png"/><br/><br/>Probability that the randomly selected point is at least one inch from the edge of the rectangle $=\frac{4 \times 3}{6 \times 5}=\frac{12}{30}=\frac{2}{5}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-point-is-chosen-at-random-inside-a-rectangle-measuring-6-inches-by-5-inches-what-is-the-probability-that-the-randomly-selected-point-is-at-least-one

slug:
a-point-is-chosen-at-random-inside-a-rectangle-measuring-6-inches-by-5-inches-what-is-the-probability-that-the-randomly-selected-point-is-at-least-one-32119

question:
<div class="question">A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{2}{5}$</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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{5}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/UB2IOrOVkA1AvTb3MnnEdYO0zie6ZTLyoUKWQkPqXZ4.original.fullsize.png"/><br/><br/>Probability that the randomly selected point is at least one inch from the edge of the rectangle $=\frac{4 \times 3}{6 \times 5}=\frac{12}{30}=\frac{2}{5}$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$z=3$</span> </div>
source_db: MarksBatch1_P1.db
title:
a-point-p-1-2-3-is-one-vertex-of-a-cuboid-formed-bythe-coordinate-planes-and-the-planes-passing-through-p-and-parallel-to-the-coordinate-planes-what-i-1

slug:
a-point-p-1-2-3-is-one-vertex-of-a-cuboid-formed-bythe-coordinate-planes-and-the-planes-passing-through-p-and-parallel-to-the-coordinate-planes-what-i-1-89835

question:
<div class="question">A point $P(1,2,3)$ is one vertex of a cuboid formed bythe coordinate planes and the planes passing through $P$ and parallel to the coordinate planes. What is the equation of the plane passing through $P(1,2,3)$ and parallel to $x y$ -plane?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+y=3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x-y=-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$z=3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x+2 y+3 z=14$</span> </li> </ul>

answer:
<div class="solution">Equation of plane passing through $(1,2,3)$ and parallel to $x y$ -plane is $z=3$.<br/>$\therefore \quad$ Option $(\mathrm{c})$ is correct.</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-point-p-1-2-3-is-one-vertex-of-a-cuboid-formed-bythe-coordinate-planes-and-the-planes-passing-through-p-and-parallel-to-the-coordinate-planes-what-i

slug:
a-point-p-1-2-3-is-one-vertex-of-a-cuboid-formed-bythe-coordinate-planes-and-the-planes-passing-through-p-and-parallel-to-the-coordinate-planes-what-i-64840

question:
<div class="question">A point $P(1,2,3)$ is one vertex of a cuboid formed bythe coordinate planes and the planes passing through $P$ and parallel to the coordinate planes. What is the length of one of the diagonals of the cuboid?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{14}$ units</span> </div>

answer:
<div class="solution">Length of one of the diagonal of cube<br/>$=\sqrt{(1)^{2}+(2)^{2}+(3)^{2}}$<br/>$=\sqrt{1+4+9}=\sqrt{14}$ units<br/>$\therefore \quad$ Option (b) is correct.</div>


title: a-point-p-moves-such-that-its-distances-from-1-2-and-2-3-are-equal-then-the-locus-of-p-is
slug: a-point-p-moves-such-that-its-distances-from-1-2-and-2-3-are-equal-then-the-locus-of-p-is-44293
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch1_P1.db
question:
<div class="question">A point $P$ moves such that its distances from $(1,2)$ and $(-2,3)$ are equal. Then the locus of $P$ is</div>

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

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

answer:
<div class="solution">Let moving point be $\mathrm{P}(\mathrm{x}, \mathrm{y})$<br/>$\sqrt{(\mathrm{y}-2)^{2}+(\mathrm{x}-1)^{2}}=\sqrt{(\mathrm{y}-3)^{2}+(\mathrm{x}+2)^{2}}$<br/>$\Rightarrow(\mathrm{y}+2)^{2}+(\mathrm{x}-1)^{2}=(\mathrm{y}-3)^{2}+(\mathrm{x}+2)^{2}$<br/>$\Rightarrow y^{2}+4+4 y+x^{2}+1-2 x=y^{2}+9-6 y+x^{2}+4 x+4$<br/>$\Rightarrow 10 \mathrm{y}-6 \mathrm{x}-8=0$<br/>$\therefore$ Locus of $\mathrm{P}$ is a straight line.</div>


tags: ['Mathematics', 'Hyperbola', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P1.db
title:
a-point-p-moves-such-that-the-difference-of-its-distances-from-two-given-points-c-0-and-c-0-is-constant-what-is-the-locus-of-the-point-p

slug:
a-point-p-moves-such-that-the-difference-of-its-distances-from-two-given-points-c-0-and-c-0-is-constant-what-is-the-locus-of-the-point-p-32953

question:
<div class="question">A point $P$ moves such that the difference of its distances from two given points $(c, 0)$ and $(-c, 0)$ is constant. What is the locus of the point $P$ ?</div>

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

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

answer:
<div class="solution">When a point $P$ mowes such that the difference of its distances from two given points $(c, 0)$ and $(-c, 0)$ is constant, then the locus of the point $\mathrm{P}$ is hyperbola. It is the definition of hyper bola also.</div>


tags: ['Physics', 'Oscillations', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-pole-is-floating-in-a-liquid-with-80-cm-of-its-length-immersed-it-is-pushed-down-a-certain-distance-and-then-released-the-time-period-of-vertical-os

slug:
a-pole-is-floating-in-a-liquid-with-80-cm-of-its-length-immersed-it-is-pushed-down-a-certain-distance-and-then-released-the-time-period-of-vertical-os-37804

question:
<div class="question">A pole is floating in a liquid with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo> </mo><mi>cm</mi></math> of its length immersed. It is pushed down a certain distance and then released. The time period of vertical oscillation 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"><mfrac><mrow><mn>4</mn><mtext>π</mtext></mrow><mn>7</mn></mfrac><mo> </mo><mi mathvariant="normal">s</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"><mfrac><mrow><mn>3</mn><mtext>π</mtext></mrow><mn>7</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mtext>π</mtext></mrow><mn>7</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>Ï</mtext></mrow><mn>7</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><mtext>π</mtext></mrow><mn>7</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math></span> </div>

answer:
<div class="solution">Using the standard result of the time period of floating bodies.<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>T</mi><mo>=</mo><mn>2</mn><mtext>π</mtext><msqrt><mfrac><mrow><mi>l</mi><msub><mo>⁡</mo><mtext>immersed</mtext></msub></mrow><mi>g</mi></mfrac></msqrt><mtext> s</mtext></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>T</mi><mo>=</mo><mn>2</mn><mtext>π</mtext><msqrt><mfrac><mrow><mn>0</mn><mtext>.</mtext><mn>8</mn><mn>0</mn></mrow><mrow><mn>9</mn><mtext>.</mtext><mn>8</mn></mrow></mfrac></msqrt><mo>=</mo><mfrac><mrow><mn>4</mn><mtext>π</mtext></mrow><mn>7</mn></mfrac><mo> </mo><mi mathvariant="normal">s</mi></math></div>


title: a-polygon-has-44-diagonals-the-number-of-its-sides-is
slug: a-polygon-has-44-diagonals-the-number-of-its-sides-is-32840
question: <div class="question">A polygon has 44 diagonals. The number of its sides is</div>
tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7</span> </li> </ul>

answer:
<div class="solution">No. of diagonals in a polygen $={ }^{\mathrm{n}} \mathrm{C}_{2}-\mathrm{n}$<br/>$\Rightarrow 44={ }^{\mathrm{n}} \mathrm{C}_{2}-\mathrm{n}$<br/>$\Rightarrow 44=\frac{\mathrm{n} !}{2 !(\mathrm{n}-2) !}-\mathrm{n}$<br/>$\Rightarrow \quad 44=\frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{n}$<br/>$\Rightarrow \quad 44=\frac{\mathrm{n}(\mathrm{n}-3)}{2}$<br/>$\Rightarrow \mathrm{n}^{2}-3 \mathrm{n}-88=0$<br/>$\Rightarrow(\mathrm{n}-11)(\mathrm{n}+8)=0$<br/>$\mathrm{n} \neq-8$<br/>$\mathrm{n}=11$</div>


tags: ['Biology', 'Organisms and Populations']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
a-population-of-paramecium-caudatum-was-growth-in-a-culture-medium-after-5-days-the-culture-medium-became-overcrowded-with-paramecium-and-had-depleted

slug:
a-population-of-paramecium-caudatum-was-growth-in-a-culture-medium-after-5-days-the-culture-medium-became-overcrowded-with-paramecium-and-had-depleted-41349

question:
<div class="question">A population of Paramecium caudatum was growth in a culture medium. After 5 days the culture medium became overcrowded with Paramecium and had depleted nutrients. What will happen to the population and what type of growth will the population attain? Draw the growth curve.</div>

answer:
<div class="solution">After the lag phase initially, the population will grow in an exponential manner as the nutrients and space will be abundant.<br/>When depletion of the food sources, the population density starts decreasing and ends in an asymptote phase, after which the population density reaches the carrying<br/>capacity (maximum number of individuals of a population or species that a given environment can sustain indefinitely).<br/>The population shows a logistic of pattern growth giving an S-shaped curve.<br/>Where $K=$ carrying capacity<br/>$N=$ population density at time ' $\mathrm{t}$ '<br/>$r=$ Intrinsic rate of natural increase<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/C1wqv1Nhe9iSbfGj-GXjDiU3QiXpF9DQgIDOExdA8-o.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-positive-acute-angle-is-divided-into-two-parts-whose-tangents-are-8-1-and-9-7-what-is-the-value-of-this-angle

slug:
a-positive-acute-angle-is-divided-into-two-parts-whose-tangents-are-8-1-and-9-7-what-is-the-value-of-this-angle-33192

question:
<div class="question">A positive acute angle is divided into two parts whose<br/>tangents are $\frac{1}{8}$ and $\frac{7}{9}$. What is the value of this angle?</div>

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

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

answer:
<div class="solution">Let two parts of an angle $\theta$ are $\phi$ and $\psi$. So, $\theta=\phi+\psi$ So, $\tan \theta=\tan (\phi+\psi)$<br/>$=\frac{\tan \phi+\tan \psi}{1-\tan \phi \tan \psi}=\frac{\frac{1}{8}+\frac{7}{9}}{1-\frac{1}{8} \cdot \frac{7}{9}}=\frac{\frac{9+56}{72}}{\frac{72-7}{72}}=\frac{\frac{65}{72}}{\frac{65}{72}}=1$<br/>$=\tan \frac{\pi}{4} \Rightarrow \theta=\frac{\pi}{4}$</div>


tags: ['Physics', 'Current Electricity', 'TS EAMCET', 'TS EAMCET 2023 (12 May Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-potentiometer-balances-at-44-cm-when-a-cell-of-internal-resistance-1-is-in-the-secondary-circuit-to-obtain-the-balancing-point-at-40-cm-the-resistan

slug:
a-potentiometer-balances-at-44-cm-when-a-cell-of-internal-resistance-1-is-in-the-secondary-circuit-to-obtain-the-balancing-point-at-40-cm-the-resistan-94419

question:
<div class="question">A potentiometer balances at $44 \mathrm{~cm}$ when a cell of internal resistance $1 \Omega$ is in the secondary circuit. To obtain the balancing point at $40 \mathrm{~cm}$, the resistance to be connected parallel to cell is</div>

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

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

answer:
<div class="solution">(b) Internal Resistance, $\mathrm{r}=1 \Omega$<br/><br/>Balancing Point, $\ell_1=44 \mathrm{~cm}$.<br/>Obtain Balancing Point $\ell_2=40 \mathrm{~cm}$.<br/>Internal Resistance is given by<br/>$$<br/>\mathrm{r}=\mathrm{R}\left[\frac{\ell_1-\ell_2}{\ell_2}\right]<br/>$$<br/>$\begin{aligned} &amp; \mathrm{R}=\frac{\mathrm{r} \ell_2}{\ell_1-\ell_2}=\frac{1 \times 40}{44-40} \\ &amp; \Rightarrow \mathrm{R}=\frac{40}{4}=10 \\ &amp; \Rightarrow \mathrm{R}=10 \Omega\end{aligned}$</div>


tags: ['Physics', 'Current Electricity', 'KCET', 'KCET 2020']
source_db: MarksBatch1_P1.db
title:
a-potentiometer-has-a-uniform-wire-of-length-5-m-a-battery-of-emf-10-v-and-negligible-internal-resistance-is-connected-between-its-ends-a-secondary-ce

slug:
a-potentiometer-has-a-uniform-wire-of-length-5-m-a-battery-of-emf-10-v-and-negligible-internal-resistance-is-connected-between-its-ends-a-secondary-ce-51340

question:
<div class="question">A potentiometer has a uniform wire of length $5 \mathrm{~m}$. A battery of emf $10 \mathrm{~V}$ and negligible internal resistance is connected between its ends. A secondary cell connected to the circuit gives balancing length at $200 \mathrm{~cm}$. The emf of the secondary cell is</div>

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

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

answer:
<div class="solution">Given, length of potentiometer wire<br/>$l=5 \mathrm{~m}$<br/>emf of battery, $E=10 \mathrm{~V}$<br/>Potential gradient across the potentiometer wire,<br/>$K=\frac{E}{l}=\frac{10}{5}=2 \mathrm{Vm}^{-1}$<br/>When secondary cell is connected to the circuit of potentiometer, then balancing length,<br/>$l_{1}=200 \mathrm{~cm}=2 \mathrm{~m}$<br/>$\therefore$ emf of the secondary cell, $E_{s}=K l_{1}=2 \times 2=4 \mathrm{~V}$</div>


title: a-potentiometer-is-an-ideal-device-of-measuring-potential-difference-because
slug: a-potentiometer-is-an-ideal-device-of-measuring-potential-difference-because-62237
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch1_P1.db
question:
<div class="question">A potentiometer is an ideal device of measuring potential difference because</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It does not disturb the potential difference it measures</span> </div>

answer:
<div class="solution">In balance condition, potentiometer doesn't take the current from secondary circuit.</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-potentiometer-wire-of-length-10-cm-and-resistance-32-is-connected-in-series-with-a-battery-of-emf-2-5-v-internal-resistance-3-and-external-resistanc

slug:
a-potentiometer-wire-of-length-10-cm-and-resistance-32-is-connected-in-series-with-a-battery-of-emf-2-5-v-internal-resistance-3-and-external-resistanc-23333

question:
<div class="question">A potentiometer wire of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi>cm</mi></math> and resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo> </mo><mi mathvariant="normal">Î©</mi></math> is connected in series with a battery of EMF <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">V</mi></math>, internal resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi mathvariant="normal">Î©</mi></math> and external resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>. If a secondary cell of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi mathvariant="normal">V</mi></math> is balanced at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>cm</mi></math> length. Then value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>110</mn><mo> </mo><mi mathvariant="normal">Ω</mi></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>105</mn><mo> </mo><mi mathvariant="normal">Ω</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"><mn>115</mn><mo> </mo><mi mathvariant="normal">Î©</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>115</mn><mo> </mo><mi mathvariant="normal">Î©</mi></math></span> </div>

answer:
<div class="solution"><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/nnn-123.JPG" style="width: 300px; height: 294px;"/><br/><br/>Potential gradient of potentiometer<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2.5</mn><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mi>R</mi><mo>+</mo><mn>32</mn><mo>)</mo></mrow></mfrac><mo>Ã</mo><mfrac><mn>30</mn><mn>10</mn></mfrac></math><br/>Balancing length for cell of emf <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mn>0.4</mn><mo> </mo><mi mathvariant="normal">V</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>cm</mi></math>.<br/><math><mo>â´</mo><mfrac><mrow><mn>2.5</mn></mrow><mrow><mo>(</mo><mn>35</mn><mo>+</mo><mi>R</mi><mo>)</mo></mrow></mfrac><mo>Ã</mo><mn>3</mn><mo>Ã</mo><mn>8</mn><mo>=</mo><mn>0.4</mn></math><br/><math><mo>â</mo><mn>5</mn><mo>Ã</mo><mn>3</mn><mo>Ã</mo><mn>4</mn><mo>=</mo><mfrac><mrow><mn>0.04</mn><mo>Ã</mo><mo>(</mo><mn>35</mn><mo>+</mo><mi>R</mi><mo>)</mo></mrow><mrow><mn>10</mn></mrow></mfrac></math><br/><math><mo>â</mo><mn>15</mn><mo>Ã</mo><mn>10</mn><mo>=</mo><mn>35</mn><mo>+</mo><mi>R</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><mi>R</mi><mo>=</mo><mn>115</mn><mo> </mo><mi mathvariant="normal">Î©</mi></math></div>


tags: ['Physics', 'Current Electricity', 'JEE Main', 'JEE Main 2022 (24 Jun Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">25</span> </div>
source_db: MarksBatch1_P1.db
title:
a-potentiometer-wire-of-length-10-m-and-resistance-20-is-connected-in-series-with-a-25-v-battery-and-an-external-resistance-30-a-cell-of-emf-e-in-seco

slug:
a-potentiometer-wire-of-length-10-m-and-resistance-20-is-connected-in-series-with-a-25-v-battery-and-an-external-resistance-30-a-cell-of-emf-e-in-seco-56088

question:
<div class="question">A potentiometer wire of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi></math> and resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> is connected in series with a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mi mathvariant="normal">V</mi></math> battery and an external resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math>. A cell of emf <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> in secondary circuit is balanced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>250</mn><mo> </mo><mi>cm</mi></math> long potentiometer wire. The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> (in volt) is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>10</mn></mfrac></math>. The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is _____ .</div>

answer:
<div class="solution"><p>Since the given resistance is in series, so, total resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>R</mi></mfenced><mo>=</mo><mn>20</mn><mo>+</mo><mn>30</mn><mo>=</mo><mn>50</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p>Given,</p><p>Potential difference<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>25</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p>Current<math xmlns="http://www.w3.org/1998/Math/MathML"><mi> </mi><mfenced><mi>I</mi></mfenced><mo>=</mo><mfrac><mi>V</mi><mi>R</mi></mfrac><mo>=</mo><mfrac><mn>25</mn><mn>50</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p><p>Also, Potential gradient <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msub><mi mathvariant="normal">V</mi><mi>pd</mi></msub><mi>L</mi></mfrac><mo>=</mo><mfrac><mrow><mi>I</mi><mi>R</mi></mrow><mi>L</mi></mfrac></math></p><p>Where, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow><mi>p</mi><mi>d</mi></mrow></msub><mo>=</mo></math>potential drop at wire</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo></math>Length of wire</p><p>Potential gradient<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>20</mn><mo>×</mo><mn>0</mn><mo>.5</mo></mrow><mn>10</mn></mfrac><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">V</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><br/>So, Emf of the cell is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo></math>Potential gradient<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo></math>Balancing Length</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi> </mi><mn>1</mn><mo>×</mo><mi> </mi><mfrac><mn>250</mn><mn>100</mn></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi> </mi><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>=</mo><mfrac><mn>25</mn><mn>10</mn></mfrac><mo> </mo><mi mathvariant="normal">V</mi></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>25</mn></math>.</p></div>


tags: ['Physics', 'Current Electricity', 'JEE Main', 'JEE Main 2022 (26 Jul Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20</span> </div>
source_db: MarksBatch1_P1.db
title:
a-potentiometer-wire-of-length-300-cm-is-connected-in-series-with-a-resistance-780-and-a-standard-cell-of-emf-4-v-a-constant-current-flows-through-pot

slug:
a-potentiometer-wire-of-length-300-cm-is-connected-in-series-with-a-resistance-780-and-a-standard-cell-of-emf-4-v-a-constant-current-flows-through-pot-49498

question:
<div class="question"><p>A potentiometer wire of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><mi>cm</mi></math> is connected in series with a resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>780</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> and a standard cell of emf <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">V</mi></math>. A constant current flows through potentiometer wire. The length of the null point for cell of emf <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mi>mV</mi></math> is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><mi>cm</mi></math>. The resistance of the potentiometer wire is _____ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ω</mi></math>.</p></div>

answer:
<div class="solution"><blockquote><p><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_main/a2d8145c-80af-43a7-8d01-ae284380fc98-1.jpg" style="width: 294px; height: 270px;"/></p><p>Let resistance of potentiometers wire is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>, then the current through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>=</mo><mfrac><mn>4</mn><mrow><mi>R</mi><mo>+</mo><mn>780</mn></mrow></mfrac></math></p><p>Potential difference across <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mi>R</mi></mrow><mrow><mi>R</mi><mo>+</mo><mn>780</mn></mrow></mfrac></math></p><p>Potential difference across <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AC</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mo>=</mo><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mfrac><mi>l</mi><mi>L</mi></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mi>R</mi></mrow><mfenced><mrow><mi>R</mi><mo>+</mo><mn>780</mn></mrow></mfenced></mfrac><mo>×</mo><mfrac><mn>60</mn><mn>300</mn></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mi>R</mi></mrow><mrow><mn>5</mn><mfenced><mrow><mi>R</mi><mo>+</mo><mn>780</mn></mrow></mfenced></mrow></mfrac></math></p><p>This should be equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mi>mV</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>4</mn><mi>R</mi></mrow><mrow><mn>5</mn><mfenced><mrow><mi>R</mi><mo>+</mo><mn>780</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mn>20</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>4</mn><mi>R</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>R</mi><mo>+</mo><mn>780</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>4</mn><mi>R</mi><mo>-</mo><mfrac><mi>R</mi><mn>10</mn></mfrac><mo>=</mo><mn>78</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>39</mn><mi>R</mi></mrow><mn>10</mn></mfrac><mo>=</mo><mn>78</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>R</mi><mo>=</mo><mn>20</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p></blockquote></div>


tags: ['Physics', 'Current Electricity', 'JEE Main', 'JEE Main 2020 (02 Sep Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-potentiometer-wire-pq-of-1-m-length-is-connected-to-a-standard-cell-e-1-another-cell-e-2-of-emf-1-02-v-is-connected-with-a-resistance-r-and-switch-s

slug:
a-potentiometer-wire-pq-of-1-m-length-is-connected-to-a-standard-cell-e-1-another-cell-e-2-of-emf-1-02-v-is-connected-with-a-resistance-r-and-switch-s-16573

question:
<div class="question"><p>A potentiometer wire <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PQ</mi></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mi mathvariant="normal">m</mi></math> length is connected to a standard cell <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">E</mi><mn>1</mn></msub></math>. Another cell <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">E</mi><mn>2</mn></msub></math> of emf <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>02</mn><mo> </mo><mi mathvariant="normal">V</mi></math> is connected with a resistance '<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">r</mi></math>' and switch <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi></math> (as shown in figure). With switch <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi></math> open, the null position is obtained at a distance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo> </mo><mi>cm</mi></math> from<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Q</mi></math>. The potential gradient in the potentiometer wire is :</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/bbc8fc12-8205-4aaf-bd8f-d0ebe62ce442-image.png" style="width: 260px; height: 179px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>/</mo><mi>cm</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"><mn>0</mn><mo>.</mo><mn>01</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>/</mo><mi>cm</mi></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>0</mn><mo>.</mo><mn>03</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>/</mo><mi>cm</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>04</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>/</mo><mi>cm</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>/</mo><mi>cm</mi></math></span> </div>

answer:
<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/f1a1def0-d310-44f6-a67c-d1cbff168757-image.png" style="width: 300px; height: 334px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mi>J</mi><mo>=</mo><mn>49</mn><mo> </mo><mi>cm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>P</mi><mi>J</mi><mo>=</mo><mn>51</mn><mo> </mo><mi>cm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>v</mi><mi>ℓ</mi></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>02</mn></mrow><mn>51</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>02</mn><mo> </mo><mi mathvariant="normal">v</mi><mo>/</mo><mi>cm</mi></math></p></div>


tags: ['Physics', 'Alternating Current']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
a-power-transmission-line-feeds-input-power-at-2300-v-to-a-step-down-transformer-with-its-primary-windings-having-4000-turns-what-should-be-the-number

slug:
a-power-transmission-line-feeds-input-power-at-2300-v-to-a-step-down-transformer-with-its-primary-windings-having-4000-turns-what-should-be-the-number-12323

question:
<div class="question">A power transmission line feeds input power at $2300 \mathrm{~V}$ to a step down transformer with its primary windings having $\mathbf{4 0 0 0}$ turns. What should be the number of turns in the secondary in order to get output power at $230 \mathrm{~V}$ ?</div>

answer:
<div class="solution">Here $E_p=2300 \mathrm{~V}, N_p=400$ turns, $\mathrm{E}_{\mathrm{s}}=230 \mathrm{~V}, \mathrm{~N}_{\mathrm{s}}=$ ?<br/>We know in a transformer, $\frac{E_S}{E_p}=\frac{N_S}{N_p}$<br/>$$<br/>N_s=\frac{E_s N_p}{E_p}=\frac{230 \times 4000}{2300}=400 \mathrm{turns}<br/>$$</div>


tags: ['Chemistry', 'Amines', 'JEE Main', 'JEE Main 2022 (28 Jun Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-primary-aliphatic-amine-on-reaction-with-nitrous-acid-in-cold-273-k-and-there-after-raising-temperature-of-reaction-mixture-to-room-temperature-298-

slug:
a-primary-aliphatic-amine-on-reaction-with-nitrous-acid-in-cold-273-k-and-there-after-raising-temperature-of-reaction-mixture-to-room-temperature-298-42658

question:
<div class="question">A primary aliphatic amine on reaction with nitrous acid in cold <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>273</mn><mo> </mo><mi mathvariant="normal">K</mi></mrow></mfenced></math> and there after raising temperature of reaction mixture to room temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>298</mn><mo> </mo><mi mathvariant="normal">K</mi></mrow></mfenced></math>, gives a/an</div>

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

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

answer:
<div class="solution"><p>A primary aliphatic amine on reaction with nitrous acid in cold gives a diazonium salt, but diazonium salt of aliphatic amines are not much stable, so, they dissociate on increasing temperature slightly and gives alcohol with water.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub><munderover><mo>→</mo><mrow><mn>273</mn><mo>°</mo><mi mathvariant="normal">C</mi></mrow><mrow><msub><mi>HNO</mi><mn>2</mn></msub><mfenced><mi>aq</mi></mfenced></mrow></munderover><msup><msub><mi>RN</mi><mn>2</mn></msub><mo>+</mo></msup><munderover><mo>→</mo><mrow><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></mrow><mrow><mn>283</mn><mo>°</mo><mi mathvariant="normal">C</mi></mrow></munderover><mi mathvariant="normal">R</mi><mo>-</mo><mi>OH</mi><mo>+</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub></math></p></div>


tags: ['Chemistry', 'Amines']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
a-primary-amine-rnh-2-can-be-reacted-with-ch-3-x-to-get-secondary-amine-r-nhch-3-but-the-only-disadvantage-is-that-3-amine-and-quaternary-ammonium-sal

slug:
a-primary-amine-rnh-2-can-be-reacted-with-ch-3-x-to-get-secondary-amine-r-nhch-3-but-the-only-disadvantage-is-that-3-amine-and-quaternary-ammonium-sal-95681

question:
<div class="question">A primary amine, $\mathrm{RNH}_2$ can be reacted with $\mathrm{CH}_3-\mathrm{X}$ to get secondary amine, $\mathrm{R}-\mathrm{NHCH}_3$ but the only disadvantage is that $3^{\circ}$ amine and quaternary ammonium salts are al so obtained as side products. Can you suggest a method where $\mathrm{RNH}_2$ forms only $2^{\circ}$ amine?</div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/modules/cbse/xCdgAjcDEI8CX_pXFlJ0DflvwXEcspNYBfXQ2jQ2j3Q.original.fullsize.png"/><br/><br/><br/><br/>Carbylamine reaction can be used for conversion of primary amine into secondary amine.</div>


tags: ['Chemistry', 'Amines', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-primary-amine-rnh-2-can-be-reacted-with-ch-3-x-to-get-secondary-amine-rnhch-3-but-the-only-disadvantage-is-that-3-amine-and-quaternary-ammonium-salt

slug:
a-primary-amine-rnh-2-can-be-reacted-with-ch-3-x-to-get-secondary-amine-rnhch-3-but-the-only-disadvantage-is-that-3-amine-and-quaternary-ammonium-salt-91647

question:
<div class="question">A primary amine, RNH<sub>2</sub>can be reacted with CH<sub>3</sub>âX to get secondary amine, RâNHCH<sub>3</sub>but the only disadvantage is that 3Â° amine and quaternary ammonium salts are also obtained as side products. Is any method possible where RNH<sub>2</sub>forms only 2Â° amine?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Perform the reaction under high pressure to favor the formation of secondary amines.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Use a large excess of CH<sub>3</sub>âX to drive the reaction toward the formation of secondary amines.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Carbylamine reaction is shown by 1Â° amine only which results in the replacement of two hydrogen atoms attached to the nitrogen atom of the NH<sub>2</sub>group by one carbon atom. On catalytic reduction the isocyanide will give a secondary amine with one methyl group.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Conduct the reaction at a low temperature to favor the formation of secondary amines.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Carbylamine reaction is shown by 1Â° amine only which results in the replacement of two hydrogen atoms attached to the nitrogen atom of the NH<sub>2</sub>group by one carbon atom. On catalytic reduction the isocyanide will give a secondary amine with one methyl group.</span> </div>

answer:
<div class="solution">Correct Option is : (C)<br/>Carbylamine reaction is shown by 1Â° amine only which results in the replacement of two hydrogen atoms attached to the nitrogen atom of the NH<sub>2</sub>group by one carbon atom. On catalytic reduction the isocyanide will give a secondary amine with one methyl group.</div>


tags: ['Physics', 'Electromagnetic Induction', 'AP EAMCET', 'AP EAMCET 2012']
source_db: MarksBatch1_P1.db
title:
a-primary-coil-and-secondary-coil-are-placed-close-to-each-other-a-current-which-changes-at-the-rate-of-25-a-in-a-millisecond-is-present-in-the-primar-1

slug:
a-primary-coil-and-secondary-coil-are-placed-close-to-each-other-a-current-which-changes-at-the-rate-of-25-a-in-a-millisecond-is-present-in-the-primar-1-29380

question:
<div class="question">A primary coil and secondary coil are placed close to each other. A current, which changes at the rate of $25 \mathrm{~A}$ in a millisecond, is present in the primary coil. If the mutual inductance is $92 \times 10^{-6} \mathrm{H}$, then the value of induced emf in the secondary coil is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.6 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2.3 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.368 \mathrm{mV}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.23 \mathrm{mV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.3 \mathrm{~V}$</span> </div>

answer:
<div class="solution">The induced emf in secondary coil.<br/>$e=M \frac{d i}{d t}=92 \times 10^{-6} \times \frac{25}{1 \times 10^{-3}}=2.3 \mathrm{~V}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'TS EAMCET', 'TS EAMCET 2012']
source_db: MarksBatch1_P1.db
title:
a-primary-coil-and-secondary-coil-are-placed-close-to-each-other-a-current-which-changes-at-the-rate-of-25-a-in-a-millisecond-is-present-in-the-primar

slug:
a-primary-coil-and-secondary-coil-are-placed-close-to-each-other-a-current-which-changes-at-the-rate-of-25-a-in-a-millisecond-is-present-in-the-primar-53331

question:
<div class="question">A primary coil and secondary coil are placed close to each other. A current, which changes at the rate of $25 \mathrm{~A}$ in a millisecond, is present in the primary coil. If the mutual inductance is $92 \times 10^{-6} \mathrm{H}$, then the value of induced emf in the secondary coil is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.6 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2.3 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.368 \mathrm{mV}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.23 \mathrm{mV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.3 \mathrm{~V}$</span> </div>

answer:
<div class="solution">The induced emf in secondary coil.<br/>$e=M \frac{d i}{d t}=92 \times 10^{-6} \times \frac{25}{1 \times 10^{-3}}=2.3 \mathrm{~V}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P1.db
title:
a-problem-in-statistics-is-given-to-three-students-a-b-and-c-whose-chances-of-solving-it-independently-are-2-1-3-1-and-4-1-respectively-the-probabilit

slug:
a-problem-in-statistics-is-given-to-three-students-a-b-and-c-whose-chances-of-solving-it-independently-are-2-1-3-1-and-4-1-respectively-the-probabilit-68706

question:
<div class="question">A problem in statistics is given to three students $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ whose chances of solving it independently are $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$ respectively. The probability that the problem will be solved is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{12}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{11}{12}$</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">$\frac{3}{4}$</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> </ul>

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

answer:
<div class="solution">$\quad$ Let $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{1}{3} \&amp; \mathrm{P}(\mathrm{C})=\frac{1}{4}$<br/>$\mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=1-\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\overline{\mathrm{B}}) \mathrm{P}(\overline{\mathrm{C}})$<br/>$=1-\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)$<br/>$=1-\left[\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}\right]=1-\frac{1}{4}=\frac{3}{4}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-problem-is-given-to-three-students-a-b-and-c-whose-probabilities-of-solving-the-problem-are-2-1-4-3-and-4-1-respectively-what-is-the-probability-tha

slug:
a-problem-is-given-to-three-students-a-b-and-c-whose-probabilities-of-solving-the-problem-are-2-1-4-3-and-4-1-respectively-what-is-the-probability-tha-13021

question:
<div class="question">A problem is given to three students $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ whose<br/>probabilities of solving the problem are $\frac{1}{2}, \frac{3}{4}$ and $\frac{1}{4}$ respectively. What is the probability that the problem will be solved if they all solve the problem independently?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{29}{32}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{27}{32}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{25}{32}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{23}{32}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{29}{32}$</span> </div>

answer:
<div class="solution">Given, $P(A)=\frac{1}{2}, P(B)=\frac{3}{4}, P(C)=\frac{1}{4}$<br/>$P(\bar{A})=1-\frac{1}{2}=\frac{1}{2}$<br/>$P(\bar{B})=1-\frac{3}{4}=\frac{1}{4}$<br/>$P(\bar{C})=1-\frac{1}{4}=\frac{3}{4}$<br/>$\therefore$ Probability that problem will be solved if they solve independently is<br/>$1-(P(\bar{A}) \cdot P(\bar{B}) \cdot P(\bar{C})$<br/>$=1-\left(\frac{1}{2}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)=1-\frac{3}{32}=\frac{29}{32}$</div>


title: a-quadratic-polynomial-with-two-distinct-roots-has-one-real-root-then-the-other-root-is
slug: a-quadratic-polynomial-with-two-distinct-roots-has-one-real-root-then-the-other-root-is-60195
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch1_P1.db
question:
<div class="question">A quadratic polynomial with two distinct roots has one real root. Then, the other root is</div>

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

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

answer:
<div class="solution">We know that, if a quadratic polynomial with two distinct root has one real root, then<br/>$b^{2}-4 a c&gt;0$<br/>Then other root will always real.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-question-is-given-to-three-students-a-b-and-c-whose-chances-of-solving-it-are-2-1-3-1-and-4-1-respectively-what-is-the-probability-that-the-question

slug:
a-question-is-given-to-three-students-a-b-and-c-whose-chances-of-solving-it-are-2-1-3-1-and-4-1-respectively-what-is-the-probability-that-the-question-67894

question:
<div class="question">A question is given to three students A, B and C whose<br/>chances of solving it are $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$ respectively. What is the probability that the question will be solved?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{24}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{23}{24}$</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">Required probability $=1-\left(\mathrm{P}^{\prime}(\mathrm{A}) . \mathrm{P}^{\prime}(\mathrm{B}) . \mathrm{P}^{\prime}(\mathrm{C})\right)$<br/>$=1-\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)$<br/>$=\left(1-\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}\right)$<br/>$=1-\frac{1}{4}=\frac{3}{4}$</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch1_P1.db
title:
a-radioactive-element-disintegrates-at-a-rate-proportional-to-the-quantity-of-substance-q-present-at-any-time-t-what-is-the-differential-equation-of-t

slug:
a-radioactive-element-disintegrates-at-a-rate-proportional-to-the-quantity-of-substance-q-present-at-any-time-t-what-is-the-differential-equation-of-t-82333

question:
<div class="question">A radioactive element disintegrates at a rate proportional to the quantity of substance $\mathrm{Q}$ present at any time $\mathrm{t}$. What is the differential equation of the disintegration?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{d Q}{d t}=-k Q, k&gt;0$</span> </div>

answer:
<div class="solution">A radioactive element disintegrates at a rate proportional to the quantity of substance $Q$ present at any time $\mathrm{t}$.<br/>$\Rightarrow \quad \frac{\mathrm{dQ}}{\mathrm{dt}} \propto-\mathrm{Q}$<br/>$\Rightarrow \frac{\mathrm{dQ}}{\mathrm{dt}}=-\mathrm{kQ}, \mathrm{k}&gt;0$ is a constant.<br/>This is required differential equation.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">415</span> </div>
source_db: MarksBatch1_P1.db
title:
a-random-sample-of-20-people-is-classified-in-the-followin-1-beginarraycc-hline-age-frequency-hline-1525-2-hline-2535-4-hline-3545-6-hline-4555-5-hlin

slug:
a-random-sample-of-20-people-is-classified-in-the-followin-1-beginarraycc-hline-age-frequency-hline-1525-2-hline-2535-4-hline-3545-6-hline-4555-5-hlin-97492

question:
<div class="question">A random sample of 20 people is classified in the followin, 1<br/>\begin{array}{|c|c|}<br/>\hline Age &amp; Frequency \\<br/>\hline 15-25 &amp; 2 \\<br/>\hline 25-35 &amp; 4 \\<br/>\hline 35-45 &amp; 6 \\<br/>\hline 45-55 &amp; 5 \\<br/>\hline 55-65 &amp; 3 \\<br/>\hline<br/>\end{array} the mean age of this group of people?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$41.0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">415</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">420</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$42.5$</span> </li> </ul>

answer:
<div class="solution">\begin{array}{|c|c|c|c|}<br/>\hline Age &amp; Mid value \mathbf{x}_{\mathbf{i}} &amp; Frequency \mathbf{f}_{\mathbf{i}} &amp; \mathbf{f}_{\mathbf{i}} \mathbf{x}_{\mathbf{i}} \\<br/>\hline 15-25 &amp; 20 &amp; 2 &amp; 40 \\<br/>\hline 25-35 &amp; 30 &amp; 4 &amp; 120 \\<br/>\hline 35-45 &amp; 40 &amp; 6 &amp; 240 \\<br/>\hline 45-55 &amp; 50 &amp; 5 &amp; 250 \\<br/>\hline 55-65 &amp; 60 &amp; 3 &amp; 180 \\<br/>\hline &amp; &amp; \mathbf{\Sigma} \mathbf{f}_{\mathrm{i}}=20 &amp; \mathbf{\Sigma x}_{\mathbf{i}} \mathbf{f}_{\mathbf{i}}=830 \\<br/>\hline<br/>\end{array}<br/>$\Rightarrow$ Mean age $=\frac{\Sigma x_{i} f_{i}}{\Sigma f_{i}}=\frac{830}{20}=41.5$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$21$</span> </div>
source_db: MarksBatch1_P1.db
title:
a-random-variable-x-follows-the-binomial-distribution-and-x-b-n-03-if-the-mean-of-x-is-three-times-as-large-as-the-standard-deviation-of-x-then-n-1

slug:
a-random-variable-x-follows-the-binomial-distribution-and-x-b-n-03-if-the-mean-of-x-is-three-times-as-large-as-the-standard-deviation-of-x-then-n-1-64527

question:
<div class="question">A random variable $X$ follows the binomial distribution and $X \sim B(n, 0.3)$. If the mean of $X$ is three times as large as the standard deviation of $X$, then $n=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$9$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$21$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$27$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3$</span> </li> </ul>

answer:
<div class="solution">Given, $p=0.3, q=1-p=0.7$<br/>Also, Mean $(\mu)=3 \times$ Standard deviation $(\sigma)$.<br/>$\Rightarrow \quad \mu=3(\sigma)$...(i)<br/>$\begin{aligned} &amp; \text { Now, } \mu=n p=0.3 n \\ &amp; \text { and } \sigma=\sqrt{n p q}=\sqrt{n(0 \cdot 3)(0 \cdot 7)}=\sqrt{0.21 n} \\ &amp; \text { From Eq. (i), }(0.3) n=3 \sqrt{(0.21) n}\end{aligned}$<br/>$\begin{array}{ll}\Rightarrow &amp; (0.1) n=\sqrt{(0.21) n} \\ \Rightarrow &amp; \left(\frac{n}{10}\right)^2=\left(\frac{21 n}{100}\right) \\ \Rightarrow &amp; n^2-21 n=0 \Rightarrow n(n-21)=0 \\ \Rightarrow &amp; n=0,21\end{array}$<br/>But $n \neq 0$<br/>Thus, $n=21$.</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2022 (08 Jul Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$21$</span> </div>
source_db: MarksBatch1_P1.db
title:
a-random-variable-x-follows-the-binomial-distribution-and-x-b-n-03-if-the-mean-of-x-is-three-times-as-large-as-the-standard-deviation-of-x-then-n

slug:
a-random-variable-x-follows-the-binomial-distribution-and-x-b-n-03-if-the-mean-of-x-is-three-times-as-large-as-the-standard-deviation-of-x-then-n-32290

question:
<div class="question">A random variable $X$ follows the binomial distribution and $X \sim B(n, 0.3)$. If the mean of $X$ is three times as large as the standard deviation of $X$, then $n=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$9$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$21$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$27$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3$</span> </li> </ul>

answer:
<div class="solution">Given, $p=0.3, q=1-p=0.7$<br/>Also, Mean $(\mu)=3 \times$ Standard deviation $(\sigma)$.<br/>$\Rightarrow \quad \mu=3(\sigma)$...(i)<br/>$\begin{aligned} &amp; \text { Now, } \mu=n p=0.3 n \\ &amp; \text { and } \sigma=\sqrt{n p q}=\sqrt{n(0 \cdot 3)(0 \cdot 7)}=\sqrt{0.21 n} \\ &amp; \text { From Eq. (i), }(0.3) n=3 \sqrt{(0.21) n}\end{aligned}$<br/>$\begin{array}{ll}\Rightarrow &amp; (0.1) n=\sqrt{(0.21) n} \\ \Rightarrow &amp; \left(\frac{n}{10}\right)^2=\left(\frac{21 n}{100}\right) \\ \Rightarrow &amp; n^2-21 n=0 \Rightarrow n(n-21)=0 \\ \Rightarrow &amp; n=0,21\end{array}$<br/>But $n \neq 0$<br/>Thus, $n=21$.</div>


tags: ['Mathematics', 'Probability', 'AP EAMCET', 'AP EAMCET 2018 (23 Apr Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-random-variable-x-has-the-following-distribution-beginarraylllllllll-hline-beginarrayl-values-of-xx-endarray-0-1-2-3-4-5-6-7-hlinepxx-quad-0-k-2-k-2

slug:
a-random-variable-x-has-the-following-distribution-beginarraylllllllll-hline-beginarrayl-values-of-xx-endarray-0-1-2-3-4-5-6-7-hlinepxx-quad-0-k-2-k-2-20262

question:
<div class="question">A random variable $X$ has the following distribution<br/>\begin{array}{lllllllll}<br/>\hline \begin{array}{l} <br/>Values of \\<br/>X(x)<br/>\end{array} &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 \\<br/>\hlineP(X=x) \quad 0 &amp; k &amp; 2 k &amp; 2 k &amp; 3 k &amp; k^2 &amp; 2 k^2 &amp; 7 k^2+k \\<br/>\hline<br/>\end{array}</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{9}{10}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\frac{9}{10}\right)^2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{3}{10}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{10}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{9}{10}\right)^2$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Since, } \Sigma P\left(X=x_i\right)=1 \\<br/>&amp; \Rightarrow 0+K+2 K+2 K+3 K+K^2+2 K^2 \\<br/>&amp; +7 K^2+K=1 \\<br/>&amp; \Rightarrow \quad 9 K+10 K^2=1 \\<br/>&amp; \Rightarrow \quad K=\frac{1}{10}, \quad[\because K&gt;0] \\<br/>&amp;<br/>\end{aligned}<br/>$$<br/>So,<br/>$$<br/>\begin{aligned}<br/>&amp; P(0 &lt; K &lt; 6)=K+2 K+2 K+3 K+K^2 \\<br/>&amp; =8 K+K^2=\frac{8}{10}+\frac{1}{100}=\frac{81}{100}=\left(\frac{9}{10}\right)^2<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'MHT CET', 'MHT CET 2008']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.77$</span> </div>
source_db: MarksBatch1_P1.db
title:
a-random-variable-x-has-the-probability-distribution-beginarraylcccccccc-hlinex-1-2-3-4-5-6-7-8-hlinepx-015-023-012-010-020-008-007-005-hline-endarray

slug:
a-random-variable-x-has-the-probability-distribution-beginarraylcccccccc-hlinex-1-2-3-4-5-6-7-8-hlinepx-015-023-012-010-020-008-007-005-hline-endarray-93663

question:
<div class="question">A random variable $X$ has the probability distribution<br/>\begin{array}{l|c|c|c|c|c|c|c|c}<br/>\hlinex &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 \\<br/>\hlineP(x) &amp; 0.15 &amp; 0.23 &amp; 0.12 &amp; 0.10 &amp; 0.20 &amp; 0.08 &amp; 0.07 &amp; 0.05 \\<br/>\hline<br/>\end{array}<br/>For the events $E=\{x$ is prime number $\}$ and $F=\{x &lt; 4\}$ the probability of $P(E \cup F)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.50$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.77$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$0.35$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.87$</span> </li> </ul>

answer:
<div class="solution">Given, $E=\{x$ is a prime number $\}$<br/>$$<br/>\begin{aligned}<br/>P(E) &amp;=P(2)+P(3)+P(5)+P(7) \\<br/>&amp;=0.23+0.12+0.20+0.07=0.62 \\<br/>\text { and } \quad F &amp;=\{x &lt; 4\} \\<br/>P(F) &amp;=P(1)+P(2)+P(3) \\<br/>&amp;=0.15+0.23+0.12=0.50 \\<br/>\text { and } \quad P(E \cap F) &amp;=P(2)+P(3) \\<br/>=&amp; 0.23+0.12=0.35 \\<br/>\therefore \quad P(E \cup F) &amp;=P(E)+P(F)-P(E \cap F) \\<br/>&amp;=0.62+0.50-0.35 \\<br/>&amp;=0.77<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'AP EAMCET', 'AP EAMCET 2019 (20 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.82</span> </div>
source_db: MarksBatch1_P1.db
title:
a-random-variable-x-has-the-probability-distribution-beginarraylllllll-hline-xxi-1-2-3-4-5-6-hline-pleftxxiright-02-03-012-01-02-008-hline-endarray-if

slug:
a-random-variable-x-has-the-probability-distribution-beginarraylllllll-hline-xxi-1-2-3-4-5-6-hline-pleftxxiright-02-03-012-01-02-008-hline-endarray-if-12827

question:
<div class="question">A random variable \(X\) has the probability distribution<br/>\(\begin{array}{lllllll} \hline X=x_i &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 \\ \hline P\left(X=x_i\right) &amp; 0.2 &amp; 0.3 &amp; 0.12 &amp; 0.1 &amp; 0.2 &amp; 0.08 \\ \hline \end{array}\)<br/>If \(A=\left\{x_i / x_i\right.\) is a prime number}, \(B=\left\{x_i / x_i  &lt; 4\right\}\) are two events, then \(P(A \cup B)=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.31</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.62</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0.82</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">0.41</span> </li> </ul>

answer:
<div class="solution">The given probability distribution for random variable \(x\)<br/>\(\begin{array}{lllllll}<br/>\hline x=x_i &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 \\<br/>\hline P\left(X=x_i\right) &amp; 0.2 &amp; 0.3 &amp; 0.12 &amp; 0.1 &amp; 0.2 &amp; 0.08 \\<br/>\hline<br/>\end{array}\)<br/>and two events \(A=\left\{x_i \mid x_i\right.\) is a prime number \(\}\) \(=\{2,3,5\}\)<br/>and \(B=\left\{x_i \mid x_i  &lt; 4\right\}=\{1,2,3\}\)<br/>So,<br/>\(\begin{aligned}<br/>&amp; P(A \cup B)=P(X=1)+P(X=2)+P(X=3)+P(X=5) \\<br/>&amp; =0.2+0.3+0.12+0.2=0.82<br/>\end{aligned}\)<br/>Hence, option (3) is correct.</div>


tags: ['Physics', 'Ray Optics', 'JEE Main', 'JEE Main 2021 (25 Jul Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-ray-of-laser-of-a-wavelength-630-nm-is-incident-at-an-angle-of-30-at-the-diamondair-interface-it-is-going-from-diamond-to-air-the-refractive-index-o

slug:
a-ray-of-laser-of-a-wavelength-630-nm-is-incident-at-an-angle-of-30-at-the-diamondair-interface-it-is-going-from-diamond-to-air-the-refractive-index-o-84057

question:
<div class="question">A ray of laser of a wavelength <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>630</mn><mo> </mo><mi>nm</mi></math> is incident at an angle of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math> at the diamond-air interface. It is going from diamond to air. The refractive index of diamond is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>42</mn></math> and that of air is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>. Choose the correct option.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Angle of refraction is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo>.</mo><mn>41</mn><mo>°</mo></math>.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Angle of refraction is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math>.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Refraction is not possible.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Angle of refraction is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>53</mn><mo>.</mo><mn>4</mn><mo>°</mo></math>.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Refraction is not possible.</span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>sinθ</mi><mi mathvariant="normal">C</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>μ</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>μ</mi><mn>2</mn></msub></mrow></mfrac><mo>&lt;</mo><msub><mi>sinθ</mi><mi mathvariant="normal">C</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sinθ</mi><mo>&gt;</mo><msub><mi>sinθ</mi><mi mathvariant="normal">C</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">θ</mi><mo>&gt;</mo><msub><mi mathvariant="normal">θ</mi><mi mathvariant="normal">C</mi></msub></math></p><p>Total internal reflection will happen.</p></div>


tags: ['Physics', 'Ray Optics', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-ray-of-light-travelling-in-a-denser-medium-strikes-the-boundary-of-a-rarer-medium-the-ray-gets-partially-reflected-and-refracted-the-angle-of-reflec

slug:
a-ray-of-light-travelling-in-a-denser-medium-strikes-the-boundary-of-a-rarer-medium-the-ray-gets-partially-reflected-and-refracted-the-angle-of-reflec-34242

question:
<div class="question">A ray of light travelling in a denser medium strikes the boundary of a rarer medium. The ray gets partially reflected and refracted. The angle of reflection is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> and that of refraction is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mo>'</mo></msup></math>. If the reflected and refracted rays make an angle of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>Â°</mo></math> with each other then the critical angle for the pair of media is</div>

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

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

answer:
<div class="solution"><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/WhatsApp_Image_2020-07-03_at_4.03.43_PM.jpeg" style="width: 200px; height: 198px;"/><br/><br/><math> <mrow> <msub> <mspace></mspace> <mtext>D</mtext> </msub> <msub> <mtext>Î¼</mtext> <mtext>R</mtext> </msub> <mo>=</mo> <mfrac> <mrow> <mtext>sin r</mtext> </mrow> <mrow> <msup> <mtext>sin r</mtext> <mo>â²</mo> </msup> </mrow> </mfrac> <mspace width="0.5em"></mspace><mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace><mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace><mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace><mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mtext>-----(1)</mtext> </mrow> </math><br/><br/><math> <mrow> <mtext>At critical angle</mtext> <mspace width="0.5em"></mspace> <msub> <mspace></mspace> <mtext>D</mtext> </msub> <msub> <mtext>Î¼</mtext> <mtext>R</mtext> </msub> <mo>=</mo> <mfrac> <mrow> <mtext>sin C</mtext> </mrow> <mrow> <msup> <mtext>sin 90</mtext> <mtext>o</mtext> </msup> </mrow> </mfrac> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mtext>-----(2)</mtext> </mrow> </math><br/><br/>from equation (1) &amp; (2)<br/><br/><math> <mrow> <mtext>sin C</mtext> <mo>=</mo> <mfrac> <mrow> <mtext>sin r</mtext> </mrow> <mrow> <mtext>sin</mtext> <mfenced close=")" open="(" separators=","> <mrow> <mtext>90</mtext> <mo>-</mo> <mtext>r</mtext> </mrow> </mfenced> </mrow> </mfrac> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mo>â</mo> <mspace width="0.5em"></mspace> <mtext>sin r</mtext> <mo>=</mo> <mtext>tan r</mtext> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mo>â</mo> <mspace width="0.5em"></mspace> <mtext>C</mtext> <mo>=</mo> <msup> <mtext>sin</mtext> <mn>-1</mn> </msup> <mspace width="0.5em"></mspace> <mtext>tan r</mtext> </mrow> </math></div>


tags: ['Physics', 'Ray Optics', 'MHT CET', 'MHT CET 2020 (16 Oct Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-ray-of-light-travelling-through-glass-of-refractive-index-2-is-incident-on-glass-air-boundary-at-an-angle-of-incidence-4-5-if-refractive-index-of-ai

slug:
a-ray-of-light-travelling-through-glass-of-refractive-index-2-is-incident-on-glass-air-boundary-at-an-angle-of-incidence-4-5-if-refractive-index-of-ai-85960

question:
<div class="question">A ray of light travelling through glass of refractive index $\sqrt{2}$, is incident on glass - air boundary at an angle of incidence $45^{\circ}$. If refractive index of air is 1 , then the angle of refraction will be $\left[\sin 45^{\circ}=\frac{1}{\sqrt{2}}, \quad \sin 90^{\circ}=1\right]$</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">$90^{\circ}$</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">$45^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$90^{\circ}$</span> </div>

answer:
<div class="solution">(C)<br/>$\mu=\sqrt{2} \mathrm{i}=45^{\circ}$<br/>$\mu_{i} \sin i=\mu_{r} \sin r$<br/>$\Rightarrow \sqrt{2} \sin 45^{\circ}=\sin \mathrm{r}$<br/>$\therefore \quad \mathrm{r}=90^{\circ}$</div>


tags: ['Chemistry', 'Electrochemistry', 'MHT CET', 'MHT CET 2023 (09 May Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-reaction-ni-s-cu-im-ni-im-2-cu-s-occurs-in-a-cell-calculate-e-cell-6-if-e-cu-0-0337-v-and-e-ni-o-0257-v

slug:
a-reaction-ni-s-cu-im-ni-im-2-cu-s-occurs-in-a-cell-calculate-e-cell-6-if-e-cu-0-0337-v-and-e-ni-o-0257-v-14586

question:
<div class="question">A reaction, $\mathrm{Ni}_{(\mathrm{s})}+\mathrm{Cu}_{(\mathrm{IM})}^{++} \rightarrow \mathrm{Ni}_{(\mathrm{IM} 2}^{++}+\mathrm{Cu}_{(\mathrm{s})}$ occurs in a cell. Calculate $\mathrm{E}_{\mathrm{cell}}^6$ if $\mathrm{E}_{\mathrm{Cu}}^0=0.337 \mathrm{~V}$ and $\mathrm{E}_{\mathrm{Ni}}^{\mathrm{o}}=-0.257 \mathrm{~V}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.594 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-0.594 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-0.08 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.08 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.594 \mathrm{~V}$</span> </div>

answer:
<div class="solution">The standard cell potential is given by<br/>$\begin{aligned}<br/>\mathrm{E}_{\text {cell }}^o &amp; =\mathrm{E}_{\text {cathode }}^{\circ}-\mathrm{E}_{\text {anode }}^o \\<br/>\mathrm{E}_{\text {cell }}^{\circ} &amp; =\mathrm{E}_{\mathrm{Cu}}^{\circ}-\mathrm{E}_{\mathrm{Ni}}^{\circ} \\<br/>&amp; =(0.337 \mathrm{~V})-(-0.257 \mathrm{~V}) \\<br/>&amp; =0.337 \mathrm{~V}+0.257 \mathrm{~V} \\<br/>&amp; =0.594 \mathrm{~V}<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch1_P1.db
title:
a-rectangular-box-with-a-cover-is-to-have-a-square-base-the-volume-is-to-be-10-cubic-cm-the-surface-area-of-the-box-in-terms-of-the-side-x-is-given-by

slug:
a-rectangular-box-with-a-cover-is-to-have-a-square-base-the-volume-is-to-be-10-cubic-cm-the-surface-area-of-the-box-in-terms-of-the-side-x-is-given-by-49299

question:
<div class="question">A rectangular box with a cover is to have a square base. The volume is to be 10 cubic $\mathrm{cm}$. The surface area of the box in terms of the side $x$ is given by which one of the following functions? $\quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)=(40 / x)+2 x^{2}$</span> </div>

answer:
<div class="solution">Let the height of rectangular box be $y \mathrm{~cm}$. $\therefore$ Volume $=x \times x \times y$<br/>$\Rightarrow y=\frac{10}{x^{2}}$<br/>Now, surface area of box $=2\left(x^{2}+x y+y x\right)=2\left(x^{2}+2 x y\right)$<br/>$=2\left(x^{2}+\frac{20}{x}\right) \quad$ From equation (i)<br/>$=2 x^{2}+\frac{40}{x}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-relation-r-is-defined-on-the-set-n-of-natural-numbers-as-xry-x-2-4-xy-3-y-2-0-then-which-one-of-the-following-is-correct

slug:
a-relation-r-is-defined-on-the-set-n-of-natural-numbers-as-xry-x-2-4-xy-3-y-2-0-then-which-one-of-the-following-is-correct-35457

question:
<div class="question">A relation $\mathrm{R}$ is defined on the set $\mathrm{N}$ of natural numbers as $\mathrm{xRy} \Rightarrow \mathrm{x}^{2}-4 \mathrm{xy}+3 \mathrm{y}^{2}=0$, Then which one of the following is<br/>correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{R}$ is reflexive and symmetric, but not transitive</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{R}$ is reflexive and transitive, but not symmetric</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{R}$ is reflexive, symmetric and transitive</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{R}$ is reflexive, but neither symmetric 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">$\mathrm{R}$ is reflexive, but neither symmetric nor transitive</span> </div>

answer:
<div class="solution">Given, $x R Y \Rightarrow x^{2}-4 x y+3 y^{2}=0$<br/>$\Rightarrow x^{2}-x y-3 x y+3 y^{2}=0$<br/>$\Rightarrow x(x-y)-3 y(x-y)=0$<br/>$\Rightarrow(x-y)(x-3 y)=0$<br/>Reflexive property:<br/>$x R x \Rightarrow(x-x)(-3 x)=0$<br/>So, $\mathrm{R}$ is reflexive ...(1)<br/>Symmetric property:<br/>Let us check using an example $(1,2)$ and $(2,1)$ for $(1,2) \Rightarrow(1-2)(1-6)=(-1)(-5)=10$<br/>For $(2,1) \Rightarrow(2-1)(2-3)=(1)(-1)=-1$<br/>So, $\mathrm{R}$ is not symmetric ...(2)<br/>Transitive property :<br/>$\operatorname{For}(9 \mathrm{x}, 3 \mathrm{x}) \Rightarrow(9 \mathrm{x}-3 \mathrm{x})(9 \mathrm{x}-9 \mathrm{x})=0$<br/>for $(3 x, x) \Rightarrow(3 x-3 x)(3 x-9 x)=0$<br/>For $(9 x, x) \Rightarrow(9 x-x)(9 x-3 x)$<br/>So, $(9 x, 3 x) \in R,(3 x, x) \in R$ but $(9 x, x) \notin R$<br/>So, $\mathrm{R}$ is not transitive...(3)<br/>(3) From (1), (2), (3), R is reflexive, but not symmetr ic and transitive.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P1.db
title:
a-relation-r-is-defined-on-the-set-z-of-integers-as-follows-m-r-n-m-n-is-odd-which-of-the-following-statements-isare-true-for-r-1r-is-reflexive-2r-is-

slug:
a-relation-r-is-defined-on-the-set-z-of-integers-as-follows-m-r-n-m-n-is-odd-which-of-the-following-statements-isare-true-for-r-1r-is-reflexive-2r-is-76732

question:
<div class="question">A relation $\mathrm{R}$ is defined on the set $\mathrm{Z}$ of integers as follows:<br/>$m R n \Leftrightarrow m+n$ is odd<br/>Which of the following statements is/are true for $\mathrm{R}$ ?<br/>1.R is reflexive   <br/>2.R is symmetric <br/>3.$\mathrm{R}$ is transitive 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">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</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 3</span> </li> </ul>

answer:
<div class="solution">$\because R$ is a relation defined on the set $Z$ of integers as follows:<br/>$\mathrm{mRn} \Leftrightarrow \mathrm{m}+\mathrm{n}$ is odd<br/>(1) Then, $\mathrm{mRm}=2 \mathrm{~m}$ and $\mathrm{nRn}=2 \mathrm{n}$ are not odd multiples of 2 are not odd. Thus, it is not reflexive.<br/>(2) If $\mathrm{m}$ and $\mathrm{n}$ are numberssuch that $\mathrm{mRn} \Leftrightarrow \mathrm{m}+\mathrm{n}$ is odd. Thus, $\mathrm{n} \mathrm{Rm} \Leftrightarrow \mathrm{n}+\mathrm{m}$ is odd.<br/>This relation is symmetric<br/>(3) $\mathrm{mRn}=\mathrm{m}+\mathrm{n}$, if there is third number $\mathrm{p}$ and $\mathrm{n} \mathrm{Rp}=\mathrm{n}+\mathrm{p}$ is odd. (for $\mathrm{ex}: 2+3=5$ is odd $3+4=7$ is<br/>odd. But, $2+4=6$ is notodd.) Then $\mathrm{mRp}=\mathrm{m}+\mathrm{p}$ may not be odd. So, this relation is not transitive.</div>


title: a-relation-r-is-defined-over-the-set-of-nonnegative-integers-as-xry-x-2-y-2-36-what-is-r
slug: a-relation-r-is-defined-over-the-set-of-nonnegative-integers-as-xry-x-2-y-2-36-what-is-r-76265
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch1_P1.db
question:
<div class="question">A relation $\mathrm{R}$ is defined over the set of non-negative integers<br/>as $\mathrm{xRy} \Rightarrow \mathrm{x}^{2}+\mathrm{y}^{2}=36$ what is $\mathrm{R}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\{(0,6)\}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\{(6,0),(\sqrt{11}, 5),(3,3, \sqrt{3})$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\{(6,0),(0,6)\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(\sqrt{11}, 5),(2,4 \sqrt{2}),(5 \sqrt{11)},(4 \sqrt{2,2})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\{(6,0),(0,6)\}$</span> </div>

answer:
<div class="solution">$\mathrm{R}$ is defined over the set of non negative integers, $x^{2}+y^{2}=36$<br/>$\Rightarrow y=\sqrt{36-x^{2}}=\sqrt{(6-x)(6+x)}, x=0$ or 6<br/>for $x=0, y=6$ and for $x=6, y=0$<br/>$\mathrm{So}, \mathrm{y}$ is 6 or 0<br/>$s o, R=\{(6,0),(0,6)\}$</div>


tags: ['Physics', 'Current Electricity']
source_db: MarksBatch1_P1.db
title:
a-resistance-r-is-to-be-measured-using-a-meter-bridge-student-chooses-the-standard-resistance-s-to-be-100-he-findsthenull-point-at-l-1-29-cm-he-is-tol

slug:
a-resistance-r-is-to-be-measured-using-a-meter-bridge-student-chooses-the-standard-resistance-s-to-be-100-he-findsthenull-point-at-l-1-29-cm-he-is-tol-64446

question:
<div class="question">A resistance $\mathrm{R}$ is to be measured using a meter bridge, student chooses the standard resistance $S$ to be $100 \Omega$. He findsthenull point at $l_1=2.9 \mathrm{~cm}$. He is told to attempt to improve the accuracy.<br/>Which of the following is a useful way?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>He should measure $\mathrm{I}_1$ more accurately<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>He should change 5 to $1000 \Omega$ and repeat the experiment<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>He should change $\mathrm{S}$ to $3 \Omega$ and repeat the experiment<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>He should given up hope of a more accurate measurement with a meter bridge</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>He should change $\mathrm{S}$ to $3 \Omega$ and repeat the experiment<br/></span> </div>

answer:
<div class="solution">Adjusting the blance point near the middle of the bridge, i.e.. when $l_1$ is close to $50 \mathrm{~cm}$. requires a suitable choice of $S, R$ is unknown resistance :<br/>Since, $\quad \frac{\mathrm{R}}{\mathrm{S}}=\frac{\mathrm{R} l_1}{\mathrm{R}\left(100-l_1\right)}$<br/>$\frac{\mathrm{R}}{\mathrm{S}}=\frac{l_1}{100-l_1} \quad$ or $\quad \mathrm{R}=\mathrm{S}\left[\frac{l_1}{100-l_1}\right]$<br/>$\mathrm{R}=\mathrm{S}\left[\frac{2.9}{97.1}\right]$<br/>So, here, $\mathrm{R}: \mathrm{S}=2.9: 97.1$ implies that the $\mathrm{S}$ is nearly 33 times to that of $\mathrm{R}$. In orded to make this ratio $1: 1$ it is necessary to reduce the value of $S$ nearly $\frac{1}{33}$ times i.e., nearly $3 \Omega$,</div>


tags: ['Physics', 'Rotational Motion', 'TS EAMCET', 'TS EAMCET 2009']
source_db: MarksBatch1_P1.db
title:
a-rod-of-length-l-is-held-vertically-stationary-with-its-lower-end-located-at-a-point-p-on-the-horizontal-plane-when-the-rod-is-released-to-topple-abo-1

slug:
a-rod-of-length-l-is-held-vertically-stationary-with-its-lower-end-located-at-a-point-p-on-the-horizontal-plane-when-the-rod-is-released-to-topple-abo-1-18771

question:
<div class="question">A rod of length $l$ is held vertically stationary with its lower end located at a point $P$, on the horizontal plane. When the rod is released to topple about $P$, the velocity of the upper end of the rod with which it hits the ground is</div>

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

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

answer:
<div class="solution">In this process potential energy of the metre stick will be converted into rotational kinetic energy. PE of metre stick $=\frac{m g l}{2}$<br/>Because its centre of gravity lies at the middle of the rod.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/M-KoFfnE-GdhNdATxsKZVedLda0gGS3VdBcY3LHoe-k.original.fullsize.png"/><br/><br/>Rotational kinetic energy $E=\frac{1}{2} I \omega^2$<br/>$I=$ moment of inertia of metre stick about point<br/>$A=\frac{m l^2}{3} \text {. }$<br/>By the law of conservation of energy<br/>$\begin{aligned}<br/>m g\left(\frac{l}{2}\right) &amp; =\frac{1}{2} I \omega^2 \\<br/>&amp; =\frac{1}{2} \frac{m l^2}{3}\left(\frac{v_B}{l}\right)^2<br/>\end{aligned}$<br/>By solving, we get $v_B=\sqrt{3 g l}$</div>


tags: ['Physics', 'Rotational Motion', 'AP EAMCET', 'AP EAMCET 2009']
source_db: MarksBatch1_P1.db
title:
a-rod-of-length-l-is-held-vertically-stationary-with-its-lower-end-located-at-a-point-p-on-the-horizontal-plane-when-the-rod-is-released-to-topple-abo-2

slug:
a-rod-of-length-l-is-held-vertically-stationary-with-its-lower-end-located-at-a-point-p-on-the-horizontal-plane-when-the-rod-is-released-to-topple-abo-2-86271

question:
<div class="question">A rod of length $l$ is held vertically stationary with its lower end located at a point $P$, on the horizontal plane. When the rod is released to topple about $P$, the velocity of the upper end of the rod with which it hits the ground is</div>

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

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

answer:
<div class="solution">In this process potential energy of the metre stick will be converted into rotational kinetic energy. PE of metre stick $=\frac{m g l}{2}$<br/>Because its centre of gravity lies at the middle of the rod.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/M-KoFfnE-GdhNdATxsKZVedLda0gGS3VdBcY3LHoe-k.original.fullsize.png"/><br/><br/>Rotational kinetic energy $E=\frac{1}{2} I \omega^2$<br/>$I=$ moment of inertia of metre stick about point<br/>$A=\frac{m l^2}{3} \text {. }$<br/>By the law of conservation of energy<br/>$\begin{aligned}<br/>m g\left(\frac{l}{2}\right) &amp; =\frac{1}{2} I \omega^2 \\<br/>&amp; =\frac{1}{2} \frac{m l^2}{3}\left(\frac{v_B}{l}\right)^2<br/>\end{aligned}$<br/>By solving, we get $v_B=\sqrt{3 g l}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.91$</span> </div>
source_db: MarksBatch1_P1.db
title:
a-salesman-has-a-70-chance-to-sell-a-product-to-any-customer-the-behaviour-of-successive-customers-is-independent-if-two-customers-a-and-b-enter-what-

slug:
a-salesman-has-a-70-chance-to-sell-a-product-to-any-customer-the-behaviour-of-successive-customers-is-independent-if-two-customers-a-and-b-enter-what-68075

question:
<div class="question">A salesman has a $70 \%$ chance to sell a product to any customer. The behaviour of successive customers is independent. If two customers $\mathrm{A}$ and $\mathrm{B}$ enter, what is the probability that the salesman will sell the product to customer A or B?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.98$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.91$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$0.70$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.49$</span> </li> </ul>

answer:
<div class="solution">$\quad P(A)=\frac{70}{100}=\frac{7}{10}=P(B)$<br/>$A$ and $B$ are independent.<br/>$\begin{aligned} \Rightarrow P(A \cap B) &amp;=P(A) P(B) \\ \Rightarrow P(A \cup B) &amp;=P(A)+P(B)-P(A \cap B) \\ &amp;=\frac{7}{10}+\frac{7}{10}-\frac{7}{10} \times \frac{7}{10}=0.91 \end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-sample-of-5-observations-has-mean-32-and-median-33-later-it-is-found-that-an-observation-was-recorded-incorrectly-as-40-instead-of-35-if-we-correct-

slug:
a-sample-of-5-observations-has-mean-32-and-median-33-later-it-is-found-that-an-observation-was-recorded-incorrectly-as-40-instead-of-35-if-we-correct-62409

question:
<div class="question">A sample of 5 observations has mean 32 and median 33 . Later it is found that an observation was recorded incorrectly as 40 instead of 35 . If we correct the data, then which oneot the following is correct? $\quad$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The median remains the same but the mean will decrease</span> </div>

answer:
<div class="solution">Mean $=32$, Median $=33$<br/>It was recorded as 40 instead of $35 .$ So, sum will decrease and so mean. Median remains same.</div>


title: a-secondary-amine-is
slug: a-secondary-amine-is-65655
question: <div class="question">A secondary amine is</div>
tags: ['Chemistry', 'Amines', 'KCET', 'KCET 2022']
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a compound with two carbon atoms and an $\mathrm{NH}_2$ group</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a compound with an $\mathrm{NH}_2$ group on the carbon atom in number 2 position</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">a compound in which 2 of the hydrogen of $\mathrm{NH}_3$ have been replaced by organic group</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">an organic compound with two $\mathrm{NH}_2$ groups</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a compound in which 2 of the hydrogen of $\mathrm{NH}_3$ have been replaced by organic group</span> </div>

answer:
<div class="solution">A secondary amine is a compound in which 2 of the hydrogen of $\mathrm{NH}_3$ have been replaced by organic group.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/H7JN6DYWqC4JlN8dXrLByxt8Xh-aU4HCmueH6ImtesE.original.fullsize.png"/><br/></div>


title: a-secondary-cell-is-one
slug: a-secondary-cell-is-one-35595
question: <div class="question">A secondary cell is one</div>
tags: ['Chemistry', 'Electrochemistry', 'KCET', 'KCET 2016']
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">can be recharged.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">can be recharged by passing current through it in the same direction.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">can be recharged by passing current through it in the opposite direction.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">can not recharged</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">can be recharged by passing current through it in the opposite direction.</span> </div>

answer:
<div class="solution">In secondary cells electrode reactions can be reversed by external electric energy source, therefore, these cells can<br/>berecharged by passing electric current and used again and again.<br/>The current is passed in the direction opposite to that of the flow of current generated by the cell.</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2007']
source_db: MarksBatch1_P1.db
title:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-1

slug:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-1-44690

question:
<div class="question">A segment of wire vibrates with a fundamental frequency of $450 \mathrm{~Hz}$ under a tension of $9 \mathrm{~kg} \mathrm{wt}$. Then tension at which the fundamental frequency of the same wire becomes $900 \mathrm{~Hz}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$36 \mathrm{~kg}-\mathrm{wt}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$27 \mathrm{~kg}-\mathrm{wt}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$18 \mathrm{~kg}-\mathrm{wt}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$72 \mathrm{~kg}-\mathrm{wt}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$36 \mathrm{~kg}-\mathrm{wt}$</span> </div>

answer:
<div class="solution">Fundamental frequency of wire<br/>$\begin{array}{rlrl}f &amp; =\frac{1}{2 \pi} \sqrt{\frac{T}{m}} \\ \text { or } &amp; f &amp; \propto \sqrt{T} \\ \text { or } &amp; \frac{f_2}{f_1} &amp; =\sqrt{\frac{T_2}{T_1}} \\ \text { or } &amp; \frac{900}{450} &amp; =\sqrt{\frac{T_2}{9}} \\ \text { or } &amp; T_2 &amp; =4 \times 9=36 \mathrm{~kg}-\mathrm{wt}\end{array}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2007']
source_db: MarksBatch1_P1.db
title:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-2

slug:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-2-85417

question:
<div class="question">A segment of wire vibrates with a fundamental frequency of $450 \mathrm{~Hz}$ under a tension of $9 \mathrm{~kg} \mathrm{wt}$. Then tension at which the fundamental frequency of the same wire becomes $900 \mathrm{~Hz}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$36 \mathrm{~kg}-\mathrm{wt}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$27 \mathrm{~kg}-\mathrm{wt}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$18 \mathrm{~kg}-\mathrm{wt}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$72 \mathrm{~kg}-\mathrm{wt}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$36 \mathrm{~kg}-\mathrm{wt}$</span> </div>

answer:
<div class="solution">Fundamental frequency of wire<br/>$\begin{array}{rlrl}f &amp; =\frac{1}{2 \pi} \sqrt{\frac{T}{m}} \\ \text { or } &amp; f &amp; \propto \sqrt{T} \\ \text { or } &amp; \frac{f_2}{f_1} &amp; =\sqrt{\frac{T_2}{T_1}} \\ \text { or } &amp; \frac{900}{450} &amp; =\sqrt{\frac{T_2}{9}} \\ \text { or } &amp; T_2 &amp; =4 \times 9=36 \mathrm{~kg}-\mathrm{wt}\end{array}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch1_P1.db
title:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-3

slug:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-3-67322

question:
<div class="question">A segment of wire vibrates with a fundamental frequency of $450 \mathrm{~Hz}$ under a tension of $9 \mathrm{~kg} \mathrm{wt}$. Then tension at which the fundamental frequency of the same wire becomes $900 \mathrm{~Hz}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$36 \mathrm{~kg}-\mathrm{wt}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$27 \mathrm{~kg}-\mathrm{wt}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$18 \mathrm{~kg}-\mathrm{wt}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$72 \mathrm{~kg}-\mathrm{wt}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$36 \mathrm{~kg}-\mathrm{wt}$</span> </div>

answer:
<div class="solution">Fundamental frequency of wire<br/>$\begin{array}{rlrl}f &amp; =\frac{1}{2 \pi} \sqrt{\frac{T}{m}} \\ \text { or } &amp; f &amp; \propto \sqrt{T} \\ \text { or } &amp; \frac{f_2}{f_1} &amp; =\sqrt{\frac{T_2}{T_1}} \\ \text { or } &amp; \frac{900}{450} &amp; =\sqrt{\frac{T_2}{9}} \\ \text { or } &amp; T_2 &amp; =4 \times 9=36 \mathrm{~kg}-\mathrm{wt}\end{array}$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa

slug:
a-segment-of-wire-vibrates-with-a-fundamental-frequency-of-450-hz-under-a-tension-of-9-kg-wt-then-tension-at-which-the-fundamental-frequency-of-the-sa-16927

question:
<div class="question">A segment of wire vibrates with a fundamental frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>450</mn><mo> </mo><mi>Hz</mi></math> under a tension of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo> </mo><mi>kg</mi><mo>-</mo><mi>wt</mi></math>. Then tension at which the fundamental frequency of the same wire becomes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>900</mn><mo> </mo><mi>Hz</mi></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">36 kg-wt</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">27 kg-wt</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">18 kg-wt</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">72 kg-wt</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">36 kg-wt</span> </div>

answer:
<div class="solution">Fundamental frequency of wire<br/><br/><math><mi>f</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>Ï</mi></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></mfrac></msqrt></math><br/><br/>Or<br/><br/><math><mi>f</mi><mo>â</mo><msqrt><mi>T</mi></msqrt></math><br/><br/>Or<br/><br/><math><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>=</mo><msqrt><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></msqrt></math><br/><br/>Or<br/><br/><math><mfrac><mrow><mn>900</mn></mrow><mrow><mn>450</mn></mrow></mfrac><mo>=</mo><msqrt><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>9</mn></mrow></mfrac></msqrt></math><br/><br/>Or<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub><mo>=</mo><mn>4</mn><mo>Ã</mo><mn>9</mn><mo>=</mo><mn>36</mn><mi mathvariant="normal"> </mi><mi>kg</mi><mo>-</mo><mi>wt</mi></math></div>


tags: ['Physics', 'Alternating Current', 'JEE Main', 'JEE Main 2023 (31 Jan Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">25</span> </div>
source_db: MarksBatch1_P1.db
title:
a-series-lcr-circuit-consists-of-r-80-x-l-100-and-x-c-40-the-input-voltage-is-2500-cos-100-t-v-the-amplitude-of-current-in-the-circuit-is-a

slug:
a-series-lcr-circuit-consists-of-r-80-x-l-100-and-x-c-40-the-input-voltage-is-2500-cos-100-t-v-the-amplitude-of-current-in-the-circuit-is-a-58481

question:
<div class="question">A series LCR circuit consists of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>80</mn><mo> </mo><mo>Ω</mo></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>L</mi></msub><mo>=</mo><mn>100</mn><mo> </mo><mo>Ω</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>C</mi></msub><mo>=</mo><mn>40</mn><mo> </mo><mo>Ω</mo></math>. The input voltage is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2500</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>100</mn><mi>π</mi><mi>t</mi></mrow></mfenced><mo> </mo><mi mathvariant="normal">V</mi></math>. The amplitude of current, in the circuit, is _____  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>.</div>

answer:
<div class="solution"><p>Comparing input voltage with standard expression, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><msub><mi>V</mi><mn>0</mn></msub><mi>cos</mi><mfenced><mrow><mi>ω</mi><mi>t</mi></mrow></mfenced></math>, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>100</mn><mi>π</mi></math>.</p><p>Now, impedance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><msqrt><msup><mi>R</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msub><mi>X</mi><mi>L</mi></msub><mo>-</mo><msub><mi>X</mi><mi>C</mi></msub></mrow></mfenced><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><msup><mn>80</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>100</mn><mo>-</mo><mn>40</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>100</mn><mi mathvariant="normal">Ω</mi></math></p><p>Therefore, peak current</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mn>0</mn></msub><mo>=</mo><mfrac><msub><mi>V</mi><mn>0</mn></msub><mi>Z</mi></mfrac><mo>=</mo><mfrac><mn>2500</mn><mn>100</mn></mfrac><mo> </mo><mi mathvariant="normal">A</mi><mo>=</mo><mn>25</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p></div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P1.db
title:
a-set-of-annual-numerical-data-comparable-over-the-year-is-given-for-the-last-12-years-1-the-data-is-best-represented-by-a-broken-line-graph-each-corn

slug:
a-set-of-annual-numerical-data-comparable-over-the-year-is-given-for-the-last-12-years-1-the-data-is-best-represented-by-a-broken-line-graph-each-corn-99191

question:
<div class="question">A set of annual numerical data, comparable over the year? is given for the last 12 years.<br/> 1 . The data is best represented by a broken line graph, each corner (turning point) representing the data of one yean<br/> 2 . Such a graph depicts the chronological change and also enables one to make a short-term forecast.<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">The annual numerical data for comparable for last 12 years is represented by broken line graph, where each turning point represent the data of a particular year, while such graph do not depict the chronological change. Option (c) is correct.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch1_P1.db
title:
a-set-of-n-values-x-1-x-2-x-n-has-standard-deviation-what-is-the-standard-deviation-of-n-values-x-1-k-1-x-2-k-1-x-n-k

slug:
a-set-of-n-values-x-1-x-2-x-n-has-standard-deviation-what-is-the-standard-deviation-of-n-values-x-1-k-1-x-2-k-1-x-n-k-42048

question:
<div class="question">A set of $n$ values $x_{1}, x_{2}, \ldots, x_{n}$ has standard deviation $\sigma$. What is the standard deviation of $n$ values $x_{1}+k_{1} x_{2}+k_{1, \ldots,} x_{n}+k ?$</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">$\mathrm{a}+\mathrm{k}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sigma-\bar{k}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\underline{k} \in$</span> </li> </ul>

answer:
<div class="solution">We know that, if a number is added in values, then the standard deviation remains unaltered.<br/>$\therefore$ Standard deviation of new values $=\sigma$</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
a-set-of-n-values-x-1-x-2-x-n-has-standard-deviation-what-will-be-the-standard-deviation-of-n-values-x-1-k-x-2-k-x-n-k

slug:
a-set-of-n-values-x-1-x-2-x-n-has-standard-deviation-what-will-be-the-standard-deviation-of-n-values-x-1-k-x-2-k-x-n-k-79683

question:
<div class="question">A set of $\mathrm{n}$ values $\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{\mathrm{n}}$ has standard deviation $\sigma$. What will be the standard deviation of $n$ values $\mathrm{x}_1+\mathrm{k}, \mathrm{x}_2+\mathrm{k}, \ldots, \mathrm{x}_{\mathrm{n}}+\mathrm{k} ?$</div>

answer:
<div class="solution">If each observation is increased by a constant $\mathrm{k}$, then standard deviation is unchanged. So, $\sigma$ will be the standard deviation.</div>


title: a-significant-amount-of-progesterone-is-produced-by-the
slug: a-significant-amount-of-progesterone-is-produced-by-the-76176
question: <div class="question">A significant amount of progesterone is produced by the</div>
tags: ['Biology', 'Human Reproduction', 'NEET']
source_db: MarksBatch1_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">corpus luteum</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">testis</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">placenta</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Both A and C</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

answer:
<div class="solution">Progesterone is mainly the female sex hormone responsible for maintaining the endometrium of the uterus alongwithestrogen.It is also responsible for secondary sexual characters in females like development of mammary glands. It is secreted by corpus luteum. Corpus luteum forms after ovulation. This secretes progesterone so that after ovulation if the sperm enters the fertilised egg can easily get implanted in the uterus. Once the implantation is completed it is very important to maintain the endometrium till parturition. Thus progesterone is secreted by placenta too.</div>


title: a-simple-harmonic-motion-is-represented-by-d-t-2-d-2-x-x-0-its-period-is-1
slug: a-simple-harmonic-motion-is-represented-by-d-t-2-d-2-x-x-0-its-period-is-1-64534
tags: ['Physics', 'Oscillations', 'JEE Main']
source_db: MarksBatch1_P1.db
question:
<div class="question">A simple harmonic motion is represented by $\alpha \frac{d^2 x}{d t^2}+\beta x=0$. Its period is</div>

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

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

answer:
<div class="solution">The standard equation of SHM is: $\frac{d^2 x}{d t^2}=-\omega^2 x$<br/>On comparing with equation: $\alpha \frac{d^2 x}{d t^2}+\beta x=0$<br/>The angular frequency is given by: $\omega=\frac{\beta}{\alpha}$.<br/>Therefore, the time period of SHM is:<br/>$T=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\alpha}{\beta}}$</div>


tags: ['Physics', 'Wave Optics', 'MHT CET', 'MHT CET 2021 (22 Sep Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-single-slit-diffraction-pattern-is-formed-with-white-light-for-what-wavelength-of-light-the-3-rd-secondary-maximum-in-diffraction-pattern-coincides-

slug:
a-single-slit-diffraction-pattern-is-formed-with-white-light-for-what-wavelength-of-light-the-3-rd-secondary-maximum-in-diffraction-pattern-coincides-77695

question:
<div class="question">A single slit diffraction pattern is formed with white light. For what wavelength of light, the $3^{\text {rd }}$ secondary maximum in diffraction pattern coincides with the $2^{\text {nd }}$ secondary maximum in the pattern of red light of wavelength $6000 Ã$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$4300 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$3500 Ã$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4000 Ã$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5000 Ã$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4300 Ã$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \text { Distance of nth maximum } \mathrm{y}_{\mathrm{n}}=\left(\mathrm{n}+\frac{1}{2}\right) \frac{\lambda \mathrm{D}}{\mathrm{a}} \\ &amp; \therefore \mathrm{y}_2=\left(2+\frac{1}{2}\right) \frac{\lambda_1 \mathrm{D}}{\mathrm{a}} \text { and } \mathrm{y}_3=\left(3+\frac{1}{2}\right) \frac{\lambda_2 \mathrm{D}}{\mathrm{a}} \\ &amp; \therefore 2.5 \lambda_1=3.5 \lambda_2 \\ &amp; \therefore 2.5 \times 6000=3.5 \lambda_2 \\ &amp; \therefore \lambda_2=\frac{2.5 \times 6000}{3.5} \simeq 4300 A\end{aligned}$</div>


tags: ['Physics', 'Wave Optics', 'KCET', 'KCET 2007']
source_db: MarksBatch1_P1.db
title:
a-single-slit-fraunhofer-diffraction-pattern-is-formed-with-white-light-for-what-wavelength-of-light-the-third-secondary-maximum-in-the-diffraction-pa-1

slug:
a-single-slit-fraunhofer-diffraction-pattern-is-formed-with-white-light-for-what-wavelength-of-light-the-third-secondary-maximum-in-the-diffraction-pa-1-78357

question:
<div class="question">A single slit Fraunhofer diffraction pattern is formed with white light. For what wavelength of light the third secondary maximum in the diffraction pattern coincides with the second secondary maximum in the pattern for red light of wavelength $6500 Ã$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4400 Ã$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4100 Ã$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4642.8 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$9100 Ã$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4642.8 Ã$</span> </div>

answer:
<div class="solution">$x=\frac{(2 n+1) \lambda D}{2 a}$<br/>For red light $x=\frac{(4+1) D}{2 a} \times 6500$<br/>For unknown wavelength of light<br/>$$<br/>x=\frac{(6+1) D}{2 a} \times \lambda<br/>$$<br/>Accordingly,<br/>$$<br/>5 \times 6500=7 \times \lambda=4642.8 Ã<br/>$$</div>


tags: ['Physics', 'Wave Optics', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-single-slit-fraunhofer-diffraction-pattern-is-formed-with-white-light-for-what-wavelength-of-light-the-third-secondary-maximum-in-the-diffraction-pa

slug:
a-single-slit-fraunhofer-diffraction-pattern-is-formed-with-white-light-for-what-wavelength-of-light-the-third-secondary-maximum-in-the-diffraction-pa-34459

question:
<div class="question">A single slit Fraunhofer diffraction pattern is formed with white light, for what wavelength of light, the third secondary maximum in the diffraction pattern coincides with the second secondary maximum in the pattern for the red light of wavelength <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6500</mn><mo> </mo><mover><mi mathvariant="normal">A</mi><mo>â</mo></mover></math>?</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>9100</mn><mi> </mi><mover><mi mathvariant="normal">A</mi><mo>â</mo></mover></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4642</mn><mi> </mi><mover><mi mathvariant="normal">A</mi><mo>∘</mo></mover></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4100</mn><mi> </mi><mover><mi mathvariant="normal">A</mi><mo>∘</mo></mover></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4400</mn><mi> </mi><mover><mi mathvariant="normal">A</mi><mo>∘</mo></mover></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4642</mn><mi> </mi><mover><mi mathvariant="normal">A</mi><mo>∘</mo></mover></math></span> </div>

answer:
<div class="solution"><math><mo>∵</mo><mtext>    </mtext><mi>x</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mi>λ</mi><mi>D</mi></math><br/><br/>For red light, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>×</mo><mn>6500</mn><mo> </mo><mover><mi mathvariant="normal">A</mi><mo>∘</mo></mover><mo>×</mo><mi>D</mi></math><br/><br/>For other light, <math><mi>x</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mn>6</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mi>D</mi><mo>×</mo><mi>λ</mi></math><br/><br/><math><mo>∵</mo></math> x is the same for each<br/><br/><math><mo>∴</mo><mn>5</mn><mo>×</mo><mn>6500</mn><mo>=</mo><mn>7</mn><mi>λ</mi></math><br/><br/><math><mi>λ</mi><mo>=</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>7</mn></mrow></mfrac><mo>×</mo><mn>6500</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4642.</mn><mn>8</mn><mo> </mo><mover><mi mathvariant="normal">A</mi><mo>∘</mo></mover></math></div>


tags: ['Physics', 'Wave Optics', 'JEE Main', 'JEE Main 2017 (08 Apr Online)']
source_db: MarksBatch1_P1.db
title:
a-single-slit-of-width-b-is-illuminated-by-a-coherent-monochromatic-light-of-wavelength-if-the-second-and-fourth-minima-in-the-diffraction-pattern-at--1

slug:
a-single-slit-of-width-b-is-illuminated-by-a-coherent-monochromatic-light-of-wavelength-if-the-second-and-fourth-minima-in-the-diffraction-pattern-at-1-94946

question:
<div class="question">A single slit of width <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> is illuminated by a coherent monochromatic light of wavelength <math><mi>λ</mi><mo>.</mo></math> If the second and fourth minima in the diffraction pattern at a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>cm</mi></math> from the slit are at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>cm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>cm</mi></math> respectively from the central maximum, what is the width of the central maximum? (i.e. distance between first minimum on either side of the central maximum)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cm</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cm</mi></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>6</mn><mo>.</mo><mn>0</mn><mo> </mo><mi>cm</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"><mn>3</mn><mo>.</mo><mn>0</mn><mo> </mo><mi>cm</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>0</mn><mo> </mo><mi>cm</mi></math></span> </div>

answer:
<div class="solution">For secondary minima <math><mi>d</mi><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>⁡</mo><mrow><mi>θ</mi></mrow></mrow><mo>=</mo><mi>n</mi><mi>λ</mi></math><br/><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>⁡</mo><mrow><mi>θ</mi></mrow></mrow><mo>=</mo><mfrac><mrow><mi>n</mi><mi>λ</mi></mrow><mrow><mn>d</mn></mrow></mfrac></math><br/><br/>For second minima<br/><math><mi>n</mi><mo>=</mo><mn>2</mn></math><br/><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>⁡</mo><mrow><mi>θ</mi></mrow></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>λ</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>=</mo><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>⁡</mo><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mrow><mo>=</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>D</mi></mrow></mfrac></math><br/>For fourth minima <math><mi>n</mi><mo>=</mo><mn>4</mn></math><br/><math><msub><mrow><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>⁡</mo><mrow><mi>θ</mi></mrow></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mi>λ</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>D</mi></mrow></mfrac></math><br/><br/>Distance between both minima is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mi>λ</mi></mrow><mn>d</mn></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mi>λ</mi></mrow><mn>d</mn></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>λ</mi></mrow><mn>d</mn></mfrac><mo>= 6 -</mo><mn>3 =3</mn><mo> </mo><mi>cm</mi></math><br/><br/>Width of central maxima,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>λ</mi></mrow><mn>d</mn></mfrac><mo>=</mo><mn>3</mn><mi> </mi><mtext>c</mtext><mtext>m</mtext></math></div>


tags: ['Physics', 'Wave Optics', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-single-slit-of-width-b-is-illuminated-by-a-coherent-monochromatic-light-of-wavelength-if-the-second-and-fourth-minima-in-the-diffraction-pattern-at-

slug:
a-single-slit-of-width-b-is-illuminated-by-a-coherent-monochromatic-light-of-wavelength-if-the-second-and-fourth-minima-in-the-diffraction-pattern-at-86979

question:
<div class="question">A single slit of width <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi></math> is illuminated by a coherent monochromatic light of wavelength <math><mi>Î»</mi><mo>.</mo></math> If the second and fourth minima in the diffraction pattern at a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>cm</mi></math> from the slit are at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>cm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>cm</mi></math> respectively from the central maximum, what is the width of the central maximum? (i.e. distance between first minimum on either side of the central maximum)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cm</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cm</mi></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>6</mn><mo>.</mo><mn>0</mn><mo> </mo><mi>cm</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"><mn>3</mn><mo>.</mo><mn>0</mn><mo> </mo><mi>cm</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>0</mn><mo> </mo><mi>cm</mi></math></span> </div>

answer:
<div class="solution">For secondary minima <math><mi>d</mi><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mi>Î¸</mi></mrow></mrow><mo>=</mo><mi>n</mi><mi>Î»</mi></math><br/><br/><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mi>Î¸</mi></mrow></mrow><mo>=</mo><mfrac><mrow><mi>n</mi><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac></math><br/><br/>For second minima<br/><math><mi>n</mi><mo>=</mo><mn>2</mn></math><br/><br/><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mi>Î¸</mi></mrow></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>=</mo><mrow><mrow><mi mathvariant="normal">tan</mi></mrow><mo>â¡</mo><mrow><msub><mrow><mi>Î¸</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mrow><mo>=</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>D</mi></mrow></mfrac></math><br/><br/><br/>For fourth minima <math><mi>n</mi><mo>=</mo><mn>4</mn></math><br/><br/><math><msub><mrow><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>â¡</mo><mrow><mi>Î¸</mi></mrow></mrow></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>D</mi></mrow></mfrac></math><br/><br/><math><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac><mo>= 6 -</mo><mn>3 = </mn><mi> 3</mi><mi>c</mi><mi>m</mi></math><br/><br/>Width of central <math><mrow><mrow><mi mathvariant="normal">max</mi></mrow><mo>â¡</mo><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>Î»</mi></mrow><mrow><mn>d</mn></mrow></mfrac></mrow></mrow><mo>=</mo><mn>3</mn><mi> </mi><mtext>c</mtext><mtext>m</mtext></math></div>


tags: ['Physics', 'Current Electricity']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
a-six-leadacid-type-of-secondary-cells-each-of-emf-20-v-and-internal-resistance-0015-are-joined-in-series-to-provide-a-supply-to-a-resistance-of-85-wh

slug:
a-six-leadacid-type-of-secondary-cells-each-of-emf-20-v-and-internal-resistance-0015-are-joined-in-series-to-provide-a-supply-to-a-resistance-of-85-wh-29417

question:
<div class="question">(a) Six lead-acid type of secondary cells each of emf $2.0 \mathrm{~V}$ and internal resistance $0.015 \Omega$ are joined in series to provide a supply to a resistance of $8.5$ $\boldsymbol{\Omega}$. What are the current drawn from the supply and its terminal voltage?<br/>(b) A secondary cell after long use has an emf of 1.9 $V$ and a large internal resistance $380 \Omega$. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?</div>

answer:
<div class="solution">(a) Given, number of secondary cells, $\mathrm{N}=6$,<br/>E.m.f. of each cell, $\mathrm{E}=2.0 \mathrm{~V}$<br/>Internal resistance of each cell, $r=0.015 \Omega$,<br/>External resistance, $\mathrm{R}=8.5 \Omega$<br/>By formula, current,<br/>$I=\frac{N E}{R+N r}=\frac{6 \times 2}{8.5+6 \times 0.015}$ $=\frac{12}{8.5+0.09}=1.397 \mathrm{~A}$.<br/>Terminal voltage $\mathrm{V}=\mathrm{IR}=\frac{12 \times 8.5}{8.59}$<br/>$=11.874 \mathrm{~V}$<br/>(b) Given, $\mathrm{E}=1.9 \mathrm{~V}, \mathrm{r}=380 \Omega, \mathrm{I}_{\max }=$ ?<br/>Maximum current can be drawn by short circuit, $\mathrm{I}_{\max }=\frac{\mathrm{E}}{\mathrm{r}}=\frac{1.9}{380}=0.005 \mathrm{~A}$.<br/>Which cannot start a car, because a starter motor requires a large current $(\approx 700 \mathrm{~A}$ ) for few seconds.</div>


tags: ['Physics', 'Wave Optics', 'KCET', 'KCET 2021']
source_db: MarksBatch1_P1.db
title:
a-slit-of-width-a-is-illuminated-by-red-light-of-wavelength-6500-a-if-the-first-diffraction-minimum-falls-at-3-0-then-the-value-of-a-is

slug:
a-slit-of-width-a-is-illuminated-by-red-light-of-wavelength-6500-a-if-the-first-diffraction-minimum-falls-at-3-0-then-the-value-of-a-is-16687

question:
<div class="question">A slit of width $a$ is illuminated by red light of wavelength $6500 Ã$. If the first diffraction minimum falls at $30^{\circ}$, then the value of $a$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6.5 \times 10^{-4} \mathrm{~mm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.3$ micron</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$3250 Ã$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$26 \times 10^{-4} \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.3$ micron</span> </div>

answer:
<div class="solution">Given, $\lambda=6500 Ã=6500 \times 10^{-10} \mathrm{~m}, \theta=30^{\circ}$ According to the diffraction pattern of single slit,<br/>$a \sin \theta=n \lambda$<br/>where, $n=$ order of secondary minima $=1$<br/>$\Rightarrow a \sin 30^{\circ}=1 \times\left(6500 \times 10^{-10}\right)$<br/>$\Rightarrow \quad \frac{a}{2}=6500 \times 10^{-10}$<br/>or<br/>$\begin{aligned}<br/>a &amp;=13000 \times 10^{-10} \\<br/>&amp;=1.3 \times 10^{-6} \mathrm{~m}=1.3 \text { micron }<br/>\end{aligned}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P1.db
title:
a-solenoid-is-2-m-long-and-3-cm-in-diameter-it-has-5-layers-of-winding-of-1000-turns-each-and-carries-a-current-of-5-a-what-is-the-magnetic-field-at-i

slug:
a-solenoid-is-2-m-long-and-3-cm-in-diameter-it-has-5-layers-of-winding-of-1000-turns-each-and-carries-a-current-of-5-a-what-is-the-magnetic-field-at-i-67174

question:
<div class="question">A solenoid is $2 \mathrm{~m}$ long and $3 \mathrm{~cm}$ in diameter. It has 5 layers of winding of 1000 turns each and carries a current of $5 \mathrm{~A}$. What is the magnetic field at its centre ? Use the standard value of $\mu_{0}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.57 \times 10^{-4} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.57 \times 10^{-3} \mathrm{~T}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1.57 \times 10^{-2} \mathrm{~T}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.57 \times 10^{-6} \mathrm{~T}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.57 \times 10^{-2} \mathrm{~T}$</span> </div>

answer:
<div class="solution">Length of solenoid, $l=2 \mathrm{~m}$
<br/> <br/>Total number of turns, $\mathrm{N}=5 \times 1000=5000$
<br/> <br/>Number of turns per unit length,
<br/> <br/>$\mathrm{n}=\frac{\mathrm{N}}{l}=\frac{5000}{2}=2500 \mathrm{~m}^{-1}$
<br/> <br/>Current, I = 5A
<br/> <br/>Now, B $=\mu_{0} \mathrm{nI}=4 \pi \times 10^{-7} \times 2500 \times 5 \mathrm{~T}$
<br/> <br/>$=1.57 \times 10^{-2} \mathrm{~T}$</div>


tags: ['Chemistry', 'Electrochemistry', 'AP EAMCET', 'AP EAMCET 2020 (22 Sep Shift 1)']
source_db: MarksBatch1_P1.db
title:
a-solution-containing-45-mm-of-mno-4-and-15-mm-of-mn-2-shows-ph-of-2-the-potential-of-halfcell-reaction-is-given-lo-g-15-176-and-lo-g-45-1653-and-stan

slug:
a-solution-containing-45-mm-of-mno-4-and-15-mm-of-mn-2-shows-ph-of-2-the-potential-of-halfcell-reaction-is-given-lo-g-15-176-and-lo-g-45-1653-and-stan-40768

question:
<div class="question">A solution containing $4.5 \mathrm{mM}$ of $\mathrm{MnO}_4^{-}$and $15 \mathrm{mM}$ of $\mathrm{Mn}^{2+}$ shows $\mathrm{pH}$ of 2 . The potential of half-cell reaction is ...... .<br/>(Given, $\log 15=1.76$ and $\log 45=1.653$ and standard potential of $\mathrm{MnO}_4^{-} \longrightarrow \mathrm{Mn}^{2+}$ ) is $1.51 \mathrm{~V}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.51 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.31 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.71 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.04 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.31 \mathrm{~V}$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { (b) } \mathrm{MnO}_4^{-}+8 \mathrm{H}^{+}+5 e^{-} \longrightarrow \mathrm{Mn}^{2+}+4 \mathrm{H}_2 \mathrm{O} \\<br/>&amp; E=E_{\mathrm{Mn}^{\circ}}{ }^{7+} \mathrm{Mn}^{2+} /+\frac{0.059}{5} \log \frac{\left[\mathrm{MnO}_4^{-}\right]\left[\mathrm{H}^{+}\right]^8}{\left[\mathrm{Mn}^{2+}\right]} \\<br/>&amp; E=1.51+\frac{0.059}{5} \log \frac{\left[\frac{4.5}{1000}\right]\left[10^{-2}\right]^8}{\left[\frac{15}{1000}\right]} \\<br/>&amp; \Rightarrow \quad E=1.31 \mathrm{~V}<br/>\end{aligned}<br/>$$<br/>Hence, the correct option is (2).</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch1_P1.db
title:
a-sonometer-wire-100-cm-long-has-a-fundamental-frequency-of-300-hz-the-velocity-of-propagation-of-transverse-waves-on-the-wire-is

slug:
a-sonometer-wire-100-cm-long-has-a-fundamental-frequency-of-300-hz-the-velocity-of-propagation-of-transverse-waves-on-the-wire-is-56427

question:
<div class="question">A sonometer wire <math><mn>100 </mn><mi mathvariant="normal">cm </mi></math> long has a fundamental frequency of <math><mn>300 </mn><mi mathvariant="normal">Hz </mi></math>. The velocity of propagation of transverse waves on the wire is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>330 </mn><mi mathvariant="normal">m </mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mn>660 </mn><mi mathvariant="normal">m </mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>990</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">m </mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>550</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">m </mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>660 </mn><mi mathvariant="normal">m </mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </div>

answer:
<div class="solution">Frequency<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">n</mi><mo>=</mo><mfrac><mi mathvariant="normal">v</mi><mi mathvariant="normal">λ</mi></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">v</mi><mrow><mn>2</mn><mi mathvariant="normal">l</mi></mrow></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">v</mi><mo>=</mo><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mn>2</mn><mi mathvariant="normal">l</mi></mrow></mfenced><mo>=</mo><mn>330</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>660</mn><mi mathvariant="normal"> </mi><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2023 (11 May Shift 2)']
source_db: MarksBatch1_P1.db
title:
a-sonometer-wire-a-of-diameter-d-under-tension-t-having-density-1-vibrates-with-fundamental-frequency-n-if-we-use-another-wire-b-which-vibrates-with-s

slug:
a-sonometer-wire-a-of-diameter-d-under-tension-t-having-density-1-vibrates-with-fundamental-frequency-n-if-we-use-another-wire-b-which-vibrates-with-s-62556

question:
<div class="question">A sonometer wire ' $A$ ' of diameter ' $\mathrm{d}$ ' under tension ' $T$ ' having density ' $\rho_1$ ' vibrates with fundamental frequency ' $n$ '. If we use another wire 'B' which vibrates with same frequency under tension ' $2 \mathrm{~T}$ ' and diameter ' $2 \mathrm{D}$ ' then density ' $\rho_2$ ' of wire ' $B$ ' will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\rho_2=2 \rho_1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\rho_2=\rho_1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\rho_2=\frac{\rho_1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\rho_2=\frac{\rho_1}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\rho_2=\frac{\rho_1}{2}$</span> </div>

answer:
<div class="solution">The formula for frequency of a sonometer is $\mathrm{f}=\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\pi \rho \mathrm{D}^2}}$<br/>Here $l$ is length, $\mathrm{T}$ is tension, $\mathrm{D}$ is diameter and $\rho$ is density.<br/>The frequency of both the wires is same.<br/>The frequency of the wire $\mathrm{A}$ is $\mathrm{f}_{\mathrm{A}}=\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\pi \rho_1 \mathrm{D}^2}}$<br/>The frequency of the wire $\mathrm{B}$ is $f_{\mathrm{B}}=\frac{1}{2 l} \sqrt{\frac{2 \mathrm{~T}}{\pi \rho_2(2 \mathrm{D})^2}}$<br/>Equating both the frequencies<br/>$\begin{aligned}<br/>&amp; \frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\pi \rho \mathrm{D}^2}}=\frac{1}{2 l} \sqrt{\frac{2 \mathrm{~T}}{\pi \rho_2(2 \mathrm{D})^2}} \\<br/>&amp; \sqrt{\frac{1}{\rho_1}}=\sqrt{\frac{1}{2 \rho_2}} \\<br/>&amp; \frac{1}{\rho_1}=\frac{1}{2 \rho_2} \\<br/>&amp; \therefore \quad \rho_2=\frac{\rho_1}{2}<br/>\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2010']
source_db: MarksBatch1_P1.db
title:
a-sonometer-wire-has-a-length-of-114-cm-between-two-fixed-ends-where-should-two-bridges-be-placed-so-as-to-divide-the-wire-into-three-segments-in-cm-w-1

slug:
a-sonometer-wire-has-a-length-of-114-cm-between-two-fixed-ends-where-should-two-bridges-be-placed-so-as-to-divide-the-wire-into-three-segments-in-cm-w-1-65181

question:
<div class="question">A sonometer wire has a length of $114 \mathrm{~cm}$, between two fixed ends. Where should two bridges be placed so as to divide the wire into three segments (in $\mathrm{cm}$ ) whose fundamental frequencies are in the ratio $1: 3: 4$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$l_1, l_2, l_3=18,24,72$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$l_1, l_2, l_3=24,18,72$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$l_1, l_2, l_3=72,18,24$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$l_1, l_2, l_3=72,24,18$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$l_1, l_2, l_3=72,24,18$</span> </div>

answer:
<div class="solution">If length of the wire between the two bridges is $l$, the frequency of vibration<br/>$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>$\therefore \quad n \propto \frac{1}{l}$<br/>Therefore, $n_1: n_2: n_3=\frac{1}{l_1}: \frac{1}{l_2}: \frac{1}{l_3}$<br/>$n_1: n_2: n_3=\frac{1}{\frac{114 \times 1}{8}}: \frac{1}{\frac{114 \times 3}{8}}: \frac{1}{\frac{114 \times 4}{8}}$<br/>or $\quad n_1: n_2: n_3=8: \frac{8}{3}: 2$<br/>or $\quad n_1: n_2: n_3=72: 24: 18$</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2010']
source_db: MarksBatch1_P1.db
title:
a-sonometer-wire-has-a-length-of-114-cm-between-two-fixed-ends-where-should-two-bridges-be-placed-so-as-to-divide-the-wire-into-three-segments-in-cm-w

slug:
a-sonometer-wire-has-a-length-of-114-cm-between-two-fixed-ends-where-should-two-bridges-be-placed-so-as-to-divide-the-wire-into-three-segments-in-cm-w-56079

question:
<div class="question">A sonometer wire has a length of $114 \mathrm{~cm}$, between two fixed ends. Where should two bridges be placed so as to divide the wire into three segments (in $\mathrm{cm}$ ) whose fundamental frequencies are in the ratio $1: 3: 4$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$l_1, l_2, l_3=18,24,72$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$l_1, l_2, l_3=24,18,72$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$l_1, l_2, l_3=72,18,24$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$l_1, l_2, l_3=72,24,18$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$l_1, l_2, l_3=72,24,18$</span> </div>

answer:
<div class="solution">If length of the wire between the two bridges is $l$, the frequency of vibration<br/>$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>$\therefore \quad n \propto \frac{1}{l}$<br/>Therefore, $n_1: n_2: n_3=\frac{1}{l_1}: \frac{1}{l_2}: \frac{1}{l_3}$<br/>$n_1: n_2: n_3=\frac{1}{\frac{114 \times 1}{8}}: \frac{1}{\frac{114 \times 3}{8}}: \frac{1}{\frac{114 \times 4}{8}}$<br/>or $\quad n_1: n_2: n_3=8: \frac{8}{3}: 2$<br/>or $\quad n_1: n_2: n_3=72: 24: 18$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2013 (07 Apr)']
source_db: MarksBatch1_P1.db
title:
a-sonometer-wire-of-length-1-5-m-is-made-of-steel-the-tension-in-it-produces-an-elastic-strain-of-1-what-is-the-fundamental-frequency-of-steel-if-dens

slug:
a-sonometer-wire-of-length-1-5-m-is-made-of-steel-the-tension-in-it-produces-an-elastic-strain-of-1-what-is-the-fundamental-frequency-of-steel-if-dens-93912

question:
<div class="question">A sonometer wire of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.5</mo><mi mathvariant="normal">m</mi></math> is made of steel. The tension in it produces an elastic strain of <math> <mrow> <mtext>1%</mtext> </mrow> </math>. What is the fundamental frequency of steel if density and elasticity of steel are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.7</mo><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.2</mo><mo>×</mo><msup><mn>10</mn><mn>11</mn></msup><mo> </mo><mi mathvariant="normal">N</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> respectively?</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>200</mn><mo>.5</mo><mo> </mo><mi>Hz</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>770</mn><mo> </mo><mo>Hz</mo></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>188</mn><mo>.5</mo><mo> </mo><mi>Hz</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"><mn>178</mn><mo>.2</mo><mo> </mo><mi>Hz</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>178</mn><mo>.2</mo><mo> </mo><mi>Hz</mi></math></span> </div>

answer:
<div class="solution"><math> <mrow> <math> <mi>f</mi> </math> <mo>=</mo> <mfrac> <mrow> <math> <mi>V</mi> </math> </mrow> <mrow> <mn>2</mn> <mi>â</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>â</mi> </mrow> </mfrac> <msqrt> <mfrac> <mrow> <math> <mi>T</mi> </math> </mrow> <mrow> <mi>Î¼</mi> </mrow> </mfrac> </msqrt> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>â</mi> </mrow> </mfrac> <msqrt> <mfrac> <mrow> <math> <mi>T</mi> </math> </mrow> <mrow> <math> <mrow> <mi>A</mi><mi>d</mi></mrow> </math> </mrow> </mfrac> </msqrt> </mrow> </math><br/><br/>Also, <math> <mrow> <math> <mi>Y</mi> </math> <mo>=</mo> <mfrac> <mrow> <math> <mi>T</mi> </math> <mi>â</mi> </mrow> <mrow> <math> <mi>A</mi> </math> <mo>Î</mo> <mi>â</mi> </mrow> </mfrac> <mo>      â  </mo> <mfrac> <mrow> <math> <mi>T</mi> </math> </mrow> <mrow> <math> <mi>A</mi> </math> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <math> <mi>Y</mi> </math> <mo>Î</mo> <mi>â</mi> </mrow> <mrow> <mi>â</mi> </mrow> </mfrac> <mo>      â  </mo> <math> <mi>f</mi> </math> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>â</mi> </mrow> </mfrac> <msqrt> <mfrac> <mrow> <mtext>Y</mtext> <mo>Î</mo> <mi>â</mi> </mrow> <mrow> <mi>â</mi> <mtext>d</mtext> </mrow> </mfrac> </msqrt> </mrow> </math><br/><br/><math> <mrow> <mi>â</mi> <mo>=</mo> <mtext>1.5 m, </mtext> <mfrac> <mrow> <mo>Î</mo> <mi>â</mi> </mrow> <mrow> <mi>â</mi> </mrow> </mfrac> <mo>=</mo> <mtext>0.01, d</mtext> <mo>=</mo> <mtext>7.7</mtext> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mn>3 </mn> </msup><mtext>kg</mtext> <mo>/</mo> <msup> <mtext>m</mtext> <mn>3</mn> </msup> </mrow> </math><br/><br/><math> <mrow> <mtext>Y</mtext> <mo>=</mo> <mtext>2.2</mtext> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mn>1</mn> <mn>1 </mn> </mrow> </msup> <mtext>N</mtext> <mo>/</mo> <msup> <mtext>m</mtext> <mn>2</mn> </msup> </mrow> </math><br/><br/>After solving<br/><br/><math> <mrow> <math> <mi>f</mi> </math> <mo>=</mo> <msqrt> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>7</mn> </mrow> </mfrac> </msqrt> <mo>Ã</mo> <mfrac> <mrow> <mn>1</mn> <msup> <mn>0</mn> <mn>3</mn> </msup> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <math> <mrow> <mtext>H</mtext><mtext>z</mtext></mrow> </math> </mrow> </math><br/><br/><math> <mrow> <math> <mi>f</mi> </math> <mo> â </mo> <mtext>178.2 </mtext><math> <mrow> <mtext>H</mtext><mtext>z</mtext></mrow> </math> </mrow> </math></div>


